"Mathematical Diary"
”My calling,my way to professor Mariana Montiel”
"My study of IB diploma mathematics 2008 higher level 1,and 2"
No.1 writing from 13th March to June in 2023.

March 13th, 2023.
Today, on March 13, 2023, I have begun studying mathematics in preparation for pursuing a PhD.in mathematics at the Graduate School of Georgia State University under the supervision of mathematician Mariana Montiel.
I had very little exposure to high school mathematics during my time at a music high school and an art college.
As a result, I can't even recall that for the first problem (1.1.1) in Strang's 2014 'Linear Algebra and Differential Equations' exercise, I should solve the system of equations involving the slope of the derivative of the exponential function F(x) = e^x and the intersection point of the linear function F(x).
Therefore, for now, I will start with exponential functions and then review other topics such as the quadratic formula, inequalities, and quadratic functions as needed, in order to refresh my memory of high school mathematics.
The textbook I'm using is 'Once Again High School Mathematics' by Kazuo Takahashi."

14th
Exponential.

15th.
Quadratic formula. Factorization. Quadratic inequalities.
Exponential functions.
In high school, exponential functions are not defined when the base is a negative number because they become discontinuous.
At university, there is a definition using complex numbers to define exponential functions as complex functions.
There is also a mathematical definition derived from the exponential laws that extends the concept of a^0=1 to rational and integer numbers.
In university mathematics, these topics are covered under number theory.
Mathematics is enjoyable and fascinating.
I realized that I need to study the factor theorem again.
But I'll reach Strang at last.
It will take me three years to finish studying Strang.
I need one year to begin studying Gilbert Strang.
At 58 years old,I'll be a first-year undergraduate student, and even at 66 years old, I'll earn a doctoral degree in mathematics under the supervision by Mariana Montsieur. 
That's my goal, without a doubt."

16th.
Logarithm. Common logarithm.
Today, I have been able to solve logarithmic equations.
However, I still struggle with even the commutative property of multiplication when solving practice problems, causing me to hesitate and make mistakes.
I deeply understand that in mathematics, it is crucial to work on it every day and gradually grasp the concepts that I don't understand, pinpointing them myself.
I am still in the stage of reviewing high school mathematics to prepare for Strang's "Linear Algebra and Differential Equations" in 2014.
Strang Exercise Problem 1.
I am studying again factoring quadratic equations to solve first-order linear differential equations of exponential functions with the base of Napier and to find the intersection points of their graphs.
I have come to learn for the first time that exponential functions are not defined at negative bases at the high school mathematics and are defined as complex functions using imaginary numbers at university.
I remember thinking the same when I have learned the δ-ε definition in my first year at the Art University when I was 18 years old.
Mathematics is enjoyable. Mathematics is fascinating.
Strang claims that linear algebra is more fundamental than differential equations.
Regarding patterns where logarithmic functions involve variables and constants as bases, I compromise at the level of understanding it as a pattern similar to shifting quadratic functions and move forward.
Once I can make judgments on which level of understanding to compromise and move forward, I will be able to progress on my own.
Even from Wikipedia, it is evident that quadratic functions are related to group theory mappings and representation forms.
Since I can now solve logarithmic equations, I can move on to studying Strang's book.

17th.
Rational functions. Irrational functions. Inverse functions.

19th.
Today, trigonometric ratios.
I'm gradually recalling mathematics.
While reviewing the quadratic formula, factoring quadratic equations, quadratic inequalities, and quadratic functions, I progressed to exponential functions, rational functions, inverse functions, and irrational functions, and today, I'm focusing on trigonometric functions.
Today, I will memorize the conversion formulas for sine, cosine, and tangent in each quadrant.
Today is all about trigonometric functions. Today, I will learn how to memorize the conversion formulas for sine, cosine, and tangent in each quadrant. It seems that probability, which is learned at the beginning, is more difficult than calculus.
Today, I will research the "High School Mathematics New Curriculum" and find that vectors and complex planes come last.
I have finished studying exponential functions and finally reached Strang's Exercise 1. 
I can somehow understand the meaning and solution of question 1.11, not feeling completely lost anymore.
Once I finish trigonometric functions, all I need to cover for the battle with Strang is sequences and calculus. I think I can handle vectors, complex planes, and probability later as needed, which brings me some relief.

20th,
This morning, I studied the conversion formulas for trigonometric functions. As long as I remember the two formulas stating that tangent becomes the reciprocal only at 90 degrees, the rest can be summarized as "stc stock: 'Only at 90 degrees, sine and cosine are back-to-back.'" With just these three rules, I don't need to memorize all 18 conversion formulas. It would be time-consuming and challenging to try to memorize all of them for exams, so I will stick to remembering these two.
This morning, after learning the cosine rule, I embarked on a mission to find a better high school mathematics textbook at Nagoya University Central Library.

21st.
It's the sixth day today. By the end of today, I will have covered quadratic equations, quadratic inequalities, quadratic functions, exponential functions, and trigonometric functions, including the sine rule and cosine rule, almost entirely.

22nd.
Today, I'm continuing with trigonometric functions. I will cover Heron's formula and the unit circle.
My homework is to go through the remaining 40 pages of trigonometric functions. 
Once I finish trigonometric functions, I will move on to sequences, infinite series, differentiation, and integration. Strang's book is coming into view.
Today, I will delve into the addition formulas.
It's going mostly according to my plan and expectations.
I have gained a little confidence in my own mathematics.
Although there are numerous trigonometric theorems that can seem daunting, once you firmly grasp the addition formulas, you can derive many of them. I find the proof of the addition formulas quite inspiring.
I'm almost done with the addition formulas. Once I finish trigonometric functions, I can take a breather.
It will take me about three years to fully cover the textbook "Linear Algebra and Differential Equations" by Gilbert Strang, which is still at the level of a first-year mathematics student.
So far, I can study mathematics for a maximum of five hours a day.
I can't sustain my focus for longer than that.
However, on the other hand, without this level of intense concentration, mathematics becomes difficult.
Mathematics depends on concentration.
Trigonometric functions are more difficult than calculus.
The concept of extending triangles in an algebraic-geometric coordinate system within the realm of analysis is quite difficult to grasp.
For Strang's book, it's crucial to have a solid understanding of probability, complex planes, vectors, and, above all, exponential functions, logarithmic functions, sequences, infinite series, differentiation, and integration.
For now, I will focus on covering exponential and logarithmic functions.
Today, I will learn the methods that involve the unit circle and trigonometric functions.
The remaining topics are the addition formulas, sequences, infinite series, differentiation, and integration.
Studying mathematics is enjoyable.
Studying mathematics is fascinating.
Studying mathematics offers hope.
Today, I will borrow two books, "Mouichido Yomu Koko Sūgaku" and "Kōkō Sūgaku + Alpha," from Nagoya University Central Library.
If I borrow the High School Mathematics Formula Handbook next week, I will have copied from six different high school mathematics textbooks.
Having all of these resources should be more than enough.

23rd.
I have bought a total of 400 notebooks just in case.
Right now, I have used 3 college notebooks. 
Anyway, I'm going to write down each proof in the notebooks, solving the exercises.
I enjoy the feeling of solving the puzzle-like trigonometric identities one by one and proving them.
Even at the age of 55, I'm making progress in mathematics.
In high school mathematics, the new curriculum started with the first-year students in 2022. 
Therefore, the textbooks famous"Blue Chart" "Math C" hasn't been introduced yet.
The properties of integers in the old curriculum "Math I" disappeared, and the statistical estimation of the new curriculum "Math B" was added. 
And the old curriculum topics "Math B: Vectors" and "Math III: Complex Plane" were incorporated into the revived "Math C".
The statistical estimation in the new curriculum "Math B" is not necessary for my current battle with Strang,so I decided to purchase affordable used books of the old curriculum "Math B" "Bolue Chart".
The fact that the properties of integers in the old curriculum "Math I" disappeared and were replaced by statistical estimation in the new curriculum "Math B" probably reflects the changing times.
It's also because my teacher, Mariana Montiel, specializes in statistics.
The difference of the price  between the used books of the new curriculum's blue charts and the used books of the old curriculum's blue charts is more than five times.
For aspiring students taking domestic university entrance exams, whether it's the new curriculum or the old curriculum is a matter of life and death. 
But for me, it doesn't matter as long as it helps me get into Strang.
I was surprised to see the blue charts for linear algebra for high school students as well.
By the middle of March, I will have all the used books of the old curriculum's blue charts for Math IA, Math IIB, and Math III delivered to my home. I will finish the synthesis of simple harmonic motion.
I will almost understand trigonometric functions. The study of analysis that involves combining trigonometric functions is truly enjoyable.
Mathematics is really fun.
I will soon delve into sequences, series, differentiation, integration, and eventually, the content of Math II, which is differential equations. 
I can tackle Strang's book now.
Mathematics is enjoyable. 
Even at the age of 55, I have discovered the joy of proving the additive identities one by one, feeling like solving a puzzle.
I can now handle trigonometric inequalities almost up to the triangle inequality.
Mathematics is fun.
The more difficult it is, the greater the joy when I can solve it.
From today on, I will try to study linear algebra and differential equations parallel to sequences, differentiation, and integration using Strang's "2014 Linear Algebra and Differential Equations."

24th
It has been 10 days since I started studying mathematics. 
I have encountered geometric sequences in compound interest calculations and harmonic sequences.
Harmonic sequences are sequences of reciprocals and are related to the "Pythagorean tuning" in music. Geometric sequences are used in compound interest calculations.
With a 6% interest rate, 50,000 yen increases to 60,000 yen in 5 years through compound interest.
Harmonic sequences, on the other hand, are sequences of reciprocals with a common difference.
The Pythagorean tuning is related to these harmonic sequences.
Sigma notation, discovered by Gauss when he was still a child, is a method of calculation. 
Gauss discovered the formula for the sum of an arithmetic sequence without teacher.
I am about to start studying Math III.I will transit from infinite series to Math III.
If I can solve even one exercise problem from Strang's "2014 Linear Algebra and Differential Equations" in parallel with sequences, it will be a success for the first stage of my plan.
I never knew that mathematical proofs could be so enjoyable. At the age of 55, my life has completely changed. There is nothing in the world as enjoyable as mathematical proofs. 
I am transcribing each proof in my notebook, just like solving puzzles.
There's a high school math textbook called "Shin-taisei High School Mathematics" that is not part of the official curriculum. I made a photocopy of it, but I can't seem to find it now. I'll have to go to Tsurumi tomorrow to borrow it and make a copy. I also have books on mathematical sciences, mathematics, modern mathematics, and astronomical monthly reports, as well as back issues of National Geographic.
The textbook I'm currently using, "Mouichido Koukou Suugaku" by Kazuo Takahashi, is a fairly well-known high school textbook for working adults. There are also other famous textbooks from the "Bluebacks" series for the new curriculum high school mathematics, as well as plenty of free PDF resources available.
As someone who hardly studied high school mathematics, it turned out to be the right decision for me to use this simplest textbook. Anyway, my goal is to study Strang and maintain contact with Monsieur, so collecting free PDFs and high school mathematics textbooks at this level and then diving into Strang while working through the blue charts seems to be the best approach.
Even with this simplest textbook, it feels like I'm climbing a rock face, digging my fingers into the bedrock. But the experience of mathematical proofs continuously reveals how enjoyable it can be. Since I started studying mathematics, my life has completely changed. When I realize that harmonic sequences are related to the Pythagorean tuning, I feel even more strongly that mathematics is essential to music.

25th
Today,I dive into the limits of sequences.
In 10 days and 10 hours, I will almost finish studying high school mathematics subjects IA and IB for first-year and second-year students.
It was really tough for me, as I had hardly studied high school mathematics at a music high school and an art university.
But even a small step forward becomes a significant confidence boost for myself.
Studying mathematics is truly enjoyable.
Studying mathematics is truly fascinating.
Studying mathematics gives me hope.
Today, I am dealing with recurrent sequences and the limits of sequences.
I use the sigma Σ symbol to find recurrent sequences.
First-order recurrent sequences, second-order recurrent sequences. Tomorrow, I plan to delve into differentiation from infinite series.
Mathematics is truly enjoyable.
Mathematics is truly fascinating.
Mathematics gives me hope.
My life took a 180-degree turn when I started studying mathematics at the age of 55.
I began mathematics, and my life underwent a complete transformation.
Through the study of mathematics, I have been reborn.
Mathematics is my calling. It is my vocation.

26th
Today, I started studying differentiation,as I expected
It was tough to finish studying high school mathematics I and II in 10 days and 10 hours and move on to differentiation in mathematics III.
I still don't have the luxury to learn the δ-ε method at the level of a first-year university mathematics student.
The definition of limits, continuity, and differentiability of functions is still difficult for me.
Ideally, I would like to dive into the δ-ε method right away, but for now, I'm doing my best to progress from the definition of calculus in high school mathematics to the rigorous approach in university mathematics.
However, I have a good understanding of why the high school mathematics curriculum is designed this way,thanks to studying up to infinite series.
In 10 days and 10 hours, I have made significant progress, almost reaching university-level mathematics, which is a big step forward for me.
Today, I finished studying infinite series and moved on to differentiation.
The 36th Mariana Montiel Fasting.
Today, I'm struggling with partial fraction decomposition.
I look up some exam techniques and tricks for partial fraction decomposition, such as 1/ab(bigger) = 1/b-a × 1/ab, on the internet.
But I'm not too concerned about these kinds of university entrance exam techniques because they are not the essence of mathematics, so I continue to move forward.
Although I'm having some difficulty with infinite series, I quickly regain my concentration.
However,in nearly 10 days and 10 hours, I will almost finish studying high school mathematics necessary for Strang's calculus, up to infinite series.
I will able to progress toward  Montiel forward.

27th,
Today, I will borrow a book of Mathematica and other resources from Nagoya University Central Library. 
In the past two weeks, I have completed equations, functions, and sequences, as well as High School Mathematics I and II, and started differentiation in High School Mathematics III yesterday.
Using the admired Mathematica, I find it easier to plot graphs of quadratic functions.
I should finish High School Mathematics III by the end of this week.
In a book called "Mathematics Olympiad Encyclopedia," I learned for the first time that there are few countries in the world, like Japan, where calculus and matrices are taught in high school.
The proofs of infinite series are often used in statistics.

28th,
Today, I will finish about half of the differentiation. I have almost completed differentiation of composite functions.
Mathematics is enjoyable.
As I had almost no exposure to high school mathematics at a music high school and art university, if I can finish High School Mathematics III in five days, 
Could it be said that my study of mathematics as a mathematician is progressing at a relatively high pace?

29th,
Today, I finished differentiation of composite functions and moved on to differentiation of inverse functions.
I stumbled upon the calculation method of differentiation using the chain rule, which is based on the differentiation of composite functions.
I reached the part about inverse functions and returned home for the day.
Now on my fourth volume of "Note" (a series of math textbooks).
Indeed, as teachers often say, studying mathematics comes down to diligently copying their proofs, working through calculations with a pencil, and diligently solving practice problems.
The partial derivatives known as the chain rule, which extend the differentiation of composite functions to multivariables, are at the level of university mathematics. At this stage, I haven't even started integration in high school mathematics, and I can't differentiate x with respect to y. So, I simply memorize the explanation that uses the product of d/dx x and dx/dy to calculate the differentiation of composite functions. At my current level of mathematics, I can't handle the proof of this chain differentiation calculation, so I set it aside and continue moving forward.
Once my understanding of mathematics progresses and I delve into Strang's excersise, I believe I will naturally come to understand the proof of this chain differentiation calculation.
For now, my homework is to continue advancing, at least to the level of integration and, indeed, to the essential vector and matrix concepts in linear algebra.
Finding a scholarship for mathematics at the age of 55 is quite dificult.
In any case, understanding the chain differentiation calculation, when we differentiate x with respect to y, is extremely difficult for me at this point.
I must simply trust my teacher's explanation that we use the differentiation of composite functions and calculate it as the product of d/dx x = 1 and dx/dy. I have no choice but to keep moving forward.
There's a famous quote by Freud that says, "Neurosis is the inability to tolerate ambiguity."
I must endure the ambiguity of not fully understanding the chain differentiation calculation and continue my study of differentiation, integration, matrices, and vectors towards Mariana Montiel tomorrow.

30th
Today, I will finish the logarithmic differentiation method. Once I finish the implicit differentiation method, I can drive on to integration.
The chain differentiation calculation is fascinating. As I learn different functions such as trigonometric, logarithmic, and exponential functions, and logarithmic differentiation, my understanding of the differentiation of composite functions gradually deepens.
Today, I stumbled upon expressing the area of a sector in radians. However, I quickly found the answer by searching online.
I was dismayed to realize that I had forgotten properties of logarithms and the addition theorem of trigonometric functions that I learned two weeks ago while doing the differentiation of logarithmic functions.
But as I reviewed them, I remembered and gained a clearer understanding.

31st.
Today is the greatest and best day of my life. A memorable day. 
Tonight, for the first time, I independently solved exercise 1 from Gilbert Strang's 2014 "Linear Algebra and Differential Equations".
It was about the slope at t=0 for the exponential function y=e^t -1≦t≦1 and the intersection of the linear function y=e~t.
I compared my answers to the English solution manual, and my answers were correct.
It took me two weeks to get to this point.
It's clear that to delve into Strang's excersise, I absolutely need to grasp integration, matrices, and vectors.
Strang, being the world standard textbook from MIT, is on a completely higher level than high school mathematics.
I've only solved exercise 1.1.1 so far, and I haven't touched the preceding explanations yet.
The answers are not included in the Japanese translation of Strang's book.
Self-learners like me have to download the English solutions from MIT's Strang website: https://math.mit.edu/~gs/dela/
Mathematics is a cumulative discipline. Compared to when I struggled with trigonometric addition formulas and logarithmic equations two weeks ago, my understanding and confidence in solving exercise problems are completely different now.
I've started to grasp the proof of integral formulas, which initially intimidated me for a moment.
It's clear that if I really want to tackle Strang, I can't avoid vectors and matrices. I also understand that complex numbers, complex planes, and related topics cannot be avoided.
But once I finish vectors and matrices, I'll be ready for Strang.
Integration is the reverse operation of differentiation, so if you understand differentiation, it's relatively easier.
The chain rule calculations were difficult, and it took me two days to understand them. 
Strang introduces the chain rule right from the beginning as a concept of the chain rule in the analytical study of first-order ordinary differential equations.
In my music high school, I didn't even learn how to factorize or perform long division.
But somehow, I managed to immerse myself in Strang in just two weeks. Mathematics is my vocation, my calling.
Today, I managed to overcome chain rule calculations. 
It will take another two days for complete understanding. 
I tend to forget logarithmic equations and trigonometric addition formulas easily, unlike teenage boys who excel in these areas.
However, if I keep a formula collection with me and memorize it diligently, I can overcome the handicap of my age.
Fermat, famous for Fermat's Last Theorem, was originally a lawyer and started studying mathematics with Diophantus' arithmetic at the age of 40, which is well-known. 
Today, I plan to move on to integration from implicit functions.
Today, I stumbled upon factoring and synthetic division. 
I have to look up the divisors accurately on the internet.
I didn't even learn factoring with synthetic division in high school. 
But now, at 55 years old, learning it for the first time, it's incredibly fascinating and enjoyable.
Mathematics is my vocation, my calling. I'm finally starting to understand factoring and multi-variable factorization.
I still need to review finding the cube roots of 1, ω, ω2, -1±√3/2 and the arithmetic operations with complex number i.
But for now, my focus is on progressing towards Strang.
I finally understand factoring using the factor theorem and long multiplication.
If I hadn't seriously studied mathematics in high school for the entrance exam to an art high school, I would have to go back and review factoring. 
Today, I learned that there are cases where the factorization cannot be.
Today, I finally move on to integration. The proofs of integration formulas are still difficult for me.
 My teacher advised me to focus on understanding proofs at a high school level and not to worry too much about them. 
It is said that getting accustomed to the calculations in integration is more important than the proofs. 
Therefore, I will skip the proofs and concentrate on exercise problems.
I started studying integration in High School Mathematics I,II, and III in just two weeks.
Cannot this pace be considered a worthy speed of study for a mathematician?
As for the 36th round of fasting, 
it will end tonight on the 6th day. 
My previous records for long fasting are one week twice, 11 days once, 14 days once, and 17 days once. 
This time, I decided to end the fast on the 6th day because the nausea reaction was too severe. 
I prioritized my well-being.

April 1st.
Today, I struggled with the explanation of partial fraction decomposition in the textbook. 
From the level of coefficient comparison, which is a high school mathematics technique, to the Heaviside's partial fraction expansion in university algebra, I searched online and suddenly noticed something in the explanation.
I finally learned the technique of coefficient comparison.
Taking the time to learn each mathematical problem-solving technique, such as the factor theorem, polynomial long division, and coefficient comparison in partial fraction decomposition, is truly enjoyable. 
Mathematics is truly fascinating.
I also understood the proof of integration formulas, which I couldn't grasp yesterday. 
It is crucial to understand that integration is the inverse operation of differentiation.
Particularly, the concept of chain rule in differentiation is directly related to subsequent integration calculations and advanced topics in university mathematics.
Understanding chain rule makes calculus much more enjoyable.
Regarding indefinite integrals, the relationship with the integration constant allows for alternative solutions.
For now, I will simply memorize the explanations provided by my teacher and continue progress forward.
Once I surely grasp of chain rule in differentiation and the fact that integration is the inverse operation of differentiation, 
I can consider myself to have mastered high school calculus. 
Next, I will delve into substitution method for integration.
The beauty of mathematics lies in the fact that both the derivative and the integral of the exponential function e^x are equal to e^x. 
This fact captivates me just as much as the beauty of music. 
Mathematicians, drawn to the elegance of this property of the exponential function e^x, must see it as a divine providence and devote themselves to the study of mathematics.
Mathematics is my calling, my vocation. 
Tonight, I quickly solved exercise 2 and 3 in exercise 1.1 in Strang's book. 
Drawing the graphs of y1 = e^2t and y2 = 2e^t, and comparing the values of these functions at t=0 and t=1 in exercise 1.1.2. 
Finding the slope of dy/dt for y = e^-1 at t=0 and t=1 in exercise 1.1.3. 
However, in problem 1.1.4, solving the exponential equation e^t = 4 using logarithms,
I haven't learned the logarithmic solution method yet, as it is not covered in my high school math textbook

Tonight's Strang.
Exercise 1.1.2 was solved quickly.
It involved graphing y1 = e^(2t) and y2 = 2e^t, and comparing their values at t=0 and t=1.
In Exercise 1.1.3, I had to find the slope of dy/dt at t=0 and t=1 for y = e^(-1). 
I made a mistake by not considering the derivative of the exponent -1 when finding the derivative of y = e^(-1).
Exercise 1.1.4 involved solving the exponential equation e^t = 4.
Since the method using logarithms with the same base was not covered in my high school math review textbook,I was still confused.
I searched online and found a solution that involves taking the logarithm with the same base on both sides.
By taking the natural logarithm (log base e) of both sides of e^t = 4, I get:
log e (e^t) = log e 4 (Using the logarithmic property, the base e and e^t cancel each other out.)
t log e e = log e 4 (log e e equals 1 by the definition of logarithms.)
Therefore, t = log e 4.
So far, I have won 2 exercise and lost 2 exercise.
The initial explanation in Strang's book states:
"Preface viii
Section 1 starts with the equations dy/dt = y and dy/dt = y^2. It is truly wonderful that solving these two equations leads to the following:
dy/dt = y yields y = 1 + t + 1/2t^2 + 1/6t^3 + ... which becomes y = e^t.
dy/dt = y^2 yields y = 1 + t + t^2 + t^3 + ... which becomes y = 1/(1-t).
Encountering these two most important series right from the beginning in mathematics is truly delightful. There could be no better way to start this subject."

On page 2:
"Here, let us pay particular attention to linear equations with constant coefficients such as in Examples 1 and 2. It turns out that dy/dt = y is the most important property of the function y = e^t. No function (such as y = t^n) from algebra will have its derivative equal to the function itself.
The derivative of t^n is nt^(n-1)."
From these two passages, I can somewhat grasp the meaning of Strang's explanation about the introduction of calculus through infinite series, Euler's number as an exponential function, the function y = e^t and the series y = 1/(1-t), as well as the explanation of differential equations.
To fully understand Strang, I will need to cover the remaining topics in high school mathematics, such as integration, vectors, matrices, complex plane, and operations with imaginary numbers.
These two passages in Strang's book, discussing the infinite series of Euler's number and the function 1/(1-t), along with their differential equations, truly convey the beauty of mathematics, and I am genuinely moved by it.

2nd
Today, I'm starting the calculation of definite integrals from partial integration. Although I still lack sufficient practice of exercise for substitution integration, I came up with an answer using a chain rule differentiation calculation different from the explanation given by the teacher.
Gradually, my confidence in mathematics is growing. Once I understand chain rule differentiations, I might as well consider myself done with high school mathematics. 
I can focus more on Strang and review any missing topics like vectors, matrices, complex planes, and operations with imaginary numbers as needed.
The reason is that I've started to appreciate the beauty of the exponential differential equation with Euler's number, the differential equation of the infinite series 1/n-1, dy/dt=y, and dy/dv=y^2. 
I feel like I can grasp their beauty now. Even though I barely studied High School Mathematics I, II, and III, I managed to dive into Strang within about two weeks.
I believe I can confidently say that mathematics is my true calling and vocation. 
Why? Because I've come to appreciate the beauty of these two differential equations: dy/dt=y and dy/dt=y^2. Learning mathematics has transformed me.

3rd
I managed to understand the chain rule in analysis within two weeks. I surpassed 1,500 exercise in Strang. 
Today, I am going to the Nagoya University Central Library.
I couldn't find the differential equation of Macglawh-ills, so I need to make the double of copy.
I made a triple copy of Strang, and finally, I can start working on it now. 
It took me three years. However, I have reached a point where I can solve exercise up to 1.1.6 with respect to the solutions and integrals of differential equations. Today,
I have a double copy of Springer's Brown's differential equation and Macglaw-hill. I'll search for the copy of Macglaw-hills's calculus at home, and with the combined use of this textbook,
I should be able to manage Strang somehow. I'm waiting for the Blue Chart. Since the SGC series is not very good for the δ-ε method in textbooks, 
it would be fine to make a double copy of Macglaw-hills's calculus next time as well. 
Strang is the textbook for MIT Mathematics and the first-year mathematics department at MIT, so the explanations are more difficult than any other differential equations or linear algebra textbooks.
It probably contains over 1,000 exercises, maybe even around 1,500.
However, the content is compiled by Strang based on the latest ideas in mathematics education regarding linear algebra and differential equations.
It allows you to learn the most important content that must be learned in the shortest time.
Although it may be difficult, I have decided to use Strang and give my best.
I have come to a point within these two weeks where I can manage it if I review high school mathematics a little more and use some other slightly easier textbooks.

Today's Strang:
When y(t) = f(u(t)), write down the chain measurement (chain rule) that gives the derivative dy/dt.
Since y cannot be directly differentiated with respect to t, I first differentiate df(u(t))/d(u(t)) using the composite function f(u(t)), and then differentiate d(u(t))/dt. 
This is an exercise  in university mathematical analysis that involves chain rule differentiation, which is taught in high school mathematics. dy/dt = (d(f(u))/d(u(t))) * (d(u(t))/dt).
This 1.1.5, I managed to clear it somehow while looking at the answers halfway.

1.1.6
This exercise stating that the second-order differential equation has another solution, y=e^-t, besides y=ce^t, still confuses me.
I still cannot understand the concept of a second-order differential equation having two solutions.
Therefore, for now, I will temporarily retreat from high school mathematics topics such as integration, vectors, matrices, complex planes, and imaginary operations, as well as Strang's explanations.
So far, I have had 2 wins, 2 losses, and 2 draws.

4th.
Strang's "Linear Algebra and Differential Equations" from 2014 contains a total of 1,021 exercises. I have to solve about half of them, around 500 problems, within a year.
Since I still don't grasp the solutions of differential equations, I need to review calculus a bit, particularly "Differential Equations" and "Calculus" from the McGlawhills series, as well as "δ-ε Method" from Kyoritsu Publishing. As for linear algebra, I will study Strang's "Introduction to Linear Algebra" from 2015 after one year. 
It is said that there are two major barriers to understanding university-level calculus: "δ-ε Method" and "Real Analysis." 
I haven't had the chance to delve into "Real Analysis" yet, so I am currently focusing on copying the Kyoritsu Publishing's "δ-ε Method" book.
I still find the terminology in differential equations completely confusing. For that, I will refer to McGlaw-Hill's series, Kyoritsu Publishing's "δ-ε Method," and the Springer translation. As for linear algebra, I will study Strang's "Introduction to Linear Algebra" from 2015 and the Springer translation as well.
Today, I will copy both volumes of McGlaw-Hill Calculus, the "δ-ε Method," and their binding. Tomorrow, my plan is to work on both volumes of McGraw-Hill Linear Algebra and Springer's Linear Algebra.
In any case, my most prior homework now is to understand that in Strang 1.1.6, dy^2/dt^2=y, besides y=Ce^t, there is another solution, y=e^-t. Understanding that the second-order differential equation has two solutions is the most crucial homework for me at the moment. 
Since the solutions of differential equations involve integration, I need to revisit high school mathematics topics like integration, vectors, and matrices.
Differential equations are equations that express the relationship between the unknown function y and its derivative dx/dy.
Even if the unknown function is not known, it can be determined based on the conditions of the derivative, which means that the solutions can be found. 
Today, I learned for the first time that this concept mainly progressed from the research of equations in theoretical physics.

5th.
Strang's "1.1.6." I am still confused about the meanings of exercise 1.1.6 where the second-order differential equation d2 /d2y y=y has two solutions, y=Ce^t and another solution y=e^-t. 
I have no clue yet on how to approach the meaning and solution of question 1.1.6.
Today, I received three blue charts for Mathematics IA, IIB, and III. 
I finally understood the overall picture of high school mathematics that I couldn't learn much before.
I have finally set foot on the starting line of relearning high school mathematics. 
Starting tonight, I have started bookmaking the copies of the Springer translation. Particularly, the Springer translation is essential literature for obtaining a doctoral degree in mathematics.
Among the 1,021 exercise problems in Strang's book, which is

7th.
Combat with Strang 1.1.6 is on break.
Harold Scott MacDonald Coxeter "Geometry."
Dedekind: "What is Number?"
André Weil: "Elementary Number Theory for Beginners." All of them are famous books on number theory and geometry.
The remaining homework is to copy and bookmake the translated versions of Springer books. I'm doing it while listening to music.
Mathematics is known as the queen of sciences, and within mathematics, number theory is known as the queen.
I have more than 50 copies to make of translated books from Springer.
Finally, I'm starting my study of mathematics.

11th.
Combat with Strang 1.1.6 is on hold. Currently copying and bookmaking 100 original Springer books. Combat with Strang's exercise 1.1.6 from "Linear Algebra and Differential Equations" (2014) is on hold.
I'm researching and collecting textbooks on university-level differential equations , calculus, linear algebra, delta-epsilon definition, and real analysis by Dedekind and André Weil to prepare.
It's currently on hold. After a preparation period of one to two weeks, I plan to resume with Strang 1.1.6 and the remaining high school math homework, such as definite integrals, vectors, matrices, and complex planes.
I'm having difficulty understanding the meaning of a question in the exercise that deals with the second-order derivative d2/d2y y=y and its differential equation's solutions, y=Ce^t along with y=e^-t, and their interpretations.
Although I can vaguely understand the relationship between the chain rule in calculus and definite integrals as solutions to differential equations, I'm still far from a complete understanding.
I deeply feel the need to review high school mathematics thoroughly and upgrade to university-level textbooks.
Strang's textbook, being intended for mathematics students at MIT (Massachusetts Institute of Technology), one of the top 10 universities in the QS World University Rankings, provides almost no explanations or hints. 
The questions and answers are written in English, making it even more difficult. I have to think through the solution process almost entirely on my own, referencing the English answers.

12th
Regarding Strang 1.1.6, I now understand that from the second derivative and second-order differential equation d2y/dt2=y, using the chain rule in calculus, there is another solution y=e^-t alongside y=Ce^t.
The only remaining question is why the constant C appears as a coefficient in the first solution.

13th.
I'm stuck at the point where Strang 1.1.6 deals with the second derivative dy2/dt2 =y and the second-order differential equation, where there are two solutions: y=Ce^t and y=e^-t.
However, thanks to the chain rule in calculus, I managed to understand that there are two coefficients, + and -, multiplying the unknown function y, and by differentiating the coefficient with a -, we obtain y=-(-e^-t)=e^-t, 
which is another solution alongside y=Ce^t.
I still don't understand why the first solution y=Ce^t has an arbitrary constant coefficient.
However, I find it fascinating that the chain rule introduces both + and - coefficients for the unknown function y, and that differentiation yields a negative solution in terms of t. 
I'm increasingly drawn to the study of mathematics.
I still can't figure out on my own why there is an arbitrary constant C in the first solution, but I find it very interesting.
Currently, I'm focusing on understanding this.

14th.
Still stuck on Strang, 2014, exercise 1.1.6.
Regarding the second-order derivative and its second-order differential equation, two solutions arise: d2/dt2 y=y, with solutions d/dt y=Ce^t and d/dt y=e^(-t).
Regarding the solution y=e^(-t), I understand that the coefficient of - appears in the exponential function e's variable t due to the chain rule of analysis.
However, I am unsure why the solution y=Ce^t includes an arbitrary constant C as a coefficient. I don't fully grasp the reason behind it.

16th.
Received the Bluebacks series "Introduction to University Mathematics" (Volumes 1 and 2) from Tsurumai Library. Also made a reservation for the "Introduction to Junior High School Mathematics" (Volumes 1 and 2) at Tsurumai Library.
Since I didn't properly study junior high school mathematics when I was 15 years old and in the third year of junior high school, I need to review it now.
However, studying mathematics is truly enjoyable. I increasingly feel that mathematics is my calling.
I plan to acquire and copy the two volumes of the Bluebacks series "Introduction to Junior High School Mathematics" from Tsurumai Library and then return to Strang's exercise 1.1.6 and continue with the remaining topics of high school mathematics, such as "definite integrals," "vectors," and "matrices."
Tomorrow, I plan to visit Nagoya University Central Library to make double copies of a book on Mathematica and Springer's "Introduction to Analysis" (Volumes 1 and 2).
I still can't break through exercise 1.1.6 in Strang's book. I don't understand why there are two solutions: one with y=e^(-t) and the other with an arbitrary constant C as the coefficient in y=Ce^t.
By revisiting Junior High School Mathematics and then reviewing high school mathematics, especially topics like definite integrals, vectors, and matrices, I should be able to overcome this barrier in exercise 1.1.6.
I'm still waiting for the final contact regarding the investigation and importation of "Bach and Newton literature" from the bookseller "Fujii Yosho".
Since completing the translation of Hans Keyser's musicological literature on March 13th, I have been simultaneously engaged in contact with Montiel for admission to the Department of Mathematics at Georgia State University and combatting Strang with high school mathematics.
It has been continuingsince January.
During the process of reviewing high school mathematics, I have realized the need to revisit junior high school mathematics that I didn't properly study due to focusing on the entrance examination for the High School of Fine Arts.
However, the more I review mathematics and go back in my studies, the more I see a glimmer of hope in overcoming the barrier of Strang's exercise 1.1.6.
The guidance and support of Mariana Montiel,a mathematician in the Department of Mathematics at Georgia State University, have been exceptional.
I will follow Mariana Montiel.

17th.
Double copy of the δ ε definition formation. Copying the second volume of Mathematica. If "New System Junior High School Mathematics Textbook" arrives at the Tsurumai Library tomorrow or so, I can return to definite integrals and Strang.
I found a copy of "Analytical Mechanics" by Springer. About 40 books remaining for the copying and binding process. About 10 more hours of work. My goal  for this year are admission to the State University Georgia's mathematics department.
The State University Georgia accepts 1st year students with neither the SAT nor ACT as test optional. In the world university rankings, the State University Georgia is around the 800th position, similar to Juntendo University in Japan.
However, for someone like me who still needs to review definite integrals, vectors, matrices in high school mathematics, and make copies of junior high school math textbooks, it will still take me more than three years to even get into undergraduate studies.
Conversely, if I can master Strang, which is the textbook for the mathematics department at MIT, it will open up new possibilities in one fell swoop.
I'm still stuck at 1.1.6. Strang's textbook, from the initial introduction, has a high level of sophistication and difficulty. 
It is truly a textbook of modern mathematics, which is advancing alongside theoretical physics, written for MIT's first-year mathematics students.
I plan to apply for undergraduate admission this year. However, at 58 years old, I need to anticipate an additional three years for undergraduate studies. 
I would obtain a Bachelor's degree at 62, a Master's degree at 63, and the shortest timeline for a Ph.D. would be at 66.
I am still at the stage where I need to review definite integrals, vectors, matrices, and the complex plane.
Once I break through the barrier of Strang 1.1.6, I plan to contact Monteil and Mazzola again to study abroad.
Regarding a doctoral degree in musicology,Gabriel Pareyon has suggested that I pursue a Ph.D. in musicology focusing on mathematical music theory, which utilizes statistics and number theory. 
Gabriel Pareyon also mentioned that starting from the master's program would be acceptable.
However, if I were to pursue a doctoral degree in mathematics, especially at Montiel's level of mathematics, it would be impossible to transfer from a graduate school. 
Even if I were to pursue a doctoral degree in the history of geometry or mathematics, I would have to be prepared to start over from the undergraduate mathematics department.
Strang's "Linear Algebra and Differential Equations" in 2014 is a textbook for MIT's first-year mathematics students, but if I don't solve all 1,050 exercise in Strang, it will be impossible for me to gain admission to an undergraduate program.
Georgia State University's World University Ranking is around the 800th position, similar to Juntendo University in Japan. The shortest timeline would require four years of undergraduate studies, one year for a master's degree, and three years for a Ph.D. degree.
Considering my current age of 55, it would take a minimum of eight years to obtain a Ph.D. degree.
The earliest timeschedule would be at 63 years old.
If it takes an additional two or three years for undergraduate studies, I need to be prepared to obtain a Ph.D. degree from the age of 65 to 70.

19th.
"Bluebacks: New System Junior High School Mathematics Textbook."
Today, the upper and lower volumes of "New System Junior High School Mathematics Textbook" arrived at the Tsurumai Library. The copying and bookmaking of 100 copies of the original Springer textbook will be finished soon. 
Finally, I can go back to reviewing definite integrals, vectors, and matrices. As long as I don't overcome the hurdle of 1.1.6 in Strang, I can't even consider the admission process for the mathematics department at State University of Georgia, as Montiel has instructed. 
However, if I can understand 1.1.6, I can see the prospects of the battle with Strang itself. Strang's textbook is written for MIT's first-year mathematics students, explaining the relationship between mathematics as a mathematical science and the differential equations of mechanics in theoretical physics right from the beginning. 
For someone like me who has hardly studied high school mathematics, it's very difficult. But it's also very interesting. Fermat, famous for "Fermat's Last Theorem," was originally a lawyer and became interested in mathematics at the age of 40 through Diophantus' "Arithmetica."
I also have other materials I admired, such as the two volumes of "Mathematica: A Mathematical Toolbox" and the Bluebacks' "Introduction to University Mathematics." 
Finally, the pursuit of Strang 1.1.6 is being prepared.
The recent topic of AI, Chat GPT, tends to provide incorrect answers.
It seems that the latest AI, Chat GPT, still falls short compared to the mathematical software "Mathematica" developed by mathematician and theoretical physicist Steven Wolfram.

20th
Encountered the profound words of courage and "mathematical courage" in the popular movie "The Mathematician from Wonderland," featuring a genius mathematician who escaped from North Korea.
The teachings of the movie's protagonist, Hakson, who escaped from North Korea, resonated deeply within me.
He said, "What mathematics requires is not intelligence or effort alone. It's the 'mathematical courage' that allows you to never give up and believe that you can try solving the same problem again tomorrow." 
This teaching of Hakson aligns with the process I went through when I understood the chain rule in calculus and the computation of chain differentials in high school mathematics during my two weeks of studying mathematics from March 13, 2023. 
It was the exact process Hakson, the genius mathematician and math teacher from North Korea, described. Knowing this teaching of Hakson, I gained a little confidence in the fact that mathematics is my calling and that pursuing a Ph.D. in Mathematics at Georgia State University is "my path."

21st.
Completed the absolute value of integrals.
Understood Strang 1.1.6. Today, I finally understood why the first solution of the second-order differential equation d2y/dt2=y, y=Ce^t, involves the integration constant C, which I couldn't grasp before because I hadn't covered the relationship between differentiation and definite integration, which is taught in high school mathematics. It was the process described by Haksun, the protagonist of "The Mathematician from Wonderland," and it aligned with the process I experienced when I understood the chain rule in calculus and chain differentials in high school mathematics during my two-week study period starting from April 3rd. It was correct to take a break from studying and focus on the simple task of binding 120 copies of Springer books, contemplating the chain rule and definite integration. Just as Haksun said, "Smart people give up quickly, and those who make an effort drop out next. 
Mathematics requires courage and the mental flexibility to think, 'Let's try solving it again tomorrow.'" I managed to overcome the two obstacles of chain differentials and the reason behind the integration constant in the solution of a second-order differential equation with the base of Napier's constant e.
"Mathematics Once Again: 'Definite Integration,' 'Vectors,' 'Matrices' by Kazuo Takahashi"
Finally, after completing the copying and binding work for 25 Springer books from April 3rd and 90 original Springer books from April 9th, I can start studying "Definite Integration," "Vectors," and "Matrices" again.
It's been almost three weeks since I last studied these topics.
During these three weeks, I focused on copying and bookmaking 120 Springer books to create a study environment in a room that serves as my personal library, aiming to enhance my concentration.
Since April 1st, the Nakagawa District Library I used to frequent has been closed for ceiling repairs, rendering it unavailable for the next year starting from April 2nd. 
I must decide my location of my study of mathematics.
The time required for progress and the level of concentration at the study location are crucial. In any case, I have come to realize that a review of "definite integrals," "vectors," and "matrices" is essential in Strang's 2014 book, "Linear Algebra and Differential Equations."
Once I finish studying definite integrals, I should be able to understand section 1.1.6, and if I complete the section on matrices, I might be ready to tackle Strang's 1050 exercise.

22nd.
If I can fully grasp the relationship between definite integrals and differentials,I will be able to solve Strang's 1.1.6 exercise.I finally understand that the introduction of differential equations, which I learned in high school mathematics, is connected to the relationship between definite integrals and differentials.
Despite taking a break for three weeks, I have managed to progress to the study of differential equations in just over a month.
Today, I learned about substitution in definite integrals using trigonometric ratios. 
Tomorrow or so, I will finish studying the relationship between differentials and definite integrals, and then I will delve into vectors. 
As expected, I seem to have forgotten the addition formulas for trigonometric functions, as well as the techniques for differentiation, integration, and even the formulas for quadratic inequalities, all within these three weeks. 
However, I can recall them by reviewing the textbooks. I'm starting to understand the process of studying mathematics:
Solve exercise regularly without fail.
Don't fret or get frustrated even when formulas are forgotten or problems ia difficult.
Enjoy the process of solving exercise itself.
Perhaps these are the secrets to making progress. 
Another  is diligently copying mathematical proofs. 
By faithfully transcribing example proofs from textbooks, even through forceful and vigorous means, I can reach a point where I am satisfied with my understanding.
Studying mathematics in this way feels enjoyable.
At the age of 55, I'm experiencing an entirely new journey.
Studying mathematics can completely change one's perspective on life.
My worldview underwent a 180-degree shift starting from March 13th. 
My life changed completely. 
That's how much studying mathematics can alter one's life.
When encountering a proof or calculation that I don't understand,I try to follow the teacher's guidance and move forward according to the provided proofs and calculations.
This flexibility allows me to progress in my mathematical studies based on my own comprehension and understanding.
I'm approaching mathematics with the mindset of Jiǔ, a character from "The Mathematician in the Wonderland," taught by Hakson, the mathematician from the Wonderland.
As soon as I understand the chain rule in differential calculus, mathematics becomes fascinating. 
Even if I forget the details of trigonometric addition formulas, differentiation, and integration rules and theorems, once I comprehend the fundamental concept of the chain rule, which is the basis of analysis, my confidence in mathematics increases significantly.
It is likely that the relationship between differentiation and definite integrals follows a similar pattern. 
Strang's 1.1.6 exercise is the most fundamental exercise that relates to the chain rule and the relationship between definite integrals and differentials, which is why I'm struggling with it.
On April 28th, I decided to watch "The Mathematician in the Wonderland" at Fushimi Million Theater in Nagoya.
It will be a reward for myself and a way to kickstart my serious mathematical studies. 
I would like to invite my sister, who is suffering from a mental illness, but going to the movie theater together is still not feasible.
As Haksun says, mathematics requires courage. 
Smart people give up quickly.
Next, those who make an effort drop out.
Mathematics requires courage.
It's about having the mental space to think, "Let's try solving it again tomorrow."
Hakson says that it takes courage and the joy of taking a step into a world of unfamiliar logic, an unknown logical world.
He says that the ability to perceive this courage and joy is the talent for mathematics, and it extends to life.
On April 28th, I decided to watch "The Mathematician in the Wonderland" at Fushimi Million Theater.
I have understood Strang's 1.1.6. By delving into the relationship between definite integrals and differentials, as well as the chain rule, and carefully examining the details, I will be able to progress.
I want to study mathematics purely under Montiel's guidance.
Even if I can only obtain a doctoral degree in the history of geometry, I want to pursue a pure mathematics doctoral degree.


23th
Last summer, I participated in the IMS International Musicological Society International Congress held in Greece. 
I spent over a month visiting various places in Greece, including the Acropolis, Daphni Monastery, Athens Epidauvrus Festival, Delphi ruins, Meteora monasteries, and Samos Island. 
During my solo trip to Greece, I filmed eight videos of Byzantine chants and engaged in fieldwork on Byzantine chants and Byzantine icons.
This experience of traveling alone in Greece for a month has been transforming my life in a 180-degree manner.
Now, I am preparing to pursue a Ph.D. in mathematics at Georgia State University with mathematician Mariana Montiel as my supervisor.
The study of mathematics is changing my life in a profound way.
I'm almost done with integration. I'm likely to move on to vectors from today.
I still have some areas that I need to review, such as trigonometric addition formulas and differentiation rules that I may have forgotten.
However, I have almost mastered the chain rule for differentiation, so by supplementing with exercise and studying Strang, I should be able to reinforce my understanding. 
Once I finish the section on vectors and matrices, my review of high school mathematics will be mostly complete. 
The process of chain differentiations is fascinating.
Understanding the chain rule makes mathematics much more enjoyable.
By revisiting high school mathematics, including limits, infinite series, differentiation, integration, vectors, matrices, and complex spaces, my life has changed 180 degrees.
A completely new life is unfolding.That's how enjoyable studying mathematics is. 
That's how hopeful studying mathematics is. 
Without learning mathematical concepts such as limits, infinite series, differentiation, integration, vectors, matrices, and complex spaces, I feel ashamed of my past life and what I have been living for. Mathematics is my calling.

24th.
Today, I will finish the section on vectors. I have almost completed the section on normal vectors as well. I, who had to start over from Junior High School Mathematics, such as calculating the area of a circle or prime factorization, have reached vectors in High School Mathematics III in just three weeks of substantial studying since March 13th.
These three weeks have been the best time of my life, like a dream, the most fulfilling and happiest time. 
Everyone has different callings, destinies, talents, and missions.
Once I understand the orthogonal conditions of the dot product and projection of vectors, I will move on to matrices in one or two more days. 
Mathematics is not a subject of memorization but a subject of accumulation. The study of mathematics itself is teaching me the most important thing in life. 
If I don't understand the teacher's explanations, instead of stubbornly pondering over it or wasting time, I first trust the teacher and faithfully follow what the teacher says. 
I write down the teacher's proofs without question, almost memorizing them through a rigorous and Spartan approach.
If I don't understand something, the first basic premise is to trust the teacher and diligently write down what the teacher says, Spartan-style, without stubbornly insisting on my own inadequate thoughts.
Mathematics is an accumulation.
If you can't grasp the basics, then diligently and Spartan-style, hammer the basics into your mind until you understand. 
This is the most important thing.
Once you overcome a significant obstacle, like chain differentiations, mathematics becomes even more interesting and enjoyable.
Therefore, don't think too seriously about these matters. 
Even if I think deeply about it, there is no way I, someone like me, can handle it all. 
First, trust the teacher, faithfully write down and accumulate proofs and theorems.
Once I grasp the orthogonality conditions of vector,Ican immediately delve into matrices.
To speed up this process, vectors and matrices are handled simultaneously. 
As for the equation of a circle, is it covered in High School Mathematics II? 
It can be derived directly from the Pythagorean theorem as x^2 + y^2 = r^2. I didn't know this equation for a circle, or perhaps I had forgotten it. 
Did I learn it in high school mathematics?
If you know the Pythagorean theorem taught in junior high school mathematics, which is fundamental in Cartesian algebraic geometry, you can derive this equation on your own, right? 
However, while reading the proof of the formula for finding the volume of a sphere using integral calculus,I couldn't understand it immediately.Yet, I quickly searched the internet and found the answer.
I have just finished studying the proof of the formula for deriving the volume of a sphere as an integral of the area of a circle. Integral equations are easy.
On the other hand, as expected, differential equations, like the one in Strang 1.1.6, are difficult. 
Without a solid grasp of the chain rule in calculus and the relationship between differentiation and definite integration, it is difficult to understand. I have now moved on to vectors. 
Next, vector-matrix interactions on page 80.
I still need to review the transformations of quadratic function graphs. However, in almost three weeks and 21 days of study, I have completed calculus and reached vectors and matrices. 
Mathematics is my calling.
I've learned a few tips for studying mathematics. 
1. Write down proofs and theorems meticulously.
2. Don't fret or get frustrated if you forget theorems or formulas.
It's more important to have the courage to enjoy the process of thinking about unknown logic, rather than memorizing theorems or formulas. 
Have the courage to enjoy the process of discovering unknown logic itself. 
3. Be consistent and disciplined in studying every day. These are the three tips.

25th.
Matrices and vectors, like differentiation and integration, are inverse operations. 
Limits, infinite series, differentiation, integration, vectors, matrices. When you earnestly study mathematics, the world looks completely different.
The direction of life changes 180 degrees. Inner products of vectors, orthogonal projections, component calculations, they are truly interesting.
My textbook introduces matrices as a method for solving systems of linear equations and also covers the δ-ε definition from university mathematics in calculus.
Although it is a textbook for people like me, middle-aged men from small companies who are reviewing high school mathematics to take qualification exams, the level is high. 
I had to start over from factoring in junior high school mathematics and calculating the area of circles, but I almost finished studying up to Matrix Calculus in just over three weeks.
Since trigonometric functions and exercise on differentiation and integration are significantly lacking,
I have to reinforce my learning while using Strang's book.
Today I started studying matrices. Matrices and vectors, like differentiation and integration, are inverse operations.
Mathematics has been a subject of admiration and uncertainty at the turning points of my life at the ages of 14, 16, 18, 27, 32, and 35.
But this time, at the age of 55, I learned logarithms, trigonometric functions, limits, infinite series, differentiation, integration, vectors, matrices, and even factoring in junior high school mathematics up to High School Mathematics.
When I first learned logarithms on March 13th, it felt like memories of a completely different life, as if from a previous existence. 
Matrices are directly related to linear algebra in university mathematics.
I have gone through six high school textbooks and Strang's book in three weeks. 
My textbook doesn't cover complex planes. 
Probability and statistics should be fine for now. 
I still have to learn the proof that (a^2) is equal to the absolute value of 'a' by looking at the teacher's proof.
Now I've started studying inverse matrices.
It's a proof problem of solving a system of four simultaneous equations using inverse matrices. 
I'm already studying linear algebra in university mathematics. 
Currently, I'm studying Hamilton's theorem. Matrices' identity matrices and inverse matrices give a hint of groups and rings.
Even though I haven't been able to overcome Strang 1.1.6 yet, I can sense the aroma of abstract algebra in university mathematics, and it's enjoyable. 
Currently,I'm on my sixth high school textbook and the first Strang book. 
Despite being sloppy, I meticulously organize and store language vocabulary searches, math notes, papers, memos, and analysis data backups with a level of meticulousness that is probably many times more than what an ordinary person would do.
Accumulating math notes is as enjoyable as accumulating books or translated literature because it holds hope.
Only linear transformations and rotations remain for matrices. 
Tonight, I will most likely finish studying up to Matrix Calculus, covering approximately three weeks of study from March 13th.
I will be able to solve systems of simultaneous linear equations using matrices and understand proofs by induction. 
I'm currently studying the basics of linear algebra by using identity matrices and inverse matrices to solve systems of simultaneous linear equations.
I am amazed by the exercise in elementary linear algebra that involve these matrix operations for solving systems of simultaneous linear equations. 
I'm also amazed by the elegance of chain differentials and the most magnificent function, y=e^t.
In a one-step transformation, understanding the calculations involving matrix addition can be laborious.However, if I calmly think about it, I can understand it quickly.
Basic concepts of mathematics like this become immediately clear when I calmly think about them.
In the field of musicology, there has been a significant turning point towards mathematical music theory, which has rapidly advanced in Europe and the United States over the past 40 years. 
Mathematical music theory itself is undergoing a significant transformation from Mazzola at the University of Minnesota to Mariana Montiel at State University Georgia.
Professors such as Barry Cooper at University of Manchester, William E.Caplin at McGill University,Guerino Mazzola at University of Minnesota are all over 70 years old now.
Currently, I am in the process of a transition from studying musicology to pursuing a Ph.D. in mathematics under the guidance of Mariana Montiel, who is a mathematician, a woman, and originally from Mexico.
I couldn't pursue graduate studies in musicology, so I will definitely pursue them in mathematics.
I feel even more strongly that my vocation lies in mathematics.

26th.
In the composition of the transformation gf and fg, which are symmetric with respect to the x-axis and y-axis, why does the commutative property hold? 
Is it because the original matrices contain elements similar to the identity matrix E?
I have started to vaguely understand that Euler's formula, which uses the imaginary unit i to treat trigonometric functions such as sin^2 a + cos^2 a = 1 and y = e^t as complex functions in complex space, is a theorem in the field of analysis.
Today, rotations are complete. It has been a fun three weeks. 
These three weeks have completely changed my life, as I have clearly recognized that mathematics is my vocation.
The formula for considering the third and fourth quadrants of trigonometric functions around 270 degrees, as mentioned in my textbook, is not there. 
You can calculate it even without considering 270 degrees as the center.
The tips of "STC stock goes up, halfway at 90 degrees, sin and cos are back-to-back," is in every exam reference book.
Of course, is this formula with 270 degrees as the center included in the Blue Chart?
However,I created my own formula today, calculating with 270 degrees as the center. 
I find it enjoyable to think thoroughly about these things until I am satisfied.
Mathematics is a discipline where talent is determined by whether you find joy in thinking deeply about such things or not.
That's why Haksun said, "Smart people give up quickly. Those who give up next are those who make an effort. Mathematics requires courage. Even if you can't solve it right away, it's about having the composure to think,
'Hmm, this is a difficult problem. Let's try solving it again tomorrow.'" It's not about giving up immediately or studying frantically; it's about having the courage and composure to think, 
"Let's try solving it again tomorrow," without being overwhelmed. I haven't reached the point where I can independently approach trigonometric functions as functions on  from the sin and cos axes rather than from the unit circle .
Regarding the fact that the commutative property does not hold in the composition of x-axis symmetric function f and y-axis symmetric function g, 
I understand the reasoning that the commutative property does not hold in matrix multiplication, but in the actual composition of these x-axis and y-axis symmetric functions, they become symmetric with respect to the origin O, so the commutative property holds.
In fact, even if you reverse the order of the matrices F and G representing the functions f and g, the product of the matrices representing the composition of these f
Inverse matrices and matrices, through the arithmetic of multiplication directions and order, yield completely different results, just like function compositions, where the commutative property does not hold.
Once I finish the rotation of matrices tomorrow, I will take a break.
In these three weeks, I have covered almost everything from prime factorization in junior high school mathematics to matrices in High School Mathematics III.

"Courage."
"What mathematics requires is not intelligence or effort.
Intelligent people give up quickly. Those who give up next are the ones who put in effort. What mathematics requires is courage.
Mathematical courage. Instead of getting frustrated when unable to solve a problem, having the composure to think, 'Wow, this problem is difficult. Let me try solving it again tomorrow.'
That composure is called 'mathematical courage'."
This is the teaching of Hackson, the protagonist, a genius mathematician who defected from North Korea, from the movie "The Mathematician in Wonderland"
When I understood the process of the chain rule in calculus and the chain differentiation calculation in high school mathematics during my two weeks of studying mathematics from March 13, 2023, the process of understanding the chain rule in calculus and the chain differentiation calculation in high school mathematics was exactly as the genius mathematician and math teacher described it.
I have gained some confidence in the fact that "mathematics" is my calling and my path, which is to pursue a Ph.D. program in mathematics at Georgia State University under Mariana Montiel.
The study of mathematics that I started on March 13 began again with prime factorization in junior high school mathematics, and in about three weeks of learning, I have covered up to rotations of linear transformations in High School Mathematics III. 
I have also delved into Strang more seriously.
However, there are still many hurdles that I must overcome and exercise that I must solve before I can pursue my path to the Department of Mathematics at Georgia State University for admission.

27th.
I realized that Strang 1.1.6 is not about integration constants but about arbitrary constants C. 
I also understood that y=Ce^at is the most important and crucial general solution for differential equations, and the chain rule is related to another general solution, y=e^(-t).
So, I decided to move on to the next exercise problem, 1.1.7.
I realized that once I understand high school mathematics up to matrices, I can more or less keep up with university mathematics in Strang.
I realized that as long as I thoroughly understand the introductory explanations in the university mathematics textbooks, I can manage to understand the remaining explanations.
Mathematics is all about practicing calculations with a pencil on paper, solving exercise problems, and copying the teacher's proofs with a pencil to the notebook. 
It's about diligently and consistently solving exercise every day, following a routine. I understood that this is the fastest way to improve.
Mathematics is my calling. From prime factorization in junior high school mathematics to matrices in High School Mathematics III, I covered it in three weeks and now I'm fully immersed in Strang.
However, there are still many hurdles to overcome and exercise to solve before pursuing my path to the Ph.D. program in mathematics at Georgia State University.
On April 27, I finally understood that Strang 1.1.6 is about differential equations where C is an arbitrary constant, not an integration constant.
Strang is designed for MIT freshmen, so it's difficult. 
However, there are many exercise on differential equations written by Japanese teachers that even someone like me can understand.
Moreover, they are written in a clear and easy-to-understand manner in the latest textbooks, which I have made three copies of.
The boys, first-year students at Nagoya University of Engineering, did not learn matrices in high school; 
They only studied complex planes.These boys also mentioned that they didn't understand Strang 1.1.6. 
It seems that some high schools teach differential equations while others do not.
There were also high school students preparing for entrance exams who learned trigonometric functions by looking at the unit circle, the sine axis, and the cosine axis at Nakagawa Ward Library's study room.
From these trigonometric functions, they will move on to complex function theory and complex analysis.
In my mathematics textbooks, I mainly use materials from Springer and MIT. For the difficult parts that I don't understand, I make copies of several subtextbooks, written by Japanese teachers, which contain many simple exercise and explanations that are easy to understand.
Since I skipped differential and integral calculus and went straight to Strang, I still couldn't understand the introduction to Strang. 
I learned during Strang 1.1.6 that in university mathematics textbooks, if I don't fully understand the introduction,I won't be able to understand the subsequent explanations. 
Conversely, if I fully understand the introduction, I will be able to understand the subsequent explanations.
I am looking forward to tomorrow, the premiere of the movie "The Mathematician in Wonderland."
It's been so many years since I last watched a movie. Mathematics is my calling.
For now, I plan to focus on understanding Strang 1.1.6 while using the following materials: "Introduction to Analysis" by Springer (two volumes), "Introduction to University Mathematics" by Bluebacks (two volumes), additional exercises from the Blue Chart series, other high school mathematics books, and the Mcgrawhill series.
I am aiming for admission to State University Georgia while self-studying. Topics like the δ-ε definition, Dedekind's concept of number theory, and Weil's number theory are still difficult for me.
On the other hand, I might be able to grasp David Cox's Galois theory.

28th.
"The Family of Solutions."
In Strang, the general solution of ordinary differential equations is presented without any explanation, using the term "family of solutions."
Therefore, it is on a completely different level compared to other textbooks.
It does not explain how to find solutions to differential equations, how to obtain general solutions corresponding to the number of orders, or how to integrate to obtain general solutions.
It assumes that you already know all the basics of these differential equations and introduces the concept of "family of solutions" without any explanation, including the concepts related to mappings and sets.
Indeed, Strang is incredibly advanced and difficult. It mostly skips explaining the basics of differential equations, such as general solutions, particular solutions, orders, degrees,and the calculations required to obtain general solutions or particular solutions based on arbitrary constants and initial values.
It assumes a level equivalent to first-year students in the mathematics department at MIT and therefore omits explanations of such basic knowledge of differential equations, making it extremely difficult.
It was only after reading a book on differential equations by a Japanese teacher that I started to grasp the true meaning of exercise 1.1.6.
The book doesn't explain that second-order differential equations always have two general solutions and that particular solutions are obtained by determining initial values from arbitrary constants.
Instead, it introduces more advanced concepts without explanation from the beginning, making it extremely difficult for beginners like me who have only recently covered topics from prime factorization in junior high school mathematics to matrix determinants in High School Mathematics III,within just three weeks. 
To fully understand the first topic, I need to comprehend another level of higher-level concepts.
However, I'm almost there. If I mainly focus on Strang and use.
"If I focus on Strang and study it as an extension of high school mathematics, using three of the latest textbooks written by Japanese teachers specifically for current university students, you can manage to learn Strang on your own.However, Strang is extremely difficult because it directly presents concepts like y=e^t and the beauty of functions in analysis through a limited explanation and packed with 1050 exercise problems.
But Strang excels as a textbook because it effectively conveys the beauty of functions like y=e^t and the fascinating aspects of analysis, even to beginners like me.
To solve differential equations, I need to perform integration calculations. Unfortunately, my high school mathematics textbook didn't cover differential equations. 
So, the fact that y=Ce^t and another solution is y=e^-t in Section 1.1.6 is related to integration calculations, not the chain rule as I mistakenly assumed.
The constant C is also an arbitrary constant, not an integration constant. 
I currently understand Section 1.1.6 to this extent.
Today is the day I'm going to watch the movie "The Mathematician in Wonderland." 
I wonder how many years it has been since I last watched a movie.
It's probably been almost 40 years.
I really want to watch this "Mathematician in Wonderland" no matter the cost, even if it costs 1,900 yen.

29th.
"Blue Chart." I solved exercise from the Blue Chart's Okayama University Level 1, which has a difficulty level of 2.
Being able to solve a problem from an actual entrance exam gives me confidence.
I consider the conjugate complex number 0-0i when considering the complex number 0+0i with a real part of 0, right?
The proof of the property of the conjugate complex number α-α^n being the conjugate complex number of α^n, which is not covered in the Blue Chart or the high school mathematics textbook 
I used for review, is still unknown to me. The Blue Chart deals with Cardano's formula for solving cubic equations.
There are countless things I need to do, such as researching the properties of calculations involving the powers of conjugate complex numbers.
Learning mathematics has completely changed my life. 
It turns out my textbooks skipped things aimed at working adults.
I need to dedicate myself to the Blue Chart until autumn and be prepared to spend three years on Strang.
I need to go through Blue Chart III, solve exercise on trigonometric and exponential functions in Blue Chart II,and handle other topics as needed.
It turns out that the polar form means considering complex numbers in the complex plane using trigonometric functions.
Isn't it easier to think of the complex plane using vectors? 
Today, I will start studying the complex plane in Blue Chart III.
I can already solve the exercise problems in the Blue Chart.
Instead of skipping the proofs in the Blue Chart,I will write them down in my notebook with a pencil and study them. 
Since differential equations are mentioned at the end, if I can solve all the problems properly and come up with a supplementary textbook to reinforce the difficult parts in Strang, 
I can smoothly transition from the Blue Chart to Strang.
I'm still confused about the explanation of the concept of a solution family in Strang and the fact that the arbitrary constant C in Section 1.1.6 multiplies the coefficient. 
I'm completely lost.
I now understand that the term "family" is a concept in set theory. 
It took me three weeks and copying 120 books to understand that C is an arbitrary constant and not an integration constant.
My mathematics teacher said that high school mathematics is very important.
I finally understand why the mathematics teacher says that high school mathematics is the foundation for university mathematics and why high school math is considered difficult.
Like Jiwoo and Haksun, sharpening pencils with a knife brings peace to my mind.
I learned complex planes properly for the first time. The complex plane is also called the Gaussian plane.
High school mathematics is indeed difficult, as the teacher said. It is a complex number space created by combining and considering algebra, analysis, geometry, and number theory.
Just being able to learn the complex plane properly, which I couldn't do in high school, makes all the effort of studying from the "Blue Chart" worthwhile.
Even if I can't do it, mathematics is fun.
I still need to review the conjugate complex numbers and the equivalence of complex numbers.
Studying mathematics can completely change my life.
Thanks to reaching vectors and matrices, I can now understand the relationship between the complex plane and vectors.
Fortunately, I can now understand almost all explanations in the Blue Chart for the complex plane.
After solving all the problems in the Blue Chart, it would be good to proceed with Strang's book.
I need to start over with the complex plane from the Blue Chart.
Even the explanations and exercises for Blue Chart III Differential Equations still feel difficult to me, so it becomes clear that it was impossible to jump straight into Strang without first understanding the concept of the general solution of differential equations from the notion of a family of solutions.
It will take until autumn, September, or October to complete the Blue Chart.
Then I'll move on to Strang, so it will still take three more years.
My mathematics teacher told me that if I want to study abroad as a mathematician, it is essential, at the very least, to study "Linear Algebra and Differential Equations" from the perspective of undergraduate study and admission. 
Fortunately, I can understand almost all of the explanations in the Blue Chart III for the complex plane.
Anyway, if I patiently and diligently solve all the exercises in the Blue Chart, I should be able to proceed with Strang.
When I look at Strang's exercise problems and explanations, I understand about half of them, but the other half is completely incomprehensible.
In particular, the concept of a family of solutions explained in relation to the general solution of differential equations is completely incomprehensible to me.
I still find it half incomprehensible regarding the two solutions of y = Ce^t and y = e^(-t) for the arbitrary constant C in 1.1.6. I've decided to go back to the Blue Chart and steadily solve the exercise problems.
State University Georgia's QS World University Ranking is around the level of Juntendo University in Japan, in the 800s.
By solving all the exercise in the Blue Chart and completing all of Strang's exercises, the path to studying abroad in the undergraduate program should open up.
The Blue Chart III has 480 pages. For now, I've decided to solve all the exercises.
If I can maintain the pace from March 13th, I should finish within one and a half months, by mid-June. Although matrices and vectors are not the highest priority, 
I must solve all the exercises for sequences, complex numbers, trigonometric functions, and exponential functions in Blue Charts II.
If I have to redo all of Blue Chart II, it will take twice the time, about three months. 
Finishing Blue Chart III and II, with a total of 1000 pages, will take until August.
After completing this part, I plan to move on to the next stage, which is Strang.
Especially in regular university mathematics textbooks, there is no explanation of the relationship between the "general solution" and the "family of solutions" in differential equations.
The terminology and concepts of university mathematics are completely unfamiliar, and they are used from the beginning without any explanation in Strang.
I finally understand that it is a tough path to even grasp whether I can understand it by solving just this much.
It will be a great achievement if I can start Strang in autumn, from September.
However, it will still take me three more years just to solve all 1,050 exercise problems in Strang's 2014 Linear Algebra and Differential Equations for admission to the State University of Georgia's Department of Mathematical Sciences. 
Depending on the case, I may have to solve all the problems in High School Mathematics I, such as quadratic functions and trigonometric ratios, and if I have to redo everything up to High School Mathematics I,it may take another month or two.
So, entering Strang is a tough road to finally realize.
For now,I've decided to go back to the high school mathematics study that students do with Blue Charts.
At least,I will focus on topics related to calculus and linear algebra: infinite series, differentiation, integration, matrices, and vectors.
I am starting to understand what differential equations are, that second-order differential equations have two solutions, and that integrating solves differential equations.
I have a partial understanding of definite integration and initial values. However, Strang provides a completely different level of explanation for the "family of solutions" and the general solution compared to other textbooks,making it still too difficult for me at the moment.
It's like a child who has never even studied music theory suddenly trying to play a complex piece. 
For now, it's back to the Blue Charts.
"A Mathematician in Wonderland"
I was truly moved. It's been years since I watched a movie and cried like this. 
It was an amazing movie that shook my heart to the point where my life would change 180 degrees. 
Maybe it was because I hadn't watched a movie in a theater for about 30 years? 
Among all the movies I've seen, 
I've never been so touched.
It's a story about a genius mathematician who defected and became a high school security guard, and a math underachiever from a low-income single-parent household. 
There is also a heartwarming and gentle romance with a classmate girl.
In the end of the movie, this Jiwu will become a mathematician three years later.Haksun is portrayed as having proved the Riemann hypothesis, and there are political motivations involving North Korea claiming that South Korea kidnapped Haksun because he solved the Riemann hypothesis,
which deals with prime numbers and can be used for cryptographic purposes. 
There is also a political plot where a junior who defected also tries to bring Haksun back to North Korea.
From the anecdote in the history of mathematics where Riemann manually calculated route 2, Haksun teaches Jiwu to calculate equations manually, encouraging him to become familiar with equations.
Jiwu learns mathematics from Haksun in a warehouse,listens to Bach's Cello Suites together with Haksun, and each scene is beautifully engraved in the mind. 
The scene where Haksun  thinks of  his son as father,and  struggles with the agony of losing his own son who was shot by the military while trying to cross the border,really touches the heart. Even during the third screening on the first day, the theater was nearly empty. The most beautiful equation in the world, Euler's identity, e^
The theater was empty even for the third screening on the first day.
The most beautiful equation in the world, Euler's e^iπ+1=0, was also featured. 
This movie was worth watching. 
I want to collect not only the pamphlet but also the script and other materials related to this film. 
The episode of the anguish of Hakson's son, who was shot by the military while trying to cross the border, made me cry as I thought about my own son.
The music, including Bach's Unaccompanied Cello Suites and the Pi song, was also magnificent. 
Starting with Bach and ending with Bach, using the radio brought by Hakson after his defection, Jiu listened to it on her smartphone, amplifying the sound by placing the phone in a cup, and let Hakson hear it. 
There were many foreshadowing episodes that touched the heart, it was truly wonderful. 
It was not just about mathematics; it was a truly thought-provoking film. Hakson, a North Korean mathematical genius who defected in search of freedom, and Jiu, a math underachiever from a low-income single-parent family who learns mathematics from Hakson and eventually becomes a mathematician.
Last summer, when I went to Greece, I also felt the reality of North Korean defectors, along with the fierce competition in China and South Korea's exam-oriented society. 
This film really makes me think about various things.
In the end, Hakson exposes the teacher's leaked exam questions to prevent Jiu from transferring schools.
In my 55 years of life, this may be the first time I've been so moved.

"April 30th.
"Family"
Finished studying De Morgan's laws for sets. Now able to perform calculations with De Morgan's laws, union, and intersection.
Finished Cauchy–Schwarz inequality, only the arithmetic and geometric mean remains.
Focus on binomial theorem and multinomial theorem for probability.
Tomorrow is proof by contrapositive. It will take about a week to review complex plane's polar form and arguments.
The enjoyable aspect of mathematics is that it always humbles you. Mathematics requires courage.
Even if I redo everything from the arithmetic and geometric mean, De Morgan's laws, proof by contrapositive, binomial theorem, and probability statistics, there are still 80 pages left.
When I go back and redo it, the parts that were previously unclear become precise, and the parts that I didn't understand become clear.
Today, while solving exercises on the complex plane in the Blue Chart Math III, I encountered proofs that I couldn't understand without knowing proof by contrapositive, so I went back to sets and propositions.
I explained the binomial theorem and multinomial theorem in probability statistics in my self-review high school math textbook.
I learned for the first time that 1 is not a prime number.
Of course, 0 cannot be divided, so it is not a prime number either.
I learned for the first time that prime numbers are natural numbers that have no divisors other than 1 and themselves.
Multinomial theorem is used in probability statistics, right after the binomial theorem.
In my self-review high school mathematics textbook, I explain the multinomial theorem after the binomial theorem in probability statistics.
Now I am transcribing the proof of Cauchy–Schwarz inequality.
If you can feel that mathematics is what you love the most, then you have talent in mathematics.
I'm redoing De Morgan's laws for sets and proof by contrapositive. 
Studying mathematics can turn your life around by 180 degrees.
You must diligently build up one by one, and dishonest tricks will never work. Mathematics teaches you that.
After redoing De Morgan's laws for sets, proof by contrapositive, proof by contradiction, and mathematical induction, 
I thought I could leave probability statistics for later. 
But then I learned that I need to study binomial theorem and multinomial theorem in probability statistics, and I realized it will take another three years to redo everything up to probability statistics.
By redoing the parts I tried to deceive, I can definitely make progress.
Conversely, mathematics teaches me how harsh the consequences of deception can be.
Learning mathematics can turn your life around by 180 degrees.
Cauchy–Schwarz inequality, De Morgan's laws, proof by contrapositive, proof by contradiction, and mathematical induction, these five topics on sets and propositions need to be redone.
I realize that I made a mistake in thinking that I could skip probability statistics and leave it for later.
I realize that without understanding binomial theorem and multinomial theorem in probability statistics, I won't be able to advance to the level of algebra in university-level mathematics.
If I understand the polar form and arguments in the complex plane, I can understand Euler's formula, e^iπ = sin z + i cos z, which is the original complex analysis of the most beautiful equation in the world, e^iπ+1=0.
I have come from prime factorization in middle school mathematics to the polar form and arguments in the complex plane in three weeks.
Today, my goal is to dive into the polar form and arguments of the complex plane."
I'm not just an ordinary high school boy, so if I complete the Blue Chart III and focus only on the trigonometric functions in the Blue Chart, I should be able to grasp the next steps somehow. 
Today, I want to delve into the polar form of complex numbers.
When entering the polar form of complex numbers, the term 'polar form' appears frequently in university mathematics books, which makes it feel more like advanced mathematics.
I admire differential equations and the concept of polar form.
"The process is more important than the answer.""Courage." "You won't get the right answer from a wrong question."
I will use the terminology of function families and curve families. 
It is said that the curve family represented by ax^2 + by^2 + 1 = 0 has solutions.
"The solutions of differential equations can be thought of as a family of functions. 
When considering its graph, it becomes a curve family. 
From the curve family, we can obtain the general solution of the differential equation."
There was a syllabus on differential equations at Gifu University that touched on the concept of solution families.
The level of explaining differential equations as mathematics seems to be familiar to MIT mathematics freshmen.
The level of explanation in the textbooks for reviewing high school mathematics, including the Blue Chart, is completely different. 
The use of the term "solution family" may not be directly relevant to the exercises, but it is necessary at this stage to teach what analysis and mathematics are, so it is probably used from the beginning. 
In that sense, this syllabus from Gifu University seems to be at a high level.
The term "solution family" does not refer to the general solution itself. It explains the order of the thought process in analysis and mathematics, starting from function families and curve families to obtain the general solution.

May 1st.
I finished the arithmetic mean and geometric mean. After finishing the contrapositive proof,proof by contradiction,and binomial theorem,I plan to move on to the polar form and argument of complex numbers.
If I cover factorization to matrices within three weeks,I feel like I'm finally grasping the essence of math studies. 
Finally, yesterday,I learned how to calculate the intersection and union of sets using De Morgan's laws in set theory.
Starting today,I will focus on the contrapositive proof,proof by contradiction, and the binomial and multinomial theorems for probability.
After completing the binomial and multinomial theorems,I will return to the polar form and argument of complex numbers in the Blue Chart.
Once I finish the binomial and multinomial theorems in Takaoka Kazuo's "High School Mathematics Revisited," I can officially say that I have completed all of high school mathematics and proudly move on to the Blue Chart.
After finishing the polar form and argument of complex numbers in the Blue Chart, I can finally start understanding Euler's formula, e^iπ + 1 = 0, which is considered the most beautiful equation in the world. 
It will still take me another year to fully understand the concept of solution families in Strang's introductory differential equations, and it will take me three more years to reach the level of being admitted to the Department of Mathematics at Georgia State University with Strang's book, "Linear Algebra and Differential Equations," being an extremely difficult textbook designed for MIT mathematics freshmen.

May 2nd.
"Blue Chart Calculus"
I finally realized that the term "solution family" introduced by Strang without any explanation is a mathematical topic at the level mentioned in research papers at Kyoto University's Research Institute for Mathematical Sciences. 
It took me until January to understand this."Analytical Study of Statistical Mechanical Phenomena Revealed by Probability Processes and Solution Families Behind Partial Equation".
" [2011], [2013]. -- Kyoto University Research Institute for Mathematical Sciences, 2013.1-. -- (RIMS Kokyuroku; 1823, 1919).
In the upcoming Nagoya University Partisan, I will copy this reference. Also, find an introductory book on mathematics that seems to touch on the concept of 'family' in Tsurumai Library. I will copy this book tomorrow.
Following Montiel's advice, prioritize Blue Chart Calculus. Also,prioritize Strang 1.1.6 and the Δ E method.I will make triple copies of the SGC's Δ E definition in the next round of copying.
I made triple copies of Strang as well. 
However, I stumbled upon the concept of 'family of solutions' in set theory right from the start. 
Montiel said, 'Study the δ ε definition quickly. Rather than getting caught up in Strang's family of solutions or differential equations, focus on mastering the δ ε method as soon as possible.
Hammer the δ ε definition into your mind. Immerse yourself in the δ ε definition.'
Following Montiel's advice, even if it's a long shot or confusing, I will start the δ ε definition today. While delving into the δ ε definition, I'll continue with Strang.I'll revisit topics such as complex plane,polar form,argument,binomial theorem, multinomial theorem,and other items from Blue Chart Mathematics III as needed.
For ambiguous topics like trigonometric functions in Blue Chart Mathematics II, I'll reinforce my understanding as needed. As for the concept of 'family of solutions' that Strang used right from the beginning in differential equations, I found another book at the Nagoya University Central Library that deals with it. 
Genichiro Hara, Hideaki Matsunaga's 'Complete Guide to ε δ method,' and Michiyo Nakane's 'ε δ Method and Its Formation' (this one is a book on the history of mathematics, viewing the δ ε method from the perspective of mathematical history, specifically the analysis of Cauchy and Weierstrass).
After that, the SGC series 'δ ε Method,' and if I can find it in the midst of my home copies, I'll borrow it.
To learn, a teacher is essential. Self-study inevitably leads to arrogance. One can stray from the path. If I had understood that the coefficient 'C' attached to the first solution y=Ce^t, y=e^-t of the second-order differential equation dy^2/dx^2=y is not an integration constant but an arbitrary constant of the differential equation,it wouldn't have taken me three weeks to realize it by going the long way around.
As for the remaining review of high school mathematics, I need to cover several topics: geometric mean, proof by contradiction, proof by contrapositive, binomial theorem, set theory, logic, and probability statistics.
By learning the polar form and argument of complex plane, one can understand the most beautiful equation in the world, Euler's formula e^iπ +1=0. 
Looking back at my journey in calculus and vector matrices, the world appears completely different from when I started last autumn.
I feel as if I have been reborn. Mathematics is enjoyable. Mathematics is interesting. Mathematics is my calling.

May 3rd.
Thinking about his son who was shot by the military while trying to cross the border, I have cried, feeling as if I were myself in that situation.
It made him shed tears. It's been so long since I cried while watching a movie, it must have been years. 
Hakson, silently watching the pure and youthful love of Boram and Jiwoo, a high school couple, in the background. 
"How long has it been since I last saw such a refreshing young couple? I was truly moved by 'The Mathematician in Wonderland.' Everything felt as if it were about me. Everything seemed like a movie made just for me. Even as someone like me, who is being sternly told by Monseigneur to study calculus with the Blue Chart, I could sense that mathematics is courage. In my 55 years of life, I have never been so moved by a film. How long has it been since I watched a movie and cried? Where in the world can we find a mathematician who can confidently say, 'Mathematics is courage'?
I borrowed three mathematics dictionaries of Asakura,Springer at the Nagoya University Central Library.
The basics of mathematics were already borrowed by someone else.
For someone like me who still needs to catch up with calculus using the Blue Chart,research papers on families of solutions at the Mathematical Analysis Research Institute are like "Cast peals before swine". 
Conversely, my mathematical studies have made significant progress.I'm gradually starting to understand the arbitrary constant y=Ce^t for the two general solutions of the differential equation dy^2/dt^2=y in Strang's 1.1.6.
Mathematics is an accumulation of knowledge.
Mathematics humbles people. Learning mathematics can completely change one's life.
Today, I made copies of books that explain the terms and concepts of sets in mathematics. Two books from Asakura Mathematics Dictionary and the entry on families from Iwanami Mathematics Dictionary, making a total of three books.
Tomorrow, I have to make a copy of the term 'explosion,' which I still don't understand according to Strang's explanation. 
It took me three weeks to look up the term 'family,' a concept of set theory, in the mathematics dictionary. 
If one is truly serious about learning, strict guidance from a teacher is necessary. 
But now, I finally understand the joy of copying mathematics dictionary entries, which is 100 times more enjoyable than copying MGG materials.
Today, Nagoya University Central Library is open. 
The Science Library won't open until after Golden Week.

'Family of solutions'
In Strang's 2014 'Linear Algebra and Differential Equations,' the term 'family of solutions' introduced by Strang is used without any explanation in the research papers of the Kyoto University Research Institute for Mathematical Sciences.
It took me until January.
↓ Found it in the Science Library.
'Analysis of statistical mechanical phenomena behind partial differential equations and families of solutions: Joint Research at RIMS'
[2011], [2013]. -- Kyoto University Research Institute for Mathematical Sciences, 2013.1-. -- (RIMS Kokyuroku; 1823, 1919). <WB03187502>
The book about 'family' that I looked up on Wikipedia.
↘ Found it at the Nagoya University Central Library.
Foundations of Mathematics: Sets, Numbers, and Topology / Masahiko Saito. Tokyo: University of Tokyo Press, 2002.8
↓ Found it in Tsurumai.
Japan Mathematical Society 'Iwanami Mathematics Dictionary,' Iwanami Shoten, 1985
R.J. Wilson 'Introduction to Graph Theory, 4th edition,' translated by Takao Nishizeki and Yuko Nishizeki, Kindai Kagaku Sha, 2001
It seems to be related to the terminology and concept used in set theory.
Even in Tsurumai, there are three books on the δ-ε method. Today, I made a copy of the textbook 'Working with the δ-ε Method' by hand."

4th.
I researched the recruitment of mathematics teachers for high schools and junior high schools.
It turns out that both require a mathematics teaching license for junior high school and high school.
I realized that it's still impossible for me because I don't have a mathematics teaching license.

May 6th.
The multinomial theorem is not so difficult.
I should follow Montiel's advice and do the Δ exercises in the Red Charts textbook.
I barely finished Takahashi Kazuo. I managed to cover everything from prime factorization to the multinomial theorem in three weeks.
Just barely finished the binomial theorem. Takahashi Kazuo almost made it within three weeks. 
Combinations and sigma calculations in the binomial theorem are not so difficult.
I feel more and more that mathematics is my vocation. I should listen to Montiel as well.
Whether I should skip ahead to Δ-Ε logic as Montiel suggests or follow MIT's advice and focus on linear algebra, that's the question.
I still can't understand the law of product for case distinctions. But I'll forcefully hammer it into my mind and move forward.
Today, I'm stuck before the binomial theorem. However, the calculations for combinations n C r and permutations n P r are truly fascinating. I feel even more strongly that mathematics is my calling.
Today, I'll revisit permutations and combinations (P and C) using Takaoka Kazuo.
I skipped proof by contrapositive and proof by contradiction again.
Today, I'll start from the formulas for calculating nPr.
I finally understand the definition of differentiation and integration, the concept of limits, Δy, Δx, the change in variables, and the definition and approach to limits required to find limits using the definition.
I understand for the first time the fundamental aspects of high school mathematics' calculus, and I'm convinced. 
Of course, this is followed by the discussion of Δ-Ε logic in university mathematics, Dedekind's real number theory in analysis, and the continuity of real numbers. 
Thanks to Montiel's advice, I finally have a clear understanding of the definition of calculus and limits, Δx, increments in x, and the concept of change. Without a clear understanding of this, calculus becomes nothing more than memorizing formulas and techniques for solving derivative functions or inverse integration, and it does not lead to a fundamental understanding of differentiation.
Today, in order to learn the binomial theorem, I'll go back to Takaoka Kazuo and revisit probability, combinations (C) and permutations (P) before the Red Charts textbook. If I can't understand this binomial theorem, I won't be able to comprehend the proof of formulas for power function's derivative functions.
When you study mathematics, your perspective on life completely changes.
You learn firsthand the importance of accumulation and the futility of vanity.
In high school mathematics, only High School Mathematics II covers power functions' integration.
It's not until High School Mathematics III that you learn about sequences and limits and delve into the calculus of exponential functions,logarithmic functions,and trigonometric functions.
The reason why you study vectors and matrices before High School Mathematics III,rather than after the calculus of exponential functions, logarithmic functions, and trigonometric functions in High School Mathematics II,is because linear algebra is more essential than calculus according to MIT's Strang and others, reflecting the latest trends in university mathematics education. Following Monseigneur's advice, I'll redo all the exercises in Calculus II and Calculus III from the power functions' integration to matrices in the Red Chart textbook.
Since it's about 400 pages long, even if I can maintain the pace of completing the exercises from prime factorization to matrices in three weeks, it will still take more than a month.
The complex plane in the blue chart still feels difficult for me, so it will take more time.

7th.
I finished practicing the exercises on "proposition truthfulness," "negation," "necessary and sufficient conditions," and "mathematical induction" by Kazuo Takahashi. 
I managed to understand and write down the solutions to the exercises, including proving the formula for infinite series using mathematical induction. 
I started over from prime factorization and made progress within three weeks,reaching binomial and multinomial theorems. 
At the age of 55, I realized that mathematics is my calling and mission.
I just finished Kazuo Takahashi. Starting today, I will study Blue Chart Mathematics II: Calculus.
It was tough to cover De Morgan's laws, permutations and combinations, nPr, nCr, binomial theorem, and induction in just two days.
Now I'm studying proof by contrapositive. I don't understand the proof by contrapositive that came up in the exercise of Blue Chart Math III: 
Complex Plane, so I went back to Kazuo Takahashi.
For now, I finished De Morgan's laws, nPr, nCr, binomial theorem, and multinomial theorem.
It would be embarrassing and unsettling if I didn't know these.
I'll do my best to finish the proof by contrapositive and then move on to Blue Chart Math II: Calculus following Mon Cheri's guidance.

May 8th.
In mathematics, whether it's Kazuo Takahashi, the Blue Chart, or Strang, solving even one problem without looking at the answer gives me a tremendous confidence boost.

Blue Chart Schedule:

Math II: Calculus
Math III: Calculus, Complex Plane, and others
Math II: Vectors, Matrices
Math II: Trigonometric and Exponential Functions
Strang: 1050 problems
It took Kazuo Takahashi three weeks. The above schedule is what I need for the first three cycles. However, the explanations and problems in the Blue Chart are not as difficult as I thought.
It focuses on high school mathematics exam techniques rather than abstract university mathematics.
Around March 13th, even the Blue Chart felt difficult.
Now I can understand the Blue Chart to some extent.
I realized how lightly I regarded Strang. 
As I become able to handle calculations and proof problems in the Blue Chart, perhaps I'm one step closer to tackling Strang? Mathematics is interesting and enjoyable.
When you study mathematics, you realize the importance of building a solid foundation and the consequences of skipping ahead.
My math teacher always said that starting mathematics at an older age is very challenging and self-studying mathematics is difficult. 
However,I want to demonstrate that it is possible to start learning mathematics at the age of 55 and pursue a graduate education as a mathematician.
I still have to work incredibly hard on mathematics. 
I'm still struggling with the Blue Chart. However, I have made a little progress from Kazuo Takahashi to the Blue Chart in the past three weeks. 
I'm making small advancements.

May 8th.
I'm starting Blue Chart Math II: Calculus. I'll review the proof of the derivative formula for power functions derived from the binomial theorem.

May 9th.
At Nagoya University Central Library, there are no math books to copy. I have made about 3,000 copies in the past four years.
Montiel said  "It is stil not good enough". In Blue Chart,it is often said that you should use textbooks that are suitable for your level.
Studying mathematics at an older age is incredibly difficult.
I had to make 3,000 copies and copy equations rigorously until they became familiar to my body,and only then could I start studying.
With that, I was able to reach binomial theorem in three weeks.
Now,I can work on Blue Chart at almost the same pace.
Montiel says that we still lack textbooks for other subjects in mathematics such as complex plane in high school mathematics, complex function theory in university mathematics,and other subjects like number theory,algebra, analysis, geometry, set theory, and pool algebra for studying Strang.
It took me three weeks to understand that the solution to Strang 1.1.6, y=Ce^t, is not an integration constant but an arbitrary constant.
However, when you collect books on related subjects that are suitable for your level from the library and make copies, you gradually become familiar with equations and can solve doubts.
It always comes when you can solve them.
Strang is still too difficult for me at this point. 
As Montiel suggests, once I raise my level with Blue Chart calculus, I can move on to Strang.If I enter Strang,I will encounter δ-ε arguments. 
It is important to focus on raising the textbooks that are suitable for me at the moment and proceed through each stage of understanding.
It is more important to find it enjoyable rather than considering it as an obligation.
In Blue Chart,instead of expanding the equation, I learned to differentiate y=(2x+1)^3 using the chain rule as a composite function from the beginning,and I can now solve it in three weeks without even expanding the equation.
So if I steadily solve exercise problems, I can somehow clear Montiel's advice and Blue Chart calculus.
The next challenge is δ-ε arguments. It used to be said that if you don't understand δ-ε arguments, you shouldn't study university mathematics.
The actual differentiation using formulas to find derivatives is easy.
Understanding concepts like indeterminate forms and limits is more difficult. 
Instead of using formulas, I am solving problems that require finding derivatives and derivatives using the definition of derivatives and limits.
For integration, instead of deriving formulas, I am focusing on understanding the essence of differentiation, which is not about memorizing techniques for exams, but about finding areas as the inverse operation of differentiation, using the chain rule for composite functions, and understanding the essence of differentiation that is not explained much by Strang. 
I realized that in order to understand Strang's explanations, it is necessary to firmly instill the relationship between differentiation and definite integration into my body through rigorous practice.
If definite integration does not enter easily,
I cannot solve Strang. Therefore, Montiel's advice on Blue Chart calculus, which emphasizes differentiation and definite integration, and the chain rule for composite functions, is the most accurate as he is a mathematician. Especially for Strang's explanations,I realized that it is necessary to firmly instill the relationship between differentiation and definite integration into my body through rigorous practice. 
Montiel also emphasizes Blue Chart calculus, differentiation and definite integration, and the chain rule for composite functions, saying that I should instill these concepts rigorously. 
Of course, I also have to tackle δ-ε arguments, which are the first major homework in university mathematics.
Differentiation using d/dx and f(x) from the beginning is the method used in university mathematics.
It is better to get used to the method from high school mathematics to the description method used in university mathematics.

Seems like the 'Blue Chart'consists of three types of exercise: exam techniques, prelude to university-level mathematical analysis with indefinite limits, and problems to assess fundamental understanding. 
Just immersing myself in these problems is already fascinating for someone like me who didn't study high school mathematics at all.
In that sense, taking a thorough approach might be beneficial.
However, my goal is Strang, so what I should focus on now is speeding up, doing Spartan-style transcriptions, and internalizing the concepts.
There are two important aspects to mathematics: deeply contemplating problems on your own and transcribing the teacher's proofs and solutions to internalize them through repetition.
As my teacher,who guided me through mathematics, said, studying mathematics at an older age, like 55 in my case, can be difficult.
Memorizing and building upon each theorem and formula is incredibly tough. 
Therefore,I need to consider studying methods that are suitable for my age.
However,I want to demonstrate that pursuing a doctoral program in mathematics after turning 55 is possible,just like Fermat,who started studying mathematics alongside his career as a lawyer when he turned 40.
Regarding indefinite limits, even I, with my limited knowledge from the Blue Chart, understand that it relates to the definitions of algebraic operations for zero and limits in mathematical analysis.
However,I might not truly grasp this concept until I delve into university-level mathematical analysis. For now, I'll mimic the teacher's explanations and solutions, memorizing them blindly to move forward.
The calculations involved in these problems, such as ∞ - ∞, ∞/∞, 0/0, and 0∞, are all indefinite limits.Understanding why we perform such calculations might require reaching a more advanced level in mathematical analysis,so for now, I'll follow the teacher's guidance just like I did with Takahashi Kazuo and move forward. 
Division by zero (k/0) is undefined, so lim k/0, the limit of k/0, doesn't exist.It's defficult to decide whether to deeply contemplate the reasons behind something I don't know or to blindly follow the teacher's instructions and move forward. However, without making this decision, learning becomes impossible.
Just as I managed to make this decision by progressing from prime factorization to binomial theorem in three weeks with Takahashi Kazuo, I hope to reach a similar point.

May 10th.
Today, I finally cleaned the kitchen. After three weeks of studying with Takahashi Kazuo, I finally managed to understand composite functions and the chain rule, allowing me to perform differential calculations.
Now I have some time to clean the kitchen.
I reported this progress to my sister, as sharing knowledge is the best way to make progress.
I aim to become capable of teaching my sister about composite functions, the chain rule, and families of solutions to differential equations.
As part of my work as a musicologist, I have translated literature on Christoph Wolff's musicology regarding Bach and Newton and completed the translation of texts on Rudolf Haase's musicology regarding Hindemith's "Harmony of the Universe" and Kepler's geometric studies, as advised by Mariana Monsieur. Starting from March 13th, I dedicated myself to self-study using Gilbert Strang's "Linear Algebra and Differential Equations," the textbook for MIT's first-year mathematics students.
However, considering that I never properly studied even the basic high school math topics, such as prime factorization, required for music high school and art university, it becomes clear that Strang's book, which uses set theory terminology like "families of solutions" right from the beginning, is still too challenging for me.
Therefore, it is clear that I need to start over with high school mathematics.
I will use "Mou Ichido Koko Sūgaku" by Kazuo Takahashi as my textbook,and starting from March 13th,I will spend three weeks studying and covering almost all topics of high school mathematics,including quadratic functions,logarithmic functions, exponential functions, trigonometric functions, sequences, infinite series, limits, derivatives, integrals, matrices, vectors, sets, statistics, logic, and everything except complex planes.
However, I still find it difficult as I don't understand exercise problem 1.1.6 and why second-order differential equations have arbitrary constants.
Therefore, following Mariana Montiel's advice and instructions,
I will switch to studying Blue Chart Calculus, starting from May 8th, focusing on power functions in Calculus II.

May 11th.
Recovery of Nagoya University Central Library after three years.It's impressive. Today, I copied a research paper from Kyoto University's Mathematical Analysis Research Institute that uses the term "family of solutions."
I have no choice but to keep up with Montiel and tackle Strang. It's really tough for me,an older person,to study mathematics through Strang.
Anyway, the two main learning priorities are to understand the complex plane from Blue Chart and to solve as many Strang exercises as possible.
Whenever I encounter an unclear or unsolvable problem, I will go back to the Blue Chart and review the formulas and theorems.
I will imagine myself taking Strang's lectures on linear algebra and differential equations as a first-year student in the Department of Mathematics at State University Georgia,just as Monchelle advises.
Solving Strang's exercise is the top priority. Even if I understand at the level of Kazuo Takahashi's "Mou Ichido Koko Sūgaku" or if I finally understand the differentiation of composite functions and chain rule calculations, or if I finally understand binomial theorem, it doesn't matter. 
The main learning task is to tackle Strang's "Linear Algebra and Differential Equations 2014" with its 1,050 exercise problems. Montiel's guidance is powerful, realistic, and demanding, comparable to Barry Cooper,a world-leading expert in mathematical music theory from the University of Manchester.
The research on Bach and Newton as mathematical music theory has not progressed much in the Western world since Christoph Wolff, and there is hardly any literature on it. If I want to study the op.54 2nd movement as a genuine research in mathematical music theory, I need to temporarily put aside the musical aspect and, even if I have to sacrifice something, focus on the path of studying as a mathematician. 
Even if it takes another 3 years to enter university and 12 years to obtain a doctorate. Even if I can only obtain a doctorate in mathematical history or geometry. 
Mariana Montiel, who taught me Rudolf Hasse's research on Paul Hindemith's "The Harmonie der Welt" as a music scholar, is an excellent mathematician, comparable to Barry Cooper from the University of Manchester, or even surpassing him. Compared to the stage before my trip to Greece last summer, 
I have made progress in the past year, covering all the basic topics of university mathematics, which are based on high school mathematics, such as composite functions, chain rule calculations, integration, binomial theorem, vectors, matrices, trigonometric functions, exponential functions, except for complex planes. 
However, relying on the guidance of Montiel, who is still uncertain about my understanding of Blue Chart, and the guidance of top-class mathematicians in the West, is powerful and amazing.
Nevertheless, the guidance and mentorship of truly exceptional mathematicians in the West,Mariana Montiel,who have allowed someone like me, still inexperienced with the Blue Charts, to engage with Strang,is incredibly powerful.
Mariana Montiel,who taught me Rudolf Hassé, is an outstanding mathematician, surpassing even Barry Cooper at University of Manchester. To learn, one must seek out exceptional teachers, the best in the world.
Following Mariana Montiel's advice and instructions,from junior high school mathematics and prime factorization to the binomial theorem in Kazuo Takahashi's 'High School Mathematics Revisited,'I have made progress within three weeks since March 13th.
At this point,I will deviate from the study plan and immediately return to the learning plan focused on Exercise 1.1.6 from Gilbert Strang's 2014 textbook 'Linear Algebra and Differential Equations,' which is intended for MIT's first-year math students. 
But before that, I will cover complex number planes using the Blue Charts.

May 12th.
After various modifications to the textbooks, I finally grasped that partial differential equations are differentiations of multivariable functions and their corresponding equations. 
The general solution involves arbitrary functions, rather than arbitrary constants as seen in ordinary differentiations and ordinary differential equations. 
These arbitrary functions can be anything, such as f(x) or sin(x).
Starting from May 1st, the Nagoya University Central Library, which has been closed to students due to the coronavirus pandemic,is now accessible again.
During these past three years,I have copied around 3,000 mathematics books.
It has been the most fulfilling and hopeful three years of my life, along with my journey to Greece. 
I hope to make the next three years, until my trip to Montiel, even better.
It is often said that starting to study mathematics at an older age is very difficult, and self-study in mathematics is considered quite difficult.
There is a famous anecdote about Fermat,known for Fermat's Last Theorem,who started studying mathematics at the age of 40 alongside his legal career,specifically in Diophantus' 'Arithmetica.'
However, when I was 54, I encountered Mariana Montiel, a mathematician from the State University Georgia's Mathematics Department,who recognized my mathematical talent.
This encounters completely transformed my life. 
That's why I am determined to pursue a Ph.D. in mathematics and embark on the path of studying in a doctoral program in mathematics at the university.
Last summer, I attended an international conference as a music scholar and traveled alone to Greece. 
Until March 13th this year, I was working on translating literature as a musicologist.
Finally, starting from March 13th, I managed to cover topics from junior high school's prime factorization to high school's binomial theorem within three weeks.
In art college and music high school, and even to pursue music high school, I had never properly learned mathematics, not even prime factorization in junior high school.
I feel that mathematics is my calling.
Today,I managed to remember the term 'arbitrary function' and understand 'general solutions' for 'partial differential equations.'
Even in the field of humanities and economics, partial differential equations are now indispensable."

May 13th.
Today, I'm preparing various textbooks for Strang.
For now, I plan to try out university-level mathematics textbooks on differential equations and Dedekind's continuity axiom.
Subjects like higher-level ordinary differential equations, partial differential equations,complex functions,and vector analysis can be taken slowly.
Anyway, Strang. Differential equations and linear algebra.
I'm currently at Nagoya University Central Library, making copies of secret weapons to fight against Strang. 
These secret weapons include textbooks covering basic mathematics for science and engineering students,such as ordinary differential equations, partial differential equations,vector analysis, complex functions, 
Fourier transforms, Laplace transforms, and set theory, to get a comprehensive understanding of these subjects before diving into Strang's requirements.
The engineering department's textbook by Donald A. McQuarrie is relatively new,but it seems to have many typos despite its solid mathematical content.
I went ahead and bought "Linear Algebra for Everyone" by Strang, the 2021 edition translated into Japanese, costing me 6,000 yen out of my own pocket, from at Nagoya University Co-op.
The classic textbooks by Matsusaka Kazuo and Sugiura Mitsuo are considered standard for university mathematics. However,they are a bit old, dating back to the 1960s.
The Iwanami Mathematics Series is a good textbook series that systematically covers mathematics from high school to university, translated into English by Japan in the 2000s. 
However, they are quite lengthy.
It's difficult to find other good, systematic,and comprehensive textbooks other than the Iwanami series.
I'm still in the process of selecting textbooks, especially for subjects like set theory, ordinary differential equations, partial differential equations, vector analysis, Fourier transforms, Laplace transforms, which are part of the university's basic mathematics curriculum after completing Strang.
It's hard to find the latest translated textbooks for these subjects. 
There aren't many options besides MacGrawhill,except for the Springer series covering advanced algebra, geometry, analysis, and other subjects in university mathematics departments and the textbooks for various subjects in the university's basic mathematics curriculum.
Although I copied all volumes of Bourbaki, which I admire, they are classical texts and too old to be considered as textbooks.
Whether I manage to solve all of Strang's problems perfectly or grasp the outlines of linear algebra,differential equations,complex functions,Fourier analysis, Laplace transforms, set theory, partial differential equations, ordinary differential equations, finite element methods, and other subjects in the basic mathematics curriculum for science and engineering departments, aiming to enter Montiel's undergraduate program immediately, complete self-study is still difficult because mathematics subjects themselves have become highly specialized and are progressing too rapidly in this era.
Today, I obtained three secret weapons for fighting against Strang by making copies at the Nagoya University Central Library.
These secret weapons will cover all volumes that can be copied from the library, including mathematics and theoretical physics, encompassing both my major fields.
Takahashi Kazuo's "Once Again High School Mathematics" was also a secret weapon.
It was very easy to understand, and I managed to cover topics from prime factorization in middle school to binomial theorem in just three weeks thanks to this secret weapon.
It is said that starting to study mathematics at an older age is very difficult, and self-studying mathematics is extremely difficult. 
Even though there is a famous anecdote about Fermat, who started studying mathematics at the age of 40 while practicing law, focusing on Diophantus' "Arithmetic."
However, when I was 54 years old, I had a life-changing encounter with Professor Mariana Montiel, a mathematician in the mathematics department at State University Georgia, who recognized my mathematical talent.

14th.
"Secret Weapons"
The path to Georgia State University.
I forward the path of linear algebra and differential equations, using Strang's 2014 "Linear Algebra and Differential Equations."
Once I grasp the outline of university liberal arts mathematics, including calculus, differential equations, linear algebra, vector analysis, finite element method, Fourier transform, ordinary differential equations, partial differential equations, set theory, numerical analysis, probability and statistics, complex functions,I can immediately aim for undergraduate admission.
Self-study.
Currently, mathematics subjects have become highly specialized, and progress is too rapid to obtain a doctoral degree in mathematics through self-study.
It would be reckless.
To truly obtain a doctoral degree in mathematics, it is necessary to seriously consider re-enrollment in a university.
When researching textbooks, there are not many textbooks that consistently explain high school mathematics and university mathematics. 
The textbooks I am currently collecting are such secret weapons.
I am collecting textbooks that consistently explain high school mathematics and university mathematics with coherence.
Red "Chart" books are said to cover university-level mathematics, but basically, they are not much different from blue "Chart" books.
I am currently carrying out a secret mission to find textbooks that can enable the continuous study of university mathematics from high school mathematics.
I am collecting these secret weapon textbooks for this secret mission.
Even the even and odd functions covered in high school mathematics integrals are used in calculations like Fourier transforms in university mathematics, right?
It is necessary to approach high school mathematics with the perspective of university mathematics from the beginning and learn high school mathematics as a continuous sequence leading to university mathematics.
There are still secret weapons that Nagoya University Central Library have, so during this secret mission, I might have to personally buy the secret weapons."

15th.
To surpass 20-books of secret weapons,two more cycles of partisan operations are necessary.
As I delve deeper into physics secret weapons like quantum mechanics, analytical mechanics, classical mechanics, electromagnetism,and wave oscillation,I realize that the more I try to copy the mathematical secret weapons, the more I feel the need for them.
This is the rigor of studying.
Today marks the 14th secret weapon.I have finally almost fully memorized indefinite limits thanks to this secret weapon, which explains them in a clear and understandable manner. 
Just like the movie I watched the other day,"The Mathematician in Wonderland," seems to be a film made for me,I can see that this secret weapon is also a secret weapon made for me.
By the way,I have also learned that professors from the Faculty of Engineering at Nagoya University use this secret weapon. Linear algebra, which I initially underestimated,appears deceptively simple,resembling the simultaneous linear equations and matrices in high school mathematics. However, it is actually a foundational mathematics that warns students from the beginning, kindly and patiently, that it extends all the way to Jordan canonical form. It is truly a secret weapon made for me.
Starting to study mathematics at an older age is very difficult, and self-studying mathematics is also very difficult. 
There is a famous anecdote that Fermat, known for Fermat's Last Theorem, began studying mathematics at the age of 40 while practicing law, focusing on Diophantus' "Arithmetic."
However, when I was 54 years old, I had a life-changing encounter with Mariana Montiel,a mathematician at Mathematics Department at State University Georgia, who recognized my mathematical talent.
That's why I am determined to pursue the path of obtaining a doctoral degree in mathematics, the path of studying in the doctoral program of the graduate school of mathematics, and the path to Mariana Montiel.
Last summer, I participated in an international conference as a musicologist and traveled to Greece alone.
Until March 13th this year, I worked as a literature translator as a musicologist.
Finally, from March 13th,I managed to cover topics from prime factorization in junior high school mathematics to binomial theorem in high school mathematics in just three weeks.
In art universities and music high schools, and even for the sake of entering a music high school, I had never seriously studied mathematics before, not even basic topics like prime factorization in junior high school mathematics.
I feel that mathematics is my calling.
Today, I managed to remember the term "arbitrary function" related to "general solutions of partial differential equations."
Even in the field of humanities and economics, partial differential equations have become indispensable in today's era.

16th.
As expected, the secret weapons found in the Nagoya University Co-op bookstore are also popular among students at Nagoya University Central library.
Most of them are on loan. 
Only set theory is yet to be acquired. The remaining secret weapons that must be obtained through copying are mathematical topics like finite element method, Fourier transform, Laplace transform, statistical probability, and physics topics like oscillations, waves, quantum mechanics, mechanics, electromagnetism, analytical mechanics, and statistical mechanics. 
Around 10 books. In tomorrow's partisan operation, I will acquire copies of about three-fourths of the secret weapons. 
After all, apart from linear algebra, calculus, basic mathematics, and differential equations, the rest are like a windfall to me, so I will take my time without rushing to make copies. 
However, buying second-hand books may be cheaper considering transportation expenses.
Mathematics is my calling. At the age of 55, I become aware that mathematics is my vocation, my mission

19th.
From Fujii Yousho, I received a copy of Kepler's literature and the results of an import investigation.
For now, I requested imports of three items related to Newton and Bach, costing a little over 20,000 yen.
Importing Hans Kayser's materials required around 70,000 yen last year.
I plan to return to the complex plane of blue charts today. 
While working on the complex plane, I encountered a proof by contrapositive, which I haven't done yet, so I went back to Kazuo Takahashi. 
I have covered binomial theorem for now,but there are still topics like probability and statistics, including binomial distribution.
It has been about three weeks since then. Many things have happened during these three weeks.
This mathematical diary is a reflection of my life itself. 
To be honest, there are many unfinished homework, such as Pascal's triangle. Mathematics is a cumulative discipline, with the beauty of a systematic structure and organization.
When I get stuck with Strang, the reason is usually a lack of foundational knowledge. I must always go back to the point where I don't understand and start over.

May 20th.
In general, for the Mariana Montiel trip, I think I can manage with mathematics courses on topology and group theory.
The biggest problem is, of course, my age. The shortest time table is 58 years old for undergraduate admission, 63 years old for a master's degree, and 66 years old for a doctorate.
Mathematics is my vocation and calling.
Just two months ago,I was working as a musicologist doing literature translations. 
This transition in the past two months has been extremely tough.
I really want to go to Montiel. Even if I can only obtain a Ph.D. in the history of geometry,I still want to pursue a Ph.D. in mathematics through graduate studies.
Today, I made copies of a book that explains the most fundamental technique in solving differential equations, which is variable separation form. 
It covers solution methods for first-order linear differential equations, differential operators, Laplace transforms, and various differentiation and integration formulas.
This book is quite good and easy to understand.
Afterward, by making copies of McGraw-Hill series books on group theory, Laplace transforms, Fourier analysis, statistics, and probability, I can return to Strang.
I am gradually progressing toward the most fundamental technique in solving differential equations, which is variable separation form.
If I master calculus in the blue charts of High School Mathematics III,I will have a sufficient understanding of Strang.
I also realize the importance of not ignoring the concept of solution family that is used in sets and topology,that Strang uses right from the beginning.
The textbook by Strang of MIT is the most advanced and should be used.
It is evident that the techniques for solving differential equations include variable separation form, first-order linear differential equations, differential operators, and Laplace transforms, as the main topics.
For the blue charts, it's better to include complex planes, Mathematics II and Mathematics III calculus, exponential functions, trigonometric relations, exponential functions, and logarithmic functions.
Even if I primarily focus on Strang,it would be beneficial to return to the blue charts every time I get stuck.
In the end,with a commitment of three years mainly focusing on Strang, my current primary goal is to enter the undergraduate Mathematics department at the State University Georgia at the age of 58.
I covered topics from prime factorization to the binomial theorem in three weeks.
I am gaining some confidence in my mathematical abilities."
I decided to make copies of the McGraw-Hill University exercise series, including group theory, Fourier analysis, Laplace transform, probability, statistics, and the above five books,which are old, but I have not yet found a better and more comprehensive Japanese translation of mathematics textbooks.
d dy/y=y The general solution of this differential equation can be obtained by indefinite integration y=e^ax C a=1, where C is an arbitrary constant, so C=1, C=2, or C=0. The set of functions for which this differential equation is valid is considered to be a family of solutions.
At this stage, it is still a family of solutions, but not the general solution itself.
Then, the equation is solved specifically as y=e^ax a=1 as the general solution.
In other words, the general solution of a differential equation is considered as a set of functions and considered as a family of solutions. 
Next, we obtain the general solution as a solution of the differential equation, based on the relationship between definite and indefinite integrals.
And finally,from definite integrals, special solutions and initial values are obtained as the mathematical forms required in theoretical physics. 
The explanation of these three steps is extremely difficult to understand, but once I get it, the depth, fun, and enjoyment of mathematics are beautifully explained in the first few pages.
The difference between high school mathematics and university mathematics is that the three steps are to define a family of solutions as a set of functions, not as a general solution,to solve the general solution from the relationship between definite and indefinite integrals,and to obtain a special solution as a definite integral from the relationship with theoretical physics.
The difference is that this is explained at the level of the Kyoto University Institute for Mathematical Analysis joint research strang for first-year MIT mathematics students.
Every time it rains and goes up, the temperature, goes up. 
Without money, winter is cold.Likewise, we are sensitive to the rise in temperature from spring to summer.
Today, Macglaw-hills general phase, vector analysis, and differential equations books at the Nagoya University Central Library.
This is the simplest book on differential equations, explaining only calculation problems of congruent almost linear ordinary differential equations. 
When I find out that more than half of the book for college students is a review of calculus formulas, I feel a little more confident and secure in my current math skills, and I know how to go about studying differential equations by gradually solving actual exercises of calculus problems in high school mathematics plus simple differential equations in university mathematics,
It turns out that the fastest way to do this is to solve actual exercise of high school mathematics calculus + simple university mathematics differential equations.
The way to proceed is to focus on solving such computational exercise and reading theoretical explanations of the abstract mathematics of the Δ Ε theory.
The fact that the general and special solutions of differential equations are related to indefinite and definite integrals is also really difficult, because Strang explains them as differential equations of dynamics in theoretical physics from the beginning,together with the concept of families of solutions in set theory.
　Strang has written a textbook for first-year students of the Department of Mathematics at MIT, based on the basic premise of researcher-level study of mathematical sciences, called "Stochastic Processes and Families of Solutions of Partial Differential Equations," a joint project of the Institute for Mathematical Analysis at Kyoto University. So it is very difficult.
Strang 2014 is a very good textbook for first-year MIT mathematics students, covering more than 10 years of mathematics education curriculum from high school mathematics III to graduate school doctoral program or post-doctoral level in mathematics, from the concept of solution families in set theory to derivatives of exponential functions in high school mathematics at the joint research level of the Institute for Mathematical Analysis, Kyoto University. This is a very good textbook for first-year students in the MIT Mathematics Department.
To fully understand Strang 2014, you need 10 to 15 years up to the post-doctoral level in the Department of Mathematics. The level of the MIT Math Department is too different by that much.
Still Macglaw-hills Fourier analysis copy remains,but these are too old,because these are the books that is 60 years ago. 
I have just started to copy a college introductory mathematics textbook on differential equations.
Mitsuo Sugiura and Kazuo Matsuzaka, both recommended by the Department of Mathematics at Hokkaido University, are too old,that are  textbooks 60 years ago.
Kunihiko Kodaira's Analysis and Complex Analysis is an analysis textbook barely 30 years old.
There are, of course, three ways to study calculus up to differential equations: 
1. to learn its contents as a tradition of Greek mathematical analysis from Archimedes to Newton, 
2. to memorize it as a calculation formula like high school mathematics II, 
or 3. to understand abstract arguments of analysis centering on ΔΕ theory.
I started with the second method.
I think it would be better to start with 2, enjoying 1 as a supplementary reading book, and gradually advance to 3.

May 21.
37th fasting for state university georgia
37th Monsieur Row Prayer Fast.
1st day.
I collect only 10 books each on topology and group theory. 
This makes me feel a little  more like an aspiring mathematician.
The Nagoya University Central Library partisa operation is the Springer series on solving difficult exercise. 
Calculus,complex analysis, metric theory, sequences and series, set theory, geometry, and combinations.
I was a little surprised to find Kazuo Takahashi's textbooks. 
I got a little confidence in mathematics.
There have been mathematics contests like the Mathematical Olympic in Japan for a long time, and there are even mathematics contests for university students in Hungary and elsewhere.
I was still diabetic due to the stress of starting to learn mathematics and the failure of weight control and rebounding. 
I was still doing "Kazuo Takahashi" for the limit of the sequence of numbers in High School Mathematics III at the time of the 36th fast, which started on March 25 and last 36th fasting of six days.
In the two months since then, I have finished calculus, vectors, matrices, probability, and logic, all at one point or another, and am now in full swing in Strang 2014. 
The blue chart complex plane, trigonometric functions, and exponential calculus are the focus of the strang as a review assignment. 
Anyway, I'm now working on . 37th fast to overcome type II diabetes for studying under the supervising by professor Montiel.
The record of long term fasting so far is 2 times for a week, 1 time for 11 days, 1 time for 2 week, 1 time for 17 days. diabetes when I over the upper BMI limit of almost 10 kg.
But, based on my experience of losing 40 kg once in the last 3 years, it is possible to recover by fasting.


23th 
37th fasting for state university georgia
2nd day
I am collecting texbooks for topology that is structure above set theory,
 mathematical contest in Hungary.Now I am studying complex plaine for high school students.I have a lot of texbooks for topology and group theory,because these topics are concerned with string theory in theoretical physics and my study about musicae mundane in scientific revolution.
 Of course,category theory that is the speciality of Guerino Mazzola who is one of my most important teachers of mathematics is very important in modern mathematical music theory.In these 3 years,I have collected more than 3000 books for mathematics.I have learned from　prime factorization to binomial theorem in 3 weeks.
I have stoped at the excersise 1.1.6
in Gilbert Strang,that is the excersise about differential equation d2 dt2/f(x)=f(x) solution f(x)=Ce^t and 2nd solution f(x)=e^-t from April.This excersise is too difficult for me.At first,I should begin complex plaine and differential equation　of trigonometric function and exponential in high school mathematics.
Gilbert Strang teaches the terminology of family of set theory that is only used in the article of research institute of mathematical science in Kyouto University from beginning.
Today,I have rent a book of geometry for the student studying a teacher of mathematics in high school that explains algebraic geometry and primary conversion and matrix.I want to specialize geometry in phd.program.

25th
37th fasting for state university georgia
5th day 
Calculus is actually extremely difficult in fields that trigger the power function calculation of high school mathematics II, the Riemann integral of university mathematics, 
and even the breakthrough ofresearchers.
Boys who entered the mathematics department of university read the Iwanami Mathematics Series when they were in high school,and they were children with real mathematical talent.
Even with calculus, from the second volume, I could not still understand the book at all.
There is medical data that says transgender suicide rates are 35 times higher than for healthy people.
　Peter Pesic, in his 2022 sounding bodies, ignores Boethius's ratio of plant possibilities in medieval Christianity. as he ignored musicae mundane in scientific revolution in Music and the Formation of Modern Science in 2014,and  "Polyphonic Minds" in 2019, 
On a fundamental question about the evolution of mathematics in Ancient Greece Pythagoras and Archimedes, Medieval Christianity Boethius,the Scientific Revolution, and Pythagoreanism, which runs through three eras,the fact that a medical data that the transgender suicide rate jumps 35 times. 
 raises a very serious problem as the reductionism in the scientific revolution.
Two problems,the musice mundane problem in the scientific revolution, which Peter Pesic and Guerino Mazzola ignore, and the problem of the ratio of possibilities of plants of medieval Christianity Boethius, which Pesic ignores, are the two problems that define the reductionism of the scientific revolution.
It seems to me to be the most fundamental and important problems  about the evolution of  structure of mathematics of Pythagoreanism and the cosmology of the Scientific Revolution to overcome.
However, the musicale mundane in the Scientific Revolution is closely related to the string theory of theoretical physics, and the ratio of plant possibilities of medieval Christianity Boethius is also closely related to the topology and group theory of modern abstract mathematics. difficult.
I feel that one of the keys is how Boethius' ratio of plant possibilities of 4:3 and Archimedes' ratio of angels of ancient Greece 3:1 relate to other ratios and the ratio of coefficients that appear in mathematical physics equations of scientific revolutions.
　But this is very difficult, as Mazzola says, the Greek tetracorde became irrelevant with the creation of Cartesian algebraic geometry.

26th
37th fasting for state university georgia
6th day 
Elementary geometry based on the geometry textbook of the old junior high school that Kunihiko Kodaira wrote ,Euclid's "Elements" of a condensed version (the new Japanese translation has not yet been fully published),
and Springer Difficult Problem Solution Series Probability,today I have rent three books.
This book in 1991 by Kunihiko Kodaira is the perfect textbook for one like me who have to study again elementary geometry in junior high and high school.
As "Elements",there is no books other than this condensed verision.
The geometry of Greek mathematics is all about principles. 
The original "Elements" does not prove the Pythagorean theorem not as algebraically but geometrically.
It proves as geometry from the view of point of  the sum of squares.
Bach, when I listen to during fasting, resonates especially with me.
Only with harmony, with imitation, with bass lines,Bach impresses my heart.
Op.54 2nd movement always reminds me that I have to look at this piece from Bach. 
I think it's about how to look at Lawrence Kramer's description of the early Romanticism, the "bumping nightmare" ,from Bach, and from Newton,from the view of point mathematics.

28th
37th fasting for state university georgia
8th day
"Today, I finally return to complex plane in the blue chart.
I stumbled upon solving the differential equation in Strang 1.1.6, so I need to start by revisiting the blue chart complex plane. 
I still have to learn the properties of exponentiation of complex conjugates, the relationship between the n-th power of a complex conjugate and the conjugate of the n-th power of a complex number, (C　overline)^n = (C^n) overline.
There are three methods to prove this: using the binomial theorem, using de Moivre's theorem, and using symmetric expressions. 
Even understanding the proof of the property of exponentiation of complex conjugates using the binomial theorem gives me a hard time.
But when I do understand the proof, I am truly amazed by the beauty of its logic.
I want to cherish my appreciation for the beauty of mathematics.
Essenlialy I　only need four weeks of learning, but I will somehow manage to understand the high school-level proof of the properties of exponentiation of complex conjugates needed for learning the complex plane.
Mathematics is an accumulative discipline.
Skipping or deceiving, vanity, and pretense won't work.
It has the most beautiful system of all disciplines.
When I study mathematics,my perspective on life changes 180 degrees.
However,when I go back to the unknown and struggle to start over,the more I suffer and redo,the more good things come back to me in the field of study.

1.Write down the teacher's proof diligently by hand.
2.Solve exercise problems diligently with notes and pencil.
3.If I don't understand, go back to the point of confusion without bluffing or skipping.
In mathematics these three are essential.
If I can reach the polar form used in the formula e^iπ + 1 = 0,known as the most beautiful equation in the world,I might be able to brag a little to my friends.
Now, I'm studying δ-ε arguments using the simplest textbook.
I somewhat understood that concepts such as the continuity of real numbers,differentiability,and the introduction of integration are not just about intuitive understanding in high school mathematics.
They require precise definitions and discussions,which were theoretically established by Newton and particularly by Cauchy.
But to truly understand this, a rigorous study of university mathematics, such as mappings and sets, is necessary. 
This simplest textbook is quite good, as it covers group theory as well.
If 0<|x - a|<δ, then |f(x)-α|<ε.
When I was a freshman in university. Even then, I managed to understand it by reading math books at a bookstore.
However, at that time, I was a freshman in university, and I hadn't studied trigonometric functions at all in high school,so I gave up on majoring in mathematics.
This time, I've covered trigonometric functions, vectors, matrices, and even the binomial theorem,so I somehow understand this δ-ε argument. 
I've almost grasped the overall picture of studying mathematics at university.
I want to pursue a doctorate focused on topology and group theory, even if I can only obtain a doctorate in the history of geometry.
If I understand topology and group theory, I can somehow manage string theory.
I didn't even know about Pascal's Triangle before. 
Mathematics changes your life 180 degrees."



May 28th 2023
37th fasting for state university georgia
8th day
If 0 < |x - a| < δ, then |f(x) - α| < ε
I have understood δ　ε definition at last. .Today is my best and memorable day in my life.
δ　ε　definition
"I managed to clear the δ-ε argument with the simplest and elementary textbook.
If 0 < |x - a| < δ, then |f(x) - α| < ε
When differentiating, if the absolute value of the difference between the f(x) and a is between 0 and δ, and additionally, if the absolute value of the difference between f(x) and the limit α is smaller than a positive real number ε, then f(x) and the limit value α are said to be continuous and differentiable within that interval. It is with this necessary condition that we can perform calculations for differentiation. Isn't this the essence of it?"I　am very impressed with this wonderful δ ε　definition by Chaucy.
δ　ε論法が、ワイエルシュトラウスによって完成された1860年代は、Wagnerがトリスタンやマイスタージンガアを書いていた時代。解析学が、Newton、Eulerの無限小　無限大と言う概念で破綻を来したために、Chaucy ワイエルシュトラウスによって、精密な定義から再構築されていた時代。op.54 2nd.movement sonata rondo mixture論証は、BWV918、Newton Eulerの解析学の無限大、無限小と言った、数学とBachから見ていくべき。Bonds、Pesicが言う、科学革命第2段階、Hans Christian Oerstedの電気革命とその自然主義哲学からではなく。
  

May 29th 2023
37th fasting for state university georgia
9h day
I learned for the first time that the definition of integration in high school mathematics is the Riemann integral, which is seen as the limit of an infinite sum of approximations using rectangles.
It is taught to high school students as a practical and intuitive method based on the early concepts of infinitesimals by Newton and Leibniz.
Broadly speaking,in university mathematics,we study the Riemann integral and the more abstract Lebesgue integral.
Without knowing this major flow of analysis, it is very difficult for me, at the age of 55, managing a limited liability company,to learn mathematics in a university mathematics department.
The absolute value of 0 is 0.
Today,I was wondering why they specifically write "greater than or equal to 0".
It is because in the case of 0, there can also be cases where ε is 0,which would make it impossible to define the continuity of real numbers.
It took me a whole day to finally understand that.
Mathematics is a cumulative discipline. Skipping steps, deceiving, vanity, and pretense have no place in mathematics.
The complex plane is a topic that may or may not be included in the high school mathematics curriculum.
There are times when the complex plane is not included in the curriculum solely for the sake of quadratic equations.
However,it is essential,of course, for studying complex analysis in university mathematics departments

May　30th 2023
"Today, I will solve Strang 1.1.6 using Chat GPT.
It's an exercise  to find the two general solutions of the second-order differential equation dy^2/dt^2 = y.
As a solution ,I can use the exponential function of Euler's number, which remains the same whether we integrate or differentiate it, y = e^t.
Assuming y = e^rt, we have dy/dt = re^rt and d^2y/dt^2 = r^2e^rt = e^rt.
Therefore, r^2 = 1, and r = ±1.
Thus,this differential equation may have two general solutions:y = e^t and y = e^(-t).
So far, Chat GPT correctly finds the solutions.
However,its subsequent answer is completely wrong.
It fails to recognize that the general solutions of a second-order differential equation consist of two possibilities.
I cannot rely upon Chat GPT as a math teacher.
Expecting too much from Chat GPT is highly dangerous.
However,I still haven't fully understood why the first solution has an arbitrary constant C as y = Ce^t, while the other general solution doesn't have an arbitrary constant C and is y = Ce^(-t). 
I'm still not satisfied with that.
The concept of uniform continuity and uniform convergence in analysis is truly interesting.
The 19th century is said to be the period when Newton and Leibniz's calculus was rigorously formalized. 
Correspondingly,in aesthetics, the 19th-century Romanticism emerged.
When considering 19th-century Romanticism as an aesthetics,it is crucial to understand that the concepts of infinitesimals and infinities in calculus were rigorously formalized and reconstructed through Cauchy's δ-ε method. 
Op. 54, 2nd movement is a work that lies at the core of such fundamental issues in both music and mathematics.
Lawrence Kramer also mentions Beethoven's two-movement structure and musica mundane in 1990,which,fundamentally,is the structure of the evolution of mathematics in Pythagorean cosmology,especially the infinitesimals and infinities of analysis, without considering the formation of the δ-ε method.
I was able to understand the δ-ε method relatively quickly,in about a day.
The δ-ε method is fascinating.
It rigorously defines and proves the continuity of real numbers,differentiability,and the introduction of integration.
On the other hand,Strang 1.1.6 is difficult.
I still need to study complex numbers,probability,and differential equations,at least High School Mathematics II and III, before attempting it.
There are many troublesome exercise when it comes to fully understanding the calculus of trigonometric functions,especially in the chain rule of calculus in university mathematics.
Additionally,linear algebra,at least up to Jordan canonical form,can become very difficult if taken lightly.
There is no graduate school specializing in the history of mathematics in Japan.
It is almost impossible for someone like me,who originally came from the field of musicology and aimed to go to Montiel in pure mathematics.
Now that I am gradually understanding the δ-ε method,
I have no choice but to contact Montiel, considering that I am still struggling with 1.1.6.
Sufficient knowledge of pure mathematics is required even in the study of the history of mathematics.
Pesic is strictly speaking a part of the history of science.
Michiyo Nakane's δ-ε method and its formation are well-known in the current history of mathematics in Japan.
Tonight, I am skimming through the books "δ-ε method and its formation" and "Comprehensive Guide to the δ-ε
I still find uniform convergence, uniform continuity,and the logical symbols 'any' and 'exist' difficult,but I'm starting to grasp the overall picture.
On March 13th, I was still studying calculus, vectors, matrices, and even quadratic equation solutions, the factor theorem, and trigonometric functions. So, considering I reached the δ-ε argument in just about a month of learning, I have gained some confidence in my mathematical abilities. I have made up my mind that I need to study for admission to State University of Georgia for another three years, and by then, I believe I will easily understand the δ-ε argument as if solving puzzles. Until then, I will exercise patience and work through the blue chart and Strang's materials. As a change of pace, I will skim through other books as supplementary reading. Also, since I lack knowledge in geometry, I must include Kunitaka Kodaira's Geometry alongside Strang and the blue chart.
Mathematics fundamentally should be enjoyable. Essentially, one must find joy and fascination in solving puzzles.
The key is to diligently copy the teacher's proofs and diligently solve exercise.
When I come across concepts like De Morgan's laws or proof by contradiction in textbooks aimed at university students,I sometimes think they would be excellent textbooks for certification exams for small business owners like me.
Studying mathematics boils down to daily, regular practice of diligently copying the teacher's proofs and diligently solving exercise.
I learned from the latest issue of "Mathematics for University Entrance Exams" that the concept of "linear independence" for vectors appears in entrance exams.
I discovered a set of back issues of "Mathematics for University Entrance Exams" covering the past three years on Yahoo Auctions. 
The set includes issues from April 2013 to March 2016, a total of 36 books for less than 6,000 yen. 
Each book costs less than 200 yen. 
I decided to purchase them. 
For now, I will collect the past three years' back issues of the books as second-hand books and read the latest issue at the bookstore.
If there are topics I don't understand, I will check them on a case-by-case basis. 
Rather than focusing on high school mathematics and university entrance exam mathematics, my top priority is to overcome Strang 1.1.6 as quickly as possible.
I am not a teenage boy but rather a middle-aged man running a small family-owned company.
When I was 18 years old and a first-year student in the piano department at Tokyo University of the Arts, during the summer when I returned home,I fondly remember studying the δ-ε argument, vectors, trigonometric functions, and calculus, even though I hadn't studied them in high school, by reading through this "Mathematics for University Entrance Exams" at the bookstore.
Purchasing a set of back issues for three years of "Mathematics for University Entrance Exams" at a cost of less than 200 yen per book was an excellent purchase."

31th 2023
Research theme: 
Proportion of Angel by Archimedes and D.958, Spectrum of sound and light in Moni Daphne, Greek optics, the redefinition of geometric exactness in Descartes' early modern concepts, harmonic analysis using group theory and Boethius' ""Anagogisches potential",geometric structure of string theory, 
Hindemith's "Hrmony of the world",analysis of texture and harmony counterpoint through category theory, proportion from Rameau to Messiaen, BWV918 and Newton's calculus. Pythagorean tuning, just intonation, equal temperament, tuning in ring theory, and scale theory through group theory.
The central focus is Cartesian algebraic geometry, the geometric structure of string theory, group theory, and Boethius' "Anagogisches potential".
However, this is still small in scale. Mazzola's category theory and ring theory are essential.
In American universities, even if it takes four years to complete a bachelor's degree, it is possible to graduate in two years depending on the credit system. 
Two years for a master's degree. Three years for a Ph.D. Georgia State University requires five years for a bachelor's degree.
State University Georgia Master Program includes linear algebra, real analysis, abstract algebra, analysis, matrix analysis, and more.
For the State University Georgia PhD Program, linear algebra and analysis are required for admission. Courses such as Real Analysis, Advanced Matrix Analysis, Applied Mathematics, etc., are offered.
State University Georgia Mathematics B.S. (5 years) Bachelor program includes precalculus (preparation for studying calculus), calculus (single-variable integration), calculus (multivariable integration), linear algebra, modern algebra, statistics, analysis, and more.
Precalculus is a characteristic of mathematics departments in the United States, which includes a review of elementary algebra, exponential functions, trigonometric functions, logarithmic functions, and other topics necessary for studying calculus.
With this program,I could take PhD. in mathematics.I thought there would be more subjects. Currently,I am managing with the delta-epsilon proof.
Mathematics departments in the United States include precalculus, which is a review of high school mathematics necessary for studying calculus. 
With this,it might be possible to earn a Ph.D. in mathematics in as little as 7 years (2 years for a bachelor's degree, 3 years for a master's degree, and 2 years for a Ph.D.). 
The most biggest problem for me is my age.
There are no age restrictions for Ph.D. programs. 
Even at the age of 70, it is still possible to earn a Ph.D. 
Now, with this program, I can keep up with the State University Georgia Mathematics Bachelor's program. 
In Japan, there are no universities or graduate schools that offer courses in the history of mathematics. 
However,there are approximately half the number of universities in the United States and Europe that specialize in the study of the history of mathematics.
Whether I pursue the path of history of mathematics or pure mathematics depends on the content of my research theme and the level of my mathematical ability, as determined by my supervisor,Mariana Montiel.
However, even in the study of the history of mathematics, a knowledge and understanding of pure mathematics is necessary.
Since my ideas span ancient Greece, medieval Christianity, and the scientific revolution,and involve themes of cosmology and geometry,an understanding of the mathematics of string theory is required.
It will likely encompass modern mathematical research in areas such as groups, topology, algebraic geometry, Galois theory, and the mathematics of string theory.
Originally, the research conducted by Springer's Boss that led me to discover the proportion of Angel by Archimedes was in the field of history of mathematics.
However, it does not mean that ordinary knowledge of mathematics is unnecessary in the study of the history of mathematics. 
However, just because it's the history of mathematics doesn't mean that ordinary knowledge of mathematics is not necessary.
Specialized mathematical knowledge is required. 
It was significant progress for me to improve my high school mathematics skills and reach the delta-epsilon proof in the past six months.
Even if I turn 70, I will pursue a Ph.D. in the history of mathematics under Monsieur's guidance.
In American universities, even if it takes four years to complete a bachelor's degree, it is possible to graduate in two years depending on the credit system.
Two years for a master's degree. Three years for a Ph.D. Georgia State University requires five years for a bachelor's degree."

4th June 2023
"Today, for now, I tackled with high school probability.
The multiplication theorem of independent trials was a bit difficult for me, so I initially approached it by laboriously calculating all possible outcomes using the formula for combinations with factorial and dividing it by the number of all events. 
But I was able to come up with my own solution, which was different from the teacher's explanation, although it was even more cumbersome.
I could have drawn a more detailed tree diagram if I wanted to.
In mathematics, these accumulations of concepts, including Pascal's triangle, become clearer and more meaningful as I progress further and look back later on.
Making decisions to move forward or go back for review is quite difficult, but it is also one of the joys of studying mathematics.
Now that I've finished expected value,I'm moving on to the complex plane. Polar form is coming up soon."

11th June 2023.
Today,I am copying 9th textbooks of "Pre-algebra",that is the preparation of algebra at Undergraduate mathematics study at University.
 By now,I have copied 9 textbooks of undergraduate mathematics,taht are  "Geometry","Pre-calculus","Calculus","Pre-algebra",other than more than 100 graduate textbooks of mathematics by Springer.
 I deeply thank professor Mariana Montiel at State University Georgia mathematics department,who is my supervisor of mathematics and has permitted me to admit and book the orientation of 1st year student of the undergraduate program of mathematics,immediately when I have understood the chain rule in analysis and the derivative of composite functions.

12th June,
Today,I am photocopying 12th textbooks of "Geometry","The Handbook of Linear Algebra","Calculus" for Orientation of 1st year stundents of undergraduate program of mathematics at State University Georgia as a student of my supervisor of mathematics,professor Mariana Montiel.
,"Algebra and trigonometry",in american high school and university mathematics for the perticipating in the orientation of 1st year students on undergraduate program of mathematics at georgia state university.I deeply thank my supervisor of mathematics,professor Mariana Montiel.
 Because professor Mariana Montiel has kindly taught me,I  have been able to understand the  derivative of composite functions in english textbooks of "pre-calculus".and now am able to read the textbooks of pre-algebra and elementary algebra in university and high school mathematics.
 I deeply thank my supervisor of mathematics,professor Mariana Montiel.

13th June,
Today,I am photocopying 15th textbooks of "Algebra and trigonometry" and so on.Professor Mariana Montiel is teaching my study of mathematics especially these 3 months,and my textbooks for mathematics is clearly progressing to use english one for high school mathematics and college and university one. 
 Professor Mariana Montiel these 3 months has told me to study "calculus" by "japanese blue chart" and when I have understood "chain rule in analysis" and "derivative of composite function",she has immediately invited me to the orientation for 1st years students of undergraduate program of mathematics at Georgia State University.
 Thanks to her kind and excellent advice about my study of mathematics by professor Mariana Montiel,I can use english textbook of elementary mathematics for high school student and "pre-calculus" and "pre-algebra" for 1st year students at undergraduate program of mathematics at Georgia State University.
 Again and again,I deeply thank professor Mariana Montiel for her kind and excellent teaching of my study of mathematics to enter undergraduate program of mathematics at Georgia State University by only 3 weeks study of high school mathematics.

14th June 2023
Today,I am photocopying 18th textbooks of mathematics for high school mathematics and for 1st year student at undergraduate program of mathematics at university,following my supervisor professor Mariana Montiel,for the orientation at state university georgia.I deeply thank professor Mariana Montiel for her kind and excellent advice of my study of mathematics.
Today 3 latest english textbook of  mathematics at Nagoya University Central Library.
●Single  variable essential calculas. 
●Automatic sequences.
●Elementary differential equation. 
Although  I am  very busy for participating in  the  orientation  of 1st year students  of mathematics  on the  undergraduate program at  state university  Georgia.  I  deeply  thank  my  supervisor  of  mathematics,professor Mariana  Montiel.

23th June
 Today,I have finally finisched photocopying 26 textbook of high school algebra,geometry,and pre-calculus,pre-algebra for 1st year students for mathematics.
 In american high school,trigonometry is so important as elementary algebra,because trigonometry is important in complex analysis in university.
 Textbooks of high school mathematics in america is so systematics than that in japan.
 "Rational number",for exanoke us clearly concerned with "ratio" in pythagoreanism ,and is so difficult for japangese because greek mathematics is not traditional in japan.
  I am still 3 month late to follow her supervisor,becuase I need 3 month to photocopy 26 textbooks of high school algebra,geometry pre-culculus,pre-algebra.

2th July 2023
"Combination"
n P n=n！
n P r=n!/(n-r)!
n C r=n!/r!(n-r)!｡
Above these 3 teorem is the same.
集合 かつ ⋀｢積の法則｣
　ダブってる順列を、積の法則で逆に割って、元に戻す計算。ダブってる順列を積の法則で逆に割って、元に戻すと言う数学の美しさに、ちょっと感動。

「連鎖律」
d/dy x=dx/dx・dx/dy
xをyで微分できないから、まずxで微分して、その微分係数をyで微分する。
「δ　ε論法」近傍 εと無限小δの二つの概念の要素だけで､実数の連続性を定義する数学の厳密な美しさ｡
　この３つが分かると、数学は、本当に楽しくなる。この３つの基本概念を形作る数学の美しさに、本当に感動する。人生が180度変わる｡

16th August 
7th fasting for my study of mathematics under professor Mariana Montiel. 
第７回マリアナ　モンシェール断食
3rd day 
第3日。
今までの断食記録。5日2回。1週間2回。2週間2回。11日。17日。1回ずつ。1週間未満の断食32回については、今回からカウントしないことにした。
今回も、マッギル大学の勧告に従って、第３日で断食中止。

18th August 2023
今日は、肺気腫で死にかけてる筈のもう今85歳になる父親が、果樹園の洋梨を食べて見せて、食べ滓の芯を地面に捨てて見せて、扇風機、パソコンの電源を全部切って見せて、自宅に帰っていく。
シルバーパワー恐るべし。昨日の創価学会の経本のページ開いてあったのも、先日の創価学会の聖教新聞のコラム欄、絶壁の記事も、全て肺気腫で死にかけてる筈の父親の仕業だったことが判明。今まで後期高齢者の父親にしたこと、心から悔いて反省。昨年夏のギリシャ単独渡航時も、盛んに僕の住居の離れの玄関を開け放しては、帰って行っていた。
もう85歳になる、肺気腫で死にかけてる筈の父親の生命力の強さと知恵の逞しさに、改めて心を強く打たれて感謝。
時代や人生の決定的な正念場分水嶺の天王山の戦いの局面には、男が絶対に必要なこと、男親が絶対に必要なことを、改めて痛感して、もう85歳になる肺気腫で死に掛けてる筈の父親に改めて感謝。