和同近积大

Elementary Math, Modern Math, Uncategorized Leave a comment

Here’s a detailed summary of the steps taken to find the maximum value of the function 

f(x, y) = x \cdot y

given the constraints that ( x ) and ( y ) are integers, ( x + y = 61 ), and ( x ≠ y ):

  1. Approach: Since we are dealing with integers and ( x ≠ y ), we look for two numbers that are as close as possible to each other to maximize their product. This is because the product of two numbers with a fixed sum is maximized when the numbers are equal or nearly equal.
  2. Finding Integer Solutions: We express ( y ) in terms of ( x ) using the constraint ( x + y = 61 ), which gives us ( y = 61 – x ).
  3. Substituting ( y ): We substitute ( y ) into the objective function, resulting in  f(x, 61 - x) = x \cdot (61 - x)
  4. Expanding the Function: The expanded form of the function is f(x) = 61x - x^2
  5. Finding the Derivative: To find the maximum, we take the derivative of ( f(x) ) with respect to ( x ), which gives us f'(x) = 61 - 2x
  6. Setting the Derivative to Zero: We set the derivative equal to zero to find the critical points: 61 - 2x = 0
  7. Solving for ( x ): Solving the equation gives us ( x = 30.5 ). However, since ( x ) must be an integer and ( x ≠  y ), we consider the integers closest to 30.5, which are 30 and 31.
  8. Evaluating ( f(x, y) ): We evaluate ( f(x, y) ) at ( (30, 31) ) and ( (31, 30) ) to find the maximum product. Both give us the same maximum value of ( f(30, 31) = f(31, 30) = 930 ).
  9. Conclusion: The maximum value of ( f(x, y) ) under the given constraints is 930, which occurs when ( x = 30 ) and ( y = 31 ), or vice versa.

This process demonstrates that the maximum product of two integers with a fixed sum and the condition that they cannot be equal is achieved when the integers are as close as possible to each other. The absolute difference ( |x – y| ) is minimized to 1, which is the smallest possible difference for distinct integers.

Infinite Sets Cantor-BS Theorem

Modern Math Leave a comment

For infinite Sets A & B
|A| ≤B|, |B| ≤ |A|,
is |A| = |B| ?

Cantor-BS Theorem :
if A to B Injection and
B to A Injection ,
then
Bijection between A, B
(=> |A| = |B|)

Note:
For finite sets A, B
to have bijection
f: A-> B
must fullfill 2 conditions:
f is Surjection (满射）
& Injection (1对1 映射）

Haskell Monad Courses

IT Leave a comment

I understand! Here are some online courses and resources for learning Haskell functional programming using Monad style:

  • Monad Transformers by Bartosz Milewski: A video course on YouTube covering monad transformers, monads, and functional programming in Haskell.
  • Haskell Monad Tutorial by Learn You a Haskell for Great Good!: A comprehensive tutorial on monads, including the basics, monad laws, and advanced topics.
  • Functional Programming in Haskell by University of Glasgow on FutureLearn: A free online course covering functional programming, including monads, in Haskell.
  • Haskell and Functional Programming by University of Nottingham on Coursera: A specialisation covering functional programming, including monads, in Haskell.
  • Monads in Haskell by Data61 on YouTube: A video series covering monads, including the basics, state monads, and IO monads.
  • Haskell Monads by Tutorials Point: A tutorial covering monads, including the basics, monad laws, and advanced topics.

These resources should provide a good starting point for learning Haskell functional programming using Monad style.

Quadratic Equation by Geometry

Elementary Math Leave a comment

Quadratic Equation (QE) solved by Geometry, made a PSLE kid understand Algebra – à la Singapore Math Modeling visually.

QE only taught in Sec 2.

The Babylonians already knew it before the ancient Egyptians and the Greeks. Also， Three Kingdoms 三国.吴 赵爽 (200 AD）- then later by Francois Vieta Theorem (16CE France) – discovered the 2 roots coefficient relations
(sum and product)… and mid 19CE French 19-year-old boy Evariste Galois invented Group Theory thru Lagrange’s “Résolvant” discovery.

https://vt.tiktok.com/ZSF4wya28/

Power Tower Converge Condition

Uncategorized Leave a comment
  1. Definition of Power Towers:
    A power tower is an expression of the form:                               [a^{a^{a^{a^{\cdot^{\cdot^{\cdot}}}}}} ]
  2. Convergence Condition:
    The convergence of a power tower depends on the value of (a). Specifically, if (a) lies within the interval ((e^{-e}, e^{1/e})), the power tower converges to a finite value.
  3. Proof:
  • Let’s denote the value of the infinite power tower as (x):
    [ x = a^{a^{a^{a^{\cdot^{\cdot^{\cdot}}}}}} ]
  • Taking the natural logarithm of both sides:
    [ \ln(x) = a \ln(x) ]
    (Note: We use the fact that the logarithm of a power tower is equal to the exponent multiplied by the logarithm of the base.)
  • Solving for (x):
    [ x = e^{a \ln(x)} ]
  • Now consider the function (f(x) = e^{a \ln(x)}).
  • We want to find the fixed point of this function i.e., where \boxed{ f(x) = x \implies f'(x)=1 }
  • The derivative of (f(x)) with respect to (x) is:
    [ f'(x) = a e^{a \ln(x)} = a x ]
  • At the fixed point, we have:
    [ f'(x) = 1 ]
    [ a x = 1 ]
    [ x = \frac{1}{a} ]
  • Since (a) lies within the interval [e^{-e}, e^{1/e}], we have:
    [ e^{-e} < a < e^{1/e} ]
  • Therefore:
    [ \frac{1}{e^{1/e}} < x < \frac{1}{e^{-e}} ]
    [ e^{-1/e} < x < e^{e} ]
  • Thus, the power tower converges to a finite value within this range.
  1. Conclusion:
    For (a) within the interval [e^{-e}, e^{1/e}] , the power tower converges to a specific value. Outside this range, the power tower either diverges or oscillates without settling on a single value.

Remember that the value of (x) in this case is a real number greater than 1 and less than (e^e). The proof involves advanced mathematical concepts, but it confirms the convergence behavior of power towers within the specified range. 🌟

Let

\boxed{ \sqrt{2}^{\sqrt{2} ^{\sqrt{2}^{\sqrt{2}^{\cdot^{\cdot^{\cdot}}}}}} = x }

Solve {x =\sqrt{2}^{x}}

\boxed {x= 2}

\boxed{ \sqrt{2}^{\sqrt{2} ^{\sqrt{2}^{\sqrt{2}^{\cdot^{\cdot^{\cdot}}}}}} = 2}

Note:

Power Tower (with base a = √2) converge between :

[ e^{-1/e} < x < e^{e} ]

  1. (e^{-\frac{1}{e}}) is approximately 0.6922.
  2. (e^e) is approximately 15.1543.

Answer x = 2

Ref:   https://tomcircle.wordpress.com/2021/05/12/aaa-a/

Fields Medal Talk: Hugo Duminil-Copin

Education, Modern Math Leave a comment

NTU IAS Lee Kong Chian Math Forum in Jan 2024 by French Fields Medalist (2022) Prof Hugo Duminil-Copin (1985-).

What is Mathematics ?

Mathematics is : not afraid to make “errors”.


Mathematics is “Social : Hugo was doing his PhD Math thesis on ‘De’pendent Percolation but got stuck for 4 years with no big progress…until at one social gathering with friends of no maths background, suddenly 1 of them talking about making coffee by pouring hot water over coffee grains… – this is a real- life example of ‘De’pendent percolation.

Dependent percolation refers to a theory developed by Hugo Duminil-Copin, which has significantly advanced our understanding of critical phenomena in equilibrium statistical physics¹. His work involves using deep connections between various models, such as the Ising model and the Potts model, and developing this theory to obtain transformative results on these classical models¹.

Hugo Duminil-Copin’s research focuses on the mathematical branch of statistical physics, where he applies probability theory to study the critical behavior of models on networks. This includes exploring the critical point at which phase transitions occur and the behavior of systems above and below this critical point⁶. His contributions to the field have been recognized with the awarding of the 2022 Fields Medal¹.

Percolation theory itself is a part of probability theory and mathematical physics that studies the movement and filtering of fluids through porous materials. It has become a significant area of interest as it helps to understand the behavior of complex systems and phase transitions⁷..

Nakamura’s Blue LED

Physics, Science Leave a comment

This youtube video has garnered 9 million viewership and many praises for its excellent explanation of semi conductor and PN junction.

Nakamura’s determination is admirable, despite having no (PhD) degree then, his success is :
1) dirty hand 动手;
2) stubbornness to stick to his belief;
3) challenge the stupid upper management (of Nichia).

Note:
The blue light (on LED screen in most mobile phone, iPad, LED TV) is bad for eyes. Put a layer of blue-light screen protector (S$30) to protect eyes.

Field R Endomorphism

Modern Math Leave a comment


“Le seul endomorphisme du corps R est l’identité“?

car tout morphisme d’anneaux (unitaires) d’un corps R vers un anneau A ≠ {0} est injectif ¹. En particulier, tout morphisme de corps f : R → R est injectif. Comme f(1) = 1, on a f(1) ≠ 0, donc f est un isomorphisme de corps. Ainsi, f est déterminé par f(1) et doit être l’identité ¹.

————–Why injective?—–

Bing answers:

Soit f : R → A un morphisme d’anneaux unitaires d’un corps R vers un anneau A ≠ {0}. Nous devons montrer que f est injectif.

Soit a ∈ ker(f). Alors f(a) = 0A. Comme f est un morphisme d’anneaux, nous avons f(1R) = 1A. Par conséquent, f(a) = f(a * 1R) = f(a) * f(1R) = 0A. Cela implique que f(a) = 0A pour tout a ∈ R.

Maintenant, soit a, b ∈ R tels que f(a) = f(b). Alors f(a – b) = f(a) – f(b) = 0A – 0A = 0A. Cela implique que a – b ∈ ker(f), donc a – b = 0R, ce qui implique que a = b. Par conséquent, f est injectif.

—–English Explanation —-

An endomorphism is a morphism from a mathematical object to itself ⁵. In the case of a field, an endomorphism is a function that maps the field to itself and preserves the field structure ⁵. A field endomorphism f: R → R is a function that maps the real numbers to themselves and preserves the field structure of the real numbers ¹. To prove that any endomorphism of the field of real numbers is the identity map, we can use the following intuitive argument:

Suppose f: R → R is an endomorphism of the field of real numbers. Let x be any real number. Then f(x) is also a real number. Since f is a field endomorphism, it must preserve the field structure of the real numbers. In particular, f must preserve the addition and multiplication operations of the real numbers. This means that for any two real numbers a and b, we have:

f(a + b) = f(a) + f(b) and f(ab) = f(a)f(b)

Now, let’s consider the function g: R → R defined by g(x) = f(x) – x. Since f and x are both endomorphisms of the field of real numbers, g is also an endomorphism of the field of real numbers. Moreover, we have:

g(a + b) = f(a + b) – (a + b) = (f(a) + f(b)) – (a + b) = (f(a) – a) + (f(b) – b) = g(a) + g(b)

and

g(ab) = f(ab) – ab = f(a)f(b) – ab = (f(a) – a)(f(b) – b) + ab – af(b) – bf(a) = g(a)g(b)

Thus, g is a field endomorphism that satisfies the following properties:

g(a + b) = g(a) + g(b) and g(ab) = g(a)g(b)

for all real numbers a and b. This means that g is a linear transformation of the real numbers over itself ². Since g is a linear transformation of the real numbers over itself, it must have the form g(x) = cx for some constant c ². But we know that g(1) = f(1) – 1 = 0, so c = 0. Therefore, g(x) = 0 for all real numbers x, which implies that f(x) = x for all real numbers x. Hence, any endomorphism of the field of real numbers is the identity map.

Source: Conversation with Bing, 1/30/2024
(1) Endomorphism – Wikipedia. https://en.wikipedia.org/wiki/Endomorphism.
(2) Is an automorphism of the field of real numbers the identity map?. https://math.stackexchange.com/questions/449404/is-an-automorphism-of-the-field-of-real-numbers-the-identity-map.
(3) Ring Endomorphisms of the Reals | Shawn M. O’Hare. https://www.shawnohare.com/post/2014/04/980b9501-945b-459f-ad8f-3d2d54dbb2ba/.
(4) Algebra – endomorphisms of field – Mathematics Stack Exchange. https://math.stackexchange.com/questions/3028813/algebra-endomorphisms-of-field.
(5) Prove that if A is simple every R-module endomorphism is either the …. https://math.stackexchange.com/questions/2496972/prove-that-if-a-is-simple-every-r-module-endomorphism-is-either-the-zero-map-or.
(6) Endomorphism Definition & Meaning – Merriam-Webster. https://www.merriam-webster.com/dictionary/endomorphism.
(7) Endomorphism — from Wolfram MathWorld. https://mathworld.wolfram.com/Endomorphism.html.
(8) Ring homomorphism – Wikipedia. https://en.wikipedia.org/wiki/Ring_homomorphism.
(9) 6.2: Ring Homomorphisms – Mathematics LibreTexts. https://math.libretexts.org/Courses/Mount_Royal_University/MATH_2101_Abstract_Algebra_I/Chapter_6%3A_Rings_and_fileds/6.2%3A_Ring_Homomorphisms.
(10) Morphisms of Rings | SpringerLink. https://link.springer.com/chapter/10.1007/978-3-030-66545-6_8.
(11) Morphisms of Rings and Applications to Complexity – IIT Kanpur. https://www.cse.iitk.ac.in/users/nitin/papers/thesis.pdf.
(12) Morphisms of Rings – Dalhousie University. https://www.mscs.dal.ca/~pare/SpringLamb.pdf.
(13) Jordan Canonical Form | Brilliant Math & Science Wiki. https://brilliant.org/wiki/jordan-canonical-form/.
(14) Jordan normal form – Wikipedia. https://en.wikipedia.org/wiki/Jordan_normal_form.
(15) Jordan Canonical Form – Finding the Jordan Canonical Form – BYJU’S. https://byjus.com/maths/jordan-canonical-form/.
(16) 3Morphismes d’anneaux – univ-amu.fr. https://www.i2m.univ-amu.fr/perso/benjamin.audoux/enseignements/2019-2020_L2Maths_Algebre2/Sem2Cours.pdf.
(17) Anneaux – Université Paris-Saclay. https://www.imo.universite-paris-saclay.fr/~david.harari/enseignement/agreg21/ringsbis.pdf.
(18) le produit des anneaux – u-bordeaux.fr. https://www.math.u-bordeaux.fr/~qliu/Enseignement/StructuresAlg2/2019/cours2.pdf.
(19) VII . Anneau, morphisme – Université Paris-Saclay. https://www.imo.universite-paris-saclay.fr/~pierre.lorenzon/enseignement/Algebre/MintEns/M313_Ch_VII.pdf.
(20) Morphisme d’anneaux — Wikipédia. https://fr.wikipedia.org/wiki/Morphisme_d%27anneaux.

Can Vectors Multiply ? YES!

Modern Math Leave a comment

Vector algebra : vectors can +, – but not * or /(?).

However, Vectors can
dot (u.v = scalar ) or
cross product
(u∧v= perpendicular vector w).

Actually Vectors can multiply * :
Bivector, Trivector
that was how Quaternion was invented: (1, i,j,k) where
ij= -ji, jk=-kj, ik=-ki
i²= j²= k² = ijk=-1

This is called Geometric Algebra or Clifford Algebra:
uv = u.v + u∧v


Vector division /:
{\frac{v} {w} = v * w ^{-1}}

https://homework.study.com/explanation/how-to-invert-a-vector.html

Integration By Parts

Elementary Math Leave a comment

IbP (Integration By Parts) is the reverse of Differentiation “Product Rule”.
Hence IbP Integration works ONLY if the function f, g are :
Continuous and Differentiable
(ie f’ & g’ exist)

In French Baccalauréat (A-Level) math , before integration the students must prove “Continuity” in the interval. More rigorously, also check if Differentiable. (See counter-examples here:
https://tomcircle.wordpress.com/2023/04/02/weierstrass-cid/ )

增减辗转相除法 Synthetic Division

Education, Elementary Math Leave a comment

中学代数：
增减辗转相除法
to find the remainder of f(x) when x= u

If f(u)=0,
then x=u is a root of f(x)
or
(x-u) a factor of f(x).

In Abstract Algebra (Ring Theory since Polynomial has Ring structure behaves exactly like Integers),
we note f(x)/(x-u)
where
(x-u) is the IDEAL of f(x).

Theory:
f(x)=p(x).(x-u)+r(x) …(1)
At x= u, (x-u)=0
f(u)= r(u)
r(u) being the remainder.

If r(u)=0,
from (1):
f(u)=p(u).(x-u)
then (x-u) is a factor (IDEAL) , or
x=u is a root of f(x).

Algebraic Geometry is a study of all IDEALS of the polynomials f(x). Like study D24/猫山/黑刺 durians, just enough by analysing their kernel (核)。

Note: Idéal in “Ring” is similar to Kernel in “Group”.
They are both the “essence” (aka “DNA”) of the structure Ring or Group, respectively.