As a theoretical physics PhD student, it is quite recomforting to see that not all mathematicians are absolutely unhinged proof writing machines and they also spend time having to grasp the concepts from time to time
I'm a physicist who spends time in the math department. One thing I have noticed is that in math departments it's almost compulsory to have at least one person who walks around looking deeply lost and confused and talking to themselves, picking things up then frowning and walking away muttering incessantly to themselves, whereas in a physics department you never see these people lol.
I’m an econ major and I always find math research so fascinating. The complexity of math and how you guys go on about to solve a problem is mind boggling to me. Respect to you math geniuses
Hi! Just want you to know that I really enjoy your videos. I am currently an undergrad math student with the objective to progress to a masters in pure math and being able to see what further mathematics looks like is really enjoyable to me. Keep up with the great work!
Objectives = targets, goals -- teleological. Problem, reaction, solution -- the Hegelian dialectic. Concepts are dual to percepts -- the mind duality of Immanuel Kant. All mathematical equations are dualities, Y = X. Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics! Convergence upon solutions, synthesis is teleological -- a syntropic process. Syntropy is dual to entropy. "Always two there are" -- Yoda.
How far are you in your undergrad? I was chem engineering, wish I did more pure math though. Would love to hear what you’re learning if you’re doing junior level classes or above. My furthest math class was differential EQ/matrix algebra, and I’m self teaching abstract algebra right now.
After hearing how the first problem has already been on a professor’s mind for about eight years, I thought a lot about tenacity and how there are some puzzles that do take a long time to solve
Wo wo wo... for 8 years in the breaks between teaching duties, administrative duties, applying for grants, writting reviews and so on. On top of that, some math problems just wait for a proper victim... stfu! A proper PhD student :D
Problem, reaction, solution -- the Hegelian dialectic. Concepts are dual to percepts -- the mind duality of Immanuel Kant. All mathematical equations are dualities, Y = X. Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics! Convergence upon solutions, synthesis is teleological -- a syntropic process. Syntropy is dual to entropy. "Always two there are" -- Yoda.
Listening to you talk about how you approach your Math PhD really puts me at ease for when I got for mine. I'm glad I'm not the only one who felt the first problem made sense, but also made me wanna run away screaming!
Problem, reaction, solution -- the Hegelian dialectic. Concepts are dual to percepts -- the mind duality of Immanuel Kant. All mathematical equations are dualities, Y = X. Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics! Convergence upon solutions, synthesis is teleological -- a syntropic process. Syntropy is dual to entropy. "Always two there are" -- Yoda.
Problem, reaction, solution -- the Hegelian dialectic. Concepts are dual to percepts -- the mind duality of Immanuel Kant. All mathematical equations are dualities, Y = X. Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics! Convergence upon solutions, synthesis is teleological -- a syntropic process. Syntropy is dual to entropy. "Always two there are" -- Yoda. Isomorphisms are invertible homomorphisms. Injective is dual to surjective synthesizes bijective or isomorphism. Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference). The integers or real numbers are self dual:- ruclips.net/video/AxPwhJTHxSg/видео.html Elliptic curves are dual to modular forms. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy, summations) is dual to differentiation (entropy, differences). SINE is dual to COSINE -- the word 'co' means mutual and implies duality. Duality creates reality.
Problem, reaction, solution -- the Hegelian dialectic. Concepts are dual to percepts -- the mind duality of Immanuel Kant. All mathematical equations are dualities, Y = X. Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics! Convergence upon solutions, synthesis is teleological -- a syntropic process. Syntropy is dual to entropy. "Always two there are" -- Yoda.
@@eqwerewrqwerqre When you understand Hegel you can create new laws of physics:- Decreasing the number of dimensions or states is a syntropic process -- homology. Increasing the number of dimensions or states is an entropic process -- co-homology. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy). The 4th law of thermodynamics is hardwired into mathematics and mathematical thinking. Domains are dual to co-domains -- Group theory (symmetries). Symmetries are dual to conservation -- the duality of Noether's theorem. Teleological physics (syntropy) is dual to non teleological physics (entropy). Making predictions to track targets, goals & objectives is a syntropic process -- teleological. Syntropy (prediction, convergence) is dual to increasing entropy (divergence) -- the 4th law of thermodynamics! "Through imagination and reason we turn experience into foresight (prediction)" -- Spinoza describing syntropy. All observers make predictions and therefore they are using syntropy! "Always two there are" -- Yoda. Energy is dual to mass -- Einstein. Dark energy is dual to dark matter.
@@jennifertate4397 You are welcome, there is loads more:- Injective is dual to surjective synthesizes bijection or isomorphism. Absolute truth is dual to relative truth -- Hume's fork. Syntax is dual to semantics -- languages or communication. Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory. If mathematics is a language then it is dual. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual! The integers are self dual as they are their own conjugates. "Always two there are" -- Yoda. Subgroups are dual to subfields -- the Galois correspondence. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung. Being is dual to non-being creates becoming -- Plato's cat. Alive is dual to not alive -- Schrodinger's cat. Thesis (alive, being) is dual to anti-thesis (not alive, non-being) creates the converging or syntropic thesis, synthesis (becoming) -- the time independent Hegelian dialectic or Hegel's cat. Schrodinger's cat is based upon Hegel's cat and he stole it from Plato (Socrates). Brahman (the creator God, thesis) is dual to Shiva (the destroyer God, anti-thesis) synthesizes Vishnu (the preserver God) -- the Trimurti or Hegel's cat. The Trimurti is dual to the Hegelian dialectic.
I relate to you in quite a few aways, even though I am almost a decade younger than you. Last year was my first year of Uni and all I did was trying to get really good grades. This led me to have a virtually non-existent social life, I gained a lot of weight, and just everything felt bad in all aspects besides my good grades. I reckon that it even more important to make time for networking, exercise/health and in general having a varying life which does not revolve around uni. So this year, I decided to study less, hit the gym, and socialise a lot more. I am still trying to get good grades, but it feels so good that there is some variety in my life. Anyways, I am super inspired by your channel, I like you are open about your situation and the stresses of being a grad student. I wish the best for you my friend!
For the first problem, I would recommend looking at some results from both Functional & Harmonic Analysis. You may be able to use some ingenuity with Fourier Transforms, and use some results about compact & bounded Integral Operators, at least to make some progress. Analysis/PDE was always my favorite subject
The time domain is dual to the frequency domain -- Fourier analysis. Problem, reaction, solution -- the Hegelian dialectic. Concepts are dual to percepts -- the mind duality of Immanuel Kant. All mathematical equations are dualities, Y = X. Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics! Convergence upon solutions, synthesis is teleological -- a syntropic process. Syntropy is dual to entropy. "Always two there are" -- Yoda. Isomorphisms are invertible homomorphisms. Injective is dual to surjective synthesizes bijective or isomorphism. Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference). The integers or real numbers are self dual:- ruclips.net/video/AxPwhJTHxSg/видео.html Elliptic curves are dual to modular forms. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy, summations) is dual to differentiation (entropy, differences). SINE is dual to COSINE -- the word 'co' means mutual and implies duality. Duality creates reality.
For the second problem, if you modify it to any bounded, smooth, convex body in R2, I believe the boundaries are homotopic. If you could create an ordering on such curves, then you construct the surface by piecing the slices together with to homotopy parameter giving you the z value, since the problem is essentially generalizing conic sections to convex sections.
It looks like NA PhDs are really quite different from my own PhD experience. I didn’t have any classes during my PhD’s as the essentials are covered in your undergraduate degree in Western Europe. I live in the US now, and I was initially really surprised that PhD’s had classes, and that those classes covered material that I considered undergraduate level material (in chemistry at least). Now i see the stark difference in the undergraduate degree philosophy between Europe and the US, and I understand why classes need to be taken. Ultimately we all end up with the same skills by the end of our PhD’s irregardless of whether it is in the US or Europe, but I am so glad that I did my education in Europe as, for me, it is more focused. Now, having just turned 50, I am going to University for fun to do degrees in maths and physics…but again I took the European route as the equivalent US undergraduate degrees would not have covered the same amount of material as the European degrees, and would not have been as focused on the core subjects. no one is better than the other….just different philosophies.
I really appreciate your view on each being a different philosophy to arrive at the same goal! I finished my PhD in America with 2 European advisors who basically were there for discussion rather than direction. They really tried to emphasize everyone's degree has a totally different story and timeline so comparing doesn't necessarily help your mental health. I do think in the end the student has to find what they resonate with the most making it immensely difficult to compare between the two systems and rather make it a notion about the style of approach.
im a student in the UK, just got through the first semester of my maths degree. i have no idea what i wanna do other than research and seeing this stuff is weirdly interesting and scary at the same time, thanks for the insights though. btw that universal body is gonna keep me up at night now lmao
Problem, reaction, solution -- the Hegelian dialectic. Concepts are dual to percepts -- the mind duality of Immanuel Kant. All mathematical equations are dualities, Y = X. Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics! Convergence upon solutions, synthesis is teleological -- a syntropic process. Syntropy is dual to entropy. "Always two there are" -- Yoda. Isomorphisms are invertible homomorphisms. Injective is dual to surjective synthesizes bijective or isomorphism. Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference). The integers or real numbers are self dual:- ruclips.net/video/AxPwhJTHxSg/видео.html Elliptic curves are dual to modular forms. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy, summations) is dual to differentiation (entropy, differences). SINE is dual to COSINE -- the word 'co' means mutual and implies duality. Duality creates reality.
I really want you to pull that staple out so badly, after you had that first bit of fun playing with it. 🤣🤣🤣 Enjoy your vids. You're smart and hilarious.
Love these videos! Can you give some study tips on learning + then remembering the content when’s studying new and unfamiliar topics? I find that abstract topics that aren’t able to be “intuitively visualised” don’t stick in my brain particularly well initially, do you find that this is this a common issue?
It is common, pretty much all new topics for me are too abstract. What works for me is to brute force it and write the results by hand so that I have the muscle memory. Trying to construct examples and counter examples also help.
@@PhDVlog777 Problem, reaction, solution -- the Hegelian dialectic. Concepts are dual to percepts -- the mind duality of Immanuel Kant. All mathematical equations are dualities, Y = X. Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics! Convergence upon solutions, synthesis is teleological -- a syntropic process. Syntropy is dual to entropy. "Always two there are" -- Yoda. Isomorphisms are invertible homomorphisms. Injective is dual to surjective synthesizes bijective or isomorphism. Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference). The integers or real numbers are self dual:- ruclips.net/video/AxPwhJTHxSg/видео.html Elliptic curves are dual to modular forms. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy, summations) is dual to differentiation (entropy, differences). SINE is dual to COSINE -- the word 'co' means mutual and implies duality. Duality creates reality.
Convolution came up during our signal analysis course, so I guess it is more of a applied math thing, but we also learned about it during Probability II.
Problem, reaction, solution -- the Hegelian dialectic. Concepts are dual to percepts -- the mind duality of Immanuel Kant. All mathematical equations are dualities, Y = X. Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics! Convergence upon solutions, synthesis is teleological -- a syntropic process. Syntropy is dual to entropy. "Always two there are" -- Yoda. Isomorphisms are invertible homomorphisms. Injective is dual to surjective synthesizes bijective or isomorphism. Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference). The integers or real numbers are self dual:- ruclips.net/video/AxPwhJTHxSg/видео.html Elliptic curves are dual to modular forms. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy, summations) is dual to differentiation (entropy, differences). SINE is dual to COSINE -- the word 'co' means mutual and implies duality. Duality creates reality.
I found a RUclips channel in which a maths PhD student was solving problems on a live stream as preparation for his qualifying exams. You should use RUclips to your advantage. There are amazing lectures out there on master and PhD-level topics. I'd advise you to prepare from Indian lectures and books... they are the best! If you can solve those problems, you can probably do research easily.
You can try a stochastic approach to the first problem, set N = 3 and choose the function as a 3rd degree polynomial, than see what probability you have of the statement being true for different values of the constant C. You can try doing a fit increasing the degree of the polynomial to see what happens if the degree of the polynomial becomes infinite (arbitrary function)
Convolutions are a functional analysis thing. They get used in signal processing because they are practical to compute and so let you do the relevant functional analysis stuff in a way that's computationally feasible.
@@ffc1a28c7 Maybe it's obvious to you, but in the video there is a statement that he hasn't heard much about convolution before, so it might not be that obvious. So maybe it has been tried, and maybe it hasn't. That's why my suggestion is written in a form of a question. To me it is deffinitely not that obvious, because Fourier brings more trouble to deal with into the problem, like the fact that both of the functions have to be tempered distributions, otherwise you don't know what the Fourier will give you, so getting rid of the convolution is not cheap at all. But maybe you have to find diifferent integral transformation which does not even create this issue, I don't know.
@@ffc1a28c7 that’s no attitude to have when sinking your teeth into something. Try everything you can think of. Become familiar with it. Encourage rather than dissuade
@@crabster1297 I mean it's likely been tried by the prof that gave OP the problem. Also, that's one of the core things that people are taught about fourier series.
The most interesting part of this video is around 13:35 where you see one problem as interesting and the other makes you cry, as you say. Just the idea of what attracts someone to a problem and what pushes them away. Is the geometry naturally more intriguing because almost immediately as you say your gut tells you no such body exists and you feel compelled to disprove this etc. Whereas the other problem is full of notation that maybe is not as evocative? Could the first problem be made more compelling to you maybe through different notation? What exactly do you mean when you say "bring about the worst in people" when you look at the first one. I know it's flippant but I'd be really interested in what feelings/thoughts you have when you say that. I'm just interested in the psychology of mathematics and what makes solving problems attractive. Anyway interesting stuff.
I think the convex body problem is more accessible. You can describe it to people that don’t have a math background and they can understand it. There is a nice geometric interpretation that anyone can get their heads around. I see it more as a puzzle than a math problem. When I say “bring about the worst in people” I really didn’t mean much other than it is a difficult problem that will drive you nuts trying to solve. And because it is so difficult, it will frustrate you to the point where you just become bitter and upset. Math people are sensitive about there math abilities.
also here are a few things i noticed about the first problem that could help you: x/(abs(x)^s) is eerily similar to the sign function convolutions are sometimes used with fourier transforms, so idk you could try something
Problem, reaction, solution -- the Hegelian dialectic. Concepts are dual to percepts -- the mind duality of Immanuel Kant. All mathematical equations are dualities, Y = X. Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics! Convergence upon solutions, synthesis is teleological -- a syntropic process. Syntropy is dual to entropy. "Always two there are" -- Yoda. Isomorphisms are invertible homomorphisms. Injective is dual to surjective synthesizes bijective or isomorphism. Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference). The integers or real numbers are self dual:- ruclips.net/video/AxPwhJTHxSg/видео.html Elliptic curves are dual to modular forms. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy, summations) is dual to differentiation (entropy, differences). SINE is dual to COSINE -- the word 'co' means mutual and implies duality. Duality creates reality.
Problem, reaction, solution -- the Hegelian dialectic. Concepts are dual to percepts -- the mind duality of Immanuel Kant. All mathematical equations are dualities, Y = X. Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics! Convergence upon solutions, synthesis is teleological -- a syntropic process. Syntropy is dual to entropy. "Always two there are" -- Yoda. Isomorphisms are invertible homomorphisms. Injective is dual to surjective synthesizes bijective or isomorphism. Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference). The integers or real numbers are self dual:- ruclips.net/video/AxPwhJTHxSg/видео.html Elliptic curves are dual to modular forms. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy, summations) is dual to differentiation (entropy, differences). SINE is dual to COSINE -- the word 'co' means mutual and implies duality. Duality creates reality.
To me it feels oddly familiar to a wave function (in quantum mechanics) but since the wave function is complex in nature its just an analogy from my part but since the function is smooth so why not use the integral transform of that function in the definition of that convolution (idk how convolution works tho I've here it here for the 1st time ) to find the value of phi * g without the phi in RHS and then try going for the constant... Idk if it will work tho hope it helps. Ps: I am just a physics major so i really don't have much idea how to tackle this... Hope it helps ever so little
Question 2: Ok, I know nothing about maths - was in the thicko group at school - but I do code 3d game engines and neural nets so I'm firmly in the "little bit of knowledge being a dangerous thing" stage of having practical experience in latent space. So this is in plain old R3. So the obvious approach is to make a sausage (technical term) and have the plain intersect it perpendicularly to the length. Then of course you move the plain from one end to the other and transform the shape through the range of 2d concave shapes. This breaks the problem down into steps: 1. Dealing solely in 2d, can you make a function to transform the shape through the entire set of shapes. But this in inherently nonsensical if there is no rigid definition of what a distinct shape is because otherwise it's infinite and the sausage never ends. If we just take the set of shapes with equal length sides and corners; triangles, square, etc. That never ends. So you can never fulfil the criteria. So there must be something in your mathematical definition of a 2d convex shape that makes the list finite. Perhaps you group these shapes into one category. 2. Once you have a function of all the shapes then they must be arranged in a series where the 3d shape is kept convex. This sounds a much easier problem but you don't know until you clarify what counts as a shape in the step 1. Very likely it would be with respect to the centre of the sausage and would have a positive and negative component. 3. You then must find an interpolation between the 2d shapes which preserves the concavity. 4. You must check that all other intersections of this shape are concave. Job done*. *he says! Haha.
For the third problem maybe you should talk with Carlos Gustavo Tamm moreira (gugu), he is a brazilian researcher at IMPA, and I think he worked with Cantor sets
Problem, reaction, solution -- the Hegelian dialectic. Concepts are dual to percepts -- the mind duality of Immanuel Kant. All mathematical equations are dualities, Y = X. Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics! Convergence upon solutions, synthesis is teleological -- a syntropic process. Syntropy is dual to entropy. "Always two there are" -- Yoda. Isomorphisms are invertible homomorphisms. Injective is dual to surjective synthesizes bijective or isomorphism. Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference). The integers or real numbers are self dual:- ruclips.net/video/AxPwhJTHxSg/видео.html Elliptic curves are dual to modular forms. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy, summations) is dual to differentiation (entropy, differences). SINE is dual to COSINE -- the word 'co' means mutual and implies duality. Duality creates reality.
Using this expression, we can write U as: U(x) = (phi * x/|x|)(x) = ∫R^n phi(x - y)(y/|y|) dy where the integral is taken over all vectors y in R^n, with respect to an infinitesimal dy. Note that we have replaced the fixed vector t_0 in the convolution definition with the variable vector x, which is the point at which we are evaluating U. To estimate the value of C(n,s) in terms of the maximum magnitude of U, we can use the Cauchy-Schwarz inequality to obtain: |U(x)| ≤ (∫R^n |phi(x - y)|^2 dy)^{1/2} (∫R^n |y/|y||^2 dy)^{1/2} where the first factor on the right-hand side is the L^2 norm of phi centered at x, and the second factor is the L^2 norm of the unit vector y/|y|. The second factor can be evaluated explicitly as: (∫R^n |y/|y||^2 dy)^{1/2} = (∫S^{n-1} dS)^{1/2} = √(2π(n-1)/n) where S^{n-1} is the (n-1)-dimensional sphere in R^n, and the integral is taken over the unit sphere with respect to its surface measure. This constant depends only on the dimension n and is independent of the function phi. To estimate the first factor, we can use the fact that phi has compact support and is infinitely differentiable, which implies that its Fourier transform decays rapidly. Specifically, we can write: |phi(x - y)|^2 ≤ C |φ^(2)(ξ)|/(1 + |ξ|^2)^{2s} where C is a constant that depends on phi and the size of its support, and φ^(2) is the second derivative of the Fourier transform of phi. This inequality follows from the Paley-Wiener theorem, which relates the smoothness of a function to the decay rate of its Fourier transform. Using this inequality, we can estimate the L^2 norm of phi centered at x as: (∫R^n |phi(x - y)|^2 dy)^{1/2} ≤ C' ∫R^n |φ^(2)(ξ)|/(1 + |ξ|^2)^{s} dξ where C' is a constant that depends on C and the size of the support of phi. To estimate the integral on the right-hand side, we can use the fact that the Fourier transform of the second derivative of a function is proportional to the Fourier transform of the function itself, up to a constant factor. Specifically, we can write: |φ^(2)(ξ)| ≤ C'' |φ(ξ)| where C'' is a constant that depends on phi. Using this inequality, we can further simplify the estimate for the L^2 norm of phi centered at x as: (∫R^n |phi(x - y)|^2 dy)^{1/2} ≤ C''C' ∫R^n |φ(ξ)|/(1 + |ξ|^2)^{s} dξ where we have used the fact that the constant factors C and C'' can be absorbed into C'. |U(x)|
How do you keep yourself motivated when doing research? I'm afraid that if I would pursue a phd I would make slow or no progress compared to others in the field whom all seem so much smarter than myself.
What I find interesting is in the last two examples you built structural models to build the arguments on while in the first it appears only the symbolisms of the math equations are given. A much more difficult mechanism to get insight of the problem. Its been my experience in dealing with engineering issues a model helps in arriving at a solution. It doesn't always work but it is effective when it does. Good luck with your research.
Problem, reaction, solution -- the Hegelian dialectic. Concepts are dual to percepts -- the mind duality of Immanuel Kant. All mathematical equations are dualities, Y = X. Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics! Convergence upon solutions, synthesis is teleological -- a syntropic process. Syntropy is dual to entropy. "Always two there are" -- Yoda. Isomorphisms are invertible homomorphisms. Injective is dual to surjective synthesizes bijective or isomorphism. Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference). The integers or real numbers are self dual:- ruclips.net/video/AxPwhJTHxSg/видео.html Elliptic curves are dual to modular forms. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy, summations) is dual to differentiation (entropy, differences). SINE is dual to COSINE -- the word 'co' means mutual and implies duality. Duality creates reality.
@Валера Dark energy is repulsive gravity, negative curvature or hyperbolic space (inflation). Gaussian negative curvature is defined using two dual points -- non null homotopic (duality):- en.wikipedia.org/wiki/Gaussian_curvature The big bang is an infinite negative curvature singularity -- repulsive, divergent like a pringle! Positive curvature is dual to negative curvature -- Gauss, Riemann geometry. Curvature or gravitation is dual, gravitational energy is dual. Potential energy is dual to kinetic energy. Apples fall to the ground because they are conserving duality. Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). Energy is duality, duality is energy -- the conservation of duality (energy) will be known as the 5th law of thermodynamics, Generalized Duality. We are already living inside a black hole according to this CERN physicist:- ruclips.net/video/A8bBhkhZtd8/видео.html Inside is dual to outside. Everything in physics is made from energy (duality) likewise for mathematics!
@Валера Cosine is the same function as sine but different as there is a 90 degree phase lag -- perpendicularity. Same is dual to different. Perpendicularity, orthogonality = duality (mathematics). The Christian cross is composed of two perpendicular lines -- duality. Christians have been worshipping duality for thousands of years! Points are dual to lines -- the principle of duality in geometry.
Third one: All of the endpoints of your earlier divisions end up in your final set, but thats it. If you pull 1/4 out, it goes 3/8,1/4,3/8 for lengths. Endpoints at {0/8, 3/8, 5/8, 8/8} Going to get twice the endpoints after twice {[0/8²], [3²/8²], [(3*5)/8²], [(3*8)/8²], [(5*8)/8²], [(3²+(5*8))/8²], [((3*5)+(5*8))/8²], [8²/8²]} And so on. So now if you shift by 1/8² you get all of them, and this pattern will go on forever. Just shift by 1/(8^n). Done
If you bound a 2d convex body with another 2d convex body, map each perimeter point to one another and extrude into 3 dimensions, you’re left with a 3d convex body. Idk of a function that will give every possible convex body in 2d, but if you apply the preceding logic to such a function you can make a never-ending tunnel that increases in size and has a cross section of every possible 2d convex body. My guess anyway
Problem, reaction, solution -- the Hegelian dialectic. Concepts are dual to percepts -- the mind duality of Immanuel Kant. All mathematical equations are dualities, Y = X. Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics! Convergence upon solutions, synthesis is teleological -- a syntropic process. Syntropy is dual to entropy. "Always two there are" -- Yoda. Isomorphisms are invertible homomorphisms. Injective is dual to surjective synthesizes bijective or isomorphism. Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference). The integers or real numbers are self dual:- ruclips.net/video/AxPwhJTHxSg/видео.html Elliptic curves are dual to modular forms. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy, summations) is dual to differentiation (entropy, differences). SINE is dual to COSINE -- the word 'co' means mutual and implies duality. Duality creates reality.
@hyperduality Have you considered that the identity property and associative property of morphisms in set theory, category theory etc. are dual in that they all prohibit local entropy increase? I thought it was an interesting framework to understand why rigor is a useful construct.
@@declandougan7243 I am not understanding your question, if you would care to explain it a bit more? Decreasing the number of dimensions or states is a syntropic process -- homology. Increasing the number of dimensions or states is an entropic process -- co-homology. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy). The 4th law of thermodynamics is hardwired into mathematics and mathematical thinking. Domains are dual to co-domains -- Group theory (symmetries). Symmetries are dual to conservation -- the duality of Noether's theorem. Local entropy increase? What do you mean here?
I'm about in the same position as you. I'm not sure whether to do a Phd or to focus on my business. I'm curious what you're looking to research? Have you decided your proposal?
For the second question, what is your definition of a convex body. I mean, a single point is certainly a convex set and the intersection of any plane with it is either empty or a single point
I am a Master's student, and soon I get to pick a topic for what to research for my thesis. I won't lie, it is a bit intimidating because often times in my homework sets for my classes I can't do them completely alone and work with classmates (if i have a problem set with 5 or 6 problems, then I could probably do 4 or 5 alone), so how am I expected to do research? How is the process for you? I am deciding whether or not to pursue a PhD, or if it's simply not for me.
I did a PhD and even then I had hws I had to work with my peers on. In research, it doesn't have to be alone. Talking to my peers about my research and brainstorming ideas help immensely.
Masters student here also. Nobody works alone. PhD students and masters students regularly work together on homeworks. Professors dont do research by themselves either. Shit is hard as fuck and nobody expects you to figure it all out alone
Im 15 and im learning basic maths stuff like Arithmetic Progressions, Similarities, Thales Theoram, Quadratic Equations and basic trignometry and the 3 basic identities and more now. How is this maths? There are literally no numbers only a zero in the whole equation 😨😨😨😨😨😨
@@memekun1040think about what the quadratic formula looks like to 10 year olds, to them its just letters but for you it’s manageable since youve built up to it slowly. Im 20 (2nd year undergraduate maths student) and i do stuff that looks not doable to you, but ive built up to it. same concept applies here youll build up slowly to things and eventually do stuff that makes ur 15 year old self look silly just like ill make my 20 year old self silly in 5 years :)
I’m pretty sure I’ve solved the second problem, if you accept the Axiom of Choice. The set of convex bodies in R^2 must have a cardinality of Aleph 1, since each convex body can be described by a polar function of 2 variables, angle and distance. That means you can create a bijection between the set of convex bodies in R^2 and the real line R. And since the real line R is also the same cardinality as any finite subset of R, you can create a universal convex body as a long tube of finite length with a single orthogonal slice for every convex 2D body. Again assuming the axiom of choice there should be a well ordering that makes that shape continuous
The real line definitely does not have the same cardinality as a finite subset of real numbers. Also how do ensure the tube you create is convex? Creating a tube that has all 2D convex shapes in R3 has been done but it is not convex.
I took two courses in convex geometry in my master's program and I absolutely loved it! The second problem is really interesting and I hope to see your results when you manage to solve it.
Problem, reaction, solution -- the Hegelian dialectic. Concepts are dual to percepts -- the mind duality of Immanuel Kant. All mathematical equations are dualities, Y = X. Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics! Convergence upon solutions, synthesis is teleological -- a syntropic process. Syntropy is dual to entropy. "Always two there are" -- Yoda. Isomorphisms are invertible homomorphisms. Injective is dual to surjective synthesizes bijective or isomorphism. Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference). The integers or real numbers are self dual:- ruclips.net/video/AxPwhJTHxSg/видео.html Elliptic curves are dual to modular forms. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy, summations) is dual to differentiation (entropy, differences). SINE is dual to COSINE -- the word 'co' means mutual and implies duality. Duality creates reality.
Can someone explain how they do come up with these problems, if it’s actually very hard to solve them? And who constructs these overcomplicated math problems?
It’s important to be realistic, but I really think you should treat that first problem with the belief that you might just be able to succeed, try it go for it, what if you solved it?
For #2, yes such a body does exist. It needs to be convex so we will vary its contours in order of size from top to bottom, like an icicle. For a given plane we need, we simply append it to the top and shrink it to maintain convexity. We can do this forever, isomorphisms are not impeded by scale.
I thought of a similar approach. How can you ensure that *all* convex bodies in R^2 appear? My intuition is that somewhere you run into cardinality problems trying to go through all convex shapes. Currently studying for my first degree in mathematics so take my comment with a grain of salt:)
You run into issues when either trying to maintain convexity or fitting the icicle in R^3. If you have a circle and a square "on top of each other" they need some positive distance between them in order for you to mold one into another. And if you put all closed convex sets (with arbitrary convex sets we have no hope of this being true) with some uniform positive distance from each other you run out of space in R^3, so your icicle will be too big to fit in R^3.
The 2nd problem: I may be confused, but isn't the set of all 'shapes' of convex plain figures hypercontinual? The set of all slices of the 3d convex body, however, is just continual. This might help?
Hey, I really enjoyed that video. I am thinking of maybe making use of those problems for my own mathematical research. I do not know if I need a Master's or Ph.D's. But I am happy to admit that I actually have rough-draft mathematical proofs at my apartment. Two or three of these have to do with million-dollar conjectures. No lie. I am thinking of maybe asking you for advice. Do you have an email that I could reply to? I am interested in the work that you do. And I do want to maybe "workshop" some of my original ideas with you or my advisor. Michael Doerr.
Prob2: There's an infinite number of convex shapes in R2 so htf can that be realised in one realistic body. (Not saying it's impossible just speaking my mind. But thats a pretty big part) Also (I don't know) but the convexity of a circle is greater than a polygon since the line joining any two points ON the circle is entirely within the circle whereas that's not the case with polygon. Just making that point, presumably there's already math words/definitions making that distinction. Good luck in whatever you do.
i love ur videos man you offer content that is quite hard to find on youtube 🖤 i got a question though, when u say the support of phi do u mean the essential support? (srry if my question is obvious or doesnt make sense) good luck on ur research 🖤
Thank you for the kind words. Your question is fine, the support (in fact the compact support in this case which I don’t mention) is the set on which phi returns nonzero values.
why does everything about higher mathematics always sound like a conjecture in quantum mechanics.....i would like to propose a new problem....to wit: in the playing card game known as Solitaire (standard version) is there a method or formula of playing that will always yield the maximum number of wins? there are often several different maneuvers that one can make with the cards at various 'discards' but is there a series of rules about each possible display of cards at each consecutive choice that will always yield the maximum possible number of wins? for example, if 3 aces are showing and a deuce is available then which ace should be put onto it? there are so many choices possible and things can often be done in different order with different cards etc
I'm not far into my math education so there might be some flaws in my reasoning but regarding problem 2: can't you just compare cardinality of the set of planes and the set of 2d convex bodies? The cardinality of the set of planes is aleph1 since you can describe a plane with three points in the plane (the cardinality of R^9 is Aleph1). The cardinality of the set of 2d convex bodies is Aleph2, I think, because they correspond to a functions from angles to distances. Rotations and translations might make this calculation a bit more complicated but the cardinality of this set is probably well known. This means there does not exist an onto mapping from the set of planes to the set of 2d convex objects. This would the contradict the existence of a universal body.
The cardinality of all convex bodies is the same as the power set of R which is higher than the amount of planes in R^3. So for arbitrary convex sets this is trivially untrue, but the cardinality of closed convex sets is equal to the cardinality of R, so we have a chance. Also, saying that the cardinality of the real numbers is equal to aleph_1 is known as the continuum hypothesis, which is a famously undecidable problem. I'd be careful using aleph numbers when you talk about cardinalities of sets.
@@MK-13337 ahh, thanks. I didn't know that the cardinality of the set of closed sets is the cardinality of R, interesting. And thanks for the clarification regarding the aleph numbers. I have yet to learn these subjects formally in class, just heard about them in various RUclips videos.
for 2nd problem woudnt it be enough just to prove that from the constructing A->(R^3 convex shape) with flat surface(as if cut by plane) in the direction of "growing" u can create new A'->(R^3 convex shape) with new flat surface in shape of targeted B->( R^2 convex shape) by interpolating between A and B such that it doesnt become concave(to be proven)...and then inductively conclude the rest....(i dont know if i made sense there cuz im not highly educated...had to google most terms ...like i dont even understand 1st and 3rd problems so cant even engage in thinking there)
First you read a beginner book about Discrete Mathematics (for example from Susanna). Most basic things will be covered. After that, you will have to read books about the specific fields. If you want to understand linear algebra, it is probably best to also get a book for beginners for that, etc... . Most fields overlap heavily so you might need to also get more books about different fields. After that you just work yourself through the lingo
@@hypnogri5457 do you need a certain type of brain to understand this stuff? I feel like I won’t even be able to understand the problems let alone solve them if I pursue mathematics at a higher level. Is it possible to get an MRI scan to see if I have the right brain for advanced mathematics?
Do you plan on continuing in academia or pursuing a more commercial position? Interested as I plan to start my Comp Sci PhD soon but am in two minds regarding what I want to do with it
Problem, reaction, solution -- the Hegelian dialectic. Concepts are dual to percepts -- the mind duality of Immanuel Kant. All mathematical equations are dualities, Y = X. Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics! Convergence upon solutions, synthesis is teleological -- a syntropic process. Syntropy is dual to entropy. "Always two there are" -- Yoda. Isomorphisms are invertible homomorphisms. Injective is dual to surjective synthesizes bijective or isomorphism. Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference). The integers or real numbers are self dual:- ruclips.net/video/AxPwhJTHxSg/видео.html Elliptic curves are dual to modular forms. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy, summations) is dual to differentiation (entropy, differences). SINE is dual to COSINE -- the word 'co' means mutual and implies duality. Duality creates reality.
2 questions: 1. You explained why you were asking the 3rd question: it extends an earlier result. But what was the impetus for questions 1 & 2? Why did the questions get asked in that way? 2. You have to publish. But there's no guarantee that you'll solve any given math problem. In engineering or statistics we can always do a thing and then report our results. If what we tried didn't work, I still have something to write about and an analysis I can conduct. But naively, it seems like if you try to solve one of these problems, that you might come away with nothing to show for it. So how do research mathematicians manage to hit publication targets?
If you can't make head way on a problem you move on to do something else that is more productive. You can't publish a null result in mathematics so if you can't solve anything you run out of research money and you are pretty much done for.
For the first problem: Are you trying to show the existence of C, or trying to find the optimal C? For the existence of C, my first instinct would be to try to run a contradiction argument. Regardless, it might be worth computing C for some simple explicit functions --- or maybe for a family of functions whose supports depend on some parameter --- perhaps when n = 1. Separately, it might also be worth asking what sorts of techniques are commonly used to establish inequalities of that type, and why such techniques wouldn't work here. If your colleague has worked on this problem for 8 years, then they've probably considered all this already, but this is my two cents.
I think it's an interesting question, and I'm personally really curious about *why* all of the "obvious" stuff that first comes to mind doesn't work. Lots of people have suggested stuff in the comments. Probably (hopefully?) all of it got tried over the course of 8 years. But if it's still an open question, it all had to fail. So that makes me super curious: if it all fails, then maybe there's an interesting reason for why.
Problem, reaction, solution -- the Hegelian dialectic. Concepts are dual to percepts -- the mind duality of Immanuel Kant. All mathematical equations are dualities, Y = X. Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics! Convergence upon solutions, synthesis is teleological -- a syntropic process. Syntropy is dual to entropy. "Always two there are" -- Yoda. Isomorphisms are invertible homomorphisms. Injective is dual to surjective synthesizes bijective or isomorphism. Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference). The integers or real numbers are self dual:- ruclips.net/video/AxPwhJTHxSg/видео.html Elliptic curves are dual to modular forms. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy, summations) is dual to differentiation (entropy, differences). SINE is dual to COSINE -- the word 'co' means mutual and implies duality. Duality creates reality.
This is morbidly interesting! lol. Anyway about that first problem, if the problem involves a convolution, I would guess the solution will involve Fourier Series. In my mind those two concept are intimately related. Good luck!
If the problem still hasn't been solved, fourier series probably has probably been done to death. So I'd assume that's a dead end and would start looking for alternative ways to prove such a statement. Maybe there's some kind of wavelet decomposition you could do with the function to make progress?
Have i understood the second problem right? - that we wanna find at least one "rather round" 3D object - such that when slicing it many times from different angles - that the types of slices we would get, be equal to all possible types of "rather round" 2D objects? Well.. my intuition says yes.. would be a weird potato with edges, but possible.. My first question would be, which extreme cases can we draw for the 2D shapes? What are the boundaries of what classifies as 2D-convex? Then we make gradients from one to another and stack those sliced together, which makes the shape..
Convolutions are very common in signals and systems in electrical engineering. About 40% of an undergraduate course in signals and systems is just convolutions.
The * is to denote convolution. We used it a lot in Signal & Systems, Control Systems Design and Communications Systems Design. And I mean A LOT. F(t)*g(t) is pronounced f(t) convolved with g(t). You are trying to convolve one function with another. In short, you take the g(t) function, flip it about its x-axis, and integrate it throughout f(t) as it moves from left to right about the x-axis, or the t-axis here. That result is the convolution of f*g. Might want to look for it on utube - my explanation is pitiful. In engineering it's used to relate the input signal with the system signal to analyze or predict the output signal.
Problem 2 seems trivial the way you posed it. Construct a vertical column whose horizontal cross-sections are all the convex 2D bodies, scaled to keep the 3D body convex. Suppose there's a 2D convex body not congruent to any of these cross sections; it can be inserted into the middle (or at either end) of the column and scaled to keep the whole body convex. The solution is not unique. I think the harder question is, can you construct such a body where you guarantee only one unique pi for each body K?
@@user-me7hx8zf9y aha, I think convex-ness is relatively easy to guarantee and prove if I'm a bit more detailed about how the column is constructed, but being finite is another matter altogether. I don't think you can guarantee the shape is finite, in fact the way this construction is proved to have cross-sections containing every convex body might necessitate that it be infinite.
Just trying to grasp the second problem a bit, as a non-mathematician. Is this an accurate statement of the problem? "Does there exist a 3D convex body which, if sliced by a plane at varying angles and positions on the body, would produce every possible 2d convex shape?"
Watching your videos is much like watching a Chinese movie without subtitles. I know they're saying something but bless my soul, I have know idea what it is! 🤔
6:57 I was studying Rudin's Principle of Mathematical Analysis yesterday and seeing this problem reminded me of the proof that every k cell is compact in pg 39. Not the infinite shrinking part but how for a fixed r you could get an n small enough to arrive at the contradiction. Here n is fixed and you need an 'r' or a C. Obviously this is just a philosophy and there are a lot of details you'd need to study like how x/|x| behaves. I don't have the slightest idea what you mean by phi 'supporting' the other thing. And I don't even know why the function should even be bounded (or whatever the problem is trying to state). So I guess it would be a nice idea to find parts which should be true like that, 'Half problems' I guess. I liked War for Art when I read it cause it related to math and research in general too. It was a bit spiritual but it was right when it said it's your duty to study everything you can about the problem and understand all sides given what we know and it's completely upto luck to get the inspiration you need to get the right idea.
This isn’t meant to be a rude question, what is the objective of math research? I started out studying Psychology, now dual majoring in Psychology and Neuroscience, and high level mathematics haven’t really come up in my studies thus far (just a few statistics classes). Hopefully this isn’t a completely stupid question. I’m just an insanely curious person and don’t know a whole lot about math.
For the first problem, have you tried plugging it into the quadratic formula?
Let's not get ahead of ourselves, Stephen. Before going to the quadratic formula, we ought to try out the Pythagorean theorem
Unless you can just factor it of course
Just complete the square, man
Has anyone queried the problem statement into wolframalpha already? I bet it can easily solve it 😌
@@alexcurchin2718 so pi?
As a theoretical physics PhD student, it is quite recomforting to see that not all mathematicians are absolutely unhinged proof writing machines and they also spend time having to grasp the concepts from time to time
I'm a physicist who spends time in the math department. One thing I have noticed is that in math departments it's almost compulsory to have at least one person who walks around looking deeply lost and confused and talking to themselves, picking things up then frowning and walking away muttering incessantly to themselves, whereas in a physics department you never see these people lol.
@@holliswilliams8426 yeah the physics people throw it in the air for a sec and then frown and walk away
I love your description of "unhinged proof writing machines' lol.
And of course, when we are proof writing machines it's typically with notes in hand. We are indeed mere mortals. 😆
@@abebuckingham8198 Is writing proofs and doing research mostly fun and enjoyable? If not why don't you quit?
"This weird object, which I'm going to call U"
Ouch
🤣🤣🤣
I’m an econ major and I always find math research so fascinating. The complexity of math and how you guys go on about to solve a problem is mind boggling to me. Respect to you math geniuses
I am human. Humans can use arms and hands to pick things up. There is math. Math can use functions to perform actions.
eww we have the physics envy econ major here. you dont belong here. dont touch our maths!
As an engineering student, this sounds like wizardry.
🤣🤣🤣
Hi! Just want you to know that I really enjoy your videos. I am currently an undergrad math student with the objective to progress to a masters in pure math and being able to see what further mathematics looks like is really enjoyable to me. Keep up with the great work!
Hi MIguel,
You look hot 🔥
Objectives = targets, goals -- teleological.
Problem, reaction, solution -- the Hegelian dialectic.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
All mathematical equations are dualities, Y = X.
Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics!
Convergence upon solutions, synthesis is teleological -- a syntropic process.
Syntropy is dual to entropy.
"Always two there are" -- Yoda.
miguel ! cool name where are you from ?
How far are you in your undergrad? I was chem engineering, wish I did more pure math though. Would love to hear what you’re learning if you’re doing junior level classes or above. My furthest math class was differential EQ/matrix algebra, and I’m self teaching abstract algebra right now.
After hearing how the first problem has already been on a professor’s mind for about eight years, I thought a lot about tenacity and how there are some puzzles that do take a long time to solve
Wo wo wo... for 8 years in the breaks between teaching duties, administrative duties, applying for grants, writting reviews and so on. On top of that, some math problems just wait for a proper victim... stfu! A proper PhD student :D
Problem, reaction, solution -- the Hegelian dialectic.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
All mathematical equations are dualities, Y = X.
Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics!
Convergence upon solutions, synthesis is teleological -- a syntropic process.
Syntropy is dual to entropy.
"Always two there are" -- Yoda.
@@hyperduality2838 not hegelian dialectic!!!
If you're a professional, my guess is you just occasionally think about it while you sip on your coffee.
Listening to you talk about how you approach your Math PhD really puts me at ease for when I got for mine. I'm glad I'm not the only one who felt the first problem made sense, but also made me wanna run away screaming!
Thank you for posting this and all the other math content you have so far! I've always wanted to see content like this!
Problem, reaction, solution -- the Hegelian dialectic.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
All mathematical equations are dualities, Y = X.
Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics!
Convergence upon solutions, synthesis is teleological -- a syntropic process.
Syntropy is dual to entropy.
"Always two there are" -- Yoda.
The second problem is really interesting, please keep us updated on your progress on it!
16:19 "when you work on a problem every single day for like 2 years, it becomes incorporated in you" 😂 man that's too relatable
Problem, reaction, solution -- the Hegelian dialectic.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
All mathematical equations are dualities, Y = X.
Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics!
Convergence upon solutions, synthesis is teleological -- a syntropic process.
Syntropy is dual to entropy.
"Always two there are" -- Yoda.
Isomorphisms are invertible homomorphisms.
Injective is dual to surjective synthesizes bijective or isomorphism.
Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference).
The integers or real numbers are self dual:-
ruclips.net/video/AxPwhJTHxSg/видео.html
Elliptic curves are dual to modular forms.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy, summations) is dual to differentiation (entropy, differences).
SINE is dual to COSINE -- the word 'co' means mutual and implies duality.
Duality creates reality.
I touched on something very similar to your problem 3 in my master's thesis; definitely one of the more enjoyable and 'novel' parts of my paper!
Problem, reaction, solution -- the Hegelian dialectic.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
All mathematical equations are dualities, Y = X.
Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics!
Convergence upon solutions, synthesis is teleological -- a syntropic process.
Syntropy is dual to entropy.
"Always two there are" -- Yoda.
@hyperduality this is the most sophisticated batch of nonsense I've seen in a while. Thanks for the laugh
@@eqwerewrqwerqre When you understand Hegel you can create new laws of physics:-
Decreasing the number of dimensions or states is a syntropic process -- homology.
Increasing the number of dimensions or states is an entropic process -- co-homology.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy).
The 4th law of thermodynamics is hardwired into mathematics and mathematical thinking.
Domains are dual to co-domains -- Group theory (symmetries).
Symmetries are dual to conservation -- the duality of Noether's theorem.
Teleological physics (syntropy) is dual to non teleological physics (entropy).
Making predictions to track targets, goals & objectives is a syntropic process -- teleological.
Syntropy (prediction, convergence) is dual to increasing entropy (divergence) -- the 4th law of thermodynamics!
"Through imagination and reason we turn experience into foresight (prediction)" -- Spinoza describing syntropy.
All observers make predictions and therefore they are using syntropy!
"Always two there are" -- Yoda.
Energy is dual to mass -- Einstein.
Dark energy is dual to dark matter.
@@hyperduality2838 Amazing and impressive knowledge dude. And I love those connections between philosophy and mathematics.
@@jennifertate4397 You are welcome, there is loads more:-
Injective is dual to surjective synthesizes bijection or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Syntax is dual to semantics -- languages or communication.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory.
If mathematics is a language then it is dual.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual!
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Being is dual to non-being creates becoming -- Plato's cat.
Alive is dual to not alive -- Schrodinger's cat.
Thesis (alive, being) is dual to anti-thesis (not alive, non-being) creates the converging or syntropic thesis, synthesis (becoming) -- the time independent Hegelian dialectic or Hegel's cat.
Schrodinger's cat is based upon Hegel's cat and he stole it from Plato (Socrates).
Brahman (the creator God, thesis) is dual to Shiva (the destroyer God, anti-thesis) synthesizes Vishnu (the preserver God) -- the Trimurti or Hegel's cat.
The Trimurti is dual to the Hegelian dialectic.
I accidently started the video with the volume off and thought all they did was play with their pen.
I dearly miss higher mathematics, this was beautiful. Please keep sharing your progress throughout your phd!
I relate to you in quite a few aways, even though I am almost a decade younger than you. Last year was my first year of Uni and all I did was trying to get really good grades. This led me to have a virtually non-existent social life, I gained a lot of weight, and just everything felt bad in all aspects besides my good grades. I reckon that it even more important to make time for networking, exercise/health and in general having a varying life which does not revolve around uni. So this year, I decided to study less, hit the gym, and socialise a lot more. I am still trying to get good grades, but it feels so good that there is some variety in my life. Anyways, I am super inspired by your channel, I like you are open about your situation and the stresses of being a grad student. I wish the best for you my friend!
As soon as you said convolution, as an EE, I flinched a little 😂😂😂
I like the videos, keep it up :)
For the first problem, I would recommend looking at some results from both Functional & Harmonic Analysis. You may be able to use some ingenuity with Fourier Transforms, and use some results about compact & bounded Integral Operators, at least to make some progress. Analysis/PDE was always my favorite subject
But also it could be entirely possible no such C exists.
The time domain is dual to the frequency domain -- Fourier analysis.
Problem, reaction, solution -- the Hegelian dialectic.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
All mathematical equations are dualities, Y = X.
Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics!
Convergence upon solutions, synthesis is teleological -- a syntropic process.
Syntropy is dual to entropy.
"Always two there are" -- Yoda.
Isomorphisms are invertible homomorphisms.
Injective is dual to surjective synthesizes bijective or isomorphism.
Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference).
The integers or real numbers are self dual:-
ruclips.net/video/AxPwhJTHxSg/видео.html
Elliptic curves are dual to modular forms.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy, summations) is dual to differentiation (entropy, differences).
SINE is dual to COSINE -- the word 'co' means mutual and implies duality.
Duality creates reality.
For the second problem, if you modify it to any bounded, smooth, convex body in R2, I believe the boundaries are homotopic. If you could create an ordering on such curves, then you construct the surface by piecing the slices together with to homotopy parameter giving you the z value, since the problem is essentially generalizing conic sections to convex sections.
It looks like NA PhDs are really quite different from my own PhD experience. I didn’t have any classes during my PhD’s as the essentials are covered in your undergraduate degree in Western Europe. I live in the US now, and I was initially really surprised that PhD’s had classes, and that those classes covered material that I considered undergraduate level material (in chemistry at least). Now i see the stark difference in the undergraduate degree philosophy between Europe and the US, and I understand why classes need to be taken.
Ultimately we all end up with the same skills by the end of our PhD’s irregardless of whether it is in the US or Europe, but I am so glad that I did my education in Europe as, for me, it is more focused.
Now, having just turned 50, I am going to University for fun to do degrees in maths and physics…but again I took the European route as the equivalent US undergraduate degrees would not have covered the same amount of material as the European degrees, and would not have been as focused on the core subjects.
no one is better than the other….just different philosophies.
I really appreciate your view on each being a different philosophy to arrive at the same goal! I finished my PhD in America with 2 European advisors who basically were there for discussion rather than direction. They really tried to emphasize everyone's degree has a totally different story and timeline so comparing doesn't necessarily help your mental health. I do think in the end the student has to find what they resonate with the most making it immensely difficult to compare between the two systems and rather make it a notion about the style of approach.
im a student in the UK, just got through the first semester of my maths degree. i have no idea what i wanna do other than research and seeing this stuff is weirdly interesting and scary at the same time, thanks for the insights though. btw that universal body is gonna keep me up at night now lmao
Problem, reaction, solution -- the Hegelian dialectic.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
All mathematical equations are dualities, Y = X.
Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics!
Convergence upon solutions, synthesis is teleological -- a syntropic process.
Syntropy is dual to entropy.
"Always two there are" -- Yoda.
Isomorphisms are invertible homomorphisms.
Injective is dual to surjective synthesizes bijective or isomorphism.
Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference).
The integers or real numbers are self dual:-
ruclips.net/video/AxPwhJTHxSg/видео.html
Elliptic curves are dual to modular forms.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy, summations) is dual to differentiation (entropy, differences).
SINE is dual to COSINE -- the word 'co' means mutual and implies duality.
Duality creates reality.
@@hyperduality2838nice schizo rant my guy
I really want you to pull that staple out so badly, after you had that first bit of fun playing with it. 🤣🤣🤣 Enjoy your vids. You're smart and hilarious.
Thank you, I appreciate it :)
@@PhDVlog777
Love these videos! Can you give some study tips on learning + then remembering the content when’s studying new and unfamiliar topics? I find that abstract topics that aren’t able to be “intuitively visualised” don’t stick in my brain particularly well initially, do you find that this is this a common issue?
It is common, pretty much all new topics for me are too abstract. What works for me is to brute force it and write the results by hand so that I have the muscle memory. Trying to construct examples and counter examples also help.
@@PhDVlog777 what's the meaning of " brute forcing " a concept ? in a practical sense .
@@aihamelofairi8977 i think retrying and forcing yourself to understand
@@PhDVlog777 Problem, reaction, solution -- the Hegelian dialectic.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
All mathematical equations are dualities, Y = X.
Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics!
Convergence upon solutions, synthesis is teleological -- a syntropic process.
Syntropy is dual to entropy.
"Always two there are" -- Yoda.
Isomorphisms are invertible homomorphisms.
Injective is dual to surjective synthesizes bijective or isomorphism.
Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference).
The integers or real numbers are self dual:-
ruclips.net/video/AxPwhJTHxSg/видео.html
Elliptic curves are dual to modular forms.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy, summations) is dual to differentiation (entropy, differences).
SINE is dual to COSINE -- the word 'co' means mutual and implies duality.
Duality creates reality.
@@aihamelofairi8977 reading again and again and again and again
Convolution came up during our signal analysis course, so I guess it is more of a applied math thing, but we also learned about it during Probability II.
Yeah came up in my vibrations class and had no damn clue what to make of it 😂
It came up in my calculus class when learning Laplace transforms
Integral convolutions are very important in analysis, not just applied math. Not only fourier analysis but also when mollifying functions.
Problem, reaction, solution -- the Hegelian dialectic.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
All mathematical equations are dualities, Y = X.
Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics!
Convergence upon solutions, synthesis is teleological -- a syntropic process.
Syntropy is dual to entropy.
"Always two there are" -- Yoda.
Isomorphisms are invertible homomorphisms.
Injective is dual to surjective synthesizes bijective or isomorphism.
Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference).
The integers or real numbers are self dual:-
ruclips.net/video/AxPwhJTHxSg/видео.html
Elliptic curves are dual to modular forms.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy, summations) is dual to differentiation (entropy, differences).
SINE is dual to COSINE -- the word 'co' means mutual and implies duality.
Duality creates reality.
Awesome insight. This is what I want my life to be.
I found a RUclips channel in which a maths PhD student was solving problems on a live stream as preparation for his qualifying exams. You should use RUclips to your advantage. There are amazing lectures out there on master and PhD-level topics.
I'd advise you to prepare from Indian lectures and books... they are the best! If you can solve those problems, you can probably do research easily.
whats the other yt name??
@@hoyken8887 i would also like to know
that's not a good suggestion lmao
You can try a stochastic approach to the first problem, set N = 3 and choose the function as a 3rd degree polynomial, than see what probability you have of the statement being true for different values of the constant C. You can try doing a fit increasing the degree of the polynomial to see what happens if the degree of the polynomial becomes infinite (arbitrary function)
really needed this video
FYI convolutions are used quite a bit in signal processing, so yes an applied math thing.
Convolutions are a functional analysis thing. They get used in signal processing because they are practical to compute and so let you do the relevant functional analysis stuff in a way that's computationally feasible.
Have you tried Fourier transform of the first problem? Use the fact that Fourier transforms convolution into multiplication.
When you look at these problems, if the approach is obvious, it's highly likely that it's been tried before.
Considering how convolution is a signal, it pretty much screams fourier. It doesn't elucidate the bounds.
@@ffc1a28c7 Maybe it's obvious to you, but in the video there is a statement that he hasn't heard much about convolution before, so it might not be that obvious. So maybe it has been tried, and maybe it hasn't. That's why my suggestion is written in a form of a question.
To me it is deffinitely not that obvious, because Fourier brings more trouble to deal with into the problem, like the fact that both of the functions have to be tempered distributions, otherwise you don't know what the Fourier will give you, so getting rid of the convolution is not cheap at all. But maybe you have to find diifferent integral transformation which does not even create this issue, I don't know.
@@ffc1a28c7 that’s no attitude to have when sinking your teeth into something. Try everything you can think of. Become familiar with it. Encourage rather than dissuade
@@crabster1297 I mean it's likely been tried by the prof that gave OP the problem. Also, that's one of the core things that people are taught about fourier series.
The most interesting part of this video is around 13:35 where you see one problem as interesting and the other makes you cry, as you say. Just the idea of what attracts someone to a problem and what pushes them away. Is the geometry naturally more intriguing because almost immediately as you say your gut tells you no such body exists and you feel compelled to disprove this etc. Whereas the other problem is full of notation that maybe is not as evocative? Could the first problem be made more compelling to you maybe through different notation? What exactly do you mean when you say "bring about the worst in people" when you look at the first one. I know it's flippant but I'd be really interested in what feelings/thoughts you have when you say that. I'm just interested in the psychology of mathematics and what makes solving problems attractive. Anyway interesting stuff.
I think the convex body problem is more accessible. You can describe it to people that don’t have a math background and they can understand it. There is a nice geometric interpretation that anyone can get their heads around. I see it more as a puzzle than a math problem. When I say “bring about the worst in people” I really didn’t mean much other than it is a difficult problem that will drive you nuts trying to solve. And because it is so difficult, it will frustrate you to the point where you just become bitter and upset. Math people are sensitive about there math abilities.
@@PhDVlog777 Thanks for your reply. Interesting channel.
also here are a few things i noticed about the first problem that could help you:
x/(abs(x)^s) is eerily similar to the sign function
convolutions are sometimes used with fourier transforms, so idk you could try something
Problem, reaction, solution -- the Hegelian dialectic.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
All mathematical equations are dualities, Y = X.
Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics!
Convergence upon solutions, synthesis is teleological -- a syntropic process.
Syntropy is dual to entropy.
"Always two there are" -- Yoda.
Isomorphisms are invertible homomorphisms.
Injective is dual to surjective synthesizes bijective or isomorphism.
Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference).
The integers or real numbers are self dual:-
ruclips.net/video/AxPwhJTHxSg/видео.html
Elliptic curves are dual to modular forms.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy, summations) is dual to differentiation (entropy, differences).
SINE is dual to COSINE -- the word 'co' means mutual and implies duality.
Duality creates reality.
You have a great video about what is convolution on 3b1b if you want :)
convolutions are sometimes used in physics to my knowledge (as I encountered them sometimes)
Problem, reaction, solution -- the Hegelian dialectic.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
All mathematical equations are dualities, Y = X.
Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics!
Convergence upon solutions, synthesis is teleological -- a syntropic process.
Syntropy is dual to entropy.
"Always two there are" -- Yoda.
Isomorphisms are invertible homomorphisms.
Injective is dual to surjective synthesizes bijective or isomorphism.
Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference).
The integers or real numbers are self dual:-
ruclips.net/video/AxPwhJTHxSg/видео.html
Elliptic curves are dual to modular forms.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy, summations) is dual to differentiation (entropy, differences).
SINE is dual to COSINE -- the word 'co' means mutual and implies duality.
Duality creates reality.
“No, this is a different s.”
Idk why that made me crack up. Mathematicians could kill people with their notation.
great info vid sir , btw what type of camera you use for making vids ,a gopro ?
To me it feels oddly familiar to a wave function (in quantum mechanics) but since the wave function is complex in nature its just an analogy from my part but since the function is smooth so why not use the integral transform of that function in the definition of that convolution (idk how convolution works tho I've here it here for the 1st time ) to find the value of phi * g without the phi in RHS and then try going for the constant... Idk if it will work tho hope it helps.
Ps: I am just a physics major so i really don't have much idea how to tackle this...
Hope it helps ever so little
fyi, convolutions are used mostly in signal processing, image processing (blurring) and stuff like that
Hi! what are you considering to do after your PhD ? thanks for the video!
What's the name of that pen you are holding......?
It looks amazing
Question 2:
Ok, I know nothing about maths - was in the thicko group at school - but I do code 3d game engines and neural nets so I'm firmly in the "little bit of knowledge being a dangerous thing" stage of having practical experience in latent space.
So this is in plain old R3. So the obvious approach is to make a sausage (technical term) and have the plain intersect it perpendicularly to the length. Then of course you move the plain from one end to the other and transform the shape through the range of 2d concave shapes. This breaks the problem down into steps:
1. Dealing solely in 2d, can you make a function to transform the shape through the entire set of shapes. But this in inherently nonsensical if there is no rigid definition of what a distinct shape is because otherwise it's infinite and the sausage never ends. If we just take the set of shapes with equal length sides and corners; triangles, square, etc. That never ends. So you can never fulfil the criteria. So there must be something in your mathematical definition of a 2d convex shape that makes the list finite. Perhaps you group these shapes into one category.
2. Once you have a function of all the shapes then they must be arranged in a series where the 3d shape is kept convex. This sounds a much easier problem but you don't know until you clarify what counts as a shape in the step 1. Very likely it would be with respect to the centre of the sausage and would have a positive and negative component.
3. You then must find an interpolation between the 2d shapes which preserves the concavity.
4. You must check that all other intersections of this shape are concave.
Job done*.
*he says! Haha.
For the third problem maybe you should talk with Carlos Gustavo Tamm moreira (gugu), he is a brazilian researcher at IMPA, and I think he worked with Cantor sets
Problem, reaction, solution -- the Hegelian dialectic.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
All mathematical equations are dualities, Y = X.
Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics!
Convergence upon solutions, synthesis is teleological -- a syntropic process.
Syntropy is dual to entropy.
"Always two there are" -- Yoda.
Isomorphisms are invertible homomorphisms.
Injective is dual to surjective synthesizes bijective or isomorphism.
Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference).
The integers or real numbers are self dual:-
ruclips.net/video/AxPwhJTHxSg/видео.html
Elliptic curves are dual to modular forms.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy, summations) is dual to differentiation (entropy, differences).
SINE is dual to COSINE -- the word 'co' means mutual and implies duality.
Duality creates reality.
Using this expression, we can write U as:
U(x) = (phi * x/|x|)(x) = ∫R^n phi(x - y)(y/|y|) dy
where the integral is taken over all vectors y in R^n, with respect to an infinitesimal dy. Note that we have replaced the fixed vector t_0 in the convolution definition with the variable vector x, which is the point at which we are evaluating U.
To estimate the value of C(n,s) in terms of the maximum magnitude of U, we can use the Cauchy-Schwarz inequality to obtain:
|U(x)| ≤ (∫R^n |phi(x - y)|^2 dy)^{1/2} (∫R^n |y/|y||^2 dy)^{1/2}
where the first factor on the right-hand side is the L^2 norm of phi centered at x, and the second factor is the L^2 norm of the unit vector y/|y|.
The second factor can be evaluated explicitly as:
(∫R^n |y/|y||^2 dy)^{1/2} = (∫S^{n-1} dS)^{1/2} = √(2π(n-1)/n)
where S^{n-1} is the (n-1)-dimensional sphere in R^n, and the integral is taken over the unit sphere with respect to its surface measure. This constant depends only on the dimension n and is independent of the function phi.
To estimate the first factor, we can use the fact that phi has compact support and is infinitely differentiable, which implies that its Fourier transform decays rapidly. Specifically, we can write:
|phi(x - y)|^2 ≤ C |φ^(2)(ξ)|/(1 + |ξ|^2)^{2s}
where C is a constant that depends on phi and the size of its support, and φ^(2) is the second derivative of the Fourier transform of phi. This inequality follows from the Paley-Wiener theorem, which relates the smoothness of a function to the decay rate of its Fourier transform.
Using this inequality, we can estimate the L^2 norm of phi centered at x as:
(∫R^n |phi(x - y)|^2 dy)^{1/2} ≤ C' ∫R^n |φ^(2)(ξ)|/(1 + |ξ|^2)^{s} dξ
where C' is a constant that depends on C and the size of the support of phi.
To estimate the integral on the right-hand side, we can use the fact that the Fourier transform of the second derivative of a function is proportional to the Fourier transform of the function itself, up to a constant factor. Specifically, we can write:
|φ^(2)(ξ)| ≤ C'' |φ(ξ)|
where C'' is a constant that depends on phi.
Using this inequality, we can further simplify the estimate for the L^2 norm of phi centered at x as:
(∫R^n |phi(x - y)|^2 dy)^{1/2} ≤ C''C' ∫R^n |φ(ξ)|/(1 + |ξ|^2)^{s} dξ
where we have used the fact that the constant factors C and C'' can be absorbed into C'.
|U(x)|
I will put a like, just to be the first on what might be a solution to a long standing problem 😅
when will i be able to read this in a publication?
Imagine working on this problem for 8 years only for it or be solved in hours by a random anime pfp.
chat gpt type response lmaoooooo
For free?
How do you keep yourself motivated when doing research? I'm afraid that if I would pursue a phd I would make slow or no progress compared to others in the field whom all seem so much smarter than myself.
What I find interesting is in the last two examples you built structural models to build the arguments on while in the first it appears only the symbolisms of the math equations are given. A much more difficult mechanism to get insight of the problem. Its been my experience in dealing with engineering issues a model helps in arriving at a solution. It doesn't always work but it is effective when it does. Good luck with your research.
Problem, reaction, solution -- the Hegelian dialectic.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
All mathematical equations are dualities, Y = X.
Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics!
Convergence upon solutions, synthesis is teleological -- a syntropic process.
Syntropy is dual to entropy.
"Always two there are" -- Yoda.
Isomorphisms are invertible homomorphisms.
Injective is dual to surjective synthesizes bijective or isomorphism.
Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference).
The integers or real numbers are self dual:-
ruclips.net/video/AxPwhJTHxSg/видео.html
Elliptic curves are dual to modular forms.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy, summations) is dual to differentiation (entropy, differences).
SINE is dual to COSINE -- the word 'co' means mutual and implies duality.
Duality creates reality.
@Валера Dark energy is repulsive gravity, negative curvature or hyperbolic space (inflation).
Gaussian negative curvature is defined using two dual points -- non null homotopic (duality):-
en.wikipedia.org/wiki/Gaussian_curvature
The big bang is an infinite negative curvature singularity -- repulsive, divergent like a pringle!
Positive curvature is dual to negative curvature -- Gauss, Riemann geometry.
Curvature or gravitation is dual, gravitational energy is dual.
Potential energy is dual to kinetic energy.
Apples fall to the ground because they are conserving duality.
Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought, the principle of equivalence (duality).
Energy is duality, duality is energy -- the conservation of duality (energy) will be known as the 5th law of thermodynamics, Generalized Duality.
We are already living inside a black hole according to this CERN physicist:-
ruclips.net/video/A8bBhkhZtd8/видео.html
Inside is dual to outside.
Everything in physics is made from energy (duality) likewise for mathematics!
@Валера Cosine is the same function as sine but different as there is a 90 degree phase lag -- perpendicularity.
Same is dual to different.
Perpendicularity, orthogonality = duality (mathematics).
The Christian cross is composed of two perpendicular lines -- duality.
Christians have been worshipping duality for thousands of years!
Points are dual to lines -- the principle of duality in geometry.
convolutions often come up in probability as the density of sum of random variables is the convolution of their densities
What is that first problem called? Or does it not have a name? Where can I find more about this problem
Subscribed and I have a suggestion for the first problem, try asking chat GPT
is ok the fine tuning of mono atomic mri is in the math aplied itself so relax feel the tactile sensory folw of long form digits
Third one: All of the endpoints of your earlier divisions end up in your final set, but thats it.
If you pull 1/4 out, it goes 3/8,1/4,3/8 for lengths.
Endpoints at
{0/8, 3/8, 5/8, 8/8}
Going to get twice the endpoints after twice
{[0/8²], [3²/8²], [(3*5)/8²], [(3*8)/8²], [(5*8)/8²], [(3²+(5*8))/8²], [((3*5)+(5*8))/8²], [8²/8²]}
And so on.
So now if you shift by 1/8² you get all of them, and this pattern will go on forever. Just shift by 1/(8^n). Done
I hope you end up tackling the problem of your choice! Good luck.
Can you explain your approach for solving problem 2? It sounds like a very interesting problem and I’m really curious as to what you have to say
If you bound a 2d convex body with another 2d convex body, map each perimeter point to one another and extrude into 3 dimensions, you’re left with a 3d convex body. Idk of a function that will give every possible convex body in 2d, but if you apply the preceding logic to such a function you can make a never-ending tunnel that increases in size and has a cross section of every possible 2d convex body. My guess anyway
Problem, reaction, solution -- the Hegelian dialectic.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
All mathematical equations are dualities, Y = X.
Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics!
Convergence upon solutions, synthesis is teleological -- a syntropic process.
Syntropy is dual to entropy.
"Always two there are" -- Yoda.
Isomorphisms are invertible homomorphisms.
Injective is dual to surjective synthesizes bijective or isomorphism.
Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference).
The integers or real numbers are self dual:-
ruclips.net/video/AxPwhJTHxSg/видео.html
Elliptic curves are dual to modular forms.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy, summations) is dual to differentiation (entropy, differences).
SINE is dual to COSINE -- the word 'co' means mutual and implies duality.
Duality creates reality.
@@desmondlambe1421 I don’t know shit, but wouldn’t that tunnel quickly become concave?
@hyperduality
Have you considered that the identity property and associative property of morphisms in set theory, category theory etc. are dual in that they all prohibit local entropy increase? I thought it was an interesting framework to understand why rigor is a useful construct.
@@declandougan7243 I am not understanding your question, if you would care to explain it a bit more?
Decreasing the number of dimensions or states is a syntropic process -- homology.
Increasing the number of dimensions or states is an entropic process -- co-homology.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy).
The 4th law of thermodynamics is hardwired into mathematics and mathematical thinking.
Domains are dual to co-domains -- Group theory (symmetries).
Symmetries are dual to conservation -- the duality of Noether's theorem.
Local entropy increase? What do you mean here?
I remember making use of convolutions regularly in ODE
I'm about in the same position as you. I'm not sure whether to do a Phd or to focus on my business. I'm curious what you're looking to research? Have you decided your proposal?
I haven’t decided yet. It turns out finding an advisor is more difficult than I expected lol
For the second question, what is your definition of a convex body. I mean, a single point is certainly a convex set and the intersection of any plane with it is either empty or a single point
I am a Master's student, and soon I get to pick a topic for what to research for my thesis. I won't lie, it is a bit intimidating because often times in my homework sets for my classes I can't do them completely alone and work with classmates (if i have a problem set with 5 or 6 problems, then I could probably do 4 or 5 alone), so how am I expected to do research? How is the process for you? I am deciding whether or not to pursue a PhD, or if it's simply not for me.
I did a PhD and even then I had hws I had to work with my peers on. In research, it doesn't have to be alone. Talking to my peers about my research and brainstorming ideas help immensely.
Masters student here also. Nobody works alone. PhD students and masters students regularly work together on homeworks. Professors dont do research by themselves either.
Shit is hard as fuck and nobody expects you to figure it all out alone
Im 15 and im learning basic maths stuff like Arithmetic Progressions, Similarities, Thales Theoram, Quadratic Equations and basic trignometry and the 3 basic identities and more now. How is this maths? There are literally no numbers only a zero in the whole equation 😨😨😨😨😨😨
@@memekun1040 thats why it is called elementary(what you do)
@@memekun1040think about what the quadratic formula looks like to 10 year olds, to them its just letters but for you it’s manageable since youve built up to it slowly. Im 20 (2nd year undergraduate maths student) and i do stuff that looks not doable to you, but ive built up to it. same concept applies here youll build up slowly to things and eventually do stuff that makes ur 15 year old self look silly just like ill make my 20 year old self silly in 5 years :)
I’m pretty sure I’ve solved the second problem, if you accept the Axiom of Choice. The set of convex bodies in R^2 must have a cardinality of Aleph 1, since each convex body can be described by a polar function of 2 variables, angle and distance. That means you can create a bijection between the set of convex bodies in R^2 and the real line R. And since the real line R is also the same cardinality as any finite subset of R, you can create a universal convex body as a long tube of finite length with a single orthogonal slice for every convex 2D body. Again assuming the axiom of choice there should be a well ordering that makes that shape continuous
The real line definitely does not have the same cardinality as a finite subset of real numbers. Also how do ensure the tube you create is convex? Creating a tube that has all 2D convex shapes in R3 has been done but it is not convex.
I took two courses in convex geometry in my master's program and I absolutely loved it! The second problem is really interesting and I hope to see your results when you manage to solve it.
Problem, reaction, solution -- the Hegelian dialectic.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
All mathematical equations are dualities, Y = X.
Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics!
Convergence upon solutions, synthesis is teleological -- a syntropic process.
Syntropy is dual to entropy.
"Always two there are" -- Yoda.
Isomorphisms are invertible homomorphisms.
Injective is dual to surjective synthesizes bijective or isomorphism.
Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference).
The integers or real numbers are self dual:-
ruclips.net/video/AxPwhJTHxSg/видео.html
Elliptic curves are dual to modular forms.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy, summations) is dual to differentiation (entropy, differences).
SINE is dual to COSINE -- the word 'co' means mutual and implies duality.
Duality creates reality.
Can someone explain how they do come up with these problems, if it’s actually very hard to solve them? And who constructs these overcomplicated math problems?
Cool video
It’s important to be realistic, but I really think you should treat that first problem with the belief that you might just be able to succeed, try it go for it, what if you solved it?
For #2, yes such a body does exist. It needs to be convex so we will vary its contours in order of size from top to bottom, like an icicle. For a given plane we need, we simply append it to the top and shrink it to maintain convexity. We can do this forever, isomorphisms are not impeded by scale.
I thought of a similar approach. How can you ensure that *all* convex bodies in R^2 appear? My intuition is that somewhere you run into cardinality problems trying to go through all convex shapes. Currently studying for my first degree in mathematics so take my comment with a grain of salt:)
You run into issues when either trying to maintain convexity or fitting the icicle in R^3. If you have a circle and a square "on top of each other" they need some positive distance between them in order for you to mold one into another. And if you put all closed convex sets (with arbitrary convex sets we have no hope of this being true) with some uniform positive distance from each other you run out of space in R^3, so your icicle will be too big to fit in R^3.
My man literally pull out some egyptian hieroglyphs
The 2nd problem: I may be confused, but isn't the set of all 'shapes' of convex plain figures hypercontinual? The set of all slices of the 3d convex body, however, is just continual. This might help?
convolution is a huge thing in signal processing. haven't seen it anywhere else
What's the "support" thing in the first problem?
Hey, I really enjoyed that video. I am thinking of maybe making use of those problems for my own mathematical research. I do not know if I need a Master's or Ph.D's. But I am happy to admit that I actually have rough-draft mathematical proofs at my apartment. Two or three of these have to do with million-dollar conjectures. No lie. I am thinking of maybe asking you for advice. Do you have an email that I could reply to? I am interested in the work that you do. And I do want to maybe "workshop" some of my original ideas with you or my advisor. Michael Doerr.
Prob2: There's an infinite number of convex shapes in R2 so htf can that be realised in one realistic body. (Not saying it's impossible just speaking my mind. But thats a pretty big part) Also (I don't know) but the convexity of a circle is greater than a polygon since the line joining any two points ON the circle is entirely within the circle whereas that's not the case with polygon. Just making that point, presumably there's already math words/definitions making that distinction. Good luck in whatever you do.
i love ur videos man you offer content that is quite hard to find on youtube 🖤
i got a question though, when u say the support of phi do u mean the essential support? (srry if my question is obvious or doesnt make sense)
good luck on ur research 🖤
Thank you for the kind words. Your question is fine, the support (in fact the compact support in this case which I don’t mention) is the set on which phi returns nonzero values.
@@PhDVlog777 thank you for clarifying that 🖤
why does everything about higher mathematics always sound like a conjecture in quantum mechanics.....i would like to propose a new problem....to wit: in the playing card game known as Solitaire (standard version) is there a method or formula of playing that will always yield the maximum number of wins? there are often several different maneuvers that one can make with the cards at various 'discards' but is there a series of rules about each possible display of cards at each consecutive choice that will always yield the maximum possible number of wins? for example, if 3 aces are showing and a deuce is available then which ace should be put onto it? there are so many choices possible and things can often be done in different order with different cards etc
I'm not far into my math education so there might be some flaws in my reasoning but regarding problem 2: can't you just compare cardinality of the set of planes and the set of 2d convex bodies? The cardinality of the set of planes is aleph1 since you can describe a plane with three points in the plane (the cardinality of R^9 is Aleph1). The cardinality of the set of 2d convex bodies is Aleph2, I think, because they correspond to a functions from angles to distances. Rotations and translations might make this calculation a bit more complicated but the cardinality of this set is probably well known. This means there does not exist an onto mapping from the set of planes to the set of 2d convex objects. This would the contradict the existence of a universal body.
The cardinality of all convex bodies is the same as the power set of R which is higher than the amount of planes in R^3. So for arbitrary convex sets this is trivially untrue, but the cardinality of closed convex sets is equal to the cardinality of R, so we have a chance.
Also, saying that the cardinality of the real numbers is equal to aleph_1 is known as the continuum hypothesis, which is a famously undecidable problem. I'd be careful using aleph numbers when you talk about cardinalities of sets.
@@MK-13337 ahh, thanks. I didn't know that the cardinality of the set of closed sets is the cardinality of R, interesting. And thanks for the clarification regarding the aleph numbers. I have yet to learn these subjects formally in class, just heard about them in various RUclips videos.
Listening to this with headphones was difficult. Sound kept going out in the left ear.
for 2nd problem woudnt it be enough just to prove that from the constructing A->(R^3 convex shape) with flat surface(as if cut by plane) in the direction of "growing" u can create new A'->(R^3 convex shape) with new flat surface in shape of targeted B->( R^2 convex shape) by interpolating between A and B such that it doesnt become concave(to be proven)...and then inductively conclude the rest....(i dont know if i made sense there cuz im not highly educated...had to google most terms ...like i dont even understand 1st and 3rd problems so cant even engage in thinking there)
Does someone know where to find info about symbols used to espress math statements?. I want to be able to read math proof books. Thanks
First you read a beginner book about Discrete Mathematics (for example from Susanna). Most basic things will be covered. After that, you will have to read books about the specific fields. If you want to understand linear algebra, it is probably best to also get a book for beginners for that, etc... . Most fields overlap heavily so you might need to also get more books about different fields. After that you just work yourself through the lingo
@@hypnogri5457 do you need a certain type of brain to understand this stuff? I feel like I won’t even be able to understand the problems let alone solve them if I pursue mathematics at a higher level. Is it possible to get an MRI scan to see if I have the right brain for advanced mathematics?
11:05 "so for example, we have this weird object, which I'm gonna call 'you'..."
I have seen convolution in Laplace transform
Do you plan on continuing in academia or pursuing a more commercial position? Interested as I plan to start my Comp Sci PhD soon but am in two minds regarding what I want to do with it
Problem, reaction, solution -- the Hegelian dialectic.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
All mathematical equations are dualities, Y = X.
Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics!
Convergence upon solutions, synthesis is teleological -- a syntropic process.
Syntropy is dual to entropy.
"Always two there are" -- Yoda.
Isomorphisms are invertible homomorphisms.
Injective is dual to surjective synthesizes bijective or isomorphism.
Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference).
The integers or real numbers are self dual:-
ruclips.net/video/AxPwhJTHxSg/видео.html
Elliptic curves are dual to modular forms.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy, summations) is dual to differentiation (entropy, differences).
SINE is dual to COSINE -- the word 'co' means mutual and implies duality.
Duality creates reality.
2 questions:
1. You explained why you were asking the 3rd question: it extends an earlier result. But what was the impetus for questions 1 & 2? Why did the questions get asked in that way?
2. You have to publish. But there's no guarantee that you'll solve any given math problem. In engineering or statistics we can always do a thing and then report our results. If what we tried didn't work, I still have something to write about and an analysis I can conduct. But naively, it seems like if you try to solve one of these problems, that you might come away with nothing to show for it. So how do research mathematicians manage to hit publication targets?
If you can't make head way on a problem you move on to do something else that is more productive. You can't publish a null result in mathematics so if you can't solve anything you run out of research money and you are pretty much done for.
What kind of pen are you using?
For the first problem: Are you trying to show the existence of C, or trying to find the optimal C? For the existence of C, my first instinct would be to try to run a contradiction argument. Regardless, it might be worth computing C for some simple explicit functions --- or maybe for a family of functions whose supports depend on some parameter --- perhaps when n = 1. Separately, it might also be worth asking what sorts of techniques are commonly used to establish inequalities of that type, and why such techniques wouldn't work here.
If your colleague has worked on this problem for 8 years, then they've probably considered all this already, but this is my two cents.
I think it's an interesting question, and I'm personally really curious about *why* all of the "obvious" stuff that first comes to mind doesn't work. Lots of people have suggested stuff in the comments. Probably (hopefully?) all of it got tried over the course of 8 years. But if it's still an open question, it all had to fail. So that makes me super curious: if it all fails, then maybe there's an interesting reason for why.
Problem, reaction, solution -- the Hegelian dialectic.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
All mathematical equations are dualities, Y = X.
Thesis is dual to anti-thesis creates the converging thesis or synthesis -- the time independent Hegelian dialectic.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy) -- the 4th law of thermodynamics!
Convergence upon solutions, synthesis is teleological -- a syntropic process.
Syntropy is dual to entropy.
"Always two there are" -- Yoda.
Isomorphisms are invertible homomorphisms.
Injective is dual to surjective synthesizes bijective or isomorphism.
Isomorphism (absolute sameness) is dual to homomorphism (relative sameness, difference).
The integers or real numbers are self dual:-
ruclips.net/video/AxPwhJTHxSg/видео.html
Elliptic curves are dual to modular forms.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy, summations) is dual to differentiation (entropy, differences).
SINE is dual to COSINE -- the word 'co' means mutual and implies duality.
Duality creates reality.
Have you tried the foil method on this one?
This is morbidly interesting! lol. Anyway about that first problem, if the problem involves a convolution, I would guess the solution will involve Fourier Series. In my mind those two concept are intimately related. Good luck!
If the problem still hasn't been solved, fourier series probably has probably been done to death. So I'd assume that's a dead end and would start looking for alternative ways to prove such a statement. Maybe there's some kind of wavelet decomposition you could do with the function to make progress?
Have i understood the second problem right?
- that we wanna find at least one "rather round" 3D object
- such that when slicing it many times from different angles
- that the types of slices we would get, be equal to all possible types of "rather round" 2D objects?
Well.. my intuition says yes.. would be a weird potato with edges, but possible..
My first question would be, which extreme cases can we draw for the 2D shapes? What are the boundaries of what classifies as 2D-convex? Then we make gradients from one to another and stack those sliced together, which makes the shape..
Use Laplace Transform in the first problem
Bro I feel like I'm trying to decipher a foreign language watching this 😭
did you ever solve for LaGrange?
Me, and Econ PhD who knows nothing about anything in this video: 😯😯
Convolutions are very common in signals and systems in electrical engineering. About 40% of an undergraduate course in signals and systems is just convolutions.
The * is to denote convolution. We used it a lot in Signal & Systems, Control Systems Design and Communications Systems Design. And I mean A LOT. F(t)*g(t) is pronounced f(t) convolved with g(t). You are trying to convolve one function with another. In short, you take the g(t) function, flip it about its x-axis, and integrate it throughout f(t) as it moves from left to right about the x-axis, or the t-axis here. That result is the convolution of f*g. Might want to look for it on utube - my explanation is pitiful. In engineering it's used to relate the input signal with the system signal to analyze or predict the output signal.
I just started a signals and communications intro module and saw it in the first 5 slides. Pretty cool concept and even cooler module
This is correct, convolutions are bases for control theory, circuit analysis and communications, I'd say about 40% of electronic engineering
Problem 2 seems trivial the way you posed it. Construct a vertical column whose horizontal cross-sections are all the convex 2D bodies, scaled to keep the 3D body convex. Suppose there's a 2D convex body not congruent to any of these cross sections; it can be inserted into the middle (or at either end) of the column and scaled to keep the whole body convex. The solution is not unique.
I think the harder question is, can you construct such a body where you guarantee only one unique pi for each body K?
such a shape isn't guaranteed to be finite or convex. You'd have to prove that, but this construction runs into issues.
@@user-me7hx8zf9y aha, I think convex-ness is relatively easy to guarantee and prove if I'm a bit more detailed about how the column is constructed, but being finite is another matter altogether. I don't think you can guarantee the shape is finite, in fact the way this construction is proved to have cross-sections containing every convex body might necessitate that it be infinite.
Just trying to grasp the second problem a bit, as a non-mathematician. Is this an accurate statement of the problem? "Does there exist a 3D convex body which, if sliced by a plane at varying angles and positions on the body, would produce every possible 2d convex shape?"
Watching your videos is much like watching a Chinese movie without subtitles. I know they're saying something but bless my soul, I have know idea what it is! 🤔
What are those books? Make a video about all of ur books
6:57 I was studying Rudin's Principle of Mathematical Analysis yesterday and seeing this problem reminded me of the proof that every k cell is compact in pg 39. Not the infinite shrinking part but how for a fixed r you could get an n small enough to arrive at the contradiction. Here n is fixed and you need an 'r' or a C. Obviously this is just a philosophy and there are a lot of details you'd need to study like how x/|x| behaves. I don't have the slightest idea what you mean by phi 'supporting' the other thing. And I don't even know why the function should even be bounded (or whatever the problem is trying to state). So I guess it would be a nice idea to find parts which should be true like that, 'Half problems' I guess.
I liked War for Art when I read it cause it related to math and research in general too. It was a bit spiritual but it was right when it said it's your duty to study everything you can about the problem and understand all sides given what we know and it's completely upto luck to get the inspiration you need to get the right idea.
what pen are you using?
This isn’t meant to be a rude question, what is the objective of math research? I started out studying Psychology, now dual majoring in Psychology and Neuroscience, and high level mathematics haven’t really come up in my studies thus far (just a few statistics classes). Hopefully this isn’t a completely stupid question. I’m just an insanely curious person and don’t know a whole lot about math.
watching this makes me feel like I have not been learning math
I wish I understood what you were saying, I feel like I would find it interesting lol
The area of a triangle is 1/2 bxh