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• Andrius Kulikauskas
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I am keeping this diary so that we all can see what I'm studying in mathematics. Also, you can look at my pages Recent Changes in Research and Recent Changes in Exposition.

In particular, I listen to videos while I exercise or do mundane chores.

Diary

2021.08.01 In physics, I am reading parts of Quantum Field Theory in a Nutshell by Anthony Zee and Introduction to Quantum Mechanics by David J. Griffiths. Then I will return to my investigation of Sheffer polynomials. I am trying to make sense of quantum physics, and specifically, the Schroedinger equation, by using combinatorics, algebra and analysis to investigate the fivefold classification of Sheffer orthogonal polynomials. I am interpreting this classification as a cognitive framework for decision making, for relating a subsystem and a system, based on two kinds of causality, backwards from effect to cause - every effect has had its cause, and forwards from cause to effect - not every cause has had its effects. I zoom every other Monday with John Harland.

2021.06.03 I may be nearing the end of my study of the combinatorics of the Sheffer polynomials, the orthogonal polynomials {$P_n(x)$} which can be expressed as {$\sum_{n=0}^{\infty}P_n(x)t^n = A(t)e^{xu(t)}$}. In the most general case, for the Meixner polynomials, the building blocks of {$xu(t)$} and {$\textrm{ln}A(t)$} are cycles such that the starting point is either a single step {$x$} or a double step and we take some steps {$\alpha$} away and the some steps {$\beta$} to return. In working with exponential generating functions, this means that {$A(t)e^{xu(t)}$} should be counting permutations that we get by throwing labels on a set of such cycles. But I have to work out the details. There is also a combinatorial expression by Jiang Zeng in terms of growth processes of families of trees where the branches have weight {$l=-\alpha-\beta$} or {$k=-\alpha\beta$}. I need to work out the bijection that relates these two interpretations. What's clear is that the fivefold classification is contained in {$\alpha$} and {$\beta$} and the ways that they may be specialized.

I am very intrigued as to how to derive the measure associated with an orthogonal polynomial set. The Hamburger moment problem considers whether the measure exists. But in order to actually calculate the measure, evidently I have to learn how to apply the inverse Laplace transform to the moment generating function {$m(s)=\sum_{n=0}m_n\frac{s^n}{n!}$}.

{$$\mu(t)=L^{-1}\{F\}(t)=\frac{1}{2\pi i}\lim_{T\rightarrow\infty}\int_{r-iT}^{r+iT}e^{sT}F(s)ds$$}

where {$r$} is a real number so that the contour path of integration is in the region of convergence of {$F(s)$}.

2021.05.28 While thinking about Viennot's videos on the combinatorics of orthogonal polynomials, and wondering how to interpret {$(1-\alpha t)/(1-\beta t)$} in the all-encompassing Meixner polynomials, I came upon what I suspect is the deep idea that underlies the physical and cognitive understanding I seek, namely, suspension. Given the expression

{$$1 = \frac{1-\alpha}{1-\alpha} = (1-\alpha)(1+\alpha+\alpha^2+\alpha^3+\cdots)$$}

suppose that we temporarily suspend the expression by substituting {$\beta=\alpha$} in the numerator, giving

{$$1 = \frac{1-\beta}{1-\alpha} = (1-\beta)(1+\alpha+\alpha^2+\alpha^3+\cdots)$$}

Thus the suspension makes for a division into two, a choice between two different choices. One choice is to select a walk of arbitrary length. Another choice is to do nothing {$1$} or to choose an alternating contrary weight {$-\beta$}. Taking this further, we can accentuate the parameter {$\alpha - \beta$} by writing:

{$$1 = 1 + (\alpha - \beta)(1+\alpha+\alpha^2+\alpha^3+\cdots)$$}

The suspension allow us to distinguish the possibilities for degeneration. We can end the suspension by setting {$\beta$} back to {$\alpha$}, which yields the Laguerre polynomials describing the bound state, the hydrogen atom. We can further set them both to {$0$}, yielding the Hermite polynomials for the quantum oscillator, as with empty space in quantum field theory. We can restrict the difference to the imaginary dimension, and in fact, make {$\beta$} and {$\alpha$} indistinguishable by setting {$\beta = \overline{\alpha}$}. We can temporarily set {$\beta=0$}, yielding the Charlier polynomials, suggesting that they are a temporary state. Thus the momentary distinguishing of {$\alpha$} in the numerator and {$\alpha$} in the denominator yields the five states. These are the ways that a subsystem can be identified within a system. This also suggests why a perturbation is a valid way of exploring the behavior of a system. And it makes symplectic geometry relevant as that which provides the slack that allows for suspension or perturbation. So I am curious how the quadratic fetters of orthogonal polynomials might ground symplectic geometry and the other geometries - affine, projective, conformal - by distinguishing coordinate systems centered on {$1$}, {$\alpha$} and {$\beta$}.

2021.05.27 I am overviewing my questions and answers in my investigation of Sheffer polynomials.

I also corrected a big mistake that I made in investigating the Meixner-Pollaczek polynomials. I incorrectly supposed {$(1 - \alpha)(1 - \overline{\alpha})=1+|\alpha|^2$} when in fact {$(1 - \alpha)(1 - \overline{\alpha})=1-2\textrm{Re}\alpha + |\alpha|^2$}.

2021.05.22 I gave a talk in Lithuanian on "God's Understanding and Human's Imagination Face-to-Face at the Gates of Art". It was part of the annual conference on philosophy of art and aesthetics. This year's theme was the manifestation of religious awareness. I invite you to make sense of a machine translation into English or the Lithuanian original.

At the start of my talk, I explain our powers of imagination, how they limit us and how we transcend them. I give a mathematical example of how the imagination is lazy to think step-by-step, in other words, to do algebra. Suppose that I usually buy lunch at a cafe for 9 euros because it is one-third more at the restaurant. But suppose today that everything at the restaurant is one-third off their usual price. Where is it cheaper?  In my talk, I trace a thread from God's query, Is God necessary? to the arisal of human as the one who does not understand, to their comprehension of the asymmetry between the conditional Imaginer and the unconditional Unimagineable, to their work together towards an enlightened society which includes the self-fulfillment of each and every one. It's an overview of my philosophical vision.

2021.05.17 Every other Monday, John Harland and I talk about quantum physics. I share my work on investigating the physics in the combinatorics of orthogonal polynomials, and especially their fivefold classification, which suggests five natural bases for viewing different physical paradigms.

In our conversation today I realized that the equation for the Sheffer polynomials {$\sum_{n=0}^{\infty}P_n(x)t^n=A(t)e^{xu(t)}$} asserts that each such family is a bridge that links the Lie algebra {$xu(t)$} and the real form {$A(t)$} which manifests the Lie group. There are five kinds of bridges and I believe, for example, that the Hermite polynomials are the bridge in the case when the group is compact.

This suggests to me that we should look for adjoint functors that go from the group to the algebra and also back. I believe that there should be a five-adjoint-string of such functors. And I also suspect that two are related to the two real forms of {$\mathfrak{su}(2)$} (which is the node in Dynkin diagrams) and three are related to the three real forms of {$\mathfrak{su}(3)$} (which is the link in Dynkin diagrams). And that is probably just about all that Lie theory is about.

A result that encourages such thinking are these expressions:

{$A'(t) = - ct + \textrm{echo terms}$}

{$u'(t) = 1 - lt + (l^2-k)t^2 + \textrm{echo terms}$}

and also:

{$u'(t) = 1 -(\alpha + \beta) t + (\alpha^2 + \alpha\beta + \beta^2)t^2 + \textrm{echo terms}$}

By "echo terms", I mean that the higher order terms are all determined by the first few initial terms which I have indicated.

Here, in my combinatorial interpretation based on Zeng's trees, {$l$} stands for a link in a tree, {$k$} stands for a kink in a tree, {$c$} stands for a cycle of trees. In Zeng's interpretation of the moments {$L(x^n)$} in terms of permutations in {$S_n$}, {$\alpha$} stands for an ascent, and {$\beta$} stands for a descent.

It seems as if {$u'(t)$} describes the growth of a single tree, or walk in a single cycle, and {$A'(t)$} describes how trees are linked together into cycles, or how cycles of walks are combined. Thus here differentiation seems most fundamental in that it expresses the underlying growth.

I am currently working on this on pages about the Sheffer classification and the moments.

2021.05.02 Given a family of orthogonal polynomials {$\{P_n(x)\}$}, a key fact is that any polynomial {$f(x)$} can be uniquely expressed as a linear combination of them. And so, in particular, we can consider {$xP_n(x)$}, and realize that its expression may not include any {$P_k(x)$} where {$k>n+1$} or where {$k<n-1$}. If the orthogonal polynomials are monic, such that the leading term is {$x^n$} with coefficient {$1$}, this means we have {$xP_n(x)=P_{n+1}(x)+A_nP_n(x)+B_nP_{n-1}(x)$}. Here we see that the variable {$x$} comes from a single source on the left hand side but that it impacts the polynomial just above and the one just below in the hierarchy. The upshot is that the exponential possibilities are tamed by a second order recurrence relation, or alternatively, a second order differential equation. These quadratic fetters yield a fivefold classification based on the discriminant {$\sqrt{b^2-4ac}$}, much as with conics.

I am studying how to derive Sheffer's fivefold classification of orthogonal polynomials of A-type 0. This fivefold classification variously interprets the generator for walks with two kinds of step, {$\alpha$} and {$\beta$},

{$$\frac{1}{(1-\alpha t)(1-\beta t)}=(1+\alpha t+\alpha^2t^2+\cdots)(1+\beta t+\beta^2t^2+\cdots)$$}

where we may have

• no steps {$\beta = \alpha = 0$} (Hermite)
• one kind of step
• {$\beta = 0$}, {$\alpha \neq 0$} (Charlier)
• {$\beta = \alpha \neq 0$} (Laguerre)
• two kinds of step
• both real: {$\beta < \alpha$} (Meixner)
• complex conjugates: {$\beta = \overline{\alpha}$} (Meixner-Pollaczek), so that {$\alpha=a+bi$} and {$\beta=a-bi$}

But where do {$\alpha$} and {$\beta$} come from? What do they mean mathematically, physically, cognitively? I have found helpful materials including a history and exposition of Meixner's classification and treasure trove of lectures by Xavier Viennot on the combinatorics.

2021.04.25 I suspect that strings of adjoint functors may express what I mean by divisions of everything. I am thus trying to thoroughly understand the concept of adjunction in category theory, which is also important for truly mastering the Yoneda Lemma. I am therefore studying how to relate the three definitions of adjunction in terms of homsets, units and counits, and the universal mapping property. I have gone through the six ways of going from one definition to another. They all seem to serve to invert a functor in a given context. I think they also function like pushdown automata, for example, satisfying obligations, such as matching left parentheses with right parentheses. Now I am trying to work out and understand what that means for particular adjunctions such as inclusion for preorders.

2021.02.22 I am spilling the guts of quantum mechanics with combinatorial interpretations of orthogonal polynomials that arise in solutions of the Schroedinger equation.

2020.01.21 Nobel Prize laureate Duncan Haldane gives an accessible talk on Topological Quantum Matter, Entanglement, and the Second Quantum Revolution. (The Indian Institute of Technology Roorkee has a whole series of great talks.) Haldane's talk was a good introduction for me to physical effects that are some of the prerequisites to understanding the tenfold way. I was also intrigued that the Hall effect might be related to symmetric functions and, in particular, Hall-Littlewood polynomials but these are named after Edwin Hall and Philip Hall, respectively. The concept of fractional charge or fractional quanta is important, for example, in a chain of people holding hands where the hands at either end are free and possibly entangled. Similarly, the https://en.wikipedia.org/wiki/Fractional_quantum_Hall_effect fractional quantum Hall effect may express divisions of everything. Fractionally charged quasiparticles exhibit neither bosonic nor fermionic but rather anyonic statistics. This also relates to topological order in the zero-temperature phase of matter.

2020.01.11 I listened more to the The Tenfold Way by Vijay Shenoy (Part 2 of 4) about the tenfold way, the difference between the one ordinary (usual, linear) and three non-ordinary symmetries. It was helpful to see the chart and also to realize that the labels seem to be related to the classical Lie families. He talked about Hamiltonians but that part I didn't understand and should return to.

2020.01.10 At the NYCT After Meetup, Wenbo said he was interested in simplicial sets and quasicategories as he is realizing that categories are too sterile for modeling emergence. That makes sense to me from participating in Oliver's study group in Category theory and Statistics. I had sketched an adjunction for modeling statistics from which it became clear that categories need to be graded, which is to say, we need to be able to associate a probability to a morphism so that we can consider morphisms of a particular validity. In talking with Wenbo, I also realized that we need to be able to model nonassociativity, especially for composition of perspectives, and the probabilities would allow us to speak of associativity up to a certain probability. In emergence, systems or subsystems would "pop out" when they satisfied the requirements of category theory.

I told Wenbo about the conceptual frameworks, the difference between interpreting choice syntactically-asymmetrically as with observerful simplexes, or semantically-symetrically as with observerless coordinate systems. I related this to the problem of the collapse of the wave function when nature measures itself. This suggested to me that the collapse must occur when a coordinate system is introduced and thereby separates the observer from the observed. As a subsystem emerges, at a certain point it is robust enough to indicate a coordinate system, which an observer can then leverage to indicate, for example, that something did not happen. Thus the establishment of the coordinate system should be that which creates space and perhaps time, on the one hand, and that which collapses the wave function, on the other hand. Thus this would be the place to look for a relation between general relativity and quantum mechanics.

Wenbo is interested in wreath products. He shared a related excerpt from The Topos of Music by Guerino Mazzola. It had to do with time, and also to Greimas's theory of time in narrative.

2020.01.08 I'm interested in Bott Periodicity. The most helpful videos are the ones about the Tenfold Way in condensed matter research. I'm listening to The Tenfold Way by Vijay Shenoy (Part 2 of 4). The previous episode was a wonderful overview that I learned a lot from. In this episode it seems that he had to dumb it down, go back to basics, what is a group. But that's all right by me.

Šis puslapis paskutinį kartą keistas August 01, 2021, at 03:00 PM