Exposition

Research

Contact

  • Andrius Kulikauskas
  • ms@ms.lt
  • +370 607 27 665
  • Eičiūnų km, Alytaus raj, Lithuania

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Current Focus

Andrius Kulikauskas: Week by week I am alternating between two mathematical investigations:

  • 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.
  • Adjunction I am mastering the three ways of defining adjunction in category theory, and how they are related. Then I will try to collect and classify adjoint strings and investigate how they may express cognitive frameworks that organize perspectives and contexts. Also related is my study of how the Yoneda Lemma expresses four levels of knowledge: whether, what, how, why. I zoom every other Sunday with Wenbo Gao and the New York Category Theory and Algebra Meet Up. You are welcome to join us!

In my philosophy, I am working on the big picture of everything, such as how the definable sits within the undefinable. On May 22, at the conference on "Philosophy of Art, Aesthetics and Manifestations of Religious Consciousness" in Vilnius, Lithuania, I gave a talk on "Divine Understanding and Human Imagination Face-to-face at the Gates of Art". You can read a machine translation into English or the Lithuanian original. I included an example of how the imagination is obsessed with unity and thus unable to think step-by-step, which is to say, do algebra.

Latest Research

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 {$F(s)$}.

2021.04.25 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.

For earlier news see also my diary and the recently changed pages in my Exposition and Research sections.


Let's work together!

Truthful Wisdom

{$M\Rightarrow W$}, "M 4 W" and Math 4 Wisdom stand for Mathematics for Wisdom by Andrius Kulikauskas, ms@ms.lt, aka Living by Truth.

{$M\Rightarrow W$} inspires our imagination by studying its very limits, what we can imagine or not. Let's embrace the cognitive frameworks we can't escape and leverage them for a conceptual language of wisdom that we can express, validate and apply with the help of mathematics and all manner of sciences, creative arts and moral disciplines.


Cranky Disclaimer

Crank alert

You may be at risk of dismissing these pages as an online art show, science fiction, mental poetry, psychological symptoms, products of numerology, metaphysical truths, existential crises, adventures in mysticism, primordial forms... and so on and so on and so on. Just so you know!


Keys to Math and Other Subjects

Coming sooner or later...


Profoundly Speculative Investigations

See all investigations.

Why study math? Here are mathematical concepts that I believe are relevant for wisdom.

Understandingfor modeling
MathematicsWisdom
  
The ways of figuring things out in mathematicsHouse of Knowledge for Mathematics
Four classical Lie familiesFour levels of geometry and logic
PolytopesChoice frameworks
{$SU(2)$}Emotional transformations
  
Yoneda lemmaDivision of everything into four levels of knowledge (Whether, What, How, Why)
Yates Index theoremDivision of everything into four...
The five conicsDivision of everything into five perspectives for decision making (thus space and time)
Fivefold classification of Sheffer polynomials of A-type zeroDivision of everything into five...
Adjoint string of length NDivision of everything into N perspectives
Bott periodicityEight-cycle of divisions
Grothendieck's six operationsSix representations of divisions
PerturbationIntervention, intercession, miracle
SpinorGoing beyond oneself into oneself
{$F_1$}Division of everything into 0 perspectives
The state of contradictionGod

Wonderfully Curious Phenomena

See all phenomena.


24 ways of figuring things out.

8 states of mind.

6 transformations of mood.


















Frequently Questioned Answers

In 1971, as a six-year old child, I went to God (?) with my quest to know everything (?) and apply that knowledge usefully (?)

In 2021, I am about ready to share my wisdom (?) But who would (?) how could (?) why should anybody make an effort to comprehend it (?) take it to heart (?) try it out (?) and work together (?)

Math is a standard for truth. Math is verified again and ever again. It is a universal language, accessible to millions who make the effort, regardless of their personal creed. Enthusiasts, practitioners and specialists appreciate profound results and helpful expositions. They will invest themselves to understand, if they have good reason.

I think it best to demonstrate my wisdom through math:

  • to apply my wisdom by overviewing all of math, showing how it unfolds, with all of its branches, concepts, questions and results.
  • to express in mathematical form the cognitive frameworks that capture my wisdom.
  • to develop insightful expositions of the relevant mathematics.
  • to present my work-in-progress as I pursue my ideas in life and math.

Currently, for a year or two, I have the means to spend all of my time on my philosophy and related mathematics. But then I will need to find a way to make a living. My plan is to create this website as a work-in-progress where I present the math that I have learned, am learning or will be learning which is relevant for sharing my wisdom. I think that I will find supporters, including through Patreon, who would fund me to learn and present more math because they love advanced mathematics, want new ways of understanding it, wish to see it reach more people, and are excited how it might serve to express answers to life's great questions. I can also create learning materials, crowd source projects at Kickstarter, and work as a coach for self learners. As ever, my creative work is in the Public Domain, for all to use as they think best.

Ultimately, I hope that my wisdom grows to become our wisdom, the Holy Spirit in a language, a science, a culture of self learners. This website may reach you in a way that academic peer reviewed articles never would. I may write such articles, but I fear they won't be read, understood, embraced and lived. I can write here to you in a way that I would write to myself.

It seems natural to illustrate my ideas with visual thinking, combining text, images and diagrams, as I learned from John Rogers and from Robert Horn's books. Moreover, art is fun, art is social, art holds us together, art changes the rules, so that you might give me a chance. I want to speak from all of me, my conscious and unconscious, to all of you, your conscious and unconscious. I want you to feel that I am a person, real and true, who stands alone, yet aims to commune with you.

Wisdom is why I write, math is how I write. What I write are the creative arts that speak to your attention. Community with you is whether I succeed in writing to you that Spirit which reaches you, and whether from you it reaches me, so that it dances amongst us, as we step-in and step-out, as inspired.

I am working on a video where I introduce myself along with Math 4 Wisdom.


An Introduction to Everything

Coming soon...

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{$M\Rightarrow W$}, Math 4 Wisdom and Math for Wisdom stand for Mathematics for Wisdom by Andrius Kulikauskas.

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Šis puslapis paskutinį kartą keistas June 03, 2021, at 02:25 PM