Andrius Kulikauskas

  • ms@ms.lt
  • +370 607 27 665
  • My work is in the Public Domain for all to share freely.


  • 读物 书 影片 维基百科

Introduction E9F5FC

Understandable FFFFFF

Questions FFFFC0



Clean Up


Complex numbers are more natural than real numbers or quaternions because complex numbers have simpler nondegenerate quadratic forms: {$Q(u)=u_1^2+u_2^2+\cdots +u_n^2$}. For we can insert a scalar {$i$} and that converts any minus sign into a plus sign.

Jennifer Mather. The Case for Octopus Consciousness: Unity. https://www.mdpi.com/2673-4087/2/4/30

Baez, Moeller, Trimble. Schur functors and categorified plethysm.

Posina Venkata Rayudu

nLab: Sphere spectrum The sphere spectrum is the suspension spectrum of the point. The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres. The sphere spectrum is the higher version of the ring Z of integers. See also: nLab: Suspension

Tai-Danae Bradley. The Tensor Product, Demystified

Laws of form

  • Concatenation is "saying". Saying multiple times is the same as saying once. {$a^2=a$}.
  • Cross is negating. It is crossing. Negating twice is identity. {$b^2=1$}.
  • We can have a different kind of negating or a different kind of negated for which negating twice is reversal. {$c^2=-1$}

Matematikos žinojimo rūmų sparnus sieja

  • algebra veda iš centro į sąrašą
  • analizė vedą iš sąrašo (indukcijos) į centrą (limitą)

Apsikeičia - ar tai padalinimų ratas? Kiekviename sparne požiūriai prisideda +0, +1, +2, +3.

Auxiliary loops in space-time are compatible with the rays in space-time, perhaps in this way the mind is compatible with the brain. That might be relevant for the hierarchy of agency.

Consider a Clifford algebra with infinitely many generators {$a_i$} and {$b_j$} such that {$a_i^2=1$} and {$b_j^2=-1$}. How would different interpretations satisfy an eightfold periodicity?

John Bolender. The Self-Organizing Social Mind.

Consider the ways of constructing a 2x2 matrix by adding a generator {$a_i$} and then {$b_j$}, or alternatively, {$b_j$} and then {$a_i$}.


Grammar of sociology in terms of a pattern language grounding concepts such as "generation", "the right biography", "class", "both agressors lose".

Groups and symmetries. From finite groups to Lie groups.

Geometry arises on a vector space when we place a quadratic form on it, {$Q:V\rightarrow K$}. It gives a notion of length of a vector because {$Q(av)=a^2Q(v)$}, and also a notion of distance because there is a symmetric bilinear form {$B(v,w)=\frac{1}{2}(Q(v+w)-Q(v)-Q(w))$}.

https://en.wikipedia.org/wiki/Cayley%E2%80%93Klein_metric Are there niine Cayley-Klein geometries?

Try to undersand Morita equivalence in terms of irreducible representations.



Equivalence up to isomorphism is relevant for "user requirements" as opposed to "material implementation". This is relevant for Bott periodicity.

Wave function arises when two systems interact. As given by orthogonal Sheffer polynomials.

In Lie group for rotations, SO(3), the bracket of [x,y] gives you z.

In choice frameworks, such as the simplex, the center is the basis for geometry and the vertices are the basis for matter. Together can they ground general relativity?

I dreamed of the grouping of examples from branches of mathematics by considering whether they involve, for example, aspects of mathematics, logic, semantics, and so on.

Minkowski space. Time {$-t^2$} has us step out (thus reversing direction), space {$x^2+y^2+z^2$} has us step in.

Why doesn't adding a pair of generators {$a_1 b_1$} to a Clifford algebra change it?

Generators {$a_j$} for which {$a_j^2=1$} are experienced directly whereas generators {$b_j$} for which {$b_j^2=-1$} express the change in direction upon reflection, thus stepped out.

Jane Loevinger's psychometrics = Maslow's hierarchy of needs. E8 is "worry about the needs of another". E9 is "be perfect".

Do the generators of Clifford algebras which square to +1 and -1 encode, respectively, covariant and contravariant tensors?

Equality is inherently contradictory.

John Baez. Getting to the Bottom of Noether's Theorem.

Ivan. Was or was not regional politics helpful for potential democratic transition in Russia? Novosibirsk, Tatarstan.

Partial knowledge

  • forgetful functors
  • fibrations, lenses


Counterquestions are a foundation for learnability. Each counterquestion defines a domain of new knowledge where we had no facts that we could rely on.

Mind and Life Institute. Varela. Supporting contemplative research.

Nathaniel Virgo

  • Modeling agency
  • Improving a model of the environment = Bayesian prior.
  • Controller gets new information and also the system changes over time.
  • Don't care about the previous states, just the prior (t) and the current issue (t+1)
  • Kalman filter: when the prior is a Gaussian then the posterior is also a Gaussian
  • So only the means and variances need to be stored
  • Bart Jacobs 2020: A channle based perspective on conjugate priors - this pops out of an adjunction
  • Unifilar generator. A generator is unifilar if it is deterministic given output. They form a separate category.
  • Forgetful functor: Unifilar Generator to Generator. In BorelStoch this has a right adjoint.
  • Usually forgetful functor has a left (free) adjoint. A right adjoint of a forgetful functor is cofree. In this case the forgetful functor is forgetting both a fact about the objects and the morphisms.
  • How does this relate to lenses?
  • Strongly representable Markov categories are cool.
  • Epistemic model and dynamical model. You want your model of the system to be unified.

Supermaps - holes - contexts.

Pragmatic approach. Context defines meaning. Robert Brandom. Making It Explicit.

{$\textrm{Set}^{op}$} atomic boolean algebras. Map back into Set. Map back out PowerSet.

Simplexes observe coordinate systems.

df/dx = f. Can be expressed through the notion of infinity (Taylor series) {$e^x$}. Or through periodicity (trigonometry, Euler's equation) {$e^{ix}$}

Jim in Oneonta. Adapt, improvise, overcome.

Sheaf Representation of Monoidal Categories

  • Monoidal Categories MonCat. Presheaf F: L->MonCat and Sheafs. Generalizing Stone's Theorem.

  • Lax functor - may be relevant for allowing perspectives to be not associative yet related.
  • Spivak and Kent: Ologs
  • The volume of a unit sphere in n-dimensions goes up for small n, reaches a maximum at n=5, then goes down. https://en.wikipedia.org/wiki/Volume_of_an_n-ball

Posina Venkata Rayudu


Conformal and analytic is the same.

Energy can be defined as the "separation constant" in Schroedinger's equation. If we can separate the wave function into a time dependent function and a position dependent function, then we can segregate the two sides of the equation so that one side depends on time, the other side depends on position, and both sides are constant, and that constant is the energy.

Peter Scholze - condensed math

Visual frameworks

  • Visual frameworks connects visualization with semantic contexts such as gravity. Note that gravity is based on a quadratic power law and yields the fivefold conics.

Absolutism based on relativism is good. Relativism based on absolutism is bad. Objectivity based on subjectivity is good. Subjectivity based on objectivity is bad. The mind that does not know is based on the mind that knows and not the other way around. This is the rule of consciousness and the basis for morality.

There is a gap between the quantum ether (the quantum foam) and the waves that propagate through it. Particles don't exist, particles are the medium. Waves exist in the medium.


  • Vector bundles. Jimmy told me about vector bundles having a threefold sense: Data, gluing, and 3 sets coming together. This happens in 1 degree, 2 degrees, 3 degrees.

Folk psychology. Daniel Dennett suggested studying this. Contact him.

In the book on interpretations of quantum mechanics, there is the question of what is real. For example, in electromagnetism, the gauge can be adjusted by adding any gradient. This can change whether the change is transmitted by the speed of light or whether that speed is infinite and it happens instantaneously. But these two scenarios also raise the question of the reality of the "wiggle". What is real is a moral choice. Is the medium real? Or is the wiggling real? The wiggling creates the wave that moves across the medium. The medium is made of particles and anti-particles that appear from the foam.


  • Explain why we get alternating signs for the boundary

https://researchsemin The many-worlds interpretation of quantum mechanics and the Born rule Lev Vaidman (Tel Aviv University) ( view ) Mon May 22, 19:00-20:30 (7 days from now) Abstract: I will argue that the many-worlds interpretation is the best interpretation of quantum mechanics and discuss the status of the probability assignments in this deterministic theory.ars.org/seminar/AlgebraParticlesFoundations

Applying a boundary map twice gives zero. Applying it twice removes two vertices, and this can be done in different orders, yielding different signs, canceling out.

For the Snake Lemma, add a zero vector space before the first kernel and add a zero vector space after the last cokernel. Then we have the eightfold way with seven mappings.

Can large language models work by simply transforming existing input - taken to be grammatical - to preserve grammaticality.

Choice Frameworks

  • A category with a zero object has semantic symmetry with regard to choice frameworks. A category without a zero object, but with initial objects and terminal objects which differ, such as the categories Set or Cat, have syntactic asymmetry.

Introduction to Commutative Algebra Atiyah & Macdonald. Rings, ideals, modules, dimensions

Looking at an ellipse in various ways yields all of the conic shapes. We can get a breaking at infinity. Thus this is a way to ground infinity. What about looking at a circle? We look through the point of the cone, which is where our eye is.

Mobius transformations

  • How are perspectives transformed?
  • How are triangles mapped to triangles ?

David Corfield's video. Colin McLarty: semiotics as the language of biology - logic in a biological key - trying to categorify this?


Tristan Needham. Visual Complex Analysis.

John Stillwell. Mathematics and its History. 1989.

Moebius transformations

Emotional sphere

Freedom House Report

Fields Institute


Alex Codes. Symbolic Differentation in Python from Scratch!


  • Abstract algebra. Dummit and Foote.

  • Meditations by Marcus Aurelius
  • Meditations by Renee Descartes
  • On Liberty by John Stuart Mill
  • Octavia Butler - Parable of the Sower, Parable of the Talents
  • Cixin Liu - Trijų kūnų problema (Kitos knygos)
  • Ursula Le Guin - The Left Hand of Darkness, Dispossessed

A sum of particle clocks is like a prism operator (in the proof for singular homology that homotopic maps induce the same homomorphism for the homology groups) but without the minus signs.


Wisdom distinguishes everything and slack, what is whole and what is free, holisticity and laxity.

How to fold circles



Theo Buhler. Exact categories



Exact sequence: Grad, Curl, Div

Group theory

Nathan Carter "Visual Group Theory"



Purcell. Electricity and Magnetism

90 degrees + 90 degrees can equal anything. But specifically can go from the diameter of a cube (standing on its vertex) to the vertex and back on the diameter to any point. But the same is true for 120 + 120.

Paul Lockhart's Measurement

Quaternions act like a gauge - 3 dimensions are unspecified - but identified with the complex i.


The Hilbert space that models the spin state of a system with spin 𝑠 is a 2𝑠+1 dimensional Hilbert space. And spin can be half-integered. Think of the Hilbert space as everything divided into 2s+1 perspectives.

Simplicial sets

Greg Friedman. An elementary illustrated introduction to simplicial sets.


Physics 283b: Spacetime and Quantum Mechanics, Total Positivity & Motives

Spirtes, Glymour and Scheines. Causation, Prediction and Search. (Adaptive Computation and Machine Learning).


Bell's inequalities


Causal Set Theory

Universal concepts such as universal confounders the confounder.

Ambiguity is described by equations.

  • Atmosphere has mass of 5.15×10^{18} kg
  • Person breathes 10 tons of oxygen = 10,000,000 grams of oxygen in their lifetime
  • 16 grams of oxygen has 6*10^23 atoms of oxygen
  • 1 gram of oxygen has 0,375*10^23 atoms of oxygen

Paul Humphreys. The chances of explanation : causal explanation in the social, medical, and physical sciences.

(1) formulate a hypothesis, (2) deduce a testable consequence of the hypothesis, (3) perform an experiment and collect evidence, and (4) update your belief in the hypothesis.

Modeling experience from old self to new self - Bayesian analysis - prior belief + new evidence = revised belief

Jacobi polynomials have a notion of combinatorial space that may be relevant for thermodynamics as it relateswo disjoint sets malping into their union.


Pearl seeing vs. Doing

Homology sets up potential equivalences - they may be actual equivalences, which yield identities - or nonequivalences which are generators.

Rotation accords with orientation (of a simplex) accords with an imaginary number i or j. Orientation is related to permutation as with the linearization for orthogonal polynomials.


Judea Pearl: The Fundamental Equation of Counterfactuals {$Y_X(u)=Y_{M_X}(u)$}. Relate to independent trials - throwing away a sheet of paper (a module).

Judea Pearl, Dana Mackenzie. The Book of Why: The New Science of Cause and Effect


  • Is Monty Hall problem related to quantum probability?

Stephen Wolfram. Metamathematics Foundations & Physicalization

Ravi Vakil: Main theme of mathematics - convert harder problems to linear algebra

Ergodic theorem

Chomsky: Successor function derives from the merge function applied unitarily to a single object.

Parsing hierarchy

Speculation: The difference in the measurements of the Hubble constant may relate to the history of the universe. Early in the universe the heavier particle families (in the parsing hierarchy) may have been predominant. And they may be the source of the megastructures of the universe.

Field with one element

Thomas noted the symmetry of {$x^0=1$}. Relate this to {$F_1$}, choosing one out of one, or none out of none.

Schroedinger's cat

  • Do the probabilities of a superposition evolve as per the phasor?
  • Do they run through all possibilities between wavelengths?
  • And does that mean that the decision of whether Schroedinger's cat is awake or asleep depends on the exact moment that we make the measurement?
  • And what does that say about time? and superpositions?
  • Is this a valid interpretation of superposition?

What Are The Hidden Rules Of The Universe?

http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf Fulton Curve Book


  • How is randomness related to the Riemann Hypothesis?
  • Forms of matter express geometry as uniformity and give rise to mass behavior even randomness.
  • Entropy - physics is related to symmetry - Shannon entropy is related to information. And how does that relate to randomness?
  • Symmetry breaking - choosing one possibility. From symmetry breaking randomness appears and information is constructed. Deterministic is replaced by irreversibility.
  • Randomness as derived from a wall that allows for independent events, as with the other, or with transcendence.
  • Randomness as lack of knowledge.

Multiplying by quaternion j reverses angular momentum for electron. Is spin a clock? Like a particle clock?


Linear regression

  • In multiple regression the constant {$b_0$} acts like free space, the initial compartment. The random variables are like compartments.


  • Foundation of statistics is models. Distinguish the signal and the noise. Ignore the noise.


trivial tangent bundles on spheres?

Hamiltonian is the sum of all the projections onto the energy eigenstates with the energies being the weights.


  • Vector cross product is an example of the three-cycle.

Study of variables

Kervaire-Milnor formula

  • {$\Theta = \Pi B$} where {$B=a_m2^{2m-2}(2^{2m-1}-1)B_{2m}/4m$}

In the unfolding of math

  • consider math as given by generators and relations
  • the relations are equivalence classes

https://neo4j.com Neo4J graph database management

Johan Commelin: "Breaking the one-mind-barrier in mathematics using formal verification"

  • Selcuk Bayin.

Representation theory

Generalized Linear Models


8 is special because {$\sqrt{8/4}=\sqrt{2}$} is the distance between neighbors but also the interspersed lattice in constructing the E8 lattice. 240 is the kissing number.

  • {$128=dim(\mathbb{O}\otimes\mathbb{O}^2)$}


Are the conditions for coverings the basis for completeness?

Shoelace formula for oriented area of a polygon

Dobinski's formula relates Bell numbers and e. {$B_n=\frac{1}{e}\sum_{k=0}^{n=\infty}\frac{k^n}{k!}$}

Quaternionically differentiable is linear.

18.4 penrose. Hyperbolic length is one half of the rapidity it represents.

Laws of physics are time-symmetric for particles traveling at the speed of light so time does not change

Localization arises from local shielding by local interactions. That is what weakens global interactions which othwrwise exist.

Complexity theory

https://en.wikipedia.org/wiki/PCP_theorem Every decision problem in the NP complexity class has probabilistically checkable proofs


https://en.wikipedia.org/wiki/Overton_window Window of political discourse: Unthinkable - Radical - Acceptable - Sensible - Popular - Policy

Lambda Calculus

Study choice, probability, statistics.


In the Standard Model, fermions are not their own antiparticles, but in some theories they can be. Among other things, this involves the question of whether the relevant spinor representations of the groups Spin(p,q) are complex, real (‘Majorana spinors’) or quaternionic (‘pseudo-Majorana spinors’). The options are well-understood, and follow a nice pattern depending on the dimension and signature of spacetime modulo 8.

  • Two reflections give you a rotation. So is a reflection the square root of a rotation? And does that relate to spinors?
  • Bosons - real representations, fermions - quaternionic representations.


Richard Southwell describes how mathematical functions can be visualized by: (1) elements and arrows (2) Wiring diagrams (3) fibres (4) bouquets (5) graphs (6) ontology logs (7) categories

Dirac's plate trick Plate trick

Summer of Math Exposition 2022 Results

Coxeter. Regular polytopes. Includes prehistory. Boole.

Coxeter diagram {$D_n$} symmetry group of demicube: every other vertex of a hypercube. Is that related to a coordinate space? Combinatorially, can we flip the vectors of the demicube to get a coordinate system?

Grover's algorithm

Roe Goodman. Alice through Looking Glass after Looking Glass: The Mathematics of Mirrors and Kaleidoscopes

Cube reflections given by vectors u, v, w from the center of the cube to the center of a face, the center of an edge, and the center of another edge. And the angles between the vectors are pi/2, pi/3 and pi/4. And the two edge midpoints are separated by pi/3 so rotating through six such edges gets you back. And that is the chain for the Dynkin diagram.

Conjugation is an example of reflection.

Finite field with one element

  • Choosing one out of one: Driving on a winding road, each turn is a choice of one out of one. Whereas a fork is a choice of one out of two, a usual intersection is a choice of one out of three and so on.


The 4 returns: natural return (value of landscape), economic return (restart agriculture), social return, humans return.

Locality is the whole achievement of the continuum. Local means low overhead and the actual global time frame is even lower overhead. Locality arises with orthogonality, assumes measurement, observers, space time wrapper.

Differentiation changes level. {$x^n$} number of levels of volatility, number of derivatives

Spaces of states

  • nLab: State
  • Classical bit: line segment [0,1]
  • Qubit: shaped like an American football

{$\begin{pmatrix} a & b+ic \\ b-ic & d \end{pmatrix}$}

Think of probabilities {$a, 1-a$} and mediator {$b \pm ic$}. We have {$a^2+b^2+c^2\leq a$} and {$a^2+b^2+c^2 = a$} for pure states. Rotate {$a-a^2$} from 0 to 1 around the a-axis.

It is the MacMahon Master Theorem that unifies the angular momentum properties of composite systems in the binary build-up of such systems from more elementary constituents.

Summer of Math Expostion 2022 playlist


Learning classifier system

Semilocally path connected avoids Zeno's paradox. Universal covering as naming schemes.

Robert Gilmore. Group Theory. XIV. Group Theory and Special Functions. Relates Lie groups and orthogonal polynomials.

Local - reversible, global (default) not reversible ("Not every cause has had its effects")


Quantum Magic Rectangles: Characterization and Application to Certified Randomness Expansion Sean A. Adamson∗ and Petros Wallden and generalization of the magic squares

How are games in game theory (with incomplete information, partial information) characterized by probability distributions.

Charactization and application to certified randomness expansion

In Cartesian categories you can copy and delete information. (John Baez - Rosetta Stone) How does that relate to Turing machine?

Alytaus kredito unija

  • Kapitalas 700,000 EUR, pelnas 43,000 EUR, paskolinta 4,600,000 EUR.

Bose statistics - can't assign labels. Fermi statistics - can assign labels to particles.

Information capacity is zero if probability is the same for all cases but also if one case is given 100%. Information transmission requires asymmetry. Otherwise you cannot define choice.




Creating what you can feel certain about. (Continuity.)

  • Building up levels of certainty through topological invariants.

A few of the best math explainers from this summer

Stone's theorem: continuous implies differentiable

"belt trick", aka the "Dirac scissors" or "Balinese candle dance

When two events happen (the measurement of spins) there is a frame where one happens before the other. So if they are causally connected (as with spin measurements) there needs to be a distinguished frame. But that could be the frame in which they were initially entangled. So entanglement posits the existence of such a distinguished frame.

Path integrals depend on the number of points in space, or the number of interactions. But my approach suggests that this number is actually given by the degree of x in the relevant polynomial.

Quantum computing

Amelia: [The axiom of function extensionality is] inconsistent with many axioms of a more "computational" nature. For example, "formal Church's thesis" says that for any function N→N, there is a "program" (we call it a realizer) that realizes it. You can kinda see what goes wrong: this would be able to tell e.g. "λ x → x" and "λ x → x + 0" apart. You could imagine an assignment of realizers that sidesteps this, though, so to see that it's actually inconsistent takes slightly more work.

Have all finite limits is equivalent to

  • Having terminal objects
  • Having a product for any pair of objects
  • Having an equalizer for any pair of parallel arrows

These are the building blocks for limits

The 4th movement of Beethoven's Symphony No. 5. Conducted by Arthur Nikisch. Recorded in 1913.

{$\alpha$} and {$\beta$} count ascents and descents and these are steps forwards or backwards in the unfolding of space (in time?) and so they may relate to John's picture of evolution taking us forward and backward in time.

Roelof Koekoek & René F. Swarttouw. The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue

https://ww3.math.ucla.edu/dls/emily-riehl/ video about contractibility

An isomorphism is a special morphism but truly it is a pair of morphisms that are inverses to each other. There may be many such pairs relating two objects but in each pair the inverses are unique with respect to each other. So it is similar to complex conjugation.

Mariana M. Odashima, Beatriz G. Prado, E. Vernek. Pedagogical introduction to equilibrium Green's functions: condensed matter examples with numerical implementations.



  • OpenShot eksportuoti 30 fps nes iPhone filmuoja 30 fps
  • Open Source Software to Thank: Linux, Ubuntu, OpenShot, Dia, GIMP
  • Shot with an iPhone XS Max.

Schuller on Stone's Theorem

In physics, orthogonal polynomials relate what is necessary (top down) and actual (bottom up) as with string theory, questions and answers.

The original spectral theorem: Look for subrepresentations such that S is a one-dimensional matrix eigenvalue. Induction argument.

Classical (both x, p) and quantum (x).

Bald and bankrupt Eastern Europe

The Screwing of the Average man: How the rich get richer and you get poorer

Wick's theorem - are operators of the same particles - propagator connects

Evolution is indicated by learnability and also by sparse communication and natural differences between hierarchies, different orders of magnitude, allowing for a natural hierarchy of niches. Not only the laws of physics are sparse but also the states in nature are sparse.

Rules of physics plus configuration space plus location within that space.

Source of contradiction

  • We are finite, our system is finite, but the Spirit is infinite dimensional

Uncertainty principle - has to do with representations - representation adds a perspective - so that interferes with measuring certain things.

Minimization operator mu - superhero - who clings to ledges and other such things and is stretched and blown by the wind. And the shape mu gives the shape of his body clinging to the left.

https://en.wikipedia.org/wiki/%CE%9C_operator {$\mu$}-operator

I had a dream that i was professor anthony zee... But in a quantum superposition. Was i z or not z ? Z or not z? ..... is there a third way? Yes but there is a fourth way .... Nevermind z here is m4w!

A qubit specifies the relation between affirmation and negation of probabilities. In matrix form, it provides a complex number which is the coefficient that gets multiplied to the negation (in calculating the new affirmation) and whose conjugate gets multipled to the affirmation (in calculating the new negation). In classical bits, this coefficient is simply zero.

Bekaert, Boulanger. The unitary representations of the Poincare group in any spacetime dimension

Five zones of scattering can be thought of as

Measurement establishes a quantity with regard to boundaries - it establishes the zone within which it is - identifies with a step in the algebra - whereas analysis demarcates the boundaries.

Kojin Karatani, Sabu Kohso - Architecture as Metaphor_ Language, Number, Money (1995) semi-join lattice semilattice

Quantum physics

Brody. Quantum Mechanics and Riemann Hypothesis.


Yevgeny B. Karasik. Duality revolution: Discovery of new types and mechanisms of duality that are revolutionizing science and technology as well as our ability to solve problems


Think of -1-cell as the center (of all things), the spirit. And think of 0-cell not simply as a point but as a 0-dimensional open arc (the point shell) with regard to that center (the spirit). The point shells are glued onto the spirit, and similarly, open arcs are glued onto point shells, and so on, inductively.




In what sense are Feynman diagrams relativistic given that they have directions for time and for space?

Instead of thinking of speed of light, think of a clock that doesn't tick, so that t=0 always. And this is the case for the quantum harmonic osciallator and for the particle-clocks with no steps.

One {$\exists x$}, all {$\forall x$}, many {$\neg\exists x \wedge \neg\forall x$}.

Masaki Kashiwara, Pierre Schapira. Categories and Sheaves. 2006

Bohm Pilot Wave, Thomas Spencer

Relative invariance - more global than another

Relate the three-cycle (taking a stand, following through, reflecting) to Kan extension as defined by filtering objects and morphisms, acting on them, and then taking the colimit or limit on that filter.

Alan Turing, Cybernetics and the Secrets of Life

https://en.wikipedia.org/wiki/Algorithmic_information_theory Gregory Chaitin = Shannon + Turing = Compression-Decompression as understanding.

https://en.wikipedia.org/wiki/Cristian_S._Calude Philosophy of computation

Life in life

Thinking about the expansion of the universe as a reduction of density, by which the mass of particles becomes ever less important, by which we have an increase of entropy (becoming less deliberate). And we can reverse this by starting with an increase in entropy and arriving at the expansion of the universe.

Matematika išplaukia iš (poreikių tenkinimo) algoritmų taikymo, vedančio iš duotybių į bendrybes. O tos bendrybės įkūnija, išreiškia tam tikrus prieštaravimus, juos paverčia sąvokomis, kurias galima mąstyti toliau. Pavyzdžiui, apskritimas iškyla iš begalinės simetrijos visom kryptim, arba iš virve aprėpto ploto maksimalizavimo.

Information is what you learn. What you learn grows at the boundary, has the shape of the boundary. A shape can be thought of as being created by integrating over these boundaries as they increase.

Tai-Danae Bradley: Information is on the Boundary

  • Shannon entropy: the amount of surprise

Prove that the matrix made up of eigenvectors diagonalizes a matrix.

Potential energy source {$\frac{y}{2}$} (adding free cell) is in balance with kinetic energy source {$i\frac{\partial}{\partial y}$} (half-link)

  • Santykis su Dievu yra atgarsis, kaip kad dalelytė turi santykį su savo lauku.
  • Fizikos dėsnių raida yra pavyzdys Dievo įsakymo patobulinimo.


  • Observing symmetry requires breaking symmetry.
Keisti - Įkelti - Istorija - Spausdinti - Naujausi keitimai -
Šis puslapis paskutinį kartą keistas September 28, 2023, at 01:48 PM