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59

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Introduction

## Publications

Publications (59)

We show that every irreducible integral Apollonian packing can be set in the Euclidean space so that all of its tangency spinors and all reduced coordinates and co-curvatures are integral. As a byproduct, we prove that in any integral Descartes configuration, the sum of the curvatures of two adjacent disks can be written as a sum of two squares. De...

A simple matrix formulation of the Fibonacci, Lucas, Chebyshev, and Dixon polynomials polynomials is presented. It utilizes the powers and the symmetric tensor powers of a certain matrix.

The configuration space of tricycles (triples of disks in contact) is shown to coincide with the complex plane resulting as a projective space costructed from the tangency and Pauli spinors. Remarkably, the fractal of the depth functions assumes a particularly simple and elegant form. Moreover, the factor space due to a certain symmetry group provi...

A family of partial orders in the free monoid of words, induced from a partial order in alphabet, is presented. The induced orders generalize the chronological posets that have been defined for the two-letter alphabet only, and the morphological order. We show that the induced orders are natural with respect to alphabet homomorphisms.

A remarkably simple Diophantine quadratic equation is known to generate all Apollonian integral gaskets (disk packings). A new derivation of this formula is presented here based on inversive geometry. Also occurrences of Pythagorean triples in such gaskets is discussed.

The polynomial relationship between elementary symmetric functions (Cauchy enumeration formula) is formulated via a ``raising operator" and Fock space construction. A simple graphical proof of this relation is proposed. The new operator extends the Heisenberg algebra so that the number operator becomes a Lie product. This study is motivated by natu...

An amusing connection between Ford circles, Fibonacci numbers, and golden ratio is shown. Namely, certain tangency points of Ford circles are concyclic and involve Fibonacci numbers. They form four circles that cut the x-axis at points related to the golden ratio.

The depth function of three numbers representing curvatures of three mutually tangent circles is introduced. Its 2D plot leads to a partition of the moduli space of the triples of mutually tangent circles/disks that is unexpectedly a beautiful fractal, the general form of which resembles that of an Apollonian disk packing, except that it consists o...

A parametrization of integral Descartes configurations (and effectively Apollonian disk packings) by pairs of two-dimensional integral vectors is presented. The vectors, called here tangency spinors defined for pairs of tangent disks, are spinors associated to the Clifford algebra for three-dimensional Minkowski space. A version with Pauli spinors...

The Dedekind tessellation -- the regular tessellation of the upper half-plane by the Mobius action of the modular group -- is usually viewed as a system of ideal triangles. We change the focus from triangles to circles and give their complete algebraic characterization with the help of a representation of the modular group acting by Lorentz transfo...

In his talk "Integral Apollonian disk Packings" Peter Sarnak asked if there is a "proof from the Book" of the Descartes theorem on circles. A candidate for such a proof is presented in this note

The algebra of the relativistic composition of velocities is shown to be isomorphic to an algebraic loop defined on division algebras. This makes calculations in special relativity effortless and straightforward, unlike the standard formulation, which consists of a rather convoluted algebraic equation. The elegant appearance of the new formula brin...

An intriguing correspondence between certain finite planar tessellations and the Descartes circle arrangements is presented. This correspondence may be viewed as a visualization of the spinor structure underlying Descartes circles.

A construction and algebraic characterization of two unbounded Apollonian Disk packings in the plane and the half-plane are presented. Both turn out to involve the golden ratio.

What does it mean to ``add'' velocities relativistically -- clarification of the conceptual problems, new derivations of the related formulas, and identification of the source of the non-associativity of the standard vector version of the addition formula are addressed.

We find a formula for the area of disks tangent to a given disk in an Apollonian disk packing (corona) in terms of a certain novel arithmetic Zeta function. The idea is based on "tangency spinors" defined for pairs of tangent disks.

We define spinors for pairs of tangent disks in the Euclidean plane and prove a number of theorems, one of which may be interpreted as a "square root of Descartes Theorem". In any Apollonian disk packing, spinors form a network. In the Apollonian Window, a special case of Apollonian disk packing, all spinors are integral.

The group-theoretic content of photonic coupled microring resonance phenomena is shown with, in particular, an interesting emergence of pseudo-unitary groups. The application to the resonance condition in a tri-microring configuration is solved exactly. A practical application of this work will be in the resonance frequency tuning based on the coup...

We present the formalism of phenomenological thermodynamics in terms of the even-dimensional symplectic geometry, and argue that it catches its geometric essence in a more profound and clearer way than the popular odd-dimensional contact structure description. Among the advantages are a number of conceptual clarifications: the geometric role of int...

The group-theoretic content of photonic coupled microrings resonance phenomena is shown, in particular, an interesting emergence of pseudo-unitary group. The application to the resonance condition in a tri-microring configuration is solved exactly. A practical application of this work will be in the resonance frequency tuning based on the coupling...

An alternative framework underlying connection between tensor sl2-calculus and spin networks is suggested. New sign convention for the inner product in the dual spinor space leads to a simpler and direct set of initial rules for the diagrammatic recoupling methods. Yet, it preserves the standard chromatic graph evaluations. In contrast with the sta...

Using scaling arguments and the property of self-similarity we derive the Casimir energies of Sierpinski triangles and Sierpinski rectangles. The Hausdorff-Besicovitch dimension (fractal dimension) of the Casimir energy is introduced and the Berry-Weyl conjecture is discussed for these geometries. We propose that for a class of fractals, comprising...

We present a novel representation of the Lorentz group, the geometric version of which uses "reversions" of a sphere while the algebraic version uses pseudounitary 2x2 matrices over complex numbers and quaternions, and Clifford algebras in general. A remarkably simple formula for relativistic composition of velocities and an accompanying geometric...

An interpretation of Krawtchouk matrices in terms of discrete version of the Feynman path integral is given. Also, an algebraic characterization in terms of the algebra of split quaternions is provided. The resulting properties include an easy inference of the spectral decomposition. It is also an occasion for an expository clarification of the rol...

The Ehresmann connection on a fiber bundle that is not compatible with a (possible) Lie group structure is illustrated by the geometry of a general anholonomic observer in the Minkowski space. The 3D split of Maxwell's equations induces geometric terms that are the (generalized) curvature and torque of the connection. The notion of torque is introd...

We present a geometric theorem on a porism about cyclic quadrilaterals, namely, the existence of an
infinite number of cyclic quadrilaterals through four fixed collinear points once one exists. Also, a technique of proving such properties with the use of pseudounitary traceless matrices is presented. A similar property holds for general quadrics as...

A geometric diagram that allows one to visualize the Poincar\'e formula for
relativistic addition of velocities in one dimension is presented. An analogous
diagram representing the angle sum formula for trigonometric tangent is given.

A remarkably simple Diophantine quadratic equation is known to generate all, Apollonian integral gaskets (disk packings). A new derivation of this formula is presented here based on inversive geometry. Also, occurrence of Pythagorean triples in such gaskets is discussed.

We show that the relation between the Schr\"odinger equation and diffusion
processes has an algebraic nature and can be revealed via the structure of
"duplex numbers." This helps one to clarify that quantum mechanics cannot be
reduced to diffusion theory. Also, a generalized version of quantum mechanics
where $\C$ is replaced by a normed algebra wi...

We define the harmonic evolution of states of a graph by iterative
application of the harmonic operator (Laplacian over $Z_2$). This provides
graphs with a new geometric context and leads to a new tool to analyze them.
The digraphs of evolutions are analyzed and classified. This construction can
also be viewed as a certain topological generalizatio...

A remarkable formal similarity between Koide's Lepton mass formula and a
generalized Descartes circle formula is reported.

Finding appearances of the golden ratio in various nooks and crannies of mathematics brings delight, often surprise. This note presents, in the form of a puzzle, a configuration of circles that is replete with the golden ratio. But that is only the surface. One tool to analyze such figures is the “master matrix equation” that rules circle (and n-sp...

We show that every Lie algebra is equipped with a natural (1; 1)-variant tensor eld, the \canonical endomorphism eld", determined by the Lie structure, and satisfying a certain Nijenhuis bracket condition. This observation may be considered as complementary to the Kirillov-Kostant-Souriau theorem on symplectic geometry of coadjoint orbits. We show...

We show that every Lie algebra is equipped with a natural (1, 1)-variant tensor field, the "canonical endomorphism field", determined by the Lie structure, and satisfying a certain Nijenhuis bracket condition. This observation may be considered as complementary to the Kirillov-Kostant-Souriau theorem on symplectic geometry of coadjoint orbits. We s...

When Ptolemy inscribed a triangle into a unit-diameter circle, its sides became visualizations of ratios that were later named ?sines?. His construction is known as the ?sine theorem?. We complete this idea and propose a visualization of the complementary ratios in this triangle, namely cosines, and a dual ?cosine theorem?.
A whole sequel of theore...

We show that the space of Euclid’s parameters for Pythagorean triples is endowed with a natural symplectic structure and that
it emerges as a spinor space of the Clifford algebra R21, whose minimal version may be conceptualized as a 4-dimensional real algebra of “kwaternions.” We observe that this makes
Euclid’s parametrization the earliest appeara...

A family of sequences produced by a non-homogeneous linear recurrence formula derived from the geometry of circles inscribed in lenses is introduced and studied. Mysterious ``underground'' sequences underlying them are discovered in this paper.

A formula for the radii and positions of four circles in the plane for an arbitrary linearly independent circle configuration is found. Among special cases is the recent extended Descartes Theorem on the Descartes configuration and an analytic solution to the Apollonian problem. The general theorem for n-spheres is also considered.

Krawtchouk matrices have as entries values of the Krawtchouk polynomials for nonnegative integer arguments. We show how they arise as condensed Sylvester-Hadamard matrices via a binary shuffling function. The underlying symmetric tensor algebra is then presented.
To advertise the breadth and depth of the field of Krawtchouk polynomials / matrices t...

Krawtchouk's polynomials occur classically as orthogonal polynomials with
respect to the binomial distribution. They may be also expressed in the form of
matrices, that emerge as arrays of the values that the polynomials take. The
algebraic properties of these matrices provide a very interesting and
accessible example in the approach to probability...

We investigate the structure of the Schrödinger algebra. Two constructions are given that yield the physical realization via general methods starting from the abstract Lie algebra. Representations are found on a Fock space with basis given by a canonical Appell system. Generalized coherent states are used in the construction of the Hilbert space of...

Popularity of a candidate is greater among women than men in each town, yet popularity of the candidate in the whole district is greater among men. Procedure A has greater succes than procedure B in each hospital, yet, in general, procedure B has greater success than A.

The approach of Berezin to the quantization of so(n,2)
via generalized coherent states is considered
in detail. A family of n commuting observables is found in which the
basis for an associated Fock-type representation space is expressed.
An interesting feature is that computations can be done by
explicit matrix calculations in a particular basis....

We examine the Schrodinger algebra in the framework of Berezin quantization.
First, the Heisenberg-Weyl and sl(2) algebras are studied. Then the Berezin
representation of the Schrodinger algebra is computed. In fact, the sl(2) piece
of the Schrodinger algebra can be decoupled from the Heisenberg component. This
is accomplished using a special reali...

We investigate the structure of the Schrodinger algebra and its
representations in a Fock space realized in terms of canonical Appell systems.
Generalized coherent states are used in the construction of a Hilbert space of
functions on which certain commuting elements act as self-adjoint operators.
This yields a probabilistic interpretation of these...

Elastic collision of two balls on a line is discussed in terms of their
configuration space. The optical-mechanical analogy is analyzed in this
context. In particular, the law of collision is reinterpreted as Heron's
law of light reflection.

The Ehresmann connection on a fiber bundle that is not compatible with a (pos-sible) Lie group structure is illustrated by the geometry of a general anholonomic observer in the Minkowski space. The 3D split of Maxwell's equations induces geometric terms that are the (generalized) curvature and torque of the connection. The notion of torque is intro...

The decomposition of matrix manifolds into homogeneous spaces of direct products of certain groups is described. The results are applied to derivation of the internal structure of SU (2, 2) × SU (m) and P
4 × SU (m) invariant particle models.

We find realizations of Lie algebras of "type-H" as vectors fields. These are used in a novel approach to representing group elements as products of one-parameter subgroups (splitting formula) and for finding polynomial matrix elements of representations for the Lie group, in particular irreducible representations. Special classes of polynomials, A...

Thanks to works of Caratheodory [4] and Gibbs [5] phenomenological thermodynamics of equilibrium (PTE) has become a standard axiomatic theory. Formulated in the general way [11,3], it reveals the structure which is universal in the sense that the later statistical and quantum statistical mechanics have not replaced it but rather serve as models of...

The problem of projecting spinor spaces on Minkowski space is rather old. The first formula of this type is (1.1) projecting the two-dirensional complex spinor space C2 onto the light cone in M4. One can consider (1.1) also as a projection on E3. These projection are consistent with the group in the sense that SU(2) transformations of the ξa induce...

An 'endomorphism field' on a Lie algebra is introduced. It is defined as the (1,1)-type tensor field naturally determined by the Lie structure, satisfying a certain Nijenhuis bracket condition. The Lie endomorphism field has connections with dynamical sys- tems. Here we show its relevance for Lax equations and show that the space of Lax vector fiel...

The polynomial relationship between elementary symmetric functions (Cauchy enumeration formula) is formulated via a "raising operator" and Fock space con- struction. A simple graphical proof of this relation is proposed. The new operator extends the Heisenberg algebra so that the number operator becomes a Lie product. This study is motivated by nat...

We show the connection between the Krawtchouk matri-ces (and hence, polynomials) and the Sylvester-Hadamard matrices. Then we derive the Krawtchouk matrices from the classical symmet-ric Bernoulli random walk. This approach is continued to the quan-tum probability context, leading to the construction of Krawtchouk matrices via tensor powers of the...

## Projects

Projects (5)

1. Geometry of relativity, connections with division algebras, non-associativity of velocity composition, and more.