Search

We establish a link between the new theory of |$q$| -deformed rational numbers and the classical Burau representation of the braid group |${\mathcal {B}}_{3}$|⁠. We apply this link to the open problem of classification of faithful complex specializations of this representation. As a result we provide an answer to this problem in terms of the singular set of the |$q$| -rationals and prove the faithfulness of the Burau representation specialized at complex |$t\in {\mathbb {C}}^{*}$| outside the annulus |$3-2\sqrt 2 \leq |t| \leq 3+2\sqrt 2.$| [ABSTRACT FROM AUTHOR]

Remarkably, the integers HT ht then turn out to be the quiddity coefficients on the top of the triangulation! The initial values of HT ht are shown in Table 1. It suffices to verify that (5) is the coefficient of HT ht in HT ht , or in other words, the coefficient of HT ht in HT ht . Therefore, there are HT ht triangulations containing it. [Extracted from the article]

Moreover, for some "mysterious" reasons, the new sequence HT ht turns out to be an integer sequence! Indeed, it follows directly from (4) that (5) can be rewritten without division as HT ht . If HT ht is a square root of -1, then I A i is a complex number, called a Gaussian integer. But if HT ht satisfies the condition HT ht then I A i is called a dual number. [Extracted from the article]

We introduce an analogue of the classical Markov equation that involves dual numbers a +αε with ε² = 0. This equation characterizes the "shadow Markov numbers" recently considered by one of us. We show that this equation is characterized by invariance by cluster algebra mutations. [ABSTRACT FROM AUTHOR]

q -deformed real numbers are power series with integer coefficients. We study Stieltjes and Jacobi type continued fraction expansions of q -deformed real numbers and find many new examples of such continued fractions. We also investigate the corresponding sequences of Hankel determinants and find an infinite family of power series for which several of the first sequences of Hankel determinants consist of − 1 , 0 and 1 only. These Hankel sequences satisfy Somos and Gale-Robinson recurrences. [ABSTRACT FROM AUTHOR]

The geometric and topological picture involves the exceptional spheres HT ht , HT ht , HT ht , and HT ht . The spheres HT ht also appear as fibers of the Hopf fibrations (while HT ht appear as their bases). The identity 1 HT ht Graph is as old as mathematics. [Extracted from the article]

We will see below that HT ht is a polynomial with positive integer coefficients. Every coefficient of HT ht has a combinatorial meaning. It turns out that one can express the Jones polynomial HT ht of the two-bridge knot associated to a rational HT ht as a combination of the polynomials HT ht and HT ht of the I q i -rational HT ht . The local rule then allows for the calculation of HT ht . [Extracted from the article]

We introduce a notion of q-deformed rational numbers and q-deformed continued fractions. A q-deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the q-deformed Pascal identity for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the q-rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties, q-deformation of the Farey graph, matrix presentations and q-continuants are given, as well as a relation to the Jones polynomial of rational knots. [ABSTRACT FROM AUTHOR]

We develop a version of cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables. These algebras can be described in terms of "extended quivers," which are oriented hypergraphs. We describe mutations of such objects and define a corresponding commutative superalgebra. Our construction includes the notion of weighted quivers that has already appeared in different contexts. This paper is a step towards understanding the notion of cluster superalgebra. [ABSTRACT FROM AUTHOR]

The article focuses on algorithm for the calculation of continuants related to spectral theory of difference equation and mentions polynominal which is trace of the matrix. Topics discussed include skew-symmetric matrix which is square of certain polynominal in entries known as Pfaffian, Diophantine equation which is Coxeter-Conway system, and Chebyshev polynomials which are sequences of orthogonal polynominals.

This paper is an attempt to apply the tools of supergeometry to arithmetic. Supergeometric objects are defined over supercommutative rings of coefficients. We consider an integral ring with two odd variables. In this case the even quantities, such as numbers and continued fractions, are dual integers, having both a classical and a nilpotent part. We refer to the nilpotent part as the "shadow". We investigate the notions of supersymmetric continued fractions and the orthosymplectic modular group, and make some initial steps toward studying their properties. [ABSTRACT FROM AUTHOR]

The article presents information regarding the appearance of Hopf fibration, mathematical field of topology discovered by Heinz Hopf, in other areas of mathematics and physics, wherein it is called as four fiber bundles in spheres. Topics discussed include the Hurwitz-Radon Function which is evaluated at powers of two, the history of Hurwitz-Radon Function and the theorem of Adams about the vector fields on spheres.

We consider differential operators between sections of arbitrary powers of the determinant line bundle over a contact manifold. We extend the standard notions of the Heisenberg calculus to such differential operators: noncommutative symbolic calculus, the principal symbol, and the contact order. Our first main result is an intrinsically defined "subsymbol" of a differential operator, which is a differential invariant of degree 1 lower than that of the principal symbol. In particular, this subsymbol associates a contact vector field to an arbitrary second-order linear differential operator. Our second main result is the construction of a filtration that strengthens the well-known contact-order filtration of the Heisenberg calculus. [ABSTRACT FROM AUTHOR]

The article offers information on Clifford algebra and equivalence of problems related to quadratic forms. Topics discussed include introduction of real Clifford algebra by mathematician William Kingdon Clifford to generalize complex quaternions and numbers; classification of the algebra based on their dimensions and similarities in explaining their theorems; and mathematical proof to explain equivalence of Clifford algebras.

We consider a contact manifold with a pseudo-Riemannian metric and define a contact vector field associated with this pair of structures. We call this new differential invariant the contact Riemannian curl. We show that the contact Riemannian curl vanishes if the metric is of constant curvature and the contact structure is defined by a Killing 1-form. We also show that the contact Riemannian curl has a strong similarity with the Schwarzian derivative, since it depends only on the projective equivalence class of the metric. The subsymbol of the Laplace-Beltrami operator corresponding to a metric on a contact manifold is proportional to the contact Riemannian curl. We also show that the contact Riemannian curl vanishes on the spherical (co)tangent bundle over a Riemannian manifold. [ABSTRACT FROM AUTHOR]

We show that the space of classical Coxeter’s frieze patterns can be viewed as a discrete version of a coadjoint orbit of the Virasoro algebra. The canonical (cluster) (pre)symplectic form on the space of frieze patterns is a discretization of Kirillov’s symplectic form. We relate a continuous version of frieze patterns to conformal metrics of constant curvature in dimension 2. [ABSTRACT FROM AUTHOR]

Orthogonal designs are fundamental mathematical notions used in the construction of space time block codes for wireless transmissions. Designs have two important parameters, the rate and the decoding delay; the main problem of the theory is to construct designs maximizing the rate and minimizing the decoding delay. All known constructions of CODs are inductive or algorithmic. In this paper, we present an explicit construction of optimal CODs. We do not apply recurrent procedures and do calculate the matrix elements directly. Our formula is based on a cubic function in two binary n-vectors. In our previous work (Comm. Math. Phys., 2010, and J. Pure and Appl. Algebra, 2011), we used this function to define a series of non-associative algebras generalizing the classical algebra of octonions and to obtain sum of squares identities of Hurwitz-Radon type. [ABSTRACT FROM AUTHOR]

Abstract: We define the notions of trace, determinant and, more generally, Berezinian of matrices over a -graded commutative associative algebra . The applications include a new approach to the classical theory of matrices with coefficients in a Clifford algebra, in particular of quaternionic matrices. In a special case, we recover the classical Dieudonné determinant of quaternionic matrices, but in general our quaternionic determinant is different. We show that the graded determinant of purely even -graded matrices of degree is polynomial in its entries. In the case of the algebra of quaternions, we calculate the formula for the Berezinian in terms of a product of quasiminors in the sense of Gelfand, Retakh, and Wilson. The graded trace is related to the graded Berezinian (and determinant) by a -graded version of Liouville’s formula. [Copyright &y& Elsevier]

We study non-associative twisted group algebras over $${(\mathbb{Z}_2)^n}$$ with cubic twisting functions. We construct a series of algebras that extend the classical algebra of octonions in the same way as the Clifford algebras extend the algebra of quaternions. We study their properties, give several equivalent definitions and prove their uniqueness within some natural assumptions. We then prove a simplicity criterion. We present two applications of the constructed algebras and the developed technique. The first application is a simple explicit formula for the following famous square identity: $${(a_1^2+\cdots+a_{N}^2)\,(b_1^2+\cdots+b_{\rho(N)}^2)= c_1^2+\cdots+c_{N}^2}$$, where c are bilinear functions of the a and b and where ρ( N) is the Hurwitz-Radon function. The second application is the relation to Moufang loops and, in particular, to the code loops. To illustrate this relation, we provide an explicit coordinate formula for the factor set of the Parker loop. [ABSTRACT FROM AUTHOR]

The pentagram map is a projectively natural transformation defined on (twisted) polygons. A twisted polygon is a map from $${\mathbb Z}$$ into $${{\mathbb{RP}}^2}$$ that is periodic modulo a projective transformation called the monodromy. We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation. A research announcement of this work appeared in [16]. [ABSTRACT FROM AUTHOR]

Abstract: We study the notion of Γ-graded commutative algebra for an arbitrary abelian group Γ. The main examples are the Clifford algebras already treated in H. Albuquerque and S. Majid (2002) . We prove that the Clifford algebras are the only simple finite-dimensional associative graded commutative algebras over or . Our approach also leads to non-associative graded commutative algebras extending the Clifford algebras. [Copyright &y& Elsevier]

The article reports on the investigation of the mathematical fundamental relations of Sir William Rowan Hamilton. It is said that Hamilton's fundamental relations has referred the canonical basis of the associative algebra of quaternions H. It notes that algebra H is not commutative in which entails that the elements i, j, k of Hamilton is anti-commute with each other.

Abstract: The Hurwitz problem of composition of quadratic forms, or of “sum of squares identity” is tackled with the help of a particular class of -graded non-associative algebras generalizing the octonions. This method provides explicit formulas for the classical Hurwitz–Radon identities and leads to new solutions in a neighborhood of the Hurwitz–Radon identities. [Copyright &y& Elsevier]

We consider the supercircle S 1|1 equipped with the standard contact structure. The Lie superalgebra $${\fancyscript{K}(1)}$$ of contact vector fields contains the Möbius superalgebra osp(1|2). We study the space of linear differential operators on weighted densities as a module over osp(1|2). We introduce the canonical isomorphism between this space and the corresponding space of symbols and find all cases where such an isomorphism does not exist. [ABSTRACT FROM AUTHOR]