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In this paper, we discuss the generalized Hamming weights of a class of linear codes associated with non-degenerate quadratic forms. In order to do so, we study the quadratic forms over subspaces of finite field and obtain some interesting…
In this article, we investigate factorization problems for twisted triple product Galois representations over real quadratic fields, arising from families of Hilbert cusp forms. Specifically, we address the factorization in two distinct…
The rank of a bimatrix game is defined as the rank of the sum of the payoff matrices of the two players. The rank of a game is known to impact both the most suitable computation methods for determining a solution and the expressive power of…
In this paper, a new criterion is given to determine the $p-$rationality of some complex cubic number fields in terms of $ p-$divisibility of certain terms of a third-order recurrence sequence, several illustrated examples are…
We study varieties of complexes of projective modules with fixed ranks, and relate these varieties to the varieties of their homologies. We show that for an algebra of global dimension at most two, these two varieties are related by a pair…
This paper concerns the problem of classifying finite-dimensional real solvable Lie algebras whose derived algebras are of codimension 1 or 2. On the one hand, we present an effective method to classify all $(n+1)$-dimensional real solvable…
Lifting problem for universal quadratic forms asks for totally real number fields $K$ that admit a positive definite quadratic form with coefficients in $\mathbb{Z}$ that is universal over the ring of integers of $K$. In this paper, we show…
We give explicit structure of the graded ring of modular forms with respect to Gamma(N) (N=1,2,3,4,5,6,7,8,9,10,12,16,18) and for some other congruence groups. We also study the modular forms of half-integer weight for certain groups.
We address the problem of the maximal finite number of real points of a real algebraic curve (of a given degree and, sometimes, genus) in the projective plane. We improve the known upper and lower bounds and construct close to optimal…
Quantum hamiltonian reduction of affine superalgebras is studied in the twisted case. The Ramond sector of "minimal" superconformal W-algebras is described in detail, the determinant formula is obtained. Extensive list of examples includes…
We obtain a computable structure of Scott rank omega_1^{CK} (call this ock), and give a general coding procedure that transforms any hyperarithmetical structure A into a computable structure A' such that the rank of A is ock, ock+1, or <…
In 2010, Cassidy and Vancliff extended the notion of a quadratic form on n generators to the noncommutative setting. In this article, we suggest a notion of rank for such noncommutative quadratic forms, where n = 2 or 3. Since writing an…
A classification of all four-dimensional power-commutative real division algebras is given. It is shown that every four-dimensional power-commutative real division algebra is an isotope of a particular kind of a quadratic division algebra.…
We highlight some facts about continued fractions of real cubic irrationalities. This may be thought as a small section in a textbook on continued fractions.
Let $R(x)=g(x)/h(x)$ be a rational expression of degree three over the finite field $\mathbb{F}_q$. We count the irreducible polynomials in $\mathbb{F}_q[x]$, of a given degree, which have the form $h(x)^{\mathrm{deg}\, f}\cdot…
We consider commuting squares of finite dimensional von Neumann algebras having the algebra of complex numbers in the lower left corner. Examples include the vertex models, the spin models (in the sense of subfactor theory) and the…
If a linear differential operator with rational function coefficients is reducible, its factors may have coefficients with numerators and denominatorsof very high degree. When the base field is $\mathbb C$, we give a completely explicit…
The real roots of the cubic and quartic polynomials are studied geometrically with the help of their respective Siebeck--Marden--Northshield equilateral triangle and regular tetrahedron. The Vi\`ete trigonometric formulae for the roots of…
It is well-known that every commutative separable unital C*-algebra of real rank zero is a quotient of the C*-algebra of all compex continous functions defined on the Cantor cube. We prove a non-commutative version of this result by showing…
We investigate the representations and the structure of Hecke algebras associated to certain finite complex reflection groups. We first describe computational methods for the construction of irreducible representations of these algebras,…