Related papers: About connection of one matrix of composite number…

A new proof for adjoint systems of linear equations is presented. The argument is built on the principles of Algorithmic Differentiation. Application to scalar multiplication sets the base line. Generalization yields adjoint inner vector,…

Suppose that $p$ is an odd prime and $\genfrac{(}{)}{}{}{\cdot}{p}$ denotes the Legendre symbol modulo $p$. If $p$ is has the form $p= n^2+1$ then one easily verifies that $\genfrac{(}{)}{}{}{a}{p} = \genfrac{(}{)}{}{}{-a}{p}$ for all $a\in…

We associate to every matroid M a polynomial with integer coefficients, which we call the Kazhdan-Lusztig polynomial of M, in analogy with Kazhdan-Lusztig polynomials in representation theory. We conjecture that the coefficients are always…

A conjecture by Higman asserts that the number of conjugacy classes in the unipotent group of upper triangular matrices over a finite field depends polynomially on the number of elements of the field. We will study several alternative…

A supercongruence is a congruence between rational numbers modulo a power of a prime. In this paper, we give a technique for finding and algorithmically proving supercongruences by expressing terms as infinite series involving certain…

For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions…

We find that the solution of the polar angular differential equation can be written as the universal associated Legendre polynomials. Its generating function is applied to obtain an analytical result for a class of interesting integrals…

We define a relation that describes the ternary commutator for congruence modular varieties. Properties of this relation are used to investigate the theory of the higher commutator for congruence modular varieties.

A class of second-order differential equations commonly arising in physics applications are considered, and their explicit hypergeometric solutions are provided. Further, the relationship with the Generalized and Universal Associated…

In this paper, we clarify and build connections between various conjectures largely motivated by the works of Jean-Pierre Serre and John Tate. We closely study the Tate conjecture for algebraic cycles as well as their motivic…

We address a special case of a conjecture of M. Talagrand relating two notions of "threshold" for an increasing family $\mathcal F$ of subsets of a finite set $V$. The full conjecture implies equivalence of the "Fractional…

We discuss recent progress many problems in random matrix theory of a combinatorial nature, including several breakthroughs that solve long standing famous conjectures.

First, we calculate the Ehrhart polynomial associated to an arbitrary cube with integer coordinates for its vertices. Then, we use this result to derive relationships between the Ehrhart polynomials for regular lattice tetrahedrons and…

Let $\Lambda$ be a Legendrian in the jet space of some manifold $X$. To a generating family presentation of $\Lambda$, we associate a constructible sheaf on $X \times \mathbb{R}$ whose singular support at infinity is $\Lambda$, and such…

We study the Mathieu Conjecture for $SU(2)$ using the matrix elements of its unitary irreducible representations. We state a conjecture for the particular case $SU(2)$ implying the Mathieu Conjecture for $SU(2)$.

Let $1\leq a<q$ be a pair of small integers such that $\gcd(a,q)=1$ and let $x>1$ be a large number. This note discusses the existence of a short sequence of primes $p\equiv a\bmod q$ between two squares $x^2$ and $(x+1)^2$.

Differential graded algebra invariants are constructed for Legendrian links in the 1-jet space of the circle. In parallel to the theory for R^3, Poincare-Chekanov polynomials and characteristic algebras can be associated to such links. The…

It is shown that a normalized complex Hadamard matrix of order $6$ having three distinct columns, each containing at least one $-1$ entry necessarily belongs to the transposed Fourier family, or to the family of $2$-circulant complex…

We study an extension of Montgomery's pair-correlation conjecture and its relevance in some problems on the distribution of prime numbers.

Any link that is the closure of a positive braid has a natural Legendrian representative. These were introduced in an earlier paper, where their Chekanov--Eliashberg contact homology was also evaluated. In this paper we re-phrase and…