Related papers: The kernel of a rational Eisenstein prime at non-s…

A $k$-nucleus of a normal rational curve in PG$(n,F)$ is the intersection over all $k$-dimensional osculating subspaces of the curve ($k\in\{-1,0,...,n-1\}$). It is well known that for characteristic zero all nuclei are empty. In case of…

For a prime level $N$, we compute the index of the anemic Hecke algebra of weight 2 level $\Gamma_0(N)$ forms inside the full Hecke algebra. We prove that the index is fully described by the presence of weight 1 Katz forms of the same…

Let $R$ be a prime ring with center $Z(R)$ and with involution $*$. Given an additive subgroup $A$ of $R$, let $T(A):=\{x+x^*\mid x\in A\}$ and $K_0(A):=\{x-x^*\mid x\in A\}$. Let $L$ be a non-abelian Lie ideal of $R$. It is proved that if…

In 1975 prof. Don Zagier derived a preliminary formula for the trace of the Hecke operators acting on the space of cusp forms (\cite{5}, \cite{6}). Actually, it is an expression in terms of an integral over a fundamental domain of…

Let $\langle K,\nu \rangle$ be a real closed valued field, and let $S\subseteq K^n$ be an open semi-algebraic set. Using tools from model theory, we find an algebraic characterization of rational functions which admit, on $S$, only values…

In a previous paper, the authors studied the radical filtration of a Weyl module $\Delta_\zeta(\lambda)$ for quantum enveloping algebras $U_\zeta(\overset\circ{\mathfrak g})$ associated to a finite dimensional complex semisimple Lie algebra…

This paper consists of three parts: (I) To develop general theory of a (large) class of central simple finite dimensional algebras and answering some natural questions about them (that in general situation it is not even clear how to…

We elaborate the idea of quantum computation through measuring the correlation of a gapped ground state, while the bulk Hamiltonian is utilized to stabilize the resource. A simple computational primitive, by pulling out a single spin…

We use Heegner points to prove the existence of nontorsion rational points on the elliptic curve $y^2 = x^3 + D$ for any rational number $D=a/b$ such that $a$ and $b$ are squarefree integers for which $6$, $a$, and $b$ are pairwise…

Let $b \geq 2$ be an integer and $S$ be a finite non-empty set of primes not containing divisors of $b$. For any non-dense set $A \subset [0,1)$ such that $A \cap \mathbb{Q}$ is invariant under $\times b$ operation, we prove the finiteness…

For a finite set of non-zero natural numbers that contains at least one element different from 1 and the least common multiple of any of its subsets, there exists a subset of at least half of its members which has a common divisor larger…

In this paper, we prove: Let A be a nonnegative primitive tensor with order m and dimension n. Then its primitive degree R(A)\leq (n-1)^2+1, and the upper bound is sharp. This confirms a conjecture of Shao [7].

Let $\mathbf{G}=\mathrm{U}(2,n)$ be the unitary group associated to a Hermitian space over a quadratic imaginary number field $E$. We assume that 2 is unramified in $E$, and the Hermitian space splits at all finite places and has signature…

Let $K = \Q(\theta)$ be an algebraic number field with $\theta$ satisfying an irreducible polynomial $x^{9} - a$ over the field $\Q$ of rationals and $\Z_K$ denote the ring of algebraic integers of $K$. In this article, we provide the exact…

Let $K$ be a number field with ring of integers $\mathbb{Z}_K$. We prove two asymptotic formulas connected with the distribution of irreducible elements in $\mathbb{Z}_K$. First, we estimate the maximum number of nonassociated irreducibles…

We consider the rationality of the Fourier coefficients of a particular paramodular Siegel Eisenstein series of level $N^2$ with weight $k\geq 4$. We show that the coefficients lie in a number field.

Let $R$ be a real closed field and let ${\mathcal S}(M)$ be the ring of (continuous) semialgebraic functions on a semialgebraic set $M\subset R^n$ and let ${\mathcal S}^*(M)$ be its subring of bounded semialgebraic functions. In this work…

We develop a framework for function classes generated by parametric ridge kernels: one-dimensional kernels composed with affine projections and averaged over a parameter measure. The induced kernels are positive definite, and the resulting…

We determine the structure of the partition algebra $P_n(Q)$ (a generalized Temperley-Lieb algebra) for specific values of $Q \in \C$, focusing on the quotient which gives rise to the partition function of $n$ site $Q$-state Potts models…

Suppose that $O_L$ is the ring of integers of a number field $L$, and suppose that $f(z)=\sum_{n=1}^\infty a_f(n)q^n\in S_k\cap O_L[[q]]$ (note: $q := e^{2\pi iz}$) is a normalized Hecke eigenform for $\mathrm{SL}_2(\mathbb{Z})$. We say…