A note on quadratic congruences

The notion of quadratic congruences was introduced recently in [1]. In the present note we revise the original definition and present an explanation of the the phenomena.     

A note about congruences on subsemigroups of groups

A well known elementary argument shows that a totally disconnected normal subgroup of a connected topological group is central, in [2] K. H. Hofmann has shown that by a skillful application of this argument (or variations of it) a much wider class of interesting centrality results can be obtained. In the present paper we offer a modification of the argument also applying to congruences on subsemigroups of topological groups.     

A note on representation of lattices by weak congruences

A weak congruence is a symmetric, transitive, and compatible relation. An element u of an algebraic lattice L is ??-suitable if there is an isomorphism ?? from L to the lattice of weak congruences of an algebra such that ??(u) is the diagonal relation. Some conditions implying the ??-suitability of u are presented.     

A note on congruences of semilattices with sectionally finite height

We present a bijective correspondence between congruences of semilattices with sectionally finite height (i.e., meet-semilattices whose principal downsets have finite length) and certain special subsets of their universes. We characterize these subsets from a purely order-theoretic point of view and prove that the bijection coincides with the Leibniz operator of abstract algebraic logic.     

A note on the local theta correspondence for unitary similitude dual pairs

Following Roberts? work in the case of orthogonal-symplectic similitude dual pairs, we study the local theta correspondence for unitary similitude dual pairs over a p-adic field.     

A short note on harmonic functions and zero divisors on the Sierpinski fractal

We show that harmonic functions are not zero divisors in the algebra of real-valued continuous functions on the Sierpinski fractal.     

The curve of ``Prym canonical' Gauss divisors on a Prym theta divisor

The Gauss linear system on the theta divisor of the Jacobian of a nonhyperelliptic curve has two striking properties:

1) the branch divisor of the Gauss map on the theta divisor is dual to the canonical model of the curve;

2) those divisors in the Gauss system parametrized by the canonical curve are reducible.

In contrast, Beauville and Debarre prove on a general Prym theta divisor of dimension all Gauss divisors are irreducible and normal. One is led to ask whether properties 1) and 2) may characterize the Gauss system of the theta divisor of a Jacobian. Since for a Prym theta divisor, the most distinguished curve in the Gauss system is the Prym canonical curve, the natural analog of the canonical curve for a Jacobian, in the present paper we analyze whether the analogs of properties 1) or 2) can ever hold for the Prym canonical curve. We note that both those properties would imply that the general Prym canonical Gauss divisor would be nonnormal. Then we find an explicit geometric model for the Prym canonical Gauss divisors and prove the following results using Beauville's singularities criterion for special subvarieties of Prym varieties:

Theorem. For all smooth doubly covered nonhyperelliptic curves of genus , the general Prym canonical Gauss divisor is normal and irreducible.

Corollary. For all smooth doubly covered nonhyperelliptic curves of genus , the Prym canonical curve is not dual to the branch divisor of the Gauss map.

     

A note on a generalized circular summation formula of theta functions

In this note, we make a correction of the imaginary transformation formula of Chan and Liu?s circular formula of theta functions. We also get the imaginary transformation formulaes for a type of generalized cubic theta functions.     

A note on shifted convolution of cusp-forms with $$\theta $$-series

Recently Lachterman, Schayer, and Younger published an elegant proof of the Ramanujan congruences for the partition function p(n). Their proof uses only the classical theory of modular forms as well as a beautiful result of Choie, Kohnen, and Ono, without the need for Hecke operators. In this paper, we give a method for generalizing Lachterman, Schayer, and Younger’s proof to include Ramanujan congruences for multipartition functions \(p_k(n)\) and Ramanujan congruences for p(n) modulo certain prime powers.     

A note on the lattice of weak L-valued congruences on the group of prime order

Weak L-valued (fuzzy) congruences on an algebra, where L is a lattice, were induced in [4]. In this note, we give a representation theorem for one subdirect power of L, by means of the lattice of L-valued congruences on any cyclic group of prime order.     

Ampleness of intersections of translates of theta divisors in an abelian fourfold

We prove the ampleness of the cotangent bundle of the intersection of two general translates of a theta divosor of the Jacobian of a general curve of genus . From this, we deduce the same result in a general, principally polarized abelian variety of dimension .

     

A note on the congruences of the nakano superlattice and some properties of the associated quotients

In this paper we explore theNakano superlattice (H, ⊔, ⊓), where ⊔, ⊓ are the Nakanohyperoperations x⊔y={z:x∨z=y∨z=x∨y},x⊓y={z:x∧z=y∧z=x∧y}. In particular, we study the properties of congruences on the Nakano superlattice and the associated quotients. New hyperoperations are introduced on the quotient and their properties studied.