This paper aims at solving the Hermitian SDC problem, i.e., that of \textit{simultaneously diagonalizing via $*$-congruence} a collection finitely many (not need pairwise commute) matrices. Theoretically, we provide some equivalent conditions for such matrix can be simultaneously diagonalized $^*$-congruence.% by nonsingular matrix. Interestingly, one leads to existence positive definite soluti...

The paper investigates the non-symbolic algebraic semantics of the weak bisimulation congruenceson finite pi processes. The weak bisimulation congruences are studied both in the absence and inthe presence of the mismatch operator. Some interesting phenomena about the open congruences arerevealed. Several new tau laws are discovered and their relationship is discussed. The contributi...

OBJECTIVE Research has suggested that congruence between patient characteristics and contextual characteristics is a more robust predictor of outcomes than either patient or context characteristics alone. The goal of the present study was to examine the degree of congruence between patient preferences for the clinical encounter and reports of analogous dimensions of provider behavior and the ef...

Let f be an integer greater than one. We study three progressively finer equivalence relations on closed 3-manifolds generated by Dehn surgery with denominator f : weak f -congruence, f -congruence, and strong f -congruence. If f is odd, weak f -congruence preserves the ring structure on cohomology with Zf -coefficients. We show that strong f -congruence coincides with a relation previously stu...

A subgroup of Γ which contains some principal congruence subgroup is called a congruence subgroup. The level of a congruence subgroup G is the smallest N such that Γ(N) ⊂ G. The literature on congruence subgroups is vast, and the subject remains very active. Rademacher conjectured that there are only finitely many genus 0 congruence subgroups. This problem was studied by Knopp and Newman [Knopp...

In 1916 Ramanujan observed a remarkable congruence: τ(n) ≡ σ11(n) mod 691. The modern point of view is to interpret the Ramanujan congruence as a congruence between the Fourier coefficients of the unique normalized cusp form of weight 12 and the Eisenstein series of the same weight modulo the numerator of the Bernoulli number B12. In this paper we give a simple proof of the Ramanujan congruence...

If 91 and 23 are algebras, where 9Is23, and 6 is a congruence relation on 91, let 0® be the smallest congruence relation on 93 containing 0. 91 is called a congruence subalgebra of 93 if 9ls93 and, for every congruence relation 0 on 91, 0*n|9I|2=0. Elementary subalgebras are congruence subalgebras, and there are Directed Union and Loewenheim-Skolem Theorems for congruence subalgebras analogous ...