A frame $L$ is called {it coz-dense} if $Sigma_{coz(alpha)}=emptyset$ implies $alpha=mathbf 0$. Let $mathcal RL$ be the ring of real-valued continuous functions on a coz-dense and completely regular frame $L$. We present a description of the socle of the ring $mathcal RL$ based on minimal ideals of $mathcal RL$ and zero sets in pointfree topology. We show that socle of $mathcal RL$ is an essent...

community detection is a task of fundamental importance in social network analysis. community structures enable us to discover the hidden interactions among the network entities and summarize the network information that can be applied in many applied domains such as bioinformatics, finance, e-commerce and forensic science. there exist a variety of methods for community detection based on diffe...

Let S be an unramified regular local ring of mixed characteristic two and R the integral closure in a biquadratic extension its quotient field obtained by adjoining roots sufficiently general square free elements f,g?S. S2 denote subring lifting to image Frobenius map on S/2S. When at least one f,g?S2, we characterize Cohen-Macaulayness show that admits birational small Cohen-Macaulay module. I...

Suppose that G is a finite group and k field of characteristic $$p>0$$ . We consider the complete cohomology ring $${\mathcal {E}}_M^* = \sum _{n \in {\mathbb Z}} \widehat{{\text {Ext}}}^n_{kG}(M,M)$$ show has two distinguished ideals $$I^* \subseteq J^* {\mathcal {E}}_M^*$$ such $$I^*$$ bounded above in degrees, {E}}_M^*/J^*$$ below degree $$J^*/I^*$$ eventually periodic with terms dimension. ...

ly isomorphic as rings are conjugate in B. 146 BRIAN CONRAD (APPENDIX BY W. R. MANN) Remark A.10. The order ( A mn A A ) in (2) is the intersection of the maximal orders M2(A) and γnM2(A)γ−1 n = ( A mn m−n A ) , where γn = ( 0 πn 1 0 ) . This example is called a standard Eichler order. Proof. For the first part, it suffices to show that the set R of elements of B integral over R is an order. Th...

The strong ring R is a commutative ring generated by finite abstract simplicial complexes. To every G ∈ R belongs a Hodge Laplacian H = H(G) = D = (d + d∗)2 determining the cohomology and a unimodular connection operator L = L(G). The sum of the matrix entries of g = L−1 is the Euler characteristic χ(G). For any A,B ∈ R the spectra of H satisfy σ(H(A × B)) = σ(H(A)) + σ(H(B)) and the spectra of...

A graded K-algebra R has property Np if it is generated in degree 1, has relations in degree 2 and the syzygies of order ≤ p on the relations are linear. The GreenLazarsfeld index of R is the largest p such that it satisfies the property Np. Our main results assert that (under a mild assumption on the base field) the c-th Veronese subring of a polynomial ring has Green-Lazarsfeld index ≥ c+1. T...

Using Fourier–Mukai transformations, we prove some results about the ring of unipotent vector bundles on elliptic curves in positive characteristics. This ring was determined by Atiyah in characteristic zero, who showed that it is a polynomial ring in one variable. It turns out that the situation in characteristic p > 0 is completely different and rather bizarre: the ring is nonnoetherian and c...

0. Introduction. 1. Let G be a group of linear transformations on a finite dimensional real or complex vector space X. Assume X is completely reducible as a G-module. Let 5 be the ring of all complexvalued polynomials on X, regarded as a G-module in the obvious way, and let J C 5 be the subring of all G-invariant polynomials on X. Now let J be the set of all ƒ £ J having zero constant term and ...

Julia Robinson has given a first-order definition of the rational integers Z in the rational numbers Q by a formula (∀∃∀∃)(F = 0) where the ∀quantifiers run over a total of 8 variables, and where F is a polynomial. We show that for a large class of number fields, including Q, for every ε > 0, there exists a set of primes S of natural density exceeding 1− ε, such that Z can be defined as a subse...