Knotted pictures of the Bell bases on the surface of a trivial torus

By means of the method of torus knot theory, this paper gives the complete processes of obtaining the knotted pictures of four Bell b^es from the knotted pictures of four basic two qubit states.

Nonlocality Distillation and Trivial Communication Complexity for High-Dimensional Systems

A nonlocality distillation protocol for arbitrary high-dimensional systems is proposed. We study the nonlocality distillation in the 2-input d-output bi-partite case. Firstly, we give the one-parameter nonlocal boxes and their correlated distilling protocol. Then, we generalize the one-parameter nonlocality distillation protocol to the twoparameter case. Furthermore, we introduce a contracting protocol testifying that the 2-input d-output nonlocal boxes make communication complexity trivial.

The Almost Split Sequences for Trivial Extensions of Hereditary Algebras

Let A be a basic hereditary artin algebra and R = A Q be the trivial extension of A by its minimal injective cogenerator Q. We construct some right （left） almost split morphisms and irreducible morphisms in modR through the corresponding morphisms in modA. Furthermore, we can determine its almost split sequences in modR.

C^l -G_V TRIVIALITY OF FUNCTION GERMS AND NEWTON POLYHEDRA

We provide estimates on the degree of C l GV determinacy （ G is one of Mather’s groups R or K ） of function germs which are defined on analytic variety V and satisfies a non-degeneracy condition with respect to some Newton polyhedron. The result gives an explicit order such that the C l geometrical structure of a function germ is preserved after higher order perturbations, which generalizes the result on C l G triviality of function germs given by M.A.S.Ruas.

Trivial Sources of Heterosis in Cotton?

Historically,re-selection,pedigree,and mass selection breeding methods have been used to develop open-pollinated cultivars of Upland cotton(Gossypium hirsutum L.). Due to the

Life in Triviality: Reinterpretation of The Hours Based on Philip Glass' Scores

The movie The Hours, adapted from Michael Cunningham's novel, presents us three different women's life in a day to reflect trivial people's struggle, contemplation of self-worth and death, with Philip Glass' scores and the well-designed settings. This paper focuses on Glass' scores and analyzes its connection with other elements, reflection of those people and coherence of the movie.

Liu Hong: no trivial on exhibitions

To Work for Tomorrow Liu Hong, General Manager of TUTTI Exhibition Service (Xiamen) Co., Ltd, has set up TUTTI Exhibition Service (Xiamen) Co., Ltd in 2001, when exhibition industry firstly appear in Xiamen. Furthermore, the company has developed rapidly in recent years. As Liu Hong regards, exhibition industry, as well as advertisement industry, is an industry filled with challenges and

The Category of Modules on an n-Trivial Extension: the Basic Properties

In this paper we investigate a categorical aspect of n-trivial extension of a ring by a family of modules.Namely,we introduce the right(resp.,left)n-trivial extension of a category by a family of endofunctors.Among other results,projective,injective and flat objects of this category are characterized,and two applications are presented at the end of this paper.We characterize when an n-trivial extension ring is k-perfect and establish a result on the self-injective dimension of an n-trivial extension ring.

The first cohomology group of trivial extensions of special biserial algebras

Given a finite dimensional special biserial algebra A with normed basis we obtain the dimension formulae of the first Hochschild homology groups of A and the vector space Alt(DA). As a consequence, an explicit dimension formula of the first Hochschild cohomology group of trivial extension TA = A × DA in terms of the combinatorics of the quiver and relations is determined.

ON INDECOMPOSABLE MODULES OVER A REPRESENTATION-FINITE TRIVIAL EXTENSION ALGEBRA

Let T(A)=A?D(A) be a representation-finite trivial extension algebra. We haveproved: (i) every indecomposable module of T(A) is determined by its top and socle, (ii)every indecomposable module of T(A) is determined by its Loewy factors and socal factorsrespectively.

Generic modules for trivial extension algebras

Let A be a finite-dimensional algebra over an algebraically closed field. An indecomposable (right) ,4-module M is called generic provided M is infinite k-dimensional but finite length as (left) EndA(M)-module. Let R = A DA be the trivial extension algebra of A- Generic R-modules are constructed from generic A-modules using some functors between Mod A and Mod R. it is also proved that if A is a tame hereditary algebra, then R has only two generic modules.

Generating Relations for Grothendieck Groups of Trivial Extension Algebras

In this paper we describe explicitly the generating relations for the Grothendieck groups of trivial extension algebras of tame hereditary algebras.

Trivial extension of Koszul algebras

Let A be a Koszul algebra, and let M be a graded A-bimodule. We prove that the trivial extension algebra of A by M is also a Koszul algebra whenever M is Koszul as a left A-module. Applications and examples are also provided.

The Projective Cover of the Trivial Representation for a Finite Group of Lie Type in Defining Characteristic

We give a lower bound of the Loewy length of the projective cover of the trivial module for the group algebra kG of a finite group G of Lie type defined over a finite field of odd characteristic p, where k is an arbitrary field of characteristic p. The proof uses Auslander-Reiten theory.

Paradox of voting and the trivial preference set of groupdecision making

The structure of the trivial preference set of group decision making is studied, and the essential characters and generating reasons of the paradox of voting are revealed.

EXISTENCE OF NON-TRIVIAL SOLUTION FOR SUPERLINEAR SYSTEM OF INTEGRAL EQUATIONS AND ITS APPLICATIONS

In this paper, we use cone theory and topological degree theory to study superlinear systemof integral equations, and obtain existence theorems for non-trivial solutions; moreover, we applythe results to two-point boundary problems of ordinary differential system of equations.

ERRATUM A TOPOLOGICALLY MIXING MAP WITH TRIVIAL ROTATION SET

We showed in [G. H], if f is a Cr-topologically mixing map of the circle (r ≥0),then its rotation set will be non-trivial. But this statement is only satisfied for generic(open and dense) subset of all Cr-topologically mixing maps of the circle (r≥0). Here,we present another proof of the above statement. Moreover, we introduce a topologicallymixing map of the circle with trivial rotation set.

警惕AI热潮中的娱乐化趋势

在此轮AI热潮中,AI宣传术居功至伟,但也呈现出某些负面效应。除了娱乐化宣传外,当前的AI发展的娱乐化趋势还表现在研究方向、企业组织和评论研究的娱乐化,在产业方向、认知观念和现实战略等层面均有表现。有限主义的发展进路反对AI发展的娱乐化趋势,主张遵循有益AI理念,坚持AI以人为本,在推进AI技术创新与应用的同时,有效治理AI、控制AI。最终从战略上引领AI技术为实体经济赋能,助力推进中国式现代化建设与高质量发展要求。

Second Maximal Invariant Subgroups and Solubility of Finite Groups

Let G be a finite group and assume that a group of automorphisms A is acting on G such that A and G have coprime orders.We prove that the fact of imposing specific properties on the second maximal A-invariant subgroups of G determines that G is either soluble or isomorphic to a fewnon-soluble groups such as PSL(2,5)or SL(2,5).

非交换omni-李2-代数

本文对非交换omni-李2-代数结构进行研究。首先,在直和空间gl(G)⊕G上定义G-值配对和括号运算,构造非交换omni-李2-代数,并证明它是一个严格的Leibniz 2-代数。其次,证明非交换omni-李2-代数的括号与G-值配对是相容的,具有omni-李2-代数类似的性质。最后,构造omni-李2-代数上的Nijenhuis算子,证明非交换omni-李2-代数可以看成omni-李2-代数的平凡形变。