A metrizable X with Cp(X) not homeomorphic to Cp(X) × Cp(X)

We give an example of an infinite metrizable space X such that the space C_p(X), of continuous real-valued functions on X endowed with the pointwise topology, is not homeomorphic to its own square C_p(X) × C_p(X). The space X is a zero-dimensional subspace of the real line. Our result answers a long-standing open question in the theory of function spaces posed by A. V. Arhangel’skii.     

2 × 2 Permanental Ideals of Hypermatrices

We analyze the structure of ideals generated by some classes of 2 × 2 permanents of hypermatrices, generalizing [9 Laubenbacher , R. C. , Swanson , I. ( 2000 ). Permanental ideals . J. Symbolic Comput. 30 : 195 – 205 .[Crossref], [Web of Science ®] , [Google Scholar]] on 2 × 2 permanental ideals of generic matrices. We compare the obtained structure to that of the corresponding determinantal ideals in [11 Swanson , I. , Taylor , A. ( 2013 ). Minimal primes of ideals arising from conditional independence statements . J. Algebra 392 : 299 – 314 .[Crossref], [Web of Science ®] , [Google Scholar]]: as expected, the permanental ideals have many more (minimal) components. In the last two sections, we examine a few related classes of permanental ideals.     

Constructing 2 × 2 Bricks from Unitary Numerical Semigroups

This paper investigates numerical semigroups that yield 2 × 2 bricks. We demonstrate the existence of an infinite family of 2 × 2 bricks that includes all of the perfect 2 × 2 bricks. We provide a formula for the Frobenius numbers of these semigroups as well as a necessary and sufficient condition for the semigroups to be symmetric.     

Units OF Z(G × C 2)

Abstract

In this paper, we are mainly interested in describing constructively u(Z(G × C ₂)), where u(ZG) has been described in some way. We first consider unitary units and prove that if u(ZG) is generated by unitary units, then U(Z(G × C₂)) is also generated by unitary units. Then we consider bicyclic units and ask the following question: If G has a normal complement generated by bicyclic units, does G × C ₂ also have a normal complement generated by bicyclic units? We give a negative answer to the above in general, showing that none of the normal complements of D ₈ × C ₂ × C ₂ is generated by bicyclic units, although a normal complement of D ₈ × C ₂ is indeed generated by bicyclic units.     

The Existence of (K 2 × K 6)-Designs

Direct and recursive constructions are established for graph designs for K ₂ × K ₆ grid-blocks. Using these, the existence of graph designs of index one in which the blocks are K ₂ × K ₆ grid-blocks is completely determined: A (K ₂ × K ₆)-design of order v exists if and only if ${v\equiv1\pmod{72}}$ .     

C(X)×_aZ_2的基本性质

本文考虑 C(X)×_αZ_2 的基本性质,例如忠实的*表示,投影与迹态的一般形式,等等.     

C(X)×_aZ_2的基本性质

本文考虑 C(X)×_αZ_2 的基本性质,例如忠实的*表示,投影与迹态的一般形式,等等.     

Numerical Radius Inequalities for Certain 2 × 2 Operator Matrices

We prove several numerical radius inequalities for certain 2 × 2 operator matrices. Among other inequalities, it is shown that if X, Y, Z, and W are bounded linear operators on a Hilbert space, then

$$w\left( \left[\begin{array}{cc} X &; Y \\ Z &; W \end{array} \right] \right) \geq \max \left(w(X),w(W),\frac{w(Y+Z)}{2},\frac{w(Y-Z)}{2}\right) $$

and

$$w\left( \left[\begin{array}{cc}X &; Y \\ Z &; W\end{array} \right] \right) \leq \max \left( w(X), w(W)\right)+\frac{w(Y+Z)+w(Y-Z)}{2}. $$

As an application of a special case of the second inequality, it is shown that

$$\frac{\left\Vert X\right\Vert }{2}+\frac{\left\vert \left\Vert\operatorname{Re}{X}\right\Vert -\frac{\left\Vert X\right\Vert}{2}\right\vert }{4}+\frac{ \left\vert \left\Vert \operatorname{Im}{X}\right\Vert -\frac{\left\Vert X\right\Vert}{2}\right\vert }{4} \leq w(X), $$

which is a considerable improvement of the classical inequality \({\frac{ \left\Vert X\right\Vert }{2}\leq w(X)}\) . Here w(·) and || · || are the numerical radius and the usual operator norm, respectively.

     

有向图(→C)n×(→P)2的优美性

研究了有向图(→C)n×(→P)2的优美性,利用搜索图的标号的算法与数学证明相结合的方法,证实了有向图(→C)n×(→P)2为优美图,其中n为任意正整数.     

Burgess-like Subconvex Bounds for GL2 × GL1

Let F be a number field, π an irreducible cuspidal representation of \({{\rm GL}_2(\mathbb{A}_F)}\) with unitary central character, and χ a Hecke character of analytic conductor Q. Then \({L(1/2, \pi \otimes \chi) \ll Q^{\frac{1}{2} - \frac{1}{8}(1-2\theta)+\epsilon}}\) , where \({0 \leq \theta \leq 1/2}\) is any exponent towards the Ramanujan–Petersson conjecture. The proof is based on an idea of unipotent translation originated from P. Sarnak then developed by Ph. Michel and A. Venkatesh, combined with a method of amplification.     

When is a 2 × 2 Matrix Ring Over a Commutative Local Ring Quasipolar?

A ring R is quasipolar if for any a ∈ R, there exists p ² = p ∈ R such that p ∈ comm²(a), p + a ∈ U(R) and ap ∈ R ^qnil . In this article, we determine when a 2 × 2 matrix over a commutative local ring is quasipolar. A criterion in terms of solvability of the characteristic equation is obtained for such a matrix to be quasipolar. Consequently, we obtain several equivalent conditions for the 2 × 2 matrix ring over a commutative local ring to be quasipolar. Furthermore, it is shown that the 2 × 2 matrix ring over the ring of p-adic integers is quasipolar.     

A note on the ranks of 2 × 2 × 2 and 2 × 2 × 2 × 2 tensors

Using rank-1 reduction formula and the vector space spanned by the real rank-1 matrices, we present a different way to show that the maximum possible rank of the 2?×?2?×?2 tensors over the real field is 3. Following, we obtain that the maximum rank of the 2?×?2?×?2?×?2 tensors over the real field is less than or equal to 5 and propose another way to show that the maximum rank of the 2?×?2?×?2?×?2 tensors over the complex field is 4, except one special case.     

Computing the ℤ2-Cocharacter of 3 × 3 Matrices of Odd Degree

Let F be a field of characteristic 0 and A = M _{2, 1}(F) the algebra of 3 × 3 matrices over F endowed with the only non trivial ?₂-grading. Aver'yanov in [1 Aver'yanov , I. V. ( 2009 ). Basis of graded identities of the superalgebra M _{1, 2}(F) . Mathematical Notes 85 ( 4 ): 467 – 483 .[Crossref] , [Google Scholar]] determined a set of generators for the T ₂-ideal of graded identities of A. Here we study the identities in variables of homogeneous degree 1 via the representation theory of the symmetric group, and we determine the decomposition of the corresponding character into irreducibles.     

Invariants of 2 × 2 matrices, irreducible {\hbox{SL}(2, \mathbb{C})} characters and the Magnus trace map

We obtain an explicit characterization of the stable points of the action of on the cartesian product G  ^{× n} by simultaneous conjugation on each factor in terms of the corresponding invariant functions. From this, a simple criterion for the irreducibility of representations of finitely generated groups into G is derived. We also obtain analogous results for the action of on the vector space of n-tuples of 2 × 2 complex matrices. For a free group F _n of rank n, we show how to generically reconstruct the 2 ^n-2 conjugacy classes of representations F _n → G from their values under the map considered in Magnus [Math. Zeit. 170, 91–103 (1980)], defined by certain 3n − 3 traces of words of length one and two.      

关于直积C↑→n1×C↑→n2×…×C↑→nk的哈密顿圈及哈密顿分解

设ｎ１≤ｎ２≤…≤ｎｋ是正整数，Ｄ＝Ｃ↑→ｎ１×Ｃ↑→ｎ２×…×Ｃ↑→ｎｋ是有向圈的直积。在本文中，我们证明了如果ｎｉ│ｎｋ（１≤ｉ≤ｋ－１），则Ｄ含有哈密根图。当ｎ１＝ｎ２＝…＝ｎｋ时，我们进一步得到Ｄ含有［ｋ／２」个弧不交的哈密顿圈。作为副产品，我们推出当Γ是哈密顿有向图时Γ×Γ也是哈密顿有向图。     

Mutation Classes of Skew-Symmetric 3 × 3-Matrices

In this article, we establish a bijection between the set of mutation classes of mutation-cyclic skew-symmetric integral 3 × 3-matrices and the set of triples of integers (a, b, c) such that 2 ≤ a ≤ b ≤ c and ab ≥ c. We also give an algorithm allowing to verify whether a matrix is mutation-cyclic or not. We prove that given a, b, the two cases depend on whether c is large enough or not.