Automorphisms of Coxeter groups of type <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

A Coxeter system (W, S) is said to be of type K _n if the associated Coxeter graph Γ_S is complete on n vertices and has only odd edge labels. If W satisfies either of: (1) n = 3; (2) W is rigid; then the automorphism group of W is generated by the inner automorphisms of W and any automorphisms induced by Γ_S. Indeed, Aut(W) is the semidirect product of Inn(W) and the group of diagram automorphisms, and furthermore W is strongly rigid. We also show that if W is a Coxeter group of type K _n then W has exactly one conjugacy class of involutions and hence Aut(W) = Spec(W).     

Asymptotic <Emphasis Type="Italic">T</Emphasis><Subscript><Emphasis Type="Italic">u</Emphasis></Subscript>-Toeplitzness of weighted composition operators on <Emphasis Type="Italic">H</Emphasis><Superscript>2</Superscript>

Given a unilateral forward shift S acting on a complex, separable, innite dimensional Hilbert space H, an asymptotically S-Toeplitz operator is a bounded linear operator T on H satisfying that {S* ⁿ TS ⁿ } is convergent with respect to one of the topologies commonly used in the algebra of bounded linear operators on H. In this paper, we study the asymptotic T _u -Toeplitzness of weighted composition operators on the Hardy space H², where u is a nonconstant inner function.     

Degree sum of a pair of independent edges and <Emphasis Type="Italic">Z</Emphasis><Subscript>3</Subscript>-connectivity

Let G be a 2-edge-connected simple graph on n vertices. For an edge e = uv ∈ E(G), define d(e) = d(u) + d(v). Let F denote the set of all simple 2-edge-connected graphs on n ≥ 4 vertices such that G ∈ F if and only if d(e) + d(e’) ≥ 2n for every pair of independent edges e, e’ of G. We prove in this paper that for each G ∈ F, G is not Z ₃-connected if and only if G is one of K _2,n?2, K _3,n?3, K _2,n?2 ⁺ , K _3,n?3 ⁺ or one of the 16 specified graphs, which generalizes the results of X. Zhang et al. [Discrete Math., 2010, 310: 3390–3397] and G. Fan and X. Zhou [Discrete Math., 2008, 308: 6233–6240].     

Differences of Idempotents In <Emphasis Type="Italic">C</Emphasis>*-Algebras and the Quantum Hall Effect

Let ? be a trace on the unital C*-algebra A and M _? be the ideal of the definition of the trace ?. We obtain a C*analogue of the quantum Hall effect: if P,Q ∈ A are idempotents and P ? Q ∈ M _? , then ?((P ? Q)²ⁿ⁺¹) = ?(P ? Q) ∈ R for all n ∈ N. Let the isometries U ∈ A and A = A*∈ A be such that I+A is invertible and U-A ∈ M _? with ?(U-A) ∈ R. Then I-A, I?U ∈ M _? and ?(I?U) ∈ R. Let n ∈ N, dimH = 2n + 1, the symmetry operators U, V ∈ B(H), and W = U ? V. Then the operator W is not a symmetry, and if V = V*, then the operator W is nonunitary.     

Periodic Groups Saturated with the Groups <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

Given an indexing set I and a finite field K_α for each α ∈ I, let ? = {L₂(K_α) | α ∈ I} and \(\mathfrak{N} = \{ SL_2 (K_\alpha )|\alpha \in I\}\). We prove that each periodic group G saturated with groups in \(\Re (\mathfrak{N})\) is isomorphic to L₂(P) (respectively SL₂(P)) for a suitable locally finite field P.     

On Linear Preservers of Two-Sided Gut-Majorization On <Emphasis Type="Bold">M</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">m</Emphasis></Subscript>

For X, Y ∈ M_n,m it is said that X is gut-majorized by Y, and we write X ?_gutY, if there exists an n-by-n upper triangular g-row stochastic matrix R such that X = RY. Define the relation ~_gut as follows. X ~_gutY if X is gut-majorized by Y and Y is gut-majorized by X. The (strong) linear preservers of ?_gut on ?ⁿ and strong linear preservers of this relation on M_n,m have been characterized before. This paper characterizes all (strong) linear preservers and strong linear preservers of ~_gut on ?ⁿ and M_n,m.     

On (<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)-absorbing ideals of commutative rings

Let R be a commutative ring with 1 ≠ 0 and U(R) be the set of all unit elements of R. Let m, n be positive integers such that m > n. In this article, we study a generalization of n-absorbing ideals. A proper ideal I of R is called an (m, n)-absorbing ideal if whenever a ₁?a _m ∈I for a ₁,…, a _m ∈R?U(R), then there are n of the a _i ’s whose product is in I. We investigate the stability of (m, n)-absorbing ideals with respect to various ring theoretic constructions and study (m, n)-absorbing ideals in several commutative rings. For example, in a Bézout ring or a Boolean ring, an ideal is an (m, n)-absorbing ideal if and only if it is an n-absorbing ideal, and in an almost Dedekind domain every (m, n)-absorbing ideal is a product of at most m ? 1 maximal ideals.     

Some new scales of weight characterizations of the class <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We present an equivalence theorem, which includes all known characterizations of the class B _p , i.e., the weight class of Ariño and Muckenhoupt, and also some new equivalent characterizations. We also give equivalent characterizations for the classes B _p ^* , B _∞ ^* and RB _p , and prove and apply a “gluing lemma” of independent interest.     

Note edge-colourings of <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">n,n</Emphasis></Subscript> with no long two-coloured cycles

Consider the set of all proper edge-colourings of a graph G with n colours. Among all such colourings, the minimum length of a longest two-coloured cycle is denoted L(n, G). The problem of understanding L(n, G) was posed by Häggkvist in 1978 and, specifically, L(n, K _n,n ) has received recent attention. Here we construct, for each prime power q ≥ 8, an edge-colouring of K _n,n with n colours having all two-coloured cycles of length ≤ 2q ², for integers n in a set of density 1 ? 3/(q ? 1). One consequence is that L(n, K _n,n ) is bounded above by a polylogarithmic function of n, whereas the best known general upper bound was previously 2n ? 4.     

A boundary Schwarz lemma on the classical domain of type <Emphasis Type="Italic">I</Emphasis>

Let R _I (m, n) be the classical domain of type I in ? ^m×n with 1 ≤ m ≤ n. We obtain the optimal estimates of the eigenvalues of the Fréchet derivative Df(\(\mathop Z\limits^ \circ \)) at a smooth boundary fixed point \(\mathop Z\limits^ \circ \)of R _I (m, n) for a holomorphic self-mapping f of R _I (m, n). We provide a necessary and sufficient condition such that the boundary points of R _I (m, n) are smooth, and give some properties of the smooth boundary points of R _I (m, n). Our results extend the classical Schwarz lemma at the boundary of the unit disk Δ to R _I (m, n), which may be applied to get some optimal estimates in several complex variables.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-boundedness of a boundary integral operator on a contour with a peak

Results on the solvability of boundary integral equations on a plane contour with a peak obtained in collaboration with V.G. Maz’ya are developed. Earlier, it was proved that, on a contour Γ with an outward peak, the operator of the boundary equation of the Dirichlet boundary value problem maps the space ? _{p, β + 1} (Γ) continuously onto \(\mathcal{N}_{p,\beta } (\Gamma )\). The norm of a function in ? _{p, β} (Γ) is defined as

Open image in new window

, provided that the peak is at the origin. In this case, the norms on the spaces \(\mathcal{N}_{p,\beta }^ \mp (\Gamma )\) are defined by

Open image in new window

, where q _± are the intersection points of Γ with the circle {z: |z| = |q|} and δ > 0 is a fixed small number. On a contour with an inward peak, the operator of the boundary equation of the Dirichlet problem continuously maps ? _{p, β + 1} (Γ) onto ? _{p, β}(Γ), where ? _{p, β}(Γ) is the direct sum of \(\mathcal{N}_{p,\beta }^ + (\Gamma )\) (Γ) and the space

Open image in new window

(Γ) of functions on Γ of the form p(z) = Σ _{k = 0} ^m t ^(k)Rez ^k with the parameter m = [μ ? β ? p ^?1]. The operator I ? 2W of the boundary integral equation of plane elasticity theory, where W is the elastic double-layer potential, is considered. The main result is that the operator I ? 2W continuously maps the space ? _{p, β + 1} × ? _{p, β + 1}(Γ) to the space \(\mathcal{N}_{p,\beta }^ - \times \mathcal{N}_{p,\beta }^ - (\Gamma )\).

On a contour with an inward peak, the obtained representation of the operator I ? 2W and theorems on the boundedness of auxiliary integral operators imply that the images of vector-valued functions from ? _{p, β + 1} × ? _{p, β + 1}(Γ) have components representable as sums of functions from the spaces \(\mathcal{N}_{p,\beta }^ - (\Gamma )\)(Γ) and ? _{p, β}(Γ).     

Exponentially Many Monochromatic <Emphasis Type="Italic">n</Emphasis>-Matchings in <Emphasis Type="Italic">K</Emphasis><Subscript>3<Emphasis Type="Italic">n</Emphasis>-1</Subscript>

Erdös et al and Gerencsér et al had shown that in any 2-edge-coloring of K _3n-1, there is a n-matching containing edges with the same color(we call such matching monochromatic matching). In this paper we show that for any 2-edge-coloring of K _3n-1 there exists a monochromatic subgraph H of K _3n-1 which contains exponentially many monochromatic n-matchings.     

On <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-factorization of complete bipartite multigraphs

Let λK _m,n be a complete bipartite multigraph with two partite sets having m and n vertices, respectively. A K _p,q -factorization of λK _m,n is a set of edge-disjoint K _p,q -factors of λK _m,n which partition the set of edges of λK _m,n . When p = 1 and q is a prime number, Wang, in his paper [On K _1,q -factorization of complete bipartite graph, Discrete Math., 126: (1994), 359-364], investigated the K _1,q -factorization of K _m,n and gave a sufficient condition for such a factorization to exist. In papers [K _1,k -factorization of complete bipartite graphs, Discrete Math., 259: 301-306 (2002),; K _p,q -factorization of complete bipartite graphs, Sci. China Ser. A-Math., 47: (2004), 473-479], Du and Wang extended Wang’s result to the case that p and q are any positive integers. In this paper, we give a sufficient condition for λK _m,n to have a K _p,q -factorization. As a special case, it is shown that the necessary condition for the K _p,q -factorization of λK _m,n is always sufficient when p : q = k : (k + 1) for any positive integer k.     

Regularity for weakly (K<Subscript>1</Subscript>, K<Subscript>2</Subscript>)-quasiregular mappings

In this paper, we first give the definition of weakly (K₁,K₂-quasiregular mappings, and then by using the Hodge decomposition and the weakly reverse Hölder inequality, we obtain their regularity property: For anyq ₁ that satisfies\(0< K_1 n^{(n + 4)/2} 2^{n + 1} \times 100^{n^2 } [2^{3n/2} (2^{5n} + 1)](n - q_1 )< 1\), there existsp ₁=p ₁(n,q ₁,K ₁,K ₂)>n, such that any (K₁, K₂)-quasiregular mapping\(f \in W_{loc}^{1,q_1 } (\Omega ,R^n )\) is in fact in\(W_{loc}^{1,p_1 } (\Omega , R^n )\). That is, f is (K₁,K₂)-quasiregular in the usual sense.     

Polynomial inequalities on sets with <Emphasis Type="Italic">k</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript>-concave weighted measures

Let μ be a measurewith a k-concave density W on an open convex set V in R^m, that is, W is an integrable weight satisfying the condition

$$W(ax + (1 - a)y) \geqslant {(a{W^K}(x) + (1 - a){W^K}(y))^{1/k}},k \in ( - 1/m,\infty ]$$

for all x ∈ V, y ∈ V, and α ∈ [0, 1]. In this paper, we first show that the Fradelizi μ-distributional inequalities for polynomials P of m variables are sharp for each m and k ∈ (?1/m,∞]. Classes of extremal sets V, weights W, and polynomials P for these inequalities are presented. Sharpness of the Bobkov-Nazarov-Fradelizi dilation-type inequalities is established as well. Second, we find efficient conditions for k-concavity of a weight W and obtain new sharp polynomial inequalities.

     

Properties of element orders in covers for L<Subscript>n</Subscript>(<Emphasis Type="Italic">q</Emphasis>) and U<Subscript>n</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

We show that if a finite simple group G, isomorphic to PSL_n(q) or PSU_n(q) where either n ≠ 4 or q is prime or even, acts on a vector space over a field of the defining characteristic of G; then the corresponding semidirect product contains an element whose order is distinct from every element order of G. We infer that the group PSL_n(q), n ≠ 4 or q prime or even, is recognizable by spectrum from its covers thus giving a partial positive answer to Problem 14.60 from the Kourovka Notebook.     

An extension of Weyl Schouten Theorem for Lipschitz and <Emphasis Type="Italic">H</Emphasis><Superscript>2</Superscript> manifolds

In this article, we extend the Weyl Schouten Theorem for a C ^0,1 ∩ W ^2,2 manifold only under the assumptions on scalar curvature. Also we obtain a regularity of a C ^0,1 ∩ W ^2,2 manifold by conformal transformation under some assumptions on Weyl curvature and scalar curvature.     

Asymptotics of the Codimensions <Emphasis Type="Italic">c</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> in the Algebra <Emphasis Type="Italic">F</Emphasis><Superscript>(7)</Superscript>

The paper studies the additive structure of the algebra F⁽⁷⁾, i.e., a relatively free associative countably generated algebra with the identity [x₁,..., x₇] = 0 over an infinite field of characteristic ≠ 2, 3. First, the space of proper multilinear polynomials in this algebra is investigated. As an application, estimates for the codimensions c_n = dimF_n⁽⁷⁾ are obtained, where F_n⁽⁷⁾ stands for the subspace of multilinear polynomials of degree n in the algebra F⁽⁷⁾.     

Differences of idempotents in <Emphasis Type="Italic">C</Emphasis>*-algebras

Suppose that P and Q are idempotents on a Hilbert space H, while Q = Q* and I is the identity operator in H. If U = P ? Q is an isometry then U = U* is unitary and Q = I ? P. We establish a double inequality for the infimum and the supremum of P and Q in H and P ? Q. Applications of this inequality are obtained to the characterization of a trace and ideal F-pseudonorms on a W*-algebra. Let φ be a trace on the unital C*-algebra A and let tripotents P and Q belong to A. If P ? Q belongs to the domain of definition of φ then φ(P ? Q) is a real number. The commutativity of some operators is established.     

Non-triviality of a product in the Adams <Emphasis Type="Italic">E</Emphasis> <Subscript>2</Subscript>-term

Let p be a prime greater than five and A the mod p Steenrod algebra. In this paper, we prove that \(h_n h_m \tilde \delta _{s + 4} \in Ext_A^{s + 6,t(s,n,m) + s} (Z/p,Z/p)\) is nontrivial in the Adams E₂-term when m ≥ n + 2 ≥ 7 and 0 ≤ s < p ? 4, and trivial in the Adams E₂-term when m ≥ n + 2 = 6 and 0 ≤ s < p ? 4, where \(\tilde \delta _{s + 4} \) stands for the fourth Greek letter element and t(s, n, m) = 2(p ? 1)[(s + 1) + (s + 2)p + (s + 3)p² + (s + 4)p³ + pⁿ + p^m].