Decomposition theorems for<Emphasis Type="Italic">Q</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> spaces

We study the Möbius invariant spacesQ _p andQ _{p, 0} of analytic functions. These scales of spaces include BMOA=Q₁, VMOA=Q_{1, 0} and the Dirichlet space=Q₀. Using the Bergman metric, we establish decomposition theorems for these spaces. We obtain also a fractional derivative characterization for bothQ _p andQ _{p, 0} .     

Strictly singular operators in pairs of <Emphasis Type="Italic">L</Emphasis> <Subscript> <Emphasis Type="Italic">p</Emphasis> </Subscript> space

Let E and F be Banach spaces. A linear operator from E to F is said to be strictly singular if, for any subspace Q ? E, the restriction of A to Q is not an isomorphism. A compactness criterion for any strictly singular operator from L_p to L_q is found. There exists a strictly singular but not superstrictly singular operator on L_p, provided that p ≠ 2.     

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.     

Theory of (<Emphasis Type="Italic">q</Emphasis> <Subscript>1</Subscript>, <Emphasis Type="Italic">q</Emphasis> <Subscript>2</Subscript>)-quasimetric spaces and coincidence points

We introduce (q₁, q₂)-quasimetric spaces and examine their properties. Covering mappings between (q₁, q₂)-quasimetric spaces are investigated. Sufficient conditions for the existence of a coincidence point of two mappings acting between (q₁, q₂)-quasimetric spaces such that one is a covering mapping and the other satisfies the Lipschitz condition are obtained.     

Volterra type integral operators with homogeneous kernels in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We consider multidimensional integral Volterra type operators with kernels homogeneous of degree (?n); the operators act in L _p -spaces with a submultiplicative weight. For these operators we obtain necessary and sufficient conditions of their invertibility. Besides, we describe the Banach algebra generated by the operators. For this algebra we construct the symbolic calculus, in terms of which we obtain an invertibility criterion of the operators.     

Coincidence points of multivalued mappings in (<Emphasis Type="Italic">q</Emphasis><Subscript>1</Subscript>, <Emphasis Type="Italic">q</Emphasis><Subscript>2</Subscript>)-quasimetric spaces

The properties of (q₁, q₂)-quasimetric spaces are examined. Multivalued covering mappings between (q₁, q₂)-quasimetric spaces are investigated. Given two multivalued mappings between (q₁, q₂)-quasimetric spaces such that one of them is covering and the other satisfies the Lipschitz condition, sufficient conditions for these mappings to have a coincidence point are obtained. A theorem on the stability of coincidence points with respect to small perturbations in the considered mappings is proved.     

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.     

Conical square function estimates and functional calculi for perturbed Hodge-Dirac operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">P</Emphasis></Superscript>

Perturbed Hodge-Dirac operators and their holomorphic functional calculi, as investigated in the papers by Axelsson, Keith and the second author, provided insight into the solution of the Kato square-root problem for elliptic operators in L² spaces and allowed for an extension of these estimates to other systems with applications to non-smooth boundary value problems. In this paper, we determine conditions under which such operators satisfy conical square function estimates in a range of L ^p spaces, thus allowing us to apply the theory of Hardy spaces associated with an operator to prove that they have a bounded holomorphic functional calculus in those L ^p spaces. We also obtain functional calculus results for restrictions to certain subspaces, for a larger range of p. This provides a framework for obtaining L ^p results on perturbed Hodge Laplacians, generalising known Riesz transform bounds for an elliptic operator L with bounded measurable coefficients, one Sobolev exponent below the Hodge exponent, and L ^p bounds on the square-root of L by the gradient, two Sobolev exponents below the Hodge exponent. Our proof shows that the heart of the harmonic analysis in L² extends to L ^p for all p ∈ (1,∞), while the restrictions in p come from the operator-theoretic part of the L² proof. In the course of our work, we obtain some results of independent interest about singular integral operators on tent spaces and about the relationship between conical and vertical square functions.     

<Emphasis Type="Italic">q</Emphasis>-Frequent hypercyclicity in spaces of operators

We provide conditions for a linear map of the form \(C_{R,T}(S)=RST\) to be q-frequently hypercyclic on algebras of operators on separable Banach spaces. In particular, if R is a bounded operator satisfying the q-frequent hypercyclicity criterion, then the map \(C_{R}(S)=RSR^*\) is shown to be q-frequently hypercyclic on the space \(\mathcal {K}(H)\) of all compact operators and the real topological vector space \(\mathcal {S}(H)\) of all self-adjoint operators on a separable Hilbert space H. Further we provide a condition for \(C_{R,T}\) to be q-frequently hypercyclic on the Schatten von Neumann classes \(S_p(H)\). We also characterize frequent hypercyclicity of \(C_{M^*_\varphi ,M_\psi }\) on the trace-class of the Hardy space, where the symbol \(M_\varphi \) denotes the multiplication operator associated to \(\varphi \).     

<Emphasis Type="Italic">M</Emphasis> <Subscript> <Emphasis Type="Italic">p</Emphasis> </Subscript>-supplemented subgroups of finite groups

A subgroup K of G is M_p-supplemented in G if there exists a subgroup B of G such that G = KB and TB < G for every maximal subgroup T of K with |K: T| = p^α. In this paper we prove the following: Let p be a prime divisor of |G| and let H be ap-nilpotent subgroup having a Sylow p-subgroup of G. Suppose that H has a subgroup D with D_p ≠ 1 and |H: D| = p_α. Then G is p-nilpotent if and only if every subgroup T of H with |T| = |D| is M_p-supplemented in G and N_G(T_p)/C_G(T_p) is a p-group.     

Counting extreme <Emphasis Type="Italic">U</Emphasis><Subscript>1</Subscript> matrices and characterizing quadratic doubly stochastic operators

The U ₁ matrix and extreme U ₁ matrix were successfully used to study quadratic doubly stochastic operators by R. Ganikhodzhaev and F. Shahidi [Linear Algebra Appl., 2010, 432: 24–35], where a necessary condition for a U ₁ matrix to be extreme was given. S. Yang and C. Xu [Linear Algebra Appl., 2013, 438: 3905–3912] gave a necessary and sufficient condition for a symmetric nonnegative matrix to be an extreme U ₁ matrix and investigated the structure of extreme U ₁ matrices. In this paper, we count the number of the permutation equivalence classes of the n × n extreme U ₁ matrices and characterize the structure of the quadratic stochastic operators and the quadratic doubly stochastic operators.     

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 a new property of <Emphasis Type="Italic">n</Emphasis>-poised and <Emphasis Type="Italic">G</Emphasis><Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> sets

In this paper we consider n-poised planar node sets, as well as more special ones, called G C _n sets. For the latter sets each n-fundamental polynomial is a product of n linear factors as it always holds in the univariate case. A line ? is called k-node line for a node set \(\mathcal X\) if it passes through exactly k nodes. An (n + 1)-node line is called maximal line. In 1982 M. Gasca and J. I. Maeztu conjectured that every G C _n set possesses necessarily a maximal line. Till now the conjecture is confirmed to be true for n ≤ 5. It is well-known that any maximal line M of \(\mathcal X\) is used by each node in \(\mathcal X\setminus M, \)meaning that it is a factor of the fundamental polynomial. In this paper we prove, in particular, that if the Gasca-Maeztu conjecture is true then any n-node line of G C _n set \(\mathcal {X}\) is used either by exactly \(\binom {n}{2}\) nodes or by exactly \(\binom {n-1}{2}\) nodes. We prove also similar statements concerning n-node or (n ? 1)-node lines in more general n-poised sets. This is a new phenomenon in n-poised and G C _n sets. At the end we present a conjecture concerning any k-node line.     

Influence of <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-supplemented subgroups on the structure of <Emphasis Type="Italic">p</Emphasis>-modular subgroups

A subgroup K of G is M _p -supplemented in G if there exists a subgroup B of G such that G = KB and TB < G for every maximal subgroup T of K with |K: T| = p ^α. We study the structure of the chief factor of G by using M _p -supplemented subgroups and generalize the results of Monakhov and Shnyparkov by involving the relevant results about the p-modular subgroup O ^p (G) of G.     

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">g</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis></Subscript>-colorings of nearly bipartite graphs

Let G be a simple graph, let d(v) denote the degree of a vertex v and let g be a nonnegative integer function on V (G) with 0 ≤ g(v) ≤ d(v) for each vertex v ∈ V (G). A g _c -coloring of G is an edge coloring such that for each vertex v ∈ V (G) and each color c, there are at least g(v) edges colored c incident with v. The g _c -chromatic index of G, denoted by χ′g _c (G), is the maximum number of colors such that a gc-coloring of G exists. Any simple graph G has the g _c -chromatic index equal to δ _g (G) or δ _g (G) ? 1, where \({\delta _g}\left( G \right) = \mathop {\min }\limits_{v \in V\left( G \right)} \left\lfloor {d\left( v \right)/g\left( v \right)} \right\rfloor \). A graph G is nearly bipartite, if G is not bipartite, but there is a vertex u ∈ V (G) such that G ? u is a bipartite graph. We give some new sufficient conditions for a nearly bipartite graph G to have χ′g _c (G) = δ _g (G). Our results generalize some previous results due to Wang et al. in 2006 and Li and Liu in 2011.     

Strict <Emphasis Type="Italic">K</Emphasis>-monotonicity and <Emphasis Type="Italic">K</Emphasis>-order continuity in symmetric spaces

This paper is devoted to strict K-monotonicity and K-order continuity in symmetric spaces. Using a local approach to the geometric structure in a symmetric space E we investigate a connection between strict K-monotonicity and global convergence in measure of a sequence of the maximal functions. Next, we solve an essential problem whether an existence of a point of K-order continuity in a symmetric space E on \([0,\infty )\) implies that the embedding \(E\hookrightarrow {L^1}[0,\infty )\) does not hold. We present a complete characterization of an equivalent condition to K-order continuity in a symmetric space E using a notion of order continuity and the fundamental function of E. We also investigate a relationship between strict K-monotonicity and K-order continuity in symmetric spaces and show some examples of Lorentz spaces and Marcinkiewicz spaces having these properties or not. Finally, we discuss a local version of a crucial correspondence between order continuity and the Kadec–Klee property for global convergence in measure in a symmetric space E.     

Approximating the Distribution of Customers in <Emphasis Type="Italic">M/E</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript><Emphasis Type="Italic">/s</Emphasis> Queues

Approximation formulae are suggested for the mean and variance of customers in M/E _n /s queues. It is shown that the distributions can be approximated by using the mean and variance to fit Gamma functions. A brief comment on the more general E _m /E _n /s case is given.