Positive solutions of singular p-Laplacian dynamic equations with sign changing nonlinearity

Let be a time scale such that . By the Schauder fixed-point theorem and the upper and lower solution method, we present some existence criteria of the positive solution of m-point singular p-Laplacian dynamic equation with boundary conditions , where φ_p(s)=|s|^p-2s with p>1, is continuous for i=1,2,…,m-1 and nonincreasing if . The nonlinear term may be singular in its dependent variable and is allowed to change sign. Our results are new even for the corresponding differential and difference equations . As an application, an example is given to illustrate our result.     

Norms of some operators on the Bergman and the Hardy space in the unit polydisk and the unit ball

Let H(X) be the class of all holomorphic functions on the set and uH(X). We calculate operator norms of the multiplication operators M_u(f)=uf, on the weighted Bergman space , as well as on the Hardy space H^p(X), where X is the unit polydisk or the unit ball in . We also calculate the norm of the weighted composition operator from the weighted Bergman space , and the Hardy space , to a weighted-type space on the unit polydisk.     

Region of variability for close-to-convex functions-II

For a complex number α with let be the class of analytic functions f in the unit disk with f(0)=0 satisfying in , for some convex univalent function in . For any fixed , and we shall determine the region of variability V(z₀,α,λ) for f(z₀) when f ranges over the class

In the final section we graphically illustrate the region of variability for several sets of parameters z₀ and α.     

Norm of pre-Schwarzian derivative for the class of k-uniformly convex and k-starlike functions

A quasiconformal extension for the class of k-uniformly convex functions, denoted , and for the class of k-starlike functions, denoted is provided. Also, estimation of the norm of pre-Schwarzian derivative in is given.     

Integral-type operators from Bloch-type spaces to Zygmund-type spaces

Let denote the space of all holomorphic functions on the unit ball . This paper investigates the following integral-type operator with symbol

where is the radial derivative of g. The boundedness and compactness of the operator T_g from Bloch-type spaces to Zygmund-type spaces are studied.     

Stationary twists and energy minimizers on a space of measure preserving maps

Let be a bounded Lipschitz domain, a suitably quasiconvex integrand and consider the energy functional

over the space of measure preserving maps

In this paper we discuss the question of existence of multiple strong local minimizers for over . Moreover, motivated by their significance in topology and the study of mapping class groups, we consider a class of maps, referred to as twists, and examine them in connection with the corresponding Euler–Lagrange equations and investigate various qualitative properties of the resulting solutions, the stationary twists. Particular attention is paid to the special case of the so-called p-Dirichlet energy, i.e., when .     

All-derivable points in continuous nest algebras

Let be an operator algebra on a Hilbert space. We say that an element is an all-derivable point of for the strong operator topology if every strong operator topology continuous derivable linear mapping φ at G (i.e. φ(ST)=φ(S)T+Sφ(T) for any with ST=G) is a derivation. Let be a continuous nest on a complex and separable Hilbert space H. We show in this paper that every orthogonal projection operator P(M) () is an all-derivable point of for the strong operator topology.     

Strong uniform continuity

Let be an ideal of subsets of a metric space X,d. This paper considers a strengthening of the notion of uniform continuity of a function restricted to members of which reduces to ordinary continuity when consists of the finite subsets of X and agrees with uniform continuity on members of when is either the power set of X or the family of compact subsets of X. The paper also presents new function space topologies that are well suited to this strengthening. As a consequence of the general theory, we display necessary and sufficient conditions for continuity of the pointwise limit of a net of continuous functions.     

Asymptotic behavior of large solution to elliptic equation of Bieberbach–Rademacher type with convection terms

We analyze the asymptotic behavior of solutions to nonlinear elliptic equation Δu±|u|^q=b(x)f(u) in Ω, subject to the singular boundary condition u(x)=∞ as , where Ω is a smooth bounded domain in R^N, for some , and . Our approach employs Karamata regular variation theory combined with the method of lower and supper solution.     

Polynomial-time approximation schemes for piercing and covering with applications in wireless networks

Let be a set of disks of arbitrary radii in the plane, and let be a set of points. We study the following three problems: (i) Assuming contains the set of center points of disks in , find a minimum-cardinality subset of (if exists), such that each disk in is pierced by at least h points of , where h is a given constant. We call this problem minimum h-piercing. (ii) Assuming is such that for each there exists a point in whose distance from D's center is at most αr(D), where r(D) is D's radius and 0α<1 is a given constant, find a minimum-cardinality subset of , such that each disk in is pierced by at least one point of . We call this problem minimum discrete piercing with cores. (iii) Assuming is the set of center points of disks in , and that each covers at most l points of , where l is a constant, find a minimum-cardinality subset of , such that each point of is covered by at least one disk of . We call this problem minimum center covering. For each of these problems we present a constant-factor approximation algorithm (trivial for problem (iii)), followed by a polynomial-time approximation scheme. The polynomial-time approximation schemes are based on an adapted and extended version of Chan's [T.M. Chan, Polynomial-time approximation schemes for packing and piercing fat objects, J. Algorithms 46 (2003) 178–189] separator theorem. Our PTAS for problem (ii) enables one, in practical cases, to obtain a (1+ε)-approximation for minimum discrete piercing (i.e., for arbitrary ).     

Semifields of order with left nucleus and center

In [G. Marino, O. Polverino, R. Trombetti, On -linear sets of PG(3,q³) and semifields, J. Combin. Theory Ser. A 114 (5) (2007) 769–788] it has been proven that there exist six non-isotopic families (i=0,…,5) of semifields of order q⁶ with left nucleus and center , according to the different geometric configurations of the associated -linear sets. In this paper we first prove that any semifield of order q⁶ with left nucleus , right and middle nuclei and center is isotopic to a cyclic semifield. Then, we focus on the family by proving that it can be partitioned into three further non-isotopic families: , , and we show that any semifield of order q⁶ with left nucleus , right and middle nuclei and center belongs to the family .     

Density results for Gabor systems associated with periodic subsets of the real line

The well-known density theorem for one-dimensional Gabor systems of the form , where , states that a necessary and sufficient condition for the existence of such a system whose linear span is dense in , or which forms a frame for , is that the density condition is satisfied. The main goal of this paper is to study the analogous problem for Gabor systems for which the window function g vanishes outside a periodic set which is -shift invariant. We obtain measure-theoretic conditions that are necessary and sufficient for the existence of a window g such that the linear span of the corresponding Gabor system is dense in L²(S). Moreover, we show that if this density condition holds, there exists, in fact, a measurable set with the property that the Gabor system associated with the same parameters a,b and the window g=χ_E, forms a tight frame for L²(S).     

A characterization of the natural embedding of the split Cayley hexagon in by intersection numbers

In this paper, we prove that a set of q⁵+q⁴+q³+q²+q+1 lines of with the properties that (1) every point of is incident with either 0 or q+1 elements of , (2) every plane of is incident with either 0, 1 or q+1 elements of , (3) every solid of is incident with either 0, 1, q+1 or 2q+1 elements of , and (4) every hyperplane of is incident with at most q³+3q²+3q members of , is necessarily the set of lines of a regularly embedded split Cayley generalized hexagon in .     

On ternary Kloosterman sums modulo 12

Let K(a) denote the Kloosterman sum on . It is easy to see that for all . We completely characterize those for which , and . The simplicity of the characterization allows us to count the number of the belonging to each of these three classes. As a byproduct we offer an alternative proof for a new class of quasi-perfect ternary linear codes recently presented by Danev and Dodunekov.     

On new Bloch-type spaces

We introduce a new Bloch-type space, so called, the logarithmic Bloch-type space on the unit disc , as the space of all holomorphic functions f on such that

where α>0 and β0, and present some basic properties of the space. A necessary and a sufficient condition for a function with Hadamard gaps to belong to the logarithmic Bloch-type space are given, as well as some applications of these results to a composition operator.     

Configurations in abelian categories. II. Ringel–Hall algebras

This is the second in a series on configurations in an abelian category . Given a finite poset (I,), an (I,)-configuration (σ,ι,π) is a finite collection of objects σ(J) and morphisms ι(J,K) or in satisfying some axioms, where J,KI. Configurations describe how an object X in decomposes into subobjects.The first paper defined configurations and studied moduli spaces of (I,)-configurations in , using the theory of Artin stacks. It showed well-behaved moduli stacks of objects and configurations in exist when is the abelian category coh(P) of coherent sheaves on a projective scheme P, or mod- of representations of a quiver Q.Write for the vector space of -valued constructible functions on the stack . Motivated by the idea of Ringel–Hall algebras, we define an associative multiplication * on using pushforwards and pullbacks along 1-morphisms between configuration moduli stacks, so that is a -algebra. We also study representations of , the Lie subalgebra of functions supported on indecomposables, and other algebraic structures on .Then we generalize all these ideas to stack functions , a universal generalization of constructible functions, containing more information. When Extⁱ(X,Y)=0 for all and i>1, or when for P a Calabi–Yau 3-fold, we construct (Lie) algebra morphisms from stack algebras to explicit algebras, which will be important in the sequels on invariants counting τ-semistable objects in .     

Limits of small functors

For a small category enriched over a suitable monoidal category , the free completion of under colimits is the presheaf category . If is large, its free completion under colimits is the -category of small presheaves on , where a presheaf is small if it is a left Kan extension of some presheaf with small domain. We study the existence of limits and of monoidal closed structures on .     

Integral-type operators from iterated logarithmic Bloch spaces to Zygmund-type spaces

Let denote the space of all holomorphic functions on the unit ball of and the radial derivative of h. In this paper we study the boundedness and compactness of the following integral operator:, from iterated logarithmic Bloch spaces to Zygmund-type spaces.     

Levels of multi-continued fraction expansion of multi-formal Laurent series

The multi-continued fraction expansion of a multi-formal Laurent series is a sequence pair consisting of an index sequence and a multi-polynomial sequence . We denote the set of the different indices appearing infinitely many times in by H_∞, the set of the different indices appearing in by H₊, and call |H_∞| and |H₊| the first and second levels of , respectively. In this paper, it is shown how the dimension and basis of the linear space over F(z) (F) spanned by the components of are determined by H_∞ (H₊), and how the components are linearly dependent on the mentioned basis.     

The maximum edit distance from hereditary graph properties

For a graph property , the edit distance of a graph G from , denoted , is the minimum number of edge modifications (additions or deletions) one needs to apply to G in order to turn it into a graph satisfying . What is the largest possible edit distance of a graph on n vertices from ? Denote this distance by .A graph property is hereditary if it is closed under removal of vertices. In a previous work, the authors show that for any hereditary property, a random graph essentially achieves the maximal distance from , proving: with high probability. The proof implicitly asserts the existence of such , but it does not supply a general tool for determining its value or the edit distance.In this paper, we determine the values of and for some subfamilies of hereditary properties including sparse hereditary properties, complement invariant properties, (r,s)-colorability and more. We provide methods for analyzing the maximum edit distance from the graph properties of being induced H-free for some graphs H, and use it to show that in some natural cases G(n,1/2) is not the furthest graph. Throughout the paper, the various tools let us deduce the asymptotic maximum edit distance from some well studied hereditary graph properties, such as being Perfect, Chordal, Interval, Permutation, Claw-Free, Cograph and more. We also determine the edit distance of G(n,1/2) from any hereditary property, and investigate the behavior of as a function of p.The proofs combine several tools in Extremal Graph Theory, including strengthened versions of the Szemerédi Regularity Lemma, Ramsey Theory and properties of random graphs.