Highly connected coloured subgraphs via the regularity lemma

For integers , n≥k and r≥s, let m(n,r,s,k) be the largest (in order) k-connected component with at most s colours one can find in any r-colouring of the edges of the complete graph K_n on n vertices. Bollobás asked for the determination of m(n,r,s,k).Here, bounds are obtained in the cases s=1,2 and k=o(n), which extend results of Liu, Morris and Prince. Our techniques use Szemerédi’s Regularity Lemma for many colours.We shall also study a similar question for bipartite graphs.     

Rates of -statistical convergence of positive linear operators

In this paper we study the rates of A-statistical convergence of sequences of positive linear operators mapping the weighted space C_ρ1 into the weighted space B_ρ2.     

Norm of weighted composition operators from -Bloch spaces to weighted-type spaces

We calculate in an elegant way operator norm of the weighted composition operator from the α-Bloch space, with α(0,∞){1}, to a weighted-type space on the unit ball. This result can be regarded as a complement to our recent result regarding the same problem for the case α=1.     

Minimum degree conditions for -linked graphs

For a fixed multigraph H with vertices w₁,…,w_m, a graph G is H-linked if for every choice of vertices v₁,…,v_m in G, there exists a subdivision of H in G such that v_i is the branch vertex representing w_i (for all i). This generalizes the notions of k-linked, k-connected, and k-ordered graphs.Given a connected multigraph H with k edges and minimum degree at least two and n7.5k, we determine the least integer d such that every n-vertex simple graph with minimum degree at least d is H-linked. This value D(H,n) appears to equal the least integer d^′ such that every n-vertex graph with minimum degree at least d^′ is b(H)-connected, where b(H) is the maximum number of edges in a bipartite subgraph of H.     

Index theory for boundary value problems via continuous fields of -algebras

We prove an index theorem for boundary value problems in Boutet de Monvel's calculus on a compact manifold X with boundary. The basic tool is the tangent semi-groupoid generalizing the tangent groupoid defined by Connes in the boundaryless case, and an associated continuous field of C^*-algebras over [0,1]. Its fiber in =0, , can be identified with the symbol algebra for Boutet de Monvel's calculus; for ≠0 the fibers are isomorphic to the algebra of compact operators. We therefore obtain a natural map . Using deformation theory we show that this is the analytic index map. On the other hand, using ideas from noncommutative geometry, we construct the topological index map and prove that it coincides with the analytic index map.     

Finitely additive representation of spaces

Let be any atomless and countably additive probability measure on the product space with the usual σ-algebra. Then there is a purely finitely additive probability measure λ on the power set of a countable subset such that can be isometrically isomorphically embedded as a closed subspace of L^p(λ). The embedding is strict. It is also ‘canonical,’ in the sense that it maps simple and continuous functions on to their restrictions to T.     

Spectral theory for algebraic combinations of Toeplitz and composition operators

We determine the essential spectra of algebraic combinations of Toeplitz operators with continuous symbol and composition operators induced by a class of linear-fractional non-automorphisms of the unit disk. The operators in question act on the Hardy space H² on the unit disk. Our method is to realize the C^*-algebra that they generate as an extension of the compact operators by a concrete C^*-algebra whose invertible elements are easily characterized.     

Three solutions for a differential inclusion problem involving the -Laplacian

In this paper we consider differential inclusion problem involving the p(x)-Laplacian of the type

Applying a version of the non-smooth three-critical-points theorem we obtain the existence of three solutions of the problem in .     

-adic refinable functions and MRA-based wavelets

The main goal of this paper is the development of the MRA theory in . We described a wide class of p-adic refinement equations generating p-adic multiresolution analyses. A method for the construction of p-adic orthogonal wavelet bases within the framework of the MRA theory is suggested. A realization of this method is illustrated by an example which gives a new 3-adic wavelet basis. Another realization leads to the p-adic Haar bases which were known before.     

Corrigendum to “Extending homeomorphisms from punctured surfaces to handlebodies” [Topology Appl. 155 (6) (2008) 610–621]

We correct the statements of Theorems 9 and 10 of [A. Cattabriga, M. Mulazzani, Extending homeomorphisms from punctured surfaces to handlebodies, Topology Appl. 155 (2008) 610–621], by adding missing generators, and improve the statement of Theorem 10, by removing some redundant generators.     

Infinitely many non-negative solutions for a Dirichlet problem involving -Laplacian

In this paper, we consider a Dirichlet problem involving the p(x)-Laplacian of the type

We prove the existence of infinitely many non-negative solutions of the problem by applying a general variational principle due to B. Ricceri and the theory of the variable exponent Sobolev spaces.     

Strongly indexable graphs and applications

In 1990, Acharya and Hegde introduced the concept of strongly k-indexable graphs: A (p,q)-graph G=(V,E) is said to be strongly k-indexable if its vertices can be assigned distinct numbers 0,1,2,…,p−1 so that the values of the edges, obtained as the sums of the numbers assigned to their end vertices form an arithmetic progression k,k+1,k+2,…,k+(q−1). When k=1, a strongly k-indexable graph is simply called a strongly indexable graph. In this paper, we report some results on strongly k-indexable graphs and give an application of strongly k-indexable graphs to plane geometry, viz; construction of polygons of same internal angles and sides of distinct lengths.     

Reduced free products of finite dimensional -algebras

We find a necessary and sufficient conditions for the simplicity and uniqueness of trace for reduced free products of finite families of finite dimensional C^*-algebras with specified traces on them.     

Completions of -algebras

A μ-algebra is a model of a first-order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms (f,μ_x.f) where μ_x.f is axiomatized as the least prefixed point of f, whose axioms are equations or equational implications.Standard μ-algebras are complete meaning that their lattice reduct is a complete lattice. We prove that any nontrivial quasivariety of μ-algebras contains a μ-algebra that has no embedding into a complete μ-algebra.We then focus on modal μ-algebras, i.e. algebraic models of the propositional modal μ-calculus. We prove that free modal μ-algebras satisfy a condition–reminiscent of Whitman’s condition for free lattices–which allows us to prove that (i) modal operators are adjoints on free modal μ-algebras, (ii) least prefixed points of Σ₁-operations satisfy the constructive relation μ_x.f=_n≥0fⁿ(). These properties imply the following statement: the MacNeille–Dedekind completion of a free modal μ-algebra is a complete modal μ-algebra and moreover the canonical embedding preserves all the operations in the class of the fixed point alternation hierarchy.     

Four and more

We isolate several large classes of definable proper forcings and show how they include many partial orderings used in practice.     

Accelerating gradient projection methods for -constrained signal recovery by steplength selection rules

We propose a new gradient projection algorithm that compares favorably with the fastest algorithms available to date for ℓ₁-constrained sparse recovery from noisy data, both in the compressed sensing and inverse problem frameworks. The method exploits a line-search along the feasible direction and an adaptive steplength selection based on recent strategies for the alternation of the well-known Barzilai–Borwein rules. The convergence of the proposed approach is discussed and a computational study on both well conditioned and ill-conditioned problems is carried out for performance evaluations in comparison with five other algorithms proposed in the literature.     

Stability analysis of the zero solution for two class evolution equations

By constructing some suitable Lyapunov-type functionals and applying the theory of the definite-quadratic form, we obtain the stability of the zero solution of two classes of evolution equations with delays, including reaction–diffusion equations and damped wave equations. Our criteria depend on the derivatives of delays. Consequently, when the delays are constants, these criteria are independent of the magnitudes of the delays, so the delays are harmless for the stability of the zero solution.     

Anti-periodic solutions for a class of nonlinear th-order differential equations with delays

In this paper, we use the Leray–Schauder degree theory to establish new results on the existence and uniqueness of anti-periodic solutions for a class of nonlinear nth-order differential equations with delays of the form

x⁽ⁿ⁾(t)+f(t,x⁽ⁿ⁻¹⁾(t))+g(t,x(t−τ(t)))=e(t).