Identities for the Widder transform and transforms with Bessel function kernels

In the present paper, we consider the classical Widder transform, the H_ν-transform, the K_ν-transform, and the Y_ν-transform. Some identities involving these transforms and many others are given. By making use of these identities, a number of new Parseval-Goldstein type identities are obtained for these and other well-known integral transforms.     

Noncommutative maximal ergodic theorems for positive contractions

Let M be a von Neumann algebra equipped with a normal semifinite faithful trace τ. Let T be a positive linear contraction on M such that τ○T?τ and such that the numerical range of T as an operator on L₂(M) is contained in a Stoltz region with vertex 1. We show that Junge and Xu's noncommutative Stein maximal ergodic inequality holds for the powers of T on Lp(M), 1<p?∞. We apply this result to obtain the noncommutative analogue of a recent result of Cohen concerning the iterates of the product of a finite number of conditional expectations.     

Filling polygonal holes with minimal energy surfaces on Powell-Sabin type triangulations

In this paper we present two different methods for filling in a hole in an explicit 3D surface, defined by a smooth function f in a part of a polygonal domain D⊂R². We obtain the final reconstructed surface over the whole domain D. We do the filling in two different ways: discontinuous and continuous. In the discontinuous case, we fill the hole with a function in a Powell-Sabin spline space that minimizes a linear combination of the usual seminorms in an adequate Sobolev space, and approximates (in the least squares sense) the values of f and those of its normal derivatives at an adequate set of points. In the continuous case, we will first replace f outside the hole by a smoothing bivariate spline s_f, and then we fill the hole also with a Powell-Sabin spline minimizing a linear combination of given seminorms. In both cases, we obtain existence and uniqueness of solutions and we present some graphical examples, and, in the continuous case, we also give a local convergence result.     

Labeling graphs with two distance constraints

Given a graph G and integers p,q,d₁ and d₂, with p>q, d₂>d₁?1, an L(d₁,d₂;p,q)-labeling of G is a function f:V(G)→{0,1,2,…,n} such that |f(u)−f(v)|?p if d_G(u,v)?d₁ and |f(u)−f(v)|?q if d_G(u,v)?d₂. A k-L(d₁,d₂;p,q)-labeling is an L(d₁,d₂;p,q)-labeling f such that max_v∈V(G)f(v)?k. The L(d₁,d₂;p,q)-labeling number ofG, denoted by , is the smallest number k such that G has a k-L(d₁,d₂;p,q)-labeling. In this paper, we give upper bounds and lower bounds of the L(d₁,d₂;p,q)-labeling number for general graphs and some special graphs. We also discuss the L(d₁,d₂;p,q)-labeling number of G, when G is a path, a power of a path, or Cartesian product of two paths.     

Endomorphisms of the shift dynamical system, discrete derivatives, and applications

All continuous endomorphisms f_∞ of the shift dynamical system S on the 2-adic integers Z₂ are induced by some , where n is a positive integer, B_n is the set of n-blocks over {0, 1}, and f_∞(x)=y₀y₁y₂… where for all i∈N, y_i=f(x_ix_i+1…x_i+n−1). Define D:Z₂→Z₂ to be the endomorphism of S induced by the map {(00,0),(01,1),(10,1),(11,0)} and V:Z₂→Z₂ by V(x)=−1−x. We prove that D, V°D, S, and V°S are conjugate to S and are the only continuous endomorphisms of S whose parity vector function is solenoidal. We investigate the properties of D as a dynamical system, and use D to construct a conjugacy from the 3x+1 function T:Z₂→Z₂ to a parity-neutral dynamical system. We also construct a conjugacy R from D to T. We apply these results to establish that, in order to prove the 3x+1 conjecture, it suffices to show that for any m∈Z⁺, there exists some n∈N such that R⁻¹(m) has binary representation of the form or .     

Positive Quaternion Kähler Manifolds with fourth Betti number equal to one

Positive Quaternion Kähler Manifolds are Riemannian manifolds with holonomy contained in Sp(n)Sp(1) and with positive scalar curvature. Conjecturally, they are symmetric spaces. In this article we are mainly concerned with Positive Quaternion Kähler Manifolds M satisfying b₄(M)=1. Generalising a result of Galicki and Salamon we prove that M4n in this case is homothetic to a quaternionic projective space if 2≠n?6.     

On symmetric graphs of valency five

A graph X, with a subgroup G of the automorphism group of X, is said to be (G,s)-transitive, for some s≥1, if G is transitive on s-arcs but not on (s+1)-arcs, and s-transitive if it is -transitive. Let X be a connected (G,s)-transitive graph, and G_v the stabilizer of a vertex v∈V(X) in G. If X has valency 5 and G_v is solvable, Weiss [R.M. Weiss, An application of p-factorization methods to symmetric graphs, Math. Proc. Camb. Phil. Soc. 85 (1979) 43-48] proved that s≤3, and in this paper we prove that G_v is isomorphic to the cyclic group Z₅, the dihedral group D₁₀ or the dihedral group D₂₀ for s=1, the Frobenius group F₂₀ or F₂₀×Z₂ for s=2, or F₂₀×Z₄ for s=3. Furthermore, it is shown that for a connected 1-transitive Cayley graph of valency 5 on a non-abelian simple group G, the automorphism group of is the semidirect product , where R(G) is the right regular representation of G and .     

On certain spaces of vector measures of bounded variation

If (Σ,X) is a measurable space and X a Banach space we investigate the X-inheritance of copies of ?_∞ in certain subspaces Δ(Σ,X) of bvca(Σ,X), the Banach space of all X-valued countable additive measures of bounded variation equipped with the variation norm. Among the consequences of our main theorem we get a theorem of J. Mendoza on the X-inheritance of copies of ?_∞ in the Bochner space L₁(μ,X) and other of the author on the X-inheritance of copies of ?_∞ in bvca(Σ,X).     

More on “Connected (n, m)-graphs with minimum and maximum zeroth-order general Randi? index”

Let G be a graph and d(u) denote the degree of a vertex u in G. The zeroth-order general Randi? index ⁰R_α(G) of the graph G is defined as ∑_u∈V(G)d(u)^α, where the summation goes over all vertices of G and α is an arbitrary real number. In this paper we correct the proof of the main Theorem 3.5 of the paper by Hu et al. [Y. Hu, X. Li, Y. Shi, T. Xu, Connected (n,m)-graphs with minimum and maximum zeroth-order general Randi? index, Discrete Appl. Math. 155 (8) (2007) 1044-1054] and give a more general Theorem. We finally characterize ¹ for α<0 the connected G(n,m)-graphs with maximum value ⁰R_α(G(n,m)), where G(n,m) is a simple connected graph with n vertices and m edges.     

Eventual differentiability of functional differential equations in Banach spaces

This paper concerns the regularity of a functional differential equation in the form: , t>0, where A is the generator of an analytic semigroup on a Banach space X, and B₁,B₂ are α(γ−A)-bounded linear operator for 0<α<1. By spectral analysis, it is shown that the associated solution semigroup of this equation is eventually differentiable.     

On time-relaxed broadcasting networks

Given a communication network (modelled as a graph), a message is transmitted to at one vertex transmits to all other vertices in such a way that each message transmission takes one time unit and each vertex participates in at most one transmission to its neighbor per time step. We call this process broadcasting. For t≥0, let B_t(n) be the number of edges in the sparsest possible graph on n vertices in which broadcasting can be accomplished in ⌈log₂n⌉+t steps regardless of the originator. Shastri [A. Shastri, Time-relaxed broadcasting in communication networks, Discrete Applied mathematics 83 (1998) 263-278] conjectured that B₁(22)=24 and B₂(n)=n+1 for 25≤n≤29. In this paper, we show that B₁(22)=24, B₂(n)=n for 25≤n≤28 and 37≤n≤44, B₂(n)≤n+1 for 45≤n≤49, B₂(n)≤n+4 for 50≤n≤56, and B₃(n)=n for 55≤n≤64.     

Linear-time certifying algorithms for near-graphical sequences

A sequence 〈d₁,d₂,…,d_n〉 of non-negative integers is graphical if it is the degree sequence of some graph, that is, there exists a graph G on n vertices whose ith vertex has degree d_i, for 1≤i≤n. The notion of a graphical sequence has a natural reformulation and generalization in terms of factors of complete graphs.If H=(V,E) is a graph and g and f are integer-valued functions on the vertex set V, then a (g,f)-factor of H is a subgraph G=(V,F) of H whose degree at each vertex v∈V lies in the interval [g(v),f(v)]. Thus, a (0,1)-factor is just a matching of H and a (1, 1)-factor is a perfect matching of H. If H is complete then a (g,f)-factor realizes a degree sequence that is consistent with the sequence of intervals 〈[g(v₁),f(v₁)],[g(v₂),f(v₂)],…,[g(v_n),f(v_n)]〉.Graphical sequences have been extensively studied and admit several elegant characterizations. We are interested in extending these characterizations to non-graphical sequences by introducing a natural measure of “near-graphical”. We do this in the context of minimally deficient (g,f)-factors of complete graphs. Our main result is a simple linear-time greedy algorithm for constructing minimally deficient (g,f)-factors in complete graphs that generalizes the method of Hakimi and Havel (for constructing (f,f)-factors in complete graphs, when possible). It has the added advantage of producing a certificate of minimum deficiency (through a generalization of the Erdös-Gallai characterization of (f,f)-factors in complete graphs) at no additional cost.     

A unique decomposition result for HT factors with torsion free core

We prove that II₁ factors M have a unique (up to unitary conjugacy) cross-product type decomposition around “core subfactors” N⊂M satisfying the property HT of [S. Popa, On a class of type II₁ factors with Betti numbers invariants, Ann. of Math. (2) 163 (2006) 809-899] and a certain “torsion freeness” condition. In particular, this shows that isomorphism of factors of the form Lαi(Z²)?Fni, i=1,2, for Fni⊂SL(2,Z) free groups of rank ni and αj=e2πitj, tj∉Q, implies n₁=n₂.     

Absolutely convergent Fourier series and function classes

We study the smoothness property of a function f with absolutely convergent Fourier series, and give best possible sufficient conditions in terms of its Fourier coefficients to ensure that f belongs either to one of the Lipschitz classes Lip(α) and lip(α) for some 0<α?1, or to one of the Zygmund classes Λ_∗(1) and λ_∗(1). Our theorems generalize some of those by Boas [R.P. Boas Jr., Fourier series with positive coefficients, J. Math. Anal. Appl. 17 (1967) 463-483] and one by Németh [J. Németh, Fourier series with positive coefficients and generalized Lipschitz classes, Acta Sci. Math. (Szeged) 54 (1990) 291-304]. We also prove a localized version of a theorem by Paley [R.E.A.C. Paley, On Fourier series with positive coefficients, J. London Math. Soc. 7 (1932) 205-208] on the existence and continuity of the derivative of f.     

Well-posedness of a class of perturbed optimization problems in Banach spaces

Let X be a Banach space and Z a nonempty subset of X. Let J:Z→R be a lower semicontinuous function bounded from below and p?1. This paper is concerned with the perturbed optimization problem of finding z₀∈Z such that ‖x−z₀p^‖+J(z₀)=infz∈Z{‖x−zp^‖+J(z)}, which is denoted by minJ(x,Z). The notions of the J-strictly convex with respect to Z and of the Kadec with respect to Z are introduced and used in the present paper. It is proved that if X is a Kadec Banach space with respect to Z and Z is a closed relatively boundedly weakly compact subset, then the set of all x∈X for which every minimizing sequence of the problem minJ(x,Z) has a converging subsequence is a dense Gδ-subset of X?Z₀, where Z₀ is the set of all points z∈Z such that z is a solution of the problem minJ(z,Z). If additionally p>1 and X is J-strictly convex with respect to Z, then the set of all x∈X for which the problem minJ(x,Z) is well-posed is a dense Gδ-subset of X?Z₀.     

Constructions for cyclic Moebius ladder systems

J.A. Gallian has proved [J.A. Gallian, Labeling prisms and prism related graphs, Congr. Numer. 59 (1987) 89-100] that every cubic graph M_2k obtainable from a 2k-cycle by adding its k diameters (the so-called Moebius Ladder of order 2k) is graceful. Here, in the case of k even, we propose a new graceful labeling that besides being simpler than Gallian’s one is able to give, at the same time, a graceful labeling of the prism of order 2k. Most importantly in the case of k odd, namely in the bipartite case, we prove that M_2k also admits an α-labeling. This implies that there exists a cyclic decomposition of the complete graph K_6kt+1 into copies of M_2k for every pair of positive integers k and t with k odd.In some cases we are able to give such decompositions also when k is even. Apart from the case of t=1 that is an obvious consequence of the gracefulness of M_2k, this happens, for instance, when k≡2 (mod 4) and 6kt+1 is a prime.     

Least and most colored bases

Consider a matroid M=(E,B), where B denotes the family of bases of M, and assign a color c(e) to every element e∈E (the same color can go to more than one element). The palette of a subset F of E, denoted by c(F), is the image of F under c. Assume also that colors have prices (in the form of a function π(?), where ? is the label of a color), and define the chromatic price as: π(F)=∑_?∈c(F)π(?). We consider the following problem: find a base B∈B such that π(B) is minimum. We show that the greedy algorithm delivers a lnr(M)-approximation of the unknown optimal value, where r(M) is the rank of matroid M. By means of a reduction from SETCOVER, we prove that the lnr(M) ratio cannot be further improved, even in the special case of partition matroids, unless . The results apply to the special case where M is a graphic matroid and where the prices π(?) are restricted to be all equal. This special case was previously known as the minimum label spanning tree (MLST) problem. For the MLST, our results improve over the ln(n-1)+1 ratio achieved by Wan, Chen and Xu in 2002. Inspired by the generality of our results, we study the approximability of coloring problems with different objective function π(F), where F is a common independent set on matroids M₁,…,M_k and, more generally, to independent systems characterized by the k-for-1 property.     

Delay-dependent stability of symmetric schemes in Boundary Value Methods for DDEs

We consider the symmetric schemes in Boundary Value Methods (BVMs) applied to delay differential equations y^′(t)=ay(t)+by(t-τ) with real coefficients a and b. If the numerical solution tends to zero whenever the exact solution does, the symmetric scheme with (k₁+m,k₂)-boundary conditions is called τ_k1,k2(0)-stable. Three families of symmetric schemes, namely the Extended Trapezoidal Rules of first (ETRs) and second (ETR₂s) kind, and the Top Order Methods (TOMs), are considered in this paper.By using the boundary locus technology, the delay-dependent stability region of the symmetric schemes are analyzed and their boundaries are found. Then by using a necessary and sufficient condition, the considered symmetric schemes are proved to be τ_ν,ν-1(0)-stable.     

A distance-labelling problem for hypercubes

Let i₁≥i₂≥i₃≥1 be integers. An L(i₁,i₂,i₃)-labelling of a graph G=(V,E) is a mapping ?:V→{0,1,2,…} such that |?(u)−?(v)|≥i_t for any u,v∈V with d(u,v)=t, t=1,2,3, where d(u,v) is the distance in G between u and v. The integer ?(v) is called the label assigned to v under ?, and the difference between the largest and the smallest labels is called the span of ?. The problem of finding the minimum span, λ_i1,i2,i3(G), over all L(i₁,i₂,i₃)-labellings of G arose from channel assignment in cellular communication systems, and the related problem of finding the minimum number of labels used in an L(i₁,i₂,i₃)-labelling was originated from recent studies on the scalability of optical networks. In this paper we study the L(i₁,i₂,i₃)-labelling problem for hypercubes Q_d (d≥3) and obtain upper and lower bounds on λ_i1,i2,i3(Q_d) for any (i₁,i₂,i₃).     

Constructions of generalized Sidon sets

We give explicit constructions of sets S with the property that for each integer k, there are at most g solutions to k=s₁+s₂,s_i∈S; such sets are called Sidon sets if g=2 and generalized Sidon sets if g?3. We extend to generalized Sidon sets the Sidon-set constructions of Singer, Bose, and Ruzsa. We also further optimize Kolountzakis’ idea of interleaving several copies of a Sidon set, extending the improvements of Cilleruelo, Ruzsa and Trujillo, Jia, and Habsieger and Plagne. The resulting constructions yield the largest known generalized Sidon sets in virtually all cases.