Decay properties of linear thermoelastic plates: Cattaneo versus Fourier law

In this article, we investigate the decay properties of the linear thermoelastic plate equations in the whole space for both Fourier and Cattaneo's laws of heat conduction. We point out that while the paradox of infinite propagation speed inherent in Fourier's law is removed by changing to the Cattaneo law, the latter always leads to a loss of regularity of the solution. The main tool used to prove our results is the energy method in the Fourier space together with some integral estimates. We prove the decay estimates for initial data U ₀?∈?H ^s (?)?∩?L ¹(?). In addition, by restricting the initial data to U ₀?∈?H ^s (?)?∩?L ^1,γ(?) and γ?∈?[0,?1], we can derive faster decay estimates with the decay rate improvement by a factor of t ^?γ/2.     

Relative distance between two scalar fields. Application to mathematical modelling approximation

Inspired in the relative error between two quantities, we define a functional Δ_D that operates on two non‐negative scalar fields, which are integrable in a given open connected set D. We prove that Δ_D defines a metric, but not an ultrametric nor a translation invariant metric. We show how to apply Δ_D to evaluate the performance of analytical approximations of PDEs. The principal advantage of using Δ_D instead of other distances given in the literature is that the value given by Δ_D has a very easy interpretation: a value close to 0 means relatively near, but a value close to 1 means relatively infinitely far. Copyright © 2013 John Wiley & Sons, Ltd.     

The evolution dam problem for a compressible fluid with nonlinear Darcy's law and Dirichlet boundary condition

In this paper, we consider the evolution dam problem (P) related to a compressible fluid flow governed by a generalized nonlinear Darcy's law with Dirichlet boundary conditions on some part of the boundary. We establish existence of a solution for this problem. We choose a convenient regularized problem (P_?) for which we prove the existence and uniqueness of solution using the comparison Lemma 2.1 and the Schauder fixed‐point theorem. Then, we pass to the limit, when ? goes to 0, to get a solution for our problem. Moreover, we will see another approach for the incompressible case where we pass to the limit in (P), when α goes to 0, to get a solution.     

The chromatic Ramsey number of odd wheels

We prove that the chromatic Ramsey number of every odd wheel W_{2k+ 1}, k?2 is 14. That is, for every odd wheel W_{2k+ 1}, there exists a 14‐chromatic graph F such that when the edges of F are two‐coloured, there is a monochromatic copy of W_{2k+ 1} in F, and no graph F with chromatic number 13 has the same property. We ask whether a natural extension of odd wheels to the family of generalized Mycielski graphs could help to prove the Burr–Erd?s–Lovász conjecture on the minimum possible chromatic Ramsey number of an n‐chromatic graph. © 2011 Wiley Periodicals, Inc. J Graph Theory 69:198‐205, 2012     

On Switching to H‐Free Graphs

In this article, we study the problem of deciding if, for a fixed graph H, a given graph is switching equivalent to an H‐free graph. Polynomial‐time algorithms are known for H having at most three vertices or isomorphic to P₄. We show that for H isomorphic to a claw, the problem is polynomial, too. On the other hand, we give infinitely many graphs H such that the problem is NP‐complete, thus solving an open problem [Kratochvíl, Ne?et?il and Zýka, Ann Discrete Math 51 (1992)]. Further, we give a characterization of graphs switching equivalent to a K_{1, 2}‐free graph by ten forbidden‐induced subgraphs, each having five vertices. We also give the forbidden‐induced subgraphs for graphs switching equivalent to a forest of bounded vertex degrees.     

Equivalence of weak Dirichlet's principle,the method of weak solutions and Perron's method towards classical solutions of Dirichlet's problem for harmonic functions

For boundary data ? ∈ W ^1,2(G ) (where G ? ?^N is a bounded domain) it is an easy exercise to prove the existence of weak L ²‐solutions to the Dirichlet problem “Δu = 0 in G, u |_?G = ? |_?G ”. By means of Weyl's Lemma it is readily seen that there is ? ∈ C ^∞(G ), Δ? = 0 and ? = u a.e. in G . On the contrary it seems to be a complicated task even for this simple equation to prove continuity of ? up to the boundary in a suitable domain if ? ∈ W ^1,2(G ) ∩ C ⁰( ). The purpose of this paper is to present an elementary proof of that fact in (classical) Dirichlet domains. Here the method of weak solutions (resp. Dirichlet's principle) is equivalent to the classical approaches (Poincaré's “sweeping‐out method”, Perron's method). (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Circular flow numbers of regular multigraphs

The circular flow number F_c(G) of a graph G = (V, E) is the minimum r ϵ ℚ such that G admits a flow ϕ with 1 ≤ ϕ (e) ≤ r − 1, for each e ϵ E. We determine the circular flow number of some regular multigraphs. In particular, we characterize the bipartite (2t+1)‐regular graphs (t ≥ 1). Our results imply that there are gaps for possible circular flow numbers for (2t+1)‐regular graphs, e.g., there is no cubic graph G with 3 < F_c(G) < 4. We further show that there are snarks with circular flow number arbitrarily close to 4, answering a question of X. Zhu. © 2000 John Wiley & Sons, Inc. J Graph Theory 36: 24–34, 2001     

On characterizing Vizing's edge colouring bound

We consider multigraphs G for which equality holds in Vizing's classical edge colouring bound χ′(G)≤Δ + µ, where Δ denotes the maximum degree and µ denotes the maximum edge multiplicity of G. We show that if µ is bounded below by a logarithmic function of Δ, then G attains Vizing's bound if and only if there exists an odd subset S?V(G) with |S|≥3, such that |E[S]|>((|S| ? 1)/2)(Δ + µ ? 1). The famous Goldberg–Seymour conjecture states that this should hold for all µ≥2. We also prove a similar result concerning the edge colouring bound χ′(G)≤Δ + ?µ/?g/2??, due to Steffen (here g denotes the girth of the underlying graph). Finally we give a general approximation towards the Goldberg‐Seymour conjecture in terms of Δ and µ. © 2011 Wiley Periodicals, Inc. J Graph Theory 69:160‐168, 2012     

A New Class of Generalized Thompson's Groups and Their Normal Subgroups

For a given group G and a homomorphism ?: G → G × G, we construct groups ?_?(G), 𝒯_?(G), and 𝒱_?(G) that blend Thompson's groups F, T, and V with G, respectively. Furthermore, we describe the lattice of normal subgroups of the groups ?_Δ(G), where Δ: G → G × G is the diagonal homomorphism, Δ(g) = (g, g).     

Carleson measures and conformal self‐mappings in the real unit ball

Bounded and compact Carleson measures in the unit ball B of R ⁿ , n ≥ 2, are characterized by means of global Dirichlet integrals of the conformal self‐map T_a taking a ∈ B to the origin. The same proof applies in the unit ball of C ⁿ . It is also proved that the powers of the Jacobian of T_a satisfy the weak Harnack inequality and even Harnack's inequality with a constant independent of a. As an application of these results it is shown that the two different definitions for Carleson measures in the existing literature are equivalent for a certain range of parameter values. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A direct method for a regularized least‐squares problem

We consider a linear system of the form A₁x₁+ A₂x₂+η=b₁. The vector η consists of identically distributed random variables all with mean zero. The unknowns are split into two groups x₁ and x₂. In the model usually there are more unknowns than observations and the resulting linear system is most often consistent having an infinite number of solutions. Hence some constraint on the parameter vector x is needed. One possibility is to avoid rapid variation in, e.g. the parameters x₂. We formulate the problem as a partially regularized least‐squares problem, and propose a direct solution method based on the QR decomposition of matrix blocks. Further we consider regularizing using one and two regularization parameters, respectively. We also discuss the choice of regularization parameters, and extend Reinsch's method to the case with two parameters. Also the cross‐validation technique is treated. We present test examples taken from an application in modelling of the substance transport in rivers. Copyright © 2009 John Wiley & Sons, Ltd.     

Cropper's question and Cruse's theorem about partial Latin squares

In 1974 Cruse gave necessary and sufficient conditions for an r × s partial latin square P on symbols σ₁,σ₂,…,σ_t, which may have some unfilled cells, to be completable to an n × n latin square on symbols σ₁,σ₂,…,σ_n, subject to the condition that the unfilled cells of P must be filled with symbols chosen from {σ_{t + 1},σ_{t + 2},…,σ_n}. These conditions consisted of r + s + t + 1 inequalities. Hall's condition applied to partial latin squares is a necessary condition for their completion, and is a generalization of, and in the spirit of Hall's Condition for a system of distinct representatives. Cropper asked whether Hall's Condition might also be sufficient for the completion of partial latin squares, but we give here a counterexample to Cropper's speculation. We also show that the r + s + t + 1 inequalities of Cruse's Theorem may be replaced by just four inequalities, two of which are Hall inequalities for P (i.e. two of the inequalities which constitute Hall's Condition for P), and the other two are Hall inequalities for the conjugates of P. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:268‐279, 2011     

Turán problems for integer‐weighted graphs

A multigraph is (k,r)‐dense if every k‐set spans at most r edges. What is the maximum number of edges ex_?(n,k,r) in a (k,r)‐dense multigraph on n vertices? We determine the maximum possible weight of such graphs for almost all k and r (e.g., for all r>k³) by determining a constant m=m(k,r) and showing that ex_?(n,k,r)=m +O(n), thus giving a generalization of Turán's theorem. We find exact answers in many cases, even when negative integer weights are also allowed. In fact, our main result is to determine the maximum weight of (k,r)‐dense n‐vertex multigraphs with arbitrary integer weights with an O(n) error term. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 195–225, 2002     

A generalized Turán problem in random graphs

We study a generalization of the Turán problem in random graphs. Given graphs T and H, let ex(G(n,p),T,H) be the largest number of copies of T in an H‐free subgraph of G(n,p). We study the threshold phenomena arising in the evolution of the typical value of this random variable, for every H and every 2‐balanced T. Our results in the case when m₂(H) > m₂(T) are a natural generalization of the Erd?s‐Stone theorem for G(n,p), proved several years ago by Conlon‐Gowers and Schacht; the case T = K_m was previously resolved by Alon, Kostochka, and Shikhelman. The case when m₂(H) ≤ m₂(T) exhibits a more complex behavior. Here, the location(s) of the (possibly multiple) threshold(s) are determined by densities of various coverings of H with copies of T and the typical value(s) of ex(G(n,p),T,H) are given by solutions to deterministic hypergraph Turán‐type problems that we are unable to solve in full generality.     

Serre's Condition Rℓ for Affine Semigroup Rings

In this note we characterize the affine semigroup rings K[S] over an arbitrary field K that satisfy condition R_? of Serre. Our characterization is in terms of the face lattice of the positive cone pos(S) of S. We start by reviewing some basic facts about the faces of pos(S) and consequences for the monomial primes of K[S]. After proving our characterization we turn our attention to the Rees algebras of a special class of monomial ideals in a polynomial ring over a field. In this special case, some of the characterizing criteria are always satisfied. We give examples of non-normal affine semigroup rings that satisfy R₂.     

Two graphs without planar covers

In this note we prove that two specific graphs do not have finite planar covers. The graphs are K₇–C₄ and K_4,5–4K₂. This research is related to Negami's 1‐2‐∞ Conjecture which states “A graph G has a finite planar cover if and only if it embeds in the projective plane.” In particular, Negami's Conjecture reduces to showing that 103 specific graphs do not have finite planar covers. Previous (and subsequent) work has reduced these 103 to a few specific graphs. This paper covers 2 of the remaining cases. The sole case currently remaining is to show that K_2,2,2,1 has no finite planar cover. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 318–326, 2002     

A generalization of a conservativity theorem for classical versus intuitionistic arithmetic

A basic result in intuitionism is Π⁰₂‐conservativity. Take any proof p in classical arithmetic of some Π⁰₂‐statement (some arithmetical statement ?x.?y.P(x, y), with P decidable). Then we may effectively turn p in some intuitionistic proof of the same statement. In a previous paper [1], we generalized this result: any classical proof p of an arithmetical statement ?x.?y.P(x, y), with P of degree k, may be effectively turned into some proof of the same statement, using Excluded Middle only over degree k formulas. When k = 0, we get the original conservativity result as particular case. This result was a by‐product of a semantical construction. J. Avigad of Carnegie Mellon University, found a short, direct syntactical derivation of the same result, using H. Friedman's A‐translation. His proof is included here with his permission. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Derivation of a quasi‐stationary coupled Darcy–Reynolds equation for incompressible viscous fluid flow through a thin porous medium with a fissure

We consider a non‐stationary Stokes system in a thin porous medium of thickness ε that is perforated by periodically distributed solid cylinders of size ε, and containing a fissure of width η_ε. Passing to the limit when ε goes to zero, we find a critical size in which the flow is described by a 2D quasi‐stationary Darcy law coupled with a 1D quasi‐stationary Reynolds problem. Copyright © 2017 John Wiley & Sons, Ltd.