Kernel estimates and -spectral independence of generators of -semigroups

After the appearance of W. Arendt's result that “Gaussian estimate of a semigroup implies the L^p-spectral independence of the generator,” various generalizations have been obtained. This paper shows that a certain kernel estimate of a semigroup implies the L^p-spectral independence of the generator, generalizing the case of upper Gaussian estimate and “Gaussian estimate of order α(0,1] [S. Miyajima, H. Shindoh, Gaussian estimates of order α and L^p-spectral independence of generators of C₀-semigroups, Positivity 11 (1) (2007) 15–39], Definition 3.1.” The proof uses S. Karrmann's result about the L^p-spectral independence and B.A. Barnes' theorem about the spectrum of integral operators. As an application, the L^p-spectral independence of −[(−Δ)^α+V] (α(0,1]) for a suitable V is proved with the help of a recent result by V. Liskevich, H. Vogt and J. Voigt [V. Liskevich, H. Vogt, J. Voigt, Gaussian bounds for propagators perturbed by potentials, J. Funct. Anal. 238 (2006) 245–277].     

Strong and uniform mean stability of cosine and sine operator functions

It is first observed that a uniformly bounded cosine operator function C() and the associated sine function S() are totally non-stable. Then, using a zero-one law for the Abel limit of a closed linear operator, we prove some results concerning strong mean stability and uniform mean stability of C(). Among them are: (1) C() is strongly (C,1)-mean stable (or (C,2)-mean stable, or Abel-mean stable) if and only if 0ρ(A)σ_c(A); (2) C() is uniformly (C,2)-mean stable if and only if S() is uniformly (C,1)-mean stable, if and only if , if and only if , if and only if C() is uniformly Abel-mean stable, if and only if S() is uniformly Abel-mean stable, if and only if 0ρ(A).     

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.     

ech-complete spaces and the upper topology

Let X be a topological space and let be the set of all compact subsets of X. The purpose of this note is to prove the following: if X is regular and q-space, then X is Lindelöf and ech-complete if and only if there exists a continuous map f from a Lindelöf and ech-complete space Y to the space endowed with the upper topology, such that f(Y) is cofinal in . This result extends the following result of Saint Raymond and Christensen: if X is separable metrizable, then X is a Polish space if and only if the space endowed with the Vietoris topology is the continuous image of a Polish space.     

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).

     

Coincidence and common fixed points for hybrid strict contractions without the weakly commuting condition

In this work, we establish new coincidence and common fixed point theorems for hybrid strict contraction maps by dropping the assumption “f is T-weakly commuting” for a hybrid pair (f,T) of multivalued maps in Theorem 3.10 of T. Kamran (2004) [8]. As an application, an invariant approximation result is derived.     

Numerical comparison of nonlinear subgridscale models via adaptive mesh refinement

In this paper, we provide a method of evaluating the efficacy of nonlinear subgridscale models for use in the large eddy simulation of incompressible viscous flow problems. We compare subgridscale “artificial” viscosity models using a posteriori error estimation and adaptive mesh refinement. Specifically, we compare α-Laplacian based subgridscale models and discuss the benefits and limitations of different values of α for some standard benchmark problems for the Navier–Stokes equations.     

The limiting case of Haglund's (q,t)-Schröder theorem and an involution formula

As a generalization of Haglund's statistic on Dyck paths [Conjectured statistics for the q,t-Catalan numbers, Adv. Math. 175 (2) (2003) 319–334; A positivity result in the theory of Macdonald polynomials, Proc. Nat. Acad. Sci. 98 (2001) 4313–4316], Egge et al. introduced the (q,t)-Schröder polynomial S_n,d(q,t), which evaluates to the Schröder number when q=t=1 [A Schröder generalization of Haglund's statistic on Catalan paths, Electron. J. Combin. 10 (2003) 21pp (Research Paper 16, electronic)]. In their paper, S_n,d(q,t) was conjectured to be equal to the coefficient of a hook shape on the Schur function expansion of the symmetric function e_n, which Haiman [Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math. 149 (2002) 371–407] has shown to have a representation-theoretic interpretation. This conjecture was recently proved by Haglund [A proof of the q,t-Schröder conjecture, Internat. Math. Res. Not. (11) (2004) 525–560]. However, because that proof makes heavy use of symmetric function identities and plethystic machinery, the combinatorics behind it is not understood. Therefore, it is worthwhile to study it combinatorially. This paper investigates the limiting case of the (q,t)-Schröder Theorem and obtains interesting results by looking at some special cases.     

Minimum number of below average triangles in a weighted complete graph

Let G be an edge weighted graph with n nodes, and let A(3,G) be the average weight of a triangle in G. We show that the number of triangles with weight at most equal to A(3,G) is at least (n−2) and that this bound is sharp for all n≥7. Extensions of this result to cliques of cardinality k>3 are also discussed.     

On scatteredly continuous maps between topological spaces

A map f:X→Y between topological spaces is defined to be scatteredly continuous if for each subspace AX the restriction f|A has a point of continuity. We show that for a function f:X→Y from a perfectly paracompact hereditarily Baire Preiss–Simon space X into a regular space Y the scattered continuity of f is equivalent to (i) the weak discontinuity (for each subset AX the set D(f|A) of discontinuity points of f|A is nowhere dense in A), (ii) the piecewise continuity (X can be written as a countable union of closed subsets on which f is continuous), (iii) the G_δ-measurability (the preimage of each open set is of type G_δ). Also under Martin Axiom, we construct a G_δ-measurable map f:X→Y between metrizable separable spaces, which is not piecewise continuous. This answers an old question of V. Vinokurov.     

On Banach spaces whose norm-open sets are -sets in the weak topology

We investigate non-separable Banach spaces whose norm-open sets are countable unions of sets closed in the weak topology and a narrower class of Banach spaces with a network for the norm topology which is σ-discrete in the weak topology. In particular, we answer a question of Arhangel'skii exhibiting various examples of non-separable function spaces C(K) with a σ-discrete network for the pointwise topology and (consistently) we answer some questions of Edgar and Oncina concerning Borel structures and Kadec renormings in Banach spaces.     

The topological structure of fuzzy sets with endograph metric

For a non-degenerate convex subset Y of the n-dimensional Euclidean space Rⁿ, let be the family of all fuzzy sets ofRⁿ, which are upper-semicontinuous, fuzzy convex and normal with compact supports contained in Y. We show that the space with the topology of endograph metric is homeomorphic to the Hilbert cube Q=[-1,1]^ω iff Y is compact; and the space is homeomorphic to {(x_n)Q:sup|x_n|<1} iff Y is non-compact and locally compact.     

Hyperbolic topology of normed linear spaces

In a previous paper [H. Tsuiki, Y. Hattori, Lawson topology of the space of formal balls and the hyperbolic topology of a metric space, Theoret. Comput. Sci. 405 (2008) 198–205], the authors introduced the hyperbolic topology on a metric space, which is weaker than the metric topology and naturally derived from the Lawson topology on the space of formal balls. In this paper, we characterize spaces L_p(Ω,Σ,μ) on which the hyperbolic topology induced by the norm _p coincides with the norm topology. We show the following:

(1) The hyperbolic topology and the norm topology coincide for 1<p<∞.

(2) They coincide on L₁(Ω,Σ,μ) if and only if μ(Ω)=0 or Ω has a finite partition by atoms.

(3) They coincide on L_∞(Ω,Σ,μ) if and only if μ(Ω)=0 or there is an atom in Σ.

L-fuzzy quantifiers of type determined by fuzzy measures

The aim of this paper is, first, to introduce two new types of fuzzy integrals, namely, -fuzzy integral and →-fuzzy integral. The first integral is based on a fuzzy measure of L-fuzzy sets and the second one on a complementary fuzzy measure of L-fuzzy sets, where L is a complete residuated lattice. Some of their properties and a relation to the fuzzy (Sugeno) integral are investigated. Second, using these integrals, two classes of monadic L-fuzzy quantifiers of type 1 are defined. These L-fuzzy quantifiers can be used for modeling the semantics of natural language quantifiers like “all”, “some”, “many”, “none”, “at most half”, etc. Several semantic properties of these L-fuzzy quantifiers are studied.     

Uniformly Gâteaux smooth approximations on

Every Lipschitz mapping from c₀(Γ) into a Banach space Y can be uniformly approximated by Lipschitz mappings that are simultaneously uniformly Gâteaux smooth and C^∞-Fréchet smooth.     

On reducibility of n-ary quasigroups

An n-ary operation Q:Σⁿ→Σ is called an n-ary quasigroup of order |Σ| if in the relation x₀=Q(x₁,…,x_n) knowledge of any n elements of x₀,…,x_n uniquely specifies the remaining one. Q is permutably reducible if Q(x₁,…,x_n)=P(R(x_σ(1),…,x_σ(k)),x_σ(k+1),…,x_σ(n)) where P and R are (n-k+1)-ary and k-ary quasigroups, σ is a permutation, and 1<k<n. An m-ary quasigroup S is called a retract of Q if it can be obtained from Q or one of its inverses by fixing n-m>0 arguments. We prove that if the maximum arity of a permutably irreducible retract of an n-ary quasigroup Q belongs to {3,…,n-3}, then Q is permutably reducible.     

On the vertices of the -additive core

The core of a game v on N, which is the set of additive games φ dominating v such that φ(N)=v(N), is a central notion in cooperative game theory, decision making and in combinatorics, where it is related to submodular functions, matroids and the greedy algorithm. In many cases however, the core is empty, and alternative solutions have to be found. We define the k-additive core by replacing additive games by k-additive games in the definition of the core, where k-additive games are those games whose Möbius transform vanishes for subsets of more than k elements. For a sufficiently high value of k, the k-additive core is nonempty, and is a convex closed polyhedron. Our aim is to establish results similar to the classical results of Shapley and Ichiishi on the core of convex games (corresponds to Edmonds’ theorem for the greedy algorithm), which characterize the vertices of the core.     

An algorithmic sign-reversing involution for special rim-hook tableaux

Eğecioğlu and Remmel [Linear Multilinear Algebra 26 (1990) 59–84] gave an interpretation for the entries of the inverse Kostka matrix K⁻¹ in terms of special rim-hook tableaux. They were able to use this interpretation to give a combinatorial proof that KK⁻¹=I but were unable to do the same for the equation K⁻¹K=I. We define an algorithmic sign-reversing involution on rooted special rim-hook tableaux which can be used to prove that the last column of this second product is correct. In addition, following a suggestion of Chow [preprint, math.CO/9712230, 1997] we combine our involution with a result of Gasharov [Discrete Math. 157 (1996) 193–197] to give a combinatorial proof of a special case of the (3+1)-free Conjecture of Stanley and Stembridge [J. Combin. Theory Ser. A 62 (1993) 261–279].     

Positive solutions for Robin problem involving the -Laplacian

Consider Robin problem involving the p(x)-Laplacian on a smooth bounded domain Ω as follows

Applying the sub-supersolution method and the variational method, under appropriate assumptions on f, we prove that there exists λ_*>0 such that the problem has at least two positive solutions if λ(0,λ_*), has at least one positive solution if λ=λ_*<+∞ and has no positive solution if λ>λ_*. To prove the results, we prove a norm on W^1,p(x)(Ω) without the part of ||_L^p(x)(Ω) which is equivalent to usual one and establish a special strong comparison principle for Robin problem.     

Existence of periodic solutions for a fourth-order -Laplacian equation with a deviating argument

In this paper, we study the existence of periodic solutions for a fourth-order p-Laplacian differential equation with a deviating argument as follows:

[φ_p(u^″(t))]^″+f(u^″(t))+g(u(t−τ(t)))=e(t).