Gagliardo–Nirenberg inequalities in regular Orlicz spaces involving nonlinear expressions

We consider a triple of N-functions (M,H,J) that satisfy the Δ^′-condition, and suppose that an additive variant of interpolation inequality holds

where , is an arbitrary set invariant with respect to external and internal dilations. We show that the above inequality implies its certain nonlinear variant involving the expressions and . Various generalizations of this inequality to the more general class of N-functions, measures and to higher order derivatives are also discussed and the examples are presented.     

Histopolating splines

Given an integrable function f, we are concerned with the construction of a spline H_n(f) of degree n with prescribed knots that satisfies the histopolation conditions

for some fixed . Additionally, the resulting spline operator should be local and reproduce all polynomials of degree n. Our approach of generating such a histospline is based on a local spline interpolation operator that is exact for all polynomials of degree n.     

Construction of nonseparable dual -wavelet frames in

Suppose and are two pairs of dual M-wavelet frames and N-wavelet frames in and , respectively, where M and N are s₁×s₁ and s₂×s₂ dilation matrices with a₁(|det(M)|-1) and (2a₂+1)(|det(N)|-1). Moreover, their mask symbols both satisfy mixed extension principle (MEP). Based on the mask symbols, a family of nonseparable dual Ω-wavelet frames in are constructed, where s=s₁+s₂, and with Θ and M^-1Θ both being integer matrices. Then a convolution scheme for improving regularity of wavelet frames is given. From the nonseparable dual Ω-wavelet frames, nonseparable Ω-wavelet frames with high regularity can be constructed easily. We give an algorithm for constructing nonseparable dual symmetric or antisymmetric wavelet frames in . From the dual Ω-wavelet frames, nonseparable dual Ω-wavelet frames with symmetry can be obtained easily. In the end, two examples are given.     

Analyticity of functions analytic on circles

Let Δ be the open unit disc in, let pbΔ, and let f be a continuous function on which extends holomorphically from each circle in centered at the origin and from each circle in which passes through p. Then f is holomorphic on Δ.     

Weak and point-wise convergence in for filter convergence

We study those filters on for which weak -convergence of bounded sequences in C(K) is equivalent to point-wise -convergence. We show that it is sufficient to require this property only for C[0,1] and that the filter-analogue of the Rainwater extremal test theorem arises from it. There are ultrafilters which do not have this property and under the continuum hypothesis there are ultrafilters which have it. This implies that the validity of the Lebesgue dominated convergence theorem for -convergence is more restrictive than the property which we study.     

Local convergence of the steepest descent method in Hilbert spaces

The aim of this paper is to establish the local convergence of the steepest descent method for C¹-functionals defined on an infinite-dimensional Hilbert space H, under a Palais–Smale-type condition. The functionals f under consideration are also assumed to have a locally Lipschitz continuous gradient operator f. Our approach is based on the solutions of the ordinary differential equation .     

On a class of degenerate and singular elliptic systems in bounded domains

This paper deals with the nonexistence and multiplicity of nonnegative, nontrivial solutions to a class of degenerate and singular elliptic systems of the form

where Ω is a bounded domain with smooth boundary ∂Ω in , N2, and , , h_i (i=1,2) are allowed to have “essential” zeroes at some points in Ω, (F_u,F_v)=F, and λ is a positive parameter. Our proofs rely essentially on the critical point theory tools combined with a variant of the Caffarelli–Kohn–Nirenberg inequality in [P. Caldiroli, R. Musina, On a variational degenerate elliptic problem, NoDEA Nonlinear Differential Equations Appl. 7 (2000) 189–199].     

Ore-type and Dirac-type theorems for matroids

Let M be a connected binary matroid having no -minor. Let be a collection of cocircuits of M. We prove there is a circuit intersecting all cocircuits of if either one of two things hold:

(i) For any two disjoint cocircuits and in it holds that .

(ii) For any two disjoint cocircuits and in it holds that .

Part (ii) implies Ore's Theorem, a well-known theorem giving sufficient conditions for the existence of a hamilton cycle in a graph. As an application of part (i), it is shown that if M is a k-connected regular matroid and has cocircumference c^*2k, then there is a circuit which intersects each cocircuit of size c^*−k+2 or greater.We also extend a theorem of Dirac for graphs by showing that for any k-connected binary matroid M having no -minor, it holds that for any k cocircuits of M there is a circuit which intersects them.     

Sharp estimates of bounded solutions to some semilinear second order dissipative equations

Let H, V be two real Hilbert spaces such that VH with continuous and dense imbedding, and let FC¹(V) be convex. By using differential inequalities, a close-to-optimal ultimate bound of the energy is obtained for solutions in to u^″+cu^′+bu+F(u)=f(t) whenever .     

Refined blowup criteria and nonsymmetric blowup of an aggregation equation

We consider an aggregation equation in , d2, with fractional dissipation: u_t+(uK*u)=−νΛ^γu, where ν0, 0<γ<1, and K(x)=e^−|x|. We prove a refined blowup criteria by which the global existence of solutions is controlled by its norm, for any . We prove the finite time blowup of solutions for a general class of nonsymmetric initial data. The argument presented works for both the inviscid case ν=0 and the supercritical case ν>0 and 0<γ<1. Additionally, we present new proofs of blowup which does not use free energy arguments.     

Partial regularity for solutions of a nonlinear elliptic equation with singular nonlinearity

We consider the following nonlinear elliptic equation with singular nonlinearity:

where α>β>1, a>0, and Ω is an open subset of , n2. Let uH¹(Ω) with and be a nonnegative stationary solution. If we denote the zero set of u by

we shall prove that the Hausdorff dimension of Σ is less than or equal to .     

Construction of highly stable parallel two-step Runge–Kutta methods for delay differential equations

It is shown that any A-stable two-step Runge–Kutta method of order and stage order for ordinary differential equations can be extended to the P-stable method of uniform order for delay differential equations.     

Equivalence relation groupoids associated with certain linearly ordered dimension groups

We consider linearly ordered, Archimedean dimension groups (G,G⁺,u) for which the group G/u is torsion-free. It will be shown that if, in addition, G/u is generated by a single element (i.e., ), then (G,G⁺,u) is isomorphic to for some irrational number τ(0,1). This amounts to an extension of related results where dimension groups for which G/u is torsion were considered. We will prove, in the case of the Fibonacci dimension group, that these results can be used to directly construct an equivalence relation groupoid whose C^*-algebra is the Fibonacci C^*-algebra.     

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.     

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.     

Conditional limiting distribution of beta-independent random vectors

The paper deals with random vectors in , possessing the stochastic representation , where R is a positive random radius independent of the random vector and is a non-singular matrix. If is uniformly distributed on the unit sphere of , then for any integer m<d we have the stochastic representations and , with W≥0, such that W² is a beta distributed random variable with parameters m/2,(d−m)/2 and (U₁,…,U_m),(U_m+1,…,U_d) are independent uniformly distributed on the unit spheres of and , respectively. Assuming a more general stochastic representation for in this paper we introduce the class of beta-independent random vectors. For this new class we derive several conditional limiting results assuming that R has a distribution function in the max-domain of attraction of a univariate extreme value distribution function. We provide two applications concerning the Kotz approximation of the conditional distributions and the tail asymptotic behaviour of beta-independent bivariate random vectors.     

Pointwise universal trigonometric series

A series is called a pointwise universal trigonometric series if for any , there exists a strictly increasing sequence of positive integers such that converges to f(z) pointwise on . We find growth conditions on coefficients allowing and forbidding the existence of a pointwise universal trigonometric series. For instance, if as |n|→∞ for some ε>0, then the series S_a cannot be pointwise universal. On the other hand, there exists a pointwise universal trigonometric series S_a with as |n|→∞.     

Gradient estimates of Poisson equations on Riemannian manifolds and applications

For the reflected diffusion generated by on a connected and complete Riemannian manifold M with empty or convex boundary, we establish some sharp estimates of sup_xM|G|(x) of the Poisson equation in terms of the dimension, the diameter and the lower bound of curvature. Applications to transportation-information inequality, to Cheeger's isoperimetric inequality and to Gaussian concentration inequality are given. Several examples are provided.     

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.     

On the structure of global solutions of the nonlinear heat equation in a ball

In this paper, we consider the nonlinear heat equation

(NLH)

u_t−Δu=|u|^αu,