Symmetry of some entire solutions to the Allen–Cahn equation in two dimensions

In this paper, we prove even symmetry and monotonicity of certain solutions of Allen–Cahn equation in a half plane. We also show that entire solutions with finite Morse index and four ends must be evenly symmetric with respect to two orthogonal axes. A classification scheme of general entire solutions with finite Morse index is also presented using energy quantization.     

Contributions to the theory of first‐exit times of some compound processes in queueing theory

We consider compound processes that are linear with constant slope between i.i.d. jumps at time points forming a renewal process. These processes are basic in queueing, dam and risk theory. For positive and for negative slope we derive the distribution of the first crossing time of a prespecified level. The related problem of busy periods of single‐server queueing systems is also studied. This revised version was published online in June 2006 with corrections to the Cover Date.     

Some Extensions of Optimal Interpolation in Spaces of Lorentz–Zygmund Type

The results on optimal interpolation from [7] are extended to quasinormed spaces with p<1, to spaces with varying secondary parameters , E and to spaces of functions defined on the interval (1,). As a tool for doing this, we construct special mappings which transform these cases into the basic one, considered in [7].     

On monochromatic solutions of some nonlinear equations in ℤ/pℤ

Let the set of positive integers be colored in an arbitrary way in finitely many colors (a “finite coloring”). Is it true that, in this case, there are x, y ∈ ? such that x + y, xy, and x have the same color? This well-known problem of the Ramsey theory is still unsolved. In the present paper, we answer this question in the affirmative in the group ?/p?, where p is a prime, and obtain an even stronger density result.     

Study on existence of solutions for some nonlinear functional–integral equations

Using the technique of measures of noncompactness in Banach algebra, we employ abstract fixed point theorems such as Darbo’s theorem to study the existence solution in Banach algebra C[0,a]$C[0,a]$ for some functional–integral equations which include many key integral and functional equations that arise in nonlinear analysis.     

Global Calderón–Zygmund theory for nonlinear parabolic systems

We establish a global Calderón–Zygmund theory for solutions to a large class of nonlinear parabolic systems whose model is the inhomogeneous parabolic \(p\) -Laplacian system $$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u - {{\mathrm{div}}}(|Du|^{p-2}Du) = {{\mathrm{div}}}(|F|^{p-2}F) &{}\quad \hbox {in }\quad \Omega _T:=\Omega \times (0,T)\\ u=g &{}\quad \hbox {on }\quad \partial \Omega \times (0,T)\cup {\overline{\Omega }}\times \{0\} \end{array} \right. \end{aligned}$$ with given functions \(F\) and \(g\) . Our main result states that the spatial gradient of the solution is as integrable as the data \(F\) and \(g\) up to the lateral boundary of \(\Omega _T\) , i.e. $$\begin{aligned} F,Dg\in L^q(\Omega _T),\ \partial _t g\in L^{\frac{q(n+2)}{p(n+2)-n}}(\Omega _T) \quad \Rightarrow \quad Du\in L^q(\Omega \times (\delta ,T)) \end{aligned}$$ for any \(q>p\) and \(\delta \in (0,T)\) , together with quantitative estimates. This result is proved in a much more general setting, i.e. for asymptotically regular parabolic systems.     

A moment inequality of the Marcinkiewicz–Zygmund type for some weakly dependent random fields

The goal of this note is to give a new moment inequality for sums of some weakly dependent random fields. Our result extends moment bounds given by Wu (2007) or Liu and Lin (2009) for causal autoregressive processes and follows by using recursive applications of the Burkhölder inequality for martingales.     

Smooth and non-smooth traveling wave solutions of some generalized Camassa–Holm equations

In this paper we employ two recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of a recently-derived integrable family of generalized Camassa–Holm (GCH) equations. A recent, novel application of phase-plane analysis is employed to analyze the singular traveling wave equations of three of the GCH NLPDEs, i.e. the possible non-smooth peakon and cuspon solutions. One of the considered GCH equations supports both solitary (peakon) and periodic (cuspon) cusp waves in different parameter regimes. The second equation does not support singular traveling waves and the last one supports four-segmented, non-smooth M-wave solutions.Moreover, smooth traveling waves of the three GCH equations are considered. Here, we use a recent technique to derive convergent multi-infinite series solutions for the homoclinic orbits of their traveling-wave equations, corresponding to pulse (kink or shock) solutions respectively of the original PDEs. We perform many numerical tests in different parameter regime to pinpoint real saddle equilibrium points of the corresponding GCH equations, as well as ensure simultaneous convergence and continuity of the multi-infinite series solutions for the homoclinic orbits anchored by these saddle points. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. We also show the traveling wave nature of these pulse and front solutions to the GCH NLPDEs.     

Local computations in Dempster–Shafer theory of evidence

When applying any technique of multidimensional models to problems of practice, one always has to cope with two problems: the necessity to represent the models with a ”reasonable” number of parameters and to have sufficiently efficient computational procedures at one’s disposal. When considering graphical Markov models in probability theory, both of these conditions are fulfilled; various computational procedures for decomposable models are based on the ideas of local computations, whose theoretical foundations were laid by Lauritzen and Spiegelhalter.The presented contribution studies a possibility of transferring these ideas from probability theory into Dempster-Shafer theory of evidence. The paper recalls decomposable models, discusses connection of the model structure with the corresponding system of conditional independence relations, and shows that under special additional conditions, one can locally compute specific basic assignments which can be considered to be conditional.     

Optimal constants in the Marcinkiewicz–Zygmund inequalities

We give the optimal constants in the Marcinkiewicz–Zygmund inequalities for symmetric summands. As an application we substantially improve the estimates of Ren and Liang (2001) in the Marcinkiewicz–Zygmund–Hölder inequality and identify the best possible constants in the symmetric case.     

On the Marcinkiewicz–Zygmund law of large numbers in Banach lattices

We strengthen the well-known Marcinkiewicz–Zygmund law of large numbers in the case of Banach lattices. Examples of applications to empirical distributions are presented.     

Marcinkiewicz–Zygmund type results in multivariate domains

We investigate Marcinkiewicz–Zygmund type inequalities for multivariate polynomials on various compact domains in \({\mathbb{R}^d}\). These inequalities provide a basic tool for the discretization of the L ^p norm and are widely used in the study of the convergence properties of Fourier series, interpolation processes and orthogonal expansions. Recently Marcinkiewicz–Zygmund type inequalities were verified for univariate polynomials for the general class of doubling weights, and for multivariate polynomials on the ball and sphere with doubling weights. The main goal of the present paper is to extend these considerations to more general multidimensional domains, which in particular include polytopes, cones, spherical sectors, toruses, etc. Our approach will rely on application of various polynomial inequalities, such as Bernstein–Markov, Schur and Videnskii type estimates, and also using symmetry and rotation in order to generate results on new domains.     

Estimates on solutions of some new nonlinear retarded Volterra–Fredholm type integral inequalities

Some new explicit bounds on solutions to a class of new nonlinear retarded Volterra–Fredholm type integral inequalities are established, which can be used as effective tools in the study of certain integral equations. Applications examples are also indicated.     

On some properties of generalized solutions of the wave equation in the classes L p and W p 1 for p ≥ 1

For each p ≥ 1, in closed analytic form, we establish the existence of a unique generalized solution in L _p of the mixed problem for the wave equation in the rectangle [0 ≤ x ≤ 1] × [0 ≤ t ≤ T] with zero initial conditions and with boundary conditions of the first kind, one of which is homogeneous. Next, we derive necessary conditions for this solution to belong to W _p ¹ . We present examples showing that these necessary conditions are not sufficient for any p ≥ 1.     

On the boundedness of generalized solutions of higher-order nonlinear elliptic equations with data from an Orlicz–Zygmund class

In the present paper, a 2mth-order quasilinear divergence equation is considered under the condition that its coefficients satisfy the Carathéodory condition and the standard conditions of growth and coercivity in the Sobolev space W^m,p(Ω), Ω ? Rⁿ, p > 1. It is proved that an arbitrary generalized (in the sense of distributions) solution u ∈ W₀^m,p (Ω) of this equation is bounded if m ≥ 2, n = mp, and the right-hand side of this equation belongs to the Orlicz–Zygmund space L(log L)^n?1(Ω).     

On Lipschitz solutions for some forward–backward parabolic equations

We investigate the existence and properties of Lipschitz solutions for some forward–backward parabolic equations in all dimensions. Our main approach to existence is motivated by reformulating such equations into partial differential inclusions and relies on a Baire's category method. In this way, the existence of infinitely many Lipschitz solutions to certain initial-boundary value problem of those equations is guaranteed under a pivotal density condition. Under this framework, we study two important cases of forward–backward anisotropic diffusion in which the density condition can be realized and therefore the existence results follow together with micro-oscillatory behavior of solutions. The first case is a generalization of the Perona–Malik model in image processing and the other that of Höllig's model related to the Clausius–Duhem inequality in the second law of thermodynamics.