A probabilistic approach to positive harmonic functions in a slab with alternating Dirichlet and Neumann boundary conditions

Let , , be a dimensional slab. Denote points by , where and . Denoting the boundary of the slab by , let

where is an ordered sequence of intervals on the right half line (that is, b_{n}$">). Assume that the lengths of the intervals are bounded and that the spaces between consecutive intervals are bounded and bounded away from zero. Let . Let and denote respectively the cone of bounded, positive harmonic functions in and the cone of positive harmonic functions in which satisfy the Dirichlet boundary condition on and the Neumann boundary condition on .

Letting , the main result of this paper, under a modest assumption on the sequence , may be summarized as follows when :

1. If , then and are both one-dimensional (as in the case of the Neumann boundary condition on the entire boundary). In particular, this occurs if with 2$">.

2. If and , then and is one-dimensional. In particular, this occurs if .

3. If , then and the set of minimal elements generating is isomorphic to (as in the case of the Dirichlet boundary condition on the entire boundary). In particular, this occurs if with .

When , as soon as there is at least one interval of Dirichlet boundary condition. The dichotomy for is as above.

     

Central limit theorem and infinitely divisible probabilities associated with partial differential operators

We provide

Comparison theorems of Kolmogorov type and exact values of -widths on Hardy-Sobolev classes

Let be a strip in complex plane. denotes those -periodic, real-valued functions on which are analytic in the strip and satisfy the condition , . Osipenko and Wilderotter obtained the exact values of the Kolmogorov, linear, Gel'fand, and information -widths of in , , and 2-widths of in , , .

In this paper we continue their work. Firstly, we establish a comparison theorem of Kolmogorov type on , from which we get an inequality of Landau-Kolmogorov type. Secondly, we apply these results to determine the exact values of the Gel'fand -width of in , . Finally, we calculate the exact values of Kolmogorov -width, linear -width, and information -width of in , , .

     

Zero distribution of Müntz extremal polynomials in

Let be a sequence of distinct positive numbers. Let and let denote the extremal Müntz polynomial in with exponents . We investigate the zero distribution of . In particular, we show that if

then the normalized zero counting measure of converges weakly as to

while if or , the limiting measure is a Dirac delta at or , respectively.

     

Games and general distributive laws in Boolean algebras

The games and are played by two players in -complete and max -complete Boolean algebras, respectively. For cardinals such that or , the -distributive law holds in a Boolean algebra iff Player 1 does not have a winning strategy in . Furthermore, for all cardinals , the -distributive law holds in iff Player 1 does not have a winning strategy in . More generally, for cardinals such that , the -distributive law holds in iff Player 1 does not have a winning strategy in . For regular and , implies the existence of a Suslin algebra in which is undetermined.

     

Damped wave equation with a critical nonlinearity

We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity

where 0,$"> and space dimensions . Assume that the initial data

where \frac{n}{2},$"> weighted Sobolev spaces are

Also we suppose that

0,\int u_{0}\left( x\right) dx>0, \end{displaymath}">

where

Then we prove that there exists a positive such that the Cauchy problem above has a unique global solution satisfying the time decay property

for all 0,$"> where

     

Imaginary powers of Laplace operators

We show that if is a second-order uniformly elliptic operator in divergence form on , then . We also prove that the upper bounds remain true for any operator with the finite speed propagation property.

     

A generalization of Carleman's uniqueness theorem and a discrete Phragmén-Lindelöf theorem

Let be a Borel measure on and be its moments. T. Carleman found sharp conditions on the magnitude of for to be uniquely determined by its moments. We show that the same conditions ensure a stronger property: if are the moments of another measure, with then the measure is supported on the interval This result generalizes both the Carleman theorem and a theorem of J. Mikusi\'{n}ski. We also present an application of this result by establishing a discrete version of a Phragmén-Lindelöf theorem.

     

Characterizations of regular local rings in positive characteristics

In this note, we provide several characterizations of regular local rings in positive characteristics, in terms of the Hilbert-Kunz multiplicity and its higher counterparts . We also apply the characterizations to improve a recent result by Bridgeland and Iyengar in the characteristic case. Our proof avoids using the existence of big Cohen-Macaulay modules, which is the major tool in the proof of Bridgeland and Iyengar.

     

On the Bernstein Constants of Polynomial Approximation

Assume is not an integer. In papers published in 1913 and 1938, S.~N.~Bernstein established the limit

Heat Equation with Dynamical Boundary Conditions of Locally Reactive Type

The aim of this paper is to study the well-posedness of the initial-boundary value problem

Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems

We derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations. The space discretization uses finite element spaces that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, introduced by Makridakis and Nochetto. We derive novel a posteriori estimates for the norms of and the higher order spaces, and , with optimal orders of convergence.

     

Distributional Point Values and Convergence of Fourier Series and Integrals

In this article we show that the distributional point values of a tempered distribution are characterized by their Fourier transforms in the following way: If and , and is locally integrable, then distributionally if and only if there exists k such that , for each a > 0, and similarly in the case when is a general distribution. Here means in the Cesaro sense. This result generalizes the characterization of Fourier series of distributions with a distributional point value given in [5] by . We also show that under some extra conditions, as if the sequence belongs to the space for some and the tails satisfy the estimate ,\ as , the asymmetric partial sums\ converge to . We give convergence results in other cases and we also consider the convergence of the asymmetric partial integrals. We apply these results to lacunary Fourier series of distributions.     

Stability analysis for delay differential equations with multidelays and numerical examples

In this paper we are concerned with the asymptotic stability of the delay differential equation

where are constant complex matrices, and 0$"> stand for constant delays . We obtain two criteria for stability through the evaluation of a harmonic function on the boundary of a certain region. We also get similar results for the neutral delay differential equation

where and are constant complex matrices and 0$"> stands for constant delays , . Numerical examples on various circumstances are shown to check our results which are more general than those already reported.

     

Whitney's extension problem for multivariate -functions

We prove that the trace of the space to an arbitrary closed subset is characterized by the following ``finiteness' property. A function belongs to the trace space if and only if the restriction to an arbitrary subset consisting of at most can be extended to a function such that

The constant is sharp.

The proof is based on a Lipschitz selection result which is interesting in its own right.

     

Partitions into Primes

We investigate the asymptotic behavior of the partition function defined by , where denotes the von Mangoldt function. Improving a result of Richmond, we show that , where is a positive constant and denotes the times iterated logarithm. We also show that the error term can be improved to if and only if the Riemann Hypothesis holds.

     

New irrationality measures for -logarithms

The three main methods used in diophantine analysis of -series are combined to obtain new upper bounds for irrationality measures of the values of the -logarithm function

when and .

     

Extremal Functions for Adams Inequalities in Dimension Four

Let $\Omega\subset \mathbb{R}^4$ be a smooth bounded domain, $W_0^{2,2}(\Omega)$ be the usual Sobolev space. For any positive integer $\ell$, $\lambda_{\ell}(\Omega)$ is the $\ell$-th eigenvalue of the bi-Laplacian operator. Define $E_{\ell}=E_{\lambda_1(\Omega)}\oplus E_{\lambda_2(\Omega)}\oplus\cdots\oplus E_{\lambda_{\ell}(\Omega)}$, where $E_{\lambda_i(\Omega)}$ is eigenfunction space associated with $\lambda_i(\Omega)$. $E^{\bot}_{\ell}$ denotes the orthogonal complement of $E_\ell$ in $W_0^{2,2}(\Omega)$. For $0\leq\alpha<\lambda_{\ell+1}(\Omega)$, we define a norm by $\|u\|_{2,\alpha}^{2}=\|\Delta u\|^2_2-\alpha \|u\|^2_2$ for $u\in E^\bot_{\ell}$. In this paper, using the blow-up analysis, we prove the following Adams inequalities$$\sup_{u\in E_{\ell}^{\bot},\,\| u\|_{2,\alpha}\leq 1}\int_{\Omega}e^{32\pi^2u^2}{\rm d}x<+\infty;$$moreover, the above supremum can be attained by a function $u_0\in E_{\ell}^{\bot}\cap C^4(\overline{\Omega})$ with $\|u_0\|_{2,\alpha}=1$. This result extends that of Yang (J. Differential Equations, 2015), and complements that of Lu and Yang (Adv. Math. 2009) and Nguyen (arXiv: 1701.08249, 2017).     

Cardinal spline interpolation from

Let be the discrete Hardy space, consisting of those sequences , such that , where , , is the discrete Hilbert transform of . For a sequence , let be the unique cardinal spline of degree interpolating to at the integers. The norm of this operator, , is called a Lebesgue constant from to , and it was proved that .

It is proved in this paper that

     

Extremal properties of the derivatives of the Newman polynomials

Let be a set of distinct positive numbers. The span of

over will be denoted by

Our main result of this note is the following.

Theorem. Suppose . Let be a non-negative integer. Then there are constants 0$"> and 0$"> depending only on , , and such that

where the lower bound holds for all and for all , while the upper bound holds when and and when , , and .