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We indicate how recent results of Grafakos and Teschl (J Fourier Anal Appl 19:167–179, 2013), and Estrada (J Fourier Anal Appl 20:301–320, 2014), relating the Fourier transform of a radial function in $$\mathbb R^n$$ and the Fourier transform of the same function in $$\mathbb R^{n+2}$$ and $$\mathbb R^{n+1}$$ , respectively, are located within known results on transplantation for Hankel transforms.

The authors of [KMM] consider a system of two functional inequalities for a function $$f : {\mathbb{R}} \rightarrow {\mathbb{R}}$$ , and they show that, if certain arithmetical conditions and inequalities for certain parameters are fulfilled, f has to be a polynomial provided that f is continuous at some point x0. This result is derived here under the weaker condition that for some x0 the limit $${\rm lim}_{x \rightarrow x_0} f(x)$$ exists. Moreover, another system of inequalities is given leading to the same result on the nature of f. The methods used also give natural explanations for the fact that Bernoulli numbers play an important role in this context.

We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system with Hessian driven damping and a Tikhonov regularization term in connection with the minimization of a smooth convex function in Hilbert spaces. We obtain fast convergence results for the function values along the trajectories. The Tikhonov regularization term enables the derivation of strong convergence results of the trajectory to the minimizer of the objective function of minimum norm.

In a recent paper, Ng, Tang and Tsai (Math. Ann. 2020) have found an explicit formula for the squeezing function of an annulus via the Loewner differential equation. Their result has led them to conjecture a corresponding formula for planar domains of any finite connectivity stating that the extremum in the squeezing function problem is achieved for a suitably chosen conformal mapping onto a circularly slit disk. In this paper we disprove this conjecture. We also give a conceptually simple potential-theoretic proof of the explicit formula for the squeezing function of an annulus which has the added advantage of identifying all extremal functions., Comment: Version 2: (1) a statement on the history of the notion of squeezing function has been corrected; (2) a new reference [5] (F. Deng: Levi's problem, convexity, and squeezing functions on bounded domains) has been added; (3) a small technical issue with numbering of equations has been resolved

The purpose of this paper is to give, through the second degree character, new characterizations of a part of the family of quasi-symmetric forms. In fact, thanks to the Stieltjes function and also the moments, we give necessary and sufficient conditions for a regular form to be at the same time of the second degree, quasi-symmetric and semiclassical one of class two. We focus our attention not only on the link between all these forms and the Jacobi forms $${{\mathcal {T}}}_{p, q}={{\mathcal {J}}}(p-1/2, q-1/2), \; p, q\in {\mathbb {Z}},~p+q\ge 0$$ but also on their connection with the Tchebychev form of the first kind $${{\mathcal {T}}}={\mathcal J}\left( -1/2, -1/2\right) $$ . The paper concludes by explicitly giving their characteristic elements of the structure relation and of the second order differential equation, which leads to interesting electrostatic models.

In this paper, we consider the Laplace operator in a flat band whose lower boundary periodically oscillates under the Dirichlet boundary condition. The period and the amplitude of oscillations are two independent small parameters. The main result obtained in the paper is the absence of internal lacunas in the lower part of the spectrum of the operator for sufficiently small period and amplitude. We obtain explicit upper estimates of the period and amplitude in the form of constraints with specific numerical constants. The length of the lower part of the spectrum, in which the absence of lacunas is guaranteed, is also expressed explicitly in terms of the period function and the amplitude.

In this paper, we present some new identities for multiple polylogarithm functions by using the methods of iterated integral computations of logarithm functions. Then, by applying these formulas obtained, we establish several duality formulas for Arakawa–Kaneko zeta values and Kaneko–Tsumura $$\eta $$ -values. At the end of the paper, we study a variant of Kaneko–Tsumura $$\eta $$ -function with r-complex variables and establish two formulas about the values of this variant; these two formulas were proved previously by Yamamoto.

The classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi’s result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation$$\begin{aligned} F(x + y) - F(x) - F(y) = yf(x) + xf(y) \end{aligned}$$F(x+y)-F(x)-F(y)=yf(x)+xf(y)considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation.

A dynamical system $$ {\displaystyle \begin{array}{ll}{u}_{tt}-\Delta u-\nabla 1\mathrm{n}\;\rho \cdot \nabla u=0& in\kern0.6em {\mathrm{\mathbb{R}}}_{+}^3\times \left(0,T\right),\\ {}{\left.u\right|}_{t=0}={\left.{u}_t\right|}_{t=0}=0& in\kern0.6em \overline{{\mathrm{\mathbb{R}}}_{+}^3},\\ {}{\left.{u}_z\right|}_{z=0}=f& for\kern0.36em 0\le t\le T,\end{array}} $$ is under consideration, where ρ = ρ(x, y, z) is a smooth positive function; f = f(x, y, t) is a boundary control; u = uf (x, y, z, t) is a solution. With the system one associates a response operator R : f ↦ uf|z = 0. The inverse problem is to recover the function ρ via the response operator. A short representation of the local version of the BC-method, which recovers ρ via the data given on a part of the boundary, is provided. If ρ is constant, the forward problem is solved in explicit form. In the paper, the corresponding representations for the solutions and response operator are derived. A way to use them for testing the BC-algorithm, which solves the inverse problem, is outlined. The goal of the paper is to extend the circle of the BC-method users, who are interested in numerical realization of methods for solving inverse problems.

In this paper, we consider a version of energy minimisation technique applied to images of a 2D fluid flow. The two Navier–Stokes equations describe the static flow of a 2D fluid in terms of velocity field, (u, v), pressure field, p and forcing field, f. Apart from these two Navier–Stokes equations, we have the incompressibility condition to evaluate the three parameters. While implementing this system, random noise (usually non-Gaussian) creeps into the random force field $$\underline{f}(x,y)$$ f ̲ ( x , y ) . We denote this random field by $$\delta \underline{f}(x,y)$$ δ f ̲ ( x , y ) having zero mean and non-trivial second and third moments. We assume that these two moments are known except for some unknown parameters $$\underline{\theta }$$ θ ̲ (like mean, variance, co-variance, skewness, etc.) which we wish to estimate. In the proposed technique, we first calculate the approximate shift in the average fluid energy defined as a quadratic function of the velocity field. The energy method then requires that $$\underline{\theta }$$ θ ̲ should be such that this average increases in the energy due to the random forcing component be minimised. We should, however, note that the standard statistical approach to force field estimation is to calculate the velocity field as a function of the force field and then adopt the statistical moment matching technique. Such an approach assumes spatial ergodicity of the velocity field. This approach to force field estimation is more accurate from the statistical moment matching view point but works only if velocity measurements are made. The former technique of energy minimisation does not require any velocity measurements. Both of these techniques are discussed in this paper and MATLAB simulations presented.

This work has been motivated by recent papers that quantify the density of values of generic quadratic forms and other polynomials at integer points, in particular ones that use Rogers' second moment estimates. In this paper we establish such results in a very general framework. Given any subhomogeneous function (a notion to be defined) $f: \mathbb{R}^n \to \mathbb{R}$, we derive a necessary and sufficient condition on the approximating function $\psi$ for guaranteeing that a generic element $f\circ g$ in the $G$-orbit of $f$ is $\psi$-approximable; that is, $|f\circ g(\mathbf{v})| \le \psi(\|\mathbf{v}\|)$ for infinitely many $\mathbf{v} \in \mathbb{Z}^n$. We also deduce a sufficient condition in the case of uniform approximation. Here, $G$ can be any closed subgroup of $\rm{ASL}_n(\mathbb{R})$ satisfying certain axioms that allow for the use of Rogers-type estimates., Comment: 26 pages; misprints corrected, concluding remarks added

In this paper, first and second kind Chebyshev wavelets are studied. New estimators $$E_{2^{k-1},0}^{(1)}$$ E 2 k - 1 , 0 ( 1 ) , $$E_{2^{k-1},M}^{(2)}$$ E 2 k - 1 , M ( 2 ) , $$E_{2^{k-1},0}^{(3)}$$ E 2 k - 1 , 0 ( 3 ) , $$E_{2^{k-1},M}^{(4)}$$ E 2 k - 1 , M ( 4 ) for first kind Chebyshev wavelets and estimators $$E_{2^{k},0}^{(5)}$$ E 2 k , 0 ( 5 ) , $$E_{2^{k},M}^{(6)}$$ E 2 k , M ( 6 ) , $$E_{2^{k},0}^{(7)}$$ E 2 k , 0 ( 7 ) and $$E_{2^{k},M}^{(8)}$$ E 2 k , M ( 8 ) for second kind Chebyshev wavelets for a function f belonging to generalized H$$\ddot{o}$$ o ¨ lder’s class have been obtained. Also, a method based on first and second kind Chebyshev wavelet approximations has been presented for solving integral equations. Comparison of solutions obtained by both wavelets method has been studied. It is found that second kind Chebyshev wavelet method gives better and accurate solutions as compared to first kind Chebyshev wavelet method. This is a significant achievement of this research paper in wavelet analysis.

In this paper, we provide the boundedness property of the Riesz transforms associated to the Schrodinger operator $${\cal L} = \Delta + {\bf{V}}$$ in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces. The additional potential V considered in this paper is a non-negative function satisfying the suitable reverse Holder’s inequality. Our results are new and general in many cases of problems. As an application of the boundedness property of these singular integral operators, we obtain some regularity results of solutions to Schrodinger equations in the new Morrey space.

In this paper, we introduce a new family of generalized blending-type bivariate Bernstein operators which depends on four parameters $$s_{1}$$ , $$s_{2}$$ , $$\alpha _{1}$$ and $$\alpha _{2}$$ . Approximation properties of these operators are studied, and we obtain the rate of convergence in terms of mixed and partial modulus of continuities. Moreover, we prove a Korovkin- and a Voronovskaja-type theorems for these operators. The last part of the paper is devoted to the associated GBS operators. In this part, we study degree of approximation of the GBS operators in terms of mixed modulus of continuity. GBS operators obtained here give better approximation than the original operators to the function f(x, y). Finally, approximation properties of the suggested operators and their associated GBS operators are discussed on graphs, for some numerical examples to show how GBS operator gives better approximation to f(x, y). Also, approximation properties of the suggested operators for different values of parameters $$s_{1}$$ , $$s_{2}$$ , $$\alpha _{1}$$ and $$\alpha _{2}$$ are illustrated on graphs. It should be mentioned that any increase in $$\alpha _{i}$$ values or any decrease in $$s_{i}$$ values gives better approximation of the suggested operators to f(x, y).

In this paper, the generalized Holder’s class and extended Legendre wavelet are studied. The wavelet approximations of a function whose first and second derivatives belonging to generalized Holder’s class by extended Legendre wavelets have been determined. Two corollaries have been obtained by the main theorems of this paper. Linear and non-linear differential equations have been solved by extended Legendre wavelet methods.

In a previous paper by the author, a family of iterations for computing the matrix square root was constructed by exploiting a recursion obeyed by Zolotarev’s rational minimax approximants of the function $$z^{1/2}$$. The present paper generalizes this construction by deriving rational minimax iterations for the matrix pth root, where $$p \ge 2$$ is an integer. The analysis of these iterations is considerably different from the case $$p=2$$, owing to the fact that when $$p>2$$, rational minimax approximants of the function $$z^{1/p}$$ do not obey a recursion. Nevertheless, we show that several of the salient features of the Zolotarev iterations for the matrix square root, including equioscillatory error, order of convergence, and stability, carry over to the case $$p>2$$. A key role in the analysis is played by the asymptotic behavior of rational minimax approximants on short intervals. Numerical examples are presented to illustrate the predictions of the theory.

Insolvency risk measures play important role in the theory and practice of risk management. In this paper, we provide a numerical procedure to compute vectors of their exact values and prove for them new upper and/or lower bounds which are shown to be attainable. More precisely, we investigate a general insolvency risk measure for a regime-switching Sparre Andersen model in which the distributions of claims and/or wait times are driven by a Markov chain. The measure is defined as an arbitrary increasing function of the conditional expected harm of the deficit at ruin, given the initial state of the Markov chain. A vector-valued operator L, generated by the regime-switching process, is introduced and investigated. We show a close connection between the iterations of L and the risk measure in a finite horizon. The approach assumed in the paper enables to treat in a unified way several discrete and continuous time risk models as well as a variety of important vector-valued insolvency risk measures.

In [3], we derived three results in additive combinatorics for function fields. The proofs of these results depended on a recent bound for the large sieve with sparse sets of moduli for function fields by the first and third-named authors in [1]. Unfortunately, they discovered an error in this paper and demonstrated in [2] that this result cannot hold in full generality. In the present paper, we formulate a plausible conjecture under which the said three results in [3] remain true and the method of proof goes through using the same arguments. However, these results are now only conditional and still await a full proof.

In a previous paper (Funct. Anal. Appl., 3 (2015), 205–209), we defined quantum Markov states. Here we recall this definition and present a proof of the results from that paper (which are given there without proofs). We give a definition of a quantum hidden Markov state generated by a function of a quantum Markov process and show how it is related to other definitions of such states. Our definitions work for quantum Markov fields on ℤN and on graphs. We consider an example with the Cayley tree.

The paper is devoted to the investigations of holomorphic functions on complete n-circular domains G of ℂn which are solutions of some partial differential equations in G. Our considerations concern a collection M G , k ≥ 2, of holomorphic solutions of equations corresponding to planar Sakaguchi’s conditions for starlikeness with respect to k-symmetric points. In an earlier paper of the first author some embedding theorems for M G k were given. In this paper we solve the problem of finding some sharp estimates of m-homogeneous polynomials in a power series expansion of f from M G . We obtain a formula of the extremal function which includes some special functions. Moreover, its construction is based on properties of hypergeometric functions and (j, k)-symmetric functions. The (j, k)-symmetric functions were considered in several papers of the second author and his co-author, J. Polubinski.

This paper is concerned with the time-step condition of linearized implicit finite difference method for solving the Gross-Pitaevskii equation with an angular momentum rotation term. Unlike the existing studies in the literature, where the cut-off function technique was used to establish the error estimates under some conditions of the time-step size, this paper introduces an induction argument and a ‘lifting’ technique as well as some useful inequalities to build the optimal maximum error estimate without any constraints on the time-step size. The analysis method can be directly extended to the general nonlinear Schrodinger-type equations in twoand three-dimensions and other linear implicit finite difference schemes. As a by-product, this paper defines a new type of energy functional of the grid functions by using a recursive relation to prove that the proposed scheme preserves well the total mass and energy in the discrete sense. Several numerical results are reported to verify the error estimates and conservation laws.

In this paper, we study the well-posedness and stability of a wave equation with infinitely structural memory, herein the memory kernel function does not satisfy the monotonicity. For the model, the history function space setting is a main difficulty because the usual space setting will lead the shift semigroup to be a unbounded semigroup. In the present paper, we modify the history function space setting and prove the well-posedness of the system. Further we study the stability of the system via Lyapunov function method. By constructing appropriate Lyapunov function, we show that the energy function of the system decays exponentially if the memory kernel function satisfies some conditions. Finally, we give an example of the memory kernel function that is not monotone but satisfies all conditions proposed in the present paper.

This paper is concerned with a Delsarte-type extremal problem. Denote by$${\mathcal {P}}(G)$$P(G)the set of positive definite continuous functions on a locally compact abelian groupG. We consider the function class, which was originally introduced by Gorbachev,$$\begin{aligned}&{\mathcal {G}}(W, Q)_G = \left\{ f \in {\mathcal {P}}(G) \cap L^1(G)~:\right. \\&\qquad \qquad \qquad \qquad \qquad \left. f(0) = 1, ~ {\text {supp}}{f_+} \subseteq W,~ {\text {supp}}{\widehat{f}} \subseteq Q \right\} \end{aligned}$$G(W,Q)G=f∈P(G)∩L1(G):f(0)=1,suppf+⊆W,suppf^⊆Qwhere$$W\subseteq G$$W⊆Gis closed and of finite Haar measure and$$Q\subseteq {\widehat{G}}$$Q⊆G^is compact. We also consider the related Delsarte-type problem of finding the extremal quantity$$\begin{aligned} {\mathcal {D}}(W,Q)_G = \sup \left\{ \int _{G} f(g) \mathrm{d}\lambda _G(g) ~ : ~ f \in {\mathcal {G}}(W,Q)_G\right\} . \end{aligned}$$D(W,Q)G=sup∫Gf(g)dλG(g):f∈G(W,Q)G.The main objective of the current paper is to prove the existence of an extremal function for the Delsarte-type extremal problem$${\mathcal {D}}(W,Q)_G$$D(W,Q)G. The existence of the extremal function has recently been established by Berdysheva and Révész in the most immediate case where$$G={\mathbb {R}}^d$$G=Rd. So, the novelty here is that we consider the problem in the general setting of locally compact abelian groups. In this way, our result provides a far reaching generalization of the former work of Berdysheva and Révész.

In the first part of the paper we give a new integral representation for the principal branch of the Lambert W function. Then we deduce two continued fraction expansions for this branch. At the end of the paper we study the numerical behavior of the approximants of these expansions.

In this paper we consider a nonlinear reaction–diffusion equation in which the nonlinear term is described by a potential energy function. For this class of pde, equilibria admit a variational formulation and they can be determined by a suitable finite dimensional reduction technique called Amann–Conley–Zendher (ACZ) reduction. By extension, the ACZ reduction applies also to the pde dynamics, leading to a finite dimensional description in terms of an ode. It turns out that this ode is a gradient dynamics defined by a potential. While the static case is recovered perfectly in the reduced representation, the reduced dynamics appears to be only a good approximation of the original pde dynamics. The main aim of this paper is to present a number of tools that allow to deal with this uncertainty. First of all we show that the reduced potential W is derived by the original variational functional for the equilibria, and that it contains all the relevant information about the reduced dynamics. In particular we show that the Morse Index of the equilibra of the reduced potential W is the same of the original one, therefore the reduced dynamics give a faithful representation of the stability properties of the original equilibria. Then we highlight the strict analogy between the ACZ reduced dynamics and the approximate inertial manifold description by Temam. This allows to control the discrepancy between the exact and the reduced dynamics, by showing that the orbits of the exact dynamics enter in a thin neighborhood of our approximate inertial manifold after a certain transient time. Finally, the lack of information due to the above approximation may be modelled by adding a stochastic noise term to the reduced ode, and by considering a description of this randomly perturbed system using Freidlin–Wentzell theory, we show that the Large Deviations rate function is given by the reduced potential.

We extend our result on the convergence of double recurrence Wiener-Wintner averages to the case where we have a polynomial exponent. We will show that there exists a single set of full measure for which the averages \[ \frac{1}{N} \sum_{n=1}^N f_1(T^{an}x)f_2(T^{bn}x)\phi(p(n)) \] converge for any polynomial $p$ with real coefficients, and any continuous function $\phi$ from the torus to the set of complex numbers . We also show that if either function belongs to an orthogonal complement of an appropriate Host-Kra-Ziegler factor that depends on the degree of the polynomial $p$, then the averages converge to zero uniformly for all polynomials. This paper combines the authors' previously announced work., Comment: This is the final version to appear in Journal d'Analyse mathematique. This latest version combines the previously posted papers on the arxiv website as arXiv:1408.3064 and arXiv:1409.0463

In this paper, after a brief review of the general theory concerning regularized derivatives and integrals of a function with respect to another function, we provide a peculiar fractional generalization of the $(1+1)$-dimensional Dodson's diffusion equation. For the latter we then compute the fundamental solution, which turns out to be expressed in terms of an M-Wright function of two variables. Then, we conclude the paper providing a few interesting results for some nonlinear fractional Dodson-like equations., 10 pages, 3 figures

In this paper, we propose a meshless method using radial basis functions (RBFs) method based on finite-difference method to solve a nonlinear inverse convection–reaction–diffusion problem with an unknown source function. Indeed, the aim of this paper is to show that the meshless method based on the radial basis functions approach is also suitable for the treatment of the inverse nonlinear partial differential equations. Usually, the matrices obtained from the discretization of the equations are ill-conditioned, especially in higher-dimensional problems. To overcome such difficulties, we use Tikhonov regularization method. We prove that the time discrete scheme is stable using the energy method. The effectiveness of the proposed method is illustrated by numerical examples.

For a sequence of integers $\{a(x)\}_{x \geq 1}$ we show that the distribution of the pair correlations of the fractional parts of $\{ \langle \alpha a(x) \rangle \}_{x \geq 1}$ is asymptotically Poissonian for almost all $\alpha$ if the additive energy of truncations of the sequence has a power savings improvement over the trivial estimate. Furthermore, we give an estimate for the Hausdorff dimension of the exceptional set as a function of the density of the sequence and the power savings in the energy estimate. A consequence of these results is that the Hausdorff dimension of the set of $\alpha$ such that $\{\langle \alpha x^d \rangle\}$ fails to have Poissonian pair correlation is at most $\frac{d+2}{d+3} < 1$. This strengthens a result of Rudnick and Sarnak which states that the exceptional set has zero Lebesgue measure. On the other hand, classical examples imply that the exceptional set has Hausdorff dimension at least $\frac{2}{d+1}$. An appendix by Jean Bourgain was added after the first version of this paper was written. In this appendix two problems raised in the paper are solved., Comment: 22 pages. Version 2 contains an appendix by Jean Bourgain. Version 3 with corrections suggested by the referee. The paper will be published in the Israel Journal of Mathematics

We consider the problem of approximating a smooth function from finitely many pointwise samples using $$\ell ^1$$ minimization techniques. In the first part of this paper, we introduce an infinite-dimensional approach to this problem. Three advantages of this approach are as follows. First, it provides interpolatory approximations in the absence of noise. Second, it does not require a priori bounds on the expansion tail in order to be implemented. In particular, the truncation strategy we introduce as part of this framework is independent of the function being approximated, provided the function has sufficient regularity. Third, it allows one to explain the key role weights play in the minimization, namely, that of regularizing the problem and removing aliasing phenomena. In the second part of this paper, we present a worst-case error analysis for this approach. We provide a general recipe for analyzing this technique for arbitrary deterministic sets of points. Finally, we use this tool to show that weighted $$\ell ^1$$ minimization with Jacobi polynomials leads to an optimal method for approximating smooth, one-dimensional functions from scattered data.

The paper discusses the symbolic function of the hyperbolic paraboloid surface in Preston Scott Cohen’s design of the Lightfall at Tel Aviv Museum of Art and its precedents. While analysing several examples of the use of the hyperbolic paraboloid surface in architecture, the paper proposes three modes of symbolic use of the surface: operative, demonstrative and figurative.

In this paper, we are concerned with variable coefficients plate system subjected to three partially distributed feedbacks: time-varying delay, frictional and viscoelastic dissipations. This work is devoted to, without any prior quantification of both decay rate of relaxation function and growth rate of frictional dissipation near the origin, establish a general decay result which corresponds to a certainly stable ODE. Our result extends the decay result obtained for some kind of problems with finite history to problem with infinite history. Moreover, this paper allows a wider class of kernels of infinite history, and the usual exponential and polynomial decay rates are only special cases. The proof is based on the multiplier method and some techniques about convex functionals.

It has been proven that there exists a one-to-one correspondence H(t, x) between solutions of the linear system and the nonlinear system in the previous work. However, there is no paper considering the Holder regularity of the transformation H(t, x) in the literature. This paper fills the gap. We establish a strict proof of the Holder regularity of the transformation H(t, x). We show that the conjugating function H(t, x) in the generalized Hartman–Grobman theorem is always Holder continuous.

In the paper, using the concept of weakly weighted sharing and relaxed weighted sharing, we investigate the uniqueness problems of certain type of difference polynomials that share a small function. The results of the paper improve and extend some recent results due to C. Meng [Math. Bohem., 139(2014), 89–97] and the present first author [Commun. Math. Stat., 3(2015), 227–238].

An example of solving a boundary-value problem for a homogeneous Monge–Ampere equation is given, which produces a Bellman function for an extremal problem on the space BMO. The paper contains a step-by-step instruction for calculation of this function. Cases of rather complicated foliations are considered. This illustrates the technique elaborated in a paper by Ivanishvili, Stolyarov, Vasyunin, and Zatitskiy. Bibliography: 6 titles.

In this paper, we investigate the inverse limits of two related parameterized families of upper semi-continuous set-valued functions. We include a theorem one consequence of which is that certain inverse limits with a single bonding function from one of these families are the closure of a topological ray (usually with indecomposable remainder). Also included is a theorem giving a new sufficient condition that an inverse limit with set-valued functions be an indecomposable continuum. It is shown that some, but not all, functions from these families produce chainable continua. This expands the list of examples of chainable continua produced by set-valued functions that are not mappings. The paper includes theorems on constructing subcontinua of inverse limits as well as theorems on expressing inverse limits with set-valued functions as inverse limits with mappings.

In this paper, we show that circularly polarized transverse stress waves, standing shear stress waves, and oscillatory shear stress waves can propagate in a new class of viscoelastic solid bodies which are a subclass of bodies described by implicit constitutive theories. The class of models that is being considered includes as sub-classes, the classical Kelvin–Voigt model, the new models introduced by Rajagopal wherein the Cauchy–Green tensor is a non-linear function of the stress, and the Navier–Stokes fluid model. The solutions established in this paper are generalizations of solutions that have been established within the context of nonlinear elasticity by Carroll, and Destrade and Saccomandi, to the new class of elastic and viscoelastic bodies that are being considered.

In this paper, we provide an analytical frame work for the following multi-valued parabolic variational inequality in a cylindrical domain \({Q = \Omega \times (0, \tau)}\) : Find \({{u \in K}}\) and an \({{\eta \in L^{p'}(Q)}}\) such that $$\eta \in f(\cdot,\cdot,u), \quad \langle u_t + Au, v - u\rangle + \int_Q \eta (v - u)\,{\rm d}x{\rm d}t \ge 0, \quad \forall \, v \in K,$$ where \({{K \subset X_0 = L^p(0,\tau;W_0^{1,p}(\Omega))}}\) is some closed and convex subset, A is a time-dependent quasilinear elliptic operator, and the multi-valued function \({{s \mapsto f(\cdot,\cdot,s)}}\) is assumed to be upper semicontinuous only, so that Clarke’s generalized gradient is included as a special case. Thus, parabolic variational–hemivariational inequalities are special cases of the problem considered here. The extension of parabolic variational–hemivariational inequalities to the general class of multi-valued problems considered in this paper is not only of disciplinary interest, but is motivated by the need in applications. The main goals are as follows. First, we provide an existence theory for the above-stated problem under coercivity assumptions. Second, in the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence, comparison, and enclosure results. Third, the order structure of the solution set enclosed by sub-supersolutions is revealed. In particular, it is shown that the solution set within the sector of sub-supersolutions is a directed set. As an application, a multi-valued parabolic obstacle problem is treated.

In this paper, we investigate meromorphic functions that share a small function with one of its linear differential polynomials and prove several theorems which generalize and improve the main results given by J. L. Zhang and L. Z. Yang. Some examples are provided to show that the results in this paper are sharp.

In this paper, we shall study the solutions of functional equations of the form $$\Phi \sum\limits_{\alpha \in \mathbb{Z}^s } {a(\alpha )\Phi (M \cdot - \alpha ),}$$ where Φ = (ϕ 1, ...,ϕ r ) T is an r × 1 column vector of functions on the s-dimensional Euclidean space, $$a: = (a(\alpha ))_{\alpha \in \mathbb{Z}^s }$$ is an exponentially decaying sequence of r×r complex matrices called refinement mask and M is an s × s integer matrix such that limn → ∞ M −n = 0. We are interested in the question, for a mask a with exponential decay, if there exists a solution Φ to the functional equation with each function ϕ j , j = 1, ..., r, belonging to L 2(ℝ s ) and having exponential decay in some sense? Our approach will be to consider the convergence of vector cascade algorithms in weighted L 2 spaces. The vector cascade operator Q a,M associated with mask a and matrix M is defined by $$Q_{a,M} f: = \sum\limits_{\alpha \in \mathbb{Z}^s } {a(\alpha )f(M \cdot - \alpha ), f = \left( {f_1 , \ldots f_r } \right)^T \in \left( {L_{2,\mu } \left( {\mathbb{R}^s } \right)} \right)^r .}$$ The iterative scheme (Q f)n=1,2,... is called a vector cascade algorithm or a vector subdivision scheme. The purpose of this paper is to provide some conditions for the vector cascade algorithm to converge in (L 2 (ℝ s )) r , the weighted L 2 space. Inspired by some ideas in [Jia, R. Q., Li, S.: Refinable functions with exponential decay: An approach via cascade algorithms. J. Fourier Anal. Appl., 17, 1008–1034 (2011)], we prove that if the vector cascade algorithm associated with a and M converges in (L 2(ℝ s )) r , then its limit function belongs to (L 2, μ (ℝ s )) r for some µ > 0.

This paper is concerned with the asymptotic behavior on $${\partial\Omega}$$ of boundary blow-up solutions to semilinear elliptic equations $$\left\{\begin{array}{ll} \Delta u=b(x)f(u),~~ &x\in \Omega, \\ u(x)=\infty, ~~ &x\in\partial\Omega,\end{array} \right.$$ where b(x) is a nonnegative function on $${\Omega}$$ and may vanish on $${\partial\Omega}$$ at a very degenerate rate; f is nonnegative function on [0,∞) and normalized regularly varying or rapidly varying at infinity. The main feature of this paper is to establish a unified and explicit asymptotic formula when the function f is normalized regularly varying or grows faster than any power function at infinity. The effect of the mean curvature of the nearest point on the boundary in the second-order approximation of the boundary blow-up solution is also discussed. Our analysis relies on suitable upper and lower solutions and the Karamata regular variation theory.

A simple technique for calculation of numerical integration errors for physically meaningful functions is presented. In this paper, first the function classes are introduced that relate a function studied directly to a particular physical problem. The matrix elements defining a particular class of functions are then directly incorporated into the error formula. The proposed technique relies on the extension of the concept of the Taylor series. Since the standard Taylor series can be used for calculating the errors for smooth functions, in this paper the behaviour of the error in the immediate vicinity of discontinuity is considered only. The extended Taylor series is obtained by performing the Taylor series expansion on both sides of the discontinuity, using a matrix to describe the behaviour of the function at the discontinuity position and finally summing up all terms proportional to the function value and of all its derivatives. For illustration, several basic classes of physically meaningful functions are introduced and the extended Taylor series is derived. The series is then used to calculate the local error of the numerical integration. The numerical results obtained confirm the accuracy of the derived formulae.

This paper presents a methodology for using varying sample sizes in batch-type optimization methods for large-scale machine learning problems. The first part of the paper deals with the delicate issue of dynamic sample selection in the evaluation of the function and gradient. We propose a criterion for increasing the sample size based on variance estimates obtained during the computation of a batch gradient. We establish an $${O(1/\epsilon)}$$ complexity bound on the total cost of a gradient method. The second part of the paper describes a practical Newton method that uses a smaller sample to compute Hessian vector-products than to evaluate the function and the gradient, and that also employs a dynamic sampling technique. The focus of the paper shifts in the third part of the paper to L 1-regularized problems designed to produce sparse solutions. We propose a Newton-like method that consists of two phases: a (minimalistic) gradient projection phase that identifies zero variables, and subspace phase that applies a subsampled Hessian Newton iteration in the free variables. Numerical tests on speech recognition problems illustrate the performance of the algorithms.

In this paper we consider the infinite relaxation of the corner polyhedron with 2 rows. For the 1-row case, Gomory and Johnson proved in their seminal paper a sufficient condition for a minimal function to be extreme, the celebrated 2-Slope Theorem. Despite increased interest in understanding the multiple row setting, no generalization of this theorem was known for this case. We present an extension of the 2-Slope Theorem for the case of 2 rows by showing that minimal 3-slope functions satisfying an additional regularity condition are facets (and hence extreme). Moreover, we show that this regularity condition is necessary, unveiling a structure which is only present in the multi-row setting.

Let μ be a Borel charge (i.e., a real Borel measure) on ℝ, and let $ {P_{(y)}}(t) = \frac{y}{\pi \left( {{y^2} + {t^2}} \right)},y > 0 $ , t ∈ ℝ, denote the Poisson kernel. Bourgain proved that for a nonnegative μ and for many points t ∈ ℝ, the variation of the function $ y \mapsto \left( \mu * {P_{ {(y)}}} \right)(x) $ on (0, 1] is finite. This is true, in particular, for so-called B-points x introduced in a previous author’s paper, In the present paper, we give new descriptions of B-points which are adjusted to some applications of this notion. Bibliography: 5 titles.

In this paper, we present a large-update interior-point algorithm for convex quadratic semi-definite optimization based on a new kernel function. The proposed function is strongly convex. It is not self-regular function and also the usual logarithmic function. The goal of this paper is to investigate such a kernel function and show that the algorithm has favorable complexity bound in terms of the elegant analytic properties of the kernel function. The complexity bound is shown to be \(O\left( {\sqrt n \left( {\log n} \right)^2 \log \frac{n} {\varepsilon }} \right)\). This bound is better than that by the classical primal-dual interior-point methods based on logarithmic barrier function and recent kernel functions introduced by some authors in optimization fields. Some computational results have been provided.

In this article we consider a sequence of hierarchical space model of inverse problems. The underlying function is estimated from indirect observations over a variety of error distributions including those that are heavy-tailed and may not even possess variances or means. The main contribution of this paper is that we establish some oracle inequalities for the inverse problems by using quantile coupling technique that gives a tight bound for the quantile coupling between an arbitrary sample p-quantile and a normal variable, and an automatic selection principle for the nonrandom filters. This leads to the data-driven choice of weights. We also give an algorithm for its implementation. The quantile coupling inequality developed in this paper is of independent interest, because it includes the median coupling inequality in literature as a special case.

Let T be Takagi's continuous but nowhere-differentiable function. This paper considers the size of the level sets of T both from a probabilistic point of view and from the perspective of Baire category. We first give more elementary proofs of three recently published results. The first, due to Z. Buczolich, states that almost all level sets (with respect to Lebesgue measure on the range of T) are finite. The second, due to J. Lagarias and Z. Maddock, states that the average number of points in a level set is infinite. The third result, also due to Lagarias and Maddock, states that the average number of local level sets contained in a level set is 3/2. In the second part of the paper it is shown that, in contrast to the above results, the set of ordinates y with uncountably infinite level sets is residual, and a fairly explicit description of this set is given. The paper also gives a negative answer to a question of Lagarias and Maddock by showing that most level sets (in the sense of Baire category) contain infinitely many local level sets, and that a continuum of level sets even contain uncountably many local level sets. Finally, several of the main results are extended to a version of T with arbitrary signs in the summands., Comment: Added a new Section 5 with generalization of the main results; some new and corrected proofs of the old material; 29 pages, 3 figures

In this paper, we consider the differential inclusion in $${\mathbb{R}^N}$$ involving the p(x)-Laplacian of the type $${\begin{array}{lll}-\triangle_{p(x)} u+V(x)|u|^{p(x)-2}u\in \partial F(x,u(x)),\;\;{\rm in}\;\;\mathbb{R}^N,\quad\quad\quad\quad\quad\quad ({\rm P})\end{array}}$$ where $${p: \mathbb{R}^N \to {\mathbb{R}}}$$ is Lipschitz continuous function satisfying some given assumptions. The approach used in this paper is the variational method for locally Lipschitz functions. Under suitable oscillatory assumptions on the potential F at zero or at infinity, we show the existence of infinitely many solutions of (P). We also establish a Bartsch-Wang type compact embedding theorem for variable exponent spaces.

The purpose of this paper is to study the L 2 boundedness of operators of the form f ↦ ψ(x) ∫ f (γ t (x))K(t)dt, where γ t (x) is a C ∞ function defined on a neighborhood of the origin in (t, x) ∈ ℝ N × ℝ n , satisfying γ 0(x) ≡ x, ψ is a C ∞ cut-off function supported on a small neighborhood of 0 ∈ ℝ n , and K is a “multi-parameter singular kernel” supported on a small neighborhood of 0 ∈ ℝ N . The goal is, given an appropriate class of kernels K, to give conditions on γ such that every operator of the above form is bounded on L 2. The case when K is a Calderon-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their conditions to the case when K has a “multi-parameter” structure. For example, when K is given by a “product kernel.” Even when K is a Calderon- Zygmund kernel, our methods yield some new results. This is the first paper in a three part series, the later two of which are joint with E. M. Stein. The second paper deals with the related question of L p boundedness, while the third paper deals with the special case when γ is real analytic.