Summability Condition Research Articles

Tauberian theorems for statistical logarithmic summability of strongly measurable fuzzy valued functions

We define statistical logarithmic summability of strongly measurable fuzzy valued functions and we give slowly decreasing type Tauberian conditions under which statistical limit at infinity and statistical logarithmic summability of strongly measurable fuzzy valued functions imply ordinary limit at infinity in one dimensional fuzzy number space $E^1$. Besides, we give slowly oscillating type Tauberian conditions for statistical limit and statistical logarithmic summability of strongly measurable fuzzy valued functions in $n-$dimensional fuzzy number space $E^n$.

Stability of lattice rules and polynomial lattice rules constructed by the component-by-component algorithm

We study quasi-Monte Carlo (QMC) methods for numerical integration of multivariate functions defined over the high-dimensional unit cube. Lattice rules and polynomial lattice rules, which are special classes of QMC methods, have been intensively studied and the so-called component-by-component (CBC) algorithm has been well-established to construct rules which achieve the almost optimal rate of convergence with good tractability properties for given smoothness and set of weights. Since the CBC algorithm constructs rules for given smoothness and weights, not much is known when such rules are used for function classes with different smoothness and/or weights.In this paper we prove that a lattice rule constructed by the CBC algorithm for the weighted Korobov space with given smoothness and weights achieves the almost optimal rate of convergence with good tractability properties for general classes of smoothness and weights which satisfy some summability conditions. Such a stability result also can be shown for polynomial lattice rules in weighted Walsh spaces. We further give bounds on the weighted star discrepancy and discuss the tractability properties for these QMC rules. The results are comparable to those obtained for Halton, Sobol and Niederreiter sequences.

Multi-step inertial proximal contraction algorithms for monotone variational inclusion problems

"In this article, we introduce the multi-step inertial proximal contraction algorithms (MiPCA) to approximate a zero of the sum of two monotone operators, with one of the two operators being monotone and Lipschitz continuous. The weak convergence of the MiPCA is shown under the summability condition formulated in terms of the iterative sequence in a Hilbert space setting. We also investigate the unconditional convergence of the one-step inertial proximal contraction algorithm. Finally, numerical experiments are given to illustrate the advantage of the multi-step inertial proximal contraction algorithms."

Canonical systems with discrete spectrum

We study spectral properties of two-dimensional canonical systems y′(t)=zJH(t)y(t), t∈[a,b), where the Hamiltonian H is locally integrable on [a,b), positive semidefinite, and Weyl's limit point case takes place at b. We answer the following questions explicitly in terms of H: Here asymptotic distribution means summability and limit superior conditions relative to comparison functions growing sufficiently fast. Making an analogy with complex analysis, this corresponds to convergence class and type w.r.t. proximate orders having order larger than 1. It is a surprising fact that these properties depend only on the diagonal entries of H.In 1968 L.de Branges posed the following question as a fundamental problem: We give a complete and explicit answer.

Sufficient Conditions for Absolute Cesàro Summable Factor

Quasi-f-power increasing sequence has been used for infinite series to establish a theorem on a minimal set of sufficient conditions for absolute Cesàro φ-|〖C,α;δ;l|〗_k summable factor. Further, a set of new and well-known arbitrary results have been obtained by using the main theorem. The presented main result has been validated by the previous result under suitable conditions. In this way, the Bounded Input Bounded Output (BIBO) stability of impulse response has been improved by finding a minimal set of sufficient conditions for absolute summability because absolute summable is the necessary and sufficient condition for BIBO stability.

A variational approach to the mean field planning problem

We investigate a first-order mean field planning problem of the form{−∂tu+H(x,Du)=f(x,m)in (0,T)×Rd,∂tm−∇⋅(mHp(x,Du))=0in (0,T)×Rd,m(0,⋅)=m0,m(T,⋅)=mTin Rd, associated to a convex Hamiltonian H with quadratic growth and a monotone interaction term f with polynomial growth.We exploit the variational structure of the system, which encodes the first order optimality condition of a convex dynamic optimal entropy-transport problem with respect to the unknown density m and of its dual, involving the maximization of an integral functional among all the subsolutions u of an Hamilton-Jacobi equation.Combining ideas from optimal transport, convex analysis and renormalized solutions to the continuity equation, we will prove existence and (at least partial) uniqueness of a weak solution (m,u). A crucial step of our approach relies on a careful analysis of distributional subsolutions to Hamilton-Jacobi equations of the form −∂tu+H(x,Du)≤α, under minimal summability conditions on α, and to a measure-theoretic description of the optimality via a suitable contact-defect measure. Finally, using the superposition principle, we are able to describe the solution to the system by means of a measure on the path space encoding the local behavior of the players.

Riemann Summability of Trigonometric Series and Riemann Derivatives of Real Functions

A relation between Riemann summability and Riemann derivative is established and necessary and sufficient conditions for Riemann summability of trigonometric series are obtained

QMC integration for lognormal-parametric, elliptic PDEs: local supports and product weights

We analyze convergence rates of quasi-Monte Carlo (QMC) quadratures for countably-parametric solutions of linear, elliptic partial differential equations (PDE) in divergence form with log-Gaussian diffusion coefficient, based on the error bounds in Nichols and Kuo (J Complex 30(4):444–468, 2014. https://doi.org/10.1016/j.jco.2014.02.004 ). We prove, for representations of the Gaussian random field PDE input with locally supported basis functions, and for continuous, piecewise polynomial finite element discretizations in the physical domain novel QMC error bounds in weighted spaces with product weights that exploit localization of supports of the basis elements representing the input Gaussian random field. In this case, the cost of the fast component-by-component algorithm for constructing the QMC points scales linearly in terms of the integration dimension. The QMC convergence rate $$\mathcal {O}(N^{-1+\delta })$$ (independent of the parameter space dimension s) is achieved under weak summability conditions on the expansion coefficients.

Distributed Subgradient Method With Edge-Based Event-Triggered Communication

This paper proposes a distributed subgradient method for constrained optimization with event-triggered communications. In the proposed method, each agent has an estimate of an optimal solution as a state and iteratively updates it by a consensus-based subgradient algorithm with a projection to a common constraint set. The local communications are carried out by the edge-based triggering mechanism when the difference between the current state and the last triggered state exceeds a threshold. We show that the states of all agents asymptotically converge to one of the optimal solutions under a diminishing and summability condition on a stepsize and a threshold for a trigger condition. We also investigate the convergence rate with respect to the time-averaged state of each agent. The simulation results show that the proposed event-triggered algorithm can reduce the number of communications compared to the time-triggered algorithms.

Quasi–Monte Carlo integration with product weights for elliptic PDEs with log-normal coefficients

Abstract Quasi–Monte Carlo (QMC) integration of output functionals of solutions of the diffusion problem with a log-normal random coefficient is considered. The random coefficient is assumed to be given by an exponential of a Gaussian random field that is represented by a series expansion of some system of functions. Graham et al. (2015, Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients. Numer. Math., 131, 329–368) developed a lattice-based QMC theory for this problem and established a quadrature error decay rate ≈ 1 with respect to the number of quadrature points. The key assumption there was a suitable summability condition on the aforementioned system of functions. As a consequence, product-order-dependent weights were used to construct the lattice rule. In this paper, a different assumption on the system is considered. This assumption, originally considered by Bachmayr et al. (2017c, Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients. ESAIM Math. Model. Numer. Anal., 51, 321–339) to utilise the locality of support of basis functions in the context of polynomial approximations applied to the same type of the diffusion problem, is shown to work well in the same lattice-based QMC method considered by Graham et al.: the assumption leads us to product weights, which enables the construction of the QMC method with a smaller computational cost than Graham et al. A quadrature error decay rate ≈ 1 is established, and the theory developed here is applied to a wavelet stochastic model. By a characterisation of the Besov smoothness, it is shown that a wide class of path smoothness can be treated with this framework.

Nonuniformly elliptic energy integrals with [formula omitted]-growth

We study the local boundedness of minimizers of a nonuniformly energy integral of the form ∫Ωf(x,Dv)dx under p,q-growth conditions of the type λ(x)|ξ|p≤f(x,ξ)≤μ(x)1+|ξ|qfor some exponents q≥p>1 and with nonnegative functions λ,μ satisfying some summability conditions. We use here the original notation introduced in 1971 by Trudinger [26], where λ(x) and μ(x) had the role of the minimum and the maximum eigenvalues of an n×n symmetric matrix aijx and f(x,ξ)=∑i,j=1naijxξiξjwas the energy integrand associated to a linear nonuniformly elliptic equation in divergence form. In this paper we consider a class of energy integrals, associated to nonlinear nonuniformly elliptic equations and systems, with integrands f(x,ξ) satisfying the general growth conditions above.

Summation by Riesz means of the Fourier-Laplace series

In this work, weinvestigate conditions for summability of the Fourier-Laplace series of integrable functions by Riesz means. The kernel of Riesz means is estimated through comparison with the Cesaro means. Properties of D and D* points are required in obtaining this estimation.

On the almost everywhere convergence of the eigenfunction expansions from Liouville classes

The relevance of waves in quantum mechanics naturally implies that the decomposition of arbitrary wave packets in terms of monochromatic waves plays an important role in applications of the theory. When eigenfunction expansions does not converge, then the expansions of the functions with certain smoothness should be considered. Such functions gained prominence primarily through their application in quantum mechanics. In this work we study the almost everywhere convergence of the eigenfunction expansions from Liouville classes , related to the self-adjoint extension of the Laplace operator in torus TN. The sufficient conditions for summability is obtained using the modified Poisson formula. Isomorphism properties of the elliptic differential operators is applied in order to obtain estimation for the Fourier series of the functions from the classes of Liouville .

Dynamic uniqueness for stochastic chains with unbounded memory

We say that a probability kernel exhibits dynamic uniqueness (DU) if all the stochastic chains starting from a fixed past coincide on the future tail σ-algebra. Our first theorem is a set of properties that are pairwise equivalent to DU which allow us to understand how it compares to other more classical concepts. In particular, we prove that DU is equivalent to a weak-ℓ2 summability condition on the kernel. As a corollary to this theorem, we prove that the Bramson–Kalikow and the long-range Ising models both exhibit DU if and only if their kernels are ℓ2 summable. Finally, if we weaken the condition for DU, asking for coincidence on the future σ-algebra for almost every pair of pasts, we obtain a condition that is equivalent to β-mixing (weak-Bernoullicity) of the compatible stationary chain. As a consequence, we show that a modification of the weak-ℓ2 summability condition on the kernel is equivalent to the β-mixing of the compatible stationary chain.

Characterization of Local Besov Spaces via Wavelet Basis Expansions

In this paper we deal with local Besov spaces of periodic functions of one variable. We characterize these spaces in terms of summability conditions on the coefficients in series expansions of their elements with respect to an orthogonal Schauder basis of trigonometric polynomials. We consider a Schauder basis that was constructed by using ideas of a periodic multiresolution analysis and corresponding wavelet spaces. As an interim result we obtain a characterization of local Besov spaces via operators of the orthogonal projection on the corresponding scaling and wavelet spaces. In order to achieve our new results, we substantially use a theorem on the discretization of scaling and wavelet spaces as well as a connection between local and usual classical Besov spaces. The corresponding characterizations are also given for the classical Besov spaces.

Posterior asymptotics of nonparametric location-scale mixtures for multivariate density estimation

Density estimation represents one of the most successful applications of Bayesian nonparametrics. In particular, Dirichlet process mixtures of normals are the gold standard for density estimation and their asymptotic properties have been studied extensively, especially in the univariate case. However, a gap between practitioners and the current theoretical literature is present. So far, posterior asymptotic results in the multivariate case are available only for location mixtures of Gaussian kernels with independent prior on the common covariance matrix, while in practice as well as from a conceptual point of view a location-scale mixture is often preferable. In this paper, we address posterior consistency for such general mixture models by adapting a convergence rate result which combines the usual low-entropy, high-mass sieve approach with a suitable summability condition. Specifically, we establish consistency for Dirichlet process mixtures of Gaussian kernels with various prior specifications on the covariance matrix. Posterior convergence rates are also discussed.

A Sufficient Condition for the Harmonicity of Functions of Two Variables That Satisfy a Difference Laplace Equation

In this paper we weaken the sufficient conditions of harmonicity for functions of two variables. I.I. Privalov had shown that a continuous function that satisfies the Laplace equation in each point of the domain is harmonic. For function of two variables the Privalov’s condition on continuity can be weakened. G.P. Tolstov replaced the continuity condition by the boundness condition, later the author had shown that summability is sufficient. At the same time summability condition can not be weakened substantially.In this paper, while we keep the summability condition, we provide the sufficient condition for harmonicity of the functions, that satisfy less restricted condition than the Laplace equation in all points of the domain. We assume that arbitrary close to any point ζ there exists a collection of four nodes for which a difference relation of Schwartz type for the Laplace equation can be made arbitrary small by the absolute value. Nodes are the ends of two mutually perpendicular segments, that intersect at point ζ. We need to impose a certain weakened continuous assumption on function itself, in case of function that satisfy the traditional Laplace condition this continuity condition follows from the existence of the partial derivatives.

Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients

We consider the linear elliptic equation − div(a∇u) = f on some bounded domain D, where a has the form a = exp(b) with b a random function defined as b(y) = ∑ j ≥ 1yjψj where y = (yj) ∈ ℝNare i.i.d. standard scalar Gaussian variables and (ψj)j ≥ 1 is a given sequence of functions in L∞(D). We study the summability properties of Hermite-type expansions of the solution map y → u(y) ∈ V := H01(D) , that is, expansions of the form u(y) = ∑ ν ∈ ℱuνHν(y), where Hν(y) = ∏j≥1Hνj(yj) are the tensorized Hermite polynomials indexed by the set ℱ of finitely supported sequences of nonnegative integers. Previous results [V.H. Hoang and C. Schwab, M3AS 24 (2014) 797−826] have demonstrated that, for any 0 <p ≤ 1, the ℓp summability of the sequence (j ∥ψj ∥L∞)j ≥ 1 implies ℓp summability of the sequence (∥ uν∥V)ν ∈ ℱ. Such results ensure convergence rates n− s with s = (1/p)−(1/2) of polynomial approximations obtained by best n-term truncation of Hermite series, where the error is measured in the mean-square sense, that is, in L2(ℝN,V,γ) , where γ is the infinite-dimensional Gaussian measure. In this paper we considerably improve these results by providing sufficient conditions for the ℓp summability of (∥uν∥V)ν ∈ ℱ expressed in terms of the pointwise summability properties of the sequence (|ψj|)j ≥ 1. This leads to a refined analysis which takes into account the amount of overlap between the supports of the ψj. For instance, in the case of disjoint supports, our results imply that, for all 0 <p< 2 the ℓp summability of (∥uν∥V)ν ∈ ℱfollows from the weaker assumption that (∥ψj∥L∞)j ≥ 1is ℓq summable for q := 2p/(2−p) . In the case of arbitrary supports, our results imply that the ℓp summability of (∥uν∥V)ν ∈ ℱ follows from the ℓp summability of (jβ∥ψj∥L∞)j ≥ 1 for some β>1/2 , which still represents an improvement over the condition in [V.H. Hoang and C. Schwab, M3AS 24 (2014) 797−826]. We also explore intermediate cases of functions with local yet overlapping supports, such as wavelet bases. One interesting observation following from our analysis is that for certain relevant examples, the use of the Karhunen−Loève basis for the representation of b might be suboptimal compared to other representations, in terms of the resulting summability properties of (∥uν∥V)ν ∈ ℱ. While we focus on the diffusion equation, our analysis applies to other type of linear PDEs with similar lognormal dependence in the coefficients.

Sobolev functions on varifolds

This paper introduces first order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally nonlinear class of generalised weakly differentiable functions and share key functional analytic properties with their Euclidean counterparts. Assuming the varifold to satisfy a uniform lower density bound and a dimensionally critical summability condition on its mean curvature, the following statements hold. Firstly, continuous and compact embeddings of Sobolev spaces into Lebesgue spaces and spaces of continuous functions are available. Secondly, the geodesic distance associated to the varifold is a continuous, not necessarily H\"older continuous Sobolev function with bounded derivative. Thirdly, if the varifold additionally has bounded mean curvature and finite measure, the present Sobolev spaces are isomorphic to those previously available for finite Radon measures yielding many new results for those classes as well. Suitable versions of the embedding results obtained for Sobolev functions hold in the larger class of generalised weakly differentiable functions.

Modified Unit Root Tests with Nuisance Parameter Free Asymptotic Distributions

It is well-known that the asymptotic distributions of the Dickey-Fuller (DF) tests for a unit root with linear process errors are not free of nuisance parameters. In this paper, we introduce a consistent estimator for the nuisance parameters and then use it to modify the DF tests, denoted as R-tests. Under fairly mild moment and summability conditions on the errors, we show that the asymptotic distributions of the R-tests are of the same as the Dickey-Fuller distributions. In Monte Carlo experiments, the R-tests are shown to have improved size properties.