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This paper presents smoothing schemes for obtaining approximate stationary points of unconstrained or linearly-constrained composite nonconvex-concave min-max (and hence nonsmooth) problems by applying well-known algorithms to composite smooth approximations of the original problems. More specifically, in the unconstrained (resp. constrained) case, approximate stationary points of the original problem are obtained by applying, to its composite smooth approximation, an accelerated inexact proximal point (resp. quadratic penalty) method presented in a previous paper by the authors. Iteration complexity bounds for both smoothing schemes are also established. Finally, numerical results are given to demonstrate the efficiency of the unconstrained smoothing scheme.

The paper aims to extend the theory and application of nonconvex Newton-type methods, namely trust region and cubic regularization, to the settings in which, in addition to the solution of subproblems, the gradient and the Hessian of the objective function are approximated. Using certain conditions on such approximations, the paper establishes optimal worst-case iteration complexities as the exact counterparts. This paper is part of a broader research program on designing, analyzing, and implementing efficient second-order optimization methods for large-scale machine learning applications. The authors were based at UC Berkeley when the idea of the project was conceived. The first two authors were PhD students, the third author was a postdoc, all supervised by the fourth author.

This paper examines the practice of simultaneous interpretation of theater performances, in particular between Estonian and Russian, in Estonia over 70 years. This type of interpreting has not received much scholarly attention; rather, studies have mostly focused on the sign language interpretation of theater performances for the deaf community. I conducted interdisciplinary historical research relying on the oral history method to help preserve the fast-disappearing oral heritage of theater interpreting. This paper distinguishes between two periods in theater interpreting in Estonia, as determined by two drastically different sociopolitical periods in Estonia’s history. Drawing upon a total of 88 interviews with interpreters, people who recruited interpreters, and audience members, I identified and interviewed a total of 15 theater interpreters. I also analysed newspaper articles and performance schedules, which usually yielded single-word mentions of interpretation having taken place. This paper examines answers to the questions of who interpreted what, how, and when, and reaches the conclusion that theater interpreting can be a tool to bridge a gap between two communities as well as to facilitate integration in the same cultural space.

PurposeThe purpose of this paper is to determine the structure of nilpotent (n+6)-dimensional n-Lie algebras of class 2 when n≥4.Design/methodology/approachBy dividing a nilpotent (n+6)-dimensional n-Lie algebra of class 2 by a central element, the authors arrive to a nilpotent (n+5) dimensional n-Lie algebra of class 2. Given that the authors have the structure of nilpotent (n+5)-dimensional n-Lie algebras of class 2, the authors have access to the structure of the desired algebras.FindingsIn this paper, for each n≥4, the authors have found 24 nilpotent (n+6) dimensional n-Lie algebras of class 2. Of these, 15 are non-split algebras and the nine remaining algebras are written as direct additions of n-Lie algebras of low-dimension and abelian n-Lie algebras.Originality/valueThis classification of n-Lie algebras provides a complete understanding of these algebras that are used in algebraic studies.

Purpose The purpose of this paper is to study the Hölder calmness of solutions to equilibrium problems and apply it to economics. Design/methodology/approach The authors obtain the Hölder calmness by using an effective approach. More precisely, under the key assumption of strong convexity, sufficient conditions for the Hölder continuity of solution maps to equilibrium problems are established. Findings A new result in stability analysis for equilibrium problems and applications in economics is archived. Originality/value The authors confirm that the paper has not been published previously, is not under consideration for publication elsewhere and is not being simultaneously submitted elsewhere.

The paper aims to establish new results on Tauberian theorem for Cesàro summability of double sequences of fuzzy numbers, and thus to extend and unify several results in the available literature. Further, a number of special cases, corollaries and illustrative example in support of the investigation of this paper are also presented.

At the 1900 International Congress of Mathematicians, Hilbert claimed that the Riemann zeta function is not the solution of any algebraic ordinary differential equation its region of analyticity \cite{HilbertProb}. In 2015, Van Gorder addresses the question of whether the Riemann zeta function satisfies a {\it non}-algebraic differential equation and constructs a differential equation of infinite order which zeta satisfies \cite{RHequiv}. However, as he notes in the paper, this representation is formal and Van Gorder does not attempt to claim a region or type of convergence. In this paper, we show that Van Gorder's operator applied to the zeta function does not converge pointwise at any point in the complex plane. We also investigate the accuracy of truncations of Van Gorder's operator applied to the zeta function and show that a similar operator applied to zeta and other $L$-functions does converge., Comment: This version varies from the published version in JNT in that a convergence issue has been corrected. Section 4.1 is replaced by 2 new sections to correct this. (Previous version is the published version.)

For complex ρ, the spectral properties of the Toeplitz matrix K n ( ρ ) = [ ρ | j − k | ] j , k = 1 n , often called the Kac–Murdock–Szegő matrix, have been examined in detail in two recent papers. The second paper, in particular, introduced the concept of borderline curves. These are two closed curves in the complex-ρ plane that consist of all the ρ for which K n ( ρ ) possesses some eigenvalue whose magnitude equals the matrix dimension n. The purpose of the present paper is to examine eigenvalue bifurcations in both a qualitative and a quantitative manner, and to discuss connections between bifurcations and the borderline curves.

In 1986, Proinov published an explicit lower bound for the diaphony of finite and infinite sequences of points contained in the d−dimensional unit cube [Proinov, P. D.:On irregularities of distribution, C. R. Acad. Bulgare Sci. 39 (1986), no. 9, 31–34]. However, his widely cited paper does not contain the proof of this result but simply states that this will appear elsewhere. To the best of our knowledge, this proof was so far only available in a monograph of Proinov written in Bulgarian [Proinov, P. D.: Quantitative Theory of Uniform Distribution and Integral Approximation, University of Plovdiv, Bulgaria (2000)]. The first contribution of our paper is to give a self contained version of Proinov’s proof in English. Along the way, we improve the explicit asymptotic constants implementing recent, and corrected results of [Hinrichs, A.—Markhasin, L.: On lower bounds for the ℒ2 -discrepancy, J. Complexity 27 (2011), 127–132.] and [Hinrichs, A.—Larcher, G.: An improved lower bound for the ℒ2 -discrepancy, J. Complexity 34 (2016), 68–77]. (The corrections are due to a note in [Hinrichs, A.—Larcher, G. An improved lower bound for the ℒ2 -discrepancy, J. Complexity 34 (2016), 68–77].) Finally, as a main result, we use the method of Proinov to derive an explicit lower bound for the dyadic diaphony of finite and infinite sequences in a similar fashion.

Solidification heat transfer process of billet is described by nonlinear partial differential equation (PDE). Due to the poor productive environment, the boundary condition of this nonlinear PDE is difficult to be fixed. Therefore, the identification of boundary condition of two-dimensional nonlinear PDE is considered. This paper transforms the identification of boundary condition into a PDE optimization problem. The Lipchitz continuous of the gradient of cost function is proved based on the dual equation. In order to solve this optimization problem, this paper presents a modified conjugate gradient algorithm, and the global convergence of which is analyzed. The results of the simulation experiment show that the modified conjugate gradient algorithm obviously reduces the iterative number and running time. Due to the ill-posedness of the identification of boundary condition, this paper combines regularization method with the modified conjugate gradient algorithm. The simulation experiment illustrates that regularization method can eliminate the ill-posedness of this problem. Finally, the experimental data of a steel plant illustrate the validity of this paper’s method.

Recently, in a series of papers "simple" versions of direct-injective and direct-projective modules have been investigated. These modules are termed as "simple-direct-injective" and "simple-direct-projective", respectively. In this paper, we give a complete characterization of the aforementioned modules over the ring of integers and over semilocal rings. The ring is semilocal if and only if every right module with zero Jacobson radical is simple-direct-projective. The rings whose simple-direct-injective right modules are simple-direct-projective are fully characterized. These are exactly the left perfect right $H$-rings. The rings whose simple-direct-projective right modules are simple-direct-injective are right max-rings. For a commutative Noetherian ring, we prove that simple-direct-projective modules are simple-direct-injective if and only if simple-direct-injective modules are simple-direct-projective if and only if the ring is Artinian. Various closure properties and some classes of modules that are simple-direct-injective (resp. projective) are given.

The (interior) transmission eigenvalue problems are a type of non-elliptic, non-selfadjoint and nonlinear spectral problems that arise in the theory of wave scattering. They connect to the direct and inverse scattering problems in many aspects in a delicate way. The properties of the transmission eigenvalues have been extensively and intensively studied over the years, whereas the intrinsic properties of the transmission eigenfunctions are much less studied. Recently, in a series of papers, several intriguing local and global geometric structures of the transmission eigenfunctions are discovered. Moreover, those longly unveiled geometric properties produce some interesting applications of both theoretical and practical importance to direct and inverse scattering problems. This paper reviews those developments in the literature by summarizing the results obtained so far and discussing the rationales behind them. There are some side results of this paper including the general formulations of several types of transmission eigenvalue problems, some interesting observations on the connection between the transmission eigenvalue problems and several challenging inverse scattering problems, and several conjectures on the spectral properties of transmission eigenvalues and eigenfunctions, with most of them are new to the literature.

This paper focuses on the problem of image restoration under Cauchy noise. The variational method, which constructs the data fidelity term involving the Cauchy distribution by MAP estimator, has been proven to be a successful approach. In this paper, a nonlinear diffusion equation is proposed to deal with it. The main ingredients of the proposed equation are a gray level based diffusivity that estimates the amplitude of the noise and a classical gradient based diffusivity that controls the anisotropic diffusion according to the image’s local structure. The proposed equation has the nondivergence form, and its properties, including the existence, uniqueness, and stability of solutions, are established by the notion of viscosity solution. Experimental results show the superiority of the proposed equation over variational methods in restoring small details of images.

In this paper, we consider the recovery of group sparse signals corrupted by impulsive noise. In some recent literature, researchers have utilized stable data fitting models, like l 1 -norm, Huber penalty function and Lorentzian-norm, to substitute the l 2 -norm data fidelity model to obtain more robust performance. In this paper, a stable model is developed, which exploits the generalized l p -norm as the measure for the error for sparse reconstruction. In order to address this model, we propose an efficient alternative direction method of multipliers, which includes the proximity operator of l p -norm functions to the framework of Lagrangian methods. Besides, to guarantee the convergence of the algorithm in the case of 0 ≤ p 1 (nonconvex case), we took advantage of a smoothing strategy. For both 0 ≤ p 1 (nonconvex case) and 1 ≤ p ≤ 2 (convex case), we have derived the conditions of the convergence for the proposed algorithm. Moreover, under the block restricted isometry property with constant δ τ k 0 τ / ( 4 − τ ) for 0 τ 4 / 3 and δ τ k 0 ( τ − 1 ) / τ for τ ≥ 4 / 3 , a sharp sufficient condition for group sparse recovery in the presence of impulsive noise and its associated error upper bound estimation are established. Numerical results based on the synthetic block sparse signals and the real-world FECG signals demonstrate the effectiveness and robustness of new algorithm in highly impulsive noise.

Local Principal Component Analysis can be performed over small domains of an embedded Riemannian manifold in order to relate the covariance analysis of the underlying point set with the local extrinsic and intrinsic curvature. We show that the volume of domains on a submanifold of general codimension, determined by the intersection with higher-dimensional cylinders and balls in the ambient space, have asymptotic expansions in terms of the mean and scalar curvatures. Moreover, we propose a generalization of the classical third fundamental form to general submanifolds and prove that the local eigenvalue decomposition (EVD) of the covariance matrices have asymptotic expansions that contain the curvature information encoded by the traces of this tensor. This proves the general correspondence between the local EVD integral invariants and differential-geometric curvature for arbitrary embedded Riemannian submanifolds, found so far for curves and hypersurfaces only. Thus, we establish a key theoretical bridge, via covariance matrices at scale, for potential applications in manifold learning relating the statistics of point clouds sampled from Riemannian submanifolds to the underlying geometry. We would like to thank Louis Scharf for very helpful discussions during the writing of this paper. J.Á.-V. would like to thank Miguel Dovale for helpful insights. This paper is based on research partially supported by the National Science Foundation under Grants No. DMS-1513633, and DMS-1322508.

The concept of frame multiresolution analysis (FMRA) on local fields of positive characteristic was given by Shah in his paper, Frame Multiresolution Analysis on Local Fields published by Journal of Operators. The author has studied the concept of minimum-energy wavelet frames on these prime characteristic fields. We continued the studies based on frame multiresolution analysis and minimum-energy wavelet frames on local fields of positive characteristic. In this paper, we introduce the notion of the construction of minimum-energy wavelet frames based on FMRA on local fields of positive characteristic. We provide a constructive algorithm for the existence of the minimum-energy wavelet frame on the local field of positive characteristic. An explicit construction of the frames and bases is given. In the end, we exhibit an example to illustrate our algorithm.

In many fields of engineering as well as computational physics, it is necessary to describe dynamic phenomena which are characterized by an infinitely long horizon of past state values. This infinite horizon of past data then influences the evolution of future state trajectories. Such phenomena can be characterized effectively by means of fractional-order differential equations. In contrast to classical linear ordinary differential equations linear fractional-order models have frequency domain characteristics with amplitude responses that deviate from the classical integer multiples of ±20 dB per frequency decade and, respectively, deviate from integer multiples of ±π/2 in the limit values of their corresponding phase response. Although numerous simulation approaches have been developed in recent years for the numerical evaluation of fractional-order models with point-valued initial conditions and parameters, the robustness analysis of such system representations is still a widely open area of research. This statement is especially true if interval uncertainty is considered with respect to initial states and parameters. Therefore, this paper summarizes the current state-of-the-art concerning the simulation-based analysis of fractional-order dynamics with a restriction to those approaches that can be extended to set-valued (interval) evaluations for models with bounded uncertainty. Especially, it is shown how verified simulation techniques for integer-order models with uncertain parameters can be extended toward fractional counterparts. Selected linear as well as nonlinear illustrating examples conclude this paper to visualize algorithmic properties of the suggested interval-based simulation methodology and point out directions of ongoing research.

This paper is concerned with the inverse scattering problem of time-harmonic elastic waves by a mixed-type scatterer, which is given as the union of an impenetrable obstacle and a crack. We develop the modified factorization method to determine the shape of the mixed-type scatterer from the far field data. However, the factorization of the far field operator $F$ is related to the boundary integral matrix operator $A$, which is obtained in the study of the direct scattering problem. So, in the first part, we show the well posedness of the direct scattering problem by the boundary integral equation method. Some numerical examples are presented at the end of the paper to demonstrate the feasibility and effectiveness of the inverse algorithm.

This paper addresses the three concepts of consistency, stability and convergence in the context of compact finite volume schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of “balance laws”. Such laws express the relevant physical conservation laws in the presence of discontinuities. Finite volume approximations employ this viewpoint, and the present paper can be regarded as being in this category. It is first shown that under very mild conditions a weak solution is indeed a solution to the balance law. The schemes considered here allow the computation of several quantities per mesh cell (e.g., slopes) and the notion of consistency must be extended to this framework. Then a suitable convergence theorem is established, generalizing the classical convergence theorem of Lax and Wendroff. Finally, the limit functions are shown to be entropy solutions by using a notion of “Godunov compatibility”, which serves as a substitute to the entropy condition.

In 2010, Andrews investigated a variety of parity questions in the classical partition identities of Euler, Rogers, Ramanujan and Gordon. In particular, he considered the Rogers-Ramanujan-Gordon partitions with some constraints on even and odd parts. At the end of this paper, he left fifteen open questions, of which the eleventh is to extend his parity consideration to overpartitions. The main purpose of this paper is to undertake that question. In 2013, Chen, Sang and Shi derived an overpartition analogue of the Rogers–Ramanujan–Gordon theorem. Motivated by their work, we define two kinds of Rogers–Ramanujan–Gordon type overpartitions with some parity constraints on even and odd parts. We then provide the generating functions for such partitions in some cases.

International audience; Sampled-data control systems have steadily been gaining interest for their applications in automatic engineering where they are implemented as digital controllers and recently results have been obtained in optimal control theory for nonlinear sampled-data control systems and certain generalizations. In this paper we derive a Pontryagin maximum principle for general nonlinear finite-dimensional optimal sampled-data control problems with running inequality state constraints. In particular, we obtain a nonpositive averaged Hamiltonian gradient condition with the adjoint vector being a function of bounded variations. Our proof is based on the Ekeland variational principle. In general, optimal control problems with running inequality state constraints are difficult to solve using numerical methods due to the discontinuities (the jumps and the singular part) of the adjoint vector. However in our case we find that under certain general hypotheses the adjoint vector only experiences jumps at most at the sampling times and moreover the trajectory only contacts the running inequality state constraints at most at the sampling times. We call this behavior a bouncing trajectory phenomenon and it constitutes the second major focus of this paper. Finally taking advantage of the bouncing trajectory phenomenon we numerically solve three examples with different kinds of constraints and in several dimensions.

This paper considers a classifier ensemble problem with sparsity and diversity learning, which arises in the field of machine learning, and shows that the classifier ensemble problem can be formulated as a convex stochastic optimization problem over the fixed point set of a quasi-nonexpansive mapping. Specifically, for such a problem, this paper proposes an algorithm referred to as the stochastic fixed point optimization algorithm and performs a convergence analysis for three types of step size: 1) constant step size; 2) decreasing step size; and 3) a step size computed by line searches. In the case of a constant step size, the results indicate that a sufficiently small constant step size allows a solution to the problem to be approximated. In the case of a decreasing step size, conditions are shown under which the algorithm converges in probability to a solution. For the third case, a variation of the basic proposed algorithm also achieves convergence in probability to a solution. The high classification accuracies of the proposed algorithms are demonstrated through numerical comparisons with the conventional algorithm.

We consider elements of finite order in the Riordan group $\cal R$ over a field of characteristic $0$. Viewing $\cal R$ as a semi-direct product of groups of formal power series, we solve, for all $n \geq 2$, two foundational questions posed by L. Shapiro for the case $n = 2$ (`involutions'): Given a formal power series $F(x)$ of finite compositional order and an integer $n\geq 2$, Theorem 1 states, exactly which $g(x)$ make $\big(g(x), F(x)\big)$ a Riordan element of order $n$. Theorem 2 classifies finite-order Riordan group elements up to conjugation in $\cal R$. Viewing $\cal R$ as a group of infinite lower triangular matrices, we interpret Theorem 1 in terms of existence of eigenvectors and Theorem 2 as a normal form for finite order Riordan arrays under similarity. These lead to Theorem 3, a formula for all eigenvectors of finite order Riordan arrays; and we show how this can lead to interesting combinatorial identities. We then relate our work to papers of Cheon and Kim which motivated this paper and we solve the Open question which they posed. Finally, this circle of ideas gives a new proof of C. Marshall's theorem, which finds the unique $F(x)$, given bi-invertible $g(x)$, such that $\big(g(x), F(x))$ is an involution., Comment: 25 pages. Version 4 corrects the statement of Example 5.6 in the case when $n$ is odd, and adjusts the last four lines of the proof accordingly

This paper is concerned with optimal error estimates and recovery technique of a classical mixed finite element method for the thermistor problem, which is governed by a parabolic/elliptic system with strong nonlinearity and coupling. The method is based on a popular combination of the lowest-order Raviart–Thomas mixed approximation for the electric potential/field $(\phi , \boldsymbol{\theta })$ and the linear Lagrange approximation for the temperature $u$. A common question is how the first-order approximation influences the accuracy of the second-order approximation to the temperature in such a strongly coupled system, while previous work only showed the first-order accuracy $O(h)$ for all three components in a traditional way. In this paper, we prove that the method produces the optimal second-order accuracy $O(h^2)$ for $u$ in the spatial direction, although the accuracy for the potential/field is in the order of $O(h)$. And more importantly, we propose a simple one-step recovery technique to obtain a new numerical electric potential/field of second-order accuracy. The analysis presented in this paper relies on an $H^{-1}$-norm estimate of the mixed finite element methods and analysis on a nonclassical elliptic map. We provide numerical experiments in both two- and three-dimensional spaces to confirm our theoretical analyses.

There are a lot of effective, ineffective and explicit results concerning power values and polynomial values of binomial coefficients. Also, many papers deal with generalizations of these problems, involving polygonal numbers and pyramidal numbers. In this paper we prove effective and ineffective theorems concerning polynomial values of figurate numbers. Our results yield common extensions and generalizations of several previous theorems from the literature.

Due to the complexity of the marine environment, in deep-sea drilling, all kinds of strings are corroded by different deep-sea conditions for a long time, accompanied by high temperature and high pressure, which lead to the continuous change of mechanical properties of materials. In order to solve the problem that the material mechanical parameters cannot be accurately described in the performance analysis of the casing, deep-sea simulated corrosion and material damage experiments of P110 material were carried out in this paper. Mass loss and tensile experiments on corrosion-damaged test pieces were conducted under different corrosion experimental periods. The changes in mechanical properties of the material were analyzed. Equations of the variation of the equivalent yield strength and the equivalent tensile strength were obtained. The results show that the equivalent yield strength and the equivalent tensile strength decrease with the increase of the weight loss rate. Based on the experimental results and finite element analysis, a method for establishing the material corrosion model was proposed in this paper. The deep-sea drilling corrosion performance model of P110 material was established, which greatly reduced the error caused by the material uniformity assumption in finite element analysis. This paper provides a theoretical basis for the analysis of reliability and life of P110 materials in wells.

Computing with non-linear polynomial constraints over the real numbers has been an important area within Computer Algebra since the field's inception. More recently, satisfiability for this theory has become an important problem in the SMT-solving community. Inspired by the S C 2 project, this paper combines ideas from both communities to produce a fast “partial solver” for the theory — i.e. a solver that can sometimes decide the satisfiability of formulas, but sometimes fails to make a determination. The partial solver is based on existing “black-box/white-box” simplification algorithms, which this paper extends to allow for the production of UNSAT cores (and stronger simplifications). The production of UNSAT cores is required if the partial solver is to be used in SMT-solving, which is one of the primary motivations for this work. The authors have implemented the partial solver in the system, and this paper also reports on experiments based on that implementation.

In this paper it is shown that there is an absolute constant κ > 0 with the following property. For any prime p and nonempty subsets A , B of Z / p Z such that 1 | A | p 2 , B ∩ − B = ∅ and | B | κ | A | ln ⁡ ( | A | ) , we have that max b ∈ B ⁡ | ( A + b ) ∖ A | ≥ | B | . In 2011, V. Lev proved the former statement assuming also that | B | ≤ p 8 ; in the same paper, Lev suggested that this technical condition could be eliminated or weakened. In this paper his conjecture is confirmed.

The purpose of the paper is to consider issues related to the security of state information systems in the Russian Federation. The relevance of these issues is primarily due to the fact that the number of cyber-attacks (causing significant damage to the state) on various areas of its economy, including its state information systems, is increasing every year in Russia. For example, the national initiative for Cybersecurity Education (NICE) was launched in the United States several years ago. This event has not escaped the attention of Russian and other experts in the field of information security (is). The main areas of information property protection include: protection (state, official, commercial, banking, tax, insurance, personal data (PD), etc.) of secrets and intellectual property. State information systems include information technology tools and systems that rely on advanced scientific research, such as knowledge, advanced technologies (know-how), etc. Today, state information systems are an integral part of a fairly complex system of public administration. It is also important to recall the role of the Russian FSTEC regulator in ensuring information security. The subject of the research is SIS as the basis of management of state bodies. The paper focuses on current trends in the field of SIS information security, a brief overview of SIS legislation, and the critical information infrastructure (CII) of Russia's main geopolitical opponent. The main arguments for the importance of the chosen direction of work are discussed. Further research is expected to be conducted in the field of SIS security in an uncertain environment.

In this paper we revisit the problem of performing a $QZ$ step with a so-called ‘perfect shift’, which is an ‘exact’ eigenvalue of a given regular pencil $\lambda B-A$ in unreduced Hessenberg triangular form. In exact arithmetic, the $QZ$ step moves that eigenvalue to the bottom of the pencil, while the rest of the pencil is maintained in Hessenberg triangular form, which then yields a deflation of the given eigenvalue. But in finite precision the $QZ$ step gets ‘blurred’ and precludes the deflation of the given eigenvalue. In this paper we show that when we first compute the corresponding eigenvector to sufficient accuracy, then the $QZ$ step can be constructed using this eigenvector, so that the deflation is also obtained in finite precision. An important application of this technique is the computation of the index of a system of differential algebraic equations, since an exact deflation of the infinite eigenvalues is needed to impose correctly the algebraic constraints of such differential equations.

This paper proposes a new class of difference methods with intrinsic parallelism for solving the Burgers–Fisher equation. A new class of parallel difference schemes of pure alternating segment explicit-implicit (PASE-I) and pure alternating segment implicit-explicit (PASI-E) are constructed by taking simple classical explicit and implicit schemes, combined with the alternating segment technique. The existence, uniqueness, linear absolute stability, and convergence for the solutions of PASE-I and PASI-E schemes are well illustrated. Both theoretical analysis and numerical experiments show that PASE-I and PASI-E schemes are linearly absolute stable, with 2-order time accuracy and 2-order spatial accuracy. Compared with the implicit scheme and the Crank–Nicolson (C-N) scheme, the computational efficiency of the PASE-I (PASI-E) scheme is greatly improved. The PASE-I and PASI-E schemes have obvious parallel computing properties, which show that the difference methods with intrinsic parallelism in this paper are feasible to solve the Burgers–Fisher equation.

The reflection principle for Brownian motion gives a way to calculate the joint distribution of a hitting time and a one dimensional marginal. The Carr–Nadtochiy transform is a formulation that generalizes the reflection principle in this respect. The transform originated from a way to hedge so-called barrier options in the literature of financial mathematics. The existence of the transform has been established only for one dimensional diffusion processes. In the present paper, the existence is proved for a fairly general class of Markov chains in the multi dimensional lattice $$\mathbf {Z}^d$$ . The difficulty is that the reflection boundary is not a one-point set, contrasting the one dimensional cases. It is solved in this paper by looking at the problem in an algebraic way.

The aim of the present paper is to establish four theorems connecting Stieltjes transform and Hankel transform. By applications of the theorems established in this paper four new integral formulae involving special functions are obtained. Due to the general nature of the theorems established in this paper, several other integrals involving special functions may be evaluated.

Various studies have focused on secret sharing schemes and in all of them, each shareholder is interested in a shorter share. The information rate of a secret sharing scheme shows the ratio between the size of the secret to the maximum number of shares given to each shareholder. In this regard, the researchers investigated the optimal information rate of the graph access structure. This paper aims to discover the exact values for the optimal information rates of the two graph access structures, which remained as open problems in Van Dijk’s paper. Furthermore, we introduced the developed multipartite graph and the developed cycle graph and calculated the exact value of their optimal information rate. Moreover, we presented a lower bound on the information rate for other developed graph access structures.

A subspace partition of a finite vector space F q n of dimension n over the field F q with q elements is a collection Π of subspaces of F q n consisting of subspaces with mutually zero intersection that partition the nonzero vectors in F q n . Clearly, Π satisfies the equation ∑ W ∈ Π ( q dim W − 1 ) = q n − 1 , which is called the packing condition. We say that Π contains a direct sum if there exist W 1 , … , W r ∈ Π with W 1 ⊕ ⋯ ⊕ W r = F q n . Partitions with direct sums are ubiquitous in practice and form an important subfamily of the lattice of all subspace partitions of F q n , which is a combinatorial q-analogue of the lattice of all set partitions of the set with n elements. In this paper, we show that subspace partitions with direct sums (where the number h of distinct dimensions among subspaces of Π is arbitrary) exist when their type (the multiset of subspace dimensions found in Π) is in the convex hull of certain kinds of vertices in the lattice of solutions of the packing condition. This generalizes a result in a previous paper, where we had considered subspace partitions containing at most h = 2 distinct subspace dimensions. We also construct an infinite family of Frobenius subspace partitions with h distinct dimensions that cannot contain a direct sum, due to the fact that no combination of dimensions adds up to n.

To a large extent, the load balancing algorithm affects the clustering performance of the computer. This paper illustrated the common load balancing algorithms and elaborated on the advantages and drawbacks of such algorithms. In addition, this paper provides a kind of balancing algorithm generated on the basis of the load prediction. Due to the dynamic exponential smoothing model, such an algorithm helps obtain the corresponding smoothing coefficient with the server node load time series of current phrase and allows researchers to make prediction with the load value at the next moment of this node. Subsequently, the dispatcher makes the scheduling with the serve request of users according to the load predicted value. OPNET Internet simulated software is applied to the test, and we may conclude from the results that the application of such an algorithm acquires a higher load balancing efficiency and better load balancing effect.

In this paper, with a new approach, some new Hermite-Hadamard type inequalities for preinvex functions are obtained by using only the right Riemann-Liouville fractional integrals. Our results generalize previous studies. Results proved in this paper may stimulate further research in this field.

A parallel framework software, CCFD, based on the structure grid, and suitable for parallel computing of super-large-scale structure blocks, is designed and implemented. An overdecomposition method, in which the load balancing strategy is based on the domain decomposition method, is designed for the graph subdivision algorithm. This method takes computation and communication as the limiting condition and realizes the load balance between blocks by dividing the weighted graph. The fast convergence technique of a high-efficiency parallel geometric multigrid greatly improves the parallel efficiency and convergence speed of CCFD software. This paper introduces the software structure, process invocations, and calculation method of CCFD and introduces a hybrid parallel acceleration technology based on the Sunway TaihuLight heterogeneous architecture. The results calculated by Onera-M6 and DLR-F6 standard model show that the software structure and method in this paper are feasible and can meet the requirements of a large-scale parallel solution.

We propose and analyse numerical algorithms based on weighted least squares for the approximation of a real-valued function on a general bounded domain $\Omega \subset \mathbb{R}^d$. Given any $n$-dimensional approximation space $V_n \subset L^2(\Omega)$, the analysis in [6] shows the existence of stable and optimally converging weighted least-squares estimators, using a number of function evaluations $m$ of the order $n \log n$. When an $L^2(\Omega)$-orthonormal basis of $V_n$ is available in analytic form, such estimators can be constructed using the algorithms described in [6,Section 5]. If the basis also has product form, then these algorithms have computational complexity linear in $d$ and $m$. In this paper we show that, when $\Omega$ is an irregular domain such that the analytic form of an $L^2(\Omega)$-orthonormal basis is not available, stable and quasi-optimally weighted least-squares estimators can still be constructed from $V_n$, again with $m$ of the order $n \log n$, but using a suitable surrogate basis of $V_n$ orthonormal in a discrete sense. The computational cost for the calculation of the surrogate basis depends on the Christoffel function of $\Omega$ and $V_n$. Numerical results validating our analysis are presented., Comment: Version of the paper accepted for publication

Purpose This paper aims to examine the key factors determining bank deposit growth in Turkey for the period 2000Q1–2016Q4. Design/methodology/approach The study employs the autoregressive distributed lag approach to investigate the effect of bank-level and macroeconomic factors on deposit growth. Findings The results reveal that bank stability, banking sector efficiency, broad money supply, economic growth, and inflation are significant determinants of deposit growth in the long run. The findings further show that in the short run, only branch expansion and broad money supply are relevant for bank deposit mobilization. Originality/value This paper departs from the extant empirical studies that focus on the determinants of individual savings behaviour in Turkey. Considering the short- and long-run time dimensions, the authors distinctively examine how bank characteristics influence deposit growth, thus presenting a relatively pioneering attempt in this context.

In a series of papers published since 2017 the author introduced a simple alternative of the $n$-simplex type, to enhance the accuracy of approximations of second-order boundary value problems subject to Dirichlet boundary conditions, posed on smooth curved domains. This technique is based upon trial functions consisting of piecewise polynomials defined on straight-edged triangular or tetrahedral meshes, interpolating the Dirichlet boundary conditions at points of the true boundary. In contrast, the test functions are defined by the standard degrees of freedom associated with the underlying method for polytopic domains. While the mathematical analysis of the method for Lagrange and Hermite methods for two-dimensional second- and fourth-order problems was carried out in earlier paper by the author this paper is devoted to the study of the three-dimensional case. Well-posedness, uniform stability and optimal a priori error estimates in the energy norm are proved for a tetrahedron-based Lagrange family of finite elements. Novel error estimates in the $L^2$-norm, for the class of problems considered in this work, are also proved. A series of numerical examples illustrates the potential of the new technique. In particular, its superior accuracy at equivalent cost, as compared to the isoparametric technique, is highlighted.

In 2012, Jin Qinian considered an inexact Newton–Landweber iterative method for solving nonlinear ill-posed operator equations in the Banach space setting by making use of duality mapping. The method consists of two steps; the first one is an inner iteration which gives increments by using Landweber iteration, and the second one is an outer iteration which provides increments by using Newton iteration. He has proved a convergence result for the exact data case, and for the perturbed data case, a weak convergence result has been obtained under a Morozov type stopping rule. However, no error bound has been given. In 2013, Kaltenbacher and Tomba have considered the modified version of the Newton–Landweber iterations, in which the combination of the outer Newton loop with an iteratively regularized Landweber iteration has been used. The convergence rate result has been obtained under a Hölder type source condition. In this paper, we study the modified version of inexact Newton–Landweber iteration under the approximate source condition and will obtain an order-optimal error estimate under a suitable choice of stopping rules for the inner and outer iterations. We will also show that the results proved in this paper are more general as compared to the results proved by Kaltenbacher and Tomba in 2013. Also, we will give a numerical example of a parameter identification problem to support our method.

Construction of multivariate tight framelets is known to be a challenging problem because it is linked to the difficult problem on sum of squares of multivariate polynomials in real algebraic geometry. Multivariate dual framelets with vanishing moments generalize tight framelets and are not easy to be constructed either, since their construction is related to syzygy modules and factorization of multivariate polynomials. On the other hand, compactly supported multivariate framelets with directionality or high vanishing moments are of interest and importance in both theory and applications. In this paper we introduce the notion of a quasi-tight framelet, which is a dual framelet, but behaves almost like a tight framelet. Let ϕ ∈ L 2 ( R d ) be an arbitrary compactly supported real-valued M -refinable function with a general dilation matrix M and ϕ ˆ ( 0 ) = 1 such that its underlying real-valued low-pass filter satisfies the basic sum rule. We first constructively prove by a step-by-step algorithm that we can always easily derive from the arbitrary M -refinable function ϕ a directional compactly supported real-valued quasi-tight M -framelet in L 2 ( R d ) associated with a directional quasi-tight M -framelet filter bank, each of whose high-pass filters has one vanishing moment and only two nonzero coefficients. If in addition all the coefficients of its low-pass filter are nonnegative, then such a quasi-tight M -framelet becomes a directional tight M -framelet in L 2 ( R d ) . Furthermore, we show by a constructive algorithm that we can always derive from the arbitrary M -refinable function ϕ a compactly supported quasi-tight M -framelet in L 2 ( R d ) with the highest possible order of vanishing moments. We shall also present a result on quasi-tight framelets whose associated high-pass filters are purely differencing filters with the highest order of vanishing moments. Several examples will be provided to illustrate our main theoretical results and algorithms in this paper.

Ostwald ripening is a main phenomenon responsible for growth of particles manufactured using colloidal methods. This paper considers mathematical description of the process using population balance approach solved by method of moments. However, in the case of particles’ dissolution which is a part of the process a closure problem is generated. A novel closure method for the moment – transformed population balance equations based on Gram – Charlier expansions is introduced. Due to its simplicity it allows to reconstruct the full particle size distribution in a stepwise manner giving an access to the pointwise values of the density function required to close the model. The introduced numerical scheme is further used to simulate several carefully chosen test cases including dissolution of solid particles and ageing of particles’ population. The mathematical model presented in the paper is then applied to interpret experimental results for batch precipitation of silica.

The localized method of fundamental solutions (LMFS) is an efficient meshless collocation method that combines the concept of localization and the method of fundamental solutions (MFS). The resultant system of linear algebraic equations in the LMFS is sparse and banded and thus, drastically reduces the storage and computational burden of the MFS. In the LMFS, the moving least square (MLS) approximation, based on fundamental solutions, is used to construct approximate solution at each node. In this paper, this fundamental solutions-based MLS approximation, named as an augmented MLS (AMLS) approximation, is generalized to any point in the computational domain. Computational formulas, theoretical properties and error estimates of the AMLS approximation are derived. Then, taking Laplace equation as an example, this paper sets up a framework for the theoretical error analysis of the LMFS. Finally, numerical results are presented to verify the efficiency and theoretical results of the AMLS approximation and the LMFS. Convergence and comparison researches are conducted to validate the accuracy, convergence and efficiency.

In this paper we treat realization results for operator-valued functions which are analytic in the complex sense or slice hyperholomorphic over the quaternions. In the complex setting, we prove a realization theorem for an operator-valued function analytic in a neighborhood of the origin with a coisometric state space operator thus generalizing an analogous result in the unitary case. A main difference with previous works is the use of reproducing kernel Krein spaces. We then prove the counterpart of this result in the quaternionic setting. The present work is the first paper which presents a realization theorem with a state space which is a quaternionic Krein space and may open new avenues of research in hypercomplex analysis.

In this paper, we investigate the exponential mean-square stability for both the solution of n-dimensional stochastic delay integro-differential equations (SDIDEs) with Poisson jump, as well for the split-step θ-Milstein (SSTM) scheme implemented of the proposed model. First, by virtue of Lyapunov function and continuous semi-martingale convergence theorem, we prove that the considered model has the property of exponential mean-square stability. Moreover, it is shown that the SSTM scheme can inherit the exponential mean-square stability by using the delayed difference inequality established in the paper. Eventually, three numerical examples are provided to show the effectiveness of the theoretical results.

In this paper we consider two versions of the Collatz-Wielandt quotient for a pair of nonnegative operators A,B that map a given pointed generating cone in the first space into a given pointed generating cone in the second space. If the two spaces and two cones are identical, and B is the identity operator then one version of this quotient is the spectral radius of A. In some applications, as commodity pricing, power control in wireless networks and quantum information theory, one needs to deal with the Collatz-Wielandt quotient for two nonnegative operators. In this paper we treat the two important cases: a pair of rectangular nonnegative matrices and a pair completely positive operators. We give a characterization of minimal optimal solutions and polynomially computable bounds on the Collatz-Wielandt quotient., Comment: 24 pages. To appear in Applications of Mathematics, ISSN 0862-7940

Major advances were recently obtained in the exact solution of vehicle routing problems (VRPs). Sophisticated branch-cut-and-price (BCP) algorithms for some of the most classical VRP variants now solve many instances with up to a few hundreds of customers. However, adapting and reimplementing those successful algorithms for other variants can be a very demanding task. This work proposes a BCP solver for a generic model that encompasses a wide class of VRPs. It incorporates the key elements found in the best existing VRP algorithms: ng-path relaxation, rank-1 cuts with limited memory, path enumeration, and rounded capacity cuts; all generalized through the new concepts of “packing set” and “elementarity set”. The concepts are also used to derive a branching rule based on accumulated resource consumption and to generalize the Ryan and Foster branching rule. Extensive experiments on several variants show that the generic solver has an excellent overall performance, in many problems being better than the best specific algorithms. Even some non-VRPs, like bin packing, vector packing and generalized assignment, can be modeled and effectively solved.

Purpose: In the present paper the concept of soft almost β-continuous mappings and soft almost β-open mappings in soft topological spaces have been introduced and studied. Methodology: This notion is weaker than both soft almost pre-continuous mappings, soft almost semi-continuous mapping. The diagrams of implication among these soft classes of soft mappings have been established. Main Findings: We extend the concept of almost β-continuous mappings and almost β-open mappings in soft topology. Implications: Mapping is an important and major area of topology and it can give many relationships between other scientific areas and mathematical models. This notion captures the idea of hanging-togetherness of image elements in an object by assigning strength of connectedness to every possible path between every possible pair of image elements. It is an important tool for the designing of algorithms for image segmentation. The novelty of Study: Hope that the concepts and results established in this paper will help the researcher to enhance and promote the further study on soft topology to carry out a general framework for the development of information systems.