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This paper generalizes the non-conforming FEM of Crouzeix and Raviart and its fundamental projection property by a novel mixed formulation for the Poisson problem based on the Helmholtz decomposition. The new formulation allows for ansatz spaces of arbitrary polynomial degree and its discretization coincides with the mentioned non-conforming FEM for the lowest polynomial degree. The discretization directly approximates the gradient of the solution instead of the solution itself. Besides the a priori and medius analysis, this paper proves optimal convergence rates for an adaptive algorithm for the new discretization. These are also demonstrated in numerical experiments. Furthermore, this paper focuses on extensions of this new scheme to quadrilateral meshes, mixed FEMs, and three space dimensions.

The DEM generalization is the basis of multi-dimensional observation, the basis of expressing and analyzing the terrain. DEM is also the core of building the Multi-Scale Geographic Database. Thus, many researchers have studied both the theory and the method of DEM generalization. This paper proposed a new method of generalizing terrain, which extracts feature points based on the tree model construction which considering the nested relationship of watershed characteristics. The paper used the 5 m resolution DEM of the Jiuyuan gully watersheds in the Loess Plateau as the original data and extracted the feature points in every single watershed to reconstruct the DEM. The paper has achieved generalization from 1:10000 DEM to 1:50000 DEM by computing the best threshold. The best threshold is 0.06. In the last part of the paper, the height accuracy of the generalized DEM is analyzed by comparing it with some other classic methods, such as aggregation, resample, and VIP based on the original 1:50000 DEM. The outcome shows that the method performed well. The method can choose the best threshold according to the target generalization scale to decide the density of the feature points in the watershed. Meanwhile, this method can reserve the skeleton of the terrain, which can meet the needs of different levels of generalization. Additionally, through overlapped contour contrast, elevation statistical parameters and slope and aspect analysis, we found out that the W8D algorithm performed well and effectively in terrain representation.

The main goal of this paper is to develop uniformly optimal first-order methods for convex programming (CP). By uniform optimality we mean that the first-order methods themselves do not require the input of any problem parameters, but can still achieve the best possible iteration complexity bounds. By incorporating a multi-step acceleration scheme into the well-known bundle-level method, we develop an accelerated bundle-level (ABL) method, and show that it can achieve the optimal complexity for solving a general class of black-box CP problems without requiring the input of any smoothness information, such as, whether the problem is smooth, nonsmooth or weakly smooth, as well as the specific values of Lipschitz constant and smoothness level. We then develop a more practical, restricted memory version of this method, namely the accelerated prox-level (APL) method. We investigate the generalization of the APL method for solving certain composite CP problems and an important class of saddle-point problems recently studied by Nesterov [Mathematical Programming, 103 (2005), pp 127-152]. We present promising numerical results for these new bundle-level methods applied to solve certain classes of semidefinite programming (SDP) and stochastic programming (SP) problems., A combination of the previous two papers submitted to Mathematical Programming, i.e., "Bundle-type methods uniformly optimal for smooth and nonsmooth convex optimization" (December 2010) and "Level methods uniformly optimal for composite and structured nonsmooth convex optimization (April 2011)

This paper is dedicated to the memory of Basil Nicolaenko; International audience; We investigate the stability of the travelling wave (TW) solution in a 2D Stefan problem, a simplified version of a solid-liquid interface model. It is intended as a paradigm problem to present our method based on: (i) definition of a suitable linear one dimensional operator, (ii) projection with respect to the $x$ coordinate only; (iii) Lyapunov-Schmidt method. The main issue is that we are able to derive a parabolic equation for the corrugated front $\varphi$ near the TW as a solvability condition. This equation involves two linear pseudo-differential operators, one acting on $\varphi$, the other on $(\varphi_y)^2$ and clearly appears as a generalization of the Kuramoto-Sivashinsky equation related to turbulence phenomena in chemistry and combustion. A large part of the paper is devoted to study the properties of these operators in the context of functional spaces in the $y$ and $x,y$ coordinates with periodic boundary conditions. Technical results are deferred to the appendices.

In this paper we present a method for the numerical solution of elliptic problems with multi-scale data using multiple levels of not necessarily nested grids. The method consists in calculating successive corrections to the solution in patches whose discretizations are not necessarily conforming. This paper provides proofs of the results published earlier (see C. R. Acad. Sci. Paris, Ser. I 337 (2003) 679-684), gives a generalization of the latter to more than two domains and contains extensive numerical illustrations. New results including the spectral analysis of the iteration operator and a numerical method to evaluate the constant of the strengthened Cauchy-Buniakowski-Schwarz inequality are presented.In this paper we present a method for the numerical solution of elliptic problems with multi-scale data using multiple levels of not necessarily nested grids. The method consists in calculating successive corrections to the solution in patches whose discretizations are not necessarily conforming. This paper provides proofs of the results published earlier (see C. R. Acad. Sci. Paris, Ser. I 337 (2003) 679-684), gives a generalization of the latter to more than two domains and contains extensive numerical illustrations. New results including the spectral analysis of the iteration operator and a numerical method to evaluate the constant of the strengthened Cauchy-Buniakowski-Schwarz inequality are presented.

The Bernstein operators (BO) are not orthogonal, but they have duals, which are obtained by a linear combination of BO. In recent years dual BO have been adopted in computer graphics, computer aided geometric design, and numerical analysis. This paper presents a numerical method based on the Bernstein operational matrices to solve the time–space fractional convection–diffusion equation. A generalization of the derivative matrix operator of fractional order and the error analysis are discussed. Numerical examples compare the proposed approach with previous works, showing that the method is more accurate and efficient., The authors wish to express their cordial thanks to the editor and three anonymous referees for useful suggestions and comments.

In this paper, the generalization of the Particle Swarm Optimization (PSO) algorithm is proposed. The new algorithm involves the adoption of complex-order derivatives (CD). Since the CD produce complex-valued results, conjugate pairs of CD are considered for designing the Complex-Order PSO (CoPSO). First, an extensive sensitivity analysis is carried out for studying the influence of the control parameters on the performance of CoPSO. Then, a set of classical benchmark functions are tested to verify the performance of CoPSO. Both valued- and ranked-based methods are conducted to compare the performance of the algorithm on the whole test suite. The Friedman test is applied to determine the average ranking of the algorithms based on their performances. Additionally, the mean and the standard deviation of the best results are examined in each experiment. The results indicate that the CoPSO has outstanding performance in comparison with previous algorithms, including the standard PSO, the fractional order PSO and the linear and nonlinear decreasing inertia weight PSO. The experimental results indicate the feasibility and efficiency of the CoPSO.

Cicerone and Di Stefano defined and studied the class of k-distance-hereditary graphs, i.e., graphs where the distance in each connected induced subgraph is at most k times the distance in the whole graph. The defined graphs represent a generalization of the well known distance-hereditary graphs, which actually correspond to 1-distance-hereditary graphs. In this paper we make a step forward in the study of these new graphs by providing characterizations for the class of all the k-distance-hereditary graphs such that k<2. The new characterizations are given in terms of both forbidden subgraphs and cycle-chord properties. Such results also lead to devise a polynomial-time recognition algorithm for this kind of graph that, according to the provided characterizations, simply detects the presence of quasi-holes in any given graph.

Estimation of density functions supported on general domains arises when the data is naturally restricted to a proper subset of the real space. This problem is complicated by typically intractable normalizing constants. Score matching provides a powerful tool for estimating densities with such intractable normalizing constants, but as originally proposed is limited to densities on $\mathbb{R}^m$ and $\mathbb{R}_+^m$. In this paper, we offer a natural generalization of score matching that accommodates densities supported on a very general class of domains. We apply the framework to truncated graphical and pairwise interaction models, and provide theoretical guarantees for the resulting estimators. We also generalize a recently proposed method from bounded to unbounded domains, and empirically demonstrate the advantages of our method., 50 pages, 14 figures

The most popular values in cooperative games with transferable utilities are perhaps the Shapley and the Shapley like values which are based on the notion of players’ marginal productivity. The equal division rule on the other hand, is based on egalitarianism where resource is equally divided among players, no matter how productive they are. However none of these values explicitly discuss players’ multilateral interactions with peers in deciding to form coalitions and generate worths. In this paper we study the effect of multilateral interactions of a player that accounts for her contributions with her peers not only at an individual level but also on a group level. Based on this idea, we propose a value called the MI $$^k$$ -value and characterize it by the axioms of linearity, anonymity, efficiency and a new axiom: the axiom of MN $$^k$$ -player. An MN $$^k$$ -player is one whose average marginal contribution due to her multilateral interactions upto level k is zero and can be seen as a generalization of the standard null player axiom of the Shapley value. We have shown that the MI $$^k$$ -value on a variable player set is asymptotically close to the equal division rule. Thus our value realizes solidarity among players by incorporating both their individual and group contributions.

In this paper we analyze the nearly optimal block diagonal scalings of the rows of one factor and the columns of the other factor in the triangular form of the SR decomposition. The result is a block generalization of the result of the van der Sluis about the almost optimal diagonal scalings of the general rectangular matrices.

In this paper, possible scenarios within the experimental dynamic response of a vibro-impact single-degree-of-freedom system, symmetrically constrained by deformable and dissipative bumpers, were identified and described. The different scenarios were obtained varying selected parameters, namely peak table acceleration $$\hbox {A}$$ , amplitude of the total gap between mass and bumpers $$\hbox {G}$$ and bumper’s stiffness $$\hbox {B}$$ . Subsequently, using a Simplified Nonlinear Model results in good agreement with the experimental outcomes were obtained, although the model includes only the nonlinearities due to clearance existence and impact occurrence. Further numerical analysis highlighted other scenarios that can be obtained for values of the parameters not considered in the experimental laboratory campaign. Finally, to attempt a generalization of the results, suitable dimensionless parameters were introduced.

In this paper, we present an algorithm to compute the filtered generalized \v{C}ech complex for a finite collection of disks in the plane, which don't necessarily have the same radius. The key step behind the algorithm is to calculate the minimum scale factor needed to ensure rescaled disks have a nonempty intersection, through a numerical approach, whose convergence is guaranteed by a generalization of the well-known Vietoris-Rips Lemma, which we also prove in an alternative way, using elementary geometric arguments. We present two applications of our main results. We give an algorithm for computing the 2-dimensional filtered generalized \v{C}ech complex of a finite collection of $d$-dimensional disks in $\mathbb{R}^d$. In addition, we show how the algorithm yields the minimal enclosing ball for a finite set of points in the plane., Comment: 19 pages, 10 figures. Some proofs corrected or extended, and standard algorithms were removed

Single index model is a powerful yet simple model, widely used in statistics, machine learning, and other scientific fields. It models the regression function as $g()$, where a is an unknown index vector and x are the features. This paper deals with a nonlinear generalization of this framework to allow for a regressor that uses multiple index vectors, adapting to local changes in the responses. To do so we exploit the conditional distribution over function-driven partitions, and use linear regression to locally estimate index vectors. We then regress by applying a kNN type estimator that uses a localized proxy of the geodesic metric. We present theoretical guarantees for estimation of local index vectors and out-of-sample prediction, and demonstrate the performance of our method with experiments on synthetic and real-world data sets, comparing it with state-of-the-art methods., Comment: 37 pages, 23 figures, 4 table

The desirable properties when constructing collections of subspaces often include the algebraic constraint that the projections onto the subspaces yield a resolution of the identity like the projections onto lines spanned by vectors of an orthonormal basis (the so-called tightness condition) and the geometric constraint that the subspaces form an optimal packing of the Grassmannian, again like the one-dimensional subspaces spanned by vectors in an orthonormal basis. In this article a generalization of related constructions which use known packings to build new configurations and which appear in numerous forms in the literature is given, as well as the characterization of a long list of desirable algebraic and geometric properties which the construction preserves. Another construction based on subspace complementation is similarly analyzed. While many papers on subspace packings focus only on so-called equiisoclinic or equichordal arrangements, attention is also given to other configurations like those which saturate the orthoplex bound and thus are optimal but lie outside of the parameter regime where equiisoclinic and equichordal packings can occur. Keywords: fusion frame, Grassmannian packing, simplex bound, orthoplex bound, equichordal, strongly simplicial, equiisoclinic

In this paper, we consider the different eigenvalue condition numbers for matrix polynomials used in the literature and we compare them. One of these condition numbers is a generalization of the Wilkinson condition number for the standard eigenvalue problem. This number has the disadvantage of only being defined for finite eigenvalues. In order to give a unified approach to all the eigenvalues of a matrix polynomial, both finite and infinite, two (homogeneous) condition numbers have been defined in the literature. In their definition, very different approaches are used. One of the main goals of this note is to show that, when the matrix polynomial has a moderate degree, both homogeneous numbers are essentially the same and one of them provides a geometric interpretation of the other. We also show how the homogeneous condition numbers compare with the "Wilkinson-like" eigenvalue condition number and how they extend this condition number to zero and infinite eigenvalues., 19 pages, 1 figure

In this paper we introduce a class of numerical methods for solving an equilibrium problem. This class depends on a parameter and contains the classical extragradient method and a generalization of the two-step extragradient method proposed recently by Zykina and Melen'chuk for solving variational inequality problems. Convergence of each algorithm of this class to a solution of the equilibrium problem is obtained under the condition that the equilibrium function associated with the problem is pseudomonotone and Lipschitz continuous. Some preliminary numerical results are given to compare the numerical behavior of the two-step extragradient method with respect to the other methods of the class and in particular to the extragradient method.

This paper deals with several ways to generalize the Flanders’ theorem to matrix triples. We consider six invertible matrices and try to write them as the possible products of three matrices. Initially, we describe a wide set of necessary conditions so that this system be solvable, showing that they are not sufficient. Next, we study the simultaneous solvability of two equations, selected appropriately among the matrix system. The rest of the paper is devoted to the study of a particular case, in which the six given matrices are simultaneously diagonalizable, with distinct nonzero eigenvalues. In this case, we obtain a necessary and sufficient condition for the solvability of the full matrix system. Moreover, an explicit solution to it is constructed. Certain technical results necessary for this work may be of independent interest.

In this paper, two families of explicit two-step sixth and eighth algebraic order hybrid methods with minimal phaselag are developed for the numerical integration of special second-order periodic initial-value problems. These methods have the advantage of higher algebraic accuracy and minimal phase-lag compared with some methods in [1, 2, 4–8, 11–14]. The methods proposed in this paper may be considered as a generalization of some methods in [1, 5, 7, 8, 12]. An application to the one-dimensional Schrodinger equation on the resonance problem indicates that these new methods are generally more accurate than some methods in [5–8, 2, 11, 13, 14].

This paper is devoted to a generalization of a Hadamard type inequality for the permanent of a complex square matrix. Our proof is based on a non-trivial extension of a technique used in Carlen, Lieb and Loss (Methods and Applications of Analysis 13 (1) (2006) 1-17). We give an application to coefficients of products of linear forms and show some auxiliary inequalities, which might be of independent interest., 20 pages

Huber et al. (SIAM J Comput 43:1064–1084, 2014) introduced a concept of skew bisubmodularity, as a generalization of bisubmodularity, in their complexity dichotomy theorem for valued constraint satisfaction problems over the three-value domain, and Huber and Krokhin (SIAM J Discrete Math 28:1828–1837, 2014) showed the oracle tractability of minimization of skew-bisubmodular functions. Fujishige et al. (Discrete Optim 12:1–9, 2014) also showed a min–max theorem that characterizes the skew-bisubmodular function minimization, but devising a combinatorial polynomial algorithm for skew-bisubmodular function minimization was left open. In the present paper we give first combinatorial (weakly and strongly) polynomial algorithms for skew-bisubmodular function minimization.

This paper introduces the concept of atomic subspaces with respect to a bounded linear operator. Atomic subspaces generalize fusion frames and this generalization leads to the notion of $K$-fusion frames. Characterizations of $K$-fusion frames are discussed. Various properties of $K$-fusion frames, for example, direct sums, intersection, are studied., 11 pages, To appear in Indian Journal of Pure and Applied Mathematics (2020)

Given U an n × m matrix of rank n whose columns are denoted by ( u j ) j ⩽ m , several authors have already considered the problem of finding a subset σ ⊂ { 1 , … , m } such that ( u i ) i ∈ σ span R n and Tr ( ( ∑ i ∈ σ u i u i t ) − 1 ) is minimized. In this paper, we generalize this problem by selecting arbitrary rank matrices instead of rank 1 matrices. Another generalization is considering the same problem while allowing a part of the matrix to be fixed. The methods of selection employed develop into algorithms.

This paper addresses the classical and discrete Euler-Lagrange equations for systems of $n$ particles interacting quadratically in $\mathbb{R}^d$. By highlighting the role played by the center of mass of the particles, we solve the previous systems via the classical quadratic eigenvalue problem (QEP) and its discrete transcendental generalization. The roots of classical and discrete QEP being given, we state some conditional convergence results. Next, we focus especially on periodic and choreographic solutions and we provide some numerical experiments which confirm the convergence., Comment: 10 figures 10th IMACS International Symposium on Iterative Methods in Scientific Computing

In this paper, we obtain a complete description of the class of n-tuples (n >= 2) of doubly commuting isometries. In particular, we present a several variables analogue of the Wold decomposition for isometries on Hilbert spaces. Our main result is a generalization of M. Slocinski's Wold-type decomposition of a pair of doubly commuting isometries., 10 pages, revised version. To appear in Linear Algebra and its Applications

Butcher series are combinatorial devices used in the study of numerical methods for differential equations evolving on vector spaces. More precisely, they are formal series developments of differential operators indexed over rooted trees, and can be used to represent a large class of numerical methods. The theory of backward error analysis for differential equations has a particularly nice description when applied to methods represented by Butcher series. For the study of differential equations evolving on more general manifolds, a generalization of Butcher series has been introduced, called Lie--Butcher series. This paper presents the theory of backward error analysis for methods based on Lie--Butcher series., Minor corrections and additions. Final version

SET, ERHAN/0000-0003-1364-5396; Ahmad, Farooq/0000-0001-5240-5825 WOS: 000300562100011 In this paper, we improve and further generalize some Cebysev type inequalities involving functions whose derivatives belong to L-p spaces via certain integral identities.

An increasing number of applications are based on the manipulation of higher-order tensors. In this paper, we derive a differential-geometric Newton method for computing the best rank-(R (1), R (2), R (3)) approximation of a third-order tensor. The generalization to tensors of order higher than three is straightforward. We illustrate the fast quadratic convergence of the algorithm in a neighborhood of the solution and compare it with the known higher-order orthogonal iteration (De Lathauwer et al., SIAM J Matrix Anal Appl 21(4):1324-1342, 2000). This kind of algorithms are useful for many problems.

By applying various known summation theorems to a general formula based upon Bailey's transform theorem due to Slater, Exton has obtained numerous new quadratic transformations involving hypergeometric functions of two and of higher order. Some of the results have typographical errors and have been corrected recently by Choi and Rathie. In addition, two new quadratic transformation formul{; ; ; \ae}; ; ; were also obtained in [Junesang Choi, A.K. Rathie, Quadratic transformations involving hypergeometric functions of two and higher order. EAMJ, East Asian Math. J. 22:71--77 (2006)]. The aim of this research paper is to obtain a generalization of one of the Exton's quadratic transformation. The results are derived with the help of generalized Kummer's theorem obtained earlier by Lavoie, Grondin and Rathie. As special cases, we mention six interesting results closely related to that of Exton's result.

In [Carlini, Falcone, Forcadel, and Monneau, SIAM J. Numer. Anal., 46 (2008), pp. 2920-2952], the authors have proposed a generalization of the classical fast marching method of Sethian for the eikonal equation in the case where the normal velocity depends on space and time and can change sign. The goal of this paper is to propose a modified version of the generalized fast marching method proposed in [Carlini, Falcone, Forcadel, and Monneau, SIAM J. Numer. Anal., 46 (2008), pp. 2920-2952] for which we state a general comparison principle. We also prove the convergence of the new algorithm.

International audience; This paper is devoted to Eulerian models for incompressible fluid-structure systems. These models are primarily derived for computational purposes as they allow to simulate in a rather straightforward way complex 3D systems. We first analyze the level set model of immersed membranes proposed in [Cottet and Maitre, Math. Models Methods Appl. Sci. 16 (2006) 415-438]. We in particular show that this model can be interpreted as a generalization of so-called Korteweg fluids. We then extend this model to more generic fluid-structure systems. In this framework, assuming anisotropy, the membrane model appears as a formal limit system when the elastic body width vanishes. We finally provide some numerical experiments which illustrate this claim.

In this paper we first generalize the Ostrowski inequality on time scales for k points and then unify corresponding continuous and discrete versions. We also point out some particular Ostrowski type inequalities on time scales as special cases., 10 pages

Odibat, Zaid/0000-0002-2414-7969; Momani, Shaher M./0000-0002-6326-8456 WOS: 000254254200001 In this paper we propose a new generalization of the one-dimensional differential transform method that will extend the application of the method to differential equations of fractional order. The new generalization is based on generalized Taylor's formula and Caputo fractional derivative. Several illustrative examples are given to demonstrate the effectiveness of the obtained results. The new generalization introduces a promising tool for many linear and nonlinear models containing fractional derivatives. (c) 2007 Elsevier Inc. All rights reserved.

The paper deals with a mathematical model for the electric activity of the heart at microscopic level. The membrane model used to describe the ionic currents is a generalization of the phase-I Luo–Rudy, a model widely used in 2-D and 3-D simulations of the action potential propagation. From the mathematical viewpoint the model is made up of a parabolic reaction diffusion system coupled with an ODE system. We derive existence and some regularity results. Copyright © 2006 John Wiley & Sons, Ltd.

In the present paper, a bivariate generalization of the Meyer-Konig and Zeller operators based on the q-integers is constructed. Approximation properties and rate of convergence of these operators are established with the help of a Korovkin theorem for bivariate functions and a Korovkin-type theorem given by Heping [8] and Volkov [14] respectively. Keywords: Positive linear operators, bivariate Korovkin theorem, bivariate modulus of continuity, bivariate Lipschitz class, q-integers.

In this paper we develop a set of time integrators of type fractional step Runge-Kutta methods which generalize the time integrator involved in the classical Peaceman-Rachford scheme. Combining a time semidiscretization of this type with a standard spatial discretization, we obtain a totally discrete algorithm capable of discretizing efficiently a general parabolic problem if suitable splittings of the elliptic operator are considered. We prove that our proposal is second order consistent and stable even for an operator splitting in m terms which do not necessarily commute. Finally, we illustrate the theoretical results with various applications such as alternating directions or evolutionary domain decomposition. © 2005 Elsevier B.V. All rights reserved.

This paper shows a generalization of the linear dynamic model, which is practical and easy to compute. The generalization is made by assuming a continuous elliptical joint distribution for the parameters and errors. Updated distribution and probabilistic characteristics of the current and future vector of state and observations are given. As a particular simple submodel, the one with a multidimensional exponential power initial distribution is developed. An example showing its use is given.

In this paper a generalization of Kalman's partial realization theory is developed using partial realizations defined by descriptor systems. The use of singular system realizations in contrast to regular linear systems enables us to circumvent certain technical difficulties inherent in the standard approach to partial realizations. An existence and uniqueness result for minimal partial descriptor realizations is proven and a simple rank formula for the McMillan degree is derived.