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This paper concerns late 1920 s attempts to construct unitary theories of gravity and electromagnetism. A first attempt using a non-standard connection—with torsion and zero-curvature—was carried out by Albert Einstein in a number of publications that appeared between 1928 and 1931. In 1929, Tullio Levi-Civita discussed Einstein's geometric structure and deduced a new system of differential equations in a Riemannian manifold endowed with what is nowadays known as Levi-Civita connection. He attained an important result: Maxwell's electromagnetic equations and the gravitational equations were obtained exactly, while Einstein had deduced them only as a first order approximation. A main feature of Levi-Civita's theory is the essential use of the Ricci's rotation coefficients, introduced by Gregorio Ricci Curbastro many years before. We trace the history of Ricci's coefficients that are still used today, and highlight their geometric and mechanical meaning. [ABSTRACT FROM AUTHOR]

Abstract: Ahn, Lee, Park and Yoo proved that the best constrained degree reduction of a polynomial f in -norm equals the best weighted Euclidean approximation of the Bernstein–Bézier coefficients of f in a paper, published in the journal, Computer Aided Geometric Design 21 (2) (2004) 181–191. In this note, we point out an error in their paper and give the correct result. [Copyright &y& Elsevier]

In this paper, some properties of linear functionals are studied, which are used in functional analysis, optimization theory, numerical analysis, approximation theory, various applications, etc. [ABSTRACT FROM AUTHOR]

The radiation from conventional thermal infrared sources is incoherent and unpolarized. It has been recently shown that patterning plasmonic materials into metasurfaces can enhance the coherence of thermal radiation through thermal excitation of localized resonant modes in meta-atoms, enabling realization of thermal emitting metasurfaces—metasources—with high directivity and spectral selectivity. Toward achieving flexible modulation of radiative heat using metasurfaces and realization of complex functionalities, the design rules and modeling tools should be modified to take into account the incoherent nature of thermal radiation. In this paper, we utilize a robust formulation based on discrete dipole approximation within the framework of fluctuation dissipation theorem, which can efficiently characterize the thermal emission from large-area finite thermal metasources consisting of nanostructured materials. We employ an evolutionary lattice approach for the inverse design of radiative thermal metasources through topology optimization without relying on the phase analysis based on the assumption of coherency. To this end, the interaction matrix of dipolar equations is initialized for a lattice of subwavelength blocks and the arrangement of the blocks is subsequently optimized via a genetic algorithm toward achieving a certain spatial emission pattern with minimal computational cost. This approach enables fast and efficient design of large-scale thermal metasources with arbitrary functionalities. In particular, we investigate steering, focusing, and arbitrary shaping of partially coherent thermal radiation via finite silicon carbide gratings supporting surface phonon polaritons. Moreover, we demonstrate that the use of electro-optical materials such as graphene enables active tuning of thermal emission and realization of geometrically fixed multifunctional thermal metasources. [ABSTRACT FROM AUTHOR]

Presents comments on the study "Comparison of Continuous and Cyclic Pumping From a Well," by M.W. Kawecki. Use of exact well function values in the study; Error in the results of the study; Ratio of pumping rates obtained by using the logarithmic approximation.

This paper generalizes the recently proposed approaches for calculating the derivative couplings between adiabatic states in density-functional theory (DFT) based on a Slater transition-state density to transitions such as singlet-singlet excitations, where a single-determinant ansatz is insufficient. The proposed approach is based on restricted open-shell Frank et al. [J. Chem. Phys. 108, 4060 (1998)] theory used to describe a spin-adapted Slater transition state. To treat the dependence of electron-electron interactions on the nuclear positions, variational linear-response density-functional perturbation theory is generalized to reference states with an orbital-dependent Kohn-Sham Hamiltonian and nontrivial occupation patterns. The methods proposed in this paper are not limited to the calculation of derivative coupling vectors, but can also be used for the calculation of other transition matrix elements. Moreover, they can be used to calculate the linear response of open-shell systems to arbitrary external perturbations in DFT. [ABSTRACT FROM AUTHOR]

From the eigenvalue problem theory, we see that the convergence rate of the biharmonic eigenvalues obtained by the mixed method in I. Bab u ˇ ska and J. Osborn, [Eigenvalue Problems, Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, 1991.] is h 2 k − 2 for k ≥ 2. In this paper, we give a presentation of the lowest-order virtual element method for the approximation of Kirchhoff plate vibration problem. This discrete scheme is based on a conforming H 1 (Ω) × H 1 (Ω) formulation, following the variational formulation of Ciarlet–Raviart method, which allows us to make use of simpler and lower-regularity virtual element space. By using the classical spectral approximation theory in functional analysis, we prove the spectral approximation and optimal convergence order h 2 for the eigenvalues. Finally, some numerical experiments are presented, which show that the proposed numerical scheme can achieve the optimal convergence order. [ABSTRACT FROM AUTHOR]

We study the classical problem introduced by R. Isaacs and S. Gal of minimizing the time to find a hidden point H on a network Q moving from a known starting point. Rather than adopting the traditional continuous unit speed path paradigm, we use the dynamic "expanding search" paradigm recently introduced by the authors. Here the regions S(t) that have been searched by time t are increasing from the starting point and have total length t. Roughly speaking the search follows a sequence of arcs ai such that each one starts at some point of an earlier one. This type of search is often carried out by real life search teams in the hunt for missing persons, escaped convicts, terrorists or lost airplanes. The paper which introduced this type of search solved the adversarial problem (where H is hidden to maximize the time to be found) for the cases where Q is a tree or is 2-arc-connected. This paper's main contribution is to give two strategy classes which can be used on any network and have expected search times which are within a factor close to 1 of the value of the game (minimax search time). These strategies classes are respectively optimal for trees and 2-arc-connected networks. We also solve the game for circle-and-spike networks, which can be considered as the simplest class of networks for which a solution was previously unknown. [ABSTRACT FROM AUTHOR]

In this paper, Investigation of global approximation of the generalized Szàsz–Mirakjan type operators in exponential weight spaces is discussed. The paper focuses on calculation of moments, direct results and inverse results for the saturated as well as non-saturated cases. [ABSTRACT FROM AUTHOR]

In recent years, interest in the random-phase approximation (RPA) has grown rapidly. At the same time, tensor hypercontraction has emerged as an intriguing method to reduce the computational cost of electronic structure algorithms. In this paper, we combine the particle-particle random phase approximation with tensor hypercontraction to produce the tensor-hypercontracted particle-particle RPA (THC-ppRPA) algorithm. Unlike previous implementations of ppRPA which scale as O(r6), the THC-ppRPA algorithm scales asymptotically as only O(r4), albeit with a much larger prefactor than the traditional algorithm.We apply THC-ppRPA to several model systems and show that it yields the same results as traditional ppRPA to within mH accuracy. Our method opens the door to the development of post-Kohn Sham functionals based on ppRPA without the excessive asymptotic cost of traditional ppRPA implementations. [ABSTRACT FROM AUTHOR]

This paper presents a new family of methods for evaluating matrix polynomials more efficiently than the state-of-the-art Paterson–Stockmeyer method. Examples of the application of the methods to the Taylor polynomial approximation of matrix functions like the matrix exponential and matrix cosine are given. Their efficiency is compared with that of the best existing evaluation schemes for general polynomial and rational approximations, and also with a recent method based on mixed rational and polynomial approximants. For many years, the Paterson–Stockmeyer method has been considered the most efficient general method for the evaluation of matrix polynomials. In this paper we show that this statement is no longer true. Moreover, for many years rational approximations have been considered more efficient than polynomial approximations, although recently it has been shown that often this is not the case in the computation of the matrix exponential and matrix cosine. In this paper we show that in fact polynomial approximations provide a higher order of approximation than the state-of-the-art computational methods for rational approximations for the same cost in terms of matrix products. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a class of new general iteration processes for two finite families of total asymptotically nonexpansive mappings in hyperbolic spaces, which includes asymptotically nonexpansive mapping, (generalized) nonexpansive mapping of all normed linear spaces, Hadamard manifolds and CAT(0) spaces as special cases. Some important related properties to the new general iterative processes are also given and analyzed, and Δ-convergence and strong convergence of the iteration in hyperbolic spaces are proved. Furthermore, some meaningful illustrations for clarifying our results and two open questions are proposed. The results presented in this paper extend and improve the corresponding results announced in the current literature. [ABSTRACT FROM AUTHOR]

The diagonal linewidth in two-dimensional infrared spectra is often narrower than the distribution of transition frequencies. The width along the antidiagonal is broader than predicted by the lifetime broadening. These effects arise from time-dependent fluctuations of the transition frequencies. They can be accounted for with a semiclassical approach. For systems with many coupled vibrational modes, this approach, however, becomes computationally too demanding to be practically applicable. A time-averaging approximation was suggested for linear infrared absorption spectra. In this paper, we demonstrate that the averaging can be optimized to fit a broader scale of frequency fluctuations by using a Gaussian weight function instead of the originally proposed box function. We further generalize the time-averaging method to allow the simulation of two-dimensional infrared spectra and demonstrate the method on a simple system. The approximation delivers a large speed-up of the calculation without losing significant accuracy. [ABSTRACT FROM AUTHOR]

Finding the underlying mechanism from the statistical properties of an experimental two-state trajectory generated from dynamics in a complex on-off multisubstate kinetic scheme (KS) is the aim of many experiments. Since the data explicitly shows only transitions between substates of different states, information about the KS is lost, resulting in equivalence of KSs, i.e., the occurrence of different KSs that lead to the same data, in a statistical sense. In order to deal with this phenomenon, a canonical (unique) form of reduced dimensions (RD) is built from the data. RD forms are on-off networks with connections only between substates of different states, where the connections usually have nonexponential waiting time probability density functions. In this paper, we give a list of (about 50) relationships between properties of the data, the topology of reduced dimension forms, and features of KSs. Many of these relationships involve symmetries in RD forms, KSs, and the data and irreversible transitions in KSs. These relationships are useful both in theoretical analysis of on-off KSs and in the analysis of the data. [ABSTRACT FROM AUTHOR]

The thermal Gaussian approximation (TGA) recently developed by Frantsuzov et al. [Chem. Phys. Lett. 381, 117 (2003)] has been demonstrated to be a practical way for approximating the Boltzmann operator exp(-βH⁁) for multidimensional systems. In this paper the TGA is combined with semiclassical (SC) initial value representations (IVRs) for thermal time correlation functions. Specifically, it is used with the linearized SC-IVR (LSC-IVR, equivalent to the classical Wigner model), and the “forward-backward semiclassical dynamics” approximation developed by Shao and Makri [J. Phys. Chem. A 103, 7753 (1999); 103, 9749 (1999)]. Use of the TGA with both of these approximate SC-IVRs allows the oscillatory part of the IVR to be integrated out explicitly, providing an extremely simple result that is readily applicable to large molecular systems. Calculation of the force-force autocorrelation for a strongly anharmonic oscillator demonstrates its accuracy, and calculation of the velocity autocorrelation function (and thus the diffusion coefficient) of liquid neon demonstrates its applicability. [ABSTRACT FROM AUTHOR]

Time-dependent density-functional theory in the adiabatic approximation has been very successful for calculating excitation energies in molecular systems. This paper studies nonadiabatic effects for excitation energies, using the current–density functional of Vignale and Kohn [Phys. Rev. Lett. 77, 2037 (1996)]. We derive a general analytic expression for nonadiabatic corrections to excitation energies of finite systems and calculate singlet s→s and s→p excitations of closed-shell atoms. The approach works well for s→s excitations, giving a small improvement over the adiabatic local-density approximation, but tends to overcorrect s→p excitations. We find that the observed problems with the nonadiabatic correction have two main sources: (1) the currents associated with the s→p excitations are highly nonuniform and, in particular, change direction between atomic shells, (2) the so-called exchange-correlation kernels of the homogeneous electron gas, fxcL and fxcT, are incompletely known, in particular in the high-density atomic core regions. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The projection games (aka Label Cover) problem is of great importance to the field of approximation algorithms, since most of the NP-hardness of approximation results we know today are reductions from Label Cover. In this paper we design several approximation algorithms for projection games: (1) A polynomial-time approximation algorithm that improves on the previous best approximation by Charikar et al. (Algorithmica 61(1):190-206, 2011). (2) A sub-exponential time algorithm with much tighter approximation for the case of smooth projection games. (3) A polynomial-time approximation scheme (PTAS) for projection games on planar graphs and a tight running time lower bound for such approximation schemes. The conference version of this paper had only the PTAS but not the running time lower bound. [ABSTRACT FROM AUTHOR]

In the paper, the authors established two best approximation formulas for the Bateman's G-function. Also, they studied the completely monotonicity of some functions involving G(x). Some new inequalities are deduced for the function and its derivative such as 1/2 ln [1 + 2x + a/x2 + 2x + 4/3 ] < G(x + 2) < 1/2 ln [ 1 + 2x + b/x2 + 2x + 4/3 ], x > 0 where a = 3 and b = e4-16/12 are the best possible constants. Our results improve some recent inequalities about the function G(x). [ABSTRACT FROM AUTHOR]

This paper proposes a pointwise approximation scheme on the unit sphere S2, which aims to handle spherical scattered data with high level noise by using a regularized moving least squares (RMLS) approximation with Laplace-Beltrami operator. The pointwise errors of approximation by the RMLS are estimated, and some numerical examples are designed to examine the obtained theoretical results. Also, the given numerical examples illustrate that the different choices for the order of Laplace-Beltrami operator and regularized parameter can provide different pointwise approximation results. [ABSTRACT FROM AUTHOR]

Utility-based shortfall risk measures (SR) have received increasing attention over the past few years for their potential to quantify the risk of large tail losses more effectively than conditional value at risk. In this paper, we consider a distributionally robust version of the shortfall risk measure (DRSR) where the true probability distribution is unknown and the worst distribution from an ambiguity set of distributions is used to calculate the SR. We start by showing that the DRSR is a convex risk measure and under some special circumstance a coherent risk measure. We then move on to study an optimization problem with the objective of minimizing the DRSR of a random function and investigate numerical tractability of the optimization problem with the ambiguity set being constructed through ϕ -divergence ball and Kantorovich ball. In the case when the nominal distribution in the balls is an empirical distribution constructed through iid samples, we quantify convergence of the ambiguity sets to the true probability distribution as the sample size increases under the Kantorovich metric and consequently the optimal values of the corresponding DRSR problems. Specifically, we show that the error of the optimal value is linearly bounded by the error of each of the approximate ambiguity sets and subsequently derive a confidence interval of the optimal value under each of the approximation schemes. Some preliminary numerical test results are reported for the proposed modeling and computational schemes. [ABSTRACT FROM AUTHOR]

Mathematical analysis with the numerical simulation of the newly formulated fractional version of the Adams-Bashforth method using the Atangana-Baleanu operator which has both nonlocal and nonsingular properties is considered in this paper. We adopt the fixed point theory and approximation method to prove the existence and uniqueness of the solution via general two-component time fractional differential equations. The method is tested with three nonlinear chaotic dynamical systems in which the integer-order derivative is modeled with the proposed fractional-order case. The simulation result for different α values in (0 , 1 ] is presented. [ABSTRACT FROM AUTHOR]

In this paper, for any smooth function on [0, 1]2, we give an asymptotic representation of hyperbolic cross approximations of its Fourier series whose principal part is determined by the values of the function at vertexes of [0, 1]2 and present a novel approach to estimates of the upper bounds of approximation errors. At the same time, we also give an asymptotic formula of partial sum approxima- tions whose principal part is determined by not only partial derivatives at vertexes of [0, 1]2, but also mean values on each side. Comparing asymptotic representations of these two kinds of approximation, we ¯nd that although in general the hyperbolic cross approximation is better than the partial sum ap- proximation, the partial sum approximation possibly work better under some cases, and we also give the corresponding necessary and su±cient condition to characterize these cases. [ABSTRACT FROM AUTHOR]

In the paper, the authors established two best approximation formulas for the Bateman's G-function. Also, they studied the completely monotonicity of some functions involving G(x). Some new inequalities are deduced for the function and its derivative such as 1/2 ln [1 + 2x + a/x2 + 2x + 4/3 ] < G(x + 2) < 1/2 ln [ 1 + 2x + b/x2 + 2x + 4/3 ], x > 0 where a = 3 and b = e4-16/12 are the best possible constants. Our results improve some recent inequalities about the function G(x). [ABSTRACT FROM AUTHOR]

The topic of rough set theory considers a relation to determine the lower and upper approximations of a set X. Originally, this relation was assumed to be an equivalence relation. This research focuses on using tolerance relations instead of equivalences, i.e. we do not assume the transitivity of the relations. More specifically, in this paper we investigate tolerances induced by irredundant coverings. We characterize the interrelation between the lattices of lower and upper approximations of such tolerances R and p. The theory of Formal Concept Analysis makes it possible to examine the inclusions of the resulting concepts. We also use quasiorders (denoted by ≤ and ≥ (p)) and an equivalence relation (denoted by ker p) for summarizing the connection between tolerances and lattices in a theorem. [ABSTRACT FROM AUTHOR]

In this paper we consider rational approximation of functions in the Hardy spaces on the strip with . Observing that can be regarded as the direct sum of and , we give three kinds of adaptive rational approximation in enhancing the corresponding approximations in and where and We also obtain a type of approximation for subspaces in by making use of Riesz bases in these subspaces. [ABSTRACT FROM AUTHOR]

We give a constant factor approximation algorithm for the Asymmetric Traveling Salesman Problem on shortest path metrics of directed graphs with two different edge weights. For the case of unit edge weights, the first constant factor approximation was given recently by Svensson. This was accomplished by introducing an easier problem called Local-Connectivity ATSP and showing that a good solution to this problem can be used to obtain a constant factor approximation for ATSP. In this paper, we solve Local-Connectivity ATSP for two different edge weights. The solution is based on a flow decomposition theorem for solutions of the Held-Karp relaxation, which may be of independent interest. [ABSTRACT FROM AUTHOR]

In this paper, we examine the interaction of convection, rotation and mean flows in a thermal annulus. In this system, mean flows are driven by correlations induced by rotation leading to nontrivial Reynolds stresses. The mean flows act back on the convective turbulence acting as a barrier to transport. For this system, we demonstrate that the generalized quasilinear approximation (Marston et al. 2016 Phys. Rev. Lett. 116, 214501. (doi:10.1103/PhysRevLett.116.214501)) may provide a much better approximation to the complicated full nonlinear dynamics than the widely used quasilinear approximation. This result will enable the construction of more accurate statistical theories for the description of geophysical and astrophysical flows. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to define the notions of nano strong semi-closed set, nano strong semi-continuity and nano contra strong semi-continuity in nano topological spaces and discuss some of their properties. Also, the notion of NC-set, Nic-set, ND(c; s)-set, ND(c; p)- set, ND(c; ps)-set, ND(c;α)-set and nano strong B-set are introduced and investigated. Further we define, nanoC-continuous, nano A-continuous, nano ic-continuous, nano strongly B-continuous, nano weakly-continuous, nano weak*-continuous, ND(c; p)-continuous, ND(c; s)-continuous, ND(c; ps)- continuous and ND(c;α)-continuous and study the relationships between them and other existing continuities. Further, we have given an appropriate examples to understand the abstract concept clearly. [ABSTRACT FROM AUTHOR]

This paper serves to treat boundary conditions numerically with high order accuracy in order to suit the two-stage fourth-order finite volume schemes for hyperbolic problems developed in J. Li and Z. Du (2016) [17] . As such, it is significant when capturing small scale structures near physical boundaries. Different from previous contributions in literature, the current approach constructs a fourth order accurate approximation to boundary conditions by only using the Jacobian matrix of the flux function (characteristic information) instead of its successive differentiation of governing equations leading to tensors of high ranks in the inverse Lax–Wendroff method. Technically, data in several ghost cells are constructed with interpolation so that the interior scheme can be implemented over boundary cells, and theoretical boundary condition has to be modified properly at intermediate stages so as to make the two-stage scheme over boundary cells fully consistent with that over interior cells. This is nonintuitive and highlights the fact that theoretical boundary conditions are only prescribed for continuous partial differential equations (PDEs), while they must be approximated in a consistent way (even though they could be exactly valued) when the PDEs are discretized . Several numerical examples are provided to illustrate the performance of the current approach when dealing with general boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper we investigate a relationship between fully k -rotundity properties, uniform K -monotonicity properties, reflexivity and K -order continuity in symmetric spaces E . We also answer a crucial question whether fully k -rotundity properties might be restricted in definition to E d the positive cone of all nonnegative and decreasing elements of E . We present a complete characterization of decreasing uniform K -monotonicity and K -order continuity in E . It is worth mentioning that we also establish several auxiliary results describing reflexivity in Lorentz spaces Γ p , w and K -order continuity in Orlicz spaces L ψ . Finally, we show an application of discussed geometric properties to the approximation theory. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to present some basic as well as essential results involving the notion of p-Chebyshev sets in probabilistic normed spaces. In particular, we discuss the convexity of p-Chebyshev sets, decomposition of the space into its special subspaces, and we see a characterization of p-Chebyshev sets in quotient spaces. [ABSTRACT FROM AUTHOR]

In this paper we study d -variate approximation for weighted Korobov spaces in the worst-case setting. The considered algorithms use finitely many evaluations of arbitrary linear functionals. We give matching necessary and sufficient conditions for some notions of tractability in terms of two weight parameters. Our result is an affirmative answer to a problem which is left open in a recent paper of Kritzer, Pillichshammer and Woźniakowski. [ABSTRACT FROM AUTHOR]

Let F be a compact set of a Banach space X. This paper analyzes the "generalized empirical interpolation method," which, given a function f ∊ F, builds an interpolant Jn[f] in an n dimensional subspace Xn ⊂ X with the knowledge of n outputs (σi(f))ni=1, where σi ∊ X' and X' is the dual space of X. The space Xn is built with a greedy algorithm that is adapted to F in the sense that it is generated by elements of F itself. The algorithm also selects the linear functionals (σi)ni=1 from a dictionary Σ ⊂ X'. In this paper, we study the interpolation error maxf∊F ‖f - Jn[f]‖X by comparing it with the best possible performance on an n dimensional space, i.e., the Kolmogorov n-width of F in X, dn(F,X). For polynomial or exponential decay rates of dn(F,X), we prove that the interpolation error has the same behavior modulo the norm of the interpolation operator. Sharper results are obtained in the case where X is a Hilbert space. [ABSTRACT FROM AUTHOR]

In this paper, we consider a modification of the classical Baskakov operators based on a function ϑ. Basic qualitative and quantitative Korovkin results are stated in weighted spaces. We prove a quantitative Voronovskaya-type theorem and present some results on the monotonic convergence of the sequence. Finally, we show a shape preserving property and further direct convergence theorems. Weighted modulus of continuity of first order and the notion of ϑ-convexity are used throughout the paper. [ABSTRACT FROM AUTHOR]

Inevitable perturbations have a significant impact on collaborative production (CP) cause that the complexity are greatly increased. In order to assess the impact accurately enterprises to make precise decisions, it is prerequisite to master the dynamic behaviour of CP when changes occur. Accordingly, in this paper, an evolution model for simulating CP task state to perturbations is proposed. Regard CP tasks as a directed weighted complex network (DWCPTN) firstly, and statistical properties of DWCPTN are analysed. Two perturbation cases and two modifications policies are defined followed. Based on above mentioned, evolution model based on cellular automaton (CA) and SIS (Susceptible-Infected-Susceptible) is presented to reveal the varying process by three indices (friction of resolved nodes, convergence time and economic behaviour). Finally, analytic results from a case of a chemical product CP network (CPNCP) are used to illustrate the model and method proposed in this paper. Simulation results shows that unplanned perturbations promote a negative role to CP but the improvements of self-healing can decrease the negative effects efficiently. With the analysis, it can provide insight into ways of improving the strategic and operational decision making for enterprises. [ABSTRACT FROM AUTHOR]

Weighted geometric set-cover problems arise naturally in several geometric and nongeometric settings (e.g., the breakthrough of Bansal and Pruhs [Proceedings of FOCS, 2010, pp. 407-414] reduces a wide class of machine scheduling problems to weighted geometric set cover). More than two decades of research has succeeded in settling the (1 + ∈)-approximability status for most geometric set-cover problems, except for some basic scenarios which are still lacking. One is that of weighted disks in the plane for which, after a series of papers, Varadarajan [Proceedings of STOC'10, 2010, pp. 641-648] presented a clever quasi-sampling technique, which together with improvements by Chan et al. [Proceedings of SODA, 2012, pp. 1576-1585], yielded an O(1)-approximation algorithm. Even for the unweighted case, a polynomial time approximation scheme (PTAS) for a fundamental class of objects called pseudodisks (which includes halfspaces, disks, unit-height rectangles, translates of convex sets, etc.) is currently unknown. Another fundamental case is weighted halfspaces in R³, for which a PTAS is currently lacking. In this paper, we present a quasi PTAS (QPTAS) for all these remaining problems. Our results are based on the separator framework of Adamaszek and Wiese [Proceedings of FOCS, 2013, pp. 400-409; Proceedings of SODA, 2014, pp. 645-656], who recently obtained a QPTAS for a weighted independent set of polygonal regions. This rules out the possibility that these problems are APX-hard, assuming NP ... DTIME(2polylog(n)). Together with the recent work of Chan and Grant [Comput. Geom., 47 (2014), pp. 112-124], this settles the APX-hardness status for all natural geometric set-cover problems. [ABSTRACT FROM AUTHOR]

In this paper, a novel discriminative learning method is proposed to estimate generative models for multi-class pattern classification tasks, where a discriminative objective function is formulated with separation margins according to certain discriminative learning criterion, such as large margin estimation (LME). Furthermore, the so-called approximation-maximization (AM) method is proposed to optimize the discriminative objective function w.r.t. parameters of generative models. The AM approach provides a good framework to deal with latent variables in generative models and it is flexible enough to discriminatively learn many rather complicated generative models. In this paper, we are interested in a group of generative models derived from multinomial distributions. Under some minor relaxation conditions, it is shown that the AM-based discriminative learning methods for these generative models result in linear programming (LP) problems that can be solved effectively and efficiently even for rather large-scale models. As a case study, we have studied to learn multinomial mixture models (MMMs) for text document classification based on the large margin criterion. The proposed methods have been evaluated on a standard RCV1 text corpus. Experimental results show that large margin MMMs significantly outperform the conventional MMMs as well as pure discriminative models such as support vector machines (SVM), where over 25 % relative classification error reduction is observed in three independent RCV1 test sets. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to provide an easy-to-use approximation formula for the inverse Langevin function. The mathematical complexity of this function makes it unfeasible for an analytical manipulation and inconvenient for computer simulation. This situation has motivated a series of papers directed on its approximation. The best known solution is given by Cohen. It is used in a lot of statistically based models of rubber-like materials. The formula is derived from rounded Padé approximation [3/2]. The main idea of the presented approach in this paper relies on improvement of the precision of approximation formula for the inverse Langevin function by using multipoint Padé approximation method. We focused our study strongly on obtaining a simple and accurate approximation. It is assumed that the proposed approximation formula may be considered a useful tool for verification of the results obtained in other ways. Our results are supported by investigating several numerical examples. The paper also presents a few applications of computer software named Mathematica which can be used to calculate symbolically one point Padé approximants and numerically multipoint Padé approximants. Using this software, we showed also how to compute higher order derivatives of the inverse function in a simple and elegant way. This issue was discussed by Itskov et al. [ABSTRACT FROM AUTHOR]

It is a common practice that interest rates are modeled by a CIR process, a solution of the CIR equation. The usual motivation is that such a process is mean-reverting and takes nonnegative values (for initial nonnegative values). The stochastic Verhulst (SV) equation possesses the same properties. Moreover, one can always choose the parameters of the SV equation so that its solution has the same stationary (gamma) distribution as that of the CIR equation. So, in long run, both processes behave rather similarly. Therefore, the idea to alternatively use the SV process for modeling interest rates seems to be rather natural. In this paper, we first compare SV and CIR and discuss their pros and cons. A serious disadvantage of the SV process is that explicit expressions of its moments are not known. To diminish this gap, the second part of the paper is devoted to approximation of the moments of the SV process. [ABSTRACT FROM AUTHOR]

In this paper, a kind of new Stancu-Bézier type operators is introduced. The Korovkin type approximation theorem of these operators is investigated. The rates of convergence of these operators are studied by means of modulus of continuity. Then, by using the Ditzian-Totik modulus of smoothness, a direct theorem concerned with an approximation for these operators is also obtained. [ABSTRACT FROM AUTHOR]

Multi-granulation dual hesitant fuzzy rough set is an extension of intuitionistic fuzzy multi-granulation rough sets and multi-granulation fuzzy rough sets. For further studying the theories and applications of multi-granulation dual hesitant fuzzy rough sets, in this paper, we mainly investigate reduction approaches of the multi-granulation dual hesitant fuzzy rough sets. We develop a reduction approach in multi-granulation dual hesitant fuzzy decision information systems based on multi-granulation dual hesitant fuzzy rough sets to eliminate redundant dual hesitant fuzzy granulations. And an example is provided to illustrate the validity of this approach. [ABSTRACT FROM AUTHOR]

In this paper we study the approximation of functions from generalized Morrey spaces by nice functions. We introduce a new subspace whose elements can be approximated by infinitely differentiable compactly supported functions. This provides, in particular, an explicit description of the closure of the set of such functions in generalized Morrey spaces. [ABSTRACT FROM AUTHOR]

The present paper deals with a generalization of the Baskakov operators. Some direct theorems, asymptotic formula and A-statistical convergence are established. Our results are based on a ρ function. These results include the preservation properties of the classical Baskakov operators. [ABSTRACT FROM AUTHOR]