Nonnegative Weight Function Research Articles

New approximation algorithms for the minimum cycle cover problem

Given an undirected weighted graph G=(V,E) with nonnegative weight function obeying the triangle inequality, a set {C1,C2, …,Ck} of cycles is called a cycle cover if V⊆⋃i=1kV(Ci) and its cost is given by the maximum weight of the cycles. The Minimum Cycle Cover Problem aims to find a cycle cover of cost at most λ with the minimum number of cycles. An O(n2) 24/5-approximation algorithm and an O(n5) 14/3-approximation algorithm are given by Yu and Liu (Improved approximation algorithms for some min-max cycle cover problems. Theoretical Computer Science 654 (2016) 45–58). However, the original proofs for approximation ratios are incomplete. In this paper we first present a corrected simplified analysis on the 24/5-approximation algorithm. Based on the simplified approach of analysis and some new observations, we present a new 14/3-approximation algorithm that runs in O(n3) and give an improved 32/7-approximation algorithm that runs in O(n5).

Minimax and WLS Designs of Digital FIR Filters Using SOCP for Aliasing Errors Reduction in BI-DAC

This paper presented the optimal minimax and weighted least squares (WLS) methods for designing digital finite impulse response (FIR) filters to reduce the aliasing errors generated by the non-ideality of analog filters and mixers in bandwidth interleaving digital-to-analog converter (BI-DAC). To satisfy the given expected spurious free dynamic range (SFDR), we formulated these optimal designs of digital FIR filters in BI-DAC as a convex optimization problem-second-order cone programming (SOCP) that allowed the linear equality and convex quadratic inequality constraints including the magnitude flatness and the peak aliasing errors constraints to be merged. Furthermore, we derived the computational complexity of our presented optimal design. Several design examples were given to evaluate the performance of our presented unconstrained and constrained minimax and WLS designs using SOCP including their effectiveness and computational complexity. The simulation results showed that, in our presented unconstrained minimax and WLS designs using SOCP, the maximum distortion errors were all around 0.02 dB. The maximum aliasing errors were-73.9 and -80.5 dB, which satisfied the expected SFDR of a 12-bit BI-DAC system. In addition, we analyzed the influence of different values of the nonnegative weighting function on our presented unconstrained minimax and WLS designs using SOCP, and we found that there was a tradeoff among the nonnegative weighting function's value, and the distortion and aliasing errors. Moreover, when the constraints were imposed in our presented constrained minimax and WLS designs using SOCP in the selected frequency bands, the distortion errors were equal to zero and the aliasing errors were reduced below -110 dB, but the expense was that the larger distortion and aliasing errors achieved out of these selected frequency bands. Finally, we gave the computational complexity comparisons among our presented unconstrained and constrained minimax and WLS design using SOCP, we also compared the influence of the digital FIR filters' length on our presented designs' worst-case passband ripple and stopband roll-off, and we found that there was a tradeoff among the digital FIR filters' length, the passband ripple, the stopband roll-off, the computational complexity, and the actual hardware cost.

On the submodularity of multi-depot traveling salesman games

The Steiner traveling salesman problem is the problem of finding a minimum cost tour for a salesman that must visit a set of locations while traveling along costly streets before returning to his starting point at the depot. A solution to the problem is a minimum cost Steiner tour that both starts and ends at the depot and visits all the required locations. If different players are associated with the locations to be visited, the problem induces a cooperative traveling salesman (TS) game that poses the question of how to allocate the total cost of the tour between the different players involved. This cost allocation problem can be tackled using tools and solutions from cooperative game theory.The purpose of this paper is to generalize the notion of a traveling salesman game to allow for multiple depots in the underlying traveling salesman problem (TSP) and to analyze the submodularity of such multi-depot TS games. A multi-depot TSP can be represented by a connected (di)graph in which a fixed set of vertices are denoted depots, and a non-negative weight function is defined on the edges of the graph. The submodularity of multi-depot TS games is analyzed by characterizing graphs and digraphs that induce submodular multi-depot TS games for any choice of depots and for at least one choice of depots, respectively.

Ring-shaped optical coherence lattice distribution produced by light waves on scattering

Abstract With the help of Markov approximation, a layer-structured random medium which may produce a far-zone scattered field with ring-shaped optical coherence lattice (RSOCL) distributions is discussed, and properties of the RSOCL produced by such a medium are investigated. It is shown from the distribution of the RSOCL that the diameter of each Gaussian profile lobe, the radial distance of all adjacent lobes, the number of the rings, the number of the lobe of each ring, and the radial distance of the array rings can be properly controlled by adjusting structural parameters of the scattering medium. Furthermore, by a method of superposition of several non-negative weight functions, a RSOCL with tunable lobes intensity distribution was obtained.

Orbital-enriched flat-top partition of unity method for the Schrödinger eigenproblem

Quantum mechanical calculations require the repeated solution of a Schrödinger equation for the wavefunctions of the system, from which materials properties follow. Recent work has shown the effectiveness of enriched finite element type Galerkin methods at significantly reducing the degrees of freedom required to obtain accurate solutions. However, time to solution has been adversely affected by the need to solve a generalized rather than standard eigenvalue problem and the ill-conditioning of associated system matrices. In this work, we address both issues by proposing a stable and efficient orbital-enriched partition of unity method to solve the Schrödinger boundary-value problem in a parallelepiped unit cell subject to Bloch-periodic boundary conditions. In the proposed partition of unity method, the three-dimensional domain is covered by overlapping patches, with a compactly-supported weight function associated with each patch. A key ingredient in our approach is the use of non-negative weight functions that possess the flat-top property, i.e., each weight function is identically equal to unity over some finite subset of its support. This flat-top property provides a pathway to devise a stable approximation over the whole domain. On each patch, we use pth degree orthogonal (Legendre) polynomials that ensure pth order completeness, and in addition include eigenfunctions of the radial Schrödinger equation. Furthermore, we adopt a variational lumping approach to construct a (block-)diagonal overlap matrix that yields a standard eigenvalue problem for which there exist efficient eigensolvers. The accuracy, stability, and efficiency of the proposed method is demonstrated for the Schrödinger equation with a harmonic potential as well as a localized Gaussian potential. We show that the proposed approach delivers optimal rates of convergence in the energy, and the use of orbital enrichment significantly reduces the number of degrees of freedom for a given desired accuracy in the energy eigenvalues while the stability of the enriched approach is fully maintained.

WLS optimal design for variable FIR filters with continuous parameter of fractional delay

A spectral parameterization method with discretization parameters will meet a tradeoff between the design accuracy and computational effort when designing a variable digital filter. Thus, a continuous spectral parameterization method is presented for obtaining the WLS optimal coefficients of variable fractional delay filters. Separated non-negative weighting function and some characters of matrix and matrix operations are applied to derive a closed-form solution, and Cholesky factorization is adopted to avoid the numerical problem of computing the inverses of a matrix. Its high accuracy and considerably reduced computational complexity of the proposed algorithm are experimentally verified. The given FIR filters can found their applications in high accuracy dynamic time compensation.

Form-Invariance of the Non-Regular Exponential Family of Distributions

The weighted distributions are used when the sampling mechanism records observations according to a nonnegative weight function. Sometimes the form of the weighted distribution is the same as the original distribution except possibly for a change in the parameters that is called the form-invariant weighted distribution. In this paper, by identifying a general class of weight functions, we introduce an extended class of form-invariant weighted distributions belonging to the non-regular exponential family which included two common families of distribution: exponential family and non-regular family as special cases. Some properties of this class of distributions such as the sufficient and minimal sufficient statistics, maximum likelihood estimation and the Fisher information matrix are studied.

Size Biased Inflated Beta Distribution

The standard Beta distribution is a suitable distribution for modeling the data that include proportions. In many situations which the data of proportions include a considerable number of zeros and ones, the inflated beta distributions are more appropriate. When probabilities of recording such observations are proportional to a nonnegative weight function, the recorded observations distributed as a weighted inflated Beta. This article focuses on the size biased inflated Beta distribution as a special case of weighted inflated Beta distribution with the power weight function. Some properties of this distribution is studied and its parameters are estimated using maximum likelihood and method of moments approaches. The estimators are compared via a simulation study. Finally, the real mortality data set is fitted for this model.

Stability analysis of distributed-order nonlinear dynamic systems

ABSTRACTThe problem of asymptotic stability analysis of equilibrium points in nonlinear distributed-order dynamic systems with non-negative weight functions is considered in this paper. The Lyapunov direct method is extended to be used for this stability analysis. To this end, at first, a discretisation scheme with convergence property is introduced for distributed-order dynamic systems. Then, on the basis of this tool, Lyapunov theorems are proved for asymptotic stability analysis of equilibrium points in distributed-order systems. As the order weight function assumed for the distributed-order systems is general enough, the results are applicable to a wide range of nonlinear distributed-order systems such as fractional-order systems with multiple fractional derivatives. To verify the applicability of the obtained results, these results are applied for the stability analysis of a distributed-order diffusion system and control of a fractional-order Lorenz system with multiple fractional derivatives.

Modeling for Partially Spatially Coherent Vortex Beams

We propose a method for modeling a wide class of partially spatially coherent vortex beams. It is shown that any member of such beams can be represented as an incoherent superposition of fully coherent vortex models weighted by a non-negative weighting function. Applying this method, we introduce a new type of partially coherent vortex beams and study its far-field properties.

The Nehari Manifold Approach for a p(x)-Laplacian Problem with Nonlinear Boundary Conditions

We consider a class of p(x)-Laplacian equations involving nonnegative weight functions with nonlinear boundary conditions. Our technical approach is based on the Nehari manifold, which is similar to the fibering method proposed by Drabek and Pohozaev, together with the recent idea from Brown and Wu.

General weighted Hardy type inequalities related to Baouendi-Grushin operators

ABSTRACTIn this paper, we derive a sufficient condition on a pair of nonnegative weight functions and w in so that the general weighted Hardy type inequality with a remainder termholds for all . Here and is the sub-elliptic gradient. It is worth emphasizing here that our unifying method may be readily used to recover most of the previously known sharp weighted Hardy and Heisenberg–Pauli–Weyl type inequalities as well as to construct other new inequalities with an explicit constant. Furthermore, we also obtain new results on two-weight Hardy type inequalities with remainder terms on smooth bounded domains in via a non-linear partial differential inequality.

Simple derivation and closed-form frequency-domain design of wideband length-8 interpolation kernel

This paper proposes a frequency-domain method for designing a wide-band length-8 interpolator in the weighted-least-squares (WLS) sense. The length-8 interpolator is even-symmetric and it consists of the fourth-order piecewise polynomials. In this paper, we first present a simple method for deriving its closed-form frequency response, and then formulate a frequency-domain WLS design with the C0-continuity constraint. Since the length-8 interpolator contains a total of 20 parameters, and the C0-continuity imposes 8 constraints on the design, the remaining 12 parameters can be taken as free parameters. After expressing the frequency response as the function of the 12 free parameters, the optimal free parameters can be found in the frequency-domain by minimizing the weighted squared error between the ideal and actual frequency responses, where a non-negative weighting function is adopted for emphasizing important frequency bands and ignoring unimportant bands. Consequently, the WLS design can yield an extremely accurate wide-band length-8 interpolator. Three interpolation examples are given to verify that the designed length-8 wide-band interpolator can achieve the highest accuracy in both the frequency-domain and time-domain as compared to other existing interpolators.

On the Submodularity of Multi-Depot Traveling Salesman Games

The Steiner traveling salesman problem (STSP) is the problem of finding a minimum cost tour for a salesman that must visit a set of locations while traveling along costly streets before returning to his starting point at the depot. A solution to the problem is a minimum cost tour that both starts and ends at the depot and visits all the required locations. If different players are associated with the destinations to be visited, the STSP induces a cooperative traveling salesman (TS) game that poses the question of how to allocate the total cost of the tour between the different players involved. This cost allocation problem can be tackled using tools and solutions from cooperative games. The purpose of this paper is to generalise the notion of a traveling salesman (TS) game to allow for multiple depots in the underlying STSP and to analyse the submodularity of such multi-depot TS games. A multi-depot STSP can be represented by a connected (di)graph in which a fixed set of nodes are denoted depots, and a non-negative weight function is defined on the edges of the graph. The submodularity of multi-depot TS games is analysed by characterising graphs and digraphs that induce submodular multi-depot TS games for any position of the depots and for at least one position of the depots, respectively.

Formula omitted]-widths of certain function classes defined by the modulus of continuity

Several exact inequalities are found between the best approximation by trigonometric polynomials in L2 of differentiable periodic functions, in the sense of Weyl, averaged with the weight of the modulus of continuity of arbitrary fractional order. For some classes of functions, defined by specific moduli of continuity, the exact values of the various n-widths in L2 are calculated. In particular the problem of minimizing the constants in inequalities of Jackson–Stechkin type over all subspaces of dimension N is solved. It is proved that this value is equal to the exact value of the different widths of the class L2(α)(β,p,h,φ), where φ is a non-negative integrable weight function on (0,h](0<h≤π/n).

Integer round-up property for the chromatic number of some h-perfect graphs

We prove that every h-perfect line graph and every t-perfect claw-free graph G has the integer round-up property for the chromatic number: for every non-negative integral weight function c on the vertices of G, the weighted chromatic number of (G, c) can be obtained by rounding up its fractional relaxation. As a corollary, we obtain that the weighted chromatic number can be computed in polynomial-time for these graphs. Finally, we show a new example of a graph operation which preserves the integer round-up property for the chromatic number, and use it to provide a first example of a t-perfect 3-colorable graph which does not have this property.

One-step M-estimates of scatter and the independence property

We consider a specific set of one-step M-estimates of scatter parametrized by an arbitrary non-negative and continuous weight function and show that precisely two non-trivial scatter functionals in the set have the independence property.

A Path-Compression Approach for Improving Shortest-Path Algorithms

Given a weighted directed graph G=(V;E;w), where w is non-negative weight function, G’ is a graph obtained from G by an application of path compression. Path compression reduces the graph G to a critical set of vertices and edges that affect the generation of shortest trees. The main contribution of this paper is finding shortest path between two selected vertices by applying a new algorithm that reduces number of nodes that needs to be traversed in the graph while preserving all graph properties. The main method of the algorithm is restructuring the graph in a way that only critical/relevant nodes are considered while all other neutral vertices and weights are preserved as sub paths' properties. Our algorithm can compress the graph paths into considerable improved percentage especially when the graph is sparse and hence improves performance significantly.

A note on weighted rooted trees

Let T be a tree rooted at r. Two vertices of T are related if one is a descendant of the other; otherwise, they are unrelated. Two subsets A and B of V(T) are unrelated if, for any a∈A and b∈B, a and b are unrelated. Let ω be a nonnegative weight function defined on V(T) with ∑v∈V(T)ω(v)=1. In this note, we prove that either there is an (r,u)-path P with ∑v∈V(P)ω(v)≥13 for some u∈V(T), or there exist unrelated sets A,B⊆V(T) such that ∑a∈Aω(a)≥13 and ∑b∈Bω(b)≥13. The bound 13 is tight. This answers a question posed in a very recent paper of Bonamy, Bousquet and Thomassé.

Gauss-type quadrature rule with complex nodes and weights for integrals involving Daubechies scale functions and wavelets

This paper deals with derivation of a Gauss-type quadrature rule (named as Gauss–Daubechies quadrature rule) for numerical evaluation of integrals involving product of integrable function and Daubechies scale functions/wavelets. Some of the nodes and weights of the quadrature formula may be complex and appear with their conjugates. This is in contrast with the standard Gauss-type quadrature rule for integrals involving products of integrable functions and non-negative weight functions. This quadrature rule has accuracy as good as the standard Gauss-type quadrature rule and is also found to be stable. The efficacy of the quadrature rule derived here has been tested through some numerical examples.