ELIMINATION IN WEYI ALGEBRA AND q-IDENTITIES

In this paper, the following are introduced briefly： the basic concept of q-proper-hypergeometric; an algorithmic proof theory for q-proper-hypergeometric identities; and elimination in the non- commutative Weyl algebra. We give an algorithm for proving the single-variable q-proper-hypergeometric identities that is based on Zeilberger＇s approach and the elimination in Weyl algebra. Finally, we test several examples that have been proven by D. Zeilberger and H. Will using the WZ-pair method and Gosper algorithm.     

A family of the local convergence of the improved secant methods for nonlinear equality constrained optimization subject to bounds on variables

This paper studies a family of the local convergence of the improved secant methods for solving the nonlinear equality constrained optimization subject to bounds on variables. The Hessian of the Lagrangian is approximated using the DFP or the BFGS secant updates. The improved secant methods are used to generate a search direction. Combining with a suitable step size, each iterate switches to trial step of strict interior feasibility. When the Hessian is only positive definite in an affine null subspace, one shows that the algorithms generate the sequences converging q-linearly and two-step q-superlinearly. Furthermore, under some suitable assumptions, some sequences generated by the algorithms converge locally one-step q-superlinearly. Finally, some numerical results are presented to illustrate the effectiveness of the proposed algorithms.     

Stability of a Variable Coefficient Star-Shaped Network with Distributed Delay

The paper deals with the exponential stability problem of a variable coefficient star-shaped network, whose strings are coupled at a common end in a star-shaped configuration and the common connection of all strings can be moved. Two kinds of media materials with a component of viscous and another simply elastic are distributed on each string. Under suitable hypothesis on the coefficient functions μj(x) of damping terms and the kernels ηj(s) of distributed delay terms, the well-posedness of the ...     

Long-Time Behavior and Density Function of a Stochastic Chemostat Model with Degenerate Diffusion

This paper considers a stochastic chemostat model with degenerate diffusion. Firstly, the Markov semigroup theory is used to establish sufficient criteria for the existence of a unique stable stationary distribution. The authors show that the densities of the distributions of the solutions can converge in L¹ to an invariant density. Then, conditions are obtained to guarantee the washout of the microorganism. Furthermore, through solving the corresponding Fokker-Planck equation, the authors give the exact expression of density function around the positive equilibrium of deterministic system. Finally, numerical simulations are performed to illustrate the theoretical results.

     

New Results on H∞ Control for Nonlinear Conformable Fractional Order Systems

This paper deals with H_∞ control problem for nonlinear conformable fractional order systems. The authors first derive new sufficient condition for exponential stability of nonlinear conformable fractional order systems based on Lyapunov-like function method for conformable fractional order systems and linear matrix inequalities(LMIs) approach. Then, by introducing a new concepts of H_∞ control problem for nonlinear conformable fractional order systems, the authors study H_∞ performance analysis and H_∞ state feedback controller design problems for the considered systems. In terms of LMIs, a sufficient condition is proposed to ensure the nonlinear conformable fractional order systems are not only exponentially stable, but also satisfy H_∞ performance γ. An explicit expression for state feedback controllers is also designed to make the closed-loop system is exponentially stable with H_∞performance γ. Finally, numerical examples are given to illustrate the validity and effectiveness of the proposed results.     

A Tight Approximation Algorithm for Multi-Vehicle CVRP with Unsplittable Demands on a Line

In this paper, the authors study the multi-vehicle capacitated vehicle routing problem on a line-shaped network with unsplittable demand. The objective is to find a transportation scheme to minimize the longest distance traveled by a single vehicle such that all the customers are served without violating the capacity constraint. The authors show that this problem has no polynomial-time algorithm with performance ratio less than 2 on condition that P ≠ NP, and then provide a 2-approximation algorithm.

     

Improve Robustness and Accuracy of Deep Neural Network with L2,∞ Normalization

In this paper, the L_2,∞ normalization of the weight matrices is used to enhance the robustness and accuracy of the deep neural network(DNN) with Relu as activation functions. It is shown that the L_2,∞ normalization leads to large dihedral angles between two adjacent faces of the DNN function graph and hence smoother DNN functions, which reduces over-fitting of the DNN. A global measure is proposed for the robustness of a classification DNN, which is the average radius of th...     

Curvature-Based r-Adaptive Isogeometric Analysis with Injectivity-Preserving Multi-Sided Domain Parameterization

Inspired by the r-refinement method in isogeometric analysis, in this paper, the authors propose a curvature-based r-adaptive isogeometric method for planar multi-sided computational domains parameterized by toric surface patches. The authors construct three absolute curvature metrics of isogeometric solution surface to characterize its gradient information, which is more straightforward and effective. The proposed method takes the internal weights as optimization variables and the resulting par...     

Sampled-Data H∞ Dynamic Output Feedback Controller Design for Fuzzy Markovian Jump Systems

This paper considers the issue of H_∞ dynamic output feedback controller design for T-S fuzzy Markovian jump systems under time-varying sampling with known upper bound on the sampling intervals. The main aim is to realize sampled-data fuzzy dynamic output feedback control so as to demonstrate the stochastic stability and H_∞ performance index of the closed-loop sampled-data fuzzy Markovian jump systems. Then, by making the most of the information within the sampling interval,...     

An Adjusted Gray Map Technique for Constructing Large Four-Level Uniform Designs

A uniform experimental design (UED) is an extremely used powerful and efficient methodology for designing experiments with high-dimensional inputs, limited resources and unknown underlying models. A UED enjoys the following two significant advantages: (i) It is a robust design, since it does not require to specify a model before experimenters conduct their experiments; and (ii) it provides uniformly scatter design points in the experimental domain, thus it gives a good representation of this domain with fewer experimental trials (runs). Many real-life experiments involve hundreds or thousands of active factors and thus large UEDs are needed. Constructing large UEDs using the existing techniques is an NP-hard problem, an extremely time-consuming heuristic search process and a satisfactory result is not guaranteed. This paper presents a new effective and easy technique, adjusted Gray map technique (AGMT), for constructing (nearly) UEDs with large numbers of four-level factors and runs by converting designs with s two-level factors and n runs to (nearly) UEDs with 2^t?1s four-level factors and 2^tn runs for any t ≥ 0 using two simple transformation functions. Theoretical justifications for the uniformity of the resulting four-level designs are given, which provide some necessary and/or sufficient conditions for obtaining (nearly) uniform four-level designs. The results show that the AGMT is much easier and better than the existing widely used techniques and it can be effectively used to simply generate new recommended large (nearly) UEDs with four-level factors.

     

Limited Memory BFGS Method for Least Squares Semidefinite Programming with Banded Structure

This work is intended to solve the least squares semidefinite program with a banded structure. A limited memory BFGS method is presented to solve this structured program of high dimension. In the algorithm, the inverse power iteration and orthogonal iteration are employed to calculate partial eigenvectors instead of full decomposition of n × n matrices. One key feature of the algorithm is that it is proved to be globally convergent under inexact gradient information. Preliminary numerical results indicate that the proposed algorithm is comparable with the inexact smoothing Newton method on some large instances of the structured problem.

     

The determination on weight hierarchies of <Emphasis Type="Italic">q</Emphasis>-ary linear codes of dimension 5 in class IV

The weight hierarchy of a linear [n; k; q] code C over GF(q) is the sequence (d ₁, d ₂, ···, d _k ) where d _r is the smallest support of any r-dimensional subcode of C. “Determining all possible weight hierarchies of general linear codes” is a basic theoretical issue and has important scientific significance in communication system. However, it is impossible for q-ary linear codes of dimension k when q and k are slightly larger, then a reasonable formulation of the problem is modified as: “Determine almost all weight hierarchies of general q-ary linear codes of dimension k”. In this paper, based on the finite projective geometry method, the authors study q-ary linear codes of dimension 5 in class IV, and find new necessary conditions of their weight hierarchies, and classify their weight hierarchies into 6 subclasses. The authors also develop and improve the method of the subspace set, thus determine almost all weight hierarchies of 5-dimensional linear codes in class IV. It opens the way to determine the weight hierarchies of the rest two of 5-dimensional codes (classes III and VI), and break through the difficulties. Furthermore, the new necessary conditions show that original necessary conditions of the weight hierarchies of k-dimensional codes were not enough (not most tight nor best), so, it is important to excogitate further new necessary conditions for attacking and solving the k-dimensional problem.     

Test on Stochastic Block Model: Local Smoothing and Extreme Value Theory

In this paper, to obtain a consistent estimator of the number of communities, the authors present a new sequential testing procedure, based on the locally smoothed adjacency matrix and the extreme value theory. Under the null hypothesis, the test statistic converges to the type I extreme value distribution, and otherwise, it explodes fast and the divergence rate could even reach n in the strong signal case where n is the size of the network, guaranteeing high detection power. This method is simple to use and serves as an alternative approach to the novel one in Lei (2016) using random matrix theory. To detect the change of the community structure, the authors also propose a two-sample test for the stochastic block model with two observed adjacency matrices. Simulation studies justify the theory. The authors apply the proposed method to the political blog data set and find reasonable group structures.

     

Fault Detection for Uncertain Delta Operator Systems with Two-Channel Packet Dropouts via a Switched Systems Approach

This paper utilizes a switched systems approach to deal with the problem of fault detection(FD) for uncertain delta operator networked control systems(NCSs) with packet dropouts and timevarying delays. Uncertainties exist in the matrices of the systems and are norm-bounded time-varying.Two parts of packet dropouts are considered in this paper: From sensors to controllers, and from controllers to actuators. Two independent Bernoulli distributed white sequences are introduced to account for packet dropouts. Then an FD filter is designed under an arbitrary switching law. Furthermore, the sufficient conditions for the NCSs under consideration that are exponentially stable in the mean-square sense and satisfy H∞ performance are obtained in terms of linear matrix inequalities(LMIs), multiple Lyapunov functions(MLF) and average dwell-time(ADT) approach. The explicit expression of the desired filter parameters is given. Finally, a numerical example verifies the effectiveness of the proposed method.     

Superconvergence of mixed covolume method on quadrilateral grids for elliptic problems

This paper considers the mixed covolume method for the second-order elliptic equations over quadrilaterals.Superconvergence results are established in this paper on quadrilateral grids satisfying the h~2-parallelogram condition when the lowest-order Raviart-Thomas space is employed in the mixed covolume method.The authors prove O(h~2) accuracy between the approximate velocity or pressure and a suitable projection of the real velocity or pressure in the L~2 norm.Numerical experiments illustrating the theoretical results are provided.     

High Speed Machining for Linear Paths Blended with G3 Continuous Pythagorean-Hodograph Curves

Previously, many studies have illustrated corner blend problem with different parameter curves. Only a few of them take a Pythagorean-hodograph (PH) curve as the transition arc, let alone corresponding real-time interpolation methods. In this paper, an integrated corner-transition mixing-interpolation-based scheme (ICMS) is proposed, considering transition error and machine tool kinematics. Firstly, the ICMS smooths the sharp corners in a linear path through blending the linear path with G3 continuous PH transition curves. To obtain optimal PH transition curves globally, the problem of corner smoothing is formulated as an optimization problem with constraints. In order to improve optimization efficiency, the transition error constraint is deduced analytically, so is the curvature extreme of each transition curve. After being blended with PH transition curves, a linear path has become a blend curve. Secondly, the ICMS adopts a novel mixed interpolator to process this kind of blend curves by considering machine tool kinematics. The mixed interpolator can not only implement jerk-limited feedrate scheduling with critical points detection, but also realize self-switching of two interpolation modes. Finally, two patterns are machined with a carving platform based on ICMS. Experimental results show the effectiveness of ICMS.

     

Asymptotic Normality of Maximum Quasi-Likelihood Estimators in Generalized Linear Models with Fixed Design*

In generalized linear models with fixed design, under the assumption λ↑_n→∞ and other regularity conditions, the asymptotic normality of maximum quasi-likelihood estimator ^↑βn, which is the root of the quasi-likelihood equation with natural link function ∑i=1^n Xi（yi -μ（Xi′β）） = 0, is obtained, where λ↑_n denotes the minimum eigenvalue of ∑i=1^nXiXi′, Xi are bounded p × q regressors, and yi are q × 1 responses.     

On LCD Negacyclic Codes over Finite Fields

This paper deduces the structure of LCD negacyclic codes over the finite field Fq, where q is an odd prime power. Based on the study of q-cyclotomic cosets modulo 2 n, the authors obtain the parameters of LCD negacyclic codes of lengths n =(q+1)/2,(q~m-1)/2(q-1)and q~(t·2~τ)-1/2(q~t +1), respectively. And many optimal codes are given. Moreover, the authors research two special classes of MDS LCD negacyclic codes of length n |(q-1)/2 and n |(q+1)2, respectively.     

Conditional Congruence Closure over Uninterpreted and Interpreted Symbols

A framework for generating congruence closure and conditional congruence closure of ground terms over uninterpreted as well as interpreted symbols satisfying various properties is proposed. It is based on some of the key concepts from Kapur’s congruence closure algorithm (RTA97) for ground equations based on introducing new symbols for all nonconstant subterms appearing in the equation set and using ground completion on uninterpreted constants and purified equalities over interpreted symbols belonging to different theories. In the original signature, the resulting rewrite systems may be nonterminating but they still generate canonical forms. A byproduct of this framework is a constant Horn completion algorithm using which ground canonical Horn rewrite systems can be generated for conditional ground theories.

New efficient algorithms for generating congruence closure of conditional and unconditional equations on ground terms over uninterpreted symbols are presented. The complexity of the conditional congruence closure is shown to be O(n*log(n)), which is the same as for unconditional ground equations. The proposed algorithm is motivated by our attempts to generate efficient and succinct interpolants for the quantifier-free theory of equality over uninterpreted function symbols which are often a conjunction of conditional equations and need additional simplification. A completion algorithm to generate a canonical conditional rewrite system from ground conditional equations is also presented. The framework is general and flexible and is used later to develop congruence closure algorithms for cases when function symbols satisfy simple properties such as commutativity, nilpotency, idempotency and identity as well as their combinations. Interesting outcomes include algorithms for canonical rewrite systems for ground equational and conditional theories on uninterpreted and interpreted symbols leading to generation of canonical forms for ground terms, constrained terms and Horn equations.

     

Distance-Based Regression Analysis for Measuring Associations

Distance-based regression model, as a nonparametric multivariate method, has been widely used to detect the association between variations in a distance or dissimilarity matrix for outcomes and predictor variables of interest in genetic association studies, genomic analyses, and many other research areas. Based on it, a pseudo-F statistic which partitions the variation in distance matrices is often constructed to achieve the aim. To the best of our knowledge, the statistical properties of the pseudo-F statistic has not yet been well established in the literature. To fill this gap, the authors study the asymptotic null distribution of the pseudo-F statistic and show that it is asymptotically equivalent to a mixture of chi-squared random variables. Given that the pseudo-F test statistic has unsatisfactory power when the correlations of the response variables are large, the authors propose a square-root F-type test statistic which replaces the similarity matrix with its square root. The asymptotic null distribution of the new test statistic and power of both tests are also investigated. Simulation studies are conducted to validate the asymptotic distributions of the tests and demonstrate that the proposed test has more robust power than the pseudo-F test. Both test statistics are exemplified with a gene expression dataset for a prostate cancer pathway.