一类积分不等式的机器判定

将一类积分不等式转化为Tarski模型外的齐次对称多项式不等式,该类齐次对称多项式的次数是给定的,变元个数可以是任意多个,并且多项式的系数是与变元个数相关的变系数.这些特点与杨路等人最近提出的几个公开问题密切相关,是比较有代表性的一类齐次对称多项式.然后利用Timofte关于对称多项式不等式判定的降维方法,结合不等式证明软件BOTTEMA及差分代换方法,给出对应的一类Tarski模型外的齐次对称多项式不等式的机器判定算法,从而实现原积分不等式的机器判定.当给定的积分不等式及齐次对称多项式不等式不成立时,可给出具体不成立的数值反例.应用例子表明问题的广泛性及算法的有效性.     

A class of general quartic spline curves with shape parameters

With a support on four consecutive subintervals, a class of general quartic splines are presented for a non-uniform knot vector. The splines have C² continuity at simple knots and include the cubic non-uniform B-spline as a special case. Based on the given splines, piecewise quartic spline curves with three local shape parameters are given. The given spline curves can be C²∩G³ continuous by fixing some values of the curve?s parameters. Without solving a linear system, the spline curves can also be used to interpolate sets of points with C² continuity. The effects of varying the three shape parameters on the shape of the quartic spline curves are determined and illustrated.     

一种类四次三角样条曲线的奇拐点分析

根据移动一个控制顶点而固定其他控制顶点的方法, 分析一种类四次三角样条曲线的形状。在一类控制多边形下,利用控制顶点之间距离的关系,分别得到了对应的四次三角样条曲线含有尖点、二重点以及拐点的判别条件。     

一类三角形几何不等式的自动证明

讨论了一类只含三角函数的三角形几何不等式的自动证明问题。运用代数方法将其有理化,在不新增加根式的条件下将问题转换为一个二元多项式不等式的证明,设计的基于胞腔分解和实根分离的算法实现了二元多项式不等式的自动证明,输出的证明过程可以手工验证或借助一些数学软件进行理解。实验表明上述算法对一大批具有相当难度,特别是关于三角函数的几何不等式十分高效,并且能够解决三角形内角的任意有理倍数函数的不等式机器证明问题。     

Adaptive finite elements for a certain class of variational inequalities of second kind

In this note, we extend our studies on finite element Galerkin schemes for elliptic variational inequalities of first to the one of second kind. Especially we perform the corresponding a posteriori error analysis for a simple friction problem and a model flow of a Bingham fluid.     

Sensor fault diagnostics for a class of non-linear systems using linear matrix inequalities

This paper develops a systematic approach to fault diagnostic system design for sensor health monitoring in Lipschitz non-linear systems. The methodology applies to non-linear systems with three or more sensors in which the state is observable through any one of the sensor measurements. Two major issues are addressed in the paper—observer design for the non-linear system to ensure directional growth of residues for failure identification and use of linear matrix inequalities for explicit design of the observer gain. The use of the methodology is demonstrated through an illustrative example.     

A recurrent neural network for solving a class of general variational inequalities.

This paper presents a recurrent neural-network model for solving a special class of general variational inequalities (GVIs), which includes classical VIs as special cases. It is proved that the proposed neural network (NN) for solving this class of GVIs can be globally convergent, globally asymptotically stable, and globally exponentially stable under different conditions. The proposed NN can be viewed as a modified version of the general projection NN existing in the literature. Several numerical examples are provided to demonstrate the effectiveness and performance of the proposed NN.     

G-H-KKM type theorems and their applications to a new class of minimax inequalities

First, some nonempty intersection theorems involving G-H-KKM type mappings are presented, and then a related new class of minimax inequalities is given.     

Nonzero solutions for a class of set-valued variational inequalities in reflexive Banach spaces

In this paper, we study the existence of nonzero solutions for a class of set-valued variational inequalities involving set-contractive mappings by using the fixed point index approach in reflexive Banach spaces. Some new existence theorems of nonzero solutions for this class of set-valued variational inequalities are established.     

Solutions to a class of non-standard non-linear H infinity control problems

This paper presents new solutions to certain non-standard non-linear H infinity control problems. We consider non-linear affine plants whose measurement output is of dimension larger than the dimension of the external input. This problem is, under proper assumptions, transformed to the problem of stabilization by means of output injection and solution of a Hamilton-Jacobi partial differential inequality arising in singular H infinity state-feedback control. General sufficient solvability conditions are given. Explicit solutions are available in the local and semilocal cases. The former concerns a certain neighbourhood of the origin in the closed loop state-space, whereas the latter assumes that the trajectories are restricted to a neighbourhood of an invariant manifold. The issue of the controller order is addressed and a reducedorder controller is obtained in the local case. A new generalization of the chain-scattering formalism provides a very useful framework for solving this problem.     

Solutions to a class of nonstandard stochastic control problemswith active learning

The author formulates and solves a dynamic stochastic optimization problem of a nonstandard type, whose optimal solution features active learning. The proof of optimality and the derivation of the corresponding control policies is an indirect one, which relates the original single-person optimization problem to a sequence of nested zero-sum stochastic games. Existence of saddle points for these games implies the existence of optimal policies for the original control problem, which, in turn, can be obtained from the solution of a nonlinear deterministic, optimal control problem. The author also studies the problem of existence of stationary optimal policies when the time horizon is infinite and the objective function is discounted     

Online version of the theorem of Thue

A sequence S is nonrepetitive if no two adjacent blocks of S are the same. In 1906 Thue proved that there exist arbitrarily long nonrepetitive sequences over 3 symbols. We consider the online variant of this result in which a nonrepetitive sequence is constructed during a play between two players: Bob is choosing a position in a sequence and Alice is inserting a symbol on that position taken from a fixed set A. The goal of Bob is to force Alice to create a repetition, and if he succeeds, then the game stops. The goal of Alice is naturally to avoid that and thereby to construct a nonrepetitive sequence of any given length.We prove that Alice has a strategy to play arbitrarily long provided the size of the set A is at least 12. This is the online version of the theorem of Thue. The proof is based on nonrepetitive colorings of outerplanar graphs. On the other hand, one can prove that even over 4 symbols Alice has no chance to play for too long. The minimum size of the set of symbols needed for the online version of Thue?s theorem remains unknown.     

A generalization of Thue freeness for partial words

This paper approaches the combinatorial problem of Thue freeness for partial words. Partial words are sequences over a finite alphabet that may contain a number of “holes”. First, we give an infinite word over a three-letter alphabet which avoids squares of length greater than two even after we replace an infinite number of positions with holes. Then, we give an infinite word over an eight-letter alphabet that avoids longer squares even after an arbitrary selection of its positions are replaced with holes, and show that the alphabet size is optimal. We find similar results for overlap-free partial words.     

Positivity and nonnegativity conditions of a quartic equation and related problems

In this paper explicit conditions for positivity (no real roots), nonnegativity on positive real axis (no positive real roots with odd multiplicity), and stability aperiodicity (all roots are real, and, negative and simple) of a quartic (or biquadratic) equation are given. The derived conditions from the known solution of the quartic equation are not only complete, but simpler than those derived from Sturm, extended Hurwitz, inners, and Hankel methods. Because of Abel's Theorem (no explicit solution in terms of the roots of an equation higher than quartic exists), similar simplification for higher degree polynomial equations may not be possible. Furthermore, explicit conditions for positivity and nonnegativity of equations of higher degree than four are extremely difficult to obtain and may not be possible. The results of the paper will hopefully shed some light on a century old problem and thus enhance the engineering application of the derived condition to higher order systems.     

Proof of two conjectures on correlation inequalities for one class of monotone functions

We prove two correlation inequalities conjectured in [1] for functions that are linear combinations of unimodal Boolean monotone nondecreasing functions.     

Brauer groups of diagonal quartic surfaces

We describe explicit methods of exhibiting elements of the Brauer groups of diagonal quartic surfaces. Using these methods, we compute the algebraic Brauer–Manin obstruction in two contrasting examples. In the second example, the obstruction is found to be trivial but a computer search reveals no points of small height on the surface.     

Convergence analysis and error estimates of the element-free Galerkin method for a class of parabolic evolutionary variational inequalities

This paper is presented for the convergence analysis of the element-free Galerkin method for a class of parabolic evolutionary variational inequalities arising from the heat-servo control problem. The error estimates illustrate that the convergence order depends not only on the number of basis functions in the moving least-squares approximation but also the relationship with the time step and the spatial step. Numerical examples verify the convergence analysis and the error estimates.     

Checking Simple Properties of Transition Systems Defined by Thue Specifications

In (possibly infinite) deterministic labeled transition systems defined by Thue congruences, labels are considered as functions of states into states. This paper provides a method for computing domains of such functions for a large class of transition systems. The latter are related to model checking of transition systems defined by Thue congruences.