二次曲面和平面位置关系的判式

在解析几何中有二次曲线与直线位置关系的讨论、二次曲面与直线位置关系的讨论,而二次曲面与平面相关位置关系的探讨较少.本文给出二次曲面a11x2+a22y2+a33z2+2a12xy+2a13xz+2a23yz+2a14x+2a24y+2a34z+a44=0(1)和平面Ax+By+Cz+D=0(2)的相对位置的判别式Δ=a11a12a13a14Aa21a22a23a24Ba31a32a33a34Ca41a42a43a44DA B C D0(aij=aji).(3)并证明了:若Δ>0,则二次曲面(1)与平面(2)相交;若Δ=0,则(1)和(2)相切;若Δ<0,则(1)和(2)相离.     

Toader型平均与几种经典平均的确界

给出了最佳参数α_1,α_2,α_3,β_1,β_2,β_3∈R,使得双向不等式α_1Q(a,b)+(1-α_1)G(a,b)0且a≠b成立.其中A(a,b)=(a+b)/2,H(a,b)=2ab/(a+b),G(a,b)=(ab)~(1/2),Q(a,b)=((a~2+b~2)/2)~(1/2),C(a,b)=(a~2+b~2)/(a+b),T(a,b)=2/π∫_0~(π/2)(a~2cos~2t+b~2sin~2)~(1/2)tdt分别是两个正数a和b的算术平均,调和平均,几何平均,二次平均,反调和平均和Toader平均.     

The mathematical theory of growing bodies. Finite deformations

The fundamentals of the mathematical theory of accreting bodies for finite deformations are explained using the concept of the bundle of a differentiable manifold that enables one to construct a clear classification of the accretion processes. One of the possible types of accretion, as due to the continuous addition of stressed material surfaces to a three-dimensional body, is considered. The complete system of equations of the mechanics of accreting bodies is presented. Unlike in problems for bodies of constant composition, the tensor field of the incompatible distortion, which can be found from the equilibrium condition for the boundary of growth, that is, a material surface in contact with a deformable three-dimensional body, enters into these equations. Generally speaking, a growing body does not have a stress-free configuration in three-dimensional Euclidean space. However, there is such a configuration on a certain three-dimensional manifold with a non-Euclidean affine connectedness caused by a non-zero torsion tensor that is a measure of the incompatibility of the deformation of the growing body. Mathematical models of the stress-strain state of a growing body are therefore found to be equivalent to the models of bodies with a continuous distribution of the dislocations.     

Complex affine distributions

Geometry of affine immersions is the study of hypersurfaces that are invariant under affine transformations. As with the hypersurface theory on the Euclidean space, an affine immersion can induce a torsion-free affine connection and a (pseudo)-Riemannian metric on the hypersurface. Moreover, an affine immersion can induce a statistical manifold, which plays a central role in information geometry. Recently, a statistical manifold with a complex structure is actively studied since it connects information geometry and Kähler geometry. However, a holomorphic complex affine immersion cannot induce such a statistical manifold with a Kähler structure. In this paper, we introduce complex affine distributions, which are non-integrable generalizations of complex affine immersions. We then present the fundamental theorem for a complex affine distribution, and show that a complex affine distribution can induce a statistical manifold with a Kähler structure.     

A partial differential equation connected to option pricing with stochastic volatility: Regularity results and discretization

This paper completes a previous work on a Black and Scholes equation with stochastic volatility. This is a degenerate parabolic equation, which gives the price of a European option as a function of the time, of the price of the underlying asset, and of the volatility, when the volatility is a function of a mean reverting Orstein-Uhlenbeck process, possibly correlated with the underlying asset. The analysis involves weighted Sobolev spaces. We give a characterization of the domain of the operator, which permits us to use results from the theory of semigroups. We then study a related model elliptic problem and propose a finite element method with a regular mesh with respect to the intrinsic metric associated with the degenerate operator. For the error estimate, we need to prove an approximation result.

     

Approximate evaluation of multi-location inventory models with lateral transshipments and hold back levels

We consider a continuous-time, single-echelon, multi-location inventory model with Poisson demand processes. In case of a stock-out at a local warehouse, a demand can be fulfilled via a lateral transshipment (LT). Each warehouse is assigned a pre-determined sequence of other warehouses where it will request for an LT. However, a warehouse can hold its last part(s) back from such a request. This is called a hold back pooling policy, where each warehouse has hold back levels determining whether a request for an LT by another warehouse is satisfied. We are interested in the fractions of demand satisfied from stock (fill rate), via an LT, and via an emergency procedure from an external source. From these, the average costs of a policy can be determined. We present a new approximation algorithm for the evaluation of a given policy, approximating the above mentioned fractions. Whereas algorithms currently known in the literature approximate the stream of LT requests from a warehouse by a Poisson process, we use an interrupted Poisson process. This is a process that is turned alternatingly On and Off for exponentially distributed durations. This leads to the On/Off overflow algorithm. In a numerical study we show that this algorithm is significantly more accurate than the algorithm based on Poisson processes, although it requires a longer computation time. Furthermore, we show the benefits of hold back levels, and we illustrate how our algorithm can be used in a heuristic search for the setting of the hold back levels.     

Global asymptotic properties of an SEIRS model with multiple infectious stages

The paper presents a rigorous mathematical analysis of a deterministic model, which uses a standard incidence function, for the transmission dynamics of a communicable disease with an arbitrary number of distinct infectious stages. It is shown, using a linear Lyapunov function, that the model has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction threshold is less than unity. Further, the model has a unique endemic equilibrium when the threshold exceeds unity. The equilibrium is shown to be locally-asymptotically stable, for a special case, using a Krasnoselskii sub-linearity trick. Finally, a non-linear Lyapunov function is used to show the global asymptotic stability of the endemic equilibrium (for the special case). Numerical simulation results, using parameter values relevant to the transmission dynamics of influenza, are presented to illustrate some of the main theoretical results.     

Subgradient Projectors: Extensions,Theory, and Characterizations

Subgradient projectors play an important role in optimization and for solving convex feasibility problems. For every locally Lipschitz function, we can define a subgradient projector via generalized subgradients even if the function is not convex. The paper consists of three parts. In the first part, we study basic properties of subgradient projectors and give characterizations when a subgradient projector is a cutter, a local cutter, or a quasi-nonexpansive mapping. We present global and local convergence analyses of subgradent projectors. Many examples are provided to illustrate the theory. In the second part, we investigate the relationship between the subgradient projector of a prox-regular function and the subgradient projector of its Moreau envelope. We also characterize when a mapping is the subgradient projector of a convex function. In the third part, we focus on linearity properties of subgradient projectors. We show that, under appropriate conditions, a linear operator is a subgradient projector of a convex function if and only if it is a convex combination of the identity operator and a projection operator onto a subspace. In general, neither a convex combination nor a composition of subgradient projectors of convex functions is a subgradient projector of a convex function.     

Harmonic and pulse excitations of multiply connected cylindrical bodies

The three-dimensional problem of the theory of elasticity of the harmonic oscillations of cylindrical bodies (a layer with several tunnel cavities on a cylinder of finite length) is considered for uniform mixed boundary conditions on its bases. Using the Φ-solutions constructed, the boundary-value problems are reduced to a system of well-known one-dimensional singular integral equations. The solution of the problem of the pulse excitation of a layer on the surface of a cavity is “assembled” from a packet of corresponding harmonic oscillations using an integral Fourier transformation with respect to time. The results of calculations of the dynamic stress concentration in a layer (a plate) weakened by one and two openings of different configuration are given, as well as the amplitude-frequency characteristics for a cylinder of finite length with a transverse cross section in the form of a square with rounded corners, and data of calculations for a trapeziform pulse, acting on the surface of a circular cavity, are presented.     

On a Mathematical Comparison between Hierarchy and Network with a Classification of Coordination Structures

This paper proposes a mathematical model to compare a network organization with a hierarchical organization. In order to formulate the model, we define a three-dimensional framework of the coordination structure of a network and of other typical coordination structures. In the framework, we can define a network structure by contrasting it with a hierarchy, in terms of the distribution of decision making, which is one of the main features of information processing. Based on this definition, we have developed a mathematical model for evaluating coordination structures. Using this model, we can derive two boundary conditions among the coordination structures with respect to the optimal coordination structure. The boundary conditions help us to understand why an organization changes its coordination structure from a hierarchy to a network and what factors cause this change. They enable us, for example, to find points of structural change where the optimal coordination structure shifts from a hierarchy to a hierarchy with delegation or from a hierarchy with delegation to a network, when the nature of the task changes from routine to non-routine. In conclusion, our framework and model may provide a basis for discussing the processes that occur when coordination structures change between a hierarchy and a network.     

Topological imitation of a colored link with the same Dehn surgery manifold

By the topological imitation theory, we construct, from a given colored link, a new colored link with the same Dehn surgery manifold. In particular, we construct a link with a distinguished coloring whose Dehn surgery manifold is a given closed connected oriented 3-manifold except the 3-sphere. As a result, we can naturally generalize the difference between the Gordon–Luecke theorem and the property P conjecture to a difference between a link version of the Gordon–Luecke theorem and the Poincaré conjecture. Similarly, we construct a link with a π₁-distinguished coloring whose Dehn surgery manifold is a given non-simply-connected closed connected oriented 3-manifold. We also construct a link with just two colorings whose Dehn surgery manifolds are the 3-sphere.     

组合投资β调整与效用最大化的套期比的选择

为了针对市场风险对风险资产的组合投资进行套期保值，一般认为要选择将组合投资多头和期货合同空头结合起来的头寸方差最小化的套期保值比率，也就是要选择使某一特定函数的期望效用最大化的套期保值比率。但是本认为，由于种种原因，人们更倾向于选择对简单风险最小头寸的套期保值比率。     

Poisson比为1/2材料中球形空穴突变和球形空穴萌生的分岔问题研究

研究Ｐｏｉｓｓｏｎ比为１／２的Ｈｏｏｋｅ材料中,空穴的突变和萌生现象·求解一个球对称几何非线性弹性力学的移动边界（ｍｏｖｉｎｇｂｏｕｎｄａｒｙ）问题,空穴为球形,远离空穴处为三向均匀拉伸应力状态,在当前构形上列控制方程;在当前构形边界上列边界条件·找到了这个自由边界问题的封闭解并得到空穴半径趋于零时的叉型分岔解·计算结果显示,在位移＿载荷曲线上存在一个切分岔型分岔点（或鞍结点型分岔点、极值型分岔点）,这个分岔点说明在外力作用下空穴会发生突变,即突然“长大”;当球腔半径趋于零时,这个切分岔转化为叉型分岔（或分枝型分岔）,这个叉型分岔可以解释实心球中的空穴萌生现象     

Uniqueness of transonic shock solutions in a duct for steady potential flow

We study the uniqueness of solutions with a transonic shock in a duct in a class of transonic shock solutions, which are not necessarily small perturbations of the background solution, for steady potential flow. We prove that, for given uniform supersonic upstream flow in a straight duct, there exists a unique uniform pressure at the exit of the duct such that a transonic shock solution exists in the duct, which is unique modulo translation. For any other given uniform pressure at the exit, there exists no transonic shock solution in the duct. This is equivalent to establishing a uniqueness theorem for a free boundary problem of a partial differential equation of second order in a bounded or unbounded duct. The proof is based on the maximum/comparison principle and a judicious choice of special transonic shock solutions as a comparison solution.     

Discrete neural dynamic programming in wheeled mobile robot control

In this paper we propose a discrete algorithm for a tracking control of a two-wheeled mobile robot (WMR), using an advanced Adaptive Critic Design (ACD). We used Dual-Heuristic Programming (DHP) algorithm, that consists of two parametric structures implemented as Neural Networks (NNs): an actor and a critic, both realized in a form of Random Vector Functional Link (RVFL) NNs. In the proposed algorithm the control system consists of the DHP adaptive critic, a PD controller and a supervisory term, derived from the Lyapunov stability theorem. The supervisory term guaranties a stable realization of a tracking movement in a learning phase of the adaptive critic structure and robustness in face of disturbances. The discrete tracking control algorithm works online, uses the WMR model for a state prediction and does not require a preliminary learning. Verification has been conducted to illustrate the performance of the proposed control algorithm, by a series of experiments on the WMR Pioneer 2-DX.     

A general vectorial Ekeland’s variational principle with a P-distance

In this paper, by using p-distances on uniform spaces, we establish a general vectorial Ekeland variational principle （in short EVP）, where the objective function is defined on a uniform space and taking values in a pre-ordered real linear space and the perturbation involves a p-distance and a monotone function of the objective function. Since p-distances are very extensive, such a form of the perturbation in deed contains many different forms of perturbations appeared in the previous versions of EVP. Besides, we only require the objective function has a very weak property, as a substitute for lower semi-continuity, and only require the domain space （which is a uniform space） has a very weak type of completeness, i.e., completeness with respect to a certain p-distance. Such very weak type of completeness even includes local completeness when the uniform space is a locally convex topological vector space. From the general vectorial EVP, we deduce a general vectorial Caristi＇s fixed point theorem and a general vectorial Takahashi＇s nonconvex minimization theorem. Moreover, we show that the above three theorems are equivalent to each other. We see that the above general vectorial EVP includes many particular versions of EVP, which extend and complement the related known results.     

An algebraic index theorem for orbifolds

Using the concept of a twisted trace density on a cyclic groupoid, a trace is constructed on a formal deformation quantization of a symplectic orbifold. An algebraic index theorem for orbifolds follows as a consequence of a local Riemann-Roch theorem for such densities. In the case of a reduced orbifold, this proves a conjecture by Fedosov, Schulze, and Tarkhanov. Finally, it is shown how the Kawasaki index theorem for elliptic operators on orbifolds follows from this algebraic index theorem.     

Diffraction of compatible random substitutions in one dimension

As a guiding example, the diffraction measure of a random local mixture of the two classic Fibonacci substitutions is determined and reanalysed via self-similar measures of Hutchinson type, defined by a finite family of contractions. Our revised approach yields explicit formulas for the pure point and the absolutely continuous parts, as well as a proof for the absence of singular continuous components. This approach is then extended to the family of random noble means substitutions and, as an example with an underlying 2-adic structure, to a locally randomised version of the period doubling chain. As a first step towards a more general approach, we interpret our findings in terms of a disintegration over the Kronecker factor, which is the maximal equicontinuous factor of a covering model set.     

Semirings and path spaces

This paper develops a unified algebraic theory for a class of path problems such as that of finding the shortest or, more generally, the k shortest paths in a network; the enumeration of elemementary or simple paths in a graph. It differs from most earlier work in that the algebraic structure appended to a graph or a network of a path problem is not axiomatically given as a starting point of the theory, but is derived from a novel concept called a “path space”. This concept is shown to provide a coherent framework for the analysis of path problems, and hence the development of algebraic methods for solving them.