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.     

Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology

We study the well-posedness of the bidomain model that is commonly used to simulate electrophysiological wave propagation in the heart. We base our analysis on a formulation of the bidomain model as a system of coupled parabolic and elliptic PDEs for two potentials and ODEs representing the ionic activity. We first reformulate the parabolic and elliptic PDEs into a single parabolic PDE by the introduction of a bidomain operator. We properly define and analyze this operator, basically a non-differential and non-local operator. We then present a proof of existence, uniqueness and regularity of a local solution in time through a semigroup approach, but that applies to fairly general ionic models. The bidomain model is next reformulated as a parabolic variational problem, through the introduction of a bidomain bilinear form. A proof of existence and uniqueness of a global solution in time is obtained using a compactness argument, this time for an ionic model reading as a single ODE but including polynomial nonlinearities. Finally, the hypothesis behind the existence of that global solution are verified for three commonly used ionic models, namely the FitzHugh–Nagumo, Aliev–Panfilov and MacCulloch models.     

Partnership formation based on multiple traits

A model of partnership formation based on two traits, called beauty and character, is presented. There are two classes of individual and partners must be of different classes. Individuals prefer prospective partners with a high beauty measure and of a similar character. This problem may be interpreted as e.g. a job search problem in which the classes are employer and employee, or a mate choice problem in which the classes are male and female. Beauty can be observed instantly. However, a costly date (or interview) is required to observe the character of a prospective partner. On observing the beauty of a prospective partner, an individual decides whether he/she wishes to date. During a date, the participants observe each other’s character and then decide whether to form a pair. Mutual acceptance is required both for a date to occur and pair formation. On finding a partner, an individual stops searching. Beauty has a continuous distribution on a finite interval, while character ‘forms a circle’ and has a uniform distribution. Criteria based on the concept of a subgame perfect Nash equilibrium are used to define a symmetric equilibrium of this game. It is argued that this equilibrium is unique. When dating costs are high, this equilibrium is a block separating equilibrium as in more classical formulations of two-sided job search problems. However, for sufficiently small dating costs the form of this equilibrium is essentially different.     

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.

     

Modeling of a discrete dynamic system using a modified Petri net

Together with the proposed modification connected with the introduction of a time-dependent net, we define a criterion for optimality of a sequence of initiations of transitions, and study a method of obtaining such a sequence. We given the construction of a model of a discrete dynamic system having finite parameters. to describe it we introduce a modification of the Petri net using a determination of the state of the positions, time delay of the transitions, and variation of the conditions and rules for initiating transitions. Translated fromDinamicheskie Sistemy, No. 13, 1994. pp. 39–50.     

An algebraic characterization of planar graphs

A cycle in a graph is a set of edges that covers each vertex an even number of times. A cocycle is a collection of edges that intersects each cycle in an even number of edges. A bicycle is a collection of edges that is both a cycle and a cocycle. The cycles, cocycles, and bicycles each form a vector space over the integers modulo two when addition is defined as symmetric difference of sets. In this paper we examine the relationship between the left-right paths in a planar graph and the cycle space, cocylce space, and bicycle space. We show that planar graphs are characterized by the existence of a diagonal—a double cover by tours that interacts with the cycle space, cocycle space, and bicycle space in a special manner. This generalizes a result of Rosenstiehl and Read that characterized those planar graphs with no nonempty bicycles. © 1995 John Wiley & Sons, Inc.     

A Priori and A Posteriori: A Bootstrapping Relationship

The distinction between a priori and a posteriori knowledge has been the subject of an enormous amount of discussion, but the literature is biased against recognizing the intimate relationship between these forms of knowledge. For instance, it seems to be almost impossible to find a sample of pure a priori or a posteriori knowledge. In this paper, it will be suggested that distinguishing between a priori and a posteriori is more problematic than is often suggested, and that a priori and a posteriori resources are in fact used in parallel. We will define this relationship between a priori and a posteriori knowledge as the bootstrapping relationship. As we will see, this relationship gives us reasons to seek for an altogether novel definition of a priori and a posteriori knowledge. Specifically, we will have to analyse the relationship between a priori knowledge and a priori reasoning, and it will be suggested that the latter serves as a more promising starting point for the analysis of aprioricity. We will also analyse a number of examples from the natural sciences and consider the role of a priori reasoning in these examples. The focus of this paper is the analysis of the concepts of a priori and a posteriori knowledge rather than the epistemic domain of a posteriori and a priori justification.     

Invariant Measures and the Soliton Resolution Conjecture

The soliton resolution conjecture for the focusing nonlinear Schrödinger equation (NLS) is the vaguely worded claim that a global solution of the NLS, for generic initial data, will eventually resolve into a radiation component that disperses like a linear solution, plus a localized component that behaves like a soliton or multisoliton solution. Considered to be one of the fundamental open problems in the area of nonlinear dispersive equations, this conjecture has eluded a proof or even a precise formulation to date. This paper proves a “statistical version” of this conjecture at mass‐subcritical nonlinearity, in the following sense: The uniform probability distribution on the set of all functions with a given mass and energy, if such a thing existed, would be a natural invariant measure for the NLS flow and would reflect the long‐term behavior for “generic initial data” with that mass and energy. Unfortunately, such a probability measure does not exist. We circumvent this problem by constructing a sequence of discrete measures that, in principle, approximate this fictitious probability distribution as the grid size goes to 0. We then show that a continuum limit of this sequence of probability measures does exist in a certain sense, and in agreement with the soliton resolution conjecture, the limit measure concentrates on the unique ground state soliton. Combining this with results from ergodic theory, we present a tentative formulation and proof of the soliton resolution conjecture in the discrete setting. The above results, following in the footsteps of a program of studying the long‐term behavior of nonlinear dispersive equations through their natural invariant measures initiated by Lebowitz, Rose, and Speer and carried forward by Bourgain, McKean, Tzvetkov, Oh, and others, are proved using a combination of techniques from large deviations, PDEs, harmonic analysis, and bare‐hands probability theory. It is valid in any dimension. © 2014 Wiley Periodicals, Inc.     

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.     

Solution to the variation problem for information path functional of a controlled random process

The paper introduces a new approach to dynamic modeling, using the variation principle, applied to a functional on trajectories of a controlled random process, and its connection to the process' information functional. In [V.S. Lerner, Dynamic approximation of a random information functional, J. Math. Anal. Appl. 327 (1) (2007) 494-514, available online 5-24-06], we presented the information path functional with the Lagrangian, determined by the parameters of a controlled stochastic equation. In this paper, the solution to the path functional's variation problem provides both a dynamic model of a random process and the model's optimal control, which allows us to build a two-level information model with a random process at the microlevel and a dynamic process at the macrolevel. A wide class of random objects, modeled by the Markov diffusion process and a common structure of the process' information functional, leads to a universal information structure of the dynamic model, which is specified and identified on a particular object with the applied optimal control functions. The developed mathematical formalism, based on classical methods, is aimed toward the solution of problems identification, combined with an optimal control synthesis, which is practically implemented and also demonstrated in the paper's example.     

区域函数

为了把点函数理论、区间函数理论和方法推广到任意区域,作者建立了区域的收缩和区域的保核收缩,区域的扩张和区域的保核扩张等新理论.从这些概念出发,给出了区域函数的新定义,并将区域函数的核(即不动点)与此区域函数的定义区域的稳定中心联系起来,从而建立了区域理论,和区域与区域算法. 在应用中,为了求区域的稳定中心,作者采用了由Hartfiel和其它作者建立的矩阵测度理论;并讨论了与区域相伴的线性代数方程组系数矩阵的测度理论.     

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.     

A coupled system of ODEs and quasilinear hyperbolic PDEs arising in a multiscale blood flow model

We study a coupled system of ordinary differential equations and quasilinear hyperbolic partial differential equations that models a blood circulatory system in the human body. The mathematical system is a multiscale model in which a part of the system, where the flow can be regarded as Newtonian and homogeneous, and the vessels are long and large, is modeled by a set of hyperbolic PDEs in a one-spatial-dimensional network, and in the other part, where either vessels are too thin or the flow pattern is too complicated (such as in the heart), the flow is modeled as a lumped element by a set of ordinary differential equations as an analog of an electric circuit. The mathematical system consists of pairs of PDEs, one pair for each vessel, coupled at each junction through a system of ODEs. This model is a generalization of the widely studied models of arterial networks. We give a proof of the well-posedness of the initial-boundary value problem by showing that the classical solution exists, is unique, and depends continuously on initial, boundary and forcing functions and their derivatives.     

Mean-variance optimization problems for an accumulation phase in a defined benefit plan

In this paper we deal with contribution rate and asset allocation strategies in a pre-retirement accumulation phase. We consider a single cohort of workers and investigate a retirement plan of a defined benefit type in which an accumulated fund is converted into a life annuity. Due to the random evolution of a mortality intensity, the future price of an annuity, and as a result, the liability of the fund, is uncertain. A manager has control over a contribution rate and an investment strategy and is concerned with covering the random claim. We consider two mean-variance optimization problems, which are quadratic control problems with an additional constraint on the expected value of the terminal surplus of the fund. This functional objectives can be related to the well-established financial theory of claim hedging. The financial market consists of a risk-free asset with a constant force of interest and a risky asset whose price is driven by a Lévy noise, whereas the evolution of a mortality intensity is described by a stochastic differential equation driven by a Brownian motion. Techniques from the stochastic control theory are applied in order to find optimal strategies.     

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.     

CONSENSUS ANALYSIS AND DESIGN OF LINEAR INTERCONNECTED MULTI-AGENT SYSTEMS

We deal with the state consensus problem of a general Linear Interconnected Multi-Agent System(LIMAS) under a time-invariant and directed communication topology.Firstly,we propose a linear consensus protocol in a general form,which consists of state feedback of the agent itself and feedback form of the relative states between the agent and its neighbors.Secondly,a state-linear-transformation is applied to equivalently transform the state consensus problem into a partial stability problem.Based on the partial stability theory,we derive a sufficient and necessary criterion of consensus convergence,which is expressed via the Hurwitz stability of a real matrix constructed from the parameters of the agent models and the protocols,and present an analytical formula of the consensus function.Lastly,we propose a design procedure of the gain matrices in the protocol by solving a bilinear matrix inequality.     

Banach空间中非膨胀映象和α-逆强增生算子的强收敛性

在2-一致光滑的Banach空间中,引入一种新的迭代算法研究非膨胀映象的不动点集与α-逆强增生算子的变分不等式解集的公共元素,并获得了迭代算法的强收敛性定理.而且应用这些结果考虑了非膨胀映象和严格伪压缩映象公共不动点的收敛性问题.     

Characterizations of g-frames and g-Riesz bases in Hilbert spaces

In this paper, we introduce the pre-frame operator Q for the g-frame in a complex Hilbert space, which will play a key role in studying g-frames and g-Riesz bases etc. Using the pre-frame operator Q, we give some necessary and sufficient conditions for a g-Bessel sequence, a g-frame, and a g-Riesz basis in a complex Hilbert space, which have properties similar to those of the Bessel sequence, frame, and Riesz basis respectively. We also obtain the relation between a g-frame and a g-Riesz basis, and the relation of bounds between a g-frame and a g-Riesz basis. Lastly, we consider the stability of a g-frame or a g-Riesz basis for a Hilbert space under perturbation.     

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.