On farthest points of sets

For a convex closed bounded set in a Banach space, we study the existence and uniqueness problem for a point of this set that is the farthest point from a given point in space. In terms of the existence and uniqueness of the farthest point, as well as the Lipschitzian dependence of this point on a point in space, we obtain necessary and su.cient conditions for the strong convexity of a set in several infinite-dimensional spaces, in particular, in a Hilbert space. A set representable as the intersection of closed balls of a fixed radius is called a strongly convex set. We show that the condition “for each point in space that is sufficiently far from a set, there exists a unique farthest point of the set” is a criterion for the strong convexity of a set in a finite-dimensional normed space, where the norm ball is a strongly convex set and a generating set.     

A New Pareto Optimal Solution in a Lagrange Decomposable Multi-Objective Optimization Problem

In this paper a committee decision-making process of a convex Lagrange decomposable multi-objective optimization problem, which has been decomposed into various subproblems, is studied. Each member of the committee controls only one subproblem and attempts to select the optimal solution of this subproblem most desirable to him, under the assumption that all the constraints of the total problem are satisfied. This procedure leads to a new solution concept of a Lagrange decomposable multi-objective optimization problem, called a preferred equilibrium set. A preferred equilibrium point of a problem, for a committee, may or may not be a Pareto optimal point of this problem. In some cases, a non-Pareto optimal preferred equilibrium point of a problem, for a committee, can be considered as a special type of Pareto optimal point of this problem. This fact leads to a generalization of the Pareto optimality concept in a problem.     

The weighted complexity and the determinant functions of graphs

The complexity of a graph can be obtained as a derivative of a variation of the zeta function [S. Northshield, A note on the zeta function of a graph, J. Combin. Theory Ser. B 74 (1998) 408-410] or a partial derivative of its generalized characteristic polynomial evaluated at a point [D. Kim, H.K. Kim, J. Lee, Generalized characteristic polynomials of graph bundles, Linear Algebra Appl. 429 (4) (2008) 688-697]. A similar result for the weighted complexity of weighted graphs was found using a determinant function [H. Mizuno, I. Sato, On the weighted complexity of a regular covering of a graph, J. Combin. Theory Ser. B 89 (2003) 17-26]. In this paper, we consider the determinant function of two variables and discover a condition that the weighted complexity of a weighted graph is a partial derivative of the determinant function evaluated at a point. Consequently, we simply obtain the previous results and disclose a new formula for the complexity from a variation of the Bartholdi zeta function. We also consider a new weighted complexity, for which the weights of spanning trees are taken as the sum of weights of edges in the tree, and find a similar formula for this new weighted complexity. As an application, we compute the weighted complexities of the product of the complete graphs.     

Diads and their Application to Topoi

It is well known that the category of coalgebras for a finite-limit preserving comonad on a topos is again a topos, and the category of algebras for a finite-limit preserving monad is a topos if the monad is idempotent, but not in general. A generalisation of this result (Paré et al., Bull Aus Math Soc 39(3):421–431, 1989) is that the full subcategory of fixed points for any idempotent finite-limit preserving endofunctor is again a topos (and indeed a subquotient in the category of topoi and geometric morphisms). Here, we present a common generalisation of all the above results, based on a notion which we call a diad, which is a common generalisation of a monad and a comonad. Many of the constructions that can be applied to monads and comonads can be extended to all diads. In particular, the category of algebras or coalgebras can be generalised to a category of dialgebras for a diad. The generalisation we present here is that the category of dialgebras for a finite-limit preserving left diad (for example, the diad corresponding to a comonad, or any idempotent endofunctor) on a topos is again a topos.     

Self-similar time-varying flows of an ideal gas with a change in the adiabatic exponent in a “reflected” shock wave

Self-similar one-dimensional time-varying problems are considered under the assumption that there is a change in the adiabatic exponent in a shock wave (SW) running (“reflected”) from a centre or axis of symmetry (later from a centre of symmetry, CS) or from a plane. The medium is an ideal (inviscid and non-heat-conducting) perfect gas with constant heat capacities. In problems with strong SW, the change in the adiabatic exponent in a gas approximately simulates physicochemical processes such as dissociation and ionization and, in the problem of the collapse of a spherical cavity in a liquid, the conversion of liquid into vapour. In both cases, the adiabatic exponent decreases on passing across a reflected SW. Problems of the collapse of a spherical cavity, the reflection of a strong SW from a centre of symmetry and a simpler problem with a self-similarity index of one are examined. When it is assumed that there is an increase in the adiabatic exponent, the self-similar solutions of the first two problems are rejected due to the decrease in entropy from the instant when the SW is reflected. When it is assumed that there is a decrease in the adiabatic exponent, the solutions of these problems only become unsuitable after a finite time has elapsed for the same reason. Up to this time when the decrease in the adiabatic exponent has not reached a certain threshold, the structure of the self-similar solution does not undergo qualitative changes. When the above-mentioned threshold is exceeded, a self-similar solution is possible if a cylindrical or spherical piston expands according to a special law from the instant of SW reflection from the CS. When there is no piston, the flow behind the reflected wave becomes non-self- similar. In the case of the deceleration of a plane flow, conditions are possible with the joining of SW from different sides to a centred rarefaction wave.     

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.     

A method for analytically representing area-preserving mappings

An area-preserving mapping is considered. It is assumed that the mapping has a fixed point and is analytic in a small neighbourhood near it. A constructive algorithm for obtaining a representation of the mapping in the form of a composite of two area-preserving mappings, one of which is a nearly identity mapping, while the other corresponds to the real normal form of a linearized mapping, is described. The algorithm is used in the problem of the stability of the translational motion of a rigid body in a uniform gravitational field when it undergoes collisions with a fixed horizontal plane and in the problem of the stability of one type of resonant in-plane rotations of a satellite, i.e., a rigid body, in an elliptic orbit.     

Weak Convergence Theorems for Nonexpansive Mappings and Monotone Mappings

In this paper, we introduce an iteration process of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem for an inverse strongly-monotone mapping, and then obtain a weak convergence theorem. Using this result, we obtain a weak convergence theorem for a pair of a nonexpansive mapping and a strictly pseudocontractive mapping. Further, we consider the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse strongly-monotone mapping.     

U-标算子的结构分析

类似与标型谱算子,U-标算子是否拟仿射相似于自伴算子是一“公开问题”．尽管对具纯离散谱的U-标算子答案是肯定的,但一般情况下并不成立．本文继续探讨这一问题,证明了U-标算子在一强范数拓扑意义下是Hermite算子,或者说U-标算子拟仿射相似于Hermite算子,并给出U-标算子是标型谱算子的充要条件．     

A Chart for Evaluating Product Research and Development Projects

A method of evaluating research and development projects is described. The method is based on the use of an evaluation chart. The main purpose of the chart is to calculate a project score which is a measure of a selected evaluation criterion. A method of constructing a chart is illustrated using a project index as the evaluation criterion. The chart contains a list of twelve questions, the answers to which are assumed to be the main determinants of the variables in the formula of the index. Each question is followed by a set of answers from which a choice is to be made, and underneath each answer is a numerical score. The numerical scores are logarithmic functions of the answers to which they relate, and when added together give a project score which is a logarithmic function of the index. Various methods of using the chart, and a suggested evaluation procedure, are also described.     

Small periodic solutions of nonlinear systems of differential equations with constant deviation

We consider a nonlinear system of differential equations in a general case with a singular matrix at the derivatives, with a vector deviation which depends on a parameter. We seek for a periodic solution to the system in the set of trigonometric series such that the sequences of their coefficients belong to the space l ₁.We use the method, representing a space as a direct sum of subspaces, and the method of a fixed point of a nonlinear operator as the main investigation techniques.We reduce the question on the existence of a periodic solution to that of the solvability of an operator equation, whose principal part is defined in a finite-dimensional space.     

Semi-stable fibrations of generic p-rank 0

In this paper, we proved that, for a semi-stable fibration of a proper smooth surface to a proper smooth curve over a filed of positive characteristic, if the p-rank of the generic fiber is 0, then the base change of the fibration by a sufficiently many iterative Frobenius morphism of the base curve violates the semi-positivity theorem. As an application, we suggest a statement on a distribution of p-ranks of reductions for a certain nonclosed point in the moduli space over a number field.     

Certificates of Primal or Dual Infeasibility in Linear Programming

In general if a linear program has an optimal solution, then a primal and dual optimal solution is a certificate of the solvable status. Furthermore, it is well known that in the solvable case, then the linear program always has an optimal basic solution. Similarly, when a linear program is primal or dual infeasible then by Farkas's Lemma a certificate of the infeasible status exists. However, in the primal or dual infeasible case then there is not an uniform definition of what a suitable basis certificate of the infeasible status is.In this work we present a definition of a basis certificate and develop a strongly polynomial algorithm which given a Farkas type certificate of infeasibility computes a basis certificate of infeasibility. This result is relevant for the recently developed interior-point methods because they do not compute a basis certificate of infeasibility in general. However, our result demonstrates that a basis certificate can be obtained at a moderate computational cost.     

Slice theorem and orbit type stratification in infinite dimensions

We establish a general slice theorem for the action of a locally convex Lie group on a locally convex manifold, which generalizes the classical slice theorem of Palais to infinite dimensions. We discuss two important settings under which the assumptions of this theorem are fulfilled. First, using Glöckner's inverse function theorem, we show that the linear action of a compact Lie group on a Fréchet space admits a slice. Second, using the Nash–Moser theorem, we establish a slice theorem for the tame action of a tame Fréchet Lie group on a tame Fréchet manifold. For this purpose, we develop the concept of a graded Riemannian metric, which allows the construction of a path-length metric compatible with the manifold topology and of a local addition. Finally, generalizing a classical result in finite dimensions, we prove that the existence of a slice implies that the decomposition of the manifold into orbit types of the group action is a stratification.     

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.     

Abstraction-based synthesis for stochastic systems with omega-regular objectives

This paper studies the synthesis of controllers for discrete-time, continuous state stochastic systems subject to omega-regular specifications using finite-state abstractions. Omega-regular properties allow specifying complex behaviors and encompass, for example, linear temporal logic. First, we present a synthesis algorithm for minimizing or maximizing the probability that a discrete-time switched stochastic system with a finite number of modes satisfies an omega-regular property. Our approach relies on a finite-state abstraction of the underlying dynamics in the form of a Bounded-parameter Markov Decision Process arising from a finite partition of the system’s domain. Such Markovian abstractions allow for a range of probabilities of transition between states for each selected action representing a mode of the original system. Our method is built upon an analysis of the Cartesian product between the abstraction and a Deterministic Rabin Automaton encoding the specification of interest or its complement. Specifically, we show that synthesis can be decomposed into a qualitative problem, where the so-called greatest permanent winning components of the product automaton are created, and a quantitative problem, which requires maximizing the probability of reaching this component in the worst-case instantiation of the transition intervals. Additionally, we propose a quantitative metric for measuring the quality of the designed controller with respect to the continuous abstracted states and devise a specification-guided domain partition refinement heuristic with the objective of reaching a user-defined optimality target. Next, we present a method for computing control policies for stochastic systems with a continuous set of available inputs. In this case, the system is assumed to be affine in input and disturbance, and we derive a technique for solving the qualitative and quantitative problems in the resulting finite-state abstractions of such systems. For this, we introduce a new type of abstractions called Controlled Interval-valued Markov Chains. Specifically, we show that the greatest permanent winning component of such abstractions are found by appropriately partitioning the continuous input space in order to generate a bounded-parameter Markov decision process that accounts for all possible qualitative transitions between the finite set of states. Then, the problem of maximizing the probability of reaching these components is cast as a (possibly non-convex) optimization problem over the continuous set of available inputs. A metric of quality for the synthesized controller and a partition refinement scheme are described for this framework as well. Finally, we present a detailed case study.     

Stability and Bifurcation Analysis of a Spinning Space Tether

A detailed, geometrically exact bifurcation analysis is performed for a model of a power-generating tethered device of interest to the space industries. The structure, a short electrodynamic tether, comprises a thin, long rod that is spun in a horizontal configuration from a satellite in low Earth orbit, with a massive electrically conducting disk at its free end. The system is modelled using a Cosserat formulation leading to a system of Kirchhoff equations for the rod's shape as a function of position and time. Moving to a rotating frame, incorporating the effects of internal damping, intrinsic curvature due to the deployment method and novel force and moment boundary conditions at the contactor, the problem for steady rotating solutions is formulated as a two-point boundary value problem. Using numerical continuation methods, a bifurcation analysis is carried out varying rotation speeds up to many times the critical resonance frequency. Spatial finite differences are used to formulate the stability problem for each steady state and the corresponding eigenvalues are computed. The results show excellent agreement with earlier multibody dynamics simulations of the same problem.     

Semisimple types for p-adic classical groups

We construct, for any symplectic, unitary or special orthogonal group over a locally compact nonarchimedean local field of odd residual characteristic, a type for each Bernstein component of the category of smooth representations, using Bushnell–Kutzko’s theory of covers. Moreover, for a component corresponding to a cuspidal representation of a maximal Levi subgroup, we prove that the Hecke algebra is either abelian, or a generic Hecke algebra on an infinite dihedral group, with parameters which are, at least in principle, computable via results of Lusztig. In an appendix, we make a correction to the proof of a result of the second author: that every irreducible cuspidal representation of a classical group as considered here is irreducibly compactly-induced from a type.     

Palm calculus for a process with a stationary random measure and its applications to fluid queues

We consider a process associated with a stationary random measure, which may have infinitely many jumps in a finite interval. Such a process is a generalization of a process with a stationary embedded point process, and is applicable to fluid queues. Here, fluid queue means that customers are modeled as a continuous flow. Such models naturally arise in the study of high speed digital communication networks. We first derive the rate conservation law (RCL) for them, and then introduce a process indexed by the level of the accumulated input. This indexed process can be viewed as a continuous version of a customer characteristic of an ordinary queue, e.g., of the sojourn time. It is shown that the indexed process is stationary under a certain kind of Palm probability measure, called detailed Palm. By using this result, we consider the sojourn time processes in fluid queues. We derive the continuous version of Little's formula in our framework. We give a distributional relationship between the buffer content and the sojourn time in a fluid queue with a constant release rate.