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.     

Continuous conjugation of special nonisentropic one-dimensional gas motions

The exact partially invariant solution of equations of motion of a compressible fluid describing the collapse of particles to a point and an instantaneous source from the point in a one-dimensional nonisentropic motion is cut off by the characteristics and glued into a continuous solution of a one-dimensional submodel in a finite domain. The possibility of a continuous periodic nonisentropic motion of a compressible fluid in a bounded domain under the action of a piston is shown.     

Subspaces and quotients of Banach spaces with shrinking unconditional bases

The main result is that a separable Banach space with the weak* unconditional tree property is isomorphic to a subspace as well as a quotient of a Banach space with a shrinking unconditional basis. A consequence of this is that a Banach space is isomorphic to a subspace of a space with a shrinking unconditional basis if and only if it is isomorphic to a quotient of a space with a shrinking unconditional basis, which solves a problem dating to the 1970s. The proof of the main result also yields that a uniformly convex space with the unconditional tree property is isomorphic to a subspace as well as a quotient of a uniformly convex space with an unconditional finite dimensional decomposition.     

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.     

A Barrier Function Method for the Nonconvex Quadratic Programming Problem with Box Constraints

In this paper a barrier function method is proposed for approximating a solution of the nonconvex quadratic programming problem with box constraints. The method attempts to produce a solution of good quality by following a path as the barrier parameter decreases from a sufficiently large positive number. For a given value of the barrier parameter, the method searches for a minimum point of the barrier function in a descent direction, which has a desired property that the box constraints are always satisfied automatically if the step length is a number between zero and one. When all the diagonal entries of the objective function are negative, the method converges to at least a local minimum point of the problem if it yields a local minimum point of the barrier function for a sequence of decreasing values of the barrier parameter with zero limit. Numerical results show that the method always generates a global or near global minimum point as the barrier parameter decreases at a sufficiently slow pace.     

弱辫子T-范畴（英文）

Recently Turaev generalized the notion of a tensor category to that of a crossed group category.In[5]the authors constructed the representation category Rep(H) of a T-coalgebra H.In[2]the authors introduced the notions of a weak tensor category to characterize a weak bialgebra and a weak Hopf algebra.This paper is based on these ideas to naturally introduce the notions of a weak T-category and a weak braided T-category which are not under the usual way and prove that the categories of representations of a weak T-coalgebra and a weak braided T-coalgebra are a weak T-category and a weak braided T-category respectively.Furthermore we also discuss some properties of weak T-category.     

Concept of a vertex in a matroid and 3-connected graphs

The concept of a matroid vertex is introduced. The vertices of a matroid of a 3-connected graph are in one-to-one correspondence with vertices of the graph. Thence directly follows Whitney's theorem that cyclic isomorphism of 3-connected graphs implies isomorphism. The concept of a vertex of a matroid leads to an equally simple proof of Whitney's theorem on the unique embedding of a 3-connected planar graph in the sphere. It also leads to a number of new facts about 3-connected graphs. Thus, consideration of a vertex in a matroid that is the dual of the matroid of a graph leads to a natural concept of a nonseparating cycle of a graph. Whitney's theorem on cyclic isomorphism can be strengthened (even if the nonseparating cycles of a graph are considered, the theorem is found to work) and a new criterion for planarity of 3-connected graphs is obtained (in terms of nonseparating cycles).     

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.     

Graphs with disjoint dominating and paired-dominating sets

A dominating set of a graph is a set of vertices such that every vertex not in the set is adjacent to a vertex in the set, while a paired-dominating set of a graph is a dominating set such that the subgraph induced by the dominating set contains a perfect matching. In this paper, we show that no minimum degree is sufficient to guarantee the existence of a disjoint dominating set and a paired-dominating set. However, we prove that the vertex set of every cubic graph can be partitioned into a dominating set and a paired-dominating set.     

Stability of a singularly perturbed problem

We prove the stability of the mixed problem for a system of telegraph equations under a perturbation of one of the boundary conditions by a sum of a singular perturbation (a small parameter multiplying the highest derivative) and a small regular perturbation. The solution of the problem consists of the current and voltage in a segment of a telegraph line. One of its ends is short-circuited, and a capacitor of small capacity, together with a nonlinear resistance whose volt-ampere characteristic is perturbed by a small term, is connected to the other end. We prove the convergence of the solution of the problem to the unique continuous piecewise continuously differentiable solution of the unperturbed problem bifurcating at some instant of time from its unique classical solution.     

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.     

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.     

Scalarization of Henig Proper Efficient Points in a Normed Space

In a general normed space equipped with the order induced by a closed convex cone with a base, using a family of continuous monotone Minkowski functionals and a family of continuous norms, we obtain scalar characterizations of Henig proper efficient points of a general set and a bounded set, respectively. Moreover, we give a scalar characterization of a superefficient point of a set in a normed space equipped with the order induced by a closed convex cone with a bounded base.     

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.     

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.     

Genus fields of Kummer extensions of rational function fields

In this paper we obtain the genus field of a general Kummer extension of a global rational function field. We study first the case of a general Kummer extension of degree a power of a prime. Then we prove that the genus field of a composite of two abelian extensions of a global rational function field with relatively prime degrees is equal to the composite of their respective genus fields. Our main result, the genus of a general Kummer extension of a global rational function field, is a direct consequence of this fact.     

An Inverse Problem in Elastostatics

This paper assembles a variety of methods which have been devisedfor acoustic and elastic wave propagation inverse problems andadapts them to the problem of determining the shear modulusprofile of an elastic half-space from a knowledge of the torsionaldeflection and shear stress distributions on the surface. Methodsinclude the reduction to a Gel'fand-Levitan integral equation,for which a fast numerical algorithm is presented; a methodbased on modelling the half-space as a layered medium; identificationof the medium as a member of a family for which a closed formsolution is possible; a Green's function approach for a mediumwith small variations from uniformity.     

The Relative Maximum Genus of a Graph

A relative embedding of a graph in a surface with respect to a set of closed walks is one where each of the prescribed closed walks bounds a face of the embedding In the special case where the set of closed walks is empty, this amounts to the usual concept of a graph embedding. We present a formula for the maximum (orientable) genus of the surface on which a graph has a relative embedding with respect to a set of closed walks.     

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.     

Constructing a mathematical definition: the case of the tangent

We analyze the process of constructing a definition within the theoretical framework of Abstraction in Context. Pairs of students were engaged in a task designed to engender a need for a definition of a tangent to a graph at a given point, and lead to constructing a definition following that need. The results of our analysis point to two characteristics of the process of constructing a definition: a. Constructing the concept does not necessarily include constructing its definition; in particular, students were able to use the concept before constructing its definition. b. Students’ language becomes more precise during the constructing process, not only as a characteristic of the process but also as a means of promoting the process.