Abstract:
To deal with convex programming problems with bounded variables, we introduce a new system of ordinary differential equations in this paper. It is shown that by following the trajectory of the system, an optimal solution to the convex programming problems can be located.

Abstract:
Consider the minimization problem with a convex separable objective function over a feasible region defined by linear equality constraint(s)/linear inequality constraint of the form “greater than or equal to” and bounds on the variables. A necessary and sufficient condition and a sufficient condition are proved for a feasible solution to be an optimal solution to these two problems, respectively. Iterative algorithms of polynomial complexity for solving such problems are suggested and convergence of these algorithms is proved. Some convex functions, important for problems under consideration, as well as computational results are presented.

Abstract:
We consider the motion planning problem for a point constrained to move along a smooth closed convex path of bounded curvature. The workspace of the moving point is bounded by a convex polygon with m vertices, containing an obstacle in a form of a simple polygon with $n$ vertices. We present an O(m+n) time algorithm finding the path, going around the obstacle, whose curvature is the smallest possible.

Abstract:
For a bounded and convex domain in three dimensions we show that the Maxwell constants are bounded from below and above by Friedrichs' and Poincare's constants.

Abstract:
Let C be a simple closed convex curve in the plane for which the radius of curvature is a continuous function of the arc length. Such a curve is called a convex curve of bounded type, if lies between two fixed positive bounds. Here we give a new and simpler proof of Blaschke's Rolling Theorem. We prove one new theorem and suggest a number of open problems.

Abstract:
We derive the plasticity equations for convex quadrilaterals on a complete convex surface with bounded specific curvature and prove a plasticity principle which states that: Given four shortest arcs which meet at the weighted Fermat-Torricelli point their endpoints form a convex quadrilateral and the weighted Fermat-Torricelli point belongs to the interior of this convex quadrilateral, an increase of the weight corresponding to a shortest arc causes a decrease of the two weights that correspond to the two neighboring shortest arcs and an increase of the weight corresponding to the opposite shortest arc by solving the inverse weighted Fermat-Torricelli problem for quadrilaterals on a convex surface of bounded specific curvature. Furthermore, we show a connection between the plasticity of convex quadrilaterals on a complete convex surface with bounded specific curvature with the plasticity of some generalized convex quadrilaterals on a manifold which is certainly composed by triangles. We also study some cases of symmetrization of weighted convex quadrilaterals by introducing a new symmetrization technique which transforms some classes of weighted geodesic convex quadrilaterals on a convex surface to parallelograms in the tangent plane at the weighted Fermat-Torricelli point of the corresponding quadrilateral.

Abstract:
In this paper, we present a recurrent neural network for solving convex quadratic programming problems, in the theoretical aspect, we prove that the proposed neural network can converge globally to the solution set of the problem when the matrix involved in the problem is positive semi-definite and can converge exponentially to a unique solution when the matrix is positive definite. Illustrative examples further show the good performance of the proposed neural network.

Abstract:
A linear operator $T:X \to Y$ between topological vector spaces is called strictly singular if for any infinite dimensional vector subspace $X_0$ of $X$, the restriction of $T$ to $X_0$ is not a topological isomorphism. In this note we introduced some sufficient conditions on the pair $(X,Y)$ such that any bounded linear operator in between is strictly singular, and give some examples of spaces satisfying these conditions. Furthermore, we extended some of the results to locally convex spaces.

Abstract:
Let $f$ be an entire transcendental function of finite order and $\Delta$ be a forward invariant bounded Siegel disk for $f$ with rotation number in Herman's class $\mathcal{H}$. We show that if $f$ has two singular values with bounded orbit, then the boundary of $\Delta$ contains a critical point. We also give a criterion under which the critical point in question is recurrent. We actually prove a more general theorem with less restrictive hypotheses, from which these results follow.

Abstract:
Given a convex set $C$ in a real vector space $E$ and two points $x,y\in C$, we investivate which are the possible values for the variation $f(y)-f(x)$, where $f:C\longrightarrow [m,M]$ is a bounded convex function. We then rewrite the bounds in terms of the Funk weak metric, which will imply that a bounded convex function is Lipschitz-continuous with respect to the Thompson and Hilbert metrics. The bounds are also proved to be optimal. We also exhibit the maximal subdifferential of a bounded convex function at a given point $x\in C$.