Algebraic monoids with affine unit group are affine

In this short paper we prove that any irreducible algebraic monoid whose unit group is an affine algebraic group is affine.     

On the class of infinite horizon optimal control problems affine,which are structurally stable

We prove in this article that there is in the set of all problems we consider a subset, which is residual. Every problem in this subset is shown to be structurally stable and defines a dynamical system, which looks like the graph of the figure given in Section  1 , contrary to what happens for ordinary dynamical systems, that is, the ones associated with ODEs. There, the initial value problem (in the smooth case) is uniquely solvable; the structurally stable systems look like the figure given in Section  1 , but sources are equilibria. Copyright © 2015 John Wiley & Sons, Ltd.     

Canonical decomposition of projective and affine killing vectors on the tangent bundle

For an affine connection on the tangent bundle T(M) obtained by lifting an affine connection on M, the structure of vector fields on T(M) which generate local one-parameter groups of projective and affine collineations is described. On the T(M) of a complete irreducible Riemann manifold, every projective collineation is affine. On the T(M) of a projectively Euclidean space, every affine collineation preserves the fibration of T(M), and on the T(M) of a projectively non-Duclidean space which is maximally homogeneous (in the sense of affine collineations) there exist affine collineations permuting the fibers of T(M).Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 247–258, February, 1976.     

Finite geometries which contain dual affine planes

Finite geometries in which each plane is projective or dual affine over the field of two elements, or affine over the field of three elements, are studied. It is shown that no connected geometry can mix all three species of planes, and the geometries in which projective and dual affine planes occur are classified.     

Optimal approximation of convex curves by functions which are piecewise linear

In this paper an efficient method is presented for solving the problem of approximation of convex curves by functions that are piecewise linear, in such a manner that the maximum absolute value of the approximation error is minimized. The method requires the curves to be convex on the approximation interval only. The boundary values of the approximation function can be either free or specified. The method is based on the property of the optimal solution to be such that each linear segment approximates the curve on its interval optimally while the optimal error is uniformly distributed among the linear segments of the approximation function. Using this method the optimal solution can be determined analytically to the full extent in certain cases, as it was done for functions x² and $\text{x}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. In general, the optimal solution has to be computed numerically following the procedure suggested in the paper. Using this procedure, optimal solutions were computed for functions sin x, tg x, and arc tg x. Optimal solutions to these functions were used in practical applications.     

On improving distribution function estimators which are not monotonic functions

In this paper we consider the problem of correcting distribution function estimators which are not nondecreasing functions (for example kernel type estimators). The method is based on the orthogonal projection in L₂ and guarantees improving of the integrated mean square error for each sample size.     

Infinitesimal affine transformations of the second order tangent bundle with horizontal lift connection

We obtain the canonical expansion of an infinitesimal affine transformation of the second order tangent bundle with horizontal lift connection. We establish necessary and sufficient conditions under which a vector field is an infinitesimal affine transformation. We also construct the horizontal lift of a linear connection to a second order Weil bundle.     

Ergodic affine transformations are loosely Bernoulli

We prove that an ergodic affine transformation of a compact abelian group is loosely Bernoulli, that is, it can be induced from a Bernoulli shift.     

优化Sobolev体及其仿射性质

Sobolev不等式是联系分析和几何的基础不等式之一,而优化Sobolev体是优化Sobolev范数的临界几何核.首先,证明优化Sobolev体的一些仿射性质.然后,运用Barthe的优化迁移方法研究了凸体的特征函数和多胞形仿射函数的优化Sobolev体.     

Some remarks about polynomials which are orthogonal with respect to an indefinite weight

In this note Laguerre-, Legendre- and Jacobi-type orthogonal polynomials with respect to an indefinite weight function are considered (see [LK]). In [LK] some values of the parameters are excluded, as in these cases the eigenvectors of the associated operators in the corresponding Pontrjagin spaces do not form a complete system. It is shown, that in these exceptional cases algebraically nonsimple eigenvalues appear, and in order to get a complete system, the system of eigenfunctions has to be augmented by those root functions, which are not eigenfunctions.     

Souslin trees which are hard to specialise

We construct some -Souslin trees which cannot be specialised by any forcing which preserves cardinals and cofinalities. For a regular cardinal we use the principle, and for singular we use squares and diamonds.

     

Limiting behavior of the affine scaling continuous trajectories for linear programming problems

We consider the continuous trajectories of the vector field induced by the primal affine scaling algorithm as applied to linear programming problems in standard form. By characterizing these trajectories as solutions of certain parametrized logarithmic barrier families of problems, we show that these trajectories tend to an optimal solution which in general depends on the starting point. By considering the trajectories that arise from the Lagrangian multipliers of the above mentioned logarithmic barrier families of problems, we show that the trajectories of the dual estimates associated with the affine scaling trajectories converge to the so called centered optimal solution of the dual problem. We also present results related to asymptotic direction of the affine scaling trajectories. We briefly discuss how to apply our results to linear programs formulated in formats different from the standard form. Finally, we extend the results to the primal-dual affine scaling algorithm.     

Optimal trajectories of infinite-horizon deterministic control systems

We study the infinite-horizon deterministic control problem of minimizing ₀ ^T L(z, ) dt, T, whereL(z, ·) is convex in for fixedz but not necessarily jointly convex in (z, ). We prove the existence of a solution to the infinite-horizon Bellman equation and use it to define a differential inclusion, which reduces in certain cases to an ordinary differential equation. We discuss cases where solutions of this differential inclusion (equation) provide optimal solutions (in the overtaking optimality sense) to the optimization problem.A quantity of special interest is the minimal long-run average-cost growth rate. We compute it explicitly and show that it is equal to min _x L(x, 0) in the following two cases: one is the scalar casen = 1 and the other is' when the integrand is in a separated form      

Optimal capture trajectories using multiple gravity assists

Graph theoretic methods of optimal control in the presence of uncertainty are applied to a celestial mechanics problem. We find a fuel-efficient spacecraft trajectory which starts at infinity and is captured by the smaller member of a binary system, e.g., a moon of Jupiter, using multiple gravity assists.