Fractional-Linear Transformations of Operator Balls; Applications to Dynamical Systems

The operator sets, which are the subject of this paper, have been studied in many papers where, under different restrictions on the generating operators, convexity, compactness in the weak operator topology, and nonemptiness were proved for sets of different classes under study. Then the results obtained were used in these papers to solve several applied problems. Namely, they played the key role in establishing the dichotomy of nonautonomous dynamical systems, with either continuous or discrete time. In the present paper, we generalize and sharpen the already known criteria and obtain several new criteria for convexity, compactness, and nonemptiness of several special operator sets. Then, using the assertions obtained, we construct examples of sets of the form under study which are nonconvex, noncompact in the weak operator topology, as well as empty, and are generated by ＂smooth＂ operators of a special class. The existence problem for such sets remained open until the authors of this paper announced some of its results.     

C~k连续的保形分段2k次多项式插值

1．引言在每个子区间上，通过插入至多一个内结点，Brodlie和Butt［1］给出了分段三次多项式保形插值算法，Randal［2］等讨论了分段五次多项式插值，作者［31讨论了一般分段奇次多项式的保形插值，并且给1了内结点的位置范围公式．这种插值方法完全解决了一般的分段奇次多项式的保形插值问题．关于分段偶次多项式的保形插值，大多数文献只讨论分段二次保形插值，这里要特别指出的是Shumake［4j导出了二次样条保凸的充要条件，并且给出了一个二次样条保形插值的方法．在每一个子区间上至多插入一个内结点，则一个二次插值样条就可得到．作…     

Reciprocally convex functions

We say that f is reciprocally convex if x?f(x) is concave and x?f(1/x) is convex on (0,+∞). Reciprocally convex functions generate a sequence of quasi-arithmetic means, with the first one between harmonic and arithmetic mean and others above the arithmetic mean. We present several examples related to the gamma function and we show that if f is a Stieltjes transform, then −f is reciprocally convex. An application in probability is also presented.     

Convergence rates of a global optimization algorithm

Second order conditions for the (pseudo-) convexity of a function restricted to an affine subspace are obtained by extending those already known for functions on ⁿ . These results are then used to analyse the (pseudo-) convexity of potential functions of the type introduced by Karmarkar.This research was completed while the first author was on sabbatical leave at the Département d'Informatiques et de Recherche Opérationelle, Université de Montréal, and supported by NSERC (grant Q015807). This research was also supported by NSERC (grants A8312 and A5408) and la Coopération franco-québécoise (project 20-02-13).     

What is quasiconvex analysis?

We endeavour to answer the question of the title, or rather the question of how much one can extend convex analysis to a wider framework in which some convexity features remain. Subdifferential calculus and duality are the main directions we consider     

On the Pseudoconvexity of a Quadratic Fractional Function

In this paper we give a necessary and sufficient condition for the pseudoconvexity of a function f which is the ratio of a quadratic function over an affine function. The obtained results allow to suggest a simple algorithm to test the pseudoconvexity of f and also to characterize the pseudoconvexity of the sum of a linear and a linear fractional function.     

Morse Theory And Invariant Concepts of Convexity

In optimization, the literature on invariant definitions of convexity has largely centered on the notion of invexity. However, the connection with Morse theory and related geometric trends has not been recognized and results do not appear to be as impressive as those in topology. Our purpose is to improve the situation in optimization, and to clarify the relationship between these two trends. The main result is that the problem can be reduced to the case of functions of one variable.     

Lipschitz Continuity and Interpolation Properties of Some Classes of Increasing Functions

We study local Lipschitz continuity and interpolation properties of some classes of increasing functions defined on the cone Rⁿ ₊₊ of n-vectors with positive coordinates. We also study the so-called self-conjugate increasing positively homogeneous functions.     

The diameter of a fuzzy set

In pattern recognition one often wants to measure geometric properties of imprecisely defined subsets of an image. This paper proposes definitions of intrinsic and extrinsic diameter for fuzzy subsets which reduce to the ordinary definitions when the subsets are crisp. We also define height and width for a fuzzy subset and show how they relate to the area (i.e., integral of membership). For convex fuzzy subsets the intrinsic diameter cannot exceed the extrinsic diameter, but it can be smaller. Finally, for piecewise constant convex fuzzy subsets the intrinsic diameter cannot exceed half the fuzzy perimeter, but this need not be true in the nonconvex case.