Stochastic Order and Generalized Weighted Mean Invariance

In this paper, we present order invariance theoretical results for weighted quasi-arithmetic means of a monotonic series of numbers. The quasi-arithmetic mean, or Kolmogorov–Nagumo mean, generalizes the classical mean and appears in many disciplines, from information theory to physics, from economics to traffic flow. Stochastic orders are defined on weights (or equivalently, discrete probability distributions). They were introduced to study risk in economics and decision theory, and recently have found utility in Monte Carlo techniques and in image processing. We show in this paper that, if two distributions of weights are ordered under first stochastic order, then for any monotonic series of numbers their weighted quasi-arithmetic means share the same order. This means for instance that arithmetic and harmonic mean for two different distributions of weights always have to be aligned if the weights are stochastically ordered, this is, either both means increase or both decrease. We explore the invariance properties when convex (concave) functions define both the quasi-arithmetic mean and the series of numbers, we show its relationship with increasing concave order and increasing convex order, and we observe the important role played by a new defined mirror property of stochastic orders. We also give some applications to entropy and cross-entropy and present an example of multiple importance sampling Monte Carlo technique that illustrates the usefulness and transversality of our approach. Invariance theorems are useful when a system is represented by a set of quasi-arithmetic means and we want to change the distribution of weights so that all means evolve in the same direction.     

ϕ-Informational Measures: Some Results and Interrelations

In this paper, we focus on extended informational measures based on a convex function ϕ: entropies, extended Fisher information, and generalized moments. Both the generalization of the Fisher information and the moments rely on the definition of an escort distribution linked to the (entropic) functional ϕ. We revisit the usual maximum entropy principle—more precisely its inverse problem, starting from the distribution and constraints, which leads to the introduction of state-dependent ϕ-entropies. Then, we examine interrelations between the extended informational measures and generalize relationships such the Cramér–Rao inequality and the de Bruijn identity in this broader context. In this particular framework, the maximum entropy distributions play a central role. Of course, all the results derived in the paper include the usual ones as special cases.     

Filtering on manifolds

We coasider a partially observable diffusion process (x _t,y_t)t0 whose unobservable componentx _t lives on a submanifold M ofR ⁿ . We present some general conditions under which the conditional law ofx _t, given the observationsy _s ,s [0,t], admits a density w.r.t. a given measure on M. We characterize the analytical properties of this density by using appropriate Sobolev spaces.Research supported by the Hungarian National Foundation of Scientific Research No. 2290.     

The distribution of square-full integers

Let , wheref(n) is the characteristic function of square-full integers. A formula is proved that can be used to obtain better upper bound estimate for (x). We get (x)=O(x ^5/33+), which improves the exponent 11/72 obtained by R. Blasubramania and K. Ramachandra.     

Further results on the euler and Genocchi numbers

Summary We characterize the ordinary generating functions of the Genocchi and median Genocchi numbers as unique solutions of some functional equations and give a direct algebraic proof of several continued fraction expansions for these functions. New relations between these numbers are also obtained.     

McMullen's conditions and some lower bounds for general convex polytopes

We give a lower bound for the number of vertices of a generald-dimensional polytope with a given numberm ofi-faces for eachi = 0,..., d/2 – 1. The tightness of those bounds is proved using McMullen's conditions. Form greater than a small constant, those lower bounds are attained by simpliciali-neighbourly polytopes.     

Nonlinear transformations of the canonical gauss measure on Hilbert space and absolute continuity

The papers of R. Ramer and S. Kusuoka investigate conditions under which the probability measure induced by a nonlinear transformation on abstract Wiener space(,H,B) is absolutely continuous with respect to the abstract Wiener measure. These conditions reveal the importance of the underlying Hilbert spaceH but involve the spaceB in an essential way. The present paper gives conditions solely based onH and takes as its starting point, a nonlinear transformationT=I+F onH. New sufficient conditions for absolute continuity are given which do not seem easily comparable with those of Kusuoka or Ramer but are more general than those of Buckdahn and Enchev. The Ramer-Itô integral occurring in the expression for the Radon-Nikodym derivative is studied in some detail and, in the general context of white noise theory it is shown to be an anticipative stochastic integral which, under a stronger condition on the weak Gateaux derivative of F is directly related to the Ogawa integral.Research supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620 92 J 0154 and the Army Research Office Grant No. DAAL 03 92 G 0008.     

A dielectric rod waveguide coupled with metal rectangular waveguide

Measurements have been done in the millimeter wave region on a composite waveguide which comprises a dielectric rod waveguide connecting two metal rectangular waveguides. Such a waveguide has been used by us in a Josephson harmonic mixer installed in a small metal cryostat, to prevent the thermal invasion from outside environment and to transmit both signal and LO waves with small losses. The measured transmission loss, that is caused mainly by the coupling loss between metal rectangular waveguides (TE₁₀ mode) and a dielectric rod waveguide (HE₁₁ mode), has been less than 2dB in the frequency range of 52–104 GHz.     

Diffraction grating interferometers for mechanical characterisations of advanced fabric laminates

It is demonstrated that two grating interferometers with high spatial resolutions can successfully be applied for the mechanical characterisation of the advanced fabric composite materials. Based on these two techniques, the mechanical properties of two kinds of fabric laminates are obtained without assumption of uniform strain fields to be used in the characterisation approaches using the local strain sensors. The degree of the yarn crimp effects of the two laminates is compared in terms of the out-of-plane displacement derivatives. Especially, it is shown that the grating shearing interferometer is appropriate for the crimped fabric structure requiring a three-dimensional analysis. The modification from moiré interferometer to grating shearing interferometer is performed by introducing a Michelson interferometer modified for image shearing.