On some classes of inverse problems with overdetermination data on spatial manifolds

We consider the question of well-posedness of the inverse problems of determining the righthand side (a source function) of a special form of a parabolic system of equations. The overdetermination data are the values of a solution and its normal derivatives on a system of surfaces in a spatial domain. In particular, the cross-sections of the domain can be used as these surfaces. Sharp conditions are presented for the data of the problem to ensure well-posedness.     

Notes on classifying inputs and outputs in data envelopment analysis: A comment

Cook and Zhu (2007) introduced an innovative method to deal with flexible measures. Toloo (2009) found a computational problem in their approach and tackled this issue. Amirteimoori and Emrouznejad (2012) claimed that both Cook and Zhu (2007) and Toloo (2009) models overestimate the efficiency. In this response, we prove that their claim is incorrect and there is no overestimate in these approaches.     

Excursion probabilities of isotropic and locally isotropic Gaussian random fields on manifolds

Let X = {X(p), p ∈ M} be a centered Gaussian random field, where M is a smooth Riemannian manifold. For a suitable compact subset \(D\subset M\), we obtain approximations to the excursion probabilities \(\mathbb {P}\{\sup _{p\in D} X(p) \ge u \}\), as \(u\to \infty \), for two cases: (i) X is smooth and isotropic; (ii) X is non-smooth and locally isotropic. For case (i), the expected Euler characteristic approximation is formulated explicitly; while for case (ii), it is shown that the asymptotics is similar to Pickands’ approximation on Euclidean space which involves Pickands’ constant and the volume of D. These extend the results in Cheng and Xiao (Bernoulli 22, 1113–1130 2016) from spheres to general Riemannian manifolds.     

Sum of squares manifolds: The expressibility of the Laplace-Beltrami operator on pseudo-Riemannian manifolds as a sum of squares of vector fields

In this paper, we investigate under what circumstances the Laplace-Beltrami operator on a pseudo-Riemannian manifold can be written as a sum of squares of vector fields, as is naturally the case in Euclidean space.

We show that such an expression exists globally on one-dimensional manifolds and can be found at least locally on any analytic pseudo-Riemannian manifold of dimension greater than two. For two-dimensional manifolds this is possible if and only if the manifold is flat.

These results are achieved by formulating the problem as an exterior differential system and applying the Cartan-Kähler theorem to it.

     

Testing the independence of maxima: from bivariate vectors to spatial extreme fields

Characterizing the behaviour of multivariate or spatial extreme values is of fundamental interest to understand how extreme events tend to occur. In this paper we propose to test for the asymptotic independence of bivariate maxima vectors. Our test statistic is derived from a madogram, a notion classically used in geostatistics to capture spatial structures. The test can be applied to bivariate vectors, and a generalization to the spatial context is proposed. For bivariate vectors, a comparison to the test by Falk and Michel (Ann Inst Stat Math 58:261–290, 2006) is conducted through a simulation study. In the spatial case, special attention is paid to pairwise dependence. A multiple test procedure is designed to determine at which lag asymptotic independence takes place. This new procedure is based on the bootstrap distribution of the number of times the null hypothesis is rejected. It is then tested on maxima of three classical spatial models and finally applied to two climate datasets.     

Topological transformation groups of manifolds over non-Archimedean fields,their representations,and quasi-invariant measures,II

Diffeomorphism groups and loop groups of manifolds on Banach spaces over non-Archimedean fields are defined. Moreover, for these groups, finite-and infinite-dimensional manifolds over the corresponding fields are considered. The group structure, the differential-geometric structure, and also the topological structure of diffeomorphism groups and loops groups are studied. We prove that these groups do not locally satisfy the Campbell-Hausdorff formula. The principal distinctions in the structure for the Archimedean and classical cases are found. The quasi-invariant measures on these groups with respect to dense subgroups are constructed. Stochastic processes on topological transformation groups of manifolds and, in particular, on diffeomorphism groups and on loop groups and also the corresponding transition probabilities are constructed. Regular, strongly continuous, unitary representations of dense subgroups of topological transformation groups of manifolds, in particular, those of diffeomorphism group and loop groups associated with quasi-invariant measures on groups and also on the corresponding configurational spaces are constructed. The conditions imposed on the measure and groups under which these unitary representations are irreducible are found. The induced representations of topological groups are studied by using quasi-invariant measures on topological groups. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 18, Functional Analysis, 2006.     

Topological transformation groups of manifolds over non-Archimedean fields,their representations,and quasi-invariant measures. I

Diffeomorphism groups and loop groups of manifolds on Banach spaces over non-Archimedean fields are defined. Moreover, for these groups, finite-and infinite-dimensional manifolds over the corresponding fields are considered. The group structure, the differential-geometric structure, and also the topological structure of diffeomorphism groups and loops groups are studied. We prove that these groups do not locally satisfy the Campbell-Hausdorff formula. The principal distinctions in the structure for the Archimedean and classical cases are found. The quasi-invariant measures on these groups with respect to dense subgroups are constructed. Stochastic processes on topological transformation groups of manifolds and, in particular, on diffeomorphism groups and on loop groups and also the corresponding transition probabilities are constructed. Regular, strongly continuous, unitary representations of dense subgroups of topological transformation groups of manifolds, in particular, those of diffeomorphism groups and loop groups associated with quasi-invariant measures on groups and also on the corresponding configurational spaces are constructed. The conditions imposed on the measure and groups under which these unitary representations are irreducible are found. The induced representations of topological groups are studied by using quasi-invariant measures on topological groups. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 39, Functional Analysis, 2006.     

Convergence of the Newton method and uniqueness of zeros of vector fields on Riemannian manifolds

The estimates of the radii of convergence balls of the Newton method and uniqueness balls of zeroes of vector fields on the Riemannian manifolds are given under the assumption that the covariant derivatives of the vector fields satisfy some kind of general Lipschitz conditions. Some classical results such as the Kantorovich's type theorem and the Smale's γ-theory are extended.     

Exploring interactions in high-dimensional genomic data: an overview of Logic Regression, with applications

Logic Regression is an adaptive regression methodology mainly developed to explore high-order interactions in genomic data. Logic Regression is intended for situations where most of the covariates in the data to be analyzed are binary. The goal of Logic Regression is to find predictors that are Boolean (logical) combinations of the original predictors. In this article, we give an overview of the methodology and discuss some applications. We also describe the software for Logic Regression, which is available as an R and S-Plus package.     

Variational results on flag manifolds: Harmonic maps, geodesics and Einstein metrics

In this paper, we study variational aspects for harmonic maps from M to several types of flag manifolds and the relationship with the rich Hermitian geometry of these manifolds. We consider maps that are harmonic with respect to any invariant metric on each flag manifold. They are called equiharmonic maps. We survey some recent results for the case where M is a Riemann surface or is one dimensional; i.e., we study equigeodesics on several types of flag manifolds. We also discuss some results concerning Einstein metrics on such manifolds.     

Estimation,modeling, and aggregation of missing survey data for prioritizing customer voices

It is widely acknowledged that understanding and prioritizing the voice of customer is a critical step in new product development. In this work, we propose a novel approach to handle missing and incomplete data while combining information from different surveys for prioritizing customer voices. Our new approach comprises of the following stages: estimating and representing missing and incomplete data; estimating intervals for the criteria used in analyzing data; mapping data on criteria to a common scale; modeling interval data using interval belief structure; and aggregating evidence and ranking customer voices using the interval evidential reasoning algorithm. We demonstrate our approach using a case study from automotive domain with a given criteria hierarchy for analyzing data from three different surveys. We propose new optimization formulations for estimating intervals of the criteria used in our case study and logical yet pragmatic transformation functions for mapping criteria values to a common scale.     

Intersecting invariant manifolds in spatial restricted three-body problems: Design and optimization of Earth-to-halo transfers in the Sun-Earth-Moon scenario

This work deals with the design of transfers connecting LEOs with halo orbits around libration points of the Earth-Moon CRTBP using impulsive maneuvers. Exploiting the coupled circular restricted three-body problem approximation, suitable first guess trajectories are derived detecting intersections between stable manifolds related to halo orbits of EM spatial CRTBP and Earth-escaping trajectories integrated in planar Sun-Earth CRTBP. The accuracy of the intersections in configuration space and the discontinuities in terms of Δv are controlled through the box covering structure implemented in the software GAIO. Finally first guess solutions are optimized in the bicircular four-body problem and single-impulse and two-impulse transfers are presented.     

Estimation of elastic parameters in beams and certain plates:H 1 regularization

The estimation of elastic parameters in beams and certain types of plates is discussed using anH ¹-regularization technique that easily accommodates pointwise constraints. The optimal coefficient is shown to enjoy more regularity than that assumed in the formulation of the problem. This additional smoothness is useful for analyzing the limit behavior of finite-dimensional problems. Numerical results are presented.This work was supported in part by the Air Force Office of Scientific Research, Grant AFOSR-84-0271.     

Relevance of functional flexibility for heterogeneous sales response models: A comparison of parametric and semi-nonparametric models

So far studies estimating sales response functions on the basis of store-specific data either consider heterogeneity or functional flexibility. That is why in this contribution a model is developed possessing both these features. It is a multilayer perceptron with store-specific coefficients which follows a hierarchical Bayesian framework. An appropriate Markov Chain Monte Carlo estimation technique is introduced capable to satisfy theoretical constraints (e.g. sign constraints on elasticities). The empirical study refers to a data base consisting of weekly observations of sales and prices for nine leading brands of a packaged consumer good category. The data were acquired in 81 stores over a time span of at least 61 weeks. The multilayer perceptron is compared to a strict parametric multiplicative model and turns out to be clearly superior in terms of posterior model probability. This result indicates the benefits of using a flexible model even if heterogeneity is dealt with. Estimated sales curves and elasticities demonstrate that both models differ with regard to implications on price response.     

One-step estimation of spatial dependence parameters: Properties and extensions of the APLE statistic

We consider one-step estimation of parameters that represent the strength of spatial dependence in a geostatistical or lattice spatial model. While the maximum likelihood estimators (MLE) of spatial dependence parameters are known to have various desirable properties, they do not have closed-form expressions. Therefore, we consider a one-step alternative to maximum likelihood estimation based on solving an approximate (i.e., one-step) profile likelihood estimating equation. The resulting approximate profile likelihood estimator (APLE) has a closed-form representation, making it a suitable alternative to the widely used Moran’s I statistic. Since the finite-sample and asymptotic properties of one-step estimators of covariance-function parameters have not been studied rigorously, we explore these properties for the APLE of the spatial dependence parameter in the simultaneous autoregressive (SAR) model. Motivated by the APLE statistic’s closed from, we develop exploratory spatial data analysis tools that capture regions of local clustering or the extent to which the strength of spatial dependence varies across space. We illustrate these exploratory tools using both simulated data and observed crime rates in Columbus, OH.