scholarly journals Empirical Means on Pseudo-Orthogonal Groups

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 940
Author(s):  
Jing Wang ◽  
Huafei Sun ◽  
Simone Fiori
Keyword(s):  
Orthogonal Group ◽  
Geodesic Distance ◽  
Numerical Algorithms ◽  
Riemannian Metric ◽  
Frobenius Norm ◽  
Exponential Map ◽  
Orthogonal Groups ◽  
Orthogonal Matrices ◽  
Numerical Tests ◽  
Empirical Means

The present article studies the problem of computing empirical means on pseudo-orthogonal groups. To design numerical algorithms to compute empirical means, the pseudo-orthogonal group is endowed with a pseudo-Riemannian metric that affords the computation of the exponential map in closed forms. The distance between two pseudo-orthogonal matrices, which is an essential ingredient, is computed by both the Frobenius norm and the geodesic distance. The empirical-mean computation problem is solved via a pseudo-Riemannian-gradient-stepping algorithm. Several numerical tests are conducted to illustrate the numerical behavior of the devised algorithm.

On the Group of Almost Periodic Diffeomorphisms and its Exponential Map

2019 ◽  
Author(s):  
Xu Sun ◽  
Peter Topalov
Keyword(s):  
Lie Group ◽  
Riemannian Metric ◽  
Conjugate Points ◽  
Exponential Map ◽  
Almost Periodic ◽  
Exponential Maps ◽  
Fluid Equations ◽  
Periodic Diffeomorphisms

Abstract We define the group of almost periodic diffeomorphisms on $\mathbb{R}^n$ and on an arbitrary Lie group. We then study the properties of its Riemannian and Lie group exponential maps and provide applications to fluid equations. In particular, we show that there exists a geodesic of a weak Riemannian metric on the group of almost periodic diffeomorphisms of the line that consists entirely of conjugate points.

scholarly journals Geodesic Distance on Gaussian Manifolds to Reduce the Statistical Errors in the Investigation of Complex Systems

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-24 ◽  
Author(s):  
Michele Lungaroni ◽  
Andrea Murari ◽  
Emmanuele Peluso ◽  
Pasqualino Gaudio ◽  
Michela Gelfusa
Keyword(s):  
Euclidean Distance ◽  
Additive Noise ◽  
Geodesic Distance ◽  
Type Ii ◽  
Additional Source ◽  
Numerical Tests ◽  
Important Improvement ◽  
Scientific Expertise ◽  
Type Ii Errors ◽  
Robust Statistical Methods

In the last years the reputation of medical, economic, and scientific expertise has been strongly damaged by a series of false predictions and contradictory studies. The lax application of statistical principles has certainly contributed to the uncertainty and loss of confidence in the sciences. Various assumptions, generally held as valid in statistical treatments, have proved their limits. In particular, since some time it has emerged quite clearly that even slightly departures from normality and homoscedasticity can affect significantly classic significance tests. Robust statistical methods have been developed, which can provide much more reliable estimates. On the other hand, they do not address an additional problem typical of the natural sciences, whose data are often the output of delicate measurements. The data can therefore not only be sampled from a nonnormal pdf but also be affected by significant levels of Gaussian additive noise of various amplitude. To tackle this additional source of uncertainty, in this paper it is shown how already developed robust statistical tools can be usefully complemented with the Geodesic Distance on Gaussian Manifolds. This metric is conceptually more appropriate and practically more effective, in handling noise of Gaussian distribution, than the traditional Euclidean distance. The results of a series of systematic numerical tests show the advantages of the proposed approach in all the main aspects of statistical inference, from measures of location and scale to size effects and hypothesis testing. Particularly relevant is the reduction even of 35% in Type II errors, proving the important improvement in power obtained by applying the methods proposed in the paper. It is worth emphasizing that the proposed approach provides a general framework, in which also noise of different statistical distributions can be dealt with.

scholarly journals On standard L-functions attached to automorphic forms on definite orthogonal groups

1998 ◽  
Vol 152 ◽  
pp. 57-96 ◽  
Author(s):  
Atsushi Murase ◽  
Takashi Sugano
Keyword(s):  
Functional Equation ◽  
Number Field ◽  
Algebraic Number ◽  
Orthogonal Group ◽  
Automorphic Form ◽  
Automorphic Forms ◽  
Algebraic Number Field ◽  
Constant Function ◽  
Orthogonal Groups ◽  
Totally Real

Abstract.We show an explicit functional equation of the standard L-function associated with an automorphic form on a definite orthogonal group over a totally real algebraic number field. This is a completion and a generalization of our previous paper, in which we constructed standard L-functions by using Rankin-Selberg convolution and the theory of Shintani functions under certain technical conditions. In this article we remove these conditions. Furthermore we show that the L-function of f has a pole at s = m/2 if and only if f is a constant function.

Poles ofL-functions and theta liftings for orthogonal groups

2009 ◽  
Vol 8 (4) ◽  
pp. 693-741 ◽  
Author(s):  
David Ginzburg ◽  
Dihua Jiang ◽  
David Soudry
Keyword(s):  
Eisenstein Series ◽  
Orthogonal Group ◽  
Symplectic Groups ◽  
Orthogonal Groups ◽  
Metaplectic Groups ◽  
Automorphic Representations ◽  
Domain Of Holomorphy ◽  
First Occurrence ◽  
Cuspidal Automorphic Representations

AbstractIn this paper, we prove that the first occurrence of global theta liftings from any orthogonal group to either symplectic groups or metaplectic groups can be characterized completely in terms of the location of poles of certain Eisenstein series. This extends the work of Kudla and Rallis and the work of Moeglin to all orthogonal groups. As applications, we obtain results about basic structures of cuspidal automorphic representations and the domain of holomorphy of twisted standardL-functions.

The conjugate classes of the cubic surface group in an orthogonal representation

1955 ◽  
Vol 233 (1192) ◽  
pp. 126-146 ◽  
Keyword(s):  
Orthogonal Group ◽  
Surface Group ◽  
The Other ◽  
Identity Matrix ◽  
Orthogonal Matrices ◽  
Linear Spaces ◽  
Cubic Surface ◽  
Orthogonal Representation ◽  
Conjugate Class ◽  
Group A

The cubic surface group, of order 51840, has a representation by orthogonal matrices, of 5 rows and determinant + 1, over GF (3). It can be partitioned into conjugate classes on geometrical grounds because each matrix has two skew linear spaces, S + of even and S - of odd dimension, of latent points; the matrices fall into categories A, B, C according as the join of S + and S - has dimension 4, 2, 0. Subdivisions of A, B, C rest on the relation of S + and S - to the invariant quadric of the orthogonal group. A accounts for the identity matrix and the 4 types of involutions. B falls into two parts; one of 4 classes, discussed in §§5 to 8, the other of 9 classes, discussed in §§9 to 14. §§ 15 and 16 mention criteria for checking the number of operations in a conjugate class. Those classes in category C fall into 3 subcategories of 3, 2, 2 classes and are described in §§ 18 to 25.

scholarly journals Theta series liftings from orthogonal groups to semi-simple groups

2000 ◽  
Vol 69 (1) ◽  
pp. 127-142
Author(s):  
Min Ho Lee
Keyword(s):  
Half Space ◽  
Lie Group ◽  
Orthogonal Group ◽  
Automorphic Form ◽  
Automorphic Forms ◽  
Theta Series ◽  
Symmetric Domain ◽  
Simple Groups ◽  
Orthogonal Groups ◽  
Spin Groups

AbstractWe study a correspondence between automorphic forms on an orthogonal group and automorpbic forms on a semi-simple Lie group associated to an equivariant holomorphic map of a symmetric domain into a Siegel upper half space. We construct an automorphic form on the symmetric domain thatg corresponds to an automorphic form on an orthogonal group using theta series, and prove that such a correspondence is compatible with the appropriate Hecks operator actions on the corresponding automorphic forms. As an example, we describe the case of spin groups.

scholarly journals Singular Isometries in Orthogonal Groups

1977 ◽  
Vol 20 (2) ◽  
pp. 189-198 ◽  
Author(s):  
Georg Gunther
Keyword(s):  
Orthogonal Group ◽  
Commutator Subgroup ◽  
Orthogonal Groups ◽  
Singular Subspace

In this paper, we study the behaviour of singular isometries in orthogonal groups. These are isometries whose path is a singular subspace. We shall prove that the path of such a singular isometry is always even-dimensional. We shall use this result to show that the subgroup of the orthogonal group On(K, Q) which is generated by the singular isometries is the commutator subgroup Ωn(K, Q).

A Schensted Algorithm Which Models Tensor Representations of the Orthogonal Group

1990 ◽  
Vol 42 (1) ◽  
pp. 28-49 ◽  
Author(s):  
Robert A. Proctor
Keyword(s):  
Representation Theory ◽  
Orthogonal Group ◽  
The Body ◽  
Orthogonal Groups ◽  
Tensor Representations ◽  
Combinatorial Construction ◽  
Theoretic Motivation

This paper is concerned with a combinatorial construction which mysteriously “mimics” or “models” the decomposition of certain reducible representations of orthogonal groups. Although no knowledge of representation theory is needed to understand the body of this paper, a little familiarity is necessary to understand the representation theoretic motivation given in the introduction. Details of the proofs will most easily be understood by people who have had some exposure to Schensted's algorithm or jeu de tacquin.

scholarly journals Arithmetic of Orthogonal Groups (II)

1955 ◽  
Vol 9 ◽  
pp. 129-146 ◽  
Author(s):  
Takashi Ono
Keyword(s):  
Vector Space ◽  
Orthogonal Group ◽  
The Other ◽  
Vector Spaces ◽  
Structure Theory ◽  
Orthogonal Groups

In [0], the writer proved some theorems of Hasse type for two orthogonal groups which operate on the same vector space. In this paper, we shall further generalize those results in two directions. One is to consider the propositions of that type for two orthogonal groups which operate respectively on two vector spaces whose dimensions are different from each other, and the other is to deal with some conspicuous subgroups of an orthogonal group simultaneously which play important roles in the structure theory for orthogonal groups. For this reason, the present paper consists of three steps §1, §2 and §3 which give the generalizations in the above sense of the results in the corresponding sections of [0].

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