Vectors E1 Research Articles

On Lorentzian Ricci solitons on nilpotent Lie groups

The aim of this article is to exhibit the variety of different Ricci soliton structures that a nilpotent Lie group can support when one allows for the metric tensor to be Lorentzian. In stark contrast to the Riemannian case, we show that a nilpotent Lie group can support a number of non-isometric Lorentzian Ricci soliton structures with decidedly different qualitative behaviors and that Lorentzian Ricci solitons need not be algebraic Ricci solitons. The analysis is carried out by classifying all left invariant Lorentzian metrics on the connected, simply-connected five-dimensional Lie group having a Lie algebra with basis vectors E1,E2,E3,E4 and E5 and non-trivial bracket relations E1,E5=E3 and E2,E5=E4, investigating the various curvature properties of the resulting families of metrics, and classifying all Lorentzian Ricci soliton structures.

Letter to the Editor: Joint Moments in the Joint Coordinate System, Euler or Dual Euler Basis

Letter to the Editor: Joint Moments in the Joint Coordinate System, Euler or Dual Euler Basis

A Fundamental Property of Suslin Matrices

AbstractWe describe a homomorphism from the group SUmr (R), generated by Suslin matrices, when r is even, to the special orthogonal group SO2(r+1) (R) by relating the Suslin matrix corresponding to a pair of vectors v, w, with 〈v, w〉 = 1, to the product of two reflections, one w.r.t. the vectors v, w and the other w.r.t. the vectors e1, e1 (of length one). When r is odd we can still associate a product of reflections with an element of SUmr (R), which is well defined up to a unit u, with u2 = 1. This association enables one to study the orbit space of unimodular vectors under the elementary subgroup.

Properties of continuous functions on a normed space and its sphere

Known well is the problem of finding configurations of points on the Euclidean sphere S n that can be put into one level surface of any continuous function on S n by a rotation of S n . The paper is devoted to various ways of transferring this problem to the ease of a normed space. Here is one of the results. Let f and g be two even continuous functions on an n-dimensional normed space E, and let f(0) < f(x) for all nonzero x ∈ E. Then E contains n unit vectors e 1,…,e n such that for any 1 ≤ i ≤ j ≤ n we have f(e i + e j ): f(e i − e j ) and g(e i + e j ): g(e i − e j ). Bibliography: 16 titles.

A Geometric Representation of the Morse Fan

It was proved in Jongen and Pallaschke (1988) that every piecewise smooth Morse function f defined on an open subset of IR can be represented in suitable coordinates in the neighborhood of a nondegenerate critical point as f x0 +l y1 yk − ∑k+ i=k+1y 2 i + ∑n j=k+ +1y 2 j where the piecewise linear function l∈CS y1 yk − ∑k i=1yi is a continuous selection of the coordinate functions y1 yk and their negative sum − ∑k i=1yi In this paper we study a collection of cones in IR on which the functions l∈CS y1 yk − ∑k i=1yi are linear. This collection of cones forms a complete polyhedral fan and will be called the Morse fan. It is shown that Morse fan is a refinement of the normal fan of the polytope P which is the Minkowski sum of two pyramids and − , where =conv e1 ek − ∑k i=1ei is the convex hull of the unit vectors e1 ek∈ IR and their negative sum.

Phase identification analysis using directionality and rectilinearity functions in three component seismograms

An efficient technique for phase identification analysis is introduced using newly defined directionality and conventional rectilinearity functions obtained from three component seismograms. The directionality function,D, is defined asD(t)=1-|eb(t s )·e1(t)| where eb(t s ) and e1(t) are vectors obtained from two different short time windows at timest=t s andt=t i , respectively of a three-component seismogram set. The vector eb(t s ) is the eigenvector associated with the largest eigenvalue of the covariance matrix whose elements are values of covariance between component time series with a short time duration covering a basis phase at timet=t s . The vector e1(t) is the eigenvector associated with the largest eigenvalue of the covariance matrix computed for another phase at timet=t i . Computation of the directionality is done for a fixed basis phase at timet=t s and other phases to be compared at variable timest=t si . The first arrival phase is generally used as the basis phase, because this phase could be the most uncontaminated among all the phases in seismic records. The rectilinearity function obtained from the ratio of the largest and the second-largest eigenvalues of a covariance matrix in three component seismograms is also used in this study to increase confidence and accuracy of phase identification. The directionality and rectilinearity functions are computed along the time points of three component seismograms, and resulting time series are compared with original seismograms by inspection. These functions are not used as filters to be applied to seismograms by multiplications as used in previous researches, in order to avoid identification ambiguity caused by mixing of the characteristics of the functions and amplitudes of seismograms. This approach has improved performances in accuracy, stability, and computing time compared with the previously existing ones and is especially suitable for identification of phases in weak or moderate seismicity regions where routine identifications with large amount of data is not necessary. The technique was applied to a set of local, regional and teleseismic data, and body wave and surface wave phases were effectively identified.

On angular coherent controlof photocurrent caused by photoionization in a bichromatic laserfield

A novel kind of coherent control of various atomic and molecularphotoprocesses proceeding in a bichromatic electromagnetic (EM) laserfield, angular coherent control, is proposed and demonstratedhere theoretically by studying the particular case of negative-ionphotodetachment in the field of a linearly polarized fundamental laserfrequency and its second harmonic of arbitrary spatial configuration andorientation of the unit polarization vectors e1 and e2 ofthe respective harmonic fields. Due to the inherent spatial asymmetry ofthe applied bichromatic field, the ejection of detached photoelectrons isalso generally asymmetric in respect of the detached photoelectronmomentum replacement p→-p. As follows from ourgauge-invariant analytical calculations within the framework ofperturbation theory on EM interaction and a zero-range potential modelfor the negative ion, this field asymmetry results in a photocurrent,the magnitude and direction of which are strongly dependent on both therelative phase shift Φ between the harmonics of the ionizingbichromatic field and on the angleθ between the polarization vectors e1 and e2. Thelatter strong dependence on the angle θ is the background for angular control of the direction and magnitude of the producedphotocurrent by means of an appropriate choice and proper adjustment ofthe angle between the polarization vectors of the incident bichromaticfield harmonics.

Non-orthogonalisable vector fields on spheres

A (k – l)-field on Sn-1 may be given as a section ϕ of the fibre bundlewith fibre Vn-1, k-1 or, equivalently, as a semi-orthogonal map, i.e., a mapwhich is isometric in the second variable and such that for the basis vector e1∈Rk and every x∈Rn

A note on pseudo-umbilical surfaces

We follow the notations and basic equations of Chen (2). Let M be a surface immersed in an m-dimensional space form Rm(c) of curvature c = 1, 0 or −1. We choose a local field of orthonormal frames e1, …, em in Rm(c) such that, restricted to M, the vectors e1, e2 are tangent to M. Let ω1, …, ωm be the field of dual frames. Then the structure equations of Rm(c) are given by

Condition that All Irreducible Representations of a Compact Lie Group, if Restricted to a Subgroup, Contain no Representation More than Once

One of the results of the theory of the irreducible representations of the unitary group in n dimensions Un is that these representations, if restricted to the subgroup Un-1 leaving a vector (let us say the unit vector e1 along the first coordinate axis) invariant, do not contain any irreducible representation of this Un-1 more than once (see [1, Chapter X and Equation (10.21)]; the irreducible representations of the unitary group were first determined by I. Schur in his doctoral dissertation (Berlin, 1901)). Some time ago, a criterion for this situation was derived for finite groups [3] and the purpose of the present article is to prove the aforementioned result for compact Lie groups, and to apply it to the theory of the representations of Un.