General Invariance Properties Research Articles

Invariance properties of limiting point processes and applications to clusters of extremes

Abstract Motivated by examples from extreme value theory, but without using the theory of regularly varying time series or any assumptions about the marginal distribution, we introduce the general notion of a cluster process as a limiting point process of returns of a certain event in a time series. We explore general invariance properties of cluster processes that are implied by stationarity of the underlying time series. Of particular interest in applications are the cluster size distributions, and we derive general properties and interconnections between the size of an inspected and a typical cluster. While the extremal index commonly used in extreme value theory is often interpreted as the inverse of a “mean cluster size”, we point out that this only holds true for the expected value of the typical cluster size, caused by an effect very similar to the inspection paradox in renewal theory.

A Frequency-Sweeping Framework for Stability Analysis of Time-Delay Systems

For time-delay systems, the asymptotic behavior analysis of the critical imaginary roots w.r.t. the infinitely many critical delays is an open problem. In order to find a general solution, we will exploit the link between the asymptotic behavior of critical imaginary roots and the asymptotic behavior of frequency-sweeping curves, from a new analytic curve perspective . As a consequence, we will establish a frequency-sweeping framework with three main results: (1) A finer (regularity-singularity) classification for time-delay systems will be obtained. (2) The general invariance property will be proved and hence the asymptotic behavior of the critical imaginary roots w.r.t. the infinitely many critical delays can be adequately studied. (3) The complete stability problem can be fully solved. Moreover, the frequency-sweeping framework is extended to cover a broader class of time-delay systems. Finally, the geometric insights of frequency-sweeping curves are investigated. Consequently, some deeper properties on the asymptotic behavior of time-delay systems and the link to frequency-sweeping curves are found.

Scale-invariant gauge theories of gravity: Theoretical foundations

We consider the construction of gauge theories of gravity, focussing in particular on the extension of local Poincaré invariance to include invariance under local changes of scale. We work exclusively in terms of finite transformations, which allow for a more transparent interpretation of such theories in terms of gauge fields in Minkowski spacetime. Our approach therefore differs from the usual geometrical description of locally scale-invariant Poincaré gauge theory (PGT) and Weyl gauge theory (WGT) in terms of Riemann–Cartan and Weyl–Cartan spacetimes, respectively. In particular, we reconsider the interpretation of the Einstein gauge and also the equations of motion of matter fields and test particles in these theories. Inspired by the observation that the PGT and WGT matter actions for the Dirac field and electromagnetic field have more general invariance properties than those imposed by construction, we go on to present a novel alternative to WGT by considering an “extended” form for the transformation law of the rotational gauge field under local dilations, which includes its “normal” transformation law in WGT as a special case. The resulting “extended” Weyl gauge theory (eWGT) has a number of interesting features that we describe in detail. In particular, we present a new scale-invariant gauge theory of gravity that accommodates ordinary matter and is defined by the most general parity-invariant eWGT Lagrangian that is at most quadratic in the eWGT field strengths, and we derive its field equations. We also consider the construction of PGTs that are invariant under local dilations assuming either the “normal” or “extended” transformation law for the rotational gauge field, but show that they are special cases of WGT and eWGT, respectively.

Homogeneous minimal polynomials with prescribed interpolation conditions

Given a compact set E E in C N + 1 \mathbb {C}^{N+1} , we consider the problem of finding a homogeneous polynomial of degree d d on C N + 1 \mathbb {C}^{N+1} which deviates the least from zero on E E with respect to the uniform norm among all those satisfying interpolation conditions of the form p ( a ) = f ( a ) p(a)=f(a) where a a belongs to a given finite subset of C N + 1 \mathbb {C}^{N+1} and f f is any function on such set. We show that this formalism enables one to recover several types of minimal polynomials previously studied and to prove a general invariance property of such minimal polynomials under polynomial mappings.

Some invariance properties of the minimum noise fraction transform

The Minimum Noise Fraction (MNF) transform is widely used in the remote sensing and image processing communities, because it is usually better than the Principal Components (PC) transform at compressing and ordering multispectral and hyperspectral images in terms of image “quality”. The MNF transform is also invariant to invertible (i.e. non-singular) linear transformations of multispectral/hyperspectral data, a property not shared by the PC transform. This general invariance property of the MNF transform is proved. Three examples of the general invariance property are provided and discussed: (i) invariance to scaling, (ii) invariance to certain types of background correction, and (iii) invariance to different types of noise.

Nonparametric tests of independence between random vectors

A nonparametric test of the mutual independence between many numerical random vectors is proposed. This test is based on a characterization of mutual independence defined from probabilities of half-spaces in a combinatorial formula of Möbius. As such, it is a natural generalization of tests of independence between univariate random variables using the empirical distribution function. If the number of vectors is p and there are n observations, the test is defined from a collection of processes R n , A , where A is a subset of { 1 , … , p } of cardinality | A | > 1 , which are asymptotically independent and Gaussian. Without the assumption that each vector is one-dimensional with a continuous cumulative distribution function, any test of independence cannot be distribution free. The critical values of the proposed test are thus computed with the bootstrap which is shown to be consistent. Another similar test, with the same asymptotic properties, for the serial independence of a multivariate stationary sequence is also proposed. The proposed test works when some or all of the marginal distributions are singular with respect to Lebesgue measure. Moreover, in singular cases described in Section 4, the test inherits useful invariance properties from the general affine invariance property.

Axiomatic homotopy theory for operads

We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced.

Singlet–triplet excitation spectrum of the CO–He complex. I. Potential surfaces and bound–bound CO(a 3Π←X 1Σ+) transitions

The interaction of He with metastable CO(a 3Π) gives rise to two adiabatic potential surfaces of reflection symmetry A′ and A″ which were calculated with the partially spin-restricted open-shell single and double excitation coupled cluster method with perturbative triples, RCCSD(T). Two diabatic potentials were constructed and fitted analytically; the appropriate form of the angular expansion functions was derived from general invariance properties. From variational calculations on these diabatic potential surfaces we obtained the quasibound vibration-rotation-spin levels of the CO–He complex in its lowest triplet state. Only the lower spin–orbit levels of this complex with approximate quantum number Ω=0 of the CO(a 3Π) monomer were found to be stable with respect to dissociation into He and triplet CO. The potential and the bound van der Waals levels of the ground state CO(X 1Σ+)–He complex were recalculated and used in combination with the triplet excited state wave functions to compute the line strengths and the bound–bound part of the singlet–triplet excitation spectrum of the CO–He complex. The spin-forbidden singlet–triplet transitions access mainly the higher spin–orbit levels with |Ω|=1, but these were found to undergo rapid predissociation. The companion Paper II explicitly studies this process, predicts the excited state lifetimes, and generates the bound-continuum part of the CO–He singlet–triplet spectrum.

Unitarity of induced representations from coisotropic quantum subgroups

We study unitarity of the induced representations from coisotropic quantum subgroups which were introduced by Ciccoli in math.QA/9804138. We define a real structure on coisotropic subgroups which determines an involution on the homogeneous space. We give general invariance properties for functionals on the homogeneous space which are sufficient to build a unitary representation starting from the induced one. We present the case of the one-dimensional quantum Galilei group where we have to use in all generality our definition of quasi-invariant functional.

Soliton solutions for a classical field theory withZ(3) symmetry

We construct a matrix scalar nonlinear fieldA by using the Yang-Mills SU(2) component fieldsAµa in 1+1 dimensions. Under quite general gauge invariance properties ofA and its Lagrangian density, this field can be given a very simple dynamical description in terms of two real scalar fields and a Z(3) discrete symmetry. As a matter of fact, what we present here is the Lagrangian formulation of these two scalar fields, put in interaction by a generic quartic polynomial term. The simplest localized solutions of finite energy and their interactions are thus deduced analytically or numerically. Quite general results concerning theA field and the shape of its potential self-energy are also included.

Chiral effects in the scattering of electrons by molecules

There are a number of scattering phenomena that depend on the "handedness" of the scattering system. In this paper we concentrate on the elastic scattering of polarized electrons by closed-shell molecules and consider mainly the property of electron optic dichroism. The effects can be fully analysed using the general invariance properties of the scattering matrix under spatial inversion, time reversal, and rotations. However, there is still considerable discussion about the physical mechanism causing the effects. Among the models proposed are two involving the spin–orbit interaction; in one the bound orbitals are perturbed, and in the other the incident electron. In this paper we show how these two approaches can be combined in a unified treatment. We set up the scattering equation including exchange and spin–orbit in a consistent manner and identify which terms contribute to the chiral effects.

Methods for estimating the autocorrelation function of complex Gaussian stationary processes

New methods for estimating the autocorrelation function (acf) of a complex Gaussian stationary process are presented. These methods are based on a general invariance property for the autocorrelation of a common class of the above processes. This property suggests estimation procedures based on magnitude hard limiting and phase quantization. The procedures are an extension of the relay estimator, currently employed for real processes. The computational cost and the general properties of the methods are discussed. In particular, some estimators especially suited for very simple implementations are considered. The performance of the complex hybrid sign estimator is evaluated and compared to that of the classical Direct estimator. The proposed methods are attractive for many applications in the field of digital signal processing.

Lattice distortion of an fcc crystal near a free surface

The problem of lattice distortion near a free surface is discussed for an fcc crystal using the general invariance and symmetry properties of the crystal potential energy. The atomic force tensors were determined assuming a C 4v symmetry in the surface region, when the atoms are in the ideal lattice configuration. All the anharmonic effects are neglected. From invariance requirements it results that the static displacements are directed along the normal to the surface. They can be calculated using the Green's function method. Detailed calculations were done for a finite slab in the nearest neighbour approximation.

Identities and conservation-law generators resulting from general invariance properties of classical field theories

The principal identities, superpotentials, and conservation-law generators of classical field theories, resulting from certain invariances of the action principles of these theories, are constructed by a known method. This method is reviewed and applied in greater formal generality than before, enabling the results to include, for example, co-ordinate, tetrad, and spinor trasformations as particular cases. The distinction between the invariance group of a theory and the symmetry properties of classes of particular field structures, which is vital in the construction of significant conservation laws, is discussed. The use of symmetries in conjunction with conservation-law generators is described. Further, a way of using the identities to identify useful symmetry demands is pointed out and illustrated by an example involving tetrad transformations in general relativity. In another example, a general invariance group is described which shows how the full (uncontracted) Bianchi identities may be viewed as arising from an action principle which includes that of general relativity in an especially simple way.

Odd Nuclei, the particle-phonon interaction and the conservation laws

It is shown that the particle-phonon interaction tends to restore to an odd-nucleus collective Hamiltonian the general invariance properties lost in the Hartree-Fock approximation.

A Variational Formulation of Standard General Relativity which Includes the Full Bianchi Identities in Consequence of an Extended General Invariance Property

AbstractThe central properties of the usual variational formulations of EINSTEIN's general theory of relativity are sketched and a more general variational formulation is introduced, which is more appropriate to the geometric foundations of the theory. In particular, this formulation leads to the BIANCHI identities in their non‐contracted form following in consequence of a general invariance property. This invariance property of the general variational principle relates to transformations (not, in general, coordinate transformations) that contain arbitrary fifth (or third) order tensor “generators”. These results can be interpreted as implying that the variational principle introduced here admits an “extended principle of general covariance” (i.e., a covariance principle more general than the usual principle relating to general coordinate covariance). Some of the formal implications of these results as well as their connection with general coordinate covariance are discussed briefly. In particular these results point to the existence of a fundamental transformation theory connecting all of the RIEMANNian spacetimes of general relativity.

Implications of Approximate SU3Symmetry and Mass Formulas for the Mesons

The connection between the limit of perfect SU3 symmetry and the zero mass approximation for the pseudoscalar meson octet is discussed. The well-known quadratic mass formula for the pseudoscalar meson octet and the linear mass formula for the baryons are derived. A simple model of strong interaction is presented which leads to two mass formulas for the nine spin-1 mesons: 2Mφ+Mρ+Mω=4MK* and Mρ≅Mω. The general invariance property of this model which contains, among others, a baryon octet, a pseudoscalar octet, and a vector nonet is examined.

Remarks Concerning Reciprocity in Quantum Mechanics

A new derivation of the reciprocity theorem is given. The general invariance property of the Hamiltonian which leads to symmetry of the Green's function for a quantum mechanical system is exhibited. It is found that reciprocity does not necessarily imply Hermiticity of the Hamiltonian, so that the ``complex optical model potential,'' for example, satisfies the reciprocity relations. The concept of reciprocity is then generalized to include a somewhat wider class of symmetry properties. Some properties of antiunitary transformations are discussed.

On the transformation properties of strong interactions

By extending the procedure outlined in previous papers of grouping several baryon states to form many-component spinors obeying symmetry properties which are not apparent for the separate states, a general Dirac equation satisfied by a 32-component spinor including all the known baryon states is proposed and its properties are discussed. The baryon states, apart from the normal space time co-ordinates, are described by two kinds of independent internal parameters: the isospin variables and the hypercharge variables. The K interactions are expressed by two independent terms with two different interaction constantsF andF′, each of which is invariant for rotations in a 4 dimensional hypercharge space, and the pion interactions by the usual expression invariant for rotations in a 3 dimensional isospin space, quite independent from the hypercharge space; and finally electromagnetic interactions are formulated by the combination of two terms which allow to obtain the experimentally known charge labellings of the different baryon states when a two step separation process is applied to the equation, which leads for the K transitions to selection rules expressing naturally the conservation of hypercharge or strangeness as it is formulated in the doublet approximation. The combination of the two K interaction terms allows further to obtain two kinds of coupling constantsF+F′ andF -F′ for the different interaction terms, disposed in such a way as to explain the observed different self masses of the baryons. It is further shown that the 32-component equation is invariant under boson, spinor and charge conjugation operations separately, which are rigorously valid even when the different baryon masses are included. Finally some aspects of the γ5 transformation properties of the equation are discussed. The present approach seems therefore adequate to unify under few common points-of-view the treatment of the baryon states and to put into evidence some general invariance properties which otherwise should not be revealed.

Scattering of Polarized Nucleon Beams

The polarization formulas for the general reaction $a+b\ensuremath{\rightarrow}c+d$ are given in a compact form and specialized for the scattering of polarized nucleon beams at unpolarized targets. These targets may have arbitrary spin, possibly being different before and after collision. An interesting quantity in this problem is the polarization of the scattered nucleon beam. On the basis of general invariance properties of the transition matrix, this polarization is expressed by the polarization of the incoming nucleon beam and the relative momenta before and after collision. The invariant coefficients in this relation are functions of energy and scattering angle only; they are given in terms of the parameters of the transition matrix.By using these formulas, it is shown that with triple scattering experiments one can obtain two new relations between the parameters of the transition matrix at fixed energy and angle. Quadruple scattering leads to two further relations. These relations represent information in addition to the differential cross section and the polarization resulting from unpolarized beams. The results are specialized for targets of spin zero and spin one-half, where in the latter case also the scattering of identical particles is discussed briefly.