Additional Symmetry Properties Research Articles

Matrix stretching

Abstract We consider the tensor products of square matrices of different sizes and introduce the stretching maps, which can be viewed as a generalized matricization. Stretching maps conserve algebraic properties of the tensor product, but are not necessarily injective. Dropping the injectivity condition allows us to construct examples of stretching maps with additional symmetry properties. Furthermore, this leads to the averaging of the tensor product and possibly could be used to compress the data.

A supercritical elliptic equation in the annulus

By a combination of variational and topological techniques in the presence of invariant cones, we detect a new type of positive axially symmetric solutions of the Dirichlet problem for the elliptic equation -\Delta u + u = a(x)|u|^{p-2}u in an annulus A \subset \mathbb{R}^N ( N\ge3 ). Here p>2 is allowed to be supercritical and a(x) is an axially symmetric but possibly nonradial function with additional symmetry and monotonicity properties, which are shared by the solution u we construct. In the case where a equals a positive constant, we detect conditions, only depending on the exponent p and on the inner radius of the annulus, that ensure that the solution is nonradial.

A Hidden Symmetry of Conformally Invariant Lagrangians

In this paper a hidden extra symmetry of conformally invariant Lagrangians occuring in physics is pointed out. This symmetry is most apparent in a metric independent, i.e. in a Palatini-like presentation of the variational problem. In such presentation, the usual conformal weight of fields can be encoded as local dilatation group gauge charges. The conventional conformal invariance of Lagrangians is then equivalent to dilatation gauge invariance. The claim of the paper is, that the most commonly occurring conformally invariant Lagrangians turning up in physics are not only invariant to local dilatation gauge transformations, but they are also invariant to any change of the dilatation gauge connection, meaning an additional algebraic symmetry property. In terms of dimensional analysis and differential geometry, this additional symmetry means complete insensitivity of the Lagrangian to the choice of the parallel transport rule of local measurement units throughout spacetime.

GENERELIZED CONHARMONIC CURVATURE TENSOR OF NEARLY KAHLER MANIFOLD

In this paper we study the relationship between tensor algebraic curvature tensor, and General conharmonic curvature tensor of Nearly Kahler manifold, i. e. it has a classical symmetry properties of the Riemann carvatur tensor. Relpenishing generalized Riemannian structur of certain classes of almost Hermitian manifold allows an additional symmetry properties of this tensor.

Isoperimetric problem in $H$-type groups and Grushin spaces

We study the isoperimetric problem in H -type groups and Grushin spaces, emphasizing a relation between them. Under a symmetry assumption that depends on the dimension, we prove existence, additional symmetry, and regularity properties of isoperimetric sets.

Hidden symmetry of the beam spread function resulting from the reciprocity theorem

It is shown that the optical reciprocity theorem imposes certain constraints on the radiation field structure of a unidirectional point source (beam spread function (BSF)) in a turbid medium with spatially uniform optical properties. To satisfy the reciprocal relation, the BSF should have an additional symmetry property along with axial symmetry. This paper mathematically formulates the BSF symmetry condition that follows from the reciprocity theorem and discusses test results of some approximate analytical BSF models for their compliance with the symmetry requirement. A universal method for eliminating symmetry errors of approximate BSF models is proposed.

Additional symmetry properties of atomic states with one and two open shells

Additional symmetry properties of the ground state of an atom and quantities related with it, of the configuration with two open shells with the same orbital quantum number, and of the maximal Auger amplitudes are considered. Algebraic energy expressions for terms of the highest multiplicity and terms related with them as well as for the ground and highest levels are presented. Classification of states for the n1lN1n2lN2 configuration according to their parentage in the isoelectronic sequence of configurations and the properties of such a basis are considered. Reformulation of this basis using the isospin formalism is discussed. Existence of selection rules for the maximal Auger amplitudes is indicated.

Equivalence of quenched and annealed averaging in models of disordered polymers

The equivalence of the influence of quenched and annealed disorder on the scaling properties of long flexible polymer chains is proved by analyzing the O(m)-symmetric field theory in the polymer (de Gennes) limit m → 0. Additional symmetry properties of the model in this limit are discussed.

On nonlinear amplifier modeling and identification using baseband Volterra-Parafac models

The baseband Volterra–Parafac model is a useful tool to represent a nonlinear communication channel with a parametric complexity reduced with respect to the full Volterra model. In this paper we include additional symmetry properties of real power amplifier kernels in the equivalent baseband Volterra-Parafac approach in order to gain a further reduction in the number of parameters. To illustrate the new proposal, the parameters of the equivalent baseband Volterra–Parafac representation for a power amplifier are estimated using the complex least mean square algorithm. Comparison of the measured amplifier output and the model prediction for the case of an orthogonal frequency division multiplexing input signal demonstrates a notable model performance.

INTERPRETING THE ROLE OF THE MAGNETIC FIELD FROM DUST POLARIZATION MAPS

Dust polarization observations from the Submillimeter Array (SMA) and the Caltech Submillimeter Observatory (CSO) are analyzed with the goal of providing a general tool to interpret the role of the magnetic field in molecular clouds. Magnetic field and dust emission gradient orientations are observed to show distinct patterns and features. The angle $\delta$ between these two orientations can be interpreted as a magnetic field alignment deviation, assuming the emission gradient orientation to coincide with the density gradient orientation in the magnetohydrodynamics (MHD) force equation. In SMA high-resolution (collapsing) cores, additional symmetry properties in $\delta$ can reveal accretion and outflow zones. All these observational findings suggest the angle $\delta$ to be a relevant quantity that can assess the role of the magnetic field. Indeed, when comparing this angle with the (projection-free) magnetic field significance \Sigma_B (Koch et al. 2012a), it is demonstrated that |\delta| yields an approximation to the change in \Sigma_B. Thus, changes in the magnetic field alignment deviation $\delta$ trace changes in the role of the magnetic field. The angle $\delta$ is observationally straightforward to determine, providing a tool to distinguish between zones of minor or significant magnetic field impact. This is exemplified by the CSO M+0.25+0.01, Mon R2, CO+0.02-0.02, M-0.02-0.07 sources and by the SMA high-resolution data from W51 e2, W51 North, Orion BN/KL and g5.89. Additional CSO sources are analyzed, providing further support of this result. Finally, based on the different features found in our sample of 31 sources in total, covering sizes from large-scale complexes to collapsing cores, a schematic evolutionary scenario is proposed (abridged).

The complexity of linear tensor product problems in (anti)symmetric Hilbert spaces

We study linear problems Sd defined on tensor products of Hilbert spaces with an additional (anti)symmetry property. We construct a linear algorithm that uses finitely many continuous linear functionals and show an explicit formula for its worst case error in terms of the eigenvalues λ=(λm)m∈N of the operator W1=S1†S1 of the univariate problem. Moreover, we show that this algorithm is optimal with respect to a wide class of algorithms and investigate its complexity. We clarify the influence of different (anti)symmetry conditions on the complexity, compared to the case for the classical unrestricted problem. In particular, for symmetric problems with λ1≤1 we give characterizations for polynomial tractability and strong polynomial tractability in terms of λ and the amount of the assumed symmetry. Finally, we apply our results to the approximation problem of solutions of the electronic Schrödinger equation.

Instability of nonsymmetric nonmonotone equilibria of the Vlasov-Maxwell system

We consider the 112-dimensional relativistic Vlasov-Maxwell system that describes the time-evolution of a plasma. We find a relatively simple criterion for spectral instability of a wide class of equilibria. This class includes non-homogeneous equilibria that need not satisfy any additional symmetry properties (as was the case in previous results), nor should they be monotone in the particle energy. The criterion is given in terms of the spectral properties of two Schrödinger operators that arise naturally from Maxwell's equations. The spectral analysis of these operators is quite delicate, and some general functional analytic tools are developed to treat them. These tools can be applied to similar systems in higher dimensions, as long as their domain is finite or periodic.

Group analysis of the thin film dewetting equation

The classical Lie group analysis method in addition to constructing exact solutions, provides a regular procedure for mathematical modeling by classifying differential equations with respect to arbitrary elements. In the present paper group analysis is applied for modeling in dewetting of a thin film. The paper is focused on the equation: h t + Δ Δ h = div ( m ∇ V ( h ) ) . Group classification selects the functions m( h) and V( h) such that Eq. (2) possess additional symmetry properties. The group classification separates the set of the functions m( h) and V( h) into different classes with respect to symmetries.

On the geometry of conharmonic curvature tensor for nearly Kähler manifolds

We study additional symmetry properties for the harmonic curvature tensor of a nearly Kähler manifold. An exhaustive description for conharmonically para-Kählerian, nearly Kählerian manifolds, and conharmonically flat, nearly Kählerian manifolds is obtained.

Method of A-operators and conservation laws for the equations of gas dynamics

An algorithm is proposed which allows all conservation laws for a system of differential equations to be to obtained from its one zero-order conservation law for which the general rank of the Jacobi matrix is equal to the number of independent variables of the system. The efficiency of the algorithm is shown by examples of the equations of gas dynamics, for which new conservation laws are derived. For the equations considered, additional symmetry properties related to these conservation laws are established.

Faster Algorithms for Frobenius Numbers

The Frobenius problem, also known as the postage-stamp problem or the money-changing problem, is an integer programming problem that seeks nonnegative integer solutions to $ x_1 a_1 +\cdots + x_n a_n =M$, where $ a_i $ and $M$ are positive integers. In particular, the Frobenius number $f(A)$, where $A=\{ a_i \}$, is the largest $M$ so that this equation fails to have a solution. A simple way to compute this number is to transform the problem to a shortest-path problem in a directed weighted graph; then Dijkstra's algorithm can be used. We show how one can use the additional symmetry properties of the graph in question to design algorithms that are very fast. For example, on a standard desktop computer, our methods can handle cases where $n=10$ and $ a_1 = 10^7$. We have two main new methods, one based on breadth-first search and another that uses the number theory and combinatorial structure inherent in the problem to speed up the Dijkstra approach. For both methods we conjecture that the average-case complexity is $O( a_1 \sqrt{n} )$. The previous best method is due to Böcker and Lipták and runs in time $O( a_1 n)$. These algorithms can also be used to solve auxiliary problems such as (1) find a solution to the main equation for a given value of $M$; or (2) eliminate all redundant entries from a basis. We then show how the graph theory model leads to a new upper bound on $f(A)$ that is significantly lower than existing upper bounds. We also present a conjecture, supported by many computations, that the expected value of $f(A)$ is a small constant multiple of $ \left({1\over2} \, n! \, \Pi A\right)^{1/(n-1)} -\Sigma A$.

Split algorithms for hermitian Toeplitz matrices with arbitrary rank profile

Split algorithms for Toeplitz matrices exploit besides the Toeplitz structure additional symmetry properties to reduce the number of operations. In this paper split Levinson and Schur algorithms for hermitian Toeplitz matrices are presented that work, in contrast to previous algorithms, without additional conditions like strong nonsingularity. The main contribution is the generalization of the split Levinson-type algorithms of B. Krishna/H. Krishna and H. Krishna/S. Morgera to general nonsingular hermitian Toeplitz matrices. Furthermore, a Schur-type counterpart of this algorithm is presented that is also new in the strongly nonsingular case. Some auxiliary considerations concerning the kernel structure of hermitian Toeplitz matrices might be of independent interest.

Differential geometry of quasi-Sasakian manifolds

The full system of structure equations of a quasi-Sasakian structure is obtained. The structure of the main tensors on a quasi-Sasakian manifold (the Riemann-Christoffel tensor, the Ricci tensor, and other tensors) is studied on this basis. Interesting characterizations of quasi-Sasakian Einstein manifolds are obtained. Additional symmetry properties of the Riemann-Christoffel tensor are discovered and used for distinguishing a new class of quasi-Sasakian manifolds. An exhaustive description of the local structure of manifolds in this class is given. A complete classification (up to the -transformation of the metric) is obtained for manifolds in this class having additional properties of the isotropy kind.

Isotropic discrete orientation distributions on the 3D special orthogonal group

In modeling the linear elastic behavior of a polycrystalline material on the microscopic level, a special problem is to determine a so-called discrete orientation distributions (DODs) which satisfy the isotropy condition. A DOD is a probability measure with finite support on SO(3), the special orthogonal group in three dimensions. Isotropy of a DOD can be viewed as an invariance property of a certain moment matrix of the DOD. So the problem of finding isotropic DODs resembles that of finding weakly invariant linear regression designs. In fact, methods from matrix and group theory which have been successfully applied in linear regression design can also be utilized here to construct various isotropic DODs. Of particular interest are isotropic DODs with small support. Crystal classes with additional symmetry properties are modeled by stiffness tensors having a non-trivial symmetry group. There are six possible non-trivial symmetry groups, up to conjugation. In either cases we find isotropic DODs with fairly small support, in particular for the cubic and the transversal symmetry groups.

Ornstein-Uhlenbeck processes indexed by the circle

We consider the class of stationary, zero-mean Gaussian processes, indexed by the circle, satisfying a two-point Markov property and taking values in a vector bundle over the circle with given holonomy. We establish, subject to certain additional symmetry properties, a classification of all such processes. We then propose a construction of a Brownian motion of loops, in which these processes provide the infinitesimal increments.