Direct Product Structure Research Articles

Use of indicator functions to enumerate cross-array designs without direct product structure

Use of indicator functions to enumerate cross-array designs without direct product structure

Direct product-type grid representations for angular coordinates in extended space and their application in the MCTDH approach.

Multi-configurational time-dependent Hartree (MCTDH) calculations using time-dependent grid representations can be used to accurately simulate high-dimensional quantum dynamics on general ab initio potential energy surfaces. Employing the correlation discrete variable representation, sets of direct product type grids are employed in the calculation of the required potential energy matrix elements. This direct product structure can be a problem if the coordinate system includes polar and azimuthal angles that result in singularities in the kinetic energy operator. In the present work, a new direct product-type discrete variable representation (DVR) for arbitrary sets of polar and azimuthal angles is introduced. It employs an extended coordinate space where the range of the polar angles is taken to be [-π, π]. The resulting extended space DVR resolves problems caused by the singularities in the kinetic energy operator without generating a very large spectral width. MCTDH calculations studying the F·CH4 complex are used to investigate important properties of the new scheme. The scheme is found to allow for more efficient integration of the equations of motion compared to the previously employed cot-DVR approach [G. Schiffel and U. Manthe, Chem. Phys. 374, 118 (2010)] and decreases the required central processing unit times by about an order of magnitude.

Emerging internal symmetries from effective spacetimes

Can global internal and spacetime symmetries be connected without supersymmetry? To answer this question, we investigate Minkowski spacetimes with d space-like extra dimensions and point out under which general conditions external symmetries induce internal symmetries in the effective 4-dimensional theories. We further discuss in this context how internal degrees of freedom and spacetime symmetries can mix without supersymmetry in agreement with the Coleman–Mandula theorem. We present some specific examples which rely on a direct product structure of spacetime such that orthogonal extra dimensions can have symmetries which mix with global internal symmetries. This mechanism opens up new opportunities to understand global symmetries in particle physics.

Toric 2-Fano manifolds and extremal contractions

We show that for a projective toric manifold with the ample second Chern character, if there exists a Fano contraction, then it is isomorphic to the projective space. For the case that the second Chern character is nef, the Fano contraction gives either a projective line bundle structure or a direct product structure. We also show that, for a toric weakly 2-Fano manifold, there does not exist a divisorial contraction to a point.

Assessing the utility of phase-space-localized basis functions: Exploiting direct product structure and a new basis function selection procedure.

In this paper we show that it is possible to use an iterative eigensolver in conjunction with Halverson and Poirier's symmetrized Gaussian (SG) basis [T. Halverson and B. Poirier, J. Chem. Phys. 137, 224101 (2012)] to compute accurate vibrational energy levels of molecules with as many as five atoms. This is done, without storing and manipulating large matrices, by solving a regular eigenvalue problem that makes it possible to exploit direct-product structure. These ideas are combined with a new procedure for selecting which basis functions to use. The SG basis we work with is orders of magnitude smaller than the basis made by using a classical energy criterion. We find significant convergence errors in previous calculations with SG bases. For sum-of-product Hamiltonians, SG bases large enough to compute accurate levels are orders of magnitude larger than even simple pruned bases composed of products of harmonic oscillator functions.

A ROLE OF SYMBOLS OF MINIMUM TYPE IN EXPONENTIAL CALCULUS

Abstract. We introduce formal symbols of product type and of minimumtype and show that the formal power series representation for e p is a formalsymbol of product type, where p is a formal symbol of minimum type. 1. IntroductionPseudodi erential operators are essential for studying the theory of partialdi erential equations and itself have been studied in various respects. A pseudo-di erential operator can be represented as an equivalent class of some functionde ned on the cotangent bundle. We call this function a symbol of pseudodi er-ential operator. We can represent the operations of addition, scalar multiplica-tion, composition, formal adjoint and change of variables of pseudodi erentialoperators by using these symbols and so can treat pseudodi erential operatorsconcretely. Symbolic calculus means calculating pseudodi erential operatorsby using symbols. Exponential calculus means symbolic calculus on pseudodif-ferential operators expressed as symbols of exponential functions. A symbol ofexponential function means that the symbol can be expressed as a form of expo-nential function. We can analyze (pseudo) di erential operators of in nite orderor (pseudo) di erential equations of in nite order by studying the pseudodi er-ential operators expressed as symbols of exponential functions. In particular,Sato, M., T. Kawai and M. Kashiwara (ref. [16]) won epoch-making results inthe theory of transformation of the system of linear partial (pseudo) di eren-tial equations by using pseudodi erential operators of in nite order. T. Aokiaccomplished exponential calculus of analytic pseudodi erential operators( ref.[1] { [8]). The author considers the case of a kind of the direct product structure(ref. [12] { [15]). That is, we introduce and study calculus of analytic pseudo-di erential operators of product type and generalize the theory of T. Aoki insome sense. This study is deeply connected with exponential calculus of posi-tive de nite operators of in nite order which have deep relation to the energymethod in the hyperfunction theory(ref. [9] { [11]). In this article, we introduce

Geometric properties of a2Dspacetime arising in4Dblack hole physics

The Schwarzschild exterior space-time is conformally related to a direct product space-time, $\mathcal{M}_2 \times S_2$, where $\mathcal{M}_2$ is a two-dimensional space-time. This direct product structure arises naturally when considering the wave equation on the Schwarzschild background. Motivated by this, we establish some geometrical results relating to $\mathcal{M}_2$ that are useful for black hole physics. We prove that $\mathcal{M}_2$ has the rare property of being a causal domain. Consequently, Synge's world function and the Hadamard form for the Green function on this space-time are well-defined globally. We calculate the world function and the van Vleck determinant on $\mathcal{M}_2$ numerically and point out how these results will be used to establish global properties of Green functions on the Schwarzschild black hole space-time.

Euler-Poincaré equations on semi-direct products

We study the Euler-Poincare equations on the semi-direct products of the group of orientation preserving diffeomorphisms of the circle with itself. We establish a reduction result to the direct product structure which will allow us to investigate the well-posedness, in the smooth category, using geodesic flows on infinite dimensional Lie groups.

Entanglement entropy in gauge theories and the holographic principle for electric strings

We consider quantum entanglement between gauge fields in some region of space A and its complement B. It is argued that the Hilbert space of physical states of gauge theories cannot be decomposed into a direct product H A ⊗ H B of Hilbert spaces of states localized in A and B. The reason is that elementary excitations in gauge theories—electric strings—are associated with closed loops rather than points in space, and there are closed loops which belong both to A and B. Direct product structure and hence the reduction procedure with respect to the fields in B can only be defined if the Hilbert space of physical states is extended by including the states of electric strings which can open on the boundary of A. The positions of string endpoints on this boundary are the additional degrees of freedom which also contribute to the entanglement entropy. We explicitly demonstrate this for the confining and deconfining phases of the three-dimensional Z 2 lattice gauge theory both numerically and using a simple trial ground state wave function. The entanglement entropy appears to be saturated almost completely by the entropy of string endpoints, thus reminding of a “holographic principle” in quantum gravity and AdS/CFT correspondence.

APPROXIMATING HYPERCUBES BY INDEX-SHUFFLE GRAPHS VIA DIRECT-PRODUCT EMULATIONS

Index-shuffle graphs are a family of bounded-degree hypercube-like interconnection networks for parallel computers, introduced by [Baumslag and Obrenić (1997): Index-Shuffle Graphs, …], as an efficient substitute for two standard families of hypercube derivatives: butterflies and shuffle-exchange graphs. In the theoretical framework of graph embedding and network emulations, this paper shows that the index-shuffle graph efficiently approximates the direct-product structure of the hypercube, and thereby has a unique potential to approximate efficiently all of its derivatives. One of the consequences of our results is that any member of the following group of standard bounded-degree hypercube derivatives: butterflies, shuffles, tori, meshes of trees, is emulated by the index-shuffle graph with a slowdown in the order of the logarithm of the slowdown of the most efficient emulation achieved by any other member of this group. Emulation algorithms are presented where the emulation host is the n-dimensional index-shuffle graph Ψn, having N=2n nodes. The emulated graph G is a direct product of the form: G=F0×F1×⋯×Fk-1 where k is a power of 2, and each factor Fi is an instance of any of the following three graph families: cycle, complete binary tree, X-tree. Let the size of each factor be |Fi|≤2nf, where k·nf≤n. The index-shuffle graph Ψn, emulates any factor Fi in the product G with slowdown: O( log k) + O( log nf), which is O( log n) = O( log log N). Any collection of 2ℓ copies of the product G, such that: ℓ+k·nf≤n is emulated by the index-shuffle graph Ψn simultaneously, without any additional slowdown. Relaxing the assumption that k is a power of 2 introduces an additional factor of O( lg *N) into the slowdown.

Transfer Matrix Functional Relations for the Generalized 2(tq) Model

The $N$-state chiral Potts model in lattice statistical mechanics can be obtained as a ``descendant'' of the six-vertex model, via an intermediate ``$Q$'' or ``$\tau_2 (t_q)$'' model. Here we generalize this to obtain a column-inhomogeneous $\tau_2 (t_q)$ model, and derive the functional relations satisfied by its row-to-row transfer matrix. We do {\em not} need the usual chiral Potts relations between the $N$th powers of the rapidity parameters $a_p, b_p, c_p, d_p$ of each column. This enables us to readily consider the case of fixed-spin boundary conditions on the left and right-most columns. We thereby re-derive the simple direct product structure of the transfer matrix eigenvalues of this model, which is closely related to the superintegrable chiral Potts model with fixed-spin boundary conditions.

Soft supersymmetry breaking masses in a unified model with doublet-triplet splitting

We study soft supersymmetry breaking parameters in a supersymmetric unified model which potentially solves the doublet-triplet splitting problem. In the model the doublet-triplet splitting is solved by the discrete symmetry which is allowed to be introduced due to the direct product structure of the gauge group. The messenger fields for the gauge mediated supersymmetry breaking are naturally embedded in the model. The discrete symmetry required by the doublet-triplet splitting makes the gaugino masses non-universal and also induces a different mass spectrum for the scalar masses from the ordinary minimal gauge mediation model. Independent physical CP phases can remain in the gaugino sector even after the R-transformation.

Essential imposition of Neumann condition in Galerkin–Legendre elliptic solvers

A new Galerkin–Legendre direct spectral solver for the Neumann problem associated with Laplace and Helmholtz operators in rectangular domains is presented. The algorithm differs from other Neumann spectral solvers by the high sparsity of the matrices, exploited in conjunction with the direct product structure of the problem. The homogeneous boundary condition is satisfied exactly by expanding the unknown variable into a polynomial basis of functions which are built upon the Legendre polynomials and have a zero slope at the interval extremes. A double diagonalization process is employed pivoting around the eigenstructure of the pentadiagonal mass matrices in both directions, instead of the full stiffness matrices encountered in the classical variational formulation of the problem with a weak natural imposition of the derivative boundary condition. Nonhomogeneous Neumann data are accounted for by means of a lifting. Numerical results are given to illustrate the performance of the proposed spectral elliptic solver. The algorithm extends easily to the three-dimensional problem.

The direct product structure of atomic orbital manifolds: extension to non-Euclidean permutation groups

Atomic orbital manifolds are described as Γ × Γ direct products of degenerate irreducible representations (irreps) of odd dimension in suitable groups, which may be either the polyhedral symmetry point groups (T, O, I) or larger simple non-Euclidean permutation groups containing the polyhedral symmetry point groups as subgroups. Such larger non-Euclidean groups include the simple pollakispolyhedral groups 7 O [≈L 2(7)] and 11 I [≈L 2(11)] of orders 168 and 660, respectively, and the simple alternating groups A 6 and A 7 of orders 360 and 2520, respectively. Thus the 9-dimensional (9D) direct product T × T of a 3D irrep of the polyhedral or larger point groups leads to the nine-orbital sp3d5 manifold. In icosahedral symmetry I this 9D direct product splits into separate irreps for the s, p and d orbitals, whereas in the larger non-Euclidean 7 O group this direct product splits only into a 6D irrep for the gerade orbitals (s + d) and a 3D irrep for the ungerade p orbitals. A similar method can be used to study the 25-orbital sp3d5f7g9 manifold as H × H direct products in the icosahedral group I and the two larger simple non-Euclidean groups A 6 and 11 I, in which I is a subgroup of index 6 and 11, respectively. The mathematics implicit in these direct products appears to have some connections with a few areas of chemical physics including icosahedral quasicrystals and unusual degeneracies in the Coulomb energies of terms in transition metal atomic spectra.

Spectral influence matrix for vorticity without corner pathology

This paper describes a spectral method for the uncoupled solution of the vorticity and stream function equations in which two-dimensional integral conditions for the vorticity are imposed according to the Glowinski–Pironneau method. A nonsingular influence matrix is obtained which allows to determine the trace of vorticity under no-slip conditions along the entire boundary of a rectangular domain. The method is based on a Galerkin formulation using Legendre polynomials in both directions and fully exploits the direct product structure of the two equations of the Stokes time-discretized problem. Numerical tests for the driven cavity problem are presented to demonstrate that no theoretical or numerical difficulty arises in the proposed 2D spectral approximation, which does not require any a priori regularization of the boundary condition at the corners. The algorithm is found to guarantee convergence to the correct solution for steady and unsteady problems, although spectral accuracy cannot be achieved due to the expected Gibbs' phenomenon.

Triangular norms on product lattices

In this paper, triangular norms (t-norms) are studied in the general setting of bounded partially ordered sets, with emphasis on finite chains, product lattices and the real unit square. The sets of idempotent elements, zero divisors and nilpotent elements associated to a t-norm are introduced and related to each other. The Archimedean property of t-norms is discussed, in particular its relationship to the diagonal inequality. The main subject of the paper is the direct product of t-norms on product posets. It is shown that the direct product of t-norms without zero divisors is again a t-norm without zero divisors. A weaker version of the Archimedean property is presented and it is shown that the direct product of such pseudo-Archimedean t-norms is again pseudo-Archimedean. A generalization of the cancellation law is presented, in the same spirit as the definition of the set of zero divisors. It is shown that the direct product of cancellative t-norms is again cancellative. Direct products of t-norms on a product lattice are characterized as t-norms with partial mappings that show some particular morphism behaviour. Finally, it is shown that in the case of the real unit square, transformations by means of an automorphism preserve the direct product structure.

Symmetry methods in collisionless many-body problems

We formulate an appropriate symmetry context for studying periodic solutions to equal-mass many-body problems in the plane and 3-space. In a technically tractable but unphysical case (attractive force a smooth function of squared distance, bodies permitted to coincide) we apply the equivariant Moser-Weinstein Theorem of Montaldiet al. to prove the existence of various symmetry classes of solutions. In so doing we expoit the direct product structure of the symmetry group and use recent results of Dionneet al. on ‘C-axial’ isotropy subgroups. Along the way we obtain a classification of C-axial subgroups of the symmetric group. The paper concludes with a speculative analysis of a three-dimensional solution to the 2n-body problem found by Davieset al. and some suggestion for further work.

Recursive Residuals for Multivariate Field Processes

AbstractVector recursive residuals are developed for multivariate regression models on a field. A vector response variable is observed at points on a rectangular grid, together with regression variables measured at the same points. Neighbouring values of the response vector may be correlated and simple models are considered using a direct product structure for the variance matrix. Subsequent to obtaining vector recursive residuals principal component analysis is applied to obtain an evaluation of any changes that may be occurring in the regression relationship over the field. The method is then applied to the problem of detecting zones of bush fire damage and recovery from LANDSAT data.

Some underlying manifolds in the Schwarzschild solution

This is a comment on the theorem given by Clarke (1987) which states that a spherically symmetric spacetime could topologically be not only a direct product of a typical orbit of the SO(3) group and a two-dimensional orthogonal space. Here several examples of spherically symmetric spacetimes with manifolds that do not have a direct product structure are constructed as further evidence of the result of Clarke. These spacetimes are based on the vacuum Schwarzchild solution in various standard coordinate systems.

Some remarkable spin-1/2-like algebraic properties of spin-3/2 matrices

Using for spin-3/2 matrices a direct-product structure involving the usual Pauli spin matrices, the authors derive the Dirac-Clifford matrices in terms of certain algebraic combinations of spin-3/2 matrices in a representation-independent way, thus achieving an extension of the Pauli spin matrices from the usual spin-1/2 space to the spin-3/2 space. Basing the derivation directly on this analysis, two algebras satisfied by spin-3/2 matrices are derived. One of these, which is also satisfied by spin-1/2 matrices, is directly related to the spin-3/2 algebras of Weaver (1978) and of Bhabha and Madhava Rao (for three objects). The other algebra is new and curiously is not satisfied by spin-1/2 matrices.