N-dimensional Matrix Research Articles

Kuramoto variables as eigenvalues of unitary matrices.

We generalize the Kuramoto model by interpreting the N variables on the unit circle as eigenvalues of a N-dimensional unitary matrix U in three versions: general unitary, symmetric unitary, and special orthogonal. The time evolution is generated by N^{2} coupled differential equationsfor the matrix elements of U, and synchronization happens when U evolves into a multiple of the identity. The Ott-Antonsen ansatz is related to the Poisson kernels that are so useful in quantum transport, and we prove it in the case of identical natural frequencies. When the coupling constant is a matrix, we find some surprising new dynamical behaviors.

Descriptors of dimensionality for n × n density matrices

By using the recently introduced parametrization of an n-dimensional density matrix in terms of the indices of population asymmetry and the intrinsic coherences, we define descriptors in both integer and continuous forms of the effective dimension that take place for a complete description of a density matrix, thus providing accurate information beyond the rank of the density matrix. The concepts of dimensional folding, hidden dimensional purity, and dimensional entropy are introduced and discussed in view of the new approach presented. The results are applicable to any physical system represented by a density matrix, such as n-level quantum systems, qutrits, sets of interacting pencils of radiation, classical polarization states, and to transformations of density matrices, as occurs with quantum channels.

Minimal Active Space: NOSCF and NOSI in Multistate Density Functional Theory.

In this Perspective, we introduce a minimal active space (MAS) for the lowest N eigenstates of a molecular system in the framework of multistate density functional theory (MSDFT), consisting of no more than N2 nonorthgonal Slater determinants. In comparison with some methods in wave function theory in which one seeks to expand the ever increasing size of an active space to approximate the wave functions, it is possible to have an upper bound in MSDFT because the auxiliary states in a MAS are used to represent the exact N-dimensional matrix density function D(r). Here, we partition the total Hamiltonian matrix functional into an orbital-dependent part, including multistate kinetic energy Tms and Coulomb-exchange energy EHx plus an external potential energy ∫dr v(r)D(r), and a correlation matrix density functional . The latter accounts for the part of correlation energy not explicitly included in the minimal active space. A major difference from Kohn-Sham DFT is that state interactions are necessary to represent the N-matrix density D(r) in MSDFT, rather than a noninteracting reference state for the scalar ground-state density ρo(r). Two computational approaches are highlighted. We first derive a set of nonorthogonal multistate self-consistent-field (NOSCF) equations for the variational optimization of . We introduce the multistate correlation potential, as the functional derivative of , which includes both correlation effects within the MAS and that from the correlation matrix functional. Alternatively, we describe a nonorthogonal state interaction (NOSI) procedure, in which the determinant functions are optimized separately. Both computational methods are useful for determining the exact eigenstate energies and for constructing variational diabatic states, provided that the universal correlation matrix functional is known. It is hoped that this discussion would stimulate developments of approximate multistate density functionals both for the ground and excited states.

Multistate Density Functional Theory of Excited States.

We report a rigorous formulation of density functional theory for excited states, providing a theoretical foundation for a multistate density functional theory. We prove the existence of a Hamiltonian matrix functional of the multistate matrix density D(r) in the subspace spanned by the lowest N eigenstates. Here, D(r) is an N-dimensional matrix of state densities and transition densities. Then, a variational principle of the multistate subspace energy is established, whose minimization yields both the energies and densities of the individual N eigenstates. Furthermore, we prove that the N-dimensional matrix density D(r) can be sufficiently represented by N2 nonorthogonal Slater determinants, based on which an interacting active space is introduced for practical calculations. This work establishes that the ground and excited states can be treated on an equal footing in density functional theory.

Stability and dynamics of complex order fractional difference equations

We extend the definition of n-dimensional difference equations to complex order. We investigate the stability of linear systems defined by an n-dimensional matrix and derive the conditions for the stability of zero solution of linear systems. For the one-dimensional case, we find that the stability region, if any is enclosed by a boundary curve and we obtain a parametric equation for the same. Furthermore, we find that there is no stable region if this parametric curve is self-intersecting. Even for the real eigenvalues, the solutions can be complex and dynamics in one-dimension is richer than the case for real order. These results can be extended to n-dimensions. For nonlinear systems, we observe that the stability of the linearized system determines the stability of the equilibrium point.

Application of the Transpositions matrix for obtaining n-dimensional rotation matrices

This article proposes an algorithm for generation of N-dimensional rotation matrix R, N=m+n, m=2p, n=2q, p,q  [2, 4, 8] which rotates given N-dimensional vector X in the direction of coordinate axis x1 Algorithm uses block diagonal matrix, composed by Transpositions matrices. As practical realization article gives Matlab code of functions, which creates Householder and Transpositions matrices and V matrix for given n-dimensional vector X.

Sum of Kronecker products representation and its Cholesky factorization for spatial covariance matrices from large grids

The sum of Kronecker products (SKP) representation for spatial covariance matrices from gridded observations and a corresponding adaptive-cross-approximation-based framework for building the Kronecker factors are investigated. The time cost for constructing an n-dimensional covariance matrix is O(nk2) and the total memory footprint is O(nk), where k is the number of Kronecker factors. The memory footprint under the SKP representation is compared with that under the hierarchical representation and found to be one order of magnitude smaller. A Cholesky factorization algorithm under the SKP representation is proposed and shown to factorize a one-million dimensional covariance matrix in under 600 seconds on a standard scientific workstation. With the computed Cholesky factor, simulations of Gaussian random fields in one million dimensions can be achieved at a low cost for a wide range of spatial covariance functions.

Sources of Asymmetry and the Concept of Nonregularity of n-Dimensional Density Matrices

The information contained in an n-dimensional (nD) density matrix ρ is parametrized and interpreted in terms of its asymmetry properties through the introduction of a family of components of purity that are invariant with respect to arbitrary rotations of the nD Cartesian reference frame and that are composed of two categories of meaningful parameters of different physical nature: the indices of population asymmetry and the intrinsic coherences. It is found that the components of purity coincide, up to respective simple coefficients, with the intrinsic Stokes parameters, which are also introduced in this work, and that determine two complementary sources of purity, namely the population asymmetry and the correlation asymmetry, whose weighted square average equals the overall degree of purity of ρ. A discriminating decomposition of ρ as a convex sum of three density matrices, viz. the pure, the fully random (maximally mixed) and the discriminating component, is introduced, which allows for the definition of the degree of nonregularity of ρ as the distance from ρ to a density matrix of a system composed of a pure component and a set of 2D, 3D,… and nD maximally mixed components. The chiral properties of a state ρ are analyzed and characterized from its intimate link to the degree of correlation asymmetry. The results presented constitute a generalization to nD systems of those established and exploited for polarization density matrices in a series of previous works.

Analytic Solutions of the Eigenvalues of Mathieu’s Equation

Mathieu’s eigenvalue problem −y′′(x) + 2e_0 cos(2x)y(x) = λy(x), 0 < x < ℓ is symmetric if cos(2x) = cos(2ℓ − 2x) for ℓ = k0π, k0 ∈ N, and skew-symmetric if cos(2x) = − cos(2ℓ − 2x) for ℓ = π/2. Two typical boundary conditions are considered. When the eigenfunctions are expanded by the orthonormal bases of sine functions or cosine functions, we can derive an n-dimensional matrix eigenvalue problem, endowing with a special structure of the symmetric coefficient matrix A := [a_ij], a_ij = 0 if i + j is an odd integer. Based on it, we can obtain the eigenvalues easily and analytically. When ℓ = k_0π, k_0 ∈ N, we have a_ij = 0 if |i − j| > 2k_0. Besides the diagonal band, A has two off-diagonal bands, and furthermore, a cross band appears when k_0 ≥ 2. The product formula, the recursion formulas of characteristic functions and a fictitious time integration method (FTIM) are developed to find the eigenvalues of Mathieu’s equation.

A stabilized Lagrange multiplier finite-element method for flow in porous media with fractures

In this work we introduce a stabilized, numerical method for a multi-dimensional, discrete-fracture model (DFM) for single-phase Darcy flow in fractured porous media. In the model, introduced in an earlier work, flow in the (n − 1)-dimensional fracture domain is coupled with that in the n-dimensional bulk or matrix domain by the use of Lagrange multipliers. Thus the model permits a finite element discretization in which the meshes in the fracture and matrix domains are independent so that irregular meshing and in particular the generation of small elements can be avoided. In this paper we introduce in the numerical formulation, which is a saddle-point problem based on a primal, variational formulation for flow in the matrix domain and in the fracture system, a consistent stabilizing term which penalizes discontinuities in the Lagrange multipliers. For this penalized scheme we show stability and prove convergence. With numerical experiments we analyze the performance of the method for various choices of the penalization parameter and compare with other numerical DFM's.

N-Dimensional Matrix-Based Ontology

This article describes how the assessment of semantic similarities between word pairs is an important component of understanding text which enables processing, classifying and structuring of textual resources. For this purpose, an ontology is a powerful technique when applied to compute similarity. In this article, the authors propose a novel model to represent an ontology in which an N-dimensional matrix is applied, called an N-dimensional matrix-based ontology. This matrix-based ontology attempts to decrease the time complexity of computation. Second, a new semantic similarity measure is introduced and is performed on the N-dimensional matrix-based ontology. Third, the validation of the result of the N-dimensional matrix-based ontology is compared with related studies comparing two well-known benchmarks. The results reveal that in an N-dimensional matrix-based ontology with increasing N, the accuracy of the proposed semantic similarity measure is increased. Moreover, a matrix-based ontology decreases the time complexity when compared to a graph-based ontology.

One Modification which Increases Performance of N-Dimensional Rotation Matrix Generation Algorithm

One Modification which Increases Performance of N-Dimensional Rotation Matrix Generation Algorithm

Deconvoluting interrelationships between concentrations and chemical shifts in urine provides a powerful analysis tool

The NMR chemical shifts of a substance in a complex mixture strongly depend on the composition of the mixture itself, as many weak interactions occur that are hardly predictable. Chemical shift variability is the major obstacle to automatically assigning, and subsequently quantitating, metabolite signals in body fluids, particularly urine. Here we demonstrate that the chemical shifts of signals in urine are actually predictable. This is achieved by constructing ca. 4000 artificial mixtures where the concentrations of 52 most abundant urine metabolites—including 11 inorganic ions—are varied, to sparsely but efficiently populate an N-dimensional concentration matrix. A strong relationship is established between the concentration matrix and the chemical shift matrix, so that chemical shifts of > 90 metabolite signals can be accurately predicted in real urine samples. The concentrations of the invisible inorganic ions are also accurately predicted, along with those of albumin and of several other abundant urine components.

The Extended Generalized Adaptive Biasing Force Algorithm for Multidimensional Free-Energy Calculations

Free-energy calculations in multiple dimensions constitute a challenging problem, owing to the significant computational cost incurred to achieve ergodic sampling. The generalized adaptive biasing force (gABF) algorithm calculates n one-dimensional lists of biasing forces to approximate the n-dimensional matrix by ignoring the coupling terms ordinarily taken into account in classical ABF simulations, thereby greatly accelerating sampling in the multidimensional space. This approximation may however occasionally lead to poor, incomplete exploration of the conformational space compared to classical ABF, especially when the selected coarse variables are strongly coupled. It has been found that introducing extended potentials coupled to the coarse variables of interest can virtually eliminate this shortcoming, and, thus, improve the efficiency of gABF simulations. In the present contribution, we propose a new free-energy method, coined extended generalized ABF (egABF), combining gABF with an extended Lagrangian strategy. The results for three illustrative examples indicate that (i) egABF can explore the transition coordinate much more efficiently compared with classical ABF, eABF, and gABF, in both simple and complex cases and (ii) egABF can achieve a higher accuracy than gABF, with a root mean-squared deviation between egABF and eABF free-energy profiles on the order of kBT. Furthermore, the new egABF algorithm outruns the previous ABF-based algorithms in high-dimensional free-energy calculations and, hence, represents a powerful importance-sampling alternative for the investigation of complex chemical and biological processes.

The Problem of Countability of Highest Ordinals

In this study we use the alternate point of view on the structure of ordinals, according to which each ordinal is the union of non-intersecting foregoing segments of ordinals of equal exponentiation. Each ordinal <img width="135" height="26" src="http://article.sciencepublishinggroup.com/journal/141/1411074/image001.gif" /> is seen as the union <img width="93" height="33" src="http://article.sciencepublishinggroup.com/journal/141/1411074/image002.gif" /> for any <i>j</i>=1, <i>n</i>-1 instead traditional union of foregoing intersecting segments of ordinals of consistently increasing exponentiation <img width="86" height="32" src="http://article.sciencepublishinggroup.com/journal/141/1411074/image003.gif" />. The first form corresponds to the geometric representation of ordinal <span><img width="22" height="22" src="http://article.sciencepublishinggroup.com/journal/141/1411074/image004.gif" /> as an infinite n-dimensional matrix. According traditional formulation <img width="86" height="32" src="http://article.sciencepublishinggroup.com/journal/141/1411074/image005.gif" />, thus <img width="23" height="22" src="http://article.sciencepublishinggroup.com/journal/141/1411074/image006.gif" /> is <img width="15" height="15" src="http://article.sciencepublishinggroup.com/journal/141/1411074/image007.gif" />-countable union of countable ordinals so <img width="23" height="22" src="http://article.sciencepublishinggroup.com/journal/141/1411074/image006.gif" /> is countable. According to alternate formulation <img width="88" height="33" src="http://article.sciencepublishinggroup.com/journal/141/1411074/image008.gif" /> for any <i>n,</i> thus <img width="23" height="22" src="http://article.sciencepublishinggroup.com/journal/141/1411074/image006.gif" /> is <img width="23" height="22" src="http://article.sciencepublishinggroup.com/journal/141/1411074/image006.gif" />-union of ordinals and the findings will be different. These findings are: 1) the proof of countability of countable union of countable ordinals can not be directly or inductively transferred to its first limit <img width="23" height="22" src="http://article.sciencepublishinggroup.com/journal/141/1411074/image006.gif" />-union; 2) <img width="23" height="22" src="http://article.sciencepublishinggroup.com/journal/141/1411074/image006.gif" />seems to be the first uncountable ordinal with its power is equal to continuum; 3) the subsequent ascending degrees of <img width="15" height="15" src="http://article.sciencepublishinggroup.com/journal/141/1411074/image007.gif" />-exponentiation of <img width="23" height="22" src="http://article.sciencepublishinggroup.com/journal/141/1411074/image006.gif" />, i.e. <img width="29" height="24" src="http://article.sciencepublishinggroup.com/journal/141/1411074/image009.gif" />, <img width="33" height="27" src="http://article.sciencepublishinggroup.com/journal/141/1411074/image010.gif" />,..., correspond to consecutive <img width="22" height="24" src="http://article.sciencepublishinggroup.com/journal/141/1411074/image011.gif" />, <img width="21" height="24" src="http://article.sciencepublishinggroup.com/journal/141/1411074/image012.gif" />, <img width="22" height="24" src="http://article.sciencepublishinggroup.com/journal/141/1411074/image013.gif" />, ... cardinals; 4) from here it also follows the direct justification of continuum hypothesis. Our study shows that in the domain of transfinite sets different points of view and its findings have the legal right to coexist as Nels Bohr's principle of complementarity in physics.

Inverse moment bounds for sample autocovariance matrices based on detrended time series and their applications

In this paper, we assume that observations are generated by a linear regression model with short- or long-memory dependent errors. We establish inverse moment bounds for kn-dimensional sample autocovariance matrices based on the least squares residuals (also known as the detrended time series), where kn≪n, kn→∞ and n is the sample size. These results are then used to derive the mean-square error bounds for the finite predictor coefficients of the underlying error process. Based on the detrended time series, we further estimate the inverse of the n-dimensional autocovariance matrix, Rn−1, of the error process using the banded Cholesky factorization. By making use of the aforementioned inverse moment bounds, we obtain the convergence of moments of the difference between the proposed estimator and Rn−1 under spectral norm.

Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding

It is known that the practical use of non-Hermitian (i.e., typically, PT-symmetric) phenomenological quantum Hamiltonians H≠H† requires an efficient reconstruction of an ad hoc Hilbert-space metric Θ=Θ(H) which would render the time-evolution unitary. Once one considers just the N-dimensional matrix toy models H=H(N), the matrix elements of Θ(H) may be defined via a coupled set of N2 polynomial equations. Their solution is a typical task for computer-assisted symbolic manipulations. The feasibility of such a model-completion construction is illustrated here via a discrete square well model H=p2+V endowed with a k-parametric close-to-the-boundary interaction V. The model is shown to possess (possibly, multiply degenerate) exceptional points marking the phase transitions which are attributable, due to the exact solvability of the model at any N<∞, to the loss of the regularity of the metric. In the parameter-dependence of the energy spectrum near these singularities one encounters a broad variety of alternative, topologically non-equivalent scenarios.

Rebuilding the Great Britain Historical GIS, Part 1: Building an Indefinitely Scalable Statistical Database

The Great Britain Historical Geographical Information System (GIS) has been rebuilt around a single central table holding all statistics in one column, currently containing 14,541,491 data values. This architecture enables extremely flexible data handling, but requires that the context of each data value be captured entirely as metadata. Statistical reporting areas are defined via an ontology of administrative units, in which hierarchical relationships are compulsory while boundary polygons are optional. What a number measures is recorded via a relational implementation of the Data Documentation Initiative standard, locating each value within an n-dimensional matrix, or nCube, whose dimensions are variables such as age, gender, and occupation. The data library can be extended to additional countries or more statistical topics without adding any database tables.

A discrete fracture model for two-phase flow with matrix-fracture interaction

This article introduces a model for incompressible, two-phase flow in a porous medium with a fracture. The model is an extension to the case of two-phase flow of the model for single phase flow described in [1]. The model is a discrete fracture model in which the fractures are treated as interfaces of dimension n - 1 but in which there is fluid exchange between the fracture and the surrounding rock matrix. The matrix domain is effected by the fracture flow through a Robin type boundary condition along both sides of the fracture, while the fracture takes into account the flow in the matrix by means of a source term representing the discontinuity across the fracture of the flux. Twophase flow is modeled using the global pressure formulation in which the unknowns are the global pressure and the wetting phase saturation; see [2]. The case of different rock types in the (n - 1)-dimensional fracture domain and in the n-dimensional matrix rock domain is considered.

CONTINUOUS TRACE C*-ALGEBRAS, GAUGE GROUPS AND RATIONALIZATION

Let ζ be an n-dimensional complex matrix bundle over a compact metric space X and let Aζdenote the C*-algebra of sections of this bundle. We determine the rational homotopy type as an H-space of UAζ, the group of unitaries of Aζ. The answer turns out to be independent of the bundle ζ and depends only upon n and the rational cohomology of X. We prove analogous results for the gauge group and the projective gauge group of a principal bundle over a compact metric space X.