Invariant Matrix Research Articles

Stable Algebraic Surfaces for 3D Object Representation

Linear fitting techniques by implicit algebraic models usually suffer from global stability problems. Ridge regression regularization can be used to improve the stability of algebraic surface fits. In this paper a Euclidean Invariant 3D ridge regression matrix is developed and applied to a particular linear algebraic surface fitting method. Utilization of such a regularization in fitting process makes it possible to globally stabilize 3D object fits with surfaces of any degree. Robustness to noise and moderate levels of occlusion has also been enhanced significantly. Experimental results are presented to verify the improvements in global stability and robustness of the resulting fits.

Controlling theN- andS-representability of the second-order reduced density matrix: The doublet-state case

A purification procedure that simultaneously corrects the $N$- and $S$-representability main defects of a second-order reduced density matrix (2RDM), second-order hole reduced density matrix (2HRDM), and second-order $\mathcal{G}$ matrix is presented here. In this purifying procedure, the generalized unitarily invariant second-order matrix decomposition [D. R. Alcoba, Int. J. Quantum Chem. 97, 776 (2004)] for the 2RDM and the 2HRDM as well as for the $\mathcal{G}$ matrix is combined with the $S$-representability conditions. In particular, here we will focus our attention on the RDMs corresponding to doublet states. We will thus explicitly give the $S$-representability conditions that a two-body correlation matrix has to satisfy when an $N$-electron system is in a doublet spin-state in the spin-component of maximum projection. Furthermore, as a consequence of the $\mathcal{G}$-matrix spin properties (which directly affect the $S$-representability of the 2RDM), we show that a different contracting form for the 2RDM is possible. The numerical results presented in this work confirm the efficiency of our purifying procedure.

Finite-size effects for dyonic giant magnons

We compute finite-size corrections to dyonic giant magnons in two ways. One is by examining the asymptotic behavior of helical strings of [K. Okamura, R. Suzuki, A perspective on classical strings from complex sine-Gordon solitons, Phys. Rev. D 75 (2007) 046001, hep-th/0609026] as elliptic modulus k goes to unity, and the other is by applying the generalized Lüscher formula for μ-term of [R.A. Janik, T. Łukowski, Wrapping interactions at strong coupling—the giant magnon, Phys. Rev. D 76 (2007) 126008, arXiv: 0708.2208 [hep-th]] to the situation in which incoming particles are boundstates. By careful choice of poles in the s u ( 2 | 2 ) 2 -invariant S-matrix, we find agreement of the two results, which makes possible to predict the (leading) finite-size correction for dyonic giant magnons to all orders in the 't Hooft coupling.

Globalization of the partial isometries of metric spaces and local approximation of the group of isometries of Urysohn space

We prove the equivalence of the two important facts about finite metric spaces and universal Urysohn metric spaces U , namely Theorems A and G: Theorem A (Approximation): The group of isometry ISO ( U ) contains everywhere dense locally finite subgroup; Theorem G (Globalization): For each finite metric space F there exists another finite metric space F ¯ and isometric imbedding j of F to F ¯ such that isometry j induces the imbedding of the group monomorphism of the group of isometries of the space F to the group of isometries of space F ¯ and each partial isometry of F can be extended up to global isometry in F ¯ . The fact that Theorem G, is true was announced in 2005 by author without proof, and was proved by S. Solecki in [S. Solecki, Extending partial isometries, Israel J. Math. 150 (2005) 315–332] (see also [V. Pestov, The isometry group of the Urysohn space as a Lévy group, Topology Appl. 154 (10) (2007) 2173–2184; V. Pestov, A theorem of Hrushevski–Solecki–Vershik applied to uniform and coarse embeddings of the Urysohn metric space, math/0702207]) based on the previous complicate results of other authors. The theorem is generalization of the Hrushevski's theorem about the globalization of the partial isomorphisms of finite graphs. We intend to give a constructive proof in the same spirit for metric spaces elsewhere. We also give the strengthening of homogeneity of Urysohn space and in the last paragraph we gave a short survey of the various constructions of Urysohn space including the new proof of the construction of shift invariant universal distance matrix from [P. Cameron, A. Vershik, Some isometry groups of Urysohn spaces, Ann. Pure Appl. Logic 143 (1–3) (2006) 70–78].

Euler characteristic heuristic for approximating the distribution of the largest eigenvalue of an orthogonally invariant random matrix

The Euler characteristic heuristic has been proposed as a method for approximating the upper tail probability of the maximum of a random field with smooth sample path. When the random field is Gaussian, this method is proved to be valid in the sense that the relative approximation error is exponentially smaller. However, very little is known about the validity of the method when the random field is non-Gaussian. In this paper, as a milestone to developing the general theory about the validity of the Euler characteristic heuristic, we examine the Euler characteristic heuristic for approximating the distribution of the largest eigenvalue of an orthogonally invariant non-Gaussian random matrix. In this particular example, if the probability density function of the random matrix converges to zero sufficiently fast at the boundary of its support, the approximation error of the Euler characteristic heuristic is proved to be small and the approximation is valid. Moreover, for several standard orthogonally invariant random matrices, the approximation formula for the distribution of the largest eigenvalue and its asymptotic error are obtained explicitly. Our formulas are practical enough for the purpose of numerical calculations.

The Steepest Descent Method for Orthogonal Polynomials on the Real Line with Varying Weights

We obtain Plancherel–Rotach-type asymptotics valid in all regions of the complex plane for orthogonal polynomials with varying weights of the form on the real line, assuming that V has only two Lipschitz continuous derivatives and that the corresponding equilibrium measure has typical support properties. As an application, we extend the universality class for bulk and edge asymptotics of eigenvalue statistics in unitary invariant Hermitian random matrix theory. We develop a new technique of asymptotic analysis for matrix Riemann–Hilbert problems with nonanalytic jump matrices suitable for analyzing such problems even near transition points where the solution changes from oscillatory to exponential behavior.

Critical Edge Behavior in Unitary Random Matrix Ensembles and the Thirty-Fourth Painlevé Transcendent

We describe a new universality class for unitary invariant random matrix ensembles. It arises in the double scaling limit of ensembles Z(n,N)(-1)vertical bar det M vertical bar (2 alpha)e(-NTrV(M)) dM, with alpha > -1/2, defined on n x n Hermitian matrices M. Assuming that the limiting mean eigenvalue density is regular and that the origin is a right endpoint of its support, we compute the limiting eigenvalue correlation kernel in the double scaling limit as n, N -> infinity such that n(2/3)(n/N-1) = O(1). We use the Deift-Zhou steepest descent method for the Riemann-Hilbert problem for polynomials orthogonal with respect to the weight vertical bar x vertical bar(2 alpha)e(-NV(x)). Our main attention is on the construction of a local parametrix near the origin by means of the psi-functions associated with a distinguished solution of the Painleve XXXIV equation.

Power-law eigenvalue density, scaling, and critical random-matrix ensembles

We consider a class of rotationally invariant unitary random matrix ensembles where the eigenvalue density falls off as an inverse power law. Under a scaling appropriate for such power-law densities (different from the scaling required in Gaussian random matrix ensembles), we calculate exactly the two-level kernel that determines all eigenvalue correlations. We show that such ensembles belong to the class of critical ensembles.

The antibody responses to swine influenza virus (SIV) recombinant matrix 1 (rM1), matrix 2 (M2), and hemagglutinin (HA) proteins in pigs with different SIV exposure

The influenza invariant matrix 2 (M2) protein is a potential subunit vaccine candidate to induce protective immunity against broader strains of influenza A viruses (IAV). Antibodies to M2 protein have not been well characterized in IAV natural hosts. To characterize M2-specific antibodies in pigs, an ELISA to the extracellular region of the M2 (M2e) protein was developed. Sera from pigs experimentally infected with three different swine influenza virus (SIV) subtypes, immunized with an SIV inactivated vaccine, or positive for SIV maternally derived antibodies (MDA) in the absence of SIV infection were tested in assay. Confirmation of antibody titer status of pigs, was determined using a hemagglutination-inhibition (HI) test and the presence of antibodies to matrix 1 (M1) protein was measured by a recombinant M1 (rM1)-based ELISA. The antibody titers to the HA and M2e proteins but not to the rM1 were directly correlated to the dose of virus used to infect the pigs and the level of antibodies detected by the HI assay varied according to SIV subtype. Pigs experimentally infected with SIV produced low levels of M2e antibodies compared to antibodies detected by the HI and rM1 assays. Vaccination alone followed by infection did not increase the levels of M2e antibodies in contrast to HA and rM1 antibodies. Pigs with MDA had different levels of HA antibodies and were positive to M2e antibodies, but results were not correlated to HA antibodies levels and inconsistently present.

The Zamolodchikov–Faddeev Algebra for AdS5× S5superstring

We discuss the Zamolodchikov-Faddeev algebra for the superstring sigma-model on AdS_5 x S^5. We find the canonical su(2|2)^2 invariant S-matrix satisfying the standard Yang-Baxter and crossing symmetry equations. Its near-plane-wave expansion matches exactly the leading order term recently obtained by the direct perturbative computation. We also show that the S-matrix obtained by Beisert in the gauge theory framework does not satisfy the standard Yang-Baxter equation, and, as a consequence, the corresponding ZF algebra is twisted. The S-matrices in gauge and string theories however are physically equivalent and related by a non-local transformation of the basis states which is explicitly constructed.

The large [formula omitted] limit of exceptional Jordan matrix models and [formula omitted] theory

The large N → ∞ limits of the exceptional F 4 , E 6 Jordan matrix models of Smolin and Ohwashi lead to novel Chern–Simons membrane Lagrangians which are suitable candidates for providing a nonperturbative bosonic formulation of M theory in D = 27 real and complex dimensions, respectively. Freudenthal algebras and triple Freudenthal products permit the construction of a novel E 7 × S U ( N ) invariant matrix model whose large N limit yields generalized nonlinear sigma model actions on 28-complex-dimensional backgrounds associated with a 56-real-dimensional phase space realization of the Freudenthal algebra. We argue as to why the latter matrix model, in the large N limit, might be the proper arena for a bosonic formulation of F theory. Finally, we display generalized Dirac–Nambu–Goto membrane actions in terms of 3×3×3 cubic matrix entries that match the numbers of degrees of freedom of the 27-dimensional exceptional Jordan algebra J 3 [ 0 ] .

MatG Random Matrices

A class of infinitely divisible covariance mixtures of Gaussian random matrices is introduced, and a characterization within the class of infinitely divisible left-orthogonally invariant matrix distributions is proved.

Arbitrary unitarily invariant random matrix ensembles and supersymmetry

We generalize the supersymmetry method in Random Matrix Theory to arbitrary rotation invariant ensembles. Our exact approach further extends a previous contribution in which we constructed a supersymmetric representation for the class of norm-dependent Random Matrix Ensembles. Here, we derive a supersymmetric formulation under very general circumstances. A projector is identified that provides the mapping of the probability density from ordinary to superspace. Furthermore, it is demonstrated that setting up the theory in Fourier superspace has considerable advantages. General and exact expressions for the correlation functions are given. We also show how the use of hyperbolic symmetry can be circumvented in the present context in which the non-linear sigma model is not used. We construct exact supersymmetric integral representations of the correlation functions for arbitrary positions of the imaginary increments in the Green functions.

Limiting laws of linear eigenvalue statistics for Hermitian matrix models

We study the variance and the Laplace transform of the probability law of linear eigenvalue statistics of unitary invariant Matrix Models of n×n Hermitian matrices as n→∞. Assuming that the test function of statistics is smooth enough and using the asymptotic formulas by Deift et al. [ Commun. Pure Appl. Math.52, 1325–1425 (1999)] for orthogonal polynomials with varying weights, we show first that if the support of the Density of States of the model consists of q≥2 intervals, then in the global regime the variance of statistics is a quasiperiodic function of n as n→∞ generically in the potential, determining the model. We show next that the exponent of the Laplace transform of the probability law is not, in general, 1∕2× variance, as it should be if the Central Limit Theorem would be valid, and we find the asymptotic form of the Laplace transform of the probability law in certain cases.

Symmetry and geometric evolution equations

The paper contains an analytic description of geometric flows generated by nonlinear curvature functions in terms of an invariant symmetric curvature matrix. Such an approach provides the most efficient application of ideas and methods of the modern theory of fully nonlinear, second-order, partial differential equations to the relevant geometric problems. Bibliography: 10 titles.

Double σ-multiplicative matrices

The class of σ-regular matrices was defined and characterized by Schaefer [P. Schaefer, Infinite matrices and invariant means, Proc. Amer. Math. Soc. 36 (1972) 104–110] and further studied by Mursaleen [Mursaleen, On some new invariant matrix methods of summability, Quart. J. Math. Oxford 34 (1983) 77–86], Ahmad and Mursaleen [Z.U. Ahmad, Mursaleen, An application of Banach limits, Proc. Amer. Math. Soc. 103 (1988) 244–246]. In this paper we characterize four-dimensional σ-multiplicative matrices, and establish a core theorem.

Parametric invariant random matrix model and the emergence of multifractality

We propose a random matrix modeling for the parametric evolution of eigenstates. The model is inspired by a large class of quantized chaotic systems. Its unique feature is having parametric invariance while still possessing the nonperturbative breakdown that had been discussed by Wigner 50 years ago. Of particular interest is the emergence of an additional crossover to multifractality.

All loop BMN state energies from matrices

We study a quantum corrected SO(6) invariant matrix quantum mechanics obtained from the s-wave modes of the scalars of N = 4 SYM on S^3. For commuting matrices, this model is believed to describe the 1/8 BPS states of the full SYM theory. In the large N limit the ground state corresponds to a distribution of eigenvalues on a S^5 which we identify with the sphere on the dual geometry AdS_5x S^5. We then consider non-BPS excitations by studying matrix perturbations where the off-diagonal modes are treated perturbatively. To a first approximation, these modes can be described by a free theory of "string bits" whose energies depend on the diagonal degrees of freedom. We then consider a state with two string bits and large angular momentum J on the sphere. In the large J limit we use a simple saddle point approximation to show that the energy of these states coincides precisely with the BMN spectrum to all orders in the 't Hooft coupling. We also find some new problems with the all loop Bethe Ansatz conjecture of the N=4 SYM planar spin chain model.

The [formula omitted]-convergence and [formula omitted]-core of double sequences

The σ -convergence and σ -core of a real bounded sequence were introduced in [R. Raimi, Invariant means and invariant matrix methods of summability, Duke Math. J. 30 (1963) 81–94] and [S.L. Mishra, B. Satapathy, N. Rath, Invariant means and σ -core, J. Indian Math. Soc. 60 (1984) 151–158], respectively. In this work, we extend these ideas to double sequences.

Fragmentation, nonrelativistic QCD, and NNLO factorization analysis in heavy quarkonium production

We discuss heavy quarkonium production through parton fragmentation, including a review of arguments for the factorization of high-p_T particles into fragmentation functions for hadronic initial states. We investigate the further factorization of fragmentation functions in the NRQCD formalism, and argue that this requires a modification of NRQCD octet production matrix elements to include nonabelian phases, which makes them gauge invariant. We describe the calculation of uncanceled infrared divergences in fragmentation functions that must be factorized at NNLO, and verify that they are absorbed into the new, gauge invariant matrix elements.