Non-diagonal Case Research Articles

ASYMPTOTIC BEHAVIOUR OF THE JACOBI SOBOLEV-TYPE ORTHOGONAL POLYNOMIALS. A NON DIAGONAL CASE

Asymptotic behaviour of the Jacobi Sobolev-type orthogonal polynomials. A non-diagonal case. Rev. Acad. Colomb. Cienc. 34 (133): 529-539, 2010. ISSN 0370-3908. MATEMÁTICAS 1 Universidad Nacional de Colombia, Bogotá, Colombia. Correo electrónico: haduenasr@unal.edu.co 2 Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo No. 340, Colima, Colima, México. Correo electrónico: garzaleg@gmail.com AMS Classification 2000: 33C47. Consider the following Sobolev type inner product (p, q) = ∫ p'(x)q(x)(1 - x)θ(1 + x)θdx + i[p'(1)q(1) - p(1)q'(1)], (1) where p and q are polynomials with real coefficients, 0 < θ > -1, ξ(x) = (p(x), q(x))', and A = [M0, M1; M1, M2] is a positive semidefinite matrix, with M0M1 ≥ 0, and A ∈ ℝ. The family of polynomials orthogonal with respect to (1), P_n^{(θ,θ)}, are called Jacobi Sobolev-type orthogonal polynomials. An expression that relates this family of polynomials with P_n^{(0,0)}, the usual Jacobi orthogonal polynomials, was obtained in [8]. Here, we obtain the outer relative asymptotic for P_n^{(0,0)}, as well as the corresponding Mehlér-Heine formula.

Dynamics of the General Bianchi IX Model near a Cosmological Singularity

Half a century ago, Belinsky and Khalatnikov proposed a generic solution of the Einstein equations near their cosmological singularity, basing on a generalization of the homogeneous model of Bianchi type IX. The consideration of the evolution of the most general non-diagonal case of this model is significantly simplified, if it is assumed that, when approaching the singularity t = 0, it reduces to the so-called asymptotic dynamics, at which inequality Γ1 ≫ Γ2 ≫ Γ3 holds. It has been suggested that this inequality continues to be true from the moment of its first fulfilment up to the singularity of space-time. We analyze this assumption and show that it is incorrect in the general case. However, it is shown that in any case there exists a time t0, after which this assumption becomes true. The value of t0 is the smaller, the less is the degree of non-diagonality of the model. Some details of the behavior of the non-diagonal homogeneous model of Bianchi type IX are considered at the stage of asymptotic dynamics of approaching the singularity.

Contributions to ZZV^* (V=gamma ,Z,Z') couplings from CP violating flavor changing couplings

The one-loop contributions to the trilinear neutral gauge boson couplings ZZV^* (V=gamma ,Z,Z'), parametrized in terms of one CP-conserving f_5^{V} and one CP-violating f_4^{V} form factors, are calculated in models with CP-violating flavor changing neutral current couplings mediated by the Z gauge boson and an extra neutral gauge boson Z'. Analytical results are presented in terms of Passarino-Veltman scalar functions. Constraints on the vector and axial couplings of the Z gauge boson left| g_{{VZ}}^{tu}right| < 0.0096 and left| g_{{VZ}}^{tc}right| <0.011 are obtained from the current experimental data on the trightarrow Z q decays. It is found that in the case of the ZZgamma ^* vertex the only non-vanishing form factor is f_5^{gamma }, which can be of the order of 10^{-3}, whereas for the ZZZ^* vertex both form factors f_5^{Z} and f_4^{Z} are non-vanishing and can be of the order of 10^{-6} and 10^{-5}, respectively. Our estimates for f_5^{gamma } and f_5^{Z} are smaller than those predicted by the standard model, where f_4^{Z} is absent up to the one loop level. We also estimate the ZZ{Z'}^{*} form factors arising from both diagonal and non-diagonal Z' couplings within a few extension models. It is found that in the diagonal case f_{5}^{Z'} is the only non-vanishing form factor and its real and imaginary parts can be of the order of 10^{-1}–10^{-2} and 10^{-2}–10^{-3}, respectively, with the dominant contributions arising from the light quarks and leptons. In the non-diagonal case f_{5}^{Z^prime } can be of the order of 10^{-4}, whereas f_4^{Z'} can reach values as large as 10^{-7}–10^{-8}, with the largest contributions arising from the Z'tq couplings.

A Robust Generalized Proportionate Diffusion LMS Algorithm for Distributed Estimation

This brief paper proposes a robust generalized proportionate diffusion Least Mean Square (LMS) algorithm for distributed estimation of a parameter vector in a network. The contribution of this brief is twofold. First, we generalize the concept of proportionate diffusion LMS by letting the gain matrix to be non-diagonal instead of being a diagonal matrix. Second, to achieve robustness to impulsive noise while simultaneously maintaining a fast convergence property, we use a combination of Mean Square Deviation (MSD) and disturbance incurred in the adaptation step as the objective cost function. By simplifying and optimizing the proposed cost function, a closed form formula is obtained for the gain matrix in the general non-diagonal case. Simulation results demonstrate the efficiency of the proposed method in comparison to some other state-of-the-art algorithms.

Off-diagonal Bethe Ansatz on the so(5) spin chain

The so(5) (i.e., B2) quantum integrable spin chains with both periodic and non-diagonal boundaries are studied via the off-diagonal Bethe Ansatz method. By using the fusion technique, sufficient operator product identities (comparing to those in [1]) to determine the spectrum of the transfer matrices are derived. For the periodic case, we recover the results obtained in [1], while for the non-diagonal boundary case, a new inhomogeneous T−Q relation is constructed. The present method can be directly generalized to deal with the so(2n+1) (i.e., Bn) quantum integrable spin chains with general boundaries.

Comparing the dynamics of diagonal and general Bianchi IX spacetime

We make comparison of the dynamics of the diagonal and nondiagonal Bianchi IX models in the evolution towards the cosmological singularity. Apart from the original variables, we use the Hubble normalized ones commonly applied in the examination of the dynamics of homogeneous models. Applying the dynamical systems method leads to the result that in both cases the continuous space of critical points is higher dimensional and they are of the nonhyperbolic type. This is a generic feature of the dynamics of both cases and seems to be independent on the choice of phase space variables. The topologies of the corresponding critical spaces are quite different. We conjecture that the nondiagonal case may carry a new type of chaos different from the one specific to the usually examined diagonal one.

Continuous sliding mode controllers for multi-input multi-output systems

This paper develops new continuous sliding mode controllers for multi-input multi-output mechanical systems in the presence of unknown, but bounded uncertainties in the given forces and in the masses. Assuming the absence of the uncertainties, a reference control input is first calculated using the fundamental equation of constrained motion that causes the system trajectories to exactly track the reference while minimizing a weighted $$L^{2}$$ norm of the control effort. Next, in the presence of realistic uncertainties in the given forces and in the masses, two continuous sliding mode controllers are derived according to whether the mass matrix is diagonal or not. In the diagonal case, each element of the control vector is independently designed, while in the nondiagonal case the control vector is handled as a whole because its elements are coupled to one another. The two controllers are continuous because no signum functions are used. It is also shown that various forms of control input are possible depending on the control requirements among which a simple proportional-integral-derivative-type controller is exemplified in this paper. Two numerical examples serve to demonstrate the accuracy and robustness of the control methodology suggested herein.

On the consistency of the Oppenheimer-Snyder solution for a dust star. Reply to Marshall’s criticism

The recent alternative to the Oppenheimer-Snyder (OS) solution for a dust star proposed by Marshall in the paper “Gravitational collapse without black holes” (Astrophys. Space Sci. 342:329, 2012) is analyzed. It is shown that this proposal leads to a non-diagonal metric, with which the Einstein equations become practically unsolvable. Any ansatz proposed as their exact solution turns out to be arbitrary and may be unlimited number of the such solutions. This is due to the fact that an auxiliary function $y(R,r)$ , introduced by OS as $t=M(y)$ , is unambiguously fixed by the diagonality condition and the matching on the surface, and thus in the non-diagonal case it remains arbitrary. It is also shown that the OS solution, as a description in terms of the Schwarzschild coordinates, leads to a frozen star (or frozar) picture not only for the surface, asymptotically freezing outside the gravitational radius, but for interior layers too which also freeze near their own asymptotes. At most of the inner region these asymptotes are located almost equidistantly and only for layers initially close to the surface they become denser. The reason for the such densifying is not “a gravitational repulsion”, but their later freezing and higher spatial contractions, while they remain be uniform and free falling in the comoving frames.

Refinement of Strichartz Estimates for Airy Equation in Nondiagonal Case and its Application

In this paper, we give an improvement of nondiagonal Strichartz estimates for the Airy equation by using a Morrey-type space. As its applications, we prove the small data scattering and the existence of special nonscattering solutions, which are minimal in a suitable sense, to the mass-subcritical generalized Korteweg--de Vries equation. Especially, the use of a refined nondiagonal estimate removes several technical restrictions on the previous work [S. Masaki and J. Segata, Existence of a Minimal Non-Scattering Solution to the Mass-Subcritical Generalized Korteweg-de Vries Equation, preprint, https://arxiv.org/abs/1602.05331] about the existence of the special non-scattering solution.

Stability robustness of linear quadratic regulators

Summary In quadratic optimal control theory, the multivariable linear quadratic regulator is guaranteed to have excellent stability margins if the weight on the control inputs is diagonal. However, for the non-diagonal case, it may suffer from poor robustness. In this paper, these robustness properties are studied in relation to weight selection. For general weighting matrices, a new lower bound on the minimum singular value of the return difference is presented. New guaranteed stability margins are also presented. This gives a formal mathematical basis for guidelines for weight selection. Copyright © 2015 John Wiley & Sons, Ltd.

Explicit Relations of Physical Potentials Through Generalized Hypervirial and Kramers' Recurrence Relations* *Supported in part by Project 20150964-SIP-IPN, COFAA-IPN, Mexico

Based on a Hamiltonian identity, we study one-dimensional generalized hypervirial theorem, Blanchard-like (non-diagonal case) and Kramers' (diagonal case) recurrence relations for arbitrary xκ which is independent of the central potential V(x). Some significant results in diagonal case are obtained for special κ in xκ (κ ≥ 2). In particular, we find the orthogonal relation 〈n1 | n2〉 = δn1n2 (κ = 0), 〈n1 | V′ (x)|n2〉 = (En1 − En2 )2〈n1 | x | n2〉 (κ = 1), En = 〈n | V′ (x)x/2|n〉 + 〈n | V (x)|n〉 (κ = 2) and −4En〈n| x|n〉 + 〈n|V′(x)x2|n〉 + 4〈n|V(x)x|n〉 = 0 (κ = 3). The latter two formulas can be used directly to calculate the energy levels. We present useful explicit relations for some well known physical potentials without requiring the energy spectra of quantum system.

Evaporation phenomena in f(T) gravity

We formulate evaporation phenomena in a generic model of generalized teleparallel gravity in Weitzenbock space–time with a diagonal and nondiagonal tetrad basis. We also perform perturbation analysis around the constant torsion scalar solution called the Nariai space–time, which is an exact solution of the field equations as the limiting case of the Schwarzschild – de Sitter space–time and in the limit where two black holes and their cosmological horizons coincide. By a carefully analysis of the horizon perturbation equation, we show that (anti)evaporation cannot happen if we use a diagonal tetrad basis. This result implies that a typical black hole in any generic form of generalized teleparallel gravity is frozen in its initial state if we use diagonal tetrads, but in the case of nondiagonal tetrads the analysis is completely different. With a suitable nontrivial nondiagonal tetrad basis we investigate the linear stability of the model under simultaneous perturbations of the metric and torsion. We observe that in spite of the diagonal case, both evaporation and antievaporation can happen. These phenomena depend on the initial phase of the horizon perturbation. In the first mode, when we restrict ourselves to the first lower modes (anti)evaporation takes place. So, in the nondiagonal case, the physical phenomena are reasonable. This is an important advantage of using nondiagonal tetrads instead of diagonal ones. We also see that this is a universal feature, completely independent from the form of the model.

Maximal correlation in a non-diagonal case

We present a method for the obtention of the maximal correlation coefficient that extends the simple method given by Papadatos and Xifara (2013). We illustrate our method with the calculation of the maximal correlation between the k th largest order statistics of overlapping samples.

Inhomogeneous universe in f(T) theory

We obtain the equations of motions of the $f(T)$ theory considering the Lema\^itre-Tolman-Bondi's metric for a set of diagonal and non-diagonal tetrads. In the case of diagonal tetrads the equations of motion of the $f(T)$ theory impose a constant torsion or the same equations of the General Relativity, while in the case of non-diagonal set the equations are quite different from that obtained in GR. We show a simple example of an universe dominated by the matter for the two cases. The comparison of the mass in the non-diagonal case shows a sort of increased with respect to the diagonal one. We also perform two examples for the non-diagonal case. The first concerns a black hole solution of type Sshwarzschild which presents a temperature higher than that of Schwarzschild, and a black hole in a dust-dominated universe.

The half-infinite XXZ chain in Onsagerʼs approach

The half-infinite XXZ open spin chain with general integrable boundary conditions is considered within the recently developed ‘Onsagerʼs approach’. Inspired by the finite size case, for any type of integrable boundary conditions it is shown that the transfer matrix is simply expressed in terms of the elements of a new type of current algebra recently introduced. In the massive regime −1<q<0, level one infinite dimensional representation (q-vertex operators) of the new current algebra are constructed in order to diagonalize the transfer matrix. For diagonal boundary conditions, known results of Jimbo et al. are recovered. For upper (or lower) non-diagonal boundary conditions, a solution is proposed. Vacuum and excited states are formulated within the representation theory of the current algebra using q-bosons, opening the way for the calculation of integral representations of correlation functions for a non-diagonal boundary. Finally, for q generic the long standing question of the hidden non-Abelian symmetry of the Hamiltonian is solved: it is either associated with the q-Onsager algebra (generic non-diagonal case) or the augmented q-Onsager algebra (generic diagonal case).

Jacobi–Sobolev-type orthogonal polynomials: holonomic equation and electrostatic interpretation – a non-diagonal case

Consider the Sobolev-type inner product where p and q are polynomials with real coefficients, α, β>−1, ℙ(x)=(p(x), p′(x)) t , and is a positive semidefinite matrix, with M 0, M 1≥0, and λ∈ℝ. We obtain an expression for the family of polynomials , orthogonal with respect to the above inner product, a connection formula that relates with some family of Jacobi polynomials and the holonomic equation that they satisfy, as well as an electrostatic interpretation of their zeros.

Future non-linear stability for solutions of the Einstein-Vlasov system of Bianchi types II and VI

In a recent paper [E. Nungesser, “Future non-linear stability for reflection symmetric solutions of the Einstein-Vlasov system of Bianchi types II and VI,” Annales Henri Poincare (2012)10.1007/s00023-012-0201-0], we have treated the future nonlinear stability for reflection symmetric solutions of the Einstein-Vlasov system of Bianchi types II and VI0. We have been able now to remove the reflection symmetry assumption, thus treating the non-diagonal case. Apart from the increasing complexity, the methods have been essentially the same as in the diagonal case, showing that they are thus quite powerful. Here, the challenge was to put the equations in a form that permits the use of the previous results. We are able to conclude that after a possible basis change, the future of the non-diagonal spacetimes in consideration is asymptotically diagonal.

Trace inequalities for weighted Hardy—Sobolev spaces in the non-diagonal case

We give characterizations of the positive Borel measures μ on $\mathbb{S}^n$ so that the weighted Hardy–Sobolev spaces $H_s^p(w)$ are imbedded in Lq(dμ), for a range of s &gt; 0, 0 &lt; p, q &lt; + ∞, q ≠ p, where w is a doubling weight in the unit sphere of ℂn.

The holonomic equation of the Laguerre–Sobolev-type orthogonal polynomials: a non-diagonal case

In this paper, we consider the Sobolev-type inner product where p and q are polynomials with real coefficients, and A is a positive semi-definite matrix. First, we consider a multiplication operator that is symmetric with respect to the above inner product. As a consequence, we prove that the sequence of monic polynomials orthogonal with respect to the above inner product satisfies a five-term recurrence relation. On the other hand, we obtain raising and lowering operators associated with them. As a consequence, a holonomic equation satisfied by these polynomials is given.

Description of interpolation spaces for local Morrey-type spaces

We consider the real interpolation method and prove that for local Morrey spaces, in the case when they have the same integrability parameter, the interpolation spaces are again local Morrey-type spaces with appropriately chosen parameters. Thus, in contrast to the standard Morrey-type spaces, these local spaces form an interpolation scale. In particular, this is true in the so-called nondiagonal case.