Non-constant Rank Research Articles

Poisson gauge theory

The Poisson gauge algebra is a semi-classical limit of complete non- commutative gauge algebra. In the present work we formulate the Poisson gauge theory which is a dynamical field theoretical model having the Poisson gauge algebra as a corresponding algebra of gauge symmetries. The proposed model is designed to investigate the semi-classical features of the full non-commutative gauge theory with coordinate dependent non-commutativity Θab(x), especially whose with a non-constant rank. We derive the expression for the covariant derivative of matter field. The commutator relation for the covariant derivatives defines the Poisson field strength which is covariant under the Poisson gauge transformations and reproduces the standard U(1) field strength in the commutative limit. We derive the corresponding Bianchi identities. The field equations for the gauge and the matter fields are obtained from the gauge invariant action. We consider different examples of linear in coordinates Poisson structures Θab(x), as well as non-linear ones, and obtain explicit expressions for all proposed constructions. Our model is unique up to invertible field redefinitions and coordinate transformations.

Renormalized oscillation theory for symplectic eigenvalue problems with nonlinear dependence on the spectral parameter

ABSTRACTIn this paper, we establish new renormalized oscillation theorems for discrete symplectic eigenvalue problems with Dirichlet boundary conditions. These theorems present the number of finite eigenvalues of the problem in the arbitrary interval using the number of focal points of a transformed conjoined basis associated with Wronskian of two principal solutions of the symplectic system evaluated at the endpoints a and b. We suppose that the symplectic coefficient matrix of the system depends nonlinearly on the spectral parameter and that it satisfies certain natural monotonicity assumptions. In our treatment, we admit possible oscillations in the coefficients of the symplectic system by incorporating their non-constant rank with respect to the spectral parameter.

Riemannian metrics and Laplacians for generalized smooth distributions

We show that any generalized smooth distribution on a smooth manifold, possibly of non-constant rank, admits a Riemannian metric. Using such a metric, we attach a Laplace operator to any smooth distribution as such. When the underlying manifold is compact, we show that it is essentially self-adjoint. Viewing this Laplacian in the longitudinal pseudodifferential calculus of the smallest singular foliation which includes the distribution, we prove hypoellipticity.

Local (Sub)-Finslerian Geometry for the Maximum Norms in Dimension 2

We consider specific sub-Finslerian structures in the neighborhood of 0 in $\mathbb {R}^{2}$ , defined by fixing a family of vector fields (F1,F2) and considering the norm defined on the non-constant rank distribution Δ = vect{F1, F2} by $$|G|=\inf_{u}\{\max\{|u_{1}|,|u_{2}|\} | G=u_{1} F_{1}+u_{2} F_{2}\}. $$ If F1 and F2 are not proportional at p, then we obtain a Finslerian structure; if not, the structure is sub-Finslerian on a distribution with non-constant rank. We are interested in the study of the local geometry of these Finslerian and sub-Finslerian structures: generic properties, normal form, short geodesics, cut locus, switching locus, and small spheres.

Discrete oscillation theorems for symplectic eigenvalue problems with general boundary conditions depending nonlinearly on spectral parameter

In this paper we establish new oscillation theorems for discrete symplectic eigenvalue problems with general boundary conditions. We suppose that the symplectic coefficient matrix of the system and the boundary conditions are nonlinear functions of the spectral parameter and that they satisfy certain natural monotonicity assumptions. In our new theory we admit possible oscillations in the coefficients of the symplectic system and the boundary conditions by incorporating their nonconstant rank with respect to the spectral parameter. We also prove necessary and sufficient conditions for boundedness of the real part of spectrum of these eigenvalue problems.

Some theoretical limitations of second-order algorithms for smooth constrained optimization

In second-order algorithms, we investigate the relevance of the constant rank of the full set of active constraints in ensuring global convergence to a second-order stationary point under a constraint qualification. We show that second-order stationarity is not expected in the non-constant rank case if the growth of so-called tangent AKKT2 sequences is not controlled. Since no algorithm controls their growth, we argue that there is a theoretical limitation of algorithms in finding second-order stationary points without constant rank assumptions.

On a remarkable class of paracontact metric manifolds

We study a remarkable class of paracontact metric manifolds which have no contact metric counterpart: the paracontact metric [Formula: see text]-spaces which are not paraSasakian (i.e. have [Formula: see text]). We present explicit examples with [Formula: see text] of every possible constant rank and some with non-constant rank, which were not known to exist until recently.

Local classification and examples of an important class of paracontact metric manifolds

We study paracontact metric (k,?)-spaces with k = -1, equivalent to h2=0 but not h=0. In particular, we will give an alternative proof of Theorem 3.2 of [11] and present examples of paracontact metric (-1,2)-spaces and (-1,0)-spaces of arbitrary dimension with tensor h of every possible constant rank. We will also show explicit examples of paracontact metric (-1,?)-spaces with tensor h of non-constant rank, which were not known to exist until now.

Discrete oscillation theorems and weighted focal points for Hamiltonian difference systems with nonlinear dependence on a spectral parameter

In this paper we generalize oscillation theorems for discrete Hamiltonian eigenvalue problems with nonlinear dependence on the spectral parameter λ . In our version of the discrete oscillation theorems, we incorporate the case when the block B k ( λ ) of the discrete Hamiltonian H k ( λ ) has nonconstant rank with respect to λ . We introduce a new notion of weighted focal points for conjoined bases of the Hamiltonian difference systems and we show that the number of weighted focal points plays the role of the classical number of focal points in the discrete oscillation theorems for the Hamiltonian spectral problems with the nonconstant rank of B k ( λ ) .

An extension of the Dirac theory of constraints

Constructions introduced by Dirac for singular Lagrangians are extended and reinterpreted to cover cases when kernel distributions are either nonintegrable or of nonconstant rank, and constraint sets need not be closed.

1-Inverses for polynomial matrices of non-constant rank

Let p 1, … p t be polynomials in C n with a variety V of common zeros contained in a suitable open set U. Explicit formulas are provided to construct rational functions λ 1, … λ s such that Σ i=1 s p i λ i  1, and such that the singularities of the λ i are contained in U. This result is applied to compute rational functions-valued 1-inverses of matrices with polynomial coefficients, which do not have constant rank, while retaining control over the location of the singularities of the rational functions themselves.

The principle of limiting absorption and decay of local energy for the linearized equation of magnetogasdynamics

The present paper is a continuation of [6] and [7] in which the principle of limiting absorption has been verified for symmetric systems of first order with long-range perturbations, but the operators considered there are of constant rank. On the other hand, operators with non-constant rank are also important in application as well as from a purely theoretical point of view. In this paper, we consider the linearized equation of magnetogasdynamics with long-range perturbation as an important example of such operators.