Gauge Function Research Articles

Mapping between different cosmological eras in scale-covariant formalism

In this work we consider the scale-covariant formalism proposed by Canuto et al.,[Formula: see text] in order to map different eras of the universe. This technique considers a scale gauge function that can be adjusted by using different arguments like Dirac’s large numbers hypothesis or a restriction on the particle production rate. A Chaplygin constituent shows to be a consistent idea to establish a mapping between an old decelerated–accelerated universe ruled by Einstein equations and an early universe, where a new equation of state appears together with a modified general relativity theory and a de Sitter universe then emerges. These properties are a direct consequence of the use of the scale-covariant formalism. Besides, a new discussion and remarks are presented related to the well-known barotropic constituent case.

Special Functions of Mathematical Physics: A Unified Lagrangian Formalism

Lagrangian formalism is established for differential equations with special functions of mathematical physics as solutions. Formalism is based on either standard or non-standard Lagrangians. This work shows that the procedure of deriving the standard Lagrangians leads to Lagrangians for which the Euler–Lagrange equation vanishes identically, and that only some of these Lagrangians become the null Lagrangians with the well-defined gauge functions. It is also demonstrated that the non-standard Lagrangians require that the Euler–Lagrange equations are amended by the auxiliary conditions, which is a new phenomenon in the calculus of variations. The existence of the auxiliary conditions has profound implications on the validity of the Helmholtz conditions. The obtained results are used to derive the Lagrangians for the Airy, Bessel, Legendre and Hermite equations. The presented examples clearly demonstrate that the developed Lagrangian formalism is applicable to all considered differential equations, including the Airy (and other similar) equations, and that the regular and modified Bessel equations are the only ones with the gauge functions. Possible implications of the existence of the gauge functions for these equations are discussed.

A weighted anisotropic Sobolev type inequality and its applications to Hardy inequalities

In this paper we focus our attention on an embedding result for a weighted Sobolev space that involves as weight the distance function from the boundary taken with respect to a general smooth gauge function F. Starting from this type of inequalities we prove some refined Hardy-type inequalities.

Dark energy and modified scale covariant theory of gravitation

Taking up four model universes we study the behaviour and contribution of dark energy to the accelerated expansion of the universe, in the modified scale covariant theory of gravitation. Here, it is seen that though this modified theory may be a cause of the accelerated expansion it cannot totally outcast the contribution of dark energy in causing the accelerated expansion. In one case the dark energy is found to be the sole cause of the accelerated expansion. The dark energy contained in these models come out to be of the ΛCDM type and quintessence type comparable to the modern observations. Some of the models originated with a big bang, the dark energy being prevalent inside the universe before the evolution of this era. One of the models predicts big rip singularity, though at a very distant future. It is interestingly found that the interaction between the dark energy and the other part of the universe containing different matters is enticed and enhanced by the gauge function ϕ(t) here.

Modified f(R,T) cosmology with observational constraints in Lyra’s geometry

In this paper, we have investigated modified [Formula: see text] cosmological models with observational constraints in Lyra’s geometry. We have studied the models in two cases: in the first one it is taken as constant displacement vector field and in second case it is taken as time-dependent. We have established a relationship among energy parameters [Formula: see text], respectively, called as matter, anisotropy and dark energy (DE) density parameters in Bianchi type-I space-time in Lyra’s geometry. We have compared our models with observational data sets with union 2.1 compilation SNe Ia data and [Formula: see text] data sets and have found best fit values of various energy parameters for [Formula: see text]. We have found that the model with constant displacement vector field is the best fit to the study of the whole evolution of the model and the gauge function [Formula: see text] of Lyra geometry behaves like a DE candidate. Also, the various physical and kinematical parameters have been studied in this study.

A Formula for the Superdifferential of the Distance Determined by the Gauge Function to the Complement of a Convex Set

A Formula for the Superdifferential of the Distance Determined by the Gauge Function to the Complement of a Convex Set

Fronsdal fields from gauge functions in Vasiliev\u2019s higher spin gravity

In this paper, we revisit a number of issues in Vasiliev’s theory related to gauge functions, ordering schemes, and the embedding of Fronsdal fields into master fields. First, we parametrize a broad equivalence class of linearized solutions using gauge functions and integration constants, and show explicitly how Fronsdal fields and their Weyl tensors arise from these data in accordance with Vasiliev’s central on mass shell theorem. We then gauge transform the linearized piece of exact solutions, obtained in a convenient gauge in Weyl order, to the aforementioned class, where we land in normal order. We spell out this map for massless particle and higher spin black hole modes. Our results show that Vasiliev’s equations admit the correct free-field limit for master field configurations that relax the original regularity and gauge conditions in twistor space. Moreover, they support the off-shell Frobenius-Chern-Simons formulation of higher spin gravity for which Weyl order plays a crucial role. Finally, we propose a Fefferman-Graham-like scheme for computing asymptotically anti-de Sitter master field configurations, based on the assumption that gauge function and integration constant can be adjusted perturbatively so that the full master fields approach free master fields asymptotically.

Higher spin fluctuations on spinless 4D BTZ black hole

We construct linearized solutions to Vasiliev’s four-dimensional higher spin gravity on warped AdS3 ×ξS1 which is an Sp(2) × U(1) invariant non-rotating BTZ-like black hole with ℝ2 × T2 topology. The background can be obtained from AdS4 by means of identifications along a Killing boost K in the region where ξ2 ≡ K2 ≥ 0, or, equivalently, by gluing together two Bañados-Gomberoff-Martinez eternal black holes along their past and future space-like singularities (where ξ vanishes) as to create a periodic (non-Killing) time. The fluctuations are constructed from gauge functions and initial data obtained by quantizing inverted harmonic oscillators providing an oscillator realization of K and of a commuting Killing boost tilde{K} . The resulting solution space has two main branches in which K star commutes and anti-commutes, respectively, to Vasiliev’s twisted-central closed two-form J. Each branch decomposes further into two subsectors generated from ground states with zero momentum on S1. We examine the subsector in which K anti-commutes to J and the ground state is mathrm{U}{(1)}_Ktimes mathrm{U}{(1)}_{tilde{K}} -invariant of which U(1)K is broken by momenta on S1 and mathrm{U}{(1)}_{tilde{K}} by quasi-normal modes. We show that a set of mathrm{U}{(1)}_{tilde{K}} -invariant modes (with n units of S1 momenta) are singularity-free as master fields living on a total bundle space, although the individual Fronsdal fields have membrane-like singularities at {tilde{K}}^2=1 . We interpret our findings as an example where Vasiliev’s theory completes singular classical Lorentzian geometries into smooth higher spin geometries.

Brans-Dicke scalar field cosmological model in Lyra’s geometry

In this paper, we have developed a new cosmological model in Einstein's modified gravity theory using two types of modification.(i) Geometrical modification, in which we have used Lyra's geometry in the left hand side of the Einstein field equations (EFE) and (ii) Modification in gravity (energy momentum tensor) on right hand side of EFE, as per Brans-Dicke (BD) model. With these two modifications, we have investigated a spatially homogeneous and anisotropic Bianchi type-I cosmological models of Einstein's Brans-Dicke theory of gravitation in Lyra geometry. The model represents accelerating universe at present and decelerating in past and is considered to be dominated by dark energy. Gauge function $\beta$ and BD-scalar field $\phi$ are considered as a candidate for the dark energy and is responsible for the present acceleration. The derived model agrees at par with the recent supernovae (SN Ia) observations. We have set BD-coupling constant $\omega$ to be greater than 40000, seeing the solar system tests and evidences. We have discussed the various physical and geometrical properties of the models and have compared them with the corresponding relativistic models.

On the convergence of almost minimal sets for the Hausdorff and varifold topologies

The geometric properties of (almost) minimal sets, especially the regularity, is an interesting topic in geometric measure theory, which were often studied in the literature, for example [2,4,5,8–10,19,22]. Soap films as well as solutions to Plateau's Problem could be a typical example of such kind of minimal sets. In this paper, we will investigate the convergence of a sequence of such sets, and show that Hausdorff convergence and varifold convergence coincide on the class of almost minimal sets bounded by a uniform gauge function, and we will see that a large amount of sets, including all compact C1,1 submanifolds in Rn, are almost minimal.

Noncoercive Stationary Navier–Stokes Equations of Heat-Conducting Fluids Modeled by Hemivariational Inequalities: An Equilibrium Problem Approach

In this paper, we study the existence of solutions for noncoercive stationary Navier–Stokes equations of heat-conducting fluids with nonmonotone boundary conditions modeled by hemivariational inequalities. Our method is new and differs from most of the existing techniques developed in the literature. It is based on a recent approach developed on the existence of solutions for mixed equilibrium problems described by the sum of a maximal monotone bifunction and a pseudomonotone bifunction in the sense of Brezis. We introduce a Browder–Tikhonov regularization method for mixed equilibrium problems by means of the duality mapping with gauge function $$\mu (t)$$. By using this regularization procedure and techniques from the recession analysis, we study the existence of solutions for the problem considered in this paper.

A joint multifractal analysis of vector valued non Gibbs measures

Abstract The multifractal formalism for measures holds whenever the existence of corresponding Gibbs-like measures supported on the singularities sets holds. In the present work we tried to relax such a hypothesis and introduce a more general framework of joint multifractal analysis where the measures constructed on the singularities sets are not Gibbs but controlled by an extra-function allowing the multifractal formalism to hold. We fall on the classical case by a particular choice of such a function. An answer to a question raised in [2] on which gauge function φ shall we get a finite, infinite or zero value of H μ , φ q , t ( K ) for the singularities set K is provided.

A comparative study of Kaluza–Klein model with magnetic field in Lyra manifold and general relativity

Kaluza–Klein (K–K) dark energy (DE) cosmological models in presence of magnetized anisotropic fluid have been examined in normal gauge for the field equations for Lyra’s manifold where gauge function β is taken as the function of time. We have chosen the scale factor a(t)=[tkexp(t)]1n (k ≥ 0, n ≥ 0) as the generalized Hybrid expansion law which yields the time-dependent deceleration parameter (DP), to obtain the deterministic solutions of field equations, speaking to a class of models which produce the transition of the universe from the early decelerating stage to the ongoing accelerating stage. The dark energy EoS (ω) is observed to be the time-dependent and its current range for inferred models is in great concurrence with the ongoing observations. For various estimations of (n, k), we can produce a class of physically suitable DE models in K–K theory. The anisotropy of the model vanishes and become isotropic under the suitable condition. The solutions are likewise compared with the solutions in general relativity and without magnetic field. The geometric and physical properties of both the models are likewise talked about in detail.

Dynamics of soliton solutions of the nonlocal Kundu-nonlinear Schrödinger equation.

In this paper, we investigate the nonlocal Kundu-nonlinear Schrödinger (Kundu-NLS) equation, which can be obtained from the reduction of the coupled Kundu-NLS system. Based on the analysis of the eigenfunctions, a Riemann-Hilbert problem is constructed to derive the N-soliton solutions of the coupled Kundu-NLS system. The N-soliton solutions of the nonlocal Kundu-NLS equation are then deduced with properly chosen symmetry relations on the scattering data. The dynamics of the solitons in the nonlocal Kundu-NLS equation are explored. The impact of the gauge function on the solitons is displayed for one-solitons. Compared with the dynamics of the two-solitons in the local Kundu-NLS equation, the two-solitons in the nonlocal Kundu-NLS equation display many differences. The repeated collapsing is a common feature of the singular solitons, and it seems that some of them are not the superposition of one-solitons. The singular solitons exhibit various behaviors in different eigenvalue configurations in the spectral space. Besides that, three kinds of bounded solutions are presented according to these eigenvalue configurations. In addition, two kinds of degenerate solutions are presented, and in particular, the positon solutions are discussed in detail. The decomposition of the positon solutions is analyzed and their trajectories are given approximately.

Fundamental Fields as Eigenvectors of the Metric Tensor in a 16-Dimensional Space-Time

An alternative approach to the usual Kaluza-Klein way to field unification is presented which seems conceptually more satisfactory and elegant. The main idea is that of associating each fundamental interaction and matter field with a vector potential which is an eigenvector of the metric tensor of a multidimensional space-time manifold (n-dimensional “vierbein”). We deduce a system of field equations involving both Einstein and Maxwell-like equations for the fundamental fields. Confinement of the fields within the observable 4-dimensional space-time and non-vanishing particles’ rest mass problem are shown to be related to the choice of a scalar boson field (Higgs boson) appearing in the theory as a gauge function. Physical interpretation of the results, in order that all the known fundamental interactions may be included within the metric and connection, requires that the extended space-time is 16-dimensional. Fermions are shown to be included within the additional components of the vector potentials arising because of the increased dimensionality of space-time. A cosmological solution is also presented providing a possible explanation both to space-time flatness and to dark matter and dark energy as arising from the field components hidden within the extra space dimensions. Suggestions for gravity quantization are also examined.

Polar Convolution

The Moreau envelope is one of the key convexity-preserving functional operations in convex analysis, and it is central to the development and analysis of many approaches for convex optimization. This paper develops the theory for an analogous convolution operation, called the polar envelope, specialized to gauge functions. Many important properties of the Moreau envelope and the proximal map are mirrored by the polar envelope and its corresponding proximal map. These properties include smoothness of the envelope function, uniqueness and continuity of the proximal map, which play important roles in duality and in the construction of algorithms for gauge optimization. A suite of tools with which to build algorithms for this family of optimization problems is thus established.

Noether’s Theorem and Symmetry

In Noether’s original presentation of her celebrated theorem of 1918, allowance was made for the dependence of the coefficient functions of the differential operator, which generated the infinitesimal transformation of the action integral upon the derivatives of the dependent variable(s), the so-called generalized, or dynamical, symmetries. A similar allowance is to be found in the variables of the boundary function, often termed a gauge function by those who have not read the original paper. This generality was lost after texts such as those of Courant and Hilbert or Lovelock and Rund confined attention to point transformations only. In recent decades, this diminution of the power of Noether’s theorem has been partly countered, in particular in the review of Sarlet and Cantrijn. In this Special Issue, we emphasize the generality of Noether’s theorem in its original form and explore the applicability of even more general coefficient functions by allowing for nonlocal terms. We also look for the application of these more general symmetries to problems in which parameters or parametric functions have a more general dependence on the independent variables.

Block minimum perturbation algorithm based on block Arnoldi process for nonsymmetric linear systems with multiple right-hand sides

In many applications in scientific computing and engineering one has to solve huge sparse linear systems of equations with several right-hand sides. Customarily, one uses the residual error as a stopping condition, but small residue does not imply accurate approximate solutions. Therefore, minimizing quantities other than the residual norm may be more adequate in this context. Based on the above consideration a block minimum perturbation Krylov subspace method (BMinPet) for the solution, Xm satisfying (A−ΔA)Xm=(B+ΔB) at each step, of nonsymmetric linear systems AX=B is shown such that the normwise backward error meets some optimality condition with minimizing the block joint backward perturbation norm ‖(ΔA, ΔB)‖F of the matrix (A, B), and a stopping criterion is given for consistent system with multiple right-hand sides to get a more robust way to stop the iterations. This process is a generalization of minimum perturbation algorithm (MinPert) by Kasenally and Simoncini (1997) (Analysis of a minimum perturbation algorithm for nonsymmetric linear systems. SIAM J. Numer. Anal., 34, 48–66) to multiple right-hand sides. For the benefit of the amount of calculation, we give lower and upper bounds about min ‖(ΔA, ΔB)‖F that are easier to compute. As a by-product, we come up with a conception of Function ψΦ(F, G) which is a generalization of Function ψΦ that is a symmetric gauge (SG) function. The algorithm is used to analyze the performance of BGMRES method by evaluating the proximity of their solutions to block joint backward perturbation optimality. The relationships between BMinPet method and relative methods are investigated. Numerical examples show us that BMinPert shows its superiority compared with BFGMRES-S(m, pf), GsGMRES, Bl-BiCG-rQ, BGMRES and BArnoldi in solving the large sparse ill-conditioned problems.

An angular dependent supersymmetric quantum mechanics with a Z2-invariant potential

We generalize the conformally invariant topological quantum mechanics of a particle propagating on a punctured plane by introducing a potential that breaks both the rotational and the conformal invariance down to a Z2 angular-dependent discrete symmetry. We derive a topological quantum mechanics whose localization gauge functions give interesting self-dual equations. The model contains an order parameter and exhibits a critical phenomenon with the existence of two ground states above a critical scale. Unlike the ordinary O(2)-invariant Higgs potential, an angular-dependence is found and saddle points, instead of local maxima, appear, posing subtle questions about the existence of instantons. The supersymmetric quantum mechanical model is constructed in both the path integral and the operatorial frameworks.

Nanocircuits in loop structures: Continuous waves preclude gauge invariant wavelengths

Tunnel junctions for quantum computing require discrete spectra from continuous waves on a doubly connected coordinate or loop. For an electron on a metal ring discrete spectra follow from discontinuous Bloch waves. Can both propositions be true? We find using a gauge function originating in the Lagrangian that continuity on a ring or loop violates gauge invariance of the de Broglie wavelength. This same gauge function shows that Lagrangians for the electron on a ring and the charge on a junction are mutual transforms. Thus persistent current on a metal ring and the Coulomb blockade on a tunnel junction seem to be the same dynamical theory based on discontinuous Bloch waves on the compact perimeter of a circle