Invariant Equations Research Articles

Symplectically invariant flow equations for N = 2, D = 4 gauged supergravity with hypermultiplets

We consider N=2 supergravity in four dimensions, coupled to an arbitrary number of vector- and hypermultiplets, where abelian isometries of the quaternionic hyperscalar target manifold are gauged. Using a static and spherically or hyperbolically symmetric ansatz for the fields, a one-dimensional effective action is derived whose variation yields all the equations of motion. By imposing a sort of Dirac charge quantization condition, one can express the complete scalar potential in terms of a superpotential and write the action as a sum of squares. This leads to first-order flow equations, that imply the second-order equations of motion. The first-order flow turns out to be driven by Hamilton's characteristic function in the Hamilton-Jacobi formalism, and contains among other contributions the superpotential of the scalars. We then include also magnetic gaugings and generalize the flow equations to a symplectically covariant form. Moreover, by rotating the charges in an appropriate way, an alternative set of non-BPS first-order equations is obtained that corresponds to a different squaring of the action. Finally, we use our results to derive the attractor equations for near-horizon geometries of extremal black holes.

Invariant partial differential equations with two-dimensional exotic centrally extended conformal Galilei symmetry

Conformal Galilei Algebras labeled by $d,\ell$ (where $d$ is the number of space dimensions and $\ell$ denotes a spin-${\ell}$ representation w.r.t. the $\mathfrak{sl}(2)$ subalgebra) admit two types of central extensions, the ordinary one (for any $d$ and half-integer $\ell$) and the exotic central extension which only exists for $d=2$ and ${\ell}\in\mathbb{N}$. For both types of central extensions invariant second-order PDEs with continuous spectrum were constructed in [1]. It was later proved in [2] that the ordinary central extensions also lead to oscillator-like PDEs with discrete spectrum. We close in this paper the existing gap, constructing \textcolor{black}{a new class of second-order invariant PDEs for the exotic centrally extended CGAs; they admit a discrete and bounded spectrum when applied to a lowest weight representation. These PDEs are markedly different with respect to their ordinary counterparts. The ${\ell}=1$ case (which is the prototype of this class of extensions, just like the $\ell=\frac{1}{2}$ Schr\"odinger algebra is the prototype of the ordinary centrally extended CGAs) is analyzed in detail.

Generalized second law of thermodynamics on the apparent horizon in modified Gauss–Bonnet gravity

Modified gravity (MG) and generalized second law (GSL) of thermodynamics are interesting topics in the modern cosmology. In this regard, we investigate the GSL of gravitational thermodynamics in the framework of modified Gauss–Bonnet (GB) gravity or [Formula: see text]-gravity. We consider a spatially FRW universe filled with the pressureless matter and radiation enclosed by the dynamical apparent horizon with the Hawking temperature. For two viable [Formula: see text] models, we first numerically solve the set of differential equations governing the dynamics of [Formula: see text]-gravity. Then, we obtain the evolutions of the Hubble parameter, the GB curvature invariant term, the density and equation of state (EoS) parameters as well as the deceleration parameter. In addition, we check the energy conditions for both models and finally examine the validity of the GSL. For the selected [Formula: see text] models, we conclude that both models have a stable de Sitter attractor. The EoS parameters behave quite similar to those of the [Formula: see text]CDM model in the radiation/matter dominated epochs, then they enter the phantom region before reaching the de Sitter attractor with [Formula: see text]. The deceleration parameter starts from the radiation/matter dominated eras, then transits from a cosmic deceleration to acceleration and finally approaches a de Sitter regime at late times, as expected. Furthermore, the GSL is respected for both models during the standard radiation/matter dominated epochs. Thereafter when the universe becomes accelerating, the GSL is violated in some ranges of scale factor. At late times, the evolution of the GSL predicts an adiabatic behavior for the accelerated expansion of the universe.

SMOOTHNESS OF BOUNDED INVARIANT EQUIVALENCE RELATIONS

AbstractWe generalise the main theorems from the paper “The Borel cardinality of Lascar strong types” by I. Kaplan, B. Miller and P. Simon to a wider class of bounded invariant equivalence relations. We apply them to describe relationships between fundamental properties of bounded invariant equivalence relations (such as smoothness or type-definability) which also requires finding a series of counterexamples. Finally, we apply the generalisation mentioned above to prove a conjecture from a paper by the first author and J. Gismatullin, showing that the key technical assumption of the main theorem (concerning connected components in definable group extensions) from that paper is not only sufficient but also necessary to obtain the conclusion.

Singular invariant structures for Lie algebras admitted by a system of second-order ODEs

Systems of second-order ordinary differential equations (ODEs) arise in mechanics and have several applications. Differential invariants play a key role in the construction of invariant differential equations as well as the classification of a system of ODEs. Like regular invariants, singular invariant structures also possess an important role in the algebraic analysis of a system of ODEs. In this work, singular invariant equations for a system of two second-order ODEs admitting four-dimensional Lie algebras are investigated. Moreover, by using these singular invariant equations, canonical forms for systems of two second-order ODEs are constructed. Furthermore, it is observed that the same Lie algebra admitted by a system of second-order ODEs has different type of realizations some of which are related to regular invariants and some lead to singular invariant equations. Thus realizations of four-dimensional Lie algebras are associated with a regular invariant manifold as well as to a singular invariant manifold defined by a system of second-order ODEs. In addition, a case of a Lie algebra with realization resulting in a conditional singular invariant structure is presented and those cases of singular invariant equations are discussed which do not form a system of second-order ODEs. An integration procedure for the invariant representation of these canonical forms are presented. Illustrative examples are presented with physical applications to mechanics.

Perturbative type II amplitudes for BPS interactions

We consider the perturbative contributions to the , and interactions in toroidally compactified type II string theory. These BPS interactions do not receive perturbative contributions beyond genus three. We derive Poisson equations satisfied by these moduli dependent string amplitudes. These T-duality invariant equations have eigenvalues that are completely determined by the structure of the integrands of the multi-loop amplitudes. The source terms are given by boundary terms of the moduli space of Riemann surfaces corresponding to both separating and non-separating nodes. These are determined directly from the string amplitudes, as well as from U-duality constraints and logarithmic divergences of maximal supergravity. We explicitly solve these Poisson equations in nine and eight-dimensions.

Explicit Solutions of the Invariance Equation for Means

Extending the notion of projective means we first generalize an invariance identity related to the Carlson log given in Kahlig and Matkowski (Math Inequal Appl 18(3):1143–1150, 2015), and then, more generally, given a bivariate symmetric, homogeneous and monotone mean M, we give explicit formula for a rich family of pairs of M-complementary means. We prove that this method cannot be extended for higher dimension. Some examples are given and two open questions are proposed.

Classical local U(1) gauge invariance in Weyl 2-spinor lenguage and charge quantization from irreducible representations of the gauge group

A new classical 2-spinor approach to U(1) gauge theory is presented in which the usual four-potential vector field is replaced by a symmetric second rank spinor. Following a lagrangian formulation, it is shown that the four-rank spinor representing the Maxwell field tensor has a U(1) local gauge invariance in terms of the electric and magnetic field strengths. When applied to the magnetic field of a monopole, this formulation, via the irreducible representation condition for the gauge group, leads to a quantization condition differing by a factor 2 of the one predicted by Dirac without relying on any kind of singular vector potentials. Finally, the U(1) invariant spinor equations, are applied to electron magnetic resonance which has many applications in the study of materials.

A Generalization of Robust Normal Two-Armed Bandit

We consider Normal two-armed bandit problem with a priori known variances and unknown mathematical expectations of incomes in robust (minimax) setting. This setup naturally arises in group control of data processing. We show that one can solve the problem using the main theorem of the theory of games, i.e. determine minimax strategy and minimax risk as Bayesian corresponding to the worst-case prior distribution. We obtain recursive invariant Bellman-type equation for calculation appropriate Bayesian risk and Bayesian strategy. The requirement of a priori known variances of incomes may be omitted because they may be estimated at the initial stage of control.

Suborbital graphs for a special subgroup of the SL(3,Z)

In this paper we examine some properties of suborbital graphs for the group SL*(3,Z). We first introduce an invariant equivalence relation by using the congruence subgroup SL*(3,Z) instead of ?0(n) and obtain some results for the newly constructed subgraphs Fu,n whose vertices form the block [?]. We obtain edge and circuit conditions and some relations between lengths of circuits in Fu,n and elliptic elements of ?0(n).

Conformally invariant spinorial equations in six dimensions

This work deals with the conformal transformations in six-dimensional spinorial formalism. Several conformally invariant equations are obtained and their geometrical interpretations are worked out. Finally, the integrability conditions for some of these equations are established. Moreover, in the course of the article, some useful identities involving the curvature of the spinorial connection are attained and a digression about harmonic forms and more general massless fields is made.

Plus ultra

We define a reasonably well-behaved class of ultraimaginaries, i.e. classes modulo [Formula: see text]-invariant equivalence relations, called tame, and establish some basic simplicity-theoretic facts. We also show feeble elimination of supersimple ultraimaginaries: If [Formula: see text] is an ultraimaginary definable over a tuple [Formula: see text] with [Formula: see text], then [Formula: see text] is eliminable up to rank [Formula: see text]. Finally, we prove some uniform versions of the weak canonical base property.

Exact renormalization group and loop variables: A background independent approach to string theory

This paper is a self-contained review of the loop variable approach to string theory. The Exact Renormalization Group is applied to a world sheet theory describing string propagation in a general background involving both massless and massive modes. This gives interacting equations of motion for the modes of the string. Loop variable techniques are used to obtain gauge invariant equations. Since this method is not tied to flat space–time or any particular background metric, it is manifestly background independent. The technique can be applied to both open and closed strings. Thus gauge invariant and generally covariant interacting equations of motion can be written for massive higher spin fields in arbitrary backgrounds. Some explicit examples are given.

Conformal invariance of Mei symmetry and conserved quantities of Lagrange equation of thin elastic rod

By Kirchhoff dynamic analogy, the thin elastic rod static equals to rotation of rigid body dynamic. The analytical mechanics methods reflect their advantages in the study of the modeling and equilibrium and stability of elastic rod static, especially for the constrained problems. The Lagrangian structure of the equation of motion for elastic rod is deduced from the integral variational principle. The definition of conformal invariance of Mei symmetry of elastic rod in Lagrangian form is given. The determining equation of conformal invariance of Mei symmetry is obtained based on the Lie point transformation group. The relation between conformal invariance of Mei symmetry and Mei symmetry is discussed. The structure equation and conserved quantity by using the Lagrangian structure along arc coordinate deduced from conformal invariance of Mei symmetry of elastic rod are constructed. Take rod with circular cross section as example to illustrate the application of the results get in this paper. These conserved quantities will be helpful in the study of exact solutions and stability, as well as the numerical simulation of the thin elastic rod nonlinear mechanics.

A unified model with a generalized gauge symmetry and its cosmological implications

A unified model is based on a generalized gauge symmetry with groups [SU3c]color×(SU2×U1)× [U1b×U1l]. It implies that all interactions should preserve conservation laws of baryon number, lepton number, and electric charge, etc. The baryonic U1b, leptonic U1l and color SU3c gauge transformations are generalized to involve nonintegrable phase factors. One has gauge invariant fourth-order equations for massless gauge fields, which leads to linear potentials in the [U1b×U1l] and color [SU3c] sectors. We discuss possible cosmological implications of the new baryonic gauge field. It can produce a very small constant repulsive force between two baryon galaxies (or between two anti-baryon galaxies), where the baryon force can overcome the gravitational force at very large distances and leads to an accelerated cosmic expansion. Based on conservation laws in the unified model, we discuss a simple rotating dumbbell universe with equal amounts of matter and anti-matter, which may be pictured as two gigantic rotating clusters of galaxies. Within the gigantic baryonic cluster, a galaxy will have an approximately linearly accelerated expansion due to the effective force of constant density of all baryonic matter. The same expansion happens in the gigantic anti-baryonic cluster. Physical implications of the generalized gauge symmetry on charmonium confining potentials due to new SU3c field equations, frequency shift of distant supernovae Ia and their experimental tests are discussed.

Braneworld cosmological perturbations in teleparallel gravity

In this paper we find the fully gauge invariant cosmological perturbation equations in braneworld teleparallel gravity. In this theory, perturbations are the result of small fluctuations in the pentad field. We derive the gauge invariant 'potentials' for both geometric and matter variables. In teleparallel gravity, pentad perturbations can only contain scalar and vector modes. This is in contrast to the metric fluctuations in general relativity.

Conformally invariant spin- $$\frac{3}{2}$$ 3 2 field equation in de Sitter space–time

AbstractIn the previous paper (Behroozi et al., Phys Rev D 74:124014, 2006; Dehghani et al., Phys Rev D 77:064028, 2008), conformal invariance for massless tensor fields (scalar, vector and spin-2 fields) was studied and the solutions of their wave equations and two-point functions were obtained. In the present paper, conformally invariant wave equation for massless spinor field in de Sitter space–time has been obtained. For this propose, we use Dirac’s six-cone formalism. The solutions of massless spin-12documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$frac{1}{2}$$end{document} and -32documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$frac{3}{2}$$end{document} equations, in the ambient space notation, have been calculated.

$$C^{\sigma +\alpha }$$ C σ + α regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels

We establish \(C^{\sigma +\alpha }\) interior estimates for concave nonlocal fully nonlinear equations of order \(\sigma \in (0,2)\) with rough kernels. Namely, we prove that if \(u\in C^{\alpha }(\mathbb {R}^n)\) solves in \(B_1\) a concave translation invariant equation with kernels in \(\mathcal L_0(\sigma )\), then u belongs to \(C^{\sigma +\alpha }(\overline{B_{1/2}})\), with an estimate. More generally, our results allow the equation to depend on x in a \(C^\alpha \) fashion. Our method of proof combines a Liouville theorem and a blow-up (compactness) procedure. Due to its flexibility, the same method can be useful in different regularity proofs for nonlocal equations.

Hamiltonian fluid reductions of electromagnetic drift-kinetic equations for an arbitrary number of moments

We present an infinite family of Hamiltonian electromagnetic fluid models for plasmas, derived from drift-kinetic equations. An infinite hierarchy of fluid equations is obtained from a Hamiltonian drift-kinetic system by taking moments of a generalized distribution function and using Hermite polynomials as weight functions of the velocity coordinate along the magnetic guide field. Each fluid model is then obtained by truncating the hierarchy to a finite number N+1 of equations by means of a closure relation. We show that, for any positive N, a linear closure relation between the moment of order N+1 and the moment of order N guarantees that the resulting fluid model possesses a Hamiltonian structure, thus respecting the Hamiltonian character of the parent drift-kinetic model. An orthogonal transformation is identified which maps the fluid moments to a new set of dynamical variables in terms of which the Poisson brackets of the fluid models become a direct sum and which unveils remarkable dynamical properties of the models in the two-dimensional (2D) limit. Indeed, when imposing translational symmetry with respect to the direction of the magnetic guide field, all models belonging to the infinite family can be reformulated as systems of advection equations for Lagrangian invariants transported by incompressible generalized velocities. These are reminiscent of the advection properties of the parent drift-kinetic model in the 2D limit and are related to the Casimirs of the Poisson brackets of the fluid models. The Hamiltonian structure of the generic fluid model belonging to the infinite family is illustrated treating a specific example of a fluid model retaining five moments in the electron dynamics and two in the ion dynamics. We also clarify the connection existing between the fluid models of this infinite family and some fluid models already present in the literature.

Background independence, gauge invariance and equations of motion for closed string modes

In an earlier paper, gauge invariant and background covariant equations for closed string modes were obtained from the exact Renormalization Group of the world sheet theory. The background metric (but not the physical metric) had to be flat and hence the method was not manifestly background independent. In this paper, the restrictions on the background metric are relaxed. A simple prescription for the map from loop variables to space–time fields is given whereby for arbitrary backgrounds the equations are generally covariant and gauge invariant. Extra terms involving couplings of the curvature tensor to (derivatives of) the Stueckelberg fields have to be added. The background metric can thus be chosen to be the physical metric without any restrictions. This method thus gives manifestly background independent equations of motion for both open and closed string modes.