General Set Of Solutions Research Articles

Theoretical Study of Quasi-Three-Level Frequency Doubling Laser

Considering the spatial distribution of pumping beam, inverted particle number density, fundamental frequency and photon number density in the cavity of the quasi-three-level frequency-doubled laser, the solution of the rate equation depends on four dimensionless synthetic parameters by normalized correlation parameters. By solving the rate equation numerically, a set of general solutions of the equation are obtained. With these curves and expressions, the design of a quasi-three-level Q-switched intracavity frequency-doubled laser and the prediction of the pulse characteristics could be achieved.

Cosmological horizons, Noether charges and entropy

It has recently been shown that, in the vicinity of their event horizons, black holes exhibit an infinite-dimensional symmetry. This symmetry captures relevant physical information about the black hole, and in particular about its thermodynamics. Here, we show that the same holds in a cosmological setup. More precisely, we show that around the de Sitter cosmological horizon, an infinite set of diffeomorphisms preserving a sensible set of boundary conditions emerges. These boundary conditions are similar to those considered by Price and Thorne in the context of the membrane paradigm, and permit to accommodate interesting gravity solutions. As for other boundary conditions considered previously, they are preserved by an infinite-dimensional asymptotic isometry algebra that includes supertranslations and two copies of the Virasoro algebra. This symmetry has associated a set of Noether charges that turn out to be conserved, finite, and integrable. We derive these charges explicitly using the covariant formalism and analyze their physical meaning by evaluating them to the case of de Sitter space. In this case, the zero-mode of the charge is found to account for the Gibbons–Hawking entropy of the cosmological horizon. We then consider a much more general set of solutions, including asymptotically de-Sitter black holes and asymptotically Taub–NUT–de Sitter black holes.

A multi-field coupled mechanical-electric-magnetic-chemical-thermal (MEMCT) theory for material systems

A material system may have rich physical and chemical properties; it can respond to the action of multifields. This paper reports the development of a generalized constitutive theory, based on continuous thermodynamics, to describe the multifield nature and dissipative behaviors of a material system, which couples the mechanical, electric, magnetic, chemical, and thermal (MEMCT) aspects of material characteristics. This theory also deals with chemical diffusions at different time scales subjected to an electromagnetic field and subsequent chemical reactions occurring between the diffusive species and the host material, and the dynamic evolutions of charged species subjected to thermomagnetoelectric actions and related forces.Problems pertaining to a magnetoelectroelastic (MEE) body sensitive to thermal–chemical diffusions are analytically solved based on the proposed MEMCT theory. Transversely isotropic materials, which are widely encountered in MEE designs are modeled. With the assistance of the operator theory, a set of general solutions for elastic displacements, stresses, electric/magnetic potentials, electric displacements, magnetic inductions, temperature, and chemical potentials of involved species, subjected to MEMCT coupling, is derived. The presented MEMCT theory and the general solutions provide a basis for computing and simulating interfaces of complex material systems in MEMCT multifield interactions.

Axisymmetric boundary condition problems for transversely isotropic piezoelectric materials

This paper presents an exact analysis of deformation and stress field in a finite cylinder under axisymmetric boundary conditions. The problem is formulated based on the state space formulation for transversely isotropic piezoelectric media. Upon the separation of variables method, we get a set of general solutions spanning over the complete solution space to cover all axisymmetric boundary conditions. Furthermore, a rigorous solution which satisfies the end conditions is determined by using eigenfunction expansion. Numerical results show that the local effect is significant but confined to boundary layer near the fixed end where the displacement and stress are remarkably different from those of other regions.

Geometry of spinning Ellis wormholes

We give a detailed account of the properties of spinning Ellis wormholes, supported by a phantom field. The general set of solutions depends on three parameters, associated with the size of the throat, the rotation and the symmetry of the solutions. For symmetric wormholes the global charges possess the same values in both asymptotic regions, while this is no longer the case for non-symmetric wormholes. We present mass formulae for these wormholes, study their quadrupole moments, and discuss the geometry of their throat and their ergoregion. We demonstrate, that these wormholes possess limiting configurations corresponding to an extremal Kerr black hole. Moreover, we analyze the geodesics of these wormholes, and show that they possess bound orbits.

Relativistic electromagnetic mass models in spherically symmetric spacetime

Under the static spherically symmetric Einstein-Maxwell spacetime of embedding class one we explore possibility of electromagnetic mass model where mass and other physical parameters have purely electromagnetic origin (Tiwari 1984, Gautreau 1985, Gron 1985). This work is in continuation of our earlier investigation (Maurya 2015a) where we developed an algorithm and found out three new solutions of electromagnetic mass models. In the present letter we consider different metric potentials $\nu$ and $\lambda$ and analyzed them in a systematic way. It is observed that some of the previous solutions related to electromagnetic mass models are nothing but special cases of the presently obtained generalized solution set. We further verify the solution set and show that these are extremely applicable in the case of compact stars as well as for understanding structure of the electron.

Generalized solution sets of the interval generalized Sylvester matrix equation [formula omitted] and some approaches for inner and outer estimations

In this work, we extend the concept of generalized solution sets that is introduced for the first time by Shary (1996, 1999), to the interval generalized Sylvester matrix equation ∑i=1pAiXi+∑j=1qYjBj=C. Then AE-solution sets for this equation will be characterized and we develop some approaches (called algebraic approaches) in which the inner and outer estimation problems for some special cases of AE-solution sets reduce to problems of computing algebraic solutions of some auxiliary interval equations. Also we present a numerical technique that is an extension of the well-known interval Gauss–Seidel method for outer estimation of the solution set.

Dynamics of a scalar field in Robertson-Walker spacetimes

We analyze the dynamics of a single scalar field in Friedmann-Robertson-Walker universes with spatial curvature. We obtain the fixed point solutions which are shown to be late time attractors. In particular, we determine the corresponding scalar field potentials which correspond to these stable solutions. The analysis is quite general and incorporates expanding and contracting universes with both positive and negative scalar potentials. We demonstrate that the known power law, exponential, and de Sitter solutions are certain limits of our general set of solutions.

Eigenfunctions of the curl in cylindrical geometry

Eigenfunctions of the equation ∇⃗×B⃗=λB⃗ are found for finite cylindrical geometry with normal boundary condition B⃗∙n̂=0 and nonaxisymmetric modes ∼eimθ,m≠0. The vector field B⃗ can be represented by a scalar generating function of the Chandrasekhar-Kendall type with radial Bessel functions for the nondegenerate cases. A general set of solutions can also be generated by transformation of variables. A series solution in terms of radial Bessel functions is found which has excellent convergence properties (an∼1∕n4) and a robust method of locating eigenvalues is described.

Flux compactification of M‐theory on compact manifolds with Spin(7) holonomy

AbstractAt the leading order, M‐theory admits minimal supersymmetric compactifications if the internal manifold has exceptional holonomy. The inclusion of non‐vanishing fluxes in M‐theory and string theory compactifications induce a superpotential in the lower dimensional theory, which depends on the fluxes. In this work, we check the conjectured form of this superpotential in the case of warped M‐theory compactifications on Spin(7) holonomy manifolds. We perform a Kaluza‐Klein reduction of the eleven‐dimensional supersymmetry transformation for the gravitino and we find by direct comparison the superpotential expression. We check the conjecture for the heterotic string compactified on a Calabi‐Yau three‐fold as well. The conjecture can be checked indirectly by inspecting the scalar potential obtained after the compactification of M‐theory on Spin(7) holonomy manifolds with non‐vanishing fluxes. The scalar potential can be written in terms of the superpotential and we show that this potential stabilizes all the moduli fields describing deformations of the metric except for the radial modulus.All the above analyses require the knowledge of the minimal supergravity action in three dimensions. Therefore we calculate the most general causal 𝒩 =1 three‐dimensional, gauge invariant action coupled to matter in superspace and derive its component form using Ectoplasmic integration theory. We also show that the three‐dimensional theory which results from the compactification is in agreement with the more general supergravity construction.The compactification procedure takes into account higher order quantum correction terms in the low energy effective action. We analyze the properties of these terms on a Spin(7) background. We derive a perturbative set of solutions which emerges from a warped compactification on a Spin(7) holonomy manifold with non‐vanishing flux for the M‐theory field strength and we show that in general the Ricci flatness of the internal manifold is lost, which means that the supergravity vacua are deformed away from the exceptional holonomy. Using the superpotential form we identify the supersymmetric vacua out of this general set of solutions.

A hybrid genetic programming approach for the analytical solution of differential equations

This paper presents a novel addition to the current genetic programming techniques for solving differential equations. Rather than using numerical approximation of derivatives during fitness evaluation, automatically computed analytical derivatives of the candidate solutions are employed. Because analytical derivatives are used, symbolic constants can be incorporated in the solution. This permits the development of a single solution for a range of material properties, boundary conditions or other design parameters. Additionally, for the special case of linear differential equations, a modified Gram–Schmidt algorithm is used to reduce the set of general solutions located by genetic programming to a basis set.

Two-channel constrained least squares problems: solutions using power methods and connections with canonical coordinates

The problem of two-channel constrained least squares (CLS) filtering under various sets of constraints is considered, and a general set of solutions is derived. For each set of constraints, the solution is determined by a coupled (asymmetric) generalized eigenvalue problem. This eigenvalue problem establishes a connection between two-channel CLS filtering and transform methods for resolving channel measurements into canonical or half-canonical coordinates. Based on this connection, a unified framework for reduced-rank Wiener filtering is presented. Then, various representations of reduced-rank Wiener filters in canonical and half-canonical coordinates are introduced. An alternating power method is proposed to recursively compute the canonical coordinate and half-canonical coordinate mappings. A deflation process is introduced to extract the mappings associated with the dominant coordinates. The correctness of the alternating power method is demonstrated on a synthesized data set, and conclusions are drawn.

Formula omitted]-theory vacua from warped compactifications on [formula omitted] manifolds

At the leading order, M -theory admits minimal supersymmetric compactifications if the internal manifold has exceptional holonomy. Once we take into account higher order quantum correction terms in the low energy effective action, the supergravity vacua have to be deformed away from the exceptional holonomy if we want to preserve the supersymmetry of the solution. In this paper we discuss the Spin ( 7 ) holonomy case. We derive a perturbative set of solutions which emerges from a warped compactification with non-vanishing flux for the M -theory field strength and we identify the supersymmetric vacua out of this general set of solutions. For this purpose we have to evaluate the value of the quartic polynomial J 0 on a Spin ( 7 ) holonomy manifold as well as its first variation with respect to the internal metric. We show that in general the Ricci flatness of the internal manifold is lost.

A New Technique in Systems Analysis under Interval Uncertainty and Ambiguity

The main subject of this work is mathematical and computational aspects of modeling of static systems under interval uncertainty and/or ambiguity. A cornerstone of the new approach we are advancing in the present paper is, first, the rigorous and consistent use of the logical quantifiers to characterize and distinguish different kinds of interval uncertainty that occur in the course of modeling, and, second, the systematic use of Kaucher complete interval arithmetic for the solution of problems that are minimax by their nature. As a formalization of the mathematical problem statement, concepts of generalized solution sets and AE-solution sets to an interval system of equations, inequalities, etc., are introduced. The major practical result of our paper is the development of a number of techniques for inner and outer estimation of the so-called AE-solution sets to interval systems of equations. We work out, among others, formal approach, generalized interval Gauss-Seidel iteration, generalized preconditioning and PPS-methods. Along with the general nonlinear case, the linear systems are treated more thoroughly.

Convergence and efficiency of subgradient methods for quasiconvex minimization

We study a general subgradient projection method for minimizing a quasiconvex objective subject to a convex set constraint in a Hilbert space. Our setting is very general: the objective is only upper semicontinuous on its domain, which need not be open, and various subdifferentials may be used. We extend previous results by proving convergence in objective values and to the generalized solution set for classical stepsizes t k →0, ∑t k =∞, and weak or strong convergence of the iterates to a solution for {t k }∈l2∖l1 under mild regularity conditions. For bounded constraint sets and suitable stepsizes, the method finds e-solutions with an efficiency estimate of O(e-2), thus being optimal in the sense of Nemirovskii.

Interval gauss-seidel method for generalized solution sets to interval linear systems

In the paper, we advance a numerical technique for enclosing generalized AE-solution sets to interval linear systems. The main result of the paper is an extension of the well-known interval Gauss-Seidel method to the problems of outer estimation of these generalized solution sets. We give a theoretical study of the new method, prove an optimality property for the generalized interval Gauss-Seidel iteration applied to the systems with the interval M-matrices.

Continuous Selectors of Fixed Point Sets of Multifunctions with Decomposable Values

Existence theorems of continuous selectors whose values are fixed points of multivalued contractions are proved. As an application, the existence of continuous selectors to generalized solution sets of partial differential inclusions, depending on a parameter, is obtained.

The extended linear complementarity problem

In this paper we define the Extended Linear Complementarity Problem (ELCP), an extension of the well-known Linear Complementarity Problem (LCP). We show that the ELCP can be viewed as a kind of unifying framework for the LCP and its various generalizations. We study the general solution set of an ELCP and we develop an algorithm to find all its solutions. We also show that the general ELCP is an NP-hard problem.

Comment on "Perturbation expansion of closed-time-path Green's functions"

The incoming interaction picture found by Xu and Tsai [Phys. Rev. A 41, 53 (1990)], used to develop a convenient formulation of the closed-time-path approach to Green's function theory, is shown to be a particular case of a more general set of solutions.

Anomaly-free gauged U(1)' in local supersymmetry and baryon-number violation

The supersymmetric extension of the standard model suffers from a problem of baryon-number violation. Discrete (and global) symmetries introduced to protect the proton are unstable under gravitational effects. We add a gauged U(1) X to the standard model gauge group G SM and require it to be anomaly-free. As new (chiral) superfields we only allow G SM-singlets in order to maintain the good unification predictions. We find the most general set of solutions for the rational singlet charges. We embed our models in local supersymmetry and study the breaking of supersymmetry and U(1) X to determine M X . We determine the full non-renormalizable and gauge invariant Lagrangian for the different solutions. We expect any effective theory to contain baryon- and lepton-number violating terms of dimension four suppressed by powers of M X / M Pl. The power is predicted by the U(1) X charges. We find consistency with the experimental bounds on the proton lifetime and on the neutrino masses. We also expect all supersymmetric models to have an unstable but longlived lightest supersymmetric particle. Consistency with underground experiments on upward going muons leads to stricter constraints than the proton decay experiments. These are barely satisfied.