General Local Solution Research Articles

An improved memetic algorithm for the flexible job shop scheduling problem with transportation times

In the practical production, the transportation of jobs is existed between different machines. These transportation operations directly affect the production cycle and the production efficiency. In this study, an improved memetic algorithm is proposed to solve the flexible job shop scheduling problem with transportation times, and the optimization objective is minimizing the makespan. In the improved memetic algorithm, an effective simulated annealing algorithm is adopted in the local search process, which combines the elite library and mutation operation. All the feasible solutions are divided into general solutions and local optimal solutions according to the elite library. The general solutions are executed by the simulated annealing algorithm to improve the quality, and the local optimal solutions are executed by the mutation operation to increase the diversity of the solution set. Comparison experiments with the improved genetic algorithm show that the improved memetic algorithm has better search performance and stability.

Local similarity solution for steady laminar planar jet flow of viscoelastic FENE-P fluids

Three local self-similar solutions are obtained for the laminar planar jet flow of viscoelastic fluids, described by the FENE-P constitutive equation, through an order of magnitude simplification of the governing equations. The more general solution is shown to be more accurate than two further simplified solutions, here called the delta and the Olagunju-type solutions, at least for the profiles of conformation tensor components. In the limit of vanishing viscoelasticity, all conformation tensor components reduce to the same low elasticity asymptotic behavior, and the polymer stresses become Newtonian-like. The general solution is then used to obtain the laws of the decay of the centerline velocity and of the growth of the jet half-width and to ascertain the effects of the Weissenberg number, maximum polymer extensibility parameter and the ratio of polymer to total viscosities upon the flow characteristics. The general local self-similar solution compares well with results of numerical simulations obtained by the RheoFoam module of the freeware OpenFoam code.

Dispersionless integrable hierarchies andGL(2, ℝ) geometry

AbstractParaconformal orGL(2, ℝ) geometry on ann-dimensional manifoldMis defined by a field of rational normal curves of degreen– 1 in the projectivised cotangent bundle ℙT*M. Such geometry is known to arise on solution spaces of ODEs with vanishing Wünschmann (Doubrov–Wilczynski) invariants. In this paper we discuss yet another natural source ofGL(2, ℝ) structures, namely dispersionless integrable hierarchies of PDEs such as the dispersionless Kadomtsev–Petviashvili (dKP) hierarchy. In the latter context,GL(2, ℝ) structures coincide with the characteristic variety (principal symbol) of the hierarchy.Dispersionless hierarchies provide explicit examples of particularly interesting classes of involutiveGL(2, ℝ) structures studied in the literature. Thus, we obtain torsion-freeGL(2, ℝ) structures of Bryant [5] that appeared in the context of exotic holonomy in dimension four, as well as totally geodesicGL(2, ℝ) structures of Krynski [33]. The latter possess a compatible affine connection (with torsion) and a two-parameter family of totally geodesicα-manifolds (coming from the dispersionless Lax equations), which makes them a natural generalisation of the Einstein–Weyl geometry.Our main result states that involutiveGL(2, ℝ) structures are governed by a dispersionless integrable system whose general local solution depends on 2n– 4 arbitrary functions of 3 variables. This establishes integrability of the system of Wünschmann conditions.

Global half-BPS AdS2 \xd7 S6 solutions in Type IIB

We investigate half-BPS Type IIB supergravity solutions with spacetime geometry AdS2 × S6 warped over a Riemann surface Σ. The general local solution was obtained in earlier work in terms of two holomorphic functions {mathcal{A}}_{pm } on Σ. In the first part of this paper we seek global solutions corresponding to the near-horizon behavior of (p, q)-string junctions. We identify the type of singularity in {mathcal{A}}_{pm } needed at the boundary of Σ to match the solutions locally onto the classic (p, q)-string solution. We construct solutions with multiple (p, q)-strings, but the existence of geodesically complete solutions remains unsettled. In a second part we construct multi-parameter families of non-compact globally regular and geodesically complete solutions with asymptotic regions and an AdS2 throat which caps off smoothly.

All solutions of Einstein’s equations in 2+1 dimensions: Λ-vacuum, pure radiation, or gyratons

Under a weak assumption of the existence of a geodesic null congruence, we present the general solution of the Einstein field equations in three dimensions with any value of the cosmological constant, admitting an aligned null matter field, and also gyratons (a matter field in the form of a null dust with an additional internal spin). The general local solution consists of the expanding Robinson–Trautman class and the non-expanding Kundt class. The gyratonic solutions reduce to spacetimes with a pure radiation matter field when the spin is set to zero. Without matter fields, we obtain new forms of the maximally symmetric vacuum solutions. We discuss these complete classes of solutions and their various subclasses. In particular, we identify the gravitational field of an arbitrarily accelerating source (the Kinnersley photon rocket, which reduces to a Vaidya-type non-moving object) in the Robinson–Trautman class, and pp-waves, vanishing scalar invariants (VSI) spacetimes, and constant scalar invariants (CSI) spacetimes in the Kundt class.

Propagation of electromagnetic waves through non-uniform dielectric layers

The propagation of TE- and TM-polarized electromagnetic (EM) waves through a dielectric layer with spatial variation of the refractive index is reported in this paper. The relative permittivity of the layer is assumed to be graded along the thickness direction, and its spatial variation is described by a quartic polynomial. The corresponding mode equations are Helmholtz equations with variable coefficients that can be transformed to a triconfluent Heun equation, a local exact solution of which is expressed in terms of triconfluent Heun functions. The solutions cover many particular cases owing to a variability of four optional parameters (coefficients) of the quartic polynomial. The general local solution for a TE-polarized EM wave is employed for the calculation of transmission properties of a periodic one-dimensional photonic crystal using the Floquet theory.

AdS2 \xd7 S6 versus AdS6 \xd7 S2 in Type IIB supergravity

We obtain the complete local solutions with 16 supersymmetries to Type IIB supergravity on a space-time of the form AdS2 × S6 warped over a Riemann surface Σ in terms of two locally holmorphic functions on Σ. We construct the general Ansatz for the bosonic supergravity fields and supersymmetry generators compatible with the SO(2, 1) ⊕ SO(7) isometry algebra of space-time, which extends to the corresponding real form of the exceptional Lie superalgebra F (4). We reduce the BPS equations to this Ansatz, obtain their general local solutions, and show that these local solutions solve the full Type IIB supergravity field equations and Bianchi identities. We contrast the AdS2 × S6 solution with the closely related AdS6 × S2 case and present our results for both in parallel. Finally, we present a preliminary analysis of positivity and regularity conditions for AdS2 × S6, but postpone the construction of globally regular solutions to a subsequent paper.

Warped AdS6 \xd7 S2 in Type IIB supergravity II: global solutions and five-brane webs

Motivated by the construction of holographic duals to five-dimensional superconformal quantum field theories, we obtain global solutions to Type IIB supergravity invariant under the superalgebra F (4) on a space-time of the form AdS6 × S2 warped over a two-dimensional Riemann surface Σ. In earlier work, the general local solutions were expressed in terms of two locally holomorphic functions mathcal{A} ± on Σ and global solutions were sketched when Σ is a disk. In the present paper, the physical regularity conditions on the supergravity fields required for global solutions are implemented on mathcal{A} ± for arbitrary Σ. Global solutions exist only when Σ has a non-empty boundary ∂Σ. The differentials ∂ mathcal{A} ± are allowed to have poles only on ∂Σ and each pole corresponds to a semi-infinite (p, q) five-brane. The construction for the disk is carried out in detail and the conditions for the existence of global solutions are articulated for surfaces with more than one boundary and higher genus.

Exact half-BPS string-junction solutions in six-dimensional supergravity

We construct SO(2,1) x SO(3)-invariant half-BPS solutions in six-dimensional (0,4) supergravity with m tensor multiplets. The space-time manifold of each one of these solutions consists of an AdS_2 x S^2 warped over a Riemann surface Sigma with boundary. The most general local solution is parametrized by one real harmonic function, and m+2 holomorphic functions which are subject to a quadratic constraint and a hermitian inequality, both of which are manifestly SO(2,m) invariant. Imposing suitable conditions on these harmonic and holomorphic functions, we construct globally regular supergravity solutions with N distinct AdS_3 x S^3 asymptotic regions and contractible Sigma. These solutions have an intricate moduli space, whose dimension equals 2(m+1)N -m-2 and matches the counting of three-form charge vectors and un-attracted scalars of the tensor multiplet. Exact explicit formulas for all supergravity fields are obtained in terms of the moduli. Our solutions give the near-horizon geometries for junctions of N self-dual strings in six dimensions, and are holographic duals to CFTs defined on N half-planes which share a common interface line. For m=5, the solutions lift to quarter-BPS solutions for six-dimensional (4,4) supergravity.

Exact half-BPS flux solutions in M-theory II: global solutions asymptotic toAdS7×S4

General local half-BPS solutions in M-theory, which have SO(2, 2) × SO(4) × SO(4) symmetry and are asymptotic to AdS7 × S4, were constructed in exact form by the authors in [arXiv:0806.0605]. In the present paper, suitable regularity conditions are imposed on these local solutions, and corresponding globally well-defined solutions are explicitly constructed. The physical properties of these solutions are analyzed, and interpreted in terms of the gravity duals to extended 1+1-dimensional half-BPS defects in the 6-dimensional CFT with maximal supersymmetry.

Exact half-BPS flux solutions in M-theory I local solutions

The complete eleven-dimensional supergravity solutions with 16 supersymmetries on manifolds of the form $AdS_3 \times S^3 \times S^3 \times \Sigma$, with isometry $SO(2,2) \times SO(4) \times SO(4)$, and with either $AdS_4 \times S^7$ or $AdS_7 \times S^4$ boundary behavior, are obtained in exact form. The two-dimensional parameter space $\Sigma$ is a Riemann surface with boundary, over which the product space $AdS_3 \times S^3 \times S^3$ is warped. By mapping the reduced BPS equations to an integrable system of the sine-Gordon/Liouville type, and then mapping this integrable system onto a linear equation, the general local solutions are constructed explicitly in terms of one harmonic function on $\Sigma$, and an integral transform of two further harmonic functions on $\Sigma$. The solutions to the BPS equations are shown to automatically solve the Bianchi identities and field equations for the 4-form field, as well as Einstein's equations. The solutions we obtain have non-vanishing 4-form field strength on each of the three factors of $AdS_3 \times S^3 \times S^3$, and include fully back-reacted M2-branes in $AdS_7 \times S^4$ and M5-branes in $AdS_4 \times S^7$. No interpolating solutions exist with mixed $AdS_4 \times S^7$ and $AdS_7 \times S^4$ boundary behavior. Global regularity of these local solutions, as well as the existence of further solutions with neither $AdS_4 \times S^7$ nor $AdS_7 \times S^4$ boundary behavior will be studied elsewhere.

Dual-Family Viscous Shock Waves in n Conservation Laws with Application to Multi-Phase Flow in Porous Media

We consider shock waves satisfying the viscous profile criterion in general systems of n conservation laws. We study Si, j dual-family shock waves, which are associated with a pair of characteristic families i and j. We explicitly introduce defining equations relating states and speeds of Si, j shocks, which include the Rankine–Hugoniot conditions and additional equations resulting from the viscous profile requirement. We then develop a constructive method for finding the general local solution of the defining equations for such shocks and derive formulae for the sensitivity analysis of Si, j shocks under change of problem parameters. All possible structures of solutions to the Riemann problems containing Si, j shocks and classical waves are described. As a physical application, all types of Si, j shocks with i>j are detected and studied in a family of models for multi-phase flow in porous media.

Eight-manifolds withG structure in eleven dimensional supergravity

We extend the refined G-structure classification of supersymmetric solutions of eleven dimensional supergravity. We derive necessary and sufficient conditions for the existence of an arbitrary number of Killing spinors whose common isotropy group contains a compact factor acting irreducibly in eight spatial dimensions and which embeds in $(Spin(7)\ltimes\mathbb{R}^8)\times\mathbb{R}$. We use these conditions to explicitly derive the general local bosonic solution of the Killing spinor equation admitting an N=4 SU(4) structure embedding in a $(Spin(7)\ltimes\mathbb{R}^8)\times\mathbb{R}$ structure, up to an eight-manifold of SU(4) holonomy. Subject to very mild assumptions on the form of the metric, we explicitly derive the general local bosonic solutions of the Killing spinor equation for N=6 Sp(2) structures and N=8 $SU(2)\times SU(2)$ structures embedding in a $(Spin(7)\ltimes\mathbb{R}^8)\times\mathbb{R}$ structure, again up to eight-manifolds of special holonomy. We construct several other classes of explicit solutions, including some for which the preferred local structure group defined by the Killing spinors does not correspond to any holonomy group in eleven dimensions. We also give a detailed geometrical characterisation of all supersymmetric spacetimes in eleven dimensions admitting G-structures with structure groups of the form $(G\ltimes\mathbb{R}^8)\times\mathbb{R}$.

Classification of Supersymmetric Spacetimes in Eleven Dimensions

We derive, for spacetimes admitting a Spin(7) structure, the general local bosonic solution of the Killing spinor equation of 11-dimensional supergravity. The metric, four-form, and Killing spinors are determined explicitly, up to an arbitrary eight-manifold of Spin(7) holonomy. It is sufficient to impose the Bianchi identity and one particular component of the four-form field equation to ensure that the solution of the Killing spinor equation also satisfies all the field equations, and we give these conditions explicitly.

Target-superspace in 2d dilatonic supergravity

The N=1 supersymmetric version of generalized 2d dilaton gravity can be cast into the form of a Poisson σ-model, where the target space and its Poisson bracket are graded. The target space consists of a 1+1 superspace and the dilaton, which is the generator of Lorentz boosts therein. The Poisson bracket on the target space induces the invariance of the worldsheet theory against both diffeomorphisms and local supersymmetry transformations (superdiffeomorphisms). The machinery of Poisson σ-models is then used to find the general local solution to the field equations. As a byproduct, classical equivalence between the bosonic theory and its supersymmetric extension is found.

General minimal surface solution for gravitational instantons

We construct the general instanton metric obtained from Weierstrass' general local solution for minimal surfaces using the correspondence between minimal surfaces in three-dimensional Euclidean space and gravitational instantons admitting two Killing vectors. The resulting metric contains one arbitrary analytic function and we show that it can be transformed to the Gibbons-Hawking form of an instanton metric that was reported earlier.

Self-duality equation: Monodromy matrices and algebraic curves

It is shown that the general local solution of the self-duality equation with SU(1,1) and SU(2) gauge groups is associated with some algebraic curve with moving branch points if the related “monodromy matrix” is rational. The “multisoliton” solutions including monopoles and instantons, correspond to degenerate curves when the branch cuts collapse to double points. Bibliography:18 titles.

Generalization of analytical numerical matching for structural acoustic scattering with discontinuities using a composite Green’s function approach

Analytical numerical matching (ANM), when applied to structural acoustics, efficiently models structural discontinuities by superposing high-resolution local analytical solutions at discontinuities upon a low-resolution global numerical solution. This composite solution is very accurate and computationally efficient. Previously, ANM composite solutions have been developed for several specific problem geometries. A method for treating general constraint configurations using ANM is presented using the Green’s function concept, which models general structural constraints as an integration of point force solutions of the unconstrained structure. The general local solution is found by integration of the ANM local solution derived in the case of a point force. The point force local solution is analytic, known explicitly, and independent of overall geometry. The complementary global solution is far smoother than the original problem and, therefore, can be more efficiently calculated. In short, a general discretely constrained problem is replaced by a smoothed constrained problem, plus a local analytic correction. The method is demonstrated by application to scattering from an infinitely long cylindrical shell with periodic bulkheads. Accuracy and convergence of the structural response and scattered pressure are compared using a classical modal solution, a previous ANM modal solution, and the present general method. [Work supported by ONR.]

A new form of Einstein’s equations

We derive the field equations of general relativity from a complex Lagrangian of particular simplicity. In this way we exhibit that Einstein’s vacuum equations are equivalent to the existence of certain covariantly constant vector- and bivector-valued two-forms. We discuss the implications this brings along for the exploration of new (hidden) symmetries as well as for the construction of new solutions. We even engage the equations’ simplicity for the construction of the most general (local) solution of the vacuum equations of general relativity and show that formally the general solution can be expressed in terms of Weyl’s conformal tensor and several arbitrary forms acting as ‘‘constants of integration.’’ In a 3+1 decomposition we demonstrate how the simple structure of the Lagrangian equations continues for the Hamilton equations, and compare the result with Ashtekar’s approach.

The structure of all extended supersymmetric self-dual gauge theories

Extended super self-dual systems have a structure reminiscent of a “matreoshka”: the N-extended theory has a structure largely determined by the ( N−1)-extended theory, and ultimately by the N = 0 core. Consequences of this phenomenon are explored. In particular, we present an explicit construction of local solutions of the higher- N super self-duality equations starting from any N = 0 self-dual solution. Our construction uses N = 0 solution data to produce N = 1 solution data, which in turn yields N = 2 solution data, and so on, each stage introducing a dependence of the solution on sufficiently many additional arbitrary functions to yield the most general supersymmetric solution having the initial N = 0 solution as the helicity +1 component. The problem of finding the general local solution of the N > 0 super self-duality equations therefore reduces to finding the general solution of the usual ( N = 0) self-duality equations. Another consequence of the matreoshka phenomenon is the vanishing of many observed currents, including the supercurrents, for super self-dual systems.