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In this paper we formulate a geometric theory of thermal stresses. Given a temperature distribution, we associate a Riemannian material manifold to the body, with a metric that explicitly depends on the temperature distribution. A change in temperature corresponds to a change in the material metric. In this sense, a temperature change is a concrete example of the so-called referential evolutions. We also make a concrete connection between our geometric point of view and the multiplicative decomposition of deformation gradient into thermal and elastic parts. We study the stress-free temperature distributions of the finite-deformation theory using curvature tensor of the material manifold. We find the zero-stress temperature distributions in nonlinear elasticity. Given an equilibrium configuration, we show that a change in the material manifold, i.e., a change in the material metric will change the equilibrium configuration. In the case of a temperature change, this means that given an equilibrium configuration for a given temperature distribution, a change in temperature will change the equilibrium configuration. We obtain the explicit form of the governing partial differential equations for this equilibrium change. We also show that geometric linearization of the present nonlinear theory leads to governing equations that are identical to those of the classical linear theory of thermal stresses. [ABSTRACT FROM AUTHOR]

Nonisolated focal and conjugate points along a spacelike geodesic are investigated. We give examples of nonisolated focal points and provide conditions for such phenomenon motivated by Helfer’s example of nonisolated conjugate points and Kupeli’s definitions of degenerate focal and conjugate points. [ABSTRACT FROM AUTHOR]

Penrose’s spinor calculus of four-dimensional Lorentzian geometry is extended to the case of five-dimensional Lorentzian geometry. Such fruitful ideas in Penrose’s spinor calculus as the spin covariant derivative, the curvature spinors, or the definition of the spin coefficients on a spin frame can be carried over to the spinor calculus in five-dimensional Lorentzian geometry. The algebraic and differential properties of the curvature spinors are studied in detail, and as an application, we extend the well-known four-dimensional Newman–Penrose formalism to a five-dimensional spacetime. [ABSTRACT FROM AUTHOR]

The subject of limit curve theorems in Lorentzian geometry is reviewed. A general limit curve theorem is formulated, which includes the case of converging curves with endpoints and the case in which the limit points assigned since the beginning are one, two, or at most denumerable. Some applications are considered. It is proved that in chronological spacetimes, strong causality is either everywhere verified or everywhere violated on maximizing lightlike segments with open domain. As a consequence, if in a chronological spacetime two distinct lightlike lines intersect each other then strong causality holds at their points. Finally, it is proved that two distinct components of the chronology violating set have disjoint closures or there is a lightlike line passing through each point of the intersection of the corresponding boundaries. [ABSTRACT FROM AUTHOR]

In a recently published paper in this journal [A. Cheaghlou and O. Faizy, J. Math. Phys. 49, 022104 (2008)], the authors introduce the Gazeau–Klauder coherent states for the trigonometric Rosen–Morse potential as an infinite superposition of the wavefunctions. It is shown that their proposed measure to realize the resolution of the identity condition is not positive definite. Consequently, the claimed coherencies for the trigonometric Rosen–Morse wavefunctions cannot actually exist. [ABSTRACT FROM AUTHOR]

We consider entanglement distillation from a single copy of multipartite quantum states, and instead of rates we analyze the “quality” of the distilled entanglement. This quality is quantified by the fidelity with the gigahertz state. We show that each not fully separable state σ can increase the quality of the entanglement distilled from other states, no matter how weakly entangled is σ. We also generalize this to the case where the goal is distilling states different from the gigahertz. These results provide new insights on the geometry of the set of separable states and its dual (the set of entanglement witnesses). [ABSTRACT FROM AUTHOR]

Geometry can be an unsuspected source of equations with physical relevance, as everybody is aware since Einstein formulated the general theory of relativity. However, efforts to extend a similar type of reasoning to other areas of physics, namely, electrodynamics, quantum mechanics, and particle physics, usually had very limited success; particularly in quantum mechanics the standard formalism is such that any possible relation to geometry is impossible to detect; other authors have previously trod the geometric path to quantum mechanics, some of that work being referred to in the text. In this presentation we will follow an alternate route to show that quantum mechanics has indeed a strong geometric character. The paper makes use of geometric algebra, also known as Clifford algebra, in five-dimensional space-time. The choice of this space is given the character of first principle, justified solely by the consequences that can be derived from such choice and their consistency with experimental results. Given a metric space of any dimension, one can define monogenic functions, the natural extension of analytic functions to higher dimensions; such functions have null vector derivative and have previously been shown by other authors to play a decisive role in lower dimensional spaces. All monogenic functions have null Laplacian by consequence; in a hyperbolic space this fact leads inevitably to a wave equation with planelike solutions. This is also true for five-dimensional space-time and we will explore those solutions, establishing a parallel with the solutions of the free particle Dirac equation. For this purpose we will invoke the isomorphism between the complex algebra of 4×4 matrices, also known as Dirac’s matrices. There is one problem with this isomorphism, because the solutions to Dirac’s equation are usually known as spinors (column matrices) that do not belong to the 4×4 matrix algebra and as such are excluded from the isomorphism. We will show that a solution in terms of Dirac spinors is equivalent to a plane wave solution. Just as one finds in the standard formulation, monogenic functions can be naturally split into positive/negative energy together with left/right ones. This split is provided by geometric projectors and we will show that there is a second set of projectors providing an alternate fourfold split. The possible implications of this alternate split are not yet fully understood and are presently the subject of profound research. [ABSTRACT FROM AUTHOR]

Quantum entanglement was first recognized as a feature of quantum mechanics in the famous paper of Einstein, Podolsky, and Rosen. Recently it has been realized that quantum entanglement is a key ingredient in quantum computation, quantum communication, and quantum cryptography. In this paper, we introduce algebraic sets, which are determinantal varieties in the complex projective spaces or the products of complex projective spaces, for the mixed states on bipartite or multipartite quantum systems as their invariants under local unitary transformations. These invariants are naturally arised from the physical consideration of measuring mixed states by separable pure states. Our construction has applications in the following important topics in quantum information theory: (1) separability criterion, it is proved that the algebraic sets must be a union of the linear subspaces if the mixed states are separable; (2) simulation of Hamiltonians, it is proved that the simulation of semipositive Hamiltonians of the same rank implies the projective isomorphisms of the corresponding algebraic sets; (3) construction of bound entangled mixed states, examples of the entangled mixed states which are invariant under partial transpositions (thus PPT bound entanglement) are constructed systematically from our new separability criterion. [ABSTRACT FROM AUTHOR]

Eigenfunctions of the equation ∇vector ×Bvector =λBvector are found for finite cylindrical geometry with normal boundary condition Bvector ·n⁁=0 and nonaxisymmetric modes ∼eimθ,m≠0. The vector field Bvector 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. [ABSTRACT FROM AUTHOR]

We reexamine here the issue of consistency of minimal action formulation with the minimal coupling procedure (MCP) in spaces with torsion. In Riemann­Cartan spaces, it is known that a proper use of the MCP requires that the trace of the torsion tensor be a gradient, Τµ= аµθ, and that the modified volume element τθ=еθ√gdχ¹^⋯^χn be used in the action formulation of a physical model. We rederive this result here under considerably weaker assumptions, reinforcing some recent results about the inadequacy of propagating torsion theories of gravity to explain the available observational data. The results presented here also open the door to possible applications of the modified volume element in the geometric theory of crystalline defects. [ABSTRACT FROM AUTHOR]

We present new classes of exact solutions with noncommutative symmetries constructed in vacuum Einstein gravity (in general, with nonzero cosmological constant), five-dimensional (5D) gravity and (anti) de Sitter gauge gravity. Such solutions are generated by anholonomic frame transforms and parametrized by generic off-diagonal metrics. For certain particular cases, the new classes of metrics have explicit limits with Killing symmetries but, in general, they may be characterized by certain anholonomic noncommutative matrix geometries. We argue that different classes of noncommutative symmetries can be induced by exact solutions of the field equations in commutative gravity modeled by a corresponding moving real and complex frame geometry. We analyze two classes of black ellipsoid solutions (in the vacuum case and with cosmological constant) in four-dimensional gravity and construct the analytic extensions of metrics for certain classes of associated frames with complex valued coefficients. The third class of solutions describes 5D wormholes which can be extended to complex metrics in complex gravity models defined by noncommutative geometric structures. The anholonomic noncommutative symmetries of such objects are analyzed. We also present a descriptive account how the Einstein gravity can be related to gauge models of gravity and their noncommutative extensions and discuss such constructions in relation to the Seiberg–Witten map for the gauge gravity. Finally, we consider a formalism of vielbeins deformations subjected to noncommutative symmetries in order to generate solutions for noncommutative gravity models with Moyal (star) product. [ABSTRACT FROM AUTHOR]

We consider type II superstring compactifications on the singular Spin(7) manifold constructed as a cone on SU(3)/U(1). Based on a toric realization of the projective space CP2, we discuss how the manifold can be viewed as three intersecting Calabi–Yau conifolds. The geometric transition of the manifold is then addressed in this setting. The construction is readily extended to higher dimensions where we speculate on possible higher-dimensional geometric transitions. Armed with the toric description of the Spin(7) manifold; we discuss a brane/flux duality in both type II superstring theories compactified on this manifold. [ABSTRACT FROM AUTHOR]

We complete the classification of almost commutative geometries from a particle physics point of view given by Iochum, Schücker, and Stephan, J. Math. Phys. (to be published). Four missing Krajewski diagrams will be presented after a short introduction into irreducible, nondegenerate spectral triples. [ABSTRACT FROM AUTHOR]

Using special quasigraded Lie algebras, that could be viewed as deformations of loop algebras, we obtain new hierarchies of integrable nonlinear equations admitting zero-curvature representations. In particular, we obtain integrable hierarchies that generalize the Heisenberg magnet, Landau–Lifshitz, and anisotropic chiral field hierarchies. We also obtain a new type of so(3) anisotropic chiral field equation along with its higher rank generalization. [ABSTRACT FROM AUTHOR]

To analyze linear field equations on a locally homogeneous space–time by means of separation of variables, it is necessary to set up appropriate harmonics according to its symmetry group. In this paper, the harmonics are presented for a spatially compactified Bianchi II cosmological model—the nilgeometric model. Based on the group structure of the Bianchi II group (also known as the Heisenberg group) and the compactified spatial topology, the irreducible differential regular representations and the multiplicity of each irreducible representation, as well as the explicit form of the harmonics are all completely determined. They are also extended to vector harmonics. It is demonstrated that the Klein–Gordon and Maxwell equations actually reduce to systems of ODEs, with an asymptotic solution for a special case. [ABSTRACT FROM AUTHOR]

We classify all irreducible, almost commutative geometries whose spectral action is dynamically nondegenerate. Heavy use is made of Krajewski’s diagrammatic language. The motivation for our definition of dynamical nondegeneracy stems from particle physics where the fermion masses are nondegenerate. [ABSTRACT FROM AUTHOR]

According to a theorem by Treves [Duke Math. J.DUMJAO 108, 251 (2001) ], the conserved functionals of the KdV equation vanish on each formal Laurent series 1/x2+u0+∑k=2+∞ukxk. We propose a new, very simple geometrical proof for this statement. [ABSTRACT FROM AUTHOR]

For the nonrotating Bañados, Teitelboim, and Zanelli black hole, the distributional curvature tensor field is found. It is shown to have singular parts proportional to a δ-distribution with support at the origin. This singularity is related, through Einstein field equations, to a point source. Coordinate invariance and independence on the choice of differentiable structure of the results are addressed. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Cauchy’s formula which relates the mean chord length (isotropic uniform random chords) of a convex body in R[sup n] with its volume and surface is extended to the case of nonconvex bodies in the framework of integral geometry. This allows us to generalize the extended Cauchy’s formula recently discovered by Blanco and Fournier [Europhys. Lett. 61 (2), 168 (2003)], in the field of diffusive random walks, to nonconvex bodies. Monte Carlo simulations illustrate these points in R[sup 2] and in R[sup 3].© 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The geometrical framework developed in J. Math. Phys. 44, 1709 (2003) is applied in order to obtain a geometric description of Routh's procedure. [ABSTRACT FROM AUTHOR]

We introduce the Koenigs lattice, which is a new integrable reduction of the quadrilateral lattice (discrete conjugate net) and provides natural integrable discrete analog of the Koenigs net. We construct the Darboux-type transformation of the Koenigs lattice and we show permutability of superpositions of such transformations, thus proving integrability of the Koenigs lattice. We also investigate the geometry of the discrete Koenigs transformation. In particular we characterize the Koenigs transformation in terms of an involution determined by a congruence conjugate to the lattice. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We present algorithms to evaluate two types of multiple sums, which appear in higher-order loop computations. We consider expansions of a generalized hyper-geometric-type sums, ∑n1,...,nNΓ(a1·n+c1)Γ(a2·n+c2)...Γ(aP·n+cP)Γ(b1·n+d1)Γ(b2·n+d2)...Γ(bQ·n+dQ)x1n1...xNnN with ai·n=∑j=1Naijnj, etc., in a small parameter [variant_greek_epsilon] around rational values of ci,di's. Type I sum corresponds to the case where, in the limit [variant_greek_epsilon] → 0, the summand reduces to a rational function of nj's times x1n1...xNnN; ci,di's can depend on an external integer index. Type II sum is a double sum (N = 2), where ci, di's are half-integers or integers as [variant_greek_epsilon] → 0 and xi = 1; we consider some specific cases where at most six Γ functions remain in the limit [variant_greek_epsilon] → 0. The algorithms enable evaluations of arbitrary expansion coefficients in [variant_greek_epsilon] in terms of Z-sums and multiple polylogarithms (generalized multiple zeta values). We also present applications of these algorithms. In particular, Type I sums can be used to generate a new class of relations among generalized multiple zeta values. We provide a Mathematica package, in which these algorithms are implemented. [ABSTRACT FROM AUTHOR]

In this paper, we define the q-analog Virasoro-like Lie subalgebras in x∞ = a∞(b∞, c∞, d∞). The embedding formulas into x∞ are introduced. Irreducible highest weight representations of Aq, Bq, and Cq-series of the q-analog Virasoro-like Lie algebras in terms of vertex operators are constructed. We also construct the polynomial representations of the Aq, Bq, Cq, and Dq-series of the q-analog Virasoro-like Lie algebras. [ABSTRACT FROM AUTHOR]