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We introduce a partial differential equation whose structure, in many respects, mimics that of the second Painlevé equation. For this equation we give auto-Bäcklund transformations of both ordinary differential equation and partial differential equation types. Our discussion is sufficiently generally phrased to allow us to draw some general conclusions about the connection between auto-Bäcklund transformations and the integrability of ordinary and partial differential equations. [ABSTRACT FROM AUTHOR]

We prove that all the meromorphic solutions of the nonlinear differential equation c0u‴+6u4+c1u″+c2uu′+c3u3+c4u′+c5u2+c6u+c7=0 are elliptic or degenerate elliptic, and we build them explicitly. [ABSTRACT FROM AUTHOR]

We study classes of indecomposable representations of the Diamond Lie algebra: a four-dimensional solvable Lie algebra, which is the central extension of the Poincaré Lie algebra in two dimensions. These representations are obtained first (inspired by Douglas and Premat [Commun. Algebra 35, 1433 (2007)]) by embedding the Diamond Lie algebra in the simple Lie algebras A2 and C2, and then by regarding it as a subalgebra of a truncated current Lie algebra. [ABSTRACT FROM AUTHOR]

We analyze the optimal unambiguous discrimination of two arbitrary mixed quantum states. We show that the optimal measurement is unique and we present this optimal measurement for the case where the rank of the density operator of one of the states is at most 2 (“solution in four dimensions”). The solution is illustrated by some examples. The optimality conditions proven by Eldar et al. [Phys. Rev. A 69, 062318 (2004)] are simplified to an operational form. As an application we present optimality conditions for the measurement, when only one of the two states is detected. The current status of optimal unambiguous state discrimination is summarized via a general strategy. [ABSTRACT FROM AUTHOR]

We present an exact solution of the three-dimensional Duffin–Kemmer–Petiau oscillator for spins 1 and 0 in the momentum space with the presence of minimal length uncertainty by the technique of vector spherical harmonics. The eigenfunctions are determined for both cases and the energy eigenvalues equation are obtained. The limiting case is then deduced for a small parameter of deformation. [ABSTRACT FROM AUTHOR]

In this paper we study the well posedness of solution to a certain system of nonlinear Klein–Gordon–Schrödinger equations in three space dimensions. Basing on the Strichartz estimates, we obtain the global existence, uniqueness of the solutions, and continuous dependence with respect to initial data in the Sobolev spaces of low regularities by setting appropriate contraction and taking difference estimates. [ABSTRACT FROM AUTHOR]

Recently, the first author [Adachi, “Asymptotic completeness for N-body quantum systems with long-range interactions in a time-periodic electric field,” Commun. Math. Phys. 275, 443 (2007)] proved the asymptotic completeness for N-body quantum systems with long-range interactions in a time-periodic electric field whose mean in time is nonzero by obtaining propagation estimates for the physical propagator. However, in his work, it is needed that potentials under consideration are sufficiently smooth. In this paper, when N=2, we prove the asymptotic completeness of (modified) wave operators under the assumption that the potential has the local singularity of type |x|-1+ε when d≥3 and 0<ε<1, where d is the space dimension. We also discuss the modifiers in the position representation, which are used in the definition of the modified wave operators in the long-range case. [ABSTRACT FROM AUTHOR]

Polar coordinates are used for the complex scalar free field in D=4 dimensions. The resulting nonrenormalizable theory is healed by using a recently proposed symmetric subtraction procedure. The existence of the coordinate transformation is proved by construction. [ABSTRACT FROM AUTHOR]

Starting from a four-by-four matrix spectral problem, an integrable coupling family of the Toda lattice systems is derived. A pair of discrete Hamiltonian operators is presented, and a sequence of the corresponding Hamiltonian functionals is constructed by using the discrete variational identity. Then, the bi-Hamiltonian structure of the obtained integrable coupling family is established. Finally, a nonisospectral integrable lattice family associated with the obtained integrable coupling family is given through nonisospectral discrete zero curvature representation. [ABSTRACT FROM AUTHOR]

An invariant helicity integral and a differential helicity evolution equation are found for viscous fluid flows. A geometrodynamical approach is used, which includes a vortex field. The vortex field is derivable from a vector potential A. The vector potential is then used to characterize the evolution of flow topology. The source of the helicity is found to be the topological parity k=2λω·ζ and the moving boundary surfaces of the fluid. Here, ω and ζ are the vorticity and swirl components of the vortex field {ω,ζ} and λ is a constitutive or material parameter of the fluid. Our first result using the vector calculus identifies the scalar helicity as ht=Aω. This result is then generalized using the calculus of differential forms, yielding other results including the existence of a helicity current vector proportional to ([lowercase_phi_synonym]ω-λA×ζ). [ABSTRACT FROM AUTHOR]

Globally regular (i.e., asymptotically flat and regular interior) spherically symmetric and localized (“particlelike”) solutions of the coupled Einstein Yang–Mills (EYM) equations with gauge group SU(2) have been known for more than 20 years, yet their properties are still not well understood. Spherically symmetric Yang–Mills fields are classified by a choice of isotropy generator and SO(5) is distinguished as the simplest gauge group having a model with a non-Abelian residual (little) group, SU(2)×U(1), which admits globally regular particlelike solutions. We exhibit an algebraic gauge condition which normalizes the residual gauge freedom to a finite number of discrete symmetries. This generalizes the well-known reduction to the real magnetic potential w(r,t) in the original SU(2) YM model. Reformulating using gauge-invariant polynomials dramatically simplifies the system and makes numerical search techniques feasible. We find three families of embedded SU(2) EYM equations within the SO(5) system, one of which was first detected only within the gauge-invariant polynomial reduced system. Numerical solutions representing mixtures of the three SU(2) subsystems are found, classified by a pair of positive integers. [ABSTRACT FROM AUTHOR]

We evaluate the intertwining relation for Schrödinger equations in (n+1) dimensions. The conditions for the existence of a Darboux transformation are analyzed and compared to their (1+1) dimensional counterparts. A complete solution of the conditions is given for (2+1) dimensions, and a Darboux transformation is constructed. [ABSTRACT FROM AUTHOR]

We give a short, operator-theoretic proof of the asymptotic independence (including a first correction term) of the minimal and maximal eigenvalue of the n×n Gaussian unitary ensemble in the large matrix limit n→∞. This is done by representing the joint probability distribution of the extreme eigenvalues as the Fredholm determinant of an operator matrix that asymptotically becomes diagonal. As a corollary, we get that the correlation of the extreme eigenvalues asymptotically behaves like n-2/3/4σ2, where σ2 denotes the variance of the Tracy–Widom distribution. While we conjecture that the extreme eigenvalues are asymptotically independent for Wigner random Hermitian matrix ensembles, in general, the actual constant in the asymptotic behavior of the correlation turns out to be specific and can thus be used to distinguish the Gaussian unitary ensemble statistically from certain other Wigner ensembles. [ABSTRACT FROM AUTHOR]

In this article, certain indecomposable Virasoro modules are studied. Specifically, the Virasoro mode L0 is assumed to be nondiagonalizable, possessing Jordan blocks of rank 2. Moreover, the module is further assumed to have a highest weight submodule, the “left module,” and that the quotient by this submodule yields another highest weight module, the “right module.” Such modules, which have been called staggered, have appeared repeatedly in the logarithmic conformal field theory literature, but their theory has not been explored in full generality. Here, such a theory is developed for the Virasoro algebra using rather elementary techniques. The focus centers on two different but related questions typically encountered in practical studies: How can one identify a given staggered module, and how can one demonstrate the existence of a proposed staggered module. Given just the values of the highest weights of the left and right modules, themselves subject to simple necessary conditions, invariants are defined which together with the knowledge of the left and right modules uniquely identify a staggered module. The possible values of these invariants form a vector space of dimension 0, 1, or 2, and the structures of the left and right modules limit the isomorphism classes of the corresponding staggered modules to an affine subspace (possibly empty). The number of invariants and affine restrictions is purely determined by the structures of the left and right modules. Moreover, in order to facilitate applications, the expressions for the invariants and restrictions are given by formulas as explicit as possible (they generally rely on expressions for Virasoro singular vectors). Finally, the text is liberally peppered throughout with examples illustrating the general concepts. These have been carefully chosen for their physical relevance or for the novel features they exhibit. [ABSTRACT FROM AUTHOR]

We improve the regularity criterion for the incompressible Navier–Stokes equations in the full three-dimensional space involving the gradient of one velocity component. The method is based on recent results of Cao and Titi [see “Regularity criteria for the three dimensional Navier–Stokes equations,” Indiana Univ. Math. J. 57, 2643 (2008)] and Kukavica and Ziane [see “Navier-Stokes equations with regularity in one direction,” J. Math. Phys. 48, 065203 (2007)]. In particular, for s∈[2,3], we get that the solution is regular if ∇u3∈Lt(0,T;Ls(R3)), 2/t+3/s≤2312. [ABSTRACT FROM AUTHOR]

In this paper, Rosenau and Hyman’s [Phys. Rev. Lett. 70, 564 (1993)] K(m,n) equations are studied by prolongation technique. It is proved that K(m,n) equations are Lax integrable only for certain special parameters (α,m,n). The nontrivial prolongation structures and Lax pairs for the integrable cases are given. Finally, as an example, the one and two soliton solutions for the K(-12,-12) equation are derived by means of inverse scattering method. [ABSTRACT FROM AUTHOR]

We study limiting cases of the two known integrable chiral-type models with three-dimensional configuration space. One of the initial models is the non-Abelian Toda A2(1) model and the other was found by means of the symmetry approach by Meshkov and Demskoi [Theor. Math. Phys. 134, 351 (2003)]. The C-integrability of the reduced models is established by constructing their complete sets of integrals and general solutions. A description of the generalized symmetry algebras of these models is given in terms of operator mapping integrals into symmetries. The integrals of the Liouville-type systems are known to define Miura-type transformations for their generalized symmetries. This fact allowed us to find a few new systems of the Yajima–Oikawa type. We present a recursion operator for one them. [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 main result of this article is that we show that from supersymmetry we can generate new superintegrable Hamiltonians. We consider a particular case with a third order integral and apply Mielnik’s construction in supersymmetric quantum mechanics. We obtain a new superintegrable potential separable in Cartesian coordinates with a quadratic and quintic integrals and also one with a quadratic integral and an integral of order of 7. We also construct a superintegrable system written in terms of the fourth Painlevé transcendent with a quadratic integral and an integral of order of 7. [ABSTRACT FROM AUTHOR]

We estimate the growth in time of the solutions to a class of nonlinear fractional differential equations D0+α(x-x0)=f(t,x) which includes D0+α(x-x0)=H(t)xλ with λ∈(0,1) for the case of slowly decaying coefficients H. The proof is based on the triple interpolation inequality on the real line and the growth estimate reads as x(t)=o(taα) when t→+∞ for 1>α>1-a>λ>0. Our result can be thought of as a noninteger counterpart of the classical Bihari asymptotic integration result for nonlinear ordinary differential equations. By a carefully designed example we show that in some circumstances such an estimate is optimal. [ABSTRACT FROM AUTHOR]

The governing equation is [∇2+k2-q(x)]u=0 in R3. It is shown that any desired potential q(x), vanishing outside a bounded domain D, bounded in D, Riemann integrable, can be obtained if one embeds into D many small scatterers qm(x), vanishing outside balls Bm:{x:|x-xm|Δn(x)dx[1+o(1)] as a→0, where V(a)=4πa3/3, then the limit of the function uM(x), lima→0 uM=ue(x), does exist and solves the equation [∇2+k2-q(x)]u=0 in R3, where q(x)=n(x)A(x), A(xm)=Am, and uM(x) is a solution to the equation [∇2+k2-p(x)]u=0, where p(x):pM(x) is some piecewise-constant potential. The total number M of small inhomogeneities is equal to N(D) and is of the order O(a-3) as a→0. A similar result is derived in the one-dimensional case. [ABSTRACT FROM AUTHOR]

A detailed characterization of partial positive transposition (PPT) states, both in the Heisenberg and in the Schrödinger picture, is given. Measures of entanglement are defined and discussed in details. Illustrative examples are provided. [ABSTRACT FROM AUTHOR]

In this paper, we prove the global existence of classical solutions to the Cauchy problem for the minimal surface equation in the Minkowski space R1+2 with slow decay initial value. [ABSTRACT FROM AUTHOR]

We study the final state problem for the nonlinear Klein–Gordon equation, utt+u-uxx=μu3, t∈R,x∈R, where μ∈R. We prove the existence of solutions in the neighborhood of the approximate solutions 2 Re U(t)w+(t), where U(t) is the free evolution group defined by U(t)=F-1e-it〈ξ〉F, =1+x2, F and F-1 are the direct and inverse Fourier transformations, respectively, and w+(t,x)=F-1(u⁁+(ξ)e(3/2)iμ〈ξ〉2|u⁁+(ξ)|2log t), with a given final data u+ is a real-valued function and |〈ξ〉3u⁁+(ξ)|L∞ is small. [ABSTRACT FROM AUTHOR]

The recently developed theory of quasi-Lie schemes is studied and applied to investigate several equations of Emden type and a scheme to deal with them and some of their generalizations is given. As a first result we obtain t-dependent constants of the motion for particular instances of Emden equations by means of some of their particular solutions. Previously known results are recovered from this new perspective. Some t-dependent constants of the motion for equations of Emden type satisfying certain conditions are recovered. Finally new exact particular solutions are given for certain cases of Emden equations. [ABSTRACT FROM AUTHOR]

We investigate the quantum integrability of the Landau–Lifshitz (LL) model and solve the long-standing problem of finding the local quantum Hamiltonian for the arbitrary n-particle sector. The particular difficulty of the LL model quantization, which arises due to the ill-defined operator product, is dealt with by simultaneously regularizing the operator product and constructing the self-adjoint extensions of a very particular structure. The diagonalizibility difficulties of the Hamiltonian of the LL model, due to the highly singular nature of the quantum-mechanical Hamiltonian, are also resolved in our method for the arbitrary n-particle sector. We explicitly demonstrate the consistency of our construction with the quantum inverse scattering method due to Sklyanin [Lett. Math. Phys. 15, 357 (1988)] and give a prescription to systematically construct the general solution, which explains and generalizes the puzzling results of Sklyanin for the particular two-particle sector case. Moreover, we demonstrate the S-matrix factorization and show that it is a consequence of the discontinuity conditions on the functions involved in the construction of the self-adjoint extensions. [ABSTRACT FROM AUTHOR]

We show that appropriate superpositions of motional states are a reference frame resource that enables breaking of time-reversal superselection so that two parties lacking knowledge about the other’s direction of time can still communicate. We identify the time-reversal reference frame resource states and determine the corresponding frameness monotone, which connects time-reversal frameness to entanglement. In contradistinction to other studies of reference frame quantum resources, this is the first analysis that involves an antiunitary rather than unitary representation. [ABSTRACT FROM AUTHOR]

This paper discusses the no-cloning theorem in a logicoalgebraic approach. In this approach, an orthoalgebra is considered as a general structure for propositions in a physical theory. We proved that an orthoalgebra admits cloning operation if and only if it is a Boolean algebra. That is, only classical theory admits the cloning of states. If unsharp propositions are to be included in the theory, then a notion of effect algebra is considered. We proved that an atomic Archimedean effect algebra admitting cloning operation is a Boolean algebra. This paper also presents a partial result, indicating a relation between the cloning on effect algebras and hidden variables. [ABSTRACT FROM AUTHOR]

We consider a superintegrable quantum potential in two-dimensional Euclidean space with a second and a third order integral of motion. The potential is written in terms of the fourth Painlevé transcendent. We construct for this system a cubic algebra of integrals of motion. The algebra is realized in terms of parafermionic operators and we present Fock-type representations which yield the corresponding energy spectra. We also discuss this potential from the point of view of higher order supersymmetric quantum mechanics and obtain ground state wave functions. [ABSTRACT FROM AUTHOR]

Clarifications on the optical reciprocity theorem are provided by explicitly proving the equivalence between the Lorentz lemma and the symmetry of the Green dyadic for the electromagnetic wave equation. This is achieved by explicitly including the surface term in the former so that different boundary conditions can be considered as required in the formulation of the latter. In addition, we shall also extend the theorem to include anisotropic magnetic materials with a nonlocal response, leading to a result which will be useful for the study of materials possessing such properties such as certain types of metamaterials. [ABSTRACT FROM AUTHOR]

Stochastic microlensing is a central tool in probing dark matter on galactic scales. From first principles, we initiate the development of a mathematical theory of stochastic microlensing. Beginning with the random time delay function and associated lensing map, we determine exact expressions for the mean and variance of these transformations. In addition, we derive the probability density function (pdf) of a random point-mass potential, which form the constituent of a stochastic microlens potential. We characterize the exact pdf of a normalized random time delay function at the origin, showing that it is a shifted gamma distribution, which also holds at leading order in the limit of a large number of point masses if the normalized time delay function was at a general point of the lens plane. For the large number of point-mass limit, we also prove that the asymptotic pdf of the random lensing map under a specified scaling converges to a bivariate normal distribution. We show analytically that the pdf of the random scaled lensing map at leading order depends on the magnitude of the scaled bending angle due purely to point masses as well as demonstrate explicitly how this radial symmetry is broken at the next order. Interestingly, we found at leading order a formula linking the expectation and variance of the normalized random time delay function to the first Betti number of its domain. We also determine an asymptotic pdf for the random bending angle vector and find an integral expression for the probability of a lens plane point being near a fixed point. Lastly, we show explicitly how the results are affected by location in the lens plane. The results of this paper are relevant to the theory of random fields and provide a platform for further generalizations as well as analytical limits for checking astrophysical studies of stochastic microlensing. [ABSTRACT FROM AUTHOR]

A new geometrical perturbation scheme is developed in order to calculate the electromagnetic fields produced by charged sources in prescribed motion moving in a nonstraight perfectly conducting beam pipe. The pipe is regarded as a perturbed infinitely long hollow right-circular cylinder. The perturbation maintains the pipe’s circular cross section while deforming its axis into a planar space curve with, in general, nonconstant curvature. Various charged source models are considered including a charged bunch and an off-axis point particle. In the ultrarelativistic limit this permits a calculation of the longitudinal wake potential in terms of powers of the product of the pipe radius and the arbitrarily varying curvature of the axial space curve. Analytic expressions to leading order are presented for beam pipes with piecewise defined constant curvature modeling pipes with straight segments linked by circular arcs of finite length. The language of differential forms is used throughout, and to illustrate the power of this formalism, a pedagogical introduction is developed by deriving the theory ab initio from Maxwell’s equations expressed intrinsically as a differential system on (Minkowski) space-time. [ABSTRACT FROM AUTHOR]

The structure of cones of positive and k-positive maps acting on a finite-dimensional Hilbert space is investigated. Special emphasis is given to their duality relations to the sets of superpositive and k-superpositive maps. We characterize k-positive and k-superpositive maps with regard to their properties under taking compositions. A number of results obtained for maps are also rephrased for the corresponding cones of block positive, k-block positive, separable, and k-entangled operators due to the Jamiołkowski–Choi isomorphism. Generalizations to a situation where no such simple isomorphism is available are also made, employing the idea of mapping cones. As a side result to our discussion, we show that extreme entanglement witnesses, which are optimal, should be of special interest in entanglement studies. [ABSTRACT FROM AUTHOR]

By means of a generalization of the S-expansion method we construct a procedure to obtain expanded higher-order Lie algebras. It is shown that the direct product between an Abelian semigroup S and a higher-order Lie algebra (G,[,...,]) is also a higher-order Lie algebra. From this S-expanded Lie algebra are obtained resonant submultialgebras and reduced multialgebras of a resonant submultialgebra. [ABSTRACT FROM AUTHOR]

In this paper I investigate the existence of nontrivial breather solutions of the discrete nonlinear Schrödinger equation with the unbounded potential at infinity. First I derive a discrete version of compact embedding theorem. Then combining the Nehari manifold approach and the compact embedding theorem, I show the existence of breather solutions without Palais–Smale condition. The results on the exponential decay of breather solutions are also provided in this paper. [ABSTRACT FROM AUTHOR]

We classify Lie n-algebras possessing an invariant Lorentzian inner product. [ABSTRACT FROM AUTHOR]

It is shown that the coupled Volterra system governed by two-coupled nonlinear partial differential–difference equations (PDΔEs) has many integrability properties such as Lax representation, infinitely many generalized (nonpoint) symmetries and polynomial conserved densities, and a recursion operator. A systematic investigation on the continuous point symmetries of the coupled Volterra system shows that it admits a three dimensional nilpotent Lie algebra. By exploiting the obtained continuous point symmetries a similarity reduction is obtained. [ABSTRACT FROM AUTHOR]

We consider the system x=a(z-a1x3-a2x2-bx), y=-z, ż=-b1x+y+b2z, where a and b are parameters and b1=7/10, b2=6/25, a1=44/3, and a2=41/2. We analyze the existence of local and global analytic first integrals. [ABSTRACT FROM AUTHOR]

The Lie symmetry algebra of a generalized Davey–Stewartson (GDS) system is obtained. The general element of this algebra depends on eight arbitrary functions of time, which has a Kac–Moody–Virasoro loop algebra structure and is isomorphic to that of the standard integrable Davey–Stewartson equations under certain conditions imposed on parameters and arbitrary functions. Then based on the symmetry group direct method proposed by Lou and Ma [J. Phys. A 38, L129 (2005)] the full symmetry groups of the GDS system are obtained. From the full symmetry groups, both the Lie symmetry group and a group of discrete transformations can be obtained. Finally, some exact solutions involving sech-sech2-sech2 and tanh-tanh2-tanh2 type solitary wave solutions are presented by a generalized subequation expansion method. [ABSTRACT FROM AUTHOR]

We prove that potential conservation laws have characteristics depending only on local variables if and only if they are induced by local conservation laws. Therefore, characteristics of pure potential conservation laws have to essentially depend on potential variables. This statement provides a significant generalization of results of the recent paper by Bluman et al. [J. Math. Phys. 47, 113505 (2006)]. Moreover, we present extensions to gauged potential systems, Abelian and general coverings, and general foliated systems of differential equations. An example illustrating possible applications of these results is given. A special version of the Hadamard lemma for fiber bundles and the notions of weighted jet spaces are proposed as new tools for the investigation of potential conservation laws. [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]

We investigate the singularly perturbed Boussinesq equation in terms of the approximate symmetry perturbation method and the approximate direct method. The similarity reduction solutions and similarity reduction equations of different orders display formal coincidence for both methods. Series reduction solutions are consequently derived. For the approximate symmetry perturbation method, similarity reduction equations of the zero order are equivalent to the Painlevé IV, Painlevé I, and Weierstrass elliptic equations. For the approximate direct method, similarity reduction equations of the zero order are equivalent to the Painlevé IV, Painlevé II, Painlevé I, or Weierstrass elliptic equations. The approximate direct method yields more general approximate similarity reductions than the approximate symmetry perturbation method. [ABSTRACT FROM AUTHOR]

In this note we compute Leibniz algebra deformations of the three-dimensional Heisenberg Lie algebra n3 and compare it to its Lie deformations. It turns out that there are three extra Leibniz deformations. We also describe a versal Leibniz deformation of n3 with the versal base. [ABSTRACT FROM AUTHOR]

In 1940, Ulam proposed the famous Ulam stability problem. In this paper we introduce a general Cauchy–Jensen functional equation and prove the generalized Ulam stability of C*-ternary homomorphisms and C*-ternary derivations in C*-ternary algebras for the general Cauchy–Jensen equation. [ABSTRACT FROM AUTHOR]

We introduce an extension of the character expansion method to the case of supergroups. This method allows us to calculate a superversion of the Leutwyler–Smilga integral which, to the best of our knowledge, has not been calculated before. We also use the method to generalize a previously calculated superversion of the Berezin–Karpelevich integral. Our character expansion method should also allow for the calculation of other supergroup integrals. [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 construct C-algebras for a class of surfaces that are inverse images of certain polynomials of arbitrary degree. By using the directed graph associated with a matrix, the representation theory can be understood in terms of “loop” and “string” representations, which are closely related to the dynamics of an iterated map in the plane. As a particular class of algebras, we introduce the “Hénon algebras,” for which the dynamical map is a generalized Hénon map, and give an example where irreducible representations of all dimensions exist. [ABSTRACT FROM AUTHOR]

The Dirac equation written on the boundary of the Nutku helicoid space consists of a system of ordinary differential equations. We tried to analyze this system and we found that it has a higher singularity than those of the Heun equations which give the solutions of the Dirac equation in the bulk. We also lose an independent integral of motion on the boundary. This facts explain why we could not find the solution of the system on the boundary in terms of known functions. We make the stability analysis of the helicoid and catenoid cases and end up with an Appendix which gives a new example wherein one encounters a form of the Heun equation. [ABSTRACT FROM AUTHOR]

This paper is the first in a series whose goal is to develop a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. In this paper, we discuss two different types of language that can be attached to a system S. The first is a propositional language PL(S); the second is a higher-order, typed language L(S). Both languages provide deductive systems with an intuitionistic logic. The reason for introducing PL(S) is that, as shown in Paper II of the series, it is the easiest way of understanding, and expanding on, the earlier work on topos theory and quantum physics. However, the main thrust of our program utilizes the more powerful language L(S) and its representation in an appropriate topos. [ABSTRACT FROM AUTHOR]