ALBERT

Description: We introduce a method for generating rational extensions of time-dependent potentials, such that the associated Schrödinger equation admits solutions in terms of exceptional orthogonal polynomials. Our method is applicable to position-dependent Schrödinger equations, as well as to their conventional counterparts for constant mass.

Description: In this paper, we prove the existence of the Stark-Wannier quantum resonances for one-dimensional Schrödinger operators with smooth periodic potential and small external homogeneous electric field. Such a result extends the existence result previously obtained in the case of periodic potentials with a finite number of open gaps.

Description: We give an algorithm which produces a unique element of the Clifford group on n qubits ( C n ) from an integer 0 ≤ i 〈 C n (the number of elements in the group). The algorithm involves O ( n 3 ) operations and provides, in addition to a canonical mapping from the integers to group elements g , a factorization of g into a sequence of at most 4 n symplectic transvections. The algorithm can be used to efficiently select random elements of C n which are often useful in quantum information theory and quantum computation. We also give an algorithm for the inverse map, indexing a group element in time O ( n 3 ).

Description: In the present work, we redefine and generalize the action principle for dissipative systems proposed by Riewe by fixing the mathematical inconsistencies present in the original approach. In order to formulate a quadratic Lagrangian for non-conservative systems, the Lagrangian functions proposed depend on mixed integer order and fractional order derivatives. As examples, we formulate a quadratic Lagrangian for a particle under a frictional force proportional to the velocity and to the classical problem of an accelerated point charge.

Description: We consider a very natural generalization of quantum theory by letting the dimension of the Bloch ball be not necessarily three. We analyze bipartite state spaces where each of the components has a d -dimensional Euclidean ball as state space. In addition to this, we impose two very natural assumptions: the continuity and reversibility of dynamics and the possibility of characterizing bipartite states by local measurements. We classify all these bipartite state spaces and prove that, except for the quantum two-qubit state space, none of them contains entangled states. Equivalently, in any of these non-quantum theories, interacting dynamics is impossible. This result reveals that “existence of entanglement” is the requirement with minimal logical content which singles out quantum theory from our family of theories.

Description: A covariant algorithm for deriving the conserved quantities for natural Hamiltonian systems is combined with the non-relativistic framework of Eisenhart, and of Duval, in which the classical trajectories arise as geodesics in a higher dimensional space-time, realized by Brinkmann manifolds. Conserved quantities which are polynomial in the momenta can be built using time-dependent conformal Killing tensors with flux. The latter are associated with terms proportional to the Hamiltonian in the lower dimensional theory and with spectrum generating algebras for higher dimensional quantities of order 1 and 2 in the momenta. Illustrations of the general theory include the Runge-Lenz vector for planetary motion with a time-dependent gravitational constant G ( t ), motion in a time-dependent electromagnetic field of a certain form, quantum dots, the Hénon-Heiles and Holt systems, respectively, providing us with Killing tensors of rank that ranges from one to six.

Description: Knots are commonly found in molecular chains such as DNA and proteins, and they have been considered to be useful models for structural analysis of these molecules. One interested quantity is the minimum number of monomers necessary to realize a molecular knot. The minimum lattice length Len( K ) of a knot K indicates the minimum length necessary to construct K in the cubic lattice. Another important quantity in physical knot theory is the ropelength which is one of the knot energies measuring the complexity of knot conformation. The minimum ropelength Rop( K ) is the minimum length of an ideally flexible rope necessary to tie a given knot K . Much effort has been invested in the research project for finding upper bounds on both quantities in terms of the minimum crossing number c ( K ) of the knot. It is known that Len( K ) and Rop( K ) lie between O ( c ( K ) 3 4 ) and O( c ( K )[ln ( c ( K ))] 5 ), but unknown yet whether any family of knots has superlinear growth. In this paper, we focus on 2-bridge knots and links. Linear growth upper bounds on the minimum lattice length and minimum ropelength for nontrivial 2-bridge knots or links are presented as Len( K ) ⩽ 8 c ( K ) + 2 and Rop( K ) ⩽ 11.39 c ( K ) + 12.37.

Description: We construct a spectral triple for the C * -algebra of continuous functions on the space of p -adic integers by using a rooted tree obtained from coarse-grained approximation of the space, and the forward derivative on the tree. Additionally, we verify that our spectral triple satisfies the properties of a compact spectral metric space, and we show that the metric on the space of p -adic integers induced by the spectral triple is equivalent to the usual p -adic metric.

Description: We consider the 3-body problem in 3-dimensional spaces of nonzero constant Gaussian curvature and study the relationship between the masses of the Lagrangian relative equilibria, which are orbits that form a rigidly rotating equilateral triangle at all times. There are three classes of Lagrangian relative equilibria in 3-dimensional spaces of constant nonzero curvature: positive elliptic and positive elliptic-elliptic, on 3-spheres, and negative elliptic, on hyperbolic 3-spheres. We prove that all these Lagrangian relative equilibria exist only for equal values of the masses.

Description: In this paper, we study singular systems with complete sets of involutive constraints. The aim is to establish, within the Hamilton-Jacobi theory, the relationship between the Frobenius’ theorem, the infinitesimal canonical transformations generated by constraints in involution with the Poisson brackets, and the lagrangian point (gauge) transformations of physical systems.

Description: Multi-indexed Jacobi polynomials are defined by the Wronskian of four types of eigenfunctions of the Pöschl-Teller Hamiltonian. We give a correspondence between multi-indexed Jacobi polynomials and pairs of Maya diagrams, and we show that any multi-indexed Jacobi polynomial is essentially equal to some multi-indexed Jacobi polynomial of two types of eigenfunction. As an application, we show a Wronskian-type formula of some special eigenstates of the deformed Pöschl-Teller Hamiltonian.

Description: We present a novel derivation of both the Minkowski metric and Lorentz transformations from the consistent quantification of a causally ordered set of events with respect to an embedded observer. Unlike past derivations, which have relied on assumptions such as the existence of a 4-dimensional manifold, symmetries of space-time, or the constant speed of light, we demonstrate that these now familiar mathematics can be derived as the unique means to consistently quantify a network of events. This suggests that space-time need not be physical, but instead the mathematics of space and time emerges as the unique way in which an observer can consistently quantify events and their relationships to one another. The result is a potential foundation for emergent space-time.

Description: We consider the scattering theory for discrete Schrödinger operators on Z d with long-range potentials. We prove the existence of modified wave operators constructed in terms of solutions of a Hamilton-Jacobi equation on the torus T d .

Description: Let A u t m H H ( H ) denote the set of all automorphisms of a monoidal Hopf algebra H with bijective antipode in the sense of Caenepeel and Goyvaerts [“Monoidal Hom-Hopf algebras,” Commun. Algebra39, 2216–2240 (2011)] and let G be a crossed product group A u t m H H ( H ) × A u t m H H ( H ) . The main aim of this paper is to provide new examples of braided T -category in the sense of Turaev [“Crossed group-categories,” Arabian J. Sci. Eng., Sect. C33(2C), 483–503 (2008)]. For this purpose, we first introduce a class of new categories MHYD H H ( A , B ) of ( A , B )-Yetter-Drinfeld Hom-modules with A , B ∈ A u t m H H ( H ) . Then we construct a category M H Y D ( H ) = { MHYD H H ( A , B ) } ( A , B ) ∈ G and show that such category forms a new braided T -category, generalizing the main constructions by Panaite and Staic [“Generalized (anti) Yetter-Drinfel'd modules as components of a braided T-category,” Isr. J. Math.158, 349–366 (2007)]. Finally, we compute an explicit new example of such braided T -categories.

Description: The geometry of the Lie algebroid generalized tangent bundle of a generalized Lie algebroid is developed. Formulas of Ricci type and identities of Cartan and Bianchi type are presented. Introducing the notion of geodesic of a mechanical ρ , η -system with respect to a ( ρ , η )-spray, the Berwald ( ρ , η )-derivative operator, and its mixed curvature, we obtain main results to conceptualize the Weyl’s method in this general framework. Finally, we obtain two new results of Weyl type for the geometry of mechanical ρ , η -systems. In this way, it is proved that the projectively related sprays first have the same geodesics rather to an increasing parameter transformation and second their Berwald derivatives verify a respective relation.

Description: In this paper, we prove Anderson localization for the Klein-Gordon operator on the circle

Description: We generalize the idea of “extension of Hamiltonian systems”—developed in a series of previous articles—which allows the explicit construction of Hamiltonian systems with additional non-trivial polynomial first integrals of arbitrarily high degree, as well as the determination of new superintegrable systems from old ones. The present generalization, that we call “modified extension of Hamiltonian systems,” produces the third independent first integral for the (complete) Tremblay-Turbiner-Winternitz system, as well as for the caged anisotropic oscillator in dimension two.

Description: The Pöschl-Teller (PT) potential occupies a privileged place among the anharmonic oscillator potentials due to its applications from quantum mechanics to diatomic molecules. For this potential, a polynomial su (1, 1) algebra has been constructed previously. So far, the coherent states (CSs) associated with this algebra have never appeared. In this paper, we construct the coherent states of the Barut-Girardello coherent states (BG-CSs) type for the PT potentials, which have received less attention in the scientific literature. We obtain these CSs and demonstrate that they fulfil all conditions required by the coherent state. The Mandel parameter for the pure BG-CSs and Husimi’s and P-quasi distributions (for the mixed-thermal states) is also presented. Finally, the exponential form of the BG-CSs for the PT potential has been presented and enabled us to build Perelomov type CSs for the PT potential. We point out that the BG-CSs and the Perelomov type coherent states (PCSs) are related via Laplace transform.

Description: We reformulate dimensional regularization as a regularization method in position space and show that it can be used to give a closed expression for the renormalized time-ordered products as solutions to the induction scheme of Epstein-Glaser. This closed expression, which we call the Epstein-Glaser Forest Formula, is analogous to Zimmermann’s Forest Formula for BPH renormalization. For scalar fields, the resulting renormalization method is always applicable, we compute several examples. We also analyze the Hopf algebraic aspects of the combinatorics. Our starting point is the Main Theorem of Renormalization of Stora and Popineau and the arising renormalization group as originally defined by Stückelberg and Petermann.

Description: This work is motivated by a quite general question: Under which circumstances are the capacities of information transmission systems continuous? The research is explicitly carried out on finite arbitrarily varying quantum channels (AVQCs). We give an explicit example that answers the recent question whether the transmission of messages over AVQCs can benefit from assistance by distribution of randomness between the legitimate sender and receiver in the affirmative. The specific class of channels introduced in that example is then extended to show that the unassisted capacity does have discontinuity points, while it is known that the randomness-assisted capacity is always continuous in the channel. We characterize the discontinuity points and prove that the unassisted capacity is always continuous around its positivity points. After having established shared randomness as an important resource, we quantify the interplay between the distribution of finite amounts of randomness between the legitimate sender and receiver, the (nonzero) probability of a decoding error with respect to the average error criterion and the number of messages that can be sent over a finite number of channel uses. We relate our results to the entanglement transmission capacities of finite AVQCs, where the role of shared randomness is not yet well understood, and give a new sufficient criterion for the entanglement transmission capacity with randomness assistance to vanish.

Description: The main result of this work is as follows: for arbitrary pairwise disjoint, finite intervals ( α j , β j ) ⊂ [0, ∞), j = 1, …, m , and for arbitrary n ≥ 2, we construct a family of periodic non-compact domains {Ω ε ⊂ℝ n } ε〉0 such that the spectrum of the Neumann Laplacian in Ω ε has at least m gaps when ε is small enough, moreover the first m gaps tend to the intervals ( α j , β j ) as ε → 0. The constructed domain Ω ε is obtained by removing from ℝ n a system of periodically distributed “trap-like” surfaces. The parameter ε characterizes the period of the domain Ω ε , also it is involved in a geometry of the removed surfaces.

Description: We consider the general open system problem of a charged quantum oscillator confined in a harmonic trap, whose frequency can be arbitrarily modulated in time, that interacts with both an incoherent quantized (blackbody) radiation field and with an arbitrary coherent laser field. We assume that the oscillator is initially in thermodynamic equilibrium with its environment, a non-factorized initial density matrix of the system and the environment, and that at t = 0 the modulation of the frequency, the coupling to the incoherent and the coherent radiation are switched on. The subsequent dynamics, induced by the presence of the blackbody radiation, the laser field, and the frequency modulation, is studied in the framework of the influence functional approach. This approach allows incorporating, in analytic closed formulae , the non-Markovian character of the oscillator-environment interaction at any temperature as well the non-Markovian character of the blackbody radiation and its zero-point fluctuations. Expressions for the time evolution of the covariance matrix elements of the quantum fluctuations and the reduced density-operator are obtained.

Description: We treat the N -particle zero range process whose jumping rates satisfy a certain condition. This condition is required to use the Bethe ansatz and the resulting model is the q -boson model by Sasamoto and Wadati [“Exact results for one-dimensional totally asymmetric diffusion models,” J. Phys. A31, 6057–6071 (1998)] or the q -totally asymmetric zero range process (TAZRP) by Borodin and Corwin [“Macdonald processes,” Probab. Theory Relat. Fields (to be published)]. We find the explicit formula of the transition probability of the q -TAZRP via the Bethe ansatz. By using the transition probability we find the probability distribution of the left-most particle's position at time t . To find the probability for the left-most particle's position we find a new identity corresponding to identity for the asymmetric simple exclusion process by Tracy and Widom [“Integral formulas for the asymmetric simple exclusion process,” Commun. Math. Phys.279, 815–844 (2008)]. For the initial state that all particles occupy a single site, the probability distribution of the left-most particle's position at time t is represented by the contour integral of a determinant.

Description: This paper deals with the asymptotic behavior of the first Bloch eigenvalue in a heterogeneous medium with a high contrast ɛ Y -periodic conductivity. When the conductivity is bounded in L 1 and the constant of the Poincaré-Wirtinger weighted by the conductivity is very small with respect to ɛ −2 , the first Bloch eigenvalue converges as ɛ → 0 to a limit which preserves the second-order expansion with respect to the Bloch parameter. In dimension two the expansion of the limit can be improved until the fourth-order under the same hypotheses. On the contrary, in dimension three a fibers reinforced medium combined with a L 1 -unbounded conductivity leads us to a discontinuity of the limit first Bloch eigenvalue as the Bloch parameter tends to zero but remains not orthogonal to the direction of the fibers. Therefore, the high contrast conductivity of the microstructure induces an anomalous effect, since for a given low-contrast conductivity the first Bloch eigenvalue is known to be analytic with respect to the Bloch parameter around zero.

Description: We report a new class of hyperbolic asymmetric double-well whose bound state wavefunctions can be expressed in terms of confluent Heun functions. An analytic procedure is used to obtain the energy eigenvalues and the criterion for the potential to support bound states is discussed.

Description: Given a (smooth) action φ of a Lie group G on R d we construct a differential graded algebra whose Maurer–Cartan elements are in one-to-one correspondence with some class of deformations of the (induced) G -action on C ∞ ( R d ) [ [ ℏ ] ] . In the final part of this note we discuss the cohomological obstructions to the existence and to the uniqueness (in a sense to be clarified) of such deformations.

Description: In the present paper, we build a combinatorial invariant, called the “spectral monodromy” from the spectrum of a single (non-self-adjoint) h -pseudodifferential operator with two degrees of freedom in the semi-classical limit. Our inspiration comes from the quantum monodromy defined for the joint spectrum of an integrable system of n commuting self-adjoint h -pseudodifferential operators, given by S. Vu Ngoc [“Quantum monodromy in integrable systems,” Commun. Math. Phys.203(2), 465–479 (1999)]. The first simple case that we treat in this work is a normal operator. In this case, the discrete spectrum can be identified with the joint spectrum of an integrable quantum system. The second more complex case we propose is a small perturbation of a self-adjoint operator with a classical integrability property. We show that the discrete spectrum (in a small band around the real axis) also has a combinatorial monodromy. The main difficulty in this case is that we do not know the description of the spectrum everywhere, but only in a Cantor type set. In addition, we also show that the corresponding monodromy can be identified with the classical monodromy, defined by J. Duistermaat [“On global action-angle coordinates,” Commun. Pure Appl. Math.33(6), 687–706 (1980)].

Description: It is shown that three different Lax operators in the Dym hierarchy produce three generalized coupled Harry Dym equations. These equations transform, via the reciprocal link, to the coupled two-component Korteweg de Vries (KdV) system. The first equation gives us known integrable two-component KdV system, while the second reduces to the known symmetrical two-component KdV equation. The last one reduces to the Drienfeld-Sokolov equation. This approach gives us new Lax representation for these equations.

Description: We study Killing tensors in the context of warped products and apply the results to the problem of orthogonal separation of the Hamilton-Jacobi equation. This work is motivated primarily by the case of spaces of constant curvature where warped products are abundant. We first characterize Killing tensors which have a natural algebraic decomposition in warped products. We then apply this result to show how one can obtain the Killing-Stäckel space (KS-space) for separable coordinate systems decomposable in warped products. This result in combination with Benenti's theory for constructing the KS-space of certain special separable coordinates can be used to obtain the KS-space for all orthogonal separable coordinates found by Kalnins and Miller in Riemannian spaces of constant curvature. Next we characterize when a natural Hamiltonian is separable in coordinates decomposable in a warped product by showing that the conditions originally given by Benenti can be reduced. Finally, we use this characterization and concircular tensors (a special type of torsionless conformal Killing tensor) to develop a general algorithm to determine when a natural Hamiltonian is separable in a special class of separable coordinates which include all orthogonal separable coordinates in spaces of constant curvature.

Description: We study the propagation of ultra-short short solitons in a cubic nonlinear medium modeled by nonlinear Maxwell's equations with stochastic variations of media. We consider three cases: variations of (a) the dispersion, (b) the phase velocity, (c) the nonlinear coefficient. Using a modified multi-scale expansion for stochastic systems, we derive new stochastic generalizations of the short pulse equation that approximate the solutions of stochastic nonlinear Maxwell's equations. Numerical simulations show that soliton solutions of the short pulse equation propagate stably in stochastic nonlinear Maxwell's equations and that the generalized stochastic short pulse equations approximate the solutions to the stochastic Maxwell's equations over the distances under consideration. This holds for both a pathwise comparison of the stochastic equations as well as for a comparison of the resulting probability densities.

Description: We study spectral asymptotics for a large class of differential operators on an open subset of R d with finite volume. This class includes the Dirichlet Laplacian, the fractional Laplacian, and also fractional differential operators with non-homogeneous symbols. Based on a sharp estimate for the sum of the eigenvalues we establish the first term of the semiclassical asymptotics. This generalizes Weyl's law for the Laplace operator.

Description: The present paper is devoted to the scattering theory of a class of continuum Schrödinger operators with deterministic sparse potentials. We first establish the limiting absorption principle for both modified free resolvents and modified perturbed resolvents. This actually is a weak form of the classical limiting absorption principle. We then prove the existence and completeness of local wave operators, which, in particular, imply the existence of wave operators. Under additional assumptions on the sparse potential, we prove the completeness of wave operators. In the context of continuum Schrödinger operators with sparse potentials, this paper gives the first proof of the completeness of wave operators.

Description: In our previous paper, we classified linearly compact algebraic simple N = 6 3-algebras. In the present paper, we classify their “physical” counterparts, which actually appear in the N = 6 supersymmetric 3-dimensional Chern-Simons theories.

Description: We consider C -graded vertex algebras , which are vertex algebras V with a C -grading such that V is an admissible V -module generated by “lowest weight vectors.” We show that such vertex algebras have a “good” representation theory in the sense that there is a Zhu algebra A ( V ) and a bijection between simple admissible V -modules and simple A ( V )-modules. We also consider pseudo vertex operator algebras (PVOAs), which are C -graded vertex algebras with a conformal vector such that the homogeneous subspaces of V are generalized eigenspaces for L (0); essentially, these are VOAs that lack any semisimplicity or integrality assumptions on L (0). As a motivating example, we show that deformation of the conformal structure (conformal flow) of a strongly regular VOA (e.g., a lattice theory, or Wess-Zumino-Witten model) is a path in a space whose points are PVOAs.

Description: Following the lines of the recent paper of J.-P. Gazeau and F. H. Szafraniec [J. Phys. A: Math. Theor.44, 495201 (2011)], we construct here three types of coherent states, related to the Hermite polynomials in a complex variable which are orthogonal with respect to a non-rotationally invariant measure. We investigate relations between these coherent states and obtain the relationship between them and the squeezed states of quantum optics. We also obtain a second realization of the canonical coherent states in the Bargmann space of analytic functions, in terms of a squeezed basis. All this is done in the flavor of the classical approach of V. Bargmann [Commun. Pure Appl. Math.14, 187 (1961)].

Description: Using the momentum-space approach, we obtain bound states for multiple Dirac-δ wells in the framework of space-fractional quantum mechanics. Introducing first an attractive Dirac-comb potential, i.e., Dirac comb with strength − g ( g 〉 0), in the space-fractional Schrödinger equation we show that the problem of obtaining eigenenergies of a system with N Dirac-δ wells can be reduced to a problem of obtaining the eigenvalues of an N × N matrix. As an illustration we use the present matrix formulation to derive expressions satisfied by the bound-state energies of N = 1, 2, 3 delta wells. We also obtain the corresponding wave functions and express them in terms of Fox's H -function.

Description: Let F α ( C ) be the Hilbert space generated by the orthonormal basis e n α , ν ( z ) = ( 2 ν π ) 1 / 4 e ν 2 z 2 e − π 2 ν ( n + α ) 2 + 2 i π ( n + α ) z ; n ∈ N where ν 〉 0 and α are real numbers, this space is a particular case of (Γ, χ)-theta Fock-Bargmann spaces recently constructed by Ghanmi-Intissar [J. Math. Phys.54, 063514 (2013)]. In the present work, we consider on F α ( C ) the weight shift operator M defined by M e n α , ν = ω n e n + 1 α , ν , n ∈ N where ω n = c α , ν e 2 π ν n and c α , ν = e π ν + 2 α and we show the chaoticity of the operator H p = M p D p + 1 ; p = 0 , 1 , 2 , . . . where D is adjoint of M .

Description: This paper is devoted to a scalar conservation law with a linear flux function involving discontinuous coefficients. It is clear that the delta standing wave should be introduced into the Riemann solution in some nonclassical situation. In order to study the formation of delta standing wave, we consider a regularization of the discontinuous coefficient with the Helmholtz mollifier and then obtain a regularized system which depends on a regularization parameter ɛ 〉 0. The regularization mechanism is a nonlinear bending of characteristic curves that prevents their finite-time intersection. It is proved rigorously that the solutions of regularized system converge to the delta standing wave solution in the ɛ → 0 limit. Compared with the classical method of vanishing viscosity, here it is clear to see how the delta standing wave forms naturally along the characteristics.

Description: An elliptic equation that arises from a cosmic string model with the action of the Born-Infeld nonlinear electromagnetism, is considered. We classify and establish the uniqueness of radially symmetric solutions.

Description: We show a new lower bound on the H ̇ 3 2 T 3 norm of a possible blow-up solution to the Navier-Stokes equation, and also comment on the extension of this result to the whole space. This estimate can be seen as a natural limiting result for Leray's blow-up estimates in L p R 3 , 3 〈 p 〈 ∞. We also show a lower bound on the blow-up rate of a possible blow-up solution of the Navier-Stokes equation in H ̇ 5 2 T 3 , and give the corresponding extension to the case of the whole space.

Description: The quantum graph having the form of branching strips with hexagonal (honeycomb) structure is considered. The Hamiltonian is determined as free 1D Schrödinger operator on each edge and some “boundary” conditions at each vertex. We obtain the conditions ensuring the point spectrum's existence for the Schrödinger operator of the system and relations that give us the eigenvalues.

Description: Multi-time wave functions are wave functions that have a time variable for every particle, such as ϕ ( t 1 , x 1 , ... , t N , x N ) . They arise as a relativistic analog of the wave functions of quantum mechanics but can be applied also in quantum field theory. The evolution of a wave function with N time variables is governed by N Schrödinger equations, one for each time variable. These Schrödinger equations can be inconsistent with each other, i.e., they can fail to possess a joint solution for every initial condition; in fact, the N Hamiltonians need to satisfy a certain commutator condition in order to be consistent. While this condition is automatically satisfied for non-interacting particles, it is a challenge to set up consistent multi-time equations with interaction. We prove for a wide class of multi-time Schrödinger equations that the presence of interaction potentials (given by multiplication operators) leads to inconsistency. We conclude that interaction has to be implemented instead by creation and annihilation of particles, which, in fact, can be done consistently [S. Petrat and R. Tumulka, “Multi-time wave functions for quantum field theory,” Ann. Physics (to be published)]. We also prove the following result: When a cut-off length δ 〉 0 is introduced (in the sense that the multi-time wave function is defined only on a certain set of spacelike configurations, thereby breaking Lorentz invariance), then the multi-time Schrödinger equations with interaction potentials of range δ are consistent; however, in the desired limit δ → 0 of removing the cut-off, the resulting multi-time equations are interaction-free, which supports the conclusion expressed in the title.

Description: In a recent paper, Jaradat et al. [J. Math. Phys.53, 033505 (2012)] have presented the fractional form of the electromagnetic Lagrangian density within the Riemann-Liouville fractional derivative. They claimed that the Agrawal procedure [O. P. Agrawal, J. Math. Anal. Appl.272, 368 (2002)] is used to obtain Maxwell's equations in the fractional form, and the Hamilton's equations of motion together with the conserved quantities obtained from fractional Noether's theorem are reported. In this comment, we draw the attention that there are some serious steps of the procedure used in their work are not applicable even though their final results are correct. Their work should have been done based on a formulation as reported by Baleanu and Muslih [Phys. Scr.72, 119 (2005)].

Description: In this paper, we study the dependence on initial data of solutions to the incompressible Euler equations in Besov spaces. We show that for s 〉 n / p + 1, p ∈ (1, ∞), and r ∈ [1, ∞], the solution map u 0 ↦ u is Hölder continuous in Besov space B p , r s equipped with weaker topology. When the space variable x is taken to be periodic, we obtain a family of explicit periodic solutions. Furthermore, we prove that for any s ∈ R and 1 ⩽ r ⩽ ∞, the solution map is not globally uniformly continuous in B 2 , r s ( T n ) , which extends some results of Himonas and Misiołek [Commun. Math. Phys.296(1), 285–301 (2010)].

Description: We study the number of measurements required for quantum process tomography under prior information, such as a promise that the unknown channel is unitary. We introduce the notion of an interactive observable and we show that any unitary channel acting on a d -level quantum system can be uniquely identified among all other channels (unitary or otherwise) with only O ( d 2 ) interactive observables, as opposed to the O ( d 4 ) required for tomography of arbitrary channels. This result generalizes to the problem of identifying channels with at most q Kraus operators, and slight improvements can be obtained if we wish to identify such a channel only among unital channels or among other channels with q Kraus operators. These results are proven via explicit construction of large subspaces of Hermitian matrices with various conditions on rank, eigenvalues, and partial trace. Our constructions are built upon various forms of totally nonsingular matrices.

Description: Starting from well-known expressions for the T -matrix and its derivative in standard nonrelativistic potential scattering, I rederive recent path-integral formulations due to Efimov and Barbashov et al. Some new relations follow immediately.

Description: Explicit solutions for the wave equations associated to the Dirac, Euler operators and the harmonic oscillator are given.

Description: A relativistic helicity has been formulated in the four-dimensional Minkowski space-time. Whereas the relativistic distortion of space-time violates the conservation of the conventional helicity, the newly defined relativistic helicity conserves in a barotropic fluid or plasma, dictating a fundamental topological constraint. The relation between the helicity and the vortex-line topology has been delineated by analyzing the linking number of vortex filaments which are singular differential forms representing the pure states of Banach algebra. While the dimension of space-time is four, vortex filaments link, because vorticities are primarily 2-forms and the corresponding 2-chains link in four dimension; the relativistic helicity measures the linking number of vortex filaments that are proper-time cross-sections of the vorticity 2-chains. A thermodynamic force yields an additional term in the vorticity, by which the vortex filaments on a reference-time plane are no longer pure states. However, the vortex filaments on a proper-time plane remain to be pure states, if the thermodynamic force is exact (barotropic), thus, the linking number of vortex filaments conserves.

Description: Considering successive extensions of primary translationally shape invariant potentials, we enlarge the Krein-Adler theorem to mixed chains of state adding and state-deleting Darboux-Bäcklund transformations. It allows us to establish novel bilinear Wronskian and determinantal identities for classical orthogonal polynomials.

Description: In this paper we investigate the properties of a delayed bistable system under the effect of multiplicative noise. We first prove the existence and uniqueness of the positive solution and show that its moments are uniformly bounded. Then, we study stochastic dynamics of the solution in long time, the lower and upper bounds for the paths and an estimate for the average value are provided.

Description: We examine a class of discrete Painlevé equations which present a strong asymmetry. These equations can be written as a system of two equations, the right-hand-sides of which do not have the same functional form. We limit here our investigation to two canonical families of the Quispel-Roberts-Thompson (QRT) classification both of which lead to difference equations. Several new integrable discrete systems are identified.

Description: Motivated by the paper by Sutcliffe and Woolley [“On the quantum theory of molecules,” J. Chem. Phys.137, 22A544 (2012)], we present the main ideas used by mathematicians to show the accuracy of the Born-Oppenheimer approximation for molecules. Based on mathematical works on this approximation for molecular bound states, in scattering theory, in resonance theory, and for short time evolution, we give an overview of some rigorous results obtained up to now. We also point out the main difficulties mathematicians are trying to overcome and speculate on further developments. The mathematical approach does not fit exactly to the common use of the approximation in Physics and Chemistry. We criticize the latter and comment on the differences, contributing in this way to the discussion on the Born-Oppenheimer approximation initiated by Sutcliffe and Woolley. The paper neither contains mathematical statements nor proofs. Instead, we try to make accessible mathematically rigourous results on the subject to researchers in Quantum Chemistry or Physics.

Description: A magnetic field is defined by the property that its divergence is zero in a three-dimensional oriented Riemannian manifold. Each magnetic field generates a magnetic flow whose trajectories are curves called as magnetic curves. In this paper, we give a new variational approach to study the magnetic flow associated with the Killing magnetic field in a three-dimensional oriented Riemann manifold, ( M 3 , g ). And then, we investigate the trajectories of the magnetic fields called as N-magnetic and B-magnetic curves.

Description: We show that the range of applicability of the change of representation formula derived by Bravetti et al. [J. Math. Phys.54, 033513 (2013)] is very narrow and extend it to include all physical applications, particularly, applications to black hole thermodynamics, cosmology, and fluid thermodynamics.

Description: Measurement uncertainty relations are quantitative bounds on the errors in an approximate joint measurement of two observables. They can be seen as a generalization of the error/disturbance tradeoff first discussed heuristically by Heisenberg. Here we prove such relations for the case of two canonically conjugate observables like position and momentum, and establish a close connection with the more familiar preparation uncertainty relations constraining the sharpness of the distributions of the two observables in the same state. Both sets of relations are generalized to means of order α rather than the usual quadratic means, and we show that the optimal constants are the same for preparation and for measurement uncertainty. The constants are determined numerically and compared with some bounds in the literature. In both cases, the near-saturation of the inequalities entails that the state (resp. observable) is uniformly close to a minimizing one.

Description: Blow-up behavior of positive solutions of a semi-linear parabolic system arising from thermal explosion, which subject to the homogenous Dirichlet boundary conditions, is investigated. In particular, sufficient conditions for the solutions to blow up are obtained.

Description: We study the three-dimensional magnetohydrodynamics system and obtain its regularity criteria in terms of only two velocity vector field components eliminating the condition on the third component completely. The proof consists of a new decomposition of the four nonlinear terms of the system and estimating a component of the magnetic vector field in terms of the same component of the velocity vector field. This result may be seen as a component reduction result of many previous works [C. He and Z. Xin, “On the regularity of weak solutions to the magnetohydrodynamic equations ,” J. Differ. Equ.213(2), 234–254 (2005); Y. Zhou, “Remarks on regularities for the 3D MHD equations ,” Discrete Contin. Dyn. Syst.12(5), 881–886 (2005)].

Description: In this paper, we investigate Pseudo-Z symmetric space-time manifolds. First, we deal with elementary properties showing that the associated form A k is closed: in the case the Ricci tensor results to be Weyl compatible. This notion was recently introduced by one of the present authors. The consequences of the Weyl compatibility on the magnetic part of the Weyl tensor are pointed out. This determines the Petrov types of such space times. Finally, we investigate some interesting properties of ( PZS ) 4 space-time; in particular, we take into consideration perfect fluid and scalar field space-time, and interesting properties are pointed out, including the Petrov classification. In the case of scalar field space-time, it is shown that the scalar field satisfies a generalized eikonal equation. Further, it is shown that the integral curves of the gradient field are geodesics. A classical method to find a general integral is presented.

Description: We give a microscopic derivation of perturbative quantum field theory, taking causal fermion systems and the framework of the fermionic projector as the starting point. The resulting quantum field theory agrees with standard quantum field theory on the tree level and reproduces all bosonic loop diagrams. The fermion loops are described in a different formalism in which no ultraviolet divergences occur.

Description: We present the geometric formulation of gravity based on the mathematical structure of a Lie Algebroid. We show that this framework provides the geometrical setting to describe the gauge propriety of gravity.

Description: The standard solution to time-harmonic electromagnetic scattering problems in homogeneous layered media relies on the use of the electric field dyadic Green's function. However, for small values of the governing angular frequency ω, evaluation of the electric field using this Green's function exhibits numerical instability. In this short note, we provide an alternative approach which is immune from this low-frequency breakdown as ω → 0. Our approach is based on the generalized Debye source representation of Maxwell fields. Using this formulation, the electric and magnetic fields gracefully decouple in the static limit, a behavior similar to that of the classical Lorenz-Debye-Mie representation of Maxwell fields in spherical geometries. We derive extensions of both the generalized Deybe source and Lorenz-Debye-Mie representations to planar geometries, as well as provide equations for the solution of scattering from a perfectly conducting half-space and in layered media using a Sommerfeld-like approach. These formulas are stable as ω tends to zero, and offer alternatives to the electric field dyadic Green's function.

Description: We consider axisymmetric stationary dirty black holes with regular non-extremal or extremal horizons, and compute their on-horizon Petrov types. The Petrov type (PT) in the frame of the observer crossing the horizon can be different from that formally obtained in the usual (but singular in the horizon limit) frame of an observer on a circular orbit. We call this entity the boosted Petrov type (BPT), as the corresponding frame is obtained by a singular boost from the regular one. The PT off-horizon can be more general than PT on-horizon and that can be more general than the BPT on horizon. This is valid for all regular metrics, irrespective of the extremality of the horizon. We analyze and classify the possible relations between the three characteristics and discuss the nature and features of the underlying singular boost. The three Petrov types can be the same only for space-times of PT D and O off-horizon. The mutual alignment of principal null directions and the generator in the vicinity of the horizon is studied in detail. As an example, we also analyze a special class of metrics with utra-extremal horizons (for which the regularity conditions look different from the general case) and compare their off-horizon and on-horizon algebraic structure in both frames.

Description: This paper presents a simple but nontrivial generalization of Abel's mechanical problem, based on the extended isochronicity condition and the superposition principle. There are two primary aims. The first one is to reveal the linear relation between the transit-time T and the travel-length X hidden behind the isochronicity problem that is usually discussed in terms of the nonlinear equation of motion d 2 X d t 2 + d U d X = 0 with U ( X ) being an unknown potential. Second, the isochronicity condition is extended for the possible Abel-transform approach to designing the isochronous trajectories of charged particles in spectrometers and/or accelerators for time-resolving experiments. Our approach is based on the integral formula for the oscillatory motion by Landau and Lifshitz [Mechanics (Pergamon, Oxford, 1976), pp. 27–29]. The same formula is used to treat the non-periodic motion that is driven by U ( X ). Specifically, this unknown potential is determined by the (linear) Abel transform X ( U ) ∝  A [ T ( E )], where X ( U ) is the inverse function of U ( X ), A = ( 1 / π ) ∫ 0 E d U / E − U is the so-called Abel operator, and T ( E ) is the prescribed transit-time for a particle with energy E to spend in the region of interest. Based on this Abel-transform approach, we have introduced the extended isochronicity condition: typically, τ = T A ( E ) + T N ( E ) where τ is a constant period, T A ( E ) is the transit-time in the Abel type [A-type] region spanning X 〉 0 and T N ( E ) is that in the Non-Abel type [N-type] region covering X 〈 0. As for the A-type region in X 〉 0, the unknown inverse function X A ( U ) is determined from T A ( E ) via the Abel-transform relation X A ( U ) ∝  A [ T A ( E )]. In contrast, the N-type region in X 〈 0 does not ensure this linear relation: the region is covered with a predetermined potential U N ( X ) of some arbitrary choice, not necessarily obeying the Abel-transform relation. In discussing the isochronicity problem, there has been no attempt of N-type regions that are practically of full use for the charged-particle spectrometers and/or accelerators. In this Abel-transform approach, the superposition principle simplifies the derivation of X A ( U ) satisfying the extended isochronicity condition. Although the extended isochronicity condition inevitably discards the low-energy particles, there is no problem for handling accelerated particles because they do not involve the small-amplitude oscillations around the potential minimum. We present analytic examples of X A ( U ) that are instructive. In Appendix B , Urabe's criterion is interpreted in the time domain, using the Abel-transform approach.

Description: A symmetry in quantum mechanics is described by the projective representations of a Lie symmetry group that transforms between physical quantum states such that the square of the modulus of the states is invariant. The Heisenberg commutation relations that are fundamental to quantum mechanics must be valid in all of these physical states. This paper shows that the maximal quantum symmetry group, whose projective representations preserve the Heisenberg commutation relations in this manner, is the inhomogeneous symplectic group. The projective representations are equivalent to the unitary representations of the central extension of the inhomogeneous symplectic group. This centrally extended group is the semidirect product of the cover of the symplectic group and the Weyl-Heisenberg group. Its unitary irreducible representations are computed explicitly using the Mackey representation theorems for semidirect product groups.

Description: Let M n be an n × n real (resp. complex) Wigner matrix and U n Λ n U n * be its spectral decomposition. Set ( y 1 , y 2 ⋯ , y n ) T = U n * x , where x = ( x 1 , x 2 , ⋅⋅⋅, x n ) T is a real (resp. complex) unit vector. Under the assumption that the elements of M n have 4 matching moments with those of GOE (resp. GUE), we show that the process X n ( t ) = β n 2 ∑ i = 1 ⌊ n t ⌋ ( | y i | 2 − 1 n ) converges weakly to the Brownian bridge for any x satisfying ‖ x ‖ ∞ → 0 as n → ∞, where β = 1 for the real case and β = 2 for the complex case. Such a result indicates that the orthogonal (resp. unitary) matrices with columns being the eigenvectors of Wigner matrices are asymptotically Haar distributed on the orthogonal (resp. unitary) group from a certain perspective.

Description: Following Henyey procedure [Phys. Rev. D20, 1460 (1979)], we construct examples of zero modes of the Faddeev-Popov operator in the Landau gauge in Euclidean space in D dimensions, for both SU (2) and SU (3) groups. We obtain gauge field configurations A μ a which give rise to a field strength, F μ ν a = ∂ μ A ν a − ∂ ν A μ a + f a b c A μ b A ν c , whose nonlinear term, f a b c A μ b A ν c , turns out to be non-vanishing. To our knowledge, this is the first time where such a non-abelian configuration is explicitly obtained in the case of SU (3) in 4 D .

Description: The Toda chain of nearest neighbour interacting particles on a line can be described both in terms of geodesic motion on a manifold with one extra dimension, the Eisenhart lift, or in terms of geodesic motion in a symmetric space with several extra dimensions. We examine the relationship between these two realisations and discover that the symmetric space is a generalised, multi-particle Eisenhart lift of the original problem that reduces to the standard Eisenhart lift. Such generalised Eisenhart lift acts as an inverse Kaluza-Klein reduction, promoting coupling constants to momenta in higher dimension. In particular, isometries of the generalised lift metric correspond to energy preserving transformations that mix coordinates and coupling constants. A by-product of the analysis is that the lift of the Toda Lax pair can be used to construct higher rank Killing tensors for both the standard and generalised lift metrics.

Description: We consider the monomer-dimer (MD) problem on a random planar honeycomb lattice model, namely, the random multiple chain. This is a lattice system with non-periodic boundary condition, whose generating process is inspired by the growth of single walled zigzag carbon nanotubes. By applying algebraic and combinatorial techniques we establish a calculating expression of the MD partition function for bipartite graphs, which corresponds to the permanent of a matrix. Further, by using the transfer matrix argument we show that the computing problem of the permanent of high order matrix can be converted into some lower order matrices for this family of lattices, based on which we derive an explicit recurrence formula for evaluating the MD partition function of multiple chains and random multiple chains. Finally, we analyze the expectation of the number of monomer-dimer arrangements on a random multiple chain and the asymptotic behavior of the annealed MD entropy when the multiple chain becomes infinite in width and length, respectively.

Description: We consider the Schrödinger-Poisson system: −ε 2 Δ u + V ( x ) u + ϕ( x ) u = f ( u ),−Δϕ = u 2 in R 3 , where the nonlinear term f is of critical growth. In this paper, we construct a solution ( u ɛ , ϕ ɛ ) of the above elliptic system, which concentrates at an isolated component of positive locally minimum points of V as ɛ → 0 under certain conditions on f . In particular, the monotonicity of f ( s ) s 3 and the so-called Ambrosetti-Rabinowitz condition are not required.

Description: We study a Helmholtz-type spectral problem in a two-dimensional medium consisting of a fully periodic background structure and a perturbation in form of a line defect. The defect is aligned along one of the coordinate axes, periodic in that direction (with the same periodicity as the background), and bounded in the other direction. This setting models a so-called “soft-wall” waveguide problem. We show that there are no bound states, i.e., the spectrum of the operator under study contains no point spectrum.

Description: The purpose of this work is to study some problems in statistical mechanics based on the fractional classical and quantum mechanics. At first stage we have presented the thermodynamical properties of the classical ideal gas and the system of N classical oscillators. In both cases, the Hamiltonian contains fractional exponents of the phase space (position and momentum). At the second stage, in the context of the fractional quantum mechanics, we have calculated the thermodynamical properties for the black body radiation, studied the Bose-Einstein statistics with the related problem of the condensation and the Fermi-Dirac statistics.

Description: The inverse scattering transform for the focusing nonlinear Schrödinger equation with non-zero boundary conditions at infinity is presented, including the determination of the analyticity of the scattering eigenfunctions, the introduction of the appropriate Riemann surface and uniformization variable, the symmetries, discrete spectrum, asymptotics, trace formulae and the so-called theta condition, and the formulation of the inverse problem in terms of a Riemann-Hilbert problem. In addition, the general behavior of the soliton solutions is discussed, as well as the reductions to all special cases previously discussed in the literature.

Description: We present a constructive proof that, in electrodynamics, all of the gauge-invariant Lorentz scalars and pseudoscalars can be expressed as functions of the quadratic ones.

Description: We provide a characterization of all Hamiltonian systems of the form H = ( p 1 2 + p 2 2 ) / 2 + V ( q 1 , q 2 ) , where V is a homogenous polynomial of degree 3 which are completely integrable with Darboux first integrals.

Description: In this paper, we introduce the category of Yetter-Drinfel'd Hom-modules which is a braided monoidal category and show that the category of Yetter-Drinfel'd Hom-modules is a full monoidal subcategory of the left center of left Hom-module category. Also we study the equivalent relationship between the category of Yetter-Drinfel'd Hom-modules and the category of Hom-modules over the Drinfel'd double. Finally, the Faddeev-Reshetikhin-Takhtajan (FRT) type theorem for the quantum Hom-Yang-Baxter equation is investigated.

Description: Partial differential equations of the form div N = 0 , N t + curl M = 0 involving two vector functions in R 3 depending on t , x , y , z appear in different physical contexts, including the vorticity formulation of fluid dynamics, magnetohydrodynamics (MHD) equations, and Maxwell's equations. It is shown that these equations possess an infinite family of local divergence-type conservation laws involving arbitrary functions of space and time. Moreover, it is demonstrated that the equations of interest have a rather special structure of a lower-degree (degree two) conservation law in R 4 ( t , x , y , z ) . The corresponding potential system has a clear physical meaning. For the Maxwell's equations, it gives rise to the scalar electric and the vector magnetic potentials; for the vorticity equations of fluid dynamics, the potentialization inverts the curl operator to yield the fluid dynamics equations in primitive variables; for MHD equations, the potential equations yield a generalization of the Galas-Bogoyavlenskij potential that describes magnetic surfaces of ideal MHD equilibria. The lower-degree conservation law is further shown to yield curl-type conservation laws and determined potential equations in certain lower-dimensional settings. Examples of new nonlocal conservation laws, including an infinite family of nonlocal material conservation laws of ideal time-dependent MHD equations in 2+1 dimensions, are presented.

Description: We consider the structure of algebra of operators, acting in n -fold tensor product space, which are partially transposed on the last term. Using purely algebraical methods we show that this algebra is semi-simple and then, considering its regular representation, we derive basic properties of the algebra. In particular, we describe all irreducible representations of the algebra of partially transposed operators and derive expressions for matrix elements of the representations. It appears that there are two kinds of irreducible representations of the algebra. The first one is strictly connected with the representations of the group S ( n − 1) induced by irreducible representations of the group S ( n − 2). The second kind is structurally connected with irreducible representations of the group S ( n − 1).

Description: We consider a family of genus 2 hyperelliptic curves of even order and obtain explicitly the systems of 5 linear ordinary differential equations for periods of the corresponding Abelian integrals of first, second, and third kind, as functions of some parameters of the curves. The systems can be regarded as extensions of the well-studied Picard–Fuchs equations for periods of complete integrals of first and second kind on odd hyperelliptic curves. The periods we consider are linear combinations of the action variables of several integrable systems, in particular the generalized Neumann system with polynomial separable potentials. Thus the solutions of the extended Picard–Fuchs equations can be used to study various properties of the actions.

Description: In this paper, we study a class of nonlinear magnetic Choquard type equation involving a magnetic potential and nonlocal nonlinearities. By using the method of penalization argument, we show that there exists a family of solutions having multiple concentration regions which are concentrate at the minimum points of potential V .

Description: The fundamental solution of a variant of the three-dimensional wave equation known as “unidirectional pulse propagation equation” (UPPE) and its paraxial approximation is obtained. It is shown that the fundamental solution can be presented as a projection of a fundamental solution of the wave equation to some functional subspace. We discuss the degree of equivalence of the UPPE and the wave equation in this respect. In particular, we show that the UPPE, in contrast to the common belief, describes wave propagation in both longitudinal and temporal directions, and, thereby, its fundamental solution possesses a non-causal character.

Description: The aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e., filiform Lie (super)algebras, into the theory of Lie algebras of order F . Thus, the concept of filiform Lie algebras of order F is obtained. In particular, for F = 3 it has been proved that by using infinitesimal deformations of the associated model elementary Lie algebra it can be obtained families of filiform elementary lie algebras of order 3, analogously as that occurs into the theory of Lie algebras [M. Vergne, “Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes,” Bull. Soc. Math. France98, 81–116 (1970)]. Also we give the dimension, using an adaptation of the sl ( 2 , C ) - module Method , and a basis of such infinitesimal deformations in some generic cases.

Description: Stokes' theorem is investigated in the context of the time-dependent Aharonov-Bohm effect—the two-slit quantum interference experiment with a time varying solenoid between the slits. The time varying solenoid produces an electric field which leads to an additional phase shift which is found to exactly cancel the time-dependent part of the usual magnetic Aharonov-Bohm phase shift. This electric field arises from a combination of a non-single valued scalar potential and/or a 3-vector potential. The gauge transformation which leads to the scalar and 3-vector potentials for the electric field is non-single valued. This feature is connected with the non-simply connected topology of the Aharonov-Bohm set-up. The non-single valued nature of the gauge transformation function has interesting consequences for the 4-dimensional Stokes' theorem for the time-dependent Aharonov-Bohm effect. An experimental test of these conclusions is proposed.

Description: The chiral conformal field theory of free super-bosons is generated by weight one currents whose mode algebra is the affinisation of an abelian Lie super-algebra h with non-degenerate super-symmetric pairing. The mode algebras of a single free boson and of a single pair of symplectic fermions arise for even|odd dimension 1|0 and 0|2 of h , respectively. In this paper, the representations of the untwisted mode algebra of free super-bosons are equipped with a tensor product, a braiding, and an associator. In the symplectic fermion case, i.e., if h is purely odd, the braided monoidal structure is extended to representations of the Z / 2 Z -twisted mode algebra. The tensor product is obtained by computing spaces of vertex operators. The braiding and associator are determined by explicit calculations from three- and four-point conformal blocks.

Description: We study an unorthodox variant of the Berezin-Toeplitz type of quantization scheme, on a reproducing kernel Hilbert space generated by the real Hermite polynomials and work out the associated quasi-classical asymptotics.

Description: Motivated by the desire to understand the causal structure of physical signals produced in curved spacetimes – particularly around black holes – we show how, for certain classes of geometries, one might obtain its retarded or advanced minimally coupled massless scalar Green's function by using the corresponding Green's functions in the higher dimensional Minkowski spacetime where it is embedded. Analogous statements hold for certain classes of curved Riemannian spaces, with positive definite metrics, which may be embedded in higher dimensional Euclidean spaces. The general formula is applied to ( d ≥ 2)-dimensional de Sitter spacetime, and the scalar Green's function is demonstrated to be sourced by a line emanating infinitesimally close to the origin of the ambient ( d + 1)-dimensional Minkowski spacetime and piercing orthogonally through the de Sitter hyperboloids of all finite sizes. This method does not require solving the de Sitter wave equation directly. Only the zero mode solution to an ordinary differential equation, the “wave equation” perpendicular to the hyperboloid – followed by a one-dimensional integral – needs to be evaluated. A topological obstruction to the general construction is also discussed by utilizing it to derive a generalized Green's function of the Laplacian on the ( d ≥ 2)-dimensional sphere.

Description: We study the restriction to a family of second class constrained submanifolds in the cotangent bundle of a double Lie group equipped with a 2-cocycle extended symplectic form to build the corresponding Dirac brackets. It is shown that, for 2-cocycle vanishing on each isotropic subspace of the associated Manin triple, the Dirac bracket contains no traces of the cocycle. We also investigate the restriction of the left translation action of the double Lie group on its cotangent bundle, where it fails to be a canonical transformation. However, the Hamiltonian symmetry is restored on some special submanifolds. The main application is to loop groups, showing that a WZNW-type model on the double Lie group with a quadratic Hamilton function in the momentum maps associated with the left translation action on the cotangent bundle with the canonical symplectic form, restricts to a collective system on some special submanifolds. There, the Lagrangian version coincides with the so-called Poisson-Lie σ -model.

Description: Exact solutions of the ultradiscrete Toda equation with parity variables are constructed from the soliton solutions of the discrete Toda lattice equation. The solution has a periodic phase constant and describes a traveling pulse with a periodic variation. Its behavior is different from that of the usual soliton solution.

Description: We prove results about the Hölder continuity of the spectral measures of the extended CMV matrix, given power law bounds of the solution of the eigenvalue equation. We thus arrive at a unitary analogue of the results of Damanik, Killip, and Lenz [“Uniform spectral properties of one-dimensional quasicrystals, III. α-continuity,” Commun. Math. Phys.212, 191–204 (2000)] about the spectral measure of the discrete Schrödinger operator.

Description: In both theoretical and applied mechanics, the modeling of nonlinear constitutive relations of materials is a topic of prime importance. To properly formulate consistent constitutive laws some restrictions need to be imposed on tensor functions. To that aim, representations theorems for both isotropic and anisotropic functions have been extensively investigated since the middle of the 20th century. Nevertheless, in three-dimensional physical space, most of the results are restricted to sets of tensors up to second-order. The purpose of the present paper is thus to get one step further and to provide an integrity basis for isotropic polynomial functions of a completely symmetric third-order tensor. We exploit the link between the O(3)-action on harmonic tensors and the SL ( 2 , C ) -action on the space of binary forms to explicitly construct this basis. We believe that such an integrity basis may found interesting applications both in continuum mechanics and in other fields of theoretical physics.

Description: We define a two-parameter Kashiwara algebra B r , s ( g ) associated to a semi-simple Lie algebra g and its extremal projector, and study their basic properties. Applying their properties to the category O ( B r , s ( g ) ) , whose objects are “upper bounded” B r , s ( g ) -modules, we obtain its semi-simplicity and the classification of simple modules.

Description: We give new upper and lower bounds on the concavity of quantum entropy. Comparisons are given with other results in the literature.

Description: Classification of quantum spinor fields according to quantum bilinear covariants is introduced in a context of quantum Clifford algebras on Minkowski spacetime. Once the bilinear covariants are expressed in terms of algebraic spinor fields, the duality between spinor and quantum spinor fields can be discussed. Thus, by endowing the underlying spacetime with an arbitrary bilinear form with an antisymmetric part in addition to a symmetric spacetime metric, quantum algebraic spinor fields and deformed bilinear covariants can be constructed. They are thus compared to the classical (non quantum) ones. Classes of quantum spinor fields classes are introduced and compared with Lounesto's spinor field classification. A physical interpretation of the deformed parts and the underlying Z -grading is proposed. The existence of an arbitrary bilinear form endowing the spacetime already has been explored in the literature in the context of quantum gravity [S. W. Hawking, “The unpredictability of quantum gravity ,” Commun. Math. Phys.87, 395 (1982)]. Here, it is shown further to play a prominent role in the structure of Dirac, Weyl, and Majorana spinor fields, besides the most general flagpoles and flag-dipoles. We introduce a new duality between the standard and the quantum spinor fields, by showing that when Clifford algebras over vector spaces endowed with an arbitrary bilinear form are taken into account, a mixture among the classes does occur. Consequently, novel features regarding the spinor fields can be derived.

Description: We derive the Toda's lattice, the Hirota's network, and the nonlinear Schrodinger dynamic equations on a time-space scale by extension on a time-space scale the Ablowitz-Ladik hierarchy of integrable dynamic systems.

Description: In this paper we discuss global existence of the solution of the Maxwell and Newton system of equations, describing the interaction of a rigid charge distribution with the electromagnetic field it generates. A unique solution is proved to exist (for regular charge distributions) on suitable homogeneous and non-homogeneous Sobolev spaces, for the electromagnetic field, and on coordinate and velocity space for the charge; provided initial data belong to the subspace that satisfies the divergence part of Maxwell's equations.

Description: We prove equivariant spectral asymptotics for h -pseudodifferential operators for compact orthogonal group actions generalizing results of El Houakmi and Helffer [“Comportement semi-classique en présence de symétries: Action d'un groupe de Lie compact,” Asymp. Anal.5(2), 91–113 (1991)] and Cassanas [“Reduced Gutzwiller formula with symmetry: Case of a Lie group ,” J. Math. Pures Appl.85(6), 719–742 (2006)]. Using recent results for certain oscillatory integrals with singular critical sets [P. Ramacher, “Singular equivariant asymptotics and Weyl's law: On the distribution of eigenvalues of an invariant elliptic operator,” J. Reine Angew. Math. (Crelles J.) (to be published)], we can deduce a weak equivariant Weyl law. Furthermore, we can prove a complete asymptotic expansion for the Gutzwiller trace formula without any additional condition on the group action by a suitable generalization of the dynamical assumptions on the Hamilton flow.