Volume 54, Issue 2, February 2013
Index of content:

For a twodimensional steady supersonic Euler flow past a convex cornered wall with right angle, a characteristic discontinuity (vortex sheet and/or entropy wave) is generated, which separates the supersonic flow from the quiescent gas (hence subsonic). We proved that such a transonic characteristic discontinuity is structurally stable under small perturbations of the upstream supersonic flow in BV. The existence of a weak entropy solution and Lipschitz continuous free boundary (i.e., characteristic discontinuity) is established. To achieve this, the problem is formulated as a free boundary problem for a nonstrictly hyperbolic system of conservation laws; and the free boundary problem is then solved by analyzing nonlinear wave interactions and employing the front tracking method.
 ARTICLES


Partial Differential Equations

Radiative transfer and diffusion limits for wave field correlations in locally shifted random media
View Description Hide DescriptionThe aim of this paper is to develop a mathematical framework for optoelastography. In optoelastography, a mechanical perturbation of the medium produces a decorrelation of optical speckle patterns due to the displacements of optical scatterers. To model this, we consider two optically random media, with the second medium obtained by shifting the first medium in some local region. We derive the radiative transfer equation for the crosscorrelation of the wave fields in the media. Then we derive its diffusion approximation. In both the radiative transfer and the diffusion regimes, we relate the correlation of speckle patterns to the solutions of the radiative transfer and the diffusion equations. We present numerical simulations based on our model which are in agreement with recent experimental measurements.

The equivalence of the ChernSimonsSchrödinger equations and its selfdual system
View Description Hide DescriptionIn this paper, we discuss the equivalence of the second order ChernSimonsSchrödinger equations and its first order selfdual system.

Multidimensional YamadaWatanabe theorem and its applications to particle systems
View Description Hide DescriptionWe prove a multidimensional version of the YamadaWatanabe theorem, i.e., a theorem giving conditions on coefficients of a stochastic differential equation for existence and pathwise uniqueness of strong solutions. It implies an existence and uniqueness theorem for the eigenvalue and eigenvector processes of matrixvalued stochastic processes, called a “spectral” matrix YamadaWatanabe theorem. The multidimensional YamadaWatanabe theorem is also applied to particle systems of squared Bessel processes, corresponding to matrix analogues of squared Bessel processes, Wishart and Jacobi matrix processes. The βversions of these particle systems are also considered.

New porous medium PoissonNernstPlanck equations for strongly oscillating electric potentials
View Description Hide DescriptionWe consider the PoissonNernstPlanck system which is wellaccepted for describing dilute electrolytes as well as transport of charged species in homogeneous environments. Here, we study these equations in porous media whose electric permittivities show a strong contrast compared with the electric permittivity of the electrolyte phase. Our main result is the derivation of convenient lowdimensional equations, that is, of effective macroscopic porous media PoissonNernstPlanck equations, which reliably describe ionic transport. The contrast in the electric permittivities between liquid and solid phase and the heterogeneity of the porous medium induce strongly oscillating electric potentials (fields). In order to account for this specific physical scenario, we introduce a modified asymptotic multiplescale expansion which takes advantage of the nonlinearly coupled structure of the ionic transport equations. This allows for a systematic upscaling resulting in a new effective porous medium formulation which shows a new transport term on the macroscale. Solvability of all arising equations is rigorously verified. The emergence of a new transport term indicates promising physical insights into the influence of the microscale material properties on the macroscale. Hence, systematic upscaling strategies provide a source and a prospective tool to capitalize intrinsic scale effects for scientific, engineering, and industrial applications.

Longtime dynamics for a class of Kirchhoff models with memory
View Description Hide DescriptionThis paper is concerned with a class of Kirchhoff models with memory effects defined in a bounded domain of . This nonautonomous equation corresponds to a viscoelastic version of Kirchhoff models arising in dynamics of elastoplastic flows and plate vibrations. Under assumptions that the exponent p and the growth of f(u) are up to the critical range, it turns out that the model corresponds to an autonomous dynamical system in a larger phase space, by adding a component which describes the relative displacement history. Then the existence of a global attractor is granted. Furthermore, in the subcritical case, this global attractor has finite Hausdorff and fractal dimensions.

Stability of transonic characteristic discontinuities in twodimensional steady compressible Euler flows
View Description Hide DescriptionFor a twodimensional steady supersonic Euler flow past a convex cornered wall with right angle, a characteristic discontinuity (vortex sheet and/or entropy wave) is generated, which separates the supersonic flow from the quiescent gas (hence subsonic). We proved that such a transonic characteristic discontinuity is structurally stable under small perturbations of the upstream supersonic flow in BV. The existence of a weak entropy solution and Lipschitz continuous free boundary (i.e., characteristic discontinuity) is established. To achieve this, the problem is formulated as a free boundary problem for a nonstrictly hyperbolic system of conservation laws; and the free boundary problem is then solved by analyzing nonlinear wave interactions and employing the front tracking method.

A dyadic model on a tree
View Description Hide DescriptionWe study an infinite system of nonlinear differential equations coupled in a treelike structure. This system was previously introduced in the literature and it is the model from which the dyadic shell model of turbulence was derived. It mimics 3D Euler and NavierStokes equations in a rough approximation of wavelet decomposition. We prove existence of finite energy solutions, anomalous dissipation in the inviscid unforced case, existence and uniqueness of stationary solutions (either conservative or not) in the forced case.

Representation Theory and Algebraic Methods

Generalized Bell states and principal realization of the Yangian
View Description Hide DescriptionWe prove that the action of the Yangian algebra is better described by the principal generators on the tensor product of the fundamental representation and its dual. The generalized Bell states or maximally entangled states are permuted by the principal generators in a dramatically simple manner on the tensor product. Under the Yangian symmetry the new quantum number is also explicitly computed, which gives an explanation for these maximally entangled states.

Extended trigonometric Cherednik algebras and nonstationary Schrödinger equations with deltapotentials
View Description Hide DescriptionWe realize an extended version of the trigonometric Cherednik algebra as affine Dunkl operators involving Heaviside functions. We use the quadratic Casimir element of the extended trigonometric Cherednik algebra to define an explicit nonstationary Schrödinger equation with deltapotential. We use coordinate Bethe ansatz methods to construct solutions of the nonstationary Schrödinger equation in terms of generalized Bethe wave functions. It is shown that the generalized Bethe wave functions satisfy affine difference KnizhnikZamolodchikov equations as functions of the momenta. The relation to the vector valued root system analogs of the quantum Bose gas on the circle with deltafunction interactions is indicated.

Grouptheoretical derivation of AharonovBohm phase shifts
View Description Hide DescriptionThe phase shifts of the AharonovBohm effect are generally determined by means of the partial wave decomposition of the underlying Schrödinger equation. It is shown here that they readily emerge from an calculation of the energy levels employing an added harmonic oscillator potential which discretizes the spectrum.

Quantum Mechanics

conserved effective mass Hamiltonians through first and higher order charge operator in a supersymmetric framework
View Description Hide DescriptionThis paper examines the features of a generalized positiondependent mass Hamiltonian H _{ m } in a supersymmetric framework in which the constraints of pseudoHermiticity and are naturally embedded. Different representations of the charge operator are considered that lead to new massdeformed superpotentials which are inherently symmetric. The qualitative spectral behavior of H _{ m } is studied and several interesting consequences are noted.

Quantum Information and Computation

Dimensions, lengths, and separability in finitedimensional quantum systems
View Description Hide DescriptionMany important sets of normalized states in a multipartite quantum system of finite dimension d, such as the set of all separable states, are real semialgebraic sets. We compute dimensions of many such sets in several lowdimensional systems. By using dimension arguments, we show that there exist separable states which are not convex combinations of d or less pure product states. For instance, such states exist in bipartite M⊗N systems when (M − 2)(N − 2) > 1. This solves an open problem proposed by DiVincenzo, Terhal and Thapliyal about 12 years ago. We prove that there exist a separable state ρ and a pure product state, whose mixture has smaller length than that of ρ. We show that any real , which is invariant under all partial transpose operations, is a convex sum of real pure product states. In the case of the 2⊗N system, the number r of product states can be taken to be . We also show that the general multipartite separability problem can be reduced to the case of real states. Regarding the separability problem, we propose two conjectures describing as a semialgebraic set, which may eventually lead to an analytic solution in some lowdimensional systems such as 2⊗4, 3⊗3, and 2⊗2⊗2.

Bipartite entanglement, spherical actions, and geometry of local unitary orbits
View Description Hide DescriptionWe use the geometry of the moment map to investigate properties of pure entangled states of composite quantum systems. The orbits of equally entangled states are mapped by the moment map onto coadjoint orbits of local transformations (unitary transformations which do not change entanglement). Thus, the geometry of coadjoint orbits provides a partial classification of different entanglement classes. To achieve the full classification, a further study of fibers of the moment map is needed. We show how this can be done effectively in the case of the bipartite entanglement by employing Brion's theorem. In particular, we presented the exact description of the partial symplectic structure of all local orbits for two bosons, fermions, and distinguishable particles putting a special emphasis on the generality of the approach allowing one to consider all three cases in completely parallel manners.

Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

A fully covariant informationtheoretic ultraviolet cutoff for scalar fields in expanding Friedmann Robertson Walker spacetimes
View Description Hide DescriptionWhile a natural ultraviolet cutoff, presumably at the Planck length, is widely assumed to exist in nature, it is nontrivial to implement a minimum length scale covariantly. This is because the presence of a fixed minimum length needs to be reconciled with the ability of Lorentz transformations to contract lengths. In this paper, we implement a fully covariant Planck scale cutoff by cutting off the spectrum of the d’Alembertian. In this scenario, consistent with Lorentz contractions, wavelengths that are arbitrarily smaller than the Planck length continue to exist. However, the dynamics of modes of wavelengths that are significantly smaller than the Planck length possess a very small bandwidth. This has the effect of freezing the dynamics of such modes. While both wavelengths and bandwidths are frame dependent, Lorentz contraction and time dilation conspire to make the freezing of modes of transPlanckian wavelengths covariant. In particular, we show that this ultraviolet cutoff can be implemented covariantly also in curved spacetimes. We focus on Friedmann Robertson Walker spacetimes and their muchdiscussed transPlanckian question: The physical wavelength of each comoving mode was smaller than the Planck scale at sufficiently early times. What was the mode's dynamics then? Here, we show that in the presence of the covariant UV cutoff, the dynamical bandwidth of a comoving mode is essentially zero up until its physical wavelength starts exceeding the Planck length. In particular, we show that under general assumptions, the number of dynamical degrees of freedom of each comoving mode all the way up to some arbitrary finite time is actually finite. Our results also open the way to calculating the impact of this natural UV cutoff on inflationary predictions for the cosmic microwave background.

Spectral action for a oneparameter family of Dirac operators on and
View Description Hide DescriptionThe oneparameter family of Dirac operators containing the LeviCivita, cubic, and the trivial Dirac operators on a compact Lie group is analyzed. The spectra for the oneparameter family of Dirac Laplacians on SU (2) and SU (3) are computed by considering a diagonally embedded Casimir operator. Then the asymptotic expansions of the spectral actions for SU (2) and SU (3) are computed, using the Poisson summation formula and the twodimensional EulerMaclaurin formula, respectively. The inflation potential and slowroll parameters for the corresponding pure gravity inflationary theory are generated, using the full asymptotic expansion of the spectral action on SU (2).

Conformal field theories with infinitely many conservation laws
View Description Hide DescriptionGlobally conformal invariant quantum field theories in a Ddimensional spacetime (D even) have rational correlation functions and admit an infinite number of conserved (symmetric traceless) tensor currents. In a theory of a scalar field of dimension D2 they were demonstrated to be generated by bilocal normal products of free massless scalar fields with an O(N), U(N), or Sp(2N) (global) gauge symmetry [B. Bakalov, N. M. Nikolov, K.H. Rehren, and I. Todorov, “Unitary positive energy representations of scalar bilocal fields,” Commun. Math. Phys.271, 223–246 (Year: 2007)10.1007/s0022000601822; eprint arXiv:mathph/0604069v3; B. Bakalov, N. M. Nikolov, K.H. Rehren, and I. Todorov, “Infinite dimensional Lie algebras in 4D conformal quantum field theory,” J. Phys. A Math Theor.41, 194002 (Year: 2008)10.1088/17518113/41/19/194002; eprint arXiv:0711.0627v2 [hepth]]. Recently, conformal field theories “with higher spin symmetry” were considered for D = 3 by Maldacena and Zhiboedov [“Constraining conformal field theories with higher spin symmetry,” eprint arXiv:1112.1016 [hepth]] where a similar result was obtained (exploiting earlier study of CFT correlators). We suggest that the proper generalization of the notion of a 2D chiral algebra to arbitrary (even or odd) dimension is precisely a conformal field theory (CFT) with an infinite series of conserved currents. We recast and complement (part of) the argument of Maldacena and Zhiboedov into the framework of our earlier work. We extend to D = 4 the auxiliary Weylspinor formalism developed by Giombi et al. [“A note on CFT correlators in three dimensions,” eprint arXiv:1104.4317v3 [hepth]] for D = 3. The free field construction only follows for D > 3 under additional assumptions about the operator product algebra. The problem of whether a rational CFT in 4D Minkowski space is necessarily trivial remains open.

Conformally invariant formalism for the electromagnetic field with currents in RobertsonWalker spaces
View Description Hide DescriptionWe show that the LaplaceBeltrami equation □_{6} a = j in , η ≔ diag(+ − − − − +), leads under very moderate assumptions to both the Maxwell equations and the conformal EastwoodSinger gauge condition on conformally flat spaces including the spaces with a RobertsonWalker metric. This result is obtained through a geometric formalism which gives, as byproduct, simplified calculations. In particular, we build an atlas for all the conformally flat spaces considered which allows us to fully exploit the Weyl rescalling to Minkowski space.

Higher dimensional abelian ChernSimons theories and their link invariants
View Description Hide DescriptionThe role played by DeligneBeilinson cohomology in establishing the relation between ChernSimons theory and link invariants in dimensions higher than three is investigated. DeligneBeilinson cohomology classes provide a natural abelian ChernSimons action, non trivial only in dimensions 4l + 3, whose parameter k is quantized. The generalized Wilson (2l + 1)loops are observables of the theory and their charges are quantized. The ChernSimons action is then used to compute invariants for links of (2l + 1)loops, first on closed (4l + 3)manifolds through a novel geometric computation, then on through an unconventional field theoretic computation.

SeibergWitten equations and noncommutative spectral curves in Liouville theory
View Description Hide DescriptionWe propose that there exist generalized SeibergWitten equations in the Liouville conformal field theory, which allow the computation of correlation functions from the resolution of certain Ward identities. These identities involve a multivalued spin one chiral field, which is built from the energymomentum tensor. We solve the Ward identities perturbatively in an expansion around the heavy asymptotic limit, and check that the first two terms of the Liouville threepoint function agree with the known result of Dorn, Otto, Zamolodchikov, and Zamolodchikov. We argue that such calculations can be interpreted in terms of the geometry of noncommutative spectral curves.

Wick rotation for quantum field theories on degenerate Moyal space(time)
View Description Hide DescriptionIn this paper the connection between quantum field theories on flat noncommutative space(times) in Euclidean and Lorentzian signature is studied for the case that time is still commutative. By making use of the algebraic framework of quantum field theory and an analytic continuation of the symmetry groups which are compatible with the structure of Moyal space, a general correspondence between field theories on Euclidean space satisfying a time zero condition and quantum field theories on Moyal Minkowski space is presented (“Wick rotation”). It is then shown that field theories transferred to Moyal space(time) by Rieffel deformation and warped convolution fit into this framework, and that the processes of Wick rotation and deformation commute.
