Thomae Type Formulae For Singular Z N Curves

We give an elementary and rigorous proof of the Thomae type formula for the singular curves . To derive the Thomae formula we use the traditional variational method which goes back to Riemann, Thomae and Fuchs. An important step of the proof is the use of the Szegö kernel computed explicitly in algebraic form for non-singular 1/N-periods. The proof inherits principal points of Nakayashiki’s proof (Nakayashiki in Publ. Res. Inst. Math. Sci 33(6) 987–1015, 1997) obtained for non-singular Z _N curves.     

Unsharpness,Naimark Theorem and Informational Equivalence of Quantum Observables

For each commutative POV measure F there exists (Beneduci, J. Math. Phys. 47:062104-1, 2006; Int. J. Geom. Methods Mod. Phys. 3:1559, 2006) a PV measure E such that F can be interpreted as a random diffusion of E. In its turn, the self-adjoint operator A=∫ λ  dE _λ corresponding to E, can be interpreted (Beneduci, J. Math. Phys. 48:022102-1, 2007; Nuovo Cimento B 123:43–62, 2008) as the projection of a Naimark operator corresponding to the Naimark dilation E ⁺ of F. Moreover E can be algorithmically reconstructed by F. All that suggests that, in some sense, the observables represented by E and F should have the same informational content. We introduce an equivalence relation on the set of observables which we compare with other well known equivalence relations and prove that it is the only one for which E is always equivalent to F.     

Quantum Brownian Motion on Non-Commutative Manifolds: Construction,Deformation and Exit Times

We begin with a review and analytical construction of quantum Gaussian process (and quantum Brownian motions) in the sense of Franz (The Theory of Quantum Levy Processes, [math.PR], 2009), Schürmann (White noise on bioalgebras. Volume 1544 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, 1993) and others, and then formulate and study in details (with a number of interesting examples) a definition of quantum Brownian motions on those non-commutative manifolds (a la Connes) which are quantum homogeneous spaces of their quantum isometry groups in the sense of Goswami (Commun Math Phys 285(1):141–160, 2009). We prove that bi-invariant quantum Brownian motion can be ‘deformed’ in a suitable sense. Moreover, we propose a non-commutative analogue of the well-known asymptotics of the exit time of classical Brownian motion. We explicitly analyze such asymptotics for a specific example on non-commutative two-torus A_q{\mathcal{A}_\theta} , which seems to behave like a one-dimensional manifold, perhaps reminiscent of the fact that A_q{\mathcal{A}_\theta} is a non-commutative model of the (locally one-dimensional) ‘leaf-space’ of the Kronecker foliation.     

<Emphasis Type="Italic">N</Emphasis>=2 de Sitter Super-symmetry Algebra

It was shown that N=1 super-symmetry algebra can be constructed in de Sitter space (Pahlavan et al. in Phys Lett. B 627:217–223, 2005), through calculation of charge conjugation in the ambient space notation (Moradi et al. in Phys. Lett. B 613:74, 2005; Phys. Lett. B 658:284, 2008). Calculation of N=2 super-symmetry algebra constitutes the main frame of this paper. N=2 super-symmetry algebra was presented in Pilch et al. (Commun. Math. Phys. 98:105, 1985). In this paper, we obtain an alternative N=2 super-symmetry algebra.     

Rate of Convergence Towards the Hartree–von Neumann Limit in the Mean-Field Regime

In the mean-field regime we prove convergence, with explicit bounds, of N-particle density matrices satisfying the time-dependent von Neumann equation with factorized initial data to a product of one particle density matrices satisfying the Hartree–von Neumann equation. To prove explicit bounds we generalize techniques developed by Pickl (in A simple derivation of mean field limits for quantum systems. ArXiv:0907.4464, 2009) and Knowles–Pickl (in Commun. Math. Phys. 298(1):101–138, 2010).     

Evaluating Quasilocal Energy and Solving Optimal Embedding Equation at Null Infinity

We study the limit of quasilocal energy defined in Wang and Yau (Phys Rev Lett 102(2):021101, 2009; Commun Math Phys 288(3):919–942, 2009) for a family of spacelike 2-surfaces approaching null infinity of an asymptotically flat spacetime. It is shown that Lorentzian symmetry is recovered and an energy-momentum 4-vector is obtained. In particular, the result is consistent with the Bondi–Sachs energy-momentum at a retarded time. The quasilocal mass in Wang and Yau (Phys Rev Lett 102(2):021101, 2009; Commun Math Phys 288(3):919–942, 2009) is defined by minimizing quasilocal energy among admissible isometric embeddings and observers. The solvability of the Euler-Lagrange equation for this variational problem is also discussed in both the asymptotically flat and asymptotically null cases. Assuming analyticity, the equation can be solved and the solution is locally minimizing in all orders. In particular, this produces an optimal reference hypersurface in the Minkowski space for the spatial or null exterior region of an asymptotically flat spacetime.     

A Macroscopic Model for a System of Swarming Agents Using Curvature Control

In this paper, we study the macroscopic limit of a new model of collective displacement. The model, called PTWA, is a combination of the Vicsek alignment model (Vicsek et al. in Phys. Rev. Lett. 75(6):1226–1229, 1995) and the Persistent Turning Walker (PTW) model of motion by curvature control (Degond and Motsch in J. Stat. Phys. 131(6):989–1021, 2008; Gautrais et al. in J. Math. Biol. 58(3):429–445, 2009). The PTW model was designed to fit measured trajectories of individual fish (Gautrais et al. in J. Math. Biol. 58(3):429–445, 2009). The PTWA model (Persistent Turning Walker with Alignment) describes the displacements of agents which modify their curvature in order to align with their neighbors. The derivation of its macroscopic limit uses the non-classical notion of generalized collisional invariant introduced in (Degond and Motsch in Math. Models Methods Appl. Sci. 18(1):1193–1215, 2008). The macroscopic limit of the PTWA model involves two physical quantities, the density and the mean velocity of individuals. It is a system of hyperbolic type but is non-conservative due to a geometric constraint on the velocity. This system has the same form as the macroscopic limit of the Vicsek model (Degond and Motsch in Math. Models Methods Appl. Sci. 18(1):1193–1215, 2008) (the ‘Vicsek hydrodynamics’) but for the expression of the model coefficients. The numerical computations show that the numerical values of the coefficients are very close. The ‘Vicsek Hydrodynamic model’ appears in this way as a more generic macroscopic model of swarming behavior as originally anticipated.     

Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions. Critical Line between Ferroelectric and Disordered Phases

This is a continuation of the papers of Bleher and Fokin (Commun. Math. Phys., 268:223–284, 2006) and of Bleher and Liechty (Commun. Math. Phys., 286:777–801, 2009), in which the large n asymptotics is obtained for the partition function Z _n of the six-vertex model with domain wall boundary conditions in the disordered and ferroelectric phases, respectively. In the present paper we obtain the large n asymptotics of Z _n on the critical line between these two phases. The first author is supported in part by the National Science Foundation (NSF) Grant DMS-0652005.     

Obtaining Multimode Entangled State Representation by Generalized Radon Transformation of the Wigner Operator

In a preceding paper (Fan and Lv in J. Math. Phys. 50:102108, 2009), the phase-space integration corresponding to the straight line characteristic of two different real parameters λ,τ over the Wigner operator (i.e. the Radon transformation) leads to pure-state density operator |u〉 _{λ,τ  λ,τ} 〈u|, where |u〉 _λ,τ is just the coordinate-momentum intermediate representation. In this work we show that generalized Radon transformation of the Wigner operator yields multimode density operator of continuum variables. This provides us with a new approach for obtaining multimode entangled state representation. The Weyl ordering of the Wigner operator is used in our discussions.     

Non-uniqueness Results for Critical Metrics of Regularized Determinants in Four Dimensions

The regularized determinant of the Paneitz operator arises in quantum gravity [see Connes in (Noncommutative geometry, 1994), IV.4.γ]. An explicit formula for the relative determinant of two conformally related metrics was computed by Branson in (Commun Math Phys 178:301–309, 1996). A similar formula holds for Cheeger’s half-torsion, which plays a role in self-dual field theory [see Juhl in (Families of conformally covariant differential operators, q-curvature and holography. Progress in Mathematics, vol 275, 2009)], and is defined in terms of regularized determinants of the Hodge laplacian on p-forms (p < n/2). In this article we show that the corresponding actions are unbounded (above and below) on any conformal four-manifold. We also show that the conformal class of the round sphere admits a second solution which is not given by the pull-back of the round metric by a conformal map, thus violating uniqueness up to gauge equivalence. These results differ from the properties of the determinant of the conformal Laplacian established in (Commun Math Phys 149:241–262, 1992), (Ann Math 142:171–212, 1995), (Commun Math Phys 189:655–665, 1997).     

Mobility–diffusivity relationship for heavily doped organic semiconductors

The relationship between the diffusivity D _n and the mobility μ _n of chemically doped organic n-type semiconductors exhibiting a disordered band structure is presented. These semiconductors have a Gaussian-type density of states. So, calculations have been performed to elucidate the dependence of D _n /μ _n on the various parameters of this Gaussian density of states. Y. Roichman and N. Tessler (Appl. Phys. Lett. 80:1948, 2002), and subsequently Peng et al. (Appl. Phys. A 86:225, 2007), conducted numerical simulations to study this diffusivity–mobility relationship in organic semiconductors. However, almost all other previous studies of the diffusivity–mobility relationship for inorganic semiconductors are based on Fermi–Dirac integrals. An analytical formulation has therefore been developed for the diffusivity/mobility relationship for organic semiconductors based on Fermi–Dirac integrals. The D _n /μ _n relationship is general enough to be applicable to both non-degenerate and degenerate organic semiconductors. It may be an important tool to study electrical transport in these semiconductors.     

Neutron halo isomers in stable nuclei and their possible application for the production of low energy, pulsed, polarized neutron beams of high intensity and high brilliance

We propose to search for neutron halo isomers populated via γ-capture in stable nuclei with mass numbers of about A=140–180 or A=40–60, where the 4s _1/2 or 3s _1/2 neutron shell model state reaches zero binding energy. These halo nuclei can be produced for the first time with new γ-beams of high intensity and small band width (≤0.1%) achievable via Compton back-scattering off brilliant electron beams, thus offering a promising perspective to selectively populate these isomers with small separation energies of 1 eV to a few keV. Similar to single-neutron halo states for very light, extremely neutron-rich, radioactive nuclei (Hansen et al. in Annu. Rev. Nucl. Part. Sci. 45:591–634, 1995; Tanihata in J. Phys. G., Nucl. Part. Phys. 22:158–198, 1996; Aumann et al. in Phys. Rev. Lett. 84:35, 2000), the low neutron separation energy and short-range nuclear force allow the neutron to tunnel far out into free space much beyond the nuclear core radius. This results in prolonged half-lives of the isomers for the γ-decay back to the ground state in the 100 ps-μs range. Similar to the treatment of photodisintegration of the deuteron, the neutron release from the neutron halo isomer via a second, low-energy, intense photon beam has a known much larger cross section with a typical energy threshold behavior. In the second step, the neutrons can be released as a low-energy, pulsed, polarized neutron beam of high intensity and high brilliance, possibly being much superior to presently existing beams from reactors or spallation neutron sources.     

The Mean-Field Limit for a Regularized Vlasov-Maxwell Dynamics

The present work establishes the mean-field limit of a N-particle system towards a regularized variant of the relativistic Vlasov-Maxwell system, following the work of Braun-Hepp [Commun Math Phys 56:101–113, 1977] and Dobrushin [Func Anal Appl 13:115–123, 1979] for the Vlasov-Poisson system. The main ingredients in the analysis of this system are (a) a kinetic formulation of the Maxwell equations in terms of a distribution of electromagnetic potential in the momentum variable, (b) a regularization procedure for which an analogue of the total energy—i.e. the kinetic energy of the particles plus the energy of the electromagnetic field—is conserved and (c) an analogue of Dobrushin’s stability estimate for the Monge-Kantorovich-Rubinstein distance between two solutions of the regularized Vlasov-Poisson dynamics adapted to retarded potentials.     

Endomorphisms of Quantized Weyl Algebras

Belov-Kanel and Kontsevich (Lett Math Phys 74:181–199, 2005) conjectured that the group of automorphisms of the nth Weyl algebra and the group of polynomial symplectomorphisms of \mathbbC²ⁿ{\mathbb{C}^{2n}} are canonically isomorphic. We discuss how this conjecture can be approached by means of (second) quantized Weyl algebras at roots of unity.     

Quantum Gates and Quantum Algorithms with Clifford Algebra Technique

We use the Clifford algebra technique (J. Math. Phys. 43:5782, 2002; J. Math. Phys. 44:4817, 2003), that is nilpotents and projectors which are binomials of the Clifford algebra objects γ ^a with the property {γ ^a ,γ ^b }₊=2η ^ab , for representing quantum gates and quantum algorithms needed in quantum computers in a simple and an elegant way. We identify n-qubits with the spinor representations of the group SO(1,3) for a system of n spinors. Representations are expressed in terms of products of projectors and nilpotents; we pay attention also on the nonrelativistic limit. An algorithm for extracting a particular information out of a general superposition of 2 ⁿ qubit states is presented. It reproduces for a particular choice of the initial state the Grover’s algorithm (Proc. 28th Annual ACM Symp. Theory Comput. 212, 1996).     

Degree Complexity of a Family of Birational Maps: II. Exceptional Cases

We determine the degree complexity for all elements of a family k _F of birational maps which was introduced and studied in Bedford et al. (Math Phys Anal Geom 11:53–71, 2008).