The Fundamental Solution and Strichartz Estimates for the Schrödinger Equation on Flat Euclidean Cones

We study the Schrödinger equation on a flat euclidean cone ${\mathbb{R}_+ \times \mathbb{S}^1_\rho}We study the Schr?dinger equation on a flat euclidean cone \mathbbR₊ ×\mathbbS¹_r{\mathbb{R}_+ \times \mathbb{S}^1_\rho} of cross-sectional radius ρ > 0, developing asymptotics for the fundamental solution both in the regime near the cone point and at radial infinity. These asymptotic expansions remain uniform while approaching the intersection of the “geometric front,” the part of the solution coming from formal application of the method of images, and the “diffractive front” emerging from the cone tip. As an application, we prove Strichartz estimates for the Schr?dinger propagator on this class of cones.     

A Note on the Extreme Points of Positive Quantum Operations

In this paper, an error in the proof of Theorem 4.9 in Gudder’s paper (Int. J. Theor. Phys. 47(1):268–279, 2008) is pointed out and it is proved that if such that E _i ∈ℂI∖{0} and E _j ∉ℂI for some i,j in {1,2,…,n}, then . This subject is supported by the NNSF of China (No. 10571113, 10871224).     

Exponential Control of Overlap in the Replica Method for <Emphasis Type="Italic">p</Emphasis>-Spin Sherrington-Kirkpatrick Model

In Talagrand (J. Stat. Phys. 126(4–5):837–894, 2007) the large deviations limit for the moments of the partition function Z _N in the Sherrington-Kirkpatrick model (Sherrington and Kirkpatrick in Phys. Rev. Lett. 35:1792–1796, 1972) was computed for all real a≥0. For a≥1 this result extends the classical physicist’s replica method that corresponds to integer values of a. We give a new proof for a≥1 in the case of the pure p-spin SK model that provides a strong exponential control of the overlap. This work is partially supported by NSF grant.     

Role of Fe,Mn, and Zn ions as dopants on the electrical conductivity behavior of sodium phosphate glass

Sodium phosphate glass undoped and doped with different concentrations of chlorides of iron, manganese, and zinc were prepared by melt quenching. The synthesized glasses were characterized by elemental analysis, X-ray diffraction, infrared (IR) spectroscopy, differential scanning calorimetry, and electrical conductivity studies. The undoped sodium phosphate glass (Na₂O–P₂O₅) has low electrical conductivity σ compared to all doped glasses except for 10% FeCl₃-doped samples for which σ is found to be the lowest, and the trend is

On the Spectrum of an Hamiltonian in Fock Space. Discrete Spectrum Asymptotics

A model operator H associated with the energy operator of a system describing three particles in interaction, without conservation of the number of particles, is considered. The location of the essential spectrum of H is described. The existence of infinitely many eigenvalues (resp. the finiteness of eigenvalues) below the bottom τ_ess(H) of the essential spectrum of H is proved for the case where the associated Friedrichs model has a threshold energy resonance (resp. a threshold eigenvalue). For the number N(z) of eigenvalues of H lying below z < τ_ess(H) the following asymptotics is found

Conceptual Unification of Gravity and Quanta

We present a model unifying general relativity and quantum mechanics. The model is based on the (noncommutative) algebra on the groupoid Γ=E×G where E is the total space of the frame bundle over spacetime, and G the Lorentz group. The differential geometry, based on derivations of , is constructed. The eigenvalue equation for the Einstein operator plays the role of the generalized Einstein’s equation. The algebra , when suitably represented in a bundle of Hilbert spaces, is a von Neumann algebra ℳ of random operators representing the quantum sector of the model. The Tomita–Takesaki theorem allows us to define the dynamics of random operators which depends on the state φ. The same state defines the noncommutative probability measure (in the sense of Voiculescu’s free probability theory). Moreover, the state φ satisfies the Kubo–Martin–Schwinger (KMS) condition, and can be interpreted as describing a generalized equilibrium state. By suitably averaging elements of the algebra , one recovers the standard geometry of spacetime. We show that any act of measurement, performed at a given spacetime point, makes the model to collapse to the standard quantum mechanics (on the group G). As an example we compute the noncommutative version of the closed Friedman world model. Generalized eigenvalues of the Einstein operator produce the correct components of the energy-momentum tensor. Dynamics of random operators does not “feel” singularities.     

Entanglement Entropy in Quantum Spin Chains with Finite Range Interaction

We study the entropy of entanglement of the ground state in a wide family of one-dimensional quantum spin chains whose interaction is of finite range and translation invariant. Such systems can be thought of as generalizations of the XY model. The chain is divided in two parts: one containing the first consecutive L spins; the second the remaining ones. In this setting the entropy of entanglement is the von Neumann entropy of either part. At the core of our computation is the explicit evaluation of the leading order term as L → ∞ of the determinant of a block-Toeplitz matrix with symbol

NLO contributions to <Emphasis Type="Italic">B</Emphasis>→<Emphasis Type="Italic">KK</Emphasis><Superscript>*</Superscript> decays in the pQCD approach

We calculate the important next-to-leading-order (NLO) contributions to the B→KK ^* decays from the vertex corrections, the quark loops, and the magnetic penguins in the perturbative QCD (pQCD) factorization approach. The pQCD predictions for the CP-averaged branching ratios are , , and Br(B ⁰→K ⁺ K ^*−+K ⁻ K ^*+)≈1.3×10⁻⁷, which agree well with both the experimental upper limits and the predictions based on the QCD factorization approach. Furthermore, the CP violating asymmetries of the considered decay modes are also evaluated. The NLO pQCD predictions for and decays are and .     

Holomorphic Factorization of Determinants of Laplacians using Quasi-Fuchsian Uniformization

For a quasi-Fuchsian group Γ with ordinary set Ω, and Δ _n the Laplacian on n-differentials on Γ\Ω, we define a notion of a Bers dual basis for ker Δ _n . We prove that det , is, up to an anomaly computed by Takhtajan and the second author in (Commun. Math Phys 239(1-2):183–240, 2003), the modulus squared of a holomorphic function F(n), where F(n) is a quasi-Fuchsian analogue of the Selberg zeta function Z(n). This generalizes the D’Hoker–Phong formula det , and is a quasi-Fuchsian counterpart of the result for Schottky groups proved by Takhtajan and the first author in Analysis 16, 1291–1323, 2006.      

The Ginzburg-Landau expansion and the slope of the upper critical field in disordered superconductors

It is shown that the slope of the upper critical field in superconductors with d pairing drops rapidly with increasing concentration of normal impurities, while in superconductors with anisotropic s pairing increases and reaches the well-known asymptotic level characteristic for the isotropic case. This difference makes it possible, in principle, to employ measurements of H _c 2 in disordered superconductors as an experimental method for determining the type of pairing in high-T _c superconductors and systems with heavy fermions. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 5, 347–352 (10 March 1996)     

Overcrowding estimates for zeroes of Planar and Hyperbolic Gaussian analytic functions

We consider the point process of zeroes of certain Gaussian analytic functions and find the asymptotics for the probability that there are more than m points of the process in a fixed disk of radius r, as . For the planar Gaussian analytic function, , we show that this probability is asymptotic to . For the hyperbolic Gaussian analytic functions, , we show that this probability decays like .In the planar case, we also consider the problem posed by Mikhail Sodin² on moderate and very large deviations in a disk of radius r, as . We partially solve the problem by showing that there is a qualitative change in the asymptotics of the probability as we move from the large deviation regime to the moderate.Research supported by NSF grant #DMS-0104073 and NSF-FRG grant #DMS-0244479.     

N=2 structures on solvable Lie algebras: Thec=9 classification

Let be a finite-dimensional Lie algebra (not necessarily semisimple). It is known that if is self-dual (that is, if it possesses an invariant metric) then it admits anN=1 (affine) Sugawara construction. Under certain additional hypotheses, thisN=1 structure admits anN=2 extension. If this is the case, is said to possess anN=2 structure. It is also known that anN=2 structure on a self-dual Lie algebra is equivalent to a vector space decomposition , where are isotropic Lie subalgebras. In other words,N=2 structures on in one-to-one correspondence with Manin triples . In this paper we exploit this correspondence to obtain a classification of thec=9N=2 structures on solvable Lie algebras. In the process we also give some simple proofs for a variety of Lie algebras. In the process we also give some simple proofs for a variety of Lie algebraic results concerning self-dual Lie algebras admitting symplectic or Kähler structures.     

The Green-Kubo Formula for the Spin-Fermion System

The spin-fermion model describes a two level quantum system (spin 1/2) coupled to finitely many free Fermi gas reservoirs which are in thermal equilibrium at inverse temperatures β _j . We consider non-equilibrium initial conditions where not all β _j are the same. It is known that, at small coupling, the combined system has a unique non-equilibrium steady state (NESS) characterized by strictly sitive entropy production. In this paper we study linear response in this NESS and prove the Green-Kubo formula and the Onsager reciprocity relations for heat fluxes generated by temperature differentials. Dedicated to Jean Michel Combes on the occasion of his sixtyfifth birthday     

Dilatonic Cosmology Model in the <Emphasis Type="Italic">ω</Emphasis>–<Emphasis Type="Italic">ω</Emphasis>′ Plane

In dilatonic cosmology model, we study the behavior of attractor solution in ω–ω′ plane, which is defined by the equation of state parameter for the dark energy and its derivative with respect to N (the logarithm of the scale factor a). This is a good method which is useful to the study of classifying the dynamical dark energy models including “freezing” and “thawing” model. We find that our model belongs to “freezing” type model classified in ω–ω′ plane. We show mathematically the property of attractor solutions which correspond to ω _σ =−1, Ω _σ =1. The present values of energy density parameter , and are 0.715001, 0.284972 and 0.00002706 respectively, which meet the current observations well. Finally, we can obtain that the coupling between dilaton and matter affects the evolutive process of the Universe, but not the fate of the Universe.     

On the Global Wellposedness to the 3-D Incompressible Anisotropic Navier-Stokes Equations

Corresponding to the wellposedness result [2] for the classical 3-D Navier-Stokes equations (NS _ν) with initial data in the scaling invariant Besov space, here we consider a similar problem for the 3-D anisotropic Navier-Stokes equations (ANS _ν), where the vertical viscosity is zero. In order to do so, we first introduce the Besov-Sobolev type spaces, and Then with initial data in the scaling invariant space we prove the global wellposedness for (ANS _ν) provided the norm of initial data is small enough compared to the horizontal viscosity. In particular, this result implies the global wellposedness of (ANS _ν) with high oscillatory initial data (1.2).     

Inhomogeneous contact processes on trees

We consider an inhomogeneous contact process on a tree of degreek, where the infection rate at any site isλ, the death rate at any site in isδ (with 0 <δ ⩽ 1) and that at any site in is 1. Denote by the critical value for thehomogeneous model (i.e.,δ=1) on and byϑ(δ, λ) the survival probability of the inhomogeneous model on . We prove that whenk > 4, if , a subtree embedded in , with 1 ⩽σ ⩽ √k, then three existsδ _c ^σ strictly between ( ) and 1 such that ( ) whenδ >δ _c ^σ andϑ(δ, λ _c( ) > 0 whenδ <δ _c ^σ ; ifS={o}, the origin of , then for anyδ ε (0, 1).     

General theory of renormalization of gauge theories in nonlinear gauges

We discuss the general theory of renormalization of unbroken gauge theories in the nonlinear gauges in which the gauge-fixing term is of the form We show that higher loop renormalization modifiesfα [A] to contain ghost terms of the form and show how the corresponding ghost terms are deduced fromfα [A, c, c] uniquely. We show that the theory can be renormalized while preserving a modified form of BRS invariance by multiplicative and independent renormalizations onA, c, g, η, ζ, τ. We briefly discuss the independence of the renormalized S-matrix from η,ζ, τ.     

Schwinger mechanism for gluon-pair production in the presence of arbitrary time-dependent chromo-electric field

We study the Schwinger mechanism for gluon-pair production in the presence of an arbitrary time-dependent chromo-electric background field E ^a (t) with arbitrary color index a=1,2,…,8 in SU(3) by directly evaluating the path integral. We obtain an exact expression for the probability of non-perturbative gluon-pair production per unit time per unit volume and per unit transverse momentum, , from arbitrary E ^a (t). We show that the tadpole (or single-gluon) effective action does not contribute to the non-perturbative gluon-pair production rate, . We find that the exact result for non-perturbative gluon-pair production is independent of all the time derivatives , where n=1,2,…,∞, and that it has the same functional dependence on the two Casimir invariants, [E ^a (t)E ^a (t)] and [d _abc E ^a (t)E ^b (t)E ^c (t)]², as the constant chromo-electric field E ^a result with the replacement: E ^a →E ^a (t). This result relies crucially on the validity of the shift conjecture, which has not yet been established. This result may be relevant to the study of the production of a non-perturbative quark–gluon plasma at RHIC and LHC.     

Dyson’s Constants in the Asymptotics of the Determinants of Wiener-Hopf-Hankel Operators with the Sine Kernel

Let stand for the integral operators with the sine kernels acting on L ²[0,α]. Dyson conjectured that the asymptotics of the Fredholm determinants of are given by