Bethe Algebra of Homogeneous <Emphasis Type="Italic">XXX</Emphasis> Heisenberg Model has Simple Spectrum

We show that the algebra of commuting Hamiltonians of the homogeneous XXX Heisenberg model has simple spectrum on the subspace of singular vectors of the tensor product of two-dimensional -modules. As a byproduct we show that there exist exactly two-dimensional vector subspaces with a basis such that deg f = l, deg g = n − l + 1 and f (u)g(u − 1) − f (u − 1)g(u) = (u + 1) ⁿ . Supported in part by NSF grant DMS-0601005. Supported in part by RFFI grant 08-01-00638. Supported in part by NSF grant DMS-0555327.     

Gauss Decomposition of the Yangian $$Y({\mathfrak{gl}}_{m|n})$$

We describe a Gauss decomposition for the Yangian of the general linear Lie superalgebra. This gives a connection between this Yangian and the Yangian of the classical Lie superalgebra Y(A(m − 1, n − 1)) (for m ≠ n) defined and studied in papers by Stukopin, and suggests natural definitions for the Yangians and Y(A(n, n)). We also show that the coefficients of the quantum Berezinian generate the centre of the Yangian . This was conjectured by Nazarov in 1991.     

Finite-Time Singularities of an Aggregation Equation in $${\mathbb {R}^n}$$ with Fractional Dissipation

We consider an aggregation equation in , n ≥ 2 with fractional dissipation, namely, , where 0 ≤ γ < 1 and K is a nonnegative decreasing radial kernel with a Lipschitz point at the origin, e.g. K(x) = e ^−|x|. We prove that for a class of smooth initial data, the solutions develop blow-up in finite time.     

Quenched Averages for Self-Avoiding Walks and Polygons on Deterministic Fractals

We study rooted self avoiding polygons and self avoiding walks on deterministic fractal lattices of finite ramification index. Different sites on such lattices are not equivalent, and the number of rooted open walks W _n (S), and rooted self-avoiding polygons P _n (S) of n steps depend on the root S. We use exact recursion equations on the fractal to determine the generating functions for P _n (S), and W _n(S) for an arbitrary point S on the lattice. These are used to compute the averages ,, and over different positions of S. We find that the connectivity constant μ, and the radius of gyration exponent are the same for the annealed and quenched averages. However, , and , where the exponents and , take values different from the annealed case. These are expressed as the Lyapunov exponents of random product of finite-dimensional matrices. For the 3-simplex lattice, our numerical estimation gives and , to be compared with the known annealed values and .     

Moment Bounds for the Smoluchowski Equation and their Consequences

We prove bounds on moments of the Smoluchowski coagulation equations with diffusion, in any dimension d ≥ 1. If the collision propensities α(n, m) of mass n and mass m particles grow more slowly than , and the diffusion rate is non-increasing and satisfies for some b ₁ and b ₂ satisfying 0 ≤ b ₂ < b ₁ < ∞, then any weak solution satisfies for every and T ∈(0, ∞), (provided that certain moments of the initial data are finite). As a consequence, we infer that these conditions are sufficient to ensure uniqueness of a weak solution and its conservation of mass. This work was performed while A.H. held a postdoctoral fellowship in the Department of Mathematics at U.B.C. This work is supported in part by NSF grant DMS0307021.     

On the Uniqueness of Weak Weyl Representations of the Canonical Commutation Relation

Let (T, H) be a weak Weyl representation of the canonical commutation relation (CCR) with one degree of freedom. Namely T is a symmetric operator and H is a self-adjoint operator on a complex Hilbert space satisfying the weak Weyl relation: for all (the set of real numbers), e^−itH D(T) ⊂ D(T) (i is the imaginary unit and D(T) denotes the domain of T) and . In the context of quantum theory where H is a Hamiltonian, T is called a strong time operator of H. In this paper we prove the following theorem on uniqueness of weak Weyl representations: Let be separable. Assume that H is bounded below with and , where is the set of complex numbers and, for a linear operator A on a Hilbert space, σ(A) denotes the spectrum of A. Then ( is the closure of T) is unitarily equivalent to a direct sum of the weak Weyl representation on the Hilbert space , where is the multiplication operator by the variable and with . Using this theorem, we construct a Weyl representation of the CCR from the weak Weyl representation . This work is supported by the Grant-in-Aid No.17340032 for Scientific Research from Japan Society for the Promotion of Science (JSPS).     

Torelli Theorem for the Deligne–Hitchin Moduli Space

Fix integers g ≥ 3 and r ≥ 2, with r ≥ 3 if g = 3. Given a compact connected Riemann surface X of genus g, let denote the corresponding Deligne–Hitchin moduli space. We prove that the complex analytic space determines (up to an isomorphism) the unordered pair , where is the Riemann surface defined by the opposite almost complex structure on X.     

Counting Regions with Bounded Surface Area

Define a cubical complex to be a collection of integer-aligned unit cubes in d dimensions. Lebowitz and Mazel (1998) proved that there are between and complexes containing a fixed cube with connected boundary of (d − 1)-volume n. In this paper we narrow these bounds to between and . We also show that there are connected complexes containing a fixed cube with (not necessarily connected) boundary of volume n.     

One-and-a-Half Quantum de Finetti Theorems

When n − k systems of an n-partite permutation-invariant state are traced out, the resulting state can be approximated by a convex combination of tensor product states. This is the quantum de Finetti theorem. In this paper, we show that an upper bound on the trace distance of this approximation is given by , where d is the dimension of the individual system, thereby improving previously known bounds. Our result follows from a more general approximation theorem for representations of the unitary group. Consider a pure state that lies in the irreducible representation of the unitary group U(d), for highest weights μ, ν and μ + ν. Let ξ_μ be the state obtained by tracing out U _ν. Then ξ_μ is close to a convex combination of the coherent states , where and is the highest weight vector in U _μ. For the class of symmetric Werner states, which are invariant under both the permutation and unitary groups, we give a second de Finetti-style theorem (our “half” theorem). It arises from a combinatorial formula for the distance of certain special symmetric Werner states to states of fixed spectrum, making a connection to the recently defined shifted Schur functions [1]. This formula also provides us with useful examples that allow us to conclude that finite quantum de Finetti theorems (unlike their classical counterparts) must depend on the dimension d. The last part of this paper analyses the structure of the set of symmetric Werner states and shows that the product states in this set do not form a polytope in general.     

Simple Systems with Anomalous Dissipation and Energy Cascade

We analyze a class of dynamical systems of the type where f _n (t) is a forcing term with only for and the coupling coefficients c _n satisfy a condition ensuring the formal conservation of energy . Despite being formally conservative, we show that these dynamical systems support dissipative solutions (suitably defined) and, as a result, may admit unique (statistical) steady states when the forcing term f _n (t) is nonzero. This claim is demonstrated via the complete characterization of the solutions of the system above for specific choices of the coupling coefficients c _n . The mechanism of anomalous dissipations is shown to arise via a cascade of the energy towards the modes with higher n; this is responsible for solutions with interesting energy spectra, namely scales as as n→∞. Here the exponents α depend on the coupling coefficients c _n and denotes expectation with respect to the equilibrium measure. This is reminiscent of the conjectured properties of the solutions of the Navier-Stokes equations in the inviscid limit and their accepted relationship with fully developed turbulence. Hence, these simple models illustrate some of the heuristic ideas that have been advanced to characterize turbulence, similar in that respect to the random passive scalar or random Burgers equation, but even simpler and fully solvable.     

Quantisation of Twistor Theory by Cocycle Twist

We present the main ingredients of twistor theory leading up to and including the Penrose-Ward transform in a coordinate algebra form which we can then ‘quantise’ by means of a functorial cocycle twist. The quantum algebras for the conformal group, twistor space , compactified Minkowski space and the twistor correspondence space are obtained along with their canonical quantum differential calculi, both in a local form and in a global *-algebra formulation which even in the classical commutative case provides a useful alternative to the formulation in terms of projective varieties. We outline how the Penrose-Ward transform then quantises. As an example, we show that the pull-back of the tautological bundle on pulls back to the basic instanton on and that this observation quantises to obtain the Connes-Landi instanton on θ-deformed S ⁴ as the pull-back of the tautological bundle on our θ-deformed . We likewise quantise the fibration and use it to construct the bundle on θ-deformed that maps over under the transform to the θ-deformed instanton. The work was mainly completed while S.M. was visiting July-December 2006 at the Isaac Newton Institute, Cambridge, which both authors thank for support.     

Distribution of Particles Which Produces a “Smart” Material

If A _q(β, α, k) is the scattering amplitude, corresponding to a potential , where D⊂ℝ³ is a bounded domain, and is the incident plane wave, then we call the radiation pattern the function , where the unit vector α, the incident direction, is fixed, β is the unit vector in the direction of the scattered wave, and k>0, the wavenumber, is fixed. It is shown that any function , where S ² is the unit sphere in ℝ³, can be approximated with any desired accuracy by a radiation pattern: , where ∊ >0 is an arbitrary small fixed number. The potential q, corresponding to A(β), depends on f and ∊, and can be calculated analytically. There is a one-to-one correspondence between the above potential and the density of the number of small acoustically soft particles D _m⊂ D, 1≤ m≤ M, distributed in an a priori given bounded domain D⊂ℝ³. The geometrical shape of a small particle D _m is arbitrary, the boundary S _m of D _m is Lipschitz uniformly with respect to m. The wave number k and the direction α of the incident upon D plane wave are fixed. It is shown that a suitable distribution of the above particles in D can produce the scattering amplitude , at a fixed k>0, arbitrarily close in the norm of L ²(S ²× S ²) to an arbitrary given scattering amplitude f(α ', α), corresponding to a real-valued potential q∊ L ²(D), i.e., corresponding to an arbitrary refraction coefficient in D. MSC: 35J05, 35J10, 70F10, 74J25, 81U40, 81V05, 35R30. PACS: 03.04.Kf.     

An Infinite Genus Mapping Class Group and Stable Cohomology

We exhibit a finitely generated group whose rational homology is isomorphic to the rational stable homology of the mapping class group. It is defined as a mapping class group associated to a surface of infinite genus, and contains all the pure mapping class groups of compact surfaces of genus g with n boundary components, for any g ≥ 0 and n > 0. We construct a representation of into the restricted symplectic group of the real Hilbert space generated by the homology classes of non-separating circles on , which generalizes the classical symplectic representation of the mapping class groups. Moreover, we show that the first universal Chern class in is the pull-back of the Pressley-Segal class on the restricted linear group via the inclusion . L. F. was partially supported by the ANR Repsurf:ANR-06-BLAN-0311.     

Spectrum of Time Operators

Let H be a self-adjoint operator on a complex Hilbert space . A symmetric operator T on is called a time operator of H if, for all , (D(T) denotes the domain of T) and . In this paper, spectral properties of T are investigated. The following results are obtained: (i) If H is bounded below, then σ(T), the spectrum of T, is either (the set of complex numbers) or . (ii) If H is bounded above, then is either or . (iii) If H is bounded, then . The spectrum of time operators of free Hamiltonians for both nonrelativistic and relativistic particles is exactly identified. Moreover spectral analysis is made on a generalized time operator. This work is supported by the Grant-in-Aid No.17340032 for Scientific Research from the JSPS.     

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.      

Millimeter-wave Detectors Based on Antenna-coupled Low-barrier Schottky Diodes

The principles of construction of millimeter wave detectors based on low-barrier Schottky diodes and planar antennas are discussed. The modified planar slot antenna with low beam spillover at the resonant frequency of 94 GHz has been developed. Experiments have been carried out to investigate detecting characteristics of the diodes with differential contact resistances at zero bias. Experimental data are well correspond to calculations in a simple model of detector. At the maximum of rf-to-dc voltage sensitivity - more than 10000 V/W - is obtained. At lower values of a better noise equivalent power (NEP), around 10⁻¹² W Hz^−1/2, is predicted.     

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     

Stability of Planar Stationary Solutions to the Compressible Navier-Stokes Equation on the Half Space

Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space under outflow boundary condition is investigated. It is shown that the planar stationary solution is stable with respect to small perturbations in with s≥ [n/2]+1 and the perturbations decay in L ^∞ norm as t →∞, provided that the magnitude of the stationary solution is sufficiently small. The stability result is proved by the energy method. In the proof an energy functional based on the total energy of the system plays an important role.     

Constraints from the old quasar APM 08279+5255 on two classes of Λ(<Emphasis Type="Italic">t</Emphasis>)-cosmologies

The viability of two different classes of Λ(t)CDM cosmologies is tested by using the APM 08279+5255, an old quasar at redshift z = 3.91. In the first class of models, the cosmological term scales as Λ(t) ~ R ⁻ⁿ . The particular case n = 0 describes the standard ΛCDM model whereas n = 2 stands for the Chen and Wu model. For an estimated age of 2 Gyr, it is found that the power index has a lower limit n > 0.21, whereas for 3 Gyr the limit is n > 0.6. Since n can not be so large as ~ 0.81, the ΛCDM and Chen and Wu models are also ruled out by this analysis. The second class of models is the one recently proposed by Wang and Meng which describes several Λ(t)CDM cosmologies discussed in the literature. By assuming that the true age is 2 Gyr it is found that the ε parameter satisfies the lower bound , while for 3 Gyr, a lower limit of is obtained. Such limits are slightly modified when the baryonic component is included.