Phase diagrams and cluster expansions for low temperature models

Low temperature phase diagrams of two-dimensional quantum field models are constructed. Let lie in an (r?1)-dimensional space of perturbations of a polynomial withr degenerate minima. Perform a scaling and assume λ«1. We constructk distinct states on \(\left( {\begin{array}{*{20}c} r \\ k \\ \end{array} } \right)\) hypersurfaces of codimensionk?1 in the space of perturbations. An expansion is used to exhibit exponential clustering of the Schwinger functions of each of these states. At the core of the construction is a general technique for finding the thermodynamically stable phases from a collection of competing minima. We draw on ideas of Pirogov and Sinai [24] for this problem.     

PAC study of O-H and O-D in Ta

A PAC experiment was performed on a¹⁸¹HfTa sample which contained ～ 0. 1 at. oxygen and ～1 at. hydrogen. At low temperature a new quadrupole interaction withν _Q=470 MHz, η=0.95 was found. This interaction is attributed to an O-H complex. At 55 K this interaction disappears and the well known oxygen frequency shows up, indicating a breakup of the O-H pair at 55 K. With deuterium instead of hydrogen the same frequency but a different (105 K) breakup temperature was found.     

Motions of vortex patches

An evolution equation describing the motion of vortrex patches is established. The existence of steady solutions of this equation is proved. These solutions arem-fold symmetric regions of constant vorticity ω₀ and are uniformly rotating with angular velocity Ω in the range $$\tilde \Omega _{m - 1}< \tilde \Omega \leqslant \tilde \Omega _m (\tilde \Omega = \Omega /\omega _0 ,m \geqslant 2)$$ where \(\tilde \Omega _m = (m - 1)/2m\) . We call this class, ofm-fold symmetric rotating regionsD, the class of them-waves of Kelvin. Any may be regarded as a simply connected region which is a stationary configuration of the Euler equations in two dimensions. If then any magnification, rotation or reflection is also in with the same angular velocity Ω ofD. The angular velocity \(\Omega _m = \tilde \Omega _m \omega _0 \) corresponds only to the circle solution, which is a trivial member of every class ,m?2. The class corresponds to the rotating ellipses of Kirchoff. Other properties of the class are established.     

Feynman's path integral

Feynman's integral is defined with respect to a pseudomeasure on the space of paths: for instance, letC be the space of pathsq:T?? → configuration space of the system, letC be the topological dual ofC; then Feynman's integral for a particle of massm in a potentialV can be written where $$S_{\operatorname{int} } (q) = \mathop \smallint \limits_T V(q(t)) dt$$ and wheredw is a pseudomeasure whose Fourier transform is defined by for μ∈C′. Pseudomeasures are discussed; several integrals with respect to pseudomeasures are computed.     

Dynamic Logic Assigned to Automata

A dynamic logic B can be assigned to every automaton Open image in new window without regard if Open image in new window is deterministic or nondeterministic. This logic enables us to formulate observations on Open image in new window in the form of composed propositions and, due to a transition functor T, it captures the dynamic behaviour of Open image in new window . There are formulated conditions under which the automaton Open image in new window can be recovered by means of B and T.     

Solutions of the Schrödinger equation for Dirac delta decorated linear potential

We have studied bound states of the Schrödinger equation for a linear potential together with any finite number (P) of Dirac delta functions. Forx<-0, the potential is given as

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where 0<f; 0<x ₁<x ₂<...<x _P , theσ _i are arbitrary real numbers, and the potential is infinite forx<0.

     

g-Factors of high-spin states in93Ru and95Rh

The g-factors and half lives of three isomers in the N=49 nucleus⁹³Ru and the N=50 nucleus⁹⁵Rh were measured using the PAD method. The results are: . The g-factors are discussed within the shell model and with respect to M1 core polarization and mesonic effects.     

Supersymmetric Quantum Mechanics with Lévy Disorder in One Dimension

We consider the Schrödinger equation with a random potential of the form

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where w is a Lévy noise. We focus on the problem of computing the so-called complex Lyapunov exponent

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where N is the integrated density of states of the system, and γ is the Lyapunov exponent. In the case where the Lévy process is non-decreasing, we show that the calculation of Ω reduces to a Stieltjes moment problem, we ascertain the low-energy behaviour of the density of states in some generality, and relate it to the distributional properties of the Lévy process. We review the known solvable cases—where Ω can be expressed in terms of special functions—and discover a new one.

     

On the effective field theory of heterotic vacua

The effective field theory of heterotic vacua that realise Open image in new window preserving \(\mathcal {N}{=}1\) supersymmetry is studied. The vacua in question admit large radius limits taking the form Open image in new window , with Open image in new window a smooth threefold with vanishing first Chern class and a stable holomorphic gauge bundle Open image in new window . In a previous paper we calculated the kinetic terms for moduli, deducing the moduli metric and Kähler potential. In this paper, we compute the remaining couplings in the effective field theory, correct to first order in \({\alpha ^{\backprime }\,}\). In particular, we compute the contribution of the matter sector to the Kähler potential and derive the Yukawa couplings and other quadratic fermionic couplings. From this we write down a Kähler potential Open image in new window and superpotential Open image in new window .     

Temperature Effect of Hydrogen-Like Impurity on the Ground State Energy of Strong Coupling Polaron in a RbCl Quantum Pseudodot

We study the ground state energy and the mean number of LO phonons of the strong-coupling polaron in a RbCl quantum pseudodot (QPD) with hydrogen-like impurity at the center. The variations of the ground state energy and the mean number of LO phonons with the temperature and the strength of the Coulombic impurity potential are obtained by employing the variational method of Pekar type and the quantum statistical theory (VMPTQST). Our numerical results have displayed that the absolute value of the ground state energy increases (decreases) when the temperature increases at lower (higher) temperature regime, the mean number of the LO phonons increases with increasing temperature, the absolute value of ground state energy and the mean number of LO phonons are increasing functions of the strength of the Coulombic impurity potential.     

Der Zerfall des84Rb und die Fluoreszenzausbeute von Krypton

The electron capture decay and the positron decay of⁸⁴Rb were investigated using NaJ (Tl)-detectors and a Ge (Li)-detector. Measurements of all intensities and of some informative double and triple coincidences were performed. From coincidence measurements betweenK-X-radiation and the following γ-radiation we got theK-fluorescence yield of Krypton Ω_itK=0.653 ± 0.004. Taking in consideration former measurements¹ one concludes a continuous behaviour of Ω_itK(Z) forZ=36, 37 and 38 within an uncertainty of 1%. For the branching ratios of the decay of⁸⁴Rb we obtained The half-life of⁸⁴Rb was determined to beT _1/2=(34.5 ± 0.2) d.     

On the Exact Evaluation of the Face-Centred Cubic Lattice Green Function

The mathematical properties of the lattice Green function

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are investigated, where w=w ₁+iw ₂ lies in a complex plane which is cut from w=?1 to w=3, and {? ₁,? ₂,? ₃} is a set of integers with ? ₁+? ₂+? ₃ equal to an even integer. In particular, it is proved that G(2n,0,0;w), where n=0,1,2,…, is a solution of a fourth-order linear differential equation of the Fuchsian type with four regular singular points at w=?1,0,3 and ∞. It is also shown that G(2n,0,0;w) satisfies a five-term recurrence relation with respect to the integer variable n. The limiting function

$G^{-}(2n,0,0;w_1)\equiv\lim_{\epsilon\rightarrow0+}G(2n,0,0;w_1-\mathrm{i}\epsilon) =G_{\mathrm{R}}(2n,0,0;w_1)+\mathrm{i}G_{\mathrm {I}}(2n,0,0;w_1) ,\nonumber $

where w ₁∈(?1,3), is evaluated exactly in terms of ₂ F ₁ hypergeometric functions and the special cases G ^?(2n,0,0;0), G ^?(2n,0,0;1) and G(2n,0,0;3) are analysed using singular value theory. More generally, it is demonstrated that G(? ₁,? ₂,? ₃;w) can be written in the form

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where Open image in new window are rational functions of the variable ξ, K(k _?) and E(k _?) are complete elliptic integrals of the first and second kind, respectively, with

$k_{-}^2\equiv k_{-}^2(w)={1\over2}- {2\over w} \biggl(1+{1\over w} \biggr)^{-{3\over2}}- {1\over2} \biggl(1-{1\over w} \biggr ) \biggl(1+{1\over w} \biggr)^{-{3\over2}} \biggl(1-{3\over w} \biggr)^{1\over2}\nonumber $

and the parameter ξ is defined as

$\xi\equiv\xi(w)= \biggl(1+\sqrt{1-{3\over w}} \,\biggr)^{-1} \biggl(-1+\sqrt{1+{1\over w}} \,\biggr) .\nonumber $

This result is valid for all values of w which lie in the cut plane. The asymptotic behaviour of G ^?(2n,0,0;w ₁) and G(2n,0,0;w ₁) as n→∞ is also determined. In the final section of the paper a new ₂ F ₁ product form for the anisotropic face-centred cubic lattice Green function is given.

     

Nuclear quadrupolar relaxation and the dynamics of pairs of atoms in liquids

We have developed a model calculation for the electrical field gradient correlation function on a probe atom in the liquid, c_EFG(t)=20(0)V₂₀(t)>. In this model, symmetry of the liquid is introduced explicitly and the distribution function for therelative coordinate r_i(t) between the probe atom and particle i is calculated using Smoluchowski's diffusion equation with a mean force potential Φ(r)=k_BT In g(r). The results for c_EFG(t) can be characterized by two correlation times, , the shorter one being responsible for the small values of R_Q in pure liquid metals, the longer one producing the increase of R_Q in alloys. Also good agreement is found with recent results for c_efg(t) from molecular dynamics studies.     

Action of truncated quantum groups on quasi-quantum planes and a quasi-associative differential geometry and calculus

Ifq is ap ^th root of unity there exists a quasi-coassociative truncated quantum group algebra whose indecomposable representations are the physical representations ofU _q (sl ₂), whose coproduct yields the truncated tensor product of physical representations ofU _q (sl ₂), and whoseR-matrix satisfies quasi-Yang Baxter equations. These truncated quantum group algebras are examples of weak quasitriangular quasi-Hopf algebras (quasi-quantum group algebras). We describe a space of functions on the quasi quantum plane, i.e. of polynomials in noncommuting complex coordinate functionsz _a , on which multiplication operatorsZ _a and the elements of can act, so thatz _a will transform according to some representation ^f of can be made into a quasi-associative graded algebra on which elements of act as generalized derivations. In the special case of the truncatedU _q (sl ₂) algebra we show that the subspaces of monomials inz _a ofn ^th degree vanish fornp–1, and that carries the 2J+ 1 dimensional irreducible representation of ifn=2J, J=0,1/2, ..., 1/2(p–2). Assuming that the representation ^f of the quasi-quantum group algebra gives rise to anR-matrix with two eigenvalues, we develop a quasi-associative differential calculus on. This implies construction of an exterior differentiation, a graded algebra of forms and partial derivatives. A quasi-associative generalization of noncommutative differential geometry is introduced by defining a covariant exterior differentiation of forms. It is covariant under gauge transformations.     

On the relationship between Monstrous Moonshine and the uniqueness of the Moonshine Module

We consider the relationship between the conjectured uniqueness of the Moonshine Module,, and Monstrous Moonshine, the genus zero property of the modular invariance group for each Monster group Thompson series. We first discuss a family of possibleZ _n meromorphic orbifold constructions of based on automorphisms of the Leech lattice compactified bosonic string. We reproduce the Thompson series for all 51 non-Fricke classes of the Monster groupM together with a new relationship between the centralisers of these classes and 51 corresponding Conway group centralisers (generalising a well-known relationship for 5 such classes). Assuming that is unique, we consider meromorphic orbifoldings of and show that Monstrous Moonshine holds if and onlyZ _r if the only meromorphic orbifoldings of are itself or the Leech theory. This constraint on the meromorphic orbifoldings of therefore relates Monstrous Moonshine to the uniqueness of in a new way.     

Lie algebra cohomology and the fusion rules

We prove a vanishing theorem for Lie algebra cohomology which constitutes a loop group analogue of Kostant's Lie algebra version of the Borel-Weil-Bott theorem. Consider a complex semi-simple Lie algebra and an integrable, irreducible, negative energy representation of. Givenn distinct pointsz _k in , with a finite-dimensional irreducible representationV _k of assigned to each, the Lie algebra of-valued polynomials acts on eachV _k , via evaluation atz _k . Then, the relative Lie algebra cohomologyH ^* is concentrated in one degree. As an application, based on an idea of G. Segal's, we prove that a certain homolorphic induction map from representations ofG to representations ofLG at a given level takes the ordinary tensor product into the fusion product. This result had been conjectured by R. Bott.     

with central chargeN

We study representations of the central extension of the Lie algebra of differential operators on the circle, the algebra. We obtain complete and specialized character formulas for a large class of representations, which we call primitive; these include all quasi-finite irreducible unitary representations. We show that any primitive representation with central chargeN has a canonical structure of an irreducible representation of the with the same central charge and that all irreducible representations of with central chargeN arise in this way. We also establish a duality between integral modules of and finite-dimensional irreducible modules ofgl _N , and conjecture their fusion rules.Supported by a Junior Fellowship from Harvard Society of Fellows and in part by NSF grant DMS-9205303.Supported in part by NSF grant DMS-9103792.     

Scaling for a random polymer

LetQ _n ^β be the law of then-step random walk on ?^d obtained by weighting simple random walk with a factore ^?β for every self-intersection (Domb-Joyce model of “soft polymers”). It was proved by Greven and den Hollander (1993) that ind=1 and for every β∈(0, ∞) there exist θ^*(β)∈(0,1) and such that under the lawQ _n ^β asn→∞: $$\begin{array}{l} (i) \theta ^* (\beta ) is the \lim it empirical speed of the random walk; \\ (ii) \mu _\beta ^* is the limit empirical distribution of the local times. \\ \end{array}$$ A representation was given forθ ^*(β) andµ _β ^β in terms of a largest eigenvalue problem for a certain family of ? x ? matrices. In the present paper we use this representation to prove the following scaling result as β?0: $$\begin{array}{l} (i) \beta ^{ - {\textstyle{1 \over 3}}} \theta ^* (\beta ) \to b^* ; \\ (ii) \beta ^{ - {\textstyle{1 \over 3}}} \mu _\beta ^* \left( {\left\lceil { \cdot \beta ^{ - {\textstyle{1 \over 3}}} } \right\rceil } \right) \to ^{L^1 } \eta ^* ( \cdot ) . \\ \end{array}$$ The limitsb ^*∈(0, ∞) and are identified in terms of a Sturm-Liouville problem, which turns out to have several interesting properties. The techniques that are used in the proof are functional analytic and revolve around the notion of epi-convergence of functionals onL ²(?⁺). Our scaling result shows that the speed of soft polymers ind=1 is not right-differentiable at β=0, which precludes expansion techniques that have been used successfully ind≧5 (Hara and Slade (1992a, b)). In simulations the scaling limit is seen for β≦10^?2.