We study Euler-Poincare systems (i.e., the Lagrangian analogue of Lie-Poisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler-Poincare equations for a parameter dependent Lagrangian by using a variational principle of Lagrange d'Alembert type. Then we derive an Kelvin-Noether theorem for these equations. We also explore their relation with the theory of Lie-Poisson Hamiltonian systems defined on the dual of a semidirect product Lie algebra. The Legendre transformation in such cases is often not invertible; thus, it does not produce a corresponding Euler-Poincare system on that Lie algebra. We avoid this potential difficulty by developing the theory of Euler-Poincare systems entirely within the Lagrangian framework. We apply the general theory to a number of known examples, including the heavy top, ideal compressible fluids and MHD. We also use this framework to derive higher dimensional Camassa-Holm equations, which have many potentially interesting analytical properties. These equations are Euler-Poincare equations for geodesics on diffeomorphism groups (in the sense of the Arnold program) but where the metric is H1 rather than L2. Copyright 1998 Academic Press.
This is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators. Among the advantages of this approach is that the Nevanlinna functions appear as elements of a transfer matrix and convergence of Pade approximants appears as the strong resolvent convergence of finite matrix approximations to a Jacobi matrix. As a bonus of this, we obtain new results on the convergence of certain Pade approximants for series of Hamburger Copyright 1998 Academic Press.
We consider a system of finitely many nonrelativistic, quantum mechanical electrons bound to static nuclei. The electrons are minimally coupled to the quantized electromagnetic field; but we impose an ultraviolet cutoff on the electromagnetic vector potential appearing in covariant derivatives, and the interactions between the radiation field and electrons localized very Far From the nuclei are turned off. For a class of Hamiltonians we prove exponential localization of bound states, establish the existence of a ground state, and derive sufficient conditions for its uniqueness. Furthermore, we show that excited bound states of the unperturbed system become unstable and turn into resonances when the electrons are coupled to the radiation field. To this end we develop a novel renormalization transformation which acts directly on the space of Hamiltonians. (C) 1998 Academic Press.
A new presentation of the n-string braid group Bn is studied. Using it, a new solution to the word problem in Bn is obtained which retains most of the desirable features of the Garside Thurston solution, and at the same time makes possible certain computational improvements. We also give a related solution to the conjugacy problem, but the improvements in its complexity are not clear at this writing. Copyright 1998 Academic Press.
The acoustic scattering operator on the real line is mapped to a Schrödinger operator under the Liouville transformation. The potentials in the image are characterized precisely in terms of their scattering data, and the inverse transfor- mation is obtained as a simple, linear quadrature. An existence theorem for the associated Harry Dym flows is proved, using the scattering method. The scattering problem associated with the Camassa Holm flows on the real line is solved explicitly for a special case, which is used to reduce a general class of such problems to scattering problems on finite intervals. Copyright 1998 Academic Press.
A Hopf algebra is a pair (A, ) where A is an associative algebra with identity and a homomorphism form A to AA satisfying certain conditions. If we drop the assumption that A has an identity and if we allow to have values in the so-called multiplier algebra M(AA), we get a natural extension of the notion of a Hopf algebra. We call this a multiplier Hopf algebra. The motivating example is the algebra of complex functions with finite support on a group with the comultiplication defined as dual to the product in the group. Also for these multiplier Hopf algebras, there is a natural notion of left and right invariance for linear functionals (called integrals in Hopf algebra theory). We show that, if such invariant functionals exist, they are unique (up to a scalar) and faithful. For a regular multiplier Hopf algebra (A, ) (i.e., with invertible antipode) with invariant functionals, we construct, in a canonical way, the dual (Â, ). It is again a regular multiplier Hopf algebra with invariant functionals. It is also shown that the dual of (Â, ) is canonically isomorphic with the original multiplier Hopf algebra (A, ). It is possible to generalize many aspects of harmonic analysis here. One can define the Fourier transform; one can prove Plancherel's formula. Because any finite-dimensional Hopf algebra is a regular multiplier Hopf algebra and has invariant functionals, our duality theorem applies to all finite-dimensional Hopf algebras. Then it coincides with the usual duality for such Hopf algebras. But our category of multiplier Hopf algebras also includes, in a certain way, the discrete (quantum) groups and the compact (quantum) groups. Our duality includes the duality between discrete quantum groups and compact quantum groups. In particular, it includes the duality between compact abelian groups and discrete abelian groups. One of the nice features of our theory is that we have an extension of this duality to the non-abelian case, but within one category. This is shown in the last section of our paper where we introduce the algebras of compact type and the algebras of discrete type. We prove that also these are dual to each other. We treat an example that is sufficiently general to illustrate most of the different features of our theory. It is also possible to construct the quantum double of Drinfel'd within this category. This provides a still wider class of examples. So, we obtain many more than just the compact and discrete quantum within this setting. Copyright 1998 Academic Press.
Let A be a PI-algebra over a field F. We study the asymptotic behavior of the sequence of codimensions cn (A) of A. We show that if A is finitely generated over F then Inv(A)=limn ncn(A) always exists and is an integer. We also obtain the following characterization of simple algebras: A is finite dimensional central simple over F if and only if Inv(A)=dim=A. Copyright 1998 Academic Press.
In this paper we present a self-contained and detailed exposition of the new renormalization group technique proposed in [1, 2]. Its main feature is that the renormalization group transformation acts directly on a space of operators rather than on objects such as a propagator, the partition function, or correlation functions. We apply this renormalization transformation to a Hamiltonian describing the physics of an atom interacting with the quantized electromagnetic field, and we prove that excited atomic states turn into resonances when the coupling between electrons and field is nonvanishing. (C) 1998 Academic Press.
We give a definition of weakn-categories based on the theory of operads. We work with operads having an arbitrary setSof types, or "S-operads," and given such an operadO, we denote its set of operations by elt(O). Then for anyS-operadOthere is an elt(O)-operadO.sup.+whose algebras areS-operads overO. LettingIbe the initial operad with a one-element set of types, and definingI.sup.0+=I,I.sup.(i+1)+=(I.sup.i+).sup.+, we call the operations ofI.sup.(n-1)+the "n-dimensional opetopes." Opetopes form a category, and presheaves on this category are called "opetopic sets." A weakn-category is defined as an opetopic set with certain properties, in a manner reminiscent of Street's simplicial approach to weak[omega]-categories. Similarly, starting from an arbitrary operadOinstead ofI, we define "n-coherentO-algebras," which arentimes categorified analogs of algebras ofO. Examples include "monoidaln-categories," "stablen-categories," "virtualn-functors" and "representablen-prestacks." We also describe hown-coherentO-algebra objects may be defined in any (n+1)-coherentO-algebra.
We give a definition of weak n-categories based on thetheory of operads. We work with operads having an arbitraryset S of types, or "S-operads," and given such anoperad O, we denote its set of operations by elt(O). Then forany S-operad O there is an elt(O)-operad O+ whose algebrasare S-operads over O. Letting I be the initial operad with aone-element set of types, and defining I0+=I, I(i+1)+=(Ii+)+,we call the operations of I(n-1)+ the "n-dimensionalopetopes." Opetopes form a category, and presheaves onthis category are called "opetopic sets." A weakn-category is defined as an opetopic set with certain properties,in a manner reminiscent of Street's simplicial approachto weak omega-categories. Similarly, starting from an arbitraryoperad O instead of I, we define "n-coherent O-algebras,"which are n times categorified analogs of algebras of O. Examplesinclude "monoidal n-categories," "stable n-categories,""virtual n-functors" and "representable n-prestacks."We also describe how n-coherent O-algebra objects may be definedin any (n+1)-coherent O-algebra. Copyright 1998 Academic Press.
We prove that the Anderson Hamiltonian H=-+Von the Bethe lattice has extended states for small disorder. More precisely, given any closed interval I contained in the interior of the spectrum of the Laplacian on the Bethe lattice, we prove that for small disorder H has purely absolutely continuous spectrum in I with probability one (i.e.,ac(H)I=I and pp(H)I=sc(H)I=with probability one), and its integrated density of states is continuously differentiable on the interval I. Copyright 1998 Academic Press.
We consider representations of symmetric groups Sq for large q. We give the asymptotic behaviour of the characters when the corresponding Young diagrams, rescaled by a factor q-1/2, converge to some prescribed shape. This behaviour can be expressed in terms of the free cumulants for a probability measure associated with the limit shape of the diagram. We also show that the basic operations of representation theory, like taking tensor products, restriction, or induction, have a limiting behavior which can be described using the free probability theory of D. Voiculescu. Copyright 1998 Academic Press.
Let G be a compact connected Lie group, and (M,?omega)a compact Hamiltonian G-space, with moment map J : M-< '.Under the assumption that these data are pre-quantizable, onecan construct an associated Spinc-Dirac operator , whose equivariant index yields a virtual representation ofG. We prove a conjecture of Guillemin and Sternberg that if0 is a regular value of J, the multiplicity N(0) of the trivialrepresentation in the index space , is equal to the index of the Spinc-Dirac operator for the symplecticquotient M 0 =J-1(0)/G. This generalizes previous resultsfor the case that G=T is abelian, i.e., a torus. Copyright 1998 Academic Press.