Uniformly non-1 n /(1) Orlicz spaces

A characterisation of uniformly non-1 _n ^/(1) Orlicz space is obtained intrinsically in terms of the Young function determining the Orlicz space. It is shown that a uniformally non-1 _n ^/(1) Orlicz space is reflexive. This work was supported in part by Air Force Grant G2A-152015.     

The Moduli Space of n+1 Points in Complex Hyperbolic n-Space

The structure of the space of orbits of PU(n, 1) acting on (n+1)-tuples of points in complex hyperbolic space is characterized in terms of side lengths and angular invariants. The more general situation in which some of the points lie on the boundary of complex hyperbolic space is described in terms of other basic invariants.     

Relativistic spin 1/2 particle in the total space of the principalU(1) monopole bundle

The quantum dynamics of a relativistic spin 1/2 particle moving in the total spaceP of the principalU(1) bundle associated with the Dirac monopole is described. It is completely characterized in terms of an additionalU(1) bundleP ^* that is intimately associated with the scattering problem. A representation in which the dynamics takes its simplest form is obtained. The lift of the Lorentz group into the total space is constructed.P. N. Lebedev Physics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 93, No. 1, pp. 49–66, October, 1992.     

The 1/n expansion in the Gross-Neveu model: Conformal bootstrap calculation of the exponent 1/v to the order 1/n2

The critical exponent 1/v (dimensions of temperature) in theU _N -symmetric Gross-Neveu model is calculated to the order 1/n ² (nN tr 1) for arbitrary dimension of space.St. Petersburg State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 3, pp. 364–372, December, 1993.     

Bifurcations of the Hamiltonian Fourfold 1:1 Resonance with Toroidal Symmetry

This paper deals with the analysis of Hamiltonian Hopf as well as saddle-center bifurcations in 4-DOF systems defined by perturbed isotropic oscillators (1:1:1:1 resonance), in the presence of two quadratic symmetries Ξ and L ₁. When we normalize the system with respect to the quadratic part of the energy and carry out a reduction with respect to a three-torus group we end up with a 1-DOF system with several parameters on the thrice reduced phase space. Then, we focus our analysis on the evolution of relative equilibria around singular points of this reduced phase space. In particular, dealing with the Hamiltonian Hopf bifurcation the ‘geometric approach’ is used, following the steps set up by one of the authors in the context of 3-DOF systems. In order to see the interplay between integrals and physical parameters in the analysis of bifurcations, we consider as a perturbation a one-parameter family, which in particular includes one of the classical Stark–Zeeman models (parallel case) in three dimensions.     

The Singularity Property of Banach Function Spaces and Unconditional Convergence in L 1[0, 1]

Let X be a Banach function space, L^∞ [0, 1] ⊂ X ⊂ L¹[0, 1]. It is proved that if dual space of X has singularity property in closed set E ⊂ [0, 1] then: 1) there exists no orthonormal basis in C[0, 1], which forms an unconditional basis in X in metric of L¹[0, 1] space, 2) for the Hardy-Littlewood maximal operator M we have      

SeparableL 1 preduals are quotients ofC(Δ)

It is proved that every separable predual space of anL ₁ space is a quotient space ofC(Δ). Johnson's work was supported in part by NSF GP-33578.     

On the duals of Lp spaces with 0<p<1

Given a measure space < Ω,m,μ >, a locally bounded, Hausdorff topological linear space < X, τ > and a real number 0<p<1, one can define the space L_p(Ω,m,μ,X), which is, under certain assumptions, a Fréchet space if endowed with a suitable topology. M.M. Day [1] has given a necessary and sufficient condition, in terms of the properties of the measure space < Ω,m,μ >, for the dual of L_p(Ω,m,μ,C) to be trivial. In this paper a different proof along with a slight generalization is given for this result, using standard and elementary measure theoretic arguments. This revised version was published online in June 2006 with corrections to the Cover Date.     

Abelian cocycles for nonsingular ergodic transformations and the genericity of type III1 transformations

The authors prove that in the space of nonsingular transformations of a Lebesgue probability space the type III₁ ergodic transformations form a denseG set with respect to the coarse topology. They also prove that for any locally compact second countable abelian groupH, and any ergodic type III transformationT, it is generic in the space ofH-valued cocycles for the integer action given byT that the skew product ofT with the cocycle is orbit equivalent toT. Similar results are given for ergodic measure-preserving transformations as well.Research supported in part by: Nat. Sci. and Eng. Res. Council #A7163 and # U0080 F.C.A.C. Quebec, NSF Grants # MCS-8102399 and # DMS-8418431.     

Criteria of the <Emphasis Type="Italic">k</Emphasis>-singularity and division of 1-singular systems

The concept of the k-singularity of systems of points in ℝ ^m space with l ₁ metrics is studied. A system of q points is k-singular if and only if the dimensionality of the linear space of polynomials with powers no higher than the k of the columns of the matrix of pair-wise distances (element-wise multiplication) is strictly less than q. An algebraic criterion of k-singularity is obtained. The problem of dividing a system of points into subsystems that are not 1-singular is considered. An estimate of the minimum number of such subsystems is obtained.     

Fréchet Spaces of Holomorphic Functions without Copies of l1

Let X be a Banach space. Let H_w*(X*) the Fréchet space whose elements are the holomorphic functions defined on X* whose restrictions to each multiple mB(X*), m = 1,2, …, of the closed unit ball B(X*) of X* are continuous for the weak-star topology. A fundamental system of norms for this space is the supremum of the absolute value of each element of H_w*(X*) in mB(X*), m = 1,2,…. In this paper we construct the bidual of l₁ when this space contains no copy of l¹. We also show that if X is an Asplund space, then H_w*(X*) can be represented as the projective limit of a sequence of Banach spaces that are Asplund.     

The normed and Banach envelopes of WeakL 1

The space WeakL ¹ consists of all Lebesgue measurable functions on [0,1] such thatq(f)=supcλ{t:|f(t)|>c} c>0 is finite, where λ denotes Lebesgue measure. Let ρ be the gauge functional of the convex hull of the unit ball {f:q(f)≤1} of the quasi-normq, and letN be the null space of ρ. The normed envelope of WeakL ¹, which we denote byW, is the space (WeakL ¹/N, ρ). The Banach envelope of WeakL ¹, , is the completion ofW. We show that is isometrically lattice isomorphic to a sublattice ofW. It is also shown that all rearrangement invariant Banach function spaces are isometrically lattice isomorphic to a sublattice ofW.     

On the Distributional Index 2F1-transform

In this paper the index ₂F₁-transform is defined on a space of generalized functions by using the method of adjoints. We prove that smooth functions having suitable asymptotic behaviors around zero and infinite are ₂F₁-transforms of generalized functions.     

Skewing (n+1) tensor indices where the index range is n

It is known that any totally skew quantity with (n?+?1) indices, each of which ranges over n values, vanishes identically. The aim of this short note is to show that this is equivalent to the simple fact that any (n?+?1) vectors in an n-dimensional vector space are linearly dependent.     

On the Structure of L 1 of a Vector Measure via its Integration Operator

Geometric and summability properties of the integration operator associated to a vector measure m can be translated in terms of structure properties of the space L¹(m). In this paper we study the cases of the integration operator being: (i) p-concave on L^p(m), or (ii) positive p-summing on L¹(m) (where ). We prove that (i) is equivalent to saying that L¹(m) contains continuously the L^p space of a (non-negative scalar) control measure for m. On the other hand, we show that (ii) holds if and only if L¹(m) is order isomorphic to the L¹ space of a non-negative scalar measure. J.M. Calabuig was supported by MEC and FEDER (MTM2005-08350-C03-03) and Generalitat Valenciana (GV/2007/191). J. Rodríguez was supported by MEC and FEDER (MTM2005-08379) and Generalitat Valenciana (GVPRE/2008/312). E.A. Sánchez-Pérez was supported by MEC and FEDER (MTM2006-11690-C02-01).     

Renorming in the space of Bochner integrable functionsL 1(μ,X)

A sufficient condition is given when a subspaceL⊂L ₁(μ,X) of the space of Bochner integrable function, defined on a finite and positive measure space (S, Φ, μ) with values in a Banach spaceX, is locally uniformly convex renormable in terms of the integrable evaluations {∫ _A fdμ;f∈L}. This shows the lifting property thatL ₁(μ,X) is renormable if and only ifX is, and indicates a large class of renormable subspaces even ifX does not admit and equivalent locally uniformly convex norm.     

Unicity of best one-sidedL 1-approximations

Summary This paper deals with the problem of uniqueness in one-sidedL ₁-approximation. The chief purpose is to characterize finite dimensional subspacesG of the space of continuous or differentiable functions which have a unique best one-sidedL ₁-approximation. In addition, we study a related problem in moment theory. These considerations have an important application to the uniqueness of quadrature formulae of highest possible degree of precision.     

An averaging result forl 1-sequences and applications to weakly conditionally compact sets inL x 1

We establish a theorem onl ¹-sequences obtained by averaging of semi-norms. The result is applied in the study of weakly conditionally compact subsets ofL _x ¹ , whereX is a Banach space. Aspirant N.F.W.O. Belgium     

Investigation of the <1,q> point systems

Delone introduced the concept of the (r,R) point system. In this paper we investigate the thickness of 〈1,q〉 point systems in the d-dimensional Euclidean spaces. We determine the maximum of the thickness of 〈1,q〉 point systems for some q in the non lattice case when the dimension of the space is equal to 2,3,...,7, resp This revised version was published online in June 2006 with corrections to the Cover Date.     

On the number of minimal 1-Steiner trees

We count the number of nonisomorphic geometric minimum spanning trees formed by adding a single point to ann-point set ind-dimensional space, by relating it to a family of convex decompositions of space. TheO(n ^d log^{2d 2}−d n) bound that we obtain significantly improves previously known bounds and is tight to within a polylogarithmic factor. The research of D. Eppstein was performed in part while visiting Xerox PARC.