Generalizations of the Nikodym boundedness and Vitali–Hahn–Saks theorems

Generalizations of the Nikodym boundedness and Vitali–Hahn–Saks theorems for scalar-valued measures on rings of sets that are in general not σ-rings are presented. As a consequence, the rings of subsets of N with density zero and uniform density zero are shown to have the Nikodym property. In addition, vector measure generalizations of the Vitali–Hahn–Saks theorem are given.     

Dually Degenerate Varieties and the Generalization of a Theorem of Griffiths–Harris

The dual variety X^* for a smooth n-dimensional variety X of the projective space P^N is the set of tangent hyperplanes to X. In the general case, the variety X^* is a hypersurface in the dual space (P^N)^*. If dimX^*<N–1, then the variety X is called dually degenerate. The authors refine these definitions for a variety XP^N with a degenerate Gauss map of rankr. For such a variety, in the general case, the dimension of its dual variety X^* is N–l–1, where l=n–r, and X is dually degenerate if dimX^*<N–l–1. In 1979 Griffiths and Harris proved that a smooth variety XP^N is dually degenerate if and only if all its second fundamental forms are singular. The authors generalize this theorem for a variety XP^N with a degenerate Gauss map of rankr. Mathematics Subject Classification (2000) 53A20.     

Expected cover times of random walks on symmetric graphs

We give very simple proofs for an (N–1)H _N–1 lower bound and anN ² upper bound for the expected cover time of symmetric graphs.     

Pseudo-character expansions forU(N)-invariant spin models onCP N−1

We define a set of orthogonal functions on the complex projective spaceCP ^N–1, and compute their Clebsch-Gordan coefficients as well as a large class of 6-j symbols. We also provide all the needed formulae for the generation of high-temperature expansions forU(N)-invariant spin models defined onCP ^N–1.     

Effective primality tests for integers of the formsN=k3 n +1 andN=k2 m 3 n +1

Using third roots of unity Proth's theorem for primality testing is generalized to integers of the formN=k3 ⁿ +1, avoiding the use of Lucas sequences which are more suitable ifN+1 is factored instead ofN–1. This approach has the advantage of being easily combined with Proth's test and gives polynomial time algorithms for testing integers of the formN=k2 ^m 3 ⁿ +1.     

Sur la thermodynamique des processus irréversibles

Sans résuméCf.Kyrille Popoff,Sur la thermodynamique des processus irréversibles, 1^er mémoire: ZAMP3, 42–51 (1952); 2^e mémoire: ZAMP3, 440–448 (1952);Sur les relations phénoménologiques d'Onsager, C. r. Acad. Sci. Paris253, N^o 13, 648–649 (1952);Sur l'échange de chaleur par conduction d'un système à un autre, C. r. Acad. Sci. Paris256, N^o 2, 785–786 (1953).     

The Unitary Implementation of a Locally Compact Quantum Group Action

In this paper we study actions of locally compact quantum groups on von Neumann algebras and prove that every action has a canonical unitary implementation, paralleling Haagerup's classical result on the unitary implementation of a locally compact group action. This result is an important tool in the study of quantum groups in action. We will use it in this paper to study subfactors and inclusions of von Neumann algebras. When α is an action of the locally compact quantum group (M, Δ) on the von Neumann algebra N we can give necessary and sufficient conditions under which the inclusion N^αNM_αN is a basic construction. Here N^α denotes the fixed point algebra and M_αN is the crossed product. When α is an outer and integrable action on a factor N we prove that the inclusion N^αN is irreducible, of depth 2 and regular, giving a converse to the results of M. Enock and R. Nest (1996, J. Funct. Anal.137, 466–543; 1998, J. Funct. Anal.154, 67–109). Finally we prove the equivalence of minimal and outer actions and we generalize the main theorem of Yamanouchi (1999, Math. Scand.84, 297–319): every integrable outer action with infinite fixed point algebra is a dual action.     

EfficientL 2 approximation by splines

LetS _N ^k (t) be the linear space ofk-th order splines on [0, 1] having the simple knotst _i determined from a fixed functiont by the rulet _i=t(i/N). In this paper we introduce sequences of operators {Q _N } _N =1 fromC ^k [0, 1] toS _N ^k (t) which are computationally simple and which, asN, give essentially the best possible approximations tof and its firstk–1 derivatives, in the norm ofL ₂[0, 1]. Precisely, we show thatN ^k–1((f–Q _N f) ⁱ –dist₂(f ⁽¹⁾,S _N ^k–1 (t)))0 fori=0, 1, ...,k–1. Several numerical examples are given.The research of this author was partially supported by the National Science Foundation under Grant MCS-77-02464The research of this author was partially supported by the U.S. Army Reesearch Office under Grant No. DAHC04-75-G-0816     

Large deviations for a simple closed queueing model

We consider a simple queueing model with one service station. The arrival and service processes have intensitiesa(N–Q _t) andNf(N ^–1 Q _t), where Q_t is the queue length,N is a large integer,a>0 andf(x) is a positive continuous function. We establish the large deviation principle for the sequence of the normalized queue length processq _N ^t =N ^–1Q_t,N1 for both light (a<f(0)) and heavy (af(0)) traffic and use this result for an investigation of ergodic properties ofq _N ^t ,N 1.     

Tangential chow forms — rank 1 hypersurfaces in a Grassmannian

Let X ⁿ P ^N be an n-dimensional projective variety, and N–n–1kN–1. The closure in the Grassmannian G(k+1, N+1) of the set of k-planes meeting the smooth locus of X nontransversally is a tangential Chow form (TCF) of X.TCF's are generally hypersurfaces. We show that a hypersurface is a TCF iff its conormal form has rank 1, and that a TCF is a hypersurface iff some quadric in the second fundamental form of X has rank n+k+1–N.     

Ahlfors-régularité des quasi-minima de Mumford–Shah

Let be an open set. We consider on Ω the competitors (U,K) for the reduced Mumford–Shah functional, that is to say the Mumford–Shah functional in which the -norm of U term is removed, where K is a closed subset of Ω and U is a function on ΩK with gradient in  . The main result of this paper is the following: there exists a constant c for which, whenever (U,K) is a quasi-minimizer for the reduced Mumford–Shah functional and B(x,r) is a ball centered on K and contained in Ω with bounded radius, the -measure of is bounded above by cr^N−1 and bounded below by c⁻¹r^N−1.     

Rédei-Funktionen für Zweierpotenzen

L. Rédei has introduced in 1946 a class of rational functions over finite fields of orderp ^t –p being an odd prime — and has proved some interesting results on permutations which are induced by these functions. Recently, similar results have been found for factor rings of the integers of orderp ^t ; moreover, these functions have been applied to construct public key cryptosystems. In this paper we consider functions of Rédei type over finite fields and factor rings of the integers of order 2 ^t . We show that most of the results for oddp also hold in this case.

     

Weight functions admitting repeated positive Kronrod quadrature

In this note it is shown that for weight functions of the formw(t)=(1 –t ²)^1/2/s _m (t), wheres _m is a polynomial of degreem which is positive on [–1, +1], successive Kronrod extension of a certain class ofN-point interpolation quadrature formulas, including theN-point Gauss-formula, is always possible and that each Kronrod extension has the positivity and interlacing property.     

Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations

We study the asymptotics and the global solutions of the following Emden equations: –u=e ^u in a 3-dim domain (>0) or –u=u ^q +|x|^–2 u (q>1) in anN-dim domain. Precise behaviour is obtained by the use of Simon's results on analytic geometric functionals. In the case of the first equation, or the second equation with =0 andq=(N+1)/(N–3) (N>3), we point out how the asymptotics are described via the Moebius group onS ^N–1. For a conformally invariant equation –u=|u|^4/(N–2) u+|x|^–2 u(=±1) we prove the existence of a new type of solution of the formu(x)=|x|(^2–N)/2((Ln|x|)(x/|x|)) where is defined onS ^N–1 and C (;O(N)). Finnally, we extend and simplify the results of Gidas and Spruck on semilinear elliptic equations on compact Riemannian manifolds by a systematic use of the Bochner-Licherowicz-Weitzenböck formula.     

Regular splittings and the discrete Neumann problem

Summary Iterative methods are discussed for approximating a solution to a singular but consistent square linear systemAx=b. The methods are based upon splittingA=M–N withM nonsingular. Monotonicity and the concept of regular splittings, introduced by Varga, are used to determine some necessary and some sufficient conditions in order that the iterationx ⁱ⁺¹=M^–1Nxⁱ+M^–1b converge to a solution to the linear system. Finally, applications are given to solving the discrete Neumann problem by iteration which are based upon the inherent monotonicity in the formulation.This research was supported by the U. S. Army Research Office-Durham under contract no. DAHCO4 74 C 0019.     

Construction of spherical t-designs

Spherical t-designs are Chebyshev-type averaging sets on the d-dimensional unit sphere S ^d–1, that are exact for polynomials of degree at most t. The concept of such designs was introduced by Delsarte, Goethals and Seidel in 1977. The existence of spherical t-designs for every t and d was proved by Seymour and Zaslavsky in 1984. Although some sporadic examples are known, no general construction has been given. In this paper we give an explicit construction of spherical t-designs on S ^d–1 containing N points, for every t,d and N,NN ₀, where N ₀ = C(d)t ^{O(d 3}).     

Markov Processes Related with Dunkl Operators

Dunkl operators are parameterized differential-difference operators on ^Nthat are related to finite reflection groups. They can be regarded as a generalization of partial derivatives and play a major role in the study of Calogero–Moser–Sutherland-type quantum many-body systems. Dunkl operators lead to generalizations of various analytic structures, like the Laplace operator, the Fourier transform, Hermite polynomials, and the heat semigroup. In this paper we investigate some probabilistic aspects of this theory in a systematic way. For this, we introduce a concept of homogeneity of Markov processes on ^Nthat generalizes the classical notion of processes with independent, stationary increments to the Dunkl setting. This includes analogues of Brownian motion and Cauchy processes. The generalizations of Brownian motion have the càdlàg property and form, after symmetrization with respect to the underlying reflection groups, diffusions on the Weyl chambers. A major part of the paper is devoted to the concept of modified moments of probability measures on ^Nin the Dunkl setting. This leads to several results for homogeneous Markov processes (in our extended setting), including martingale characterizations and limit theorems. Furthermore, relations to generalized Hermite polynomials, Appell systems, and Ornstein–Uhlenbeck processes are discussed.     

Pseudodifference Operators on Weighted Spaces, and Applications to Discrete Schrödinger Operators

We study pseudodifference operators on Z ^N with symbols which are bounded on Z ^N ×T ^N together with their derivatives with respect to the second variable. In the same way as partial differential operators on R ^N are included in an algebra of pseudodifferential operators, difference operators on Z ^N are included in an algebra of pseudodifference operators. Particular attention is paid to the Fredholm properties of pseudodifference operators on general exponentially weighted spaces l _w ^p (Z ^N ) and to Phragmen–Lindelöf type theorems on the exponential decay at infinity of solutions to pseudodifference equations. The results are applied to describe the essential spectrum of discrete Schrödinger operators and the decay of their eigenfunctions at infinity.     

The Two-Step Nilpotent Representations of the Extended Affine Hecke Algebra of Type A

We define a set of cell modules for the extended affine Hecke algebra of type A which are parametrised by SL_n()-conjugacy classes of pairs (s, N), where s SL_n() is semisimple and N is a nilpotent element of the Lie algebra which has at most two Jordan blocks and satisfies Ad(s)·N=q ² N. When q ²–1, each of these has irreducible head, and the irreducible representations of the affine Hecke algebra so obtained are precisely those which factor through its Temperley–Lieb quotient. When q ²=–1, the above remarks apply to a subset of the cell modules. Using our work on the cellular nature of those quotients, we are able to obtain complete information on the decomposition of the cell modules in all cases, even when q is a root of unity. They turn out to be multiplicity free, and the composition factors may be precisely described in terms of a partial order on the pairs (s, N). These results give explicit formulae for the dimensions of the irreducibles. Assuming our modules are identified with the standard modules earlier defined by Bernstein–Zelevinski, Kazhdan–Lusztig and others, our results may be interpreted as the determination of certain Kazhdan–Lusztig polynomials. [This has now been proved and will appear in a subsequent work of the authors.]The second author thanks the Australian Research Council and the Alexander von Humboldt Stiftung for support and the Universität Bielefeld for hospitality during the preparation of this work.     

New Cyclic Difference Sets with Singer Parameters Constructed from d-Homogeneous Functions

In this paper, for a prime power q, new cyclic difference sets with Singer para- meters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed by using q-ary sequences (d-homogeneous functions) of period q ⁿ –1 and the generalization of GMW difference sets is proposed by combining the generation methods of d-form sequences and extended sequences. When q is a power of 3, new cyclic difference sets with Singer parameters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed from the ternary sequences of period q ⁿ –1 with ideal autocorrelation introduced by Helleseth, Kumar, and Martinsen.