Zero-One Law for Symmetric Convolution Semigroups of Measures on Groups

Let ( _t ) _t>0 be a symmetric weakly continuous semigroup of probability measures on a nonabelien complete separable group G and let v be its Lévy measure. The purpose of this paper is to provide a relatively simple proof of the zero-one law for semigroups with the Lévy measure satisfying either v(H ^c) = or v(H ^c) = 0.     

The isometric immersion of strongly parabolic metrics in the class of strongly parabolic surfaces

It is known that a k-strongly parabolic surface in Euclidean space (index of relative nullity is at least k) inherits a k-strongly parabolic metric (index of nullity is at least k). For the definitions of the index of relative nullity (x) and the index of nullity (x) see Sh. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Publishers, New York (1969), Vol. II, pp. 347–348. The converse is not true. In particular, there exists an analytic, three-dimensional, 1-strongly parabolic metric that does not admit a local isometric immersion into Euclidean space of any dimension in the class of three-dimensional 1-strongly parabolic surfaces. The proof is carried out with the help of the method of exterior Cartan forms.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 27–37, 1990.     

Two classes of conformally flat contact metric 3-manifolds

We give a classification of 3—dimensional conformally flat contact metric manifolds satisfying: =0(=L g) orR(Y, Z)=k[(Z)Y–(Y)Z]+[(Z)hY]–(Y)hZ] wherek and are functions. It is proved that they are flat (the non-Sasakian case) or of constant curvature 1 (the Sasakian case).     

Effect of contraction on turbulent channel flow

Summary Measurements in an effectively two-dimensional channel indicate that flow acceleration at constant Reynolds Number can have appreciable effects on the turbulence structure. As in wakes, the structure approaches equilibrium exponentially after the acceleration. The effect of acceleration appears to be (at least qualitatively) of the same kind as is observed when turbulence is subjected to rapid distortion.

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Zusammenfassung Messungen in einem effektiv zwei-dimensionalen Kanal zeigen, daß eine Strömungsbeschleunigung bei konstanter Reynolds-Zahl einen wesentlichen Einfluß auf die Turbulenz-Struktur haben kann. Nach der Beschleunigung findet man eine exponentielle Annäherung an das Gleichgewicht, wie bei Dellen. Der Beschleunigungs-Effekt auf die Turbulenz erscheint, wenigstens qualitativ, von der gleichen Art zu sein wie der Einfluß von raschen Verformungen.

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Symbols c _f skin-friction coefficient = - D half-width of the channel - L length of contraction=12 inches - dp/dx pressure gradient - V mean velocity in thex-direction,U=U ₀ aty=D - U mean velocity - average of the mean velocity - U ^* friction velocity = (₀/)^± - root mean square value of the turbulent velocity fluctuations inx, y andz directions - X coordinate in the direction of the flow;X=0 at the end of contraction - Y coordinate perpendicular to the surface of the wall on which measurements are madey=0 at the wall andy=D at the centre of the channel - ₀ Wall shear stress - density of the fluid - viscosity of the fluid - kinematic viscosity of the fluid=/     

Limit Theorems for Multiplicative Processes

Let W be a non-negative random variable with EW=1, and let {W _i } be a family of independent copies of W, indexed by all the finite sequences i=i ₁i _n of positive integers. For fixed r and n the random multiplicative measure ⁿ _r has, on each r-adic interval at nth level, the density with respect to the Lebesgue measure on [0,1]. If EW log Wr, the sequence { ⁿ _r } _n converges a.s. weakly to the Mandelbrot measure _r . For each fixed 1n, we study asymptotic properties for the sequence of random measures { ⁿ _r } _r as r. We prove uniform laws of large numbers, functional central limit theorems, a functional law of iterated logarithm, and large deviation principles. The function-indexed processes is a natural extension to a tree-indexed process at nth level of the usual smoothed partial-sum process corresponding to n=1. The results extend the classical ones for { ¹ _r } _r , and the recent ones for the masses of { _r } _r established in Ref. 23.     

Future independent times and Markov chains

Summary A random timeT is a future independent time for a Markov chain (X _n ) ₀ ifT is independent of (X _T+n ) _n ^/ =0 and if (X _T+n ) _n ^/ =0 is a Markov chain with initial distribution and the same transition probabilities as (X _n ) ₀ . This concept is used (with the conditional stationary measure) to give a new and short proof of the basic limit theorem of Markov chains, improving somewhat the result in the null-recurrent case.This work was supported by the Swedish Natural Science Research Council and done while the author was visiting the Department of Statistics, Stanford University     

An infinite dimensional analogue of the Albeverio-Röckner's theorem on Fukushima's conjecture

LetX be the solution of the SDE:dX _t = (X _t)dB _t +b(X _t)dt, with andb C _b (R) such that >0 for some constant , andB a real Brownian motion. Let be the law ofX onE=C([0, 1],R) andk E* – {0}, whereE* is the topological dual space ofE. Consider the classical form: _k (u, v)=u / kv / kd, whereu andv are smooth functions onE. We prove that, if _k is closable for anyk in a dense subset ofE* and if the smooth functions are contained in the domain of the generator of the closure of _k , must be a constant function.     

Stochastic integrals on general topological measurable spaces

Summary A general theory of stochastic integral in the abstract topological measurable space is established. The martingale measure is defined as a random set function having some martingale property. All square integrable martingale measures constitute a Hilbert space M ². For each M ², a real valued measure on the predictable -algebra is constructed. The stochastic integral of a random function with respect to is defined and investigated by means of Riesz's theorem and the theory of projections. The stochastic integral operator I is an isometry from L ²() to a stable subspace of M ², its inverse is defined as a random Radon-Nikodym derivative. Some basic formulas in stochastic calculus are obtained. The results are extended to the cases of local martingale and semimartingale measures as well.     

The Existence of Kirkman Squares—Doubly Resolvable (v,3,1)-BIBDs

A Kirkman square with index , latinicity , block size k, and v points, KS _k (v;,), is a t×t array (t=(v–1)/(k–1)) defined on a v-set V such that (1) every point of V is contained in precisely cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V, and (3) the collection of blocks obtained from the non-empty cells of the array is a (v, k,)-BIBD. For =1, the existence of a KS _k (v; , ) is equivalent to the existence of a doubly resolvable (v, k, )-BIBD. The spectrum of KS ₂ (v; 1, 1) or Room squares was completed by Mullin and Wallis in 1975. In this paper, we determine the spectrum for a second class of doubly resolvable designs with =1. We show that there exist KS ₃ (v; 1, 1) for , v=3 and v27 with at most 23 possible exceptions for v.     

The imaginary zeros of a mixed Bessel function

In the present work we study the existence and monotonicity properties of the imaginary zeros of the mixed Bessel functionM _v(z)=(z²+)J_v(z)+zJ_v(z). Such a function includes as particular cases the functionsJ _v(z)(==0), J_v(z)(=–v²,=1)x andH _v(z)=J_v(z)+zJ_v(z), whereJ _v(z) is the Bessel function of the first kind and of orderv>–1 andJ _v(z), J_v(z) are the first two derivatives ofJ _v(z). Upper and lower bounds found for the imaginary zeros of the functionsJ _v(z), J_v(z) andH _v(z) improve previously known bounds.

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Zusammenfassung Dieser Artikel betrifft die Existenz und Monotonie von Eigenschaften imaginärer Nullen der gemischten BesselfunktionM _v(z)=(z²+)J_v(z)+zJ_v(z). Eine solche Funktion enthält als Spezialfall die FunktionenJ _v(z)(==0), J_v(z)(=–v²,=1) undH _v(z)=J_v(z)+zJ_v(z), woJ _v(z)die Besselfunktion von erster Art und Ordnungv>–1 andJ _v(z), J_v(z) sind die erste und zweite Ableitung vonJ _v(z). Untere und obere Schranken, die für die imaginären Nullen der FunktionenJ _v(z), J_v(z) undH _v(z) gefunden wurden, verbessern früher bekannte Resultate.

     

On the motion of visco-elastic electrically conducting liquid in the presence of a magnetic field

Zusammenfassung Es wird die Strömung einer viskoelastischen, elektrisch leitenden Flüssigkeit untersucht auf Grund der elektromagnetischen Grundgleichung und der für solche Flüssigkeiten geltenden Bewegungsgleichungen. Im besonderen wird die Strömung zwischen parallelen Platten erörtert. Sie ist charakterisiert durch zwei dimensionslose Kennzahlen, die Hartmannsche Zahl und die Elastizitätszahl. Das elastische Verhalten bewirkt eine Verflachung des Geschwindigkeitsprofils und verändert auch den Druck und die Schubspannungen.

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Notations x ₁,x ₂,x ₃ space coordinates - B magnetic induction vector - E electric intensity - H magnetic field - D displacement vector - J current density vector - _e E electrostatic force - V×B current density due to the motion of the fluid - _e V convection current - _e excess charge density - electrical conductivity - magnetic permeability - dielectric constant - =1/ magnetic diffusivity - V=v ¹ i+v ² j+v ³ k velocity vector - v ₁= ₁/ kinematic viscosity - ₁ coefficient of viscosity - _c coefficient of cross-viscosity - v _c = _c / kinematic cross-viscosity - density of the fluid - relaxation time - S _j ⁱ stress-tensor - d _j ⁱ strain-tensor - x, y, z dimensionless coordinates - L spacing between the planes, taken as the characteristic length - U ₀ characteristic velocity - f non-dimensional velocity - H non-dimensionalx-component of the magnetic field.     

Distribution-independent properties of the convex hull of random points

Denote byV _n ^(d) the expected volume of the convex hull ofn points chosen independently according to a given probability measure in Euclideand-spaceE ^d. Ifd=2 ord=3 and is the measure corresponding to the uniform distribution on a convex body inE ^d, Affentranger and Badertscher derived that

Small Ball Probabilities Around Random Centers of Gaussian Measures and Applications to Quantization

Let be a centered Gaussian measure on a separable Hilbert space (E, ). We are concerned with the logarithmic small ball probabilities around a -distributed center X. It turns out that the asymptotic behavior of –log (B(X,)) is a.s. equivalent to that of a deterministic function _R (). These new insights will be used to derive the precise asymptotics of a random quantization problem which was introduced in a former article by Dereich, Fehringer, Matoussi, and Scheutzow.⁽⁸⁾     

Attainable order of rational approximations to the exponential function with only real poles

Rational approximations of the form _i=0 ^m a _i q ⁱ / _i=1 ⁿ (1+ _i q) to exp(–q),qC, are studied with respect to order and error constant. It is shown that the maximum obtainable order ism+1 and that the approximation of orderm+1 with least absolute value of the error constant has ₁=₂=...= _n . As an application it is shown that the order of av-stage semi-implicit Runge-Kutta method cannot exceedv+1.     

The shape of buckled nonlinearly elastic rings

Zusammenfassung Man studiert das qualitative Verhalten der Lösungen der gewöhnlichen Differentialgleichungen für das Gleichgewicht dehnbarer nichtlinear-elastischer Ringe unter hydrostatischem Druck. Das Material dieser Ringe wird durch eine Verzerrungsenergiefunktion definiert, die von einer Dehnungs- und von einer Biegeverformung abhängt. Liegt die Biegeverformung () in einem gewissen Streifenv ₁<<v ₂, so wird jeder Lösungszweig von der Summe der Extrema von charakterisiert. (Die Zahlenv ₁ undv ₂ hängen vom Stoff des Ringes ab und dürfen – bzw. + sein.) Liegt in einem gewissen Intervall, das (v ₁,v ₂) enthält, so hat jede einfache Konfiguration des Ringes mindestens zwei Symmetrieachsen. Die Fälle, wo nicht in diesen Streifen liegt, werden diskutiert. Schliesslich wird die Verzweigung der Lösungen mittels der Eigenwerte des um die komprimierte kreisförmige Konfiguration linearisierten Problems untersucht.     

Sur certaines algèbres associées à une mesure de Guelfand

Let be a Guelfand measure (cf. [A, B]) on a locally compact groupG DenoteL ₁ (G)=*L ₁(G)* the commutative Banach algebra associated to . We show thatL ₁ (G) is semi-simple and give a characterization of the closed ideals ofL ₁ (G). Using the -spherical Fourier transform, we characterize all linear bounded operators inL ₁ (G) which are invariants by -translations (i.e. such that ¹(( _x f) )=( _x ((f)) for eachxG andfL ₁ (G); where _x f(y)=f(xy); x,y G). WhenG is compact, we study the algebraL ₁ (G) and obtain results analogous to ones obtained for the commutative case: we show thatL ₁ (G) is regular, all closed sets of its Guelfand spectrum are sets of synthesis and establish theorems of harmonic synthesis for functions inL _p (G) (p=1,2 or +).

     

A Moment Matrix Approach to Multivariable Cubature

We develop an approach to multivariable cubature based on positivity, extension, and completion properties of moment matrices. We obtain a matrix-based lower bound on the size of a cubature rule of degree 2n + 1; for a planar measure , the bound is based on estimating where C:=C^# [ ] is a positive matrix naturally associated with the moments of . We use this estimate to construct various minimal or near-minimal cubature rules for planar measures. In the case when C = diag(c₁,...,c_n) (including the case when is planar measure on the unit disk), (C) is at least as large as the number of gaps c_k >c_k+1.     

From the Mahler Conjecture to Gauss Linking Integrals

We establish a version of the bottleneck conjecture, which in turn implies a partial solution to the Mahler conjecture on the product v(K)  = (Vol K)(Vol K°) of the volume of a symmetric convex body and its polar body K°. The Mahler conjecture asserts that the Mahler volume v(K) is minimized (non-uniquely) when K is an n-cube. The bottleneck conjecture (in its least general form) asserts that the volume of a certain domain is minimized when K is an ellipsoid. It implies the Mahler conjecture up to a factor of (π/4) ⁿ γ _n , where γ _n is a monotonic factor that begins at 4/π and converges to . This strengthens a result of Bourgain and Milman, who showed that there is a constant c such that the Mahler conjecture is true up to a factor of c ⁿ . The proof uses a version of the Gauss linking integral to obtain a constant lower bound on Vol K ^◇, with equality when K is an ellipsoid. It applies to a more general conjecture concerning the join of any two necks of the pseudospheres of an indefinite inner product space. Because the calculations are similar, we will also analyze traditional Gauss linking integrals in the sphere S ^n-1 and in hyperbolic space H ^n-1. Received: December 2006, Accepted: January 2007     

Characterization of {v μ+1 + 2v μ,v μ + 2v μ − 1;t,q}-min · hypers and its applications to error-correcting codes

Recently, Hamada [5] characterized all {v ₂ + 2v ₁,v ₁ + 2v ₀;t,q}-min · hypers for any integert 2 and any prime powerq 3 wherev _l = (q ^l – 1)/(q – 1) for any integerl 0. The purpose of this paper is to characterize all {v _{+ 1} + 2v ,v + 2v _{– 1};t,q}-min · hypers for any integerst, and any prime powerq such thatt 3, 2 t – 1 andq 5 and to characterize all (n, k, d; q)-codes meeting the Griesmer bound (1.1) for the casek 3, d = q ^k-1 – (2q ^-1 +q ) andq 5 using the results in Hamada [3, 4, 5].     

Metrics with finite sets of primitive extensions

In this paper, we give a complete characterization for the class of rational finite metrics with the property that the set () of primitive extensions of is finite. Here, for a metric on a setT, a positive extensionm of to a setV T is calledprimitive if none of the convex combinations of other extensions of toV is less than or equal tom. Our main theorem asserts that the following the properties are equivalent: (i) () is finite; (ii) Up to an integer factor, is a submetric of the path metric d ^H of a graphH with |(d ^H )=1; (iii) A certain bipartite graph associated with contains neither isometrick-cycles withk6 nor induced subgraphsK _3,3 ^– . We then show that () is finite if and only if the dimension of the tight span of is at most two. We also present other results, discuss applications to multicommodity flows, and raise open problems.This research was supported by grant 97-01-00115 from the Russian Foundation of Basic Research and a grant from the Sonderforschungsbereich 343, Bielefeld Universität, Bielefeld, Germany.