A note on the vorticity downstream of a curved shock

Zusammenfassung Es wird ein expliziter Ausdruck für die Wirbelst?rkeω hinter einem gekrümmten Verdichtungsstoss für den Fall der zweidimensionalen Str?mung eines idealen Gases in Termen der Geschwindigkeit der ungest?rten Str?mungV _∞, ihrer Mach-ZahlM _∞, des Stosswinkelsσ und des Krümmungsradiusr _s des Verdichtungsstosses im betrachteten Punkt aufgestellt. Die dimensionslose Wirbelst?rkeω r _s /V _∞ h?ngt nur vonM _∞ undσ ab. Bei festgehaltenemM _∞ treten bei einem Stosswinkel von 90° (senkrechter Verdichtungsstoss) und auch dann, wenn der Stosswinkel dem Mach-Winkel der ungest?rten Str?mung gleich ist (Verdichtungsstoss der Intensit?t Null), keine Wirbel auf. Für einen dazwischenliegenden Winkel, der von 68° fürM _∞=1,5 bis zu 36° fürM _∞=10 variiert, wird die Wirbelst?rke ein Maximum.      

Nonexistence results of solutions of semilinear differential inequalities on the Heisenberg group

Denoting Δ_? the Laplacian operator on the (2N+1)-dimensional Heisenberg group ? ^N , we prove some nonexistence results for solutions of inequalities of the three types

Conditions for the existence of a graph with given diameter, connectivity, and ball diversity vector

Given some arbitrary integers d ≥ 2, ? ? 1 and an integer vector $ \bar \tau Given some arbitrary integers d ≥ 2, ϰ ⩾ 1 and an integer vector = (τ ₀, τ ₁, …, τ _d ) with τ ₀ ⩾ τ ₁ ⩾ … ⩾ τ _d = 1 and τ _{d − 1} ⩾ d ²ϰ + 3, the existence is proved of a graph of diameter d and connectivity ϰ whose ball diversity vector is . Moreover, the nonexistence is proved of a graph of diameter d with connectivity ϰ and ball diversity vector (τ ₀, τ ₁, …, τ _d ), where τ ₀ < (d − 1)ϰ + 2. Original Russian Text ? K.L. Rychkov, 2007, published in Diskretnyi Analiz i Issledovanie Operatsii, Ser. 1, 2007, Vol. 14, No. 4, pp. 43–56.     

New bounds on the Lieb-Thirring constants

Improved estimates on the constants L _γ,d , for 1/2<γ<3/2, d∈N, in the inequalities for the eigenvalue moments of Schr?dinger operators are established. Oblatum 18-VI-1999 & 13-I-2000?Published online: 29 March 2000     

Capture time of Brownian pursuits

Let B ₀,B ₁, ⋯ ,B _n be independent standard Brownian motions, starting at 0. We investigate the tail of the capture time

On the Rees algebra of certain codimension two perfect ideals

 Suppose ? is a set of arbitrary number of smooth points in ℙ² its defining ideal. In this paper, we study the Rees algebras of the ideals generated by I _t , t ≥α. When the points of ? are general, we give a set of defining equations for the Rees algebra . When the points of ? are arbitrary, we show that for all t≫ 0, the Rees algebra is Cohen-Macaulay and its defining ideal is generated by quadratics. A cohomological characterization for arithmetic Cohen-Macaulayness of subvarieties of a product space is also given. Received 4 April 2001     

Toric varieties and modular forms

Let N⊂ℝ^r be a lattice, and let deg:N→ℂ be a piecewise-linear function that is linear on the cones of a complete rational polyhedral fan. Under certain conditions on deg, the data (N,deg) determines a function f:ℌ→ℂ that is a holomorphic modular form of weight r for the congruence subgroup Γ₁(l). Moreover, by considering all possible pairs (N ,deg), we obtain a natural subring ? (l) of modular forms with respect to Γ₁ (l). We construct an explicit set of generators for ? (l), and show that ? (l) is stable under the action of the Hecke operators. Finally, we relate ? (l) to the Hirzebruch elliptic genera that are modular with respect to Γ₁ (l). Oblatum 22-IX-1999 & 18-X-2000?Published online: 5 March 2001     

On the Distribution of the Degrees of Prime Element Factors in Additive Arithmetical Semigroups

 Results of Arnold Knopfmacher about the distribution of the degrees of irreducible factors in the canonical decomposition of monic polynomials in ? _q [X] are generalized to additive arithmetical semigroups (G,∂) satisfying a weak condition called axiom ? (Received 25 August 1998; in revised form 7 June 1999)     

Spectral structure of Anderson type Hamiltonians

We study self adjoint operators of the form?H _ω = H ₀ + ∑λ_ω(n) <δ _n ,·>δ _n ,?where the δ _n ’s are a family of orthonormal vectors and the λ_ω(n)’s are independently distributed random variables with absolutely continuous probability distributions. We prove a general structural theorem saying that for each pair (n,m), if the cyclic subspaces corresponding to the vectors δ _n and δ _m are not completely orthogonal, then the restrictions of H _ω to these subspaces are unitarily equivalent (with probability one). This has some consequences for the spectral theory of such operators. In particular, we show that “well behaved” absolutely continuous spectrum of Anderson type Hamiltonians must be pure, and use this to prove the purity of absolutely continuous spectrum in some concrete cases. Oblatum 27-V-1999 & 6-I-2000?Published online: 8 May 2000     

Everywhere H?lder continuity of the spatial gradient of weak solutions to nonlinear parabolic systems

We consider weak solutions to the parabolic system ∂u ⁱ∂t−D _α A _i ^α (∇u)=B _i(∇u) in (i=1,...,) (Q=Ω×(0,T), R ⁿ a domain), where the functionsB _i may have a quadratic growth. Under the assumptionsn≤2 and ∇u ɛL _loc ^4+δ (Q; R ^nN ) (δ>0) we prove that ∇u is locally H?lder continuous inQ.     

On the generalised jump of a function by its Fourier coefficients

Summary Author studies the summability (C,1+α+ρ) of the sequence nB_n(x) under weaker conditions than those ofMinakishisundaram [3] and thus generalises his theorems. on the ? Jump of a function ? and by applying a tauberian theorem obtaing a criteria for the (C, α+ρ) summability of the conjugate series.     

Uniqueness of Nelsons diffusions

Let ℒ≔Δ/2+(∇φ/φ) ·∇ be a generalized Schr?dinger operator or generator of Nelsons diffusion, defined on C ^∞ ₀(D) where φ is a continuous and strictly positive function on an open domain D⊂ℝ ^d such that ∇φ∈L _loc ²(D). Some results are given about the two questions below: (i) Whether does ℒ generate a unique semigroup in L ¹(D, φ² dx)? (ii) Whether the semigroup determined by ℒ is strong Feller? Received: 21 October 1997 / Revised version: 3 September 1998     

Some remarks on a question of D. H. Fremlin regarding ε-density

We show the relative consistency of ℵ₁ satisfying a combinatorial property considered by David Fremlin (in the question DU from his list) in certain choiceless inner models. This is demonstrated by first proving the property is true for Ramsey cardinals. In contrast, we show that in ZFC, no cardinal of uncountable cofinality can satisfy a similar, stronger property. The questions considered by D. H. Fremlin are if families of finite subsets of ω₁ satisfying a certain density condition necessarily contain all finite subsets of an infinite subset of ω₁, and specifically if this and a stronger property hold under MA + ?CH. Towards this we show that if MA + ?CH holds, then for every family ? of ℵ₁ many infinite subsets of ω₁, one can find a family ? of finite subsets of ω₁ which is dense in Fremlins sense, and does not contain all finite subsets of any set in ?. We then pose some open problems related to the question. Received: 2 June 1999 / Revised version: 2 February 2000 / Published online: 18 July 2001     

On Abelian Branched Coverings of the Sphere

 We obtain an enumeration formula for the number of weak equivalence classes of the branched (?×ℬ)-covering of the sphere with m-branch points, when ? and ℬ are finite abelian groups with (|?|,|ℬ|)=1. From this, we can deduce an explicit formula for enumerating the weak equivalence classes of pseudofree spherical (ℤ _p ×ℤ _q )-actions on a given surface, when p and q are distinct primes. Received: August 10, 1999 Final version received: June 19, 2000     

The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group

Let A be an Artin group with standard generating set {σ _s :s∈S}. Tits conjectured that the only relations in A amongst the squares of the generators are consequences of the obvious ones, namely that σ _s ² and σ _t ² commute whenever σ _s and σ _t commute, for s,t∈S. In this paper we prove Tits’ conjecture for all Artin groups. In fact, given a number m _s ≥2 for each s∈S, we show that the elements {T _s =σ _s ^ms :s∈S} generate a subgroup that has a finite presentation in which the only defining relations are that T _s and T _t commute if σ _s and σ _t commute. Oblatum 21-III-2000 & 1-XII-2000?Published online: 5 March 2001     

Some properties of Riesz potential associated with SchrSdinger operator

Let L = −Δ + V be the Schr?dinger operator on ℝ ^d , d ≥ 3, where Δ is the Laplacian on ℝ ^d and V ≢ 0 is a nonnegative function satisfying the reverse H?lder inequality. In this article, the author investigates some properties of the Riesz potential I _α ^L associated with L on the Campanato-type spaces Λ _L ^β and the Hardy-type spaces H _L ^p .     

Möbius Isoparametric Hypersurfaces in S n+1 with Two Distinct Principal Curvatures

A hypersurface x : M → S ⁿ⁺¹ without umbilic point is called a Möbius isoparametric hypersurface if its Möbius form Φ = ?ρ ^?2∑ _i (e _i (H) + ∑ _j (h _ij ?Hδ _ij )e _j (log ρ))θ _i vanishes and its Möbius shape operator $ {\Bbb {S}}A hypersurface x : M → S ^{n +1} without umbilic point is called a M?bius isoparametric hypersurface if its M?bius form Φ = −ρ⁻²∑ _i (e _i (H) + ∑ _j (h _ij −Hδ _ij )e _j (log ρ))θ _i vanishes and its M?bius shape operator ? = ρ⁻¹(S−Hid) has constant eigenvalues. Here {e _i } is a local orthonormal basis for I = dx·dx with dual basis {θ _i }, II = ∑ _ij h _ij θ _i ⊗θ _i is the second fundamental form, and S is the shape operator of x. It is clear that any conformal image of a (Euclidean) isoparametric hypersurface in S ^{n +1} is a M?bius isoparametric hypersurface, but the converse is not true. In this paper we classify all M?bius isoparametric hypersurfaces in S ^{n +1} with two distinct principal curvatures up to M?bius transformations. By using a theorem of Thorbergsson [1] we also show that the number of distinct principal curvatures of a compact M?bius isoparametric hypersurface embedded in S ^{n +1} can take only the values 2, 3, 4, 6. Received September 7, 2001, Accepted January 30, 2002     

On the equivalence of measures on loop space

Let K be a simply-connected compact Lie Group equipped with an Ad _K -invariant inner product on the Lie Algebra ?, of K. Given this data, there is a well known left invariant “H ¹-Riemannian structure” on L(K) (the infinite dimensional group of continuous based loops in K), as well as a heat kernel ν_T(k ₀, ·) associated with the Laplace-Beltrami operator on L(K). Here T > 0, k ₀∈L(K), and ν _T (k ₀, ·) is a certain probability measure on L(K). In this paper we show that ν₁(e,·) is equivalent to Pinned Wiener Measure on K on ? _s0 ≡<x _t : t∈ [0, s ₀]> (the σ-algebra generated by truncated loops up to “time”s ₀). Recevied: 9 September 1999 / Revised version: 13 March 2000 / Published online: 22 November 2000     

Facets with fixed defect of the stable set polytope

The stable set polytope of a graph is the convex hull of the 0-1 vectors corresponding to stable sets of vertices. To any nontrivial facet ∑ _v∈V a(v)x _v ≤b of this polytope we associate an integer δ, called the defect of the facet, by δ=∑ _v∈V a(v)-2b. We show that for any fixed δ there is a finite collection of graphs (called “basis”) such that any graph with a facet of defect δ contains a subgraph which can be obtained from one of the graphs in the basis by a simple subdivision operation. Received: September 28, 1998 / Accepted: February 24, 2000?Published online April 20, 2000     

Equilibrium fluctuations of asymmetric simple exclusion processes in dimension d≥3

We consider an asymmetric exclusion process in dimension d≥ 3 under diffusive rescaling starting from the Bernoulli product measure with density 0 < α < 1. We prove that the density fluctuation field Y ^N _t converges to a generalized Ornstein–Uhlenbeck process, which is formally the solution of the stochastic differential equatin dY _t = ?Y _t dt + dB ^∇ _t , where ? is a second order differential operator and B ^∇ _t is a mean zero Gaussian field with known covariances. Received: 31 May 1999 / Revised version: 15 June 2000 / Published online: 24 January 2001