Steady vortex rings with swirl in an ideal fluid: asymptotics for some solutions in exterior domains

In this paper, the axisymmetric flow in an ideal fluid outside the infinite cylinder (rd) where (r, , z) denotes the cylindrical co-ordinates in ³ is considered. The motion is with swirl (i.e. the -component of the velocity of the flow is non constant). The (non-dimensional) equation governing the phenomenon is (Pd) displayed below. It is known from e.g. [9] that for the problem without swirl (f _q = 0 in (f)) in the whole space, as the flux constant k tends to 1) dist(0z, A) = O(k ^1/2); diam A = O(exp(–c ₀ k ^3/2));2) k^1/2)_k converges to a vortex cylinder U _m (see (1.2)).We show that for the problem with swirl, as k , 1) holds; if m q + 2 then 2) holds and if m > q + 2 it holds with U _q+2 instead of U _m. Moreover, these results are independent of f ₀, f _q and d > 0.     

Empirical and Poisson processes on classes of sets or functions too large for central limit theorems

Summary Let P be the uniform probability law on the unit cube I ^d in d dimensions, and P _n the corresponding empirical measure. For various classes of sets AI ^d , upper and lower bounds are found for the probable size of sup {¦P _n –P) (A)¦ A }. If is the collection of lower layers in I ², or of convex sets in I ³, an asymptotic lower bound is ((log n)/n) ^1/2(log log n)^––1/2 for any >0. Thus the law of the iterated logarithm fails for these classes.If >0, is the greatest integer <, and 0 d f(x₁,...,x _d-1)} where f has all its partial derivatives of orders bounded by K and those of order satisfy a uniform Hölder condition ¦D ^p (f(x)–f(y))¦K¦x –y¦ ^–. For 0<–/(d–1+) for a constant = (d,)>0. When = d-1 the same lower bound is obtained as for the lower layers in I ² or convex sets in I ³. For 0 – 1 there is also an upper bound equal to a power of log n times the lower bound, so the powers of n are sharp.This research was partially supported by National Science Foundation Grant MCS-79-04474     

Decomposing symmetric exchanges in matroid bases

LetB,B be bases of a matroid, withX B, X B. SetsX,X are asymmetric exchange if(B – X) X and(B – X) X are bases. SetsX,X are astrong serial B-exchange if there is a bijectionf: X X, where for any ordering of the elements ofX, sayx _i ,i = 1, , m, bases are formed by the sets B₀ = B, B_i = (B_i–1 – x_i) f(x _i), fori = 1, , m. Any symmetric exchangeX,X can be decomposed by partitioning X = _i=1 ^m Y_i, X = _i=1 ^m Y_i, X, where (1) bases are formed by the setsB ₀ =B, B _i = (B _i–1 –Y _i ) Y _i ; (2) setsY _i ,Y _i are a strong serialB _i–1 -exchange; (3) properties analogous to (1) and (2) hold for baseB and setsY _i ,Y _i .     

The Milnor–Totaro Theorem for Space Polygons

If denotes the curvature and the torsion of a closed, generic, and oriented polygonal space curve X in , then we show that _X (² + ²) ds = _X ds + _X | | ds > 4 if is positive. We also show that _X (² + ²) ds 2n if no four consecutive vertices lie in a plane and X has linking number n with a straight line. These extend theorems of Milnor and Totaro.     

Skew-symmetric association schemes with two classes and strongly regular graphs of type L 2n−1(4n−1)

A construction of a pair of strongly regular graphs ⁿ and ⁿ of type L _2n–1(4n–1) from a pair of skew-symmetric association schemes W, W of order 4n–1 is presented. Examples of graphs with the same parameters as ⁿ and ⁿ, i.e., of type L _2n–1(4n–1), were known only if 4n–1=p ³, where p is a prime. The first new graph appearing in the series has parameters (v, k, )=(225, 98, 45). A 4-vertex condition for relations of a skew-symmetric association scheme (very similar to one for the strongly regular graphs) is introduced and is proved to hold in any case. This has allowed us to check the 4-vertex condition for ⁿ and ⁿ, thus to prove that ⁿ and ⁿ are not rank three graphs if n>2.     

Complete hyperbolicity of Hartogs domain

Let X be a complex space and an upper semicontinuous function on X. Consider the Hartogs domain (X) given by (X)={(z, w)X×C: |w| < e ^–(z) }. In this article, some necessary and sufficient conditions on the complete hyperbolicity of (X) are established. Mathematics Subject Classification (2000):32A10, 32C10, 32H20, 32A17     

On the central limit theorem for geometrically ergodic Markov chains

Let X₀,X₁,... be a geometrically ergodic Markov chain with state space and stationary distribution . It is known that if h: R satisfies (|h|²⁺)< for some >0, then the normalized sums of the X_is obey a central limit theorem. Here we show, by means of a counterexample, that the condition (|h|²⁺)< cannot be weakened to only assuming a finite second moment, i.e., (h²)<.Reasearch supported by the Swedish Research Council.     

Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory

Let {X _k , 1 k n} be n independent and real-valued random variables with common subexponential distribution function, and let {k, 1 k n} be other n random variables independent of {X _k , 1 k n} and satisfying a _k b for some 0 < a b < for all 1 k n. This paper proves that the asymptotic relations P (max_{1 m n k=1} ^m _k X _k > x) P (sum _k=1 ⁿ _k X _k > x) sum _k=1 ⁿ P ( _k X _k > x) hold as x . In doing so, no any assumption is made on the dependence structure of the sequence { _k , 1 k n}. An application to ruin theory is proposed.     

ApproximatingL 2-invariants by their finite-dimensional analogues

LetX be a finite connectedCW-complex. Suppose that its fundamental group is residually finite, i.e. there is a nested sequence ... _m + 1 _m ... of in normal subgroups of finite index whose intersection is trivial. Then we show that thep-thL ²-Betti number ofX is the limit of the sequenceb _p(X_m)/[: _m ] whereb _p(X_m) is the (ordinary)p-th Betti number of the finite covering ofX associated with _m .     

Separation of two (possibly unbounded) components of the spectrum of a linear operator

LetX be a complex Banach space andA: D(A)X a densely defined closed linear operator whose resolvent set contains the real line and for which (–A)^–1 is bounded onR. We give a necessary and sufficient condition, in terms of the complex powers ofA and –A, for the existence of a decompositionX=X ₊X _–, whereX _± are closed subspaces, invariant forA, the spectra of the reduced operatorsA _± are {(A);Im>0} and {(A);Im<0} respectively, and (–A _±)^–1 is bounded forIm0.Finally we give an example of an operator in anL ^p-type space for which the decomposition exists if 1<p<+ and does not exist ifp=1.     

K-theory of reduced C*-algebras and rapidly decreasing functions on groups

We associate to any length function L on a group a space of rapidly decreasing functions on (in the l ² sense), denoted by H _L (). When H _L () is contained in the reduced C*-algebra C _r ^* () of (), then it is a dense *-subalgebra of C _r ^* () and we prove a theorem of A. Connes which asserts that under this hypothesis H _L () has the same K-theory as C _r ^* (). We introduce another space of rapidly decreasing functions on (in the l ¹ sense), denoted by H _L ^1, (), which is always a dense *-subalgebra of the Banach algebra l ¹(), and we show that H _L ^1, () has the same K-theory as l ¹().     

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     

k-regular factors and semi-k-regular factors in bipartite graphs

LetG be a graph, andk1 an integer. LetU be a subset ofV(G), and letF be a spanning subgraph ofG such that deg _F (x)=k for allx V(G)–U. If deg _F (x)k for allxU, thenF is called an upper semi-k-regular factor with defect setU, and if deg _F (x)k for allxU, thenF is called a lower semi-k-regular factor with defect setU. Now letG=(X, Y;E(G)) be a bipartite graph with bipartition (X,Y) such that X=Yk+2. We prove the following two results.(1) Suppose that for each subsetU ₁X such that U ₁=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.(2) Suppose that for each subsetU ₁X such that U ₁=X–1/k+1,G has a lower semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=X–1/k+1,G has a lower semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.     

Finite Representability of ℓp in Quotients of Banach Spaces

A typical result of the paper states that if X is a Banach space with a basis and for some 1pq, the spaces _p and _q are finitely block representable in every block subspace of X, then every block subspace of X admits a block quotient Z such that for every r[p,q], the space _r is finitely block representable in Z. Results of a similar nature are also established for ^N _p-block-sequences and asymptotic spaces.     

The Szlenk index and local ℓ<Subscript>1</Subscript>-indices

We introduce two new local ₁-indices of the same type as the Bourgain ₁-index; the ⁺₁-index and the ⁺₁-weakly null index. We show that the ⁺₁-weakly null index of a Banach space X is the same as the Szlenk index of X, provided X does not contain ₁. The ⁺₁-weakly null index has the same form as the Bourgain ₁-index: if it is countable it must take values for some <₁. The different ₁-indices are closely related and so knowing the Szlenk index of a Banach space helps us calculate its ₁-index, via the ⁺₁-weakly null index. We show that I(C(^{))=^1++1}.     

Path properties of an infinite system of Wiener processes

Let be a Poisson process on ^d of intensity and letW ₁(t),W ₂ (t),..., be a sequence of independent Wiener processes. LetW _i (t)=X _i +W _i (t) whereX ₁,X ₂,..., are the points of . Consider the processess(t)=#{i:X _i (t)1}. These and related processes are studied.     

A note on characterizations of Abelian varieties by topological invariants

In this note, for any given n3 and 2mn (when m=n, we assume n divides 3 and n6), we construct examples of smooth projective varieties X of dimension n with p_g(X)=1, ₁(X)²ⁿ and the Kodaira dimension (X)=m.Mathematics Subject Classification (2000):14H45, 14H99     

Support functions of convex compacta

The properties of the space (X_x) of all sublinear functionals, defined on a space X' (topologically adjoint to a Hausdorff locally convex barrelled space X) and continuous in the Arens topology × (X, X), equipped with topology of uniform convergence on bounded subsets of X are studied. It is shown that completeness and separability of a space X are hereditary for (X_x). Criteria for the compactness of subsets of (X_x) and conditions for the metrizability of compacta in (X_x) are given. The topological isomorphism between (X_x) and the space of all nonempty convex compacta in X with the Vietoris topology is established. The results obtained here are applied for the study of the properties of multiple-valued integrals.Translated from Matematicheskie Zametki, Vol. 22, No. 2, pp. 203–213, August, 1977.The author thanks S. S. Kutateladze for useful discussions regarding this article.     

A Remark on Bernstein,Prokhorov, Bennett,Hoeffding, and Talagrand Inequalities

Let X ₁,..., X_n be independent random variables such that {X_j 1}=1 and E X _j=0 for all j. We prove an upper bound for the tail probabilities of the sum M _n=X₁+...+ X_n. Namely, we prove the inequality {M _nx} 3.7 {S_n x}, where S _n=₁+...+ _n is a sum of centered independent identically distributed Bernoulli random variables such that E S _n ² =ME M _n ² and {_k=1}=E S _n ² /(n+E S _n ² ) for all k (we call a random variable Bernoulli if it assumes at most two values). The inequality holds for x at which the survival function x{S _nx} has a jump down. For remaining x, the inequality still holds provided that we interpolate the function between the adjacent jump points linearly or log-linearly. If necessary, in order to estimate {S _nx} one can use special bounds for binomial probabilities. Up to the factor at most 2.375, the inequality is final. The inequality improves the classical Bernstein, Prokhorov, Bennett, Hoeffding, Talagrand, and other bounds.     

Sur une conjecture de Mukai

Résumé Généralisant une question de Mukai, nous conjecturons quune variété de Fano X de nombre de Picard _X et de pseudo-indice _X vérifie $ _X ( _X - 1) dim(X). Nous démontrons cette conjecture dans plusieurs situations: X est une variété de Fano de dimension 4, X est une variété de Fano torique de dimension 7 ou X est une variété de Fano torique de dimension arbitraire avec $ _X dim(X) / 3 + 1. Enfin, nous présentons une approche nouvelle pour le cas général.