Order preserving nonexpansive operators inL 1

We study the limit behaviour ofT ^k f and of Cesaro averagesA _n f of this sequence, whenT is order preserving and nonexpansive inL ₁ ⁺ . IfT contracts also theL _∞-norm, the sequenceT ⁿ f converges in distribution, andA _n f converges weakly inL _p (1<p<∞), and also inL ₁ if the measure is finite. “Speed limit” operators are introduced to show that strong convergence ofA _n f need not hold. The concept of convergence in distribution is extended to infinite measure spaces. Much of this work was done during a visit of the first author at Ben Gurion University of the Negev in Beer Sheva, supported by the Deutsche Forschungsgemeinschaft.     

Far field velocity potential induced by a rapidly decaying vorticity distribution

We study the velocity field induced by a vorticity distribution decaying rapidly in the distancer from the origin. In the far field, the vector potential for the velocity field can be represented by a series A ⁽ⁿ⁾, withA ⁽ⁿ⁾ proportional tor ^–n–1, forn=1, 2, .... We show thatA ⁽ⁿ⁾ can be expressed as a linear combination ofM _n linearly independent vector functions. The numberM _n is equal to 3 forn=1 and 4n forn2 and the coefficient of a vector function is defined by a linear combination of nth moments of vorticity. We then show that only 2n+1 linear combinations of thoseM _n vector functions contribute to the far field velocity which is irrotational. The corresponding scalar potential ⁽ⁿ⁾ is then represented by a linear combination of 2n+1 spherical harmonics ofnth order whose coefficients are again linear combinations ofnth moments of vorticity.

---

Zusammenfassung Die vorliegende Arbeit beschreibt das Geschwindigkeitsfeld fernab einer Wirbelverteilung, welche mit dem Abstandr vom Ursprung eines geeigneten Bezugssystems hinreichend schnell abklingt. Die Geschwindigkeit besitzt ein Vektorpotential, dessen Fernfeldverhalten einer Reihenentwicklung A ⁽ⁿ⁾, genügt. Dabei istA ⁽ⁿ⁾ proportional zur ^–n–1 fürn=1, 2, .... Wir entwickeln eine explizite Darstellung vonA ⁽ⁿ⁾ als Linearkombination vonM _n linear unabhängigen Vektorfunktionen. Die auftretenden Koeffizienten sind ihrerseits Kombinationenn-ter Momente der Wirbelverteilung. Die ZahlM ₁ ist gleich 3 und es istM _n=4n fürn2, während die Gesamtzahl dernten Momente beträgt. Weiterhin zeigen wir, da\ nur 2n+1 dieser Vektorfunktionen auch zum drehungsfreien Fernfeld der Geschwindigkeitn-ter Ordnung beitragen können und identifizieren die zugehörigen Kombinationen von Wirbelmomenten. Dieselben Kombinationen liefern dann auch die Koeffizienten in einer Entwicklung desskalaren Fernfeldpotentials nach Kugelfunktionen.

     

Maximum and minimum sets for some geometric mean values

Ifv _r is ther-dimensional volume of ther-simplex formed byr+1 points taken at random from a compact setK in ⁿ , withrn, andh is a (strictly) increasing function, then the (unique) compact set that gives the minimum expected value ofh o v _r, is proved to be the ellipsoid (whenr=n) and the ball (whenr) almost everywhere. This result is established by using a single integral inequality for centrally symmetric quasiconvex functions integrated over compact rectangles.     

A Signed Generalization of the Bernoulli–Laplace Diffusion Model

We bound the rate of convergence to stationarity for a signed generalization of the Bernoulli–Laplace diffusion model; this signed generalization is a Markov chain on the homogeneous space ( ₂S _n )/(S _r ×S _n–r ). Specifically, for r not too far from n/2, we determine that, to first order in n, n log n steps are both necessary and sufficient for total variation distance to become small. Moreover, for r not too far from n/2, we show that our signed generalization also exhibits the cutoff phenomenon.     

A Remark on Characterizations of Projective Spaces among Singular Varieties

Let X be a projective variety of dimension n ≥ 2 with at worst log-terminal singularities and let be an ample vector bundle of rank r. By partially extending previous results due to Andreatta and Wiśniewski in the smooth case, we prove that if r = n then , while if r = n − 1 and X has only isolated singularities, then either or n = 2 and X is the quadric cone Q ₂. Received: April 20, 2006. Revised: April 5, 2007.     

Exact Bounds on the Sizes of Covering Codes

A code c is a covering code of X with radius r if every element of X is within Hamming distance r from at least one codeword from c. The minimum size of such a c is denoted by c _r(X). Answering a question of Hämäläinen et al. [10], we show further connections between Turán theory and constant weight covering codes. Our main tool is the theory of supersaturated hypergraphs. In particular, for n > n ₀(r) we give the exact minimum number of Hamming balls of radius r required to cover a Hamming ball of radius r + 2 in {0, 1}ⁿ. We prove that c _r(B _n(0, r + 2)) = _{1 i r + 1} ( ^{(n + i – 1) / (r + 1) 2}) + n / (r + 1) and that the centers of the covering balls B(x, r) can be obtained by taking all pairs in the parts of an (r + 1)-partition of the n-set and by taking the singletons in one of the parts.     

Locating–dominating codes in paths

Bertrand, Charon, Hudry and Lobstein studied, in their paper in 2004 [1], r-locating–dominating codes in paths P_n. They conjectured that if r≥2 is a fixed integer, then the smallest cardinality of an r-locating–dominating code in P_n, denoted by , satisfies for infinitely many values of n. We prove that this conjecture holds. In fact, we show a stronger result saying that for any r≥3 we have for all n≥n_r when n_r is large enough. In addition, we solve a conjecture on location–domination with segments of even length in the infinite path.     

The golden ratio and super central configurations of the n-body problem

In this paper, we consider the problem of central configurations of the n-body problem with the general homogeneous potential 1/rα. A configuration q=(q₁,q₂,…,qn) is called a super central configuration if there exists a positive mass vector m=(m₁,…,mn) such that q is a central configuration for m with mi attached to qi and q is also a central configuration for m^′, where m^′≠m and m^′ is a permutation of m. The main discovery in this paper is that super central configurations of the n-body problem have surprising connections with the golden ratio φ. Let r be the ratio of the collinear three-body problem with the ordered positions q₁, q₂, q₃ on a line. q is a super central configuration if and only if 1/r₁(α)<r<r₁(α) and r≠1, where r₁(α)>1 is a continuous function such that , the golden ratio. The existence and classification of super central configurations are established in the collinear three-body problem with general homogeneous potential 1/rα. Super central configurations play an important role in counting the number of central configurations for a given mass vector which may decrease the number of central configurations under geometric equivalence.     

<Emphasis Type="Italic">r</Emphasis>-Minimal submanifolds in space forms

Let be an n-dimensional compact, possibly with boundary, submanifold in an (n + p)-dimensional space form R ^n+p (c). Assume that r is even and , in this paper we introduce rth mean curvature function S _r and (r + 1)-th mean curvature vector field . We call M to be an r-minimal submanifold if on M, we note that the concept of 0-minimal submanifold is the concept of minimal submanifold. In this paper, we define a functional of , by calculation of the first variational formula of J _r we show that x is a critical point of J _r if and only if x is r-minimal. Besides, we give many examples of r-minimal submanifolds in space forms. We calculate the second variational formula of J _r and prove that there exists no compact without boundary stable r-minimal submanifold with in the unit sphere S ^n+p . When r = 0, noting S ₀ = 1, our result reduces to Simons’ result: there exists no compact without boundary stable minimal submanifold in the unit sphere S ^n+p .      

Integral Formulas for the r-Mean Curvature Linearized Operator of a Hypersurface

For a normal variation of a hypersurface M ⁿ in a space form Q _c ⁿ⁺¹ by a normal vector field fN, R. Reilly proved:

Pairs of non-intersecting random flats

Summary Let A _r,A _s,1r+sn-1, be independent isotropic uniform random r- resp. s-flats meeting a n-dimensional convex body K. It is shown that the probability that the points realizing the distance of A _rand A _sbelong to K is maximal if and only if K is a ball.     

Invariant equations defining coincident root loci

Let F(x) = xⁿ+₁ x^n-1+₂ x^n-2+ ··· +_n be a polynomial with complex coefficients, and suppose we are given a partition (₁,...,_r) of n. It is a classical problem to determine explicit algebraic conditions on the _i so that F may have roots with multiplicities ₁,...,_r. We give an invariant theoretic solution to this problem, to wit, we exhibit a set of covariants of F whose vanishing is a necessary and sufficient condition. The construction of such covariants is combinatorial, and involves associating a set of graphs on n vertices (called decisive graphs) to each .Received: 28 September 2003     

Iterates of Number Theoretic Functions with Periodic Rational Coefficients (Generalization of the 3x + 1 Problem)

Given p ∈ N = {1, 2, 3 ...}, as a generalization of the 3x + 1 problem [7], we study the behavior of the sequences s(m) = {m_n}_{n ≥ 0}, m ∈ Z (the set of the integers), defined by the iterative formula and a_r = t_r/p (t_r ∈ N), and b_r are chosen in such a way that m_n ∈ Z for every n. Our aim is to establish when these sequences are divergent, or convergent into a cycle, considering also a fixed point as a cycle. Moreover, the structure of possible cycles is investigated.     

Local solvability of a nonstationary leakage problem for an ideal incompressible fluid, 3

In this paper we prove the existence and uniqueness of solutions of the leakage problem for the Euler equations in bounded domain Ω C R³ with corners π/n, n = 2, 3… We consider the case where the tangent components of the vorticity vector are given on the part S₁ of the boundary where the fluid enters the domain. We prove the existence of an unique solution in the Sobolev space W_p^l(Ω), for arbitrary natural l and p > 1. The proof is divided on three parts: (1) the existence of solutions of the elliptic problem in the domain with corners where v – velocity vector, ω – vorticity vector and n is an unit outward vector normal to the boundary, (2) the existence of solutions of the following evolution problem for given velocity vector (3) the method of successive approximations, using solvability of problems (1) and (2).     

The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent

LetH be a fixed graph of chromatic numberr. It is shown that the number of graphs onn vertices and not containingH as a subgraph is . Leth _r (n) denote the maximum number of edges in anr-uniform hypergraph onn vertices and in which the union of any three edges has size greater than 3r – 3. It is shown thath _r (n) =o(n ²) although for every fixedc < 2 one has lim _n h _r (n)/n ^c = .     

On the optimum capacity of capacity expansion problems

In this paper we consider problems of the following type: Let E = { e ₁, e ₂,..., e _n } be a finite set and be a family of subsets of E. For each element e _i in E, c _i is a given capacity and _i is the cost of increasing capacity c _i by one unit. It is assumed that we can expand the capacity of each element in E so that the capacity of family can be expanded to a level r. For each r, let f (r) be the efficient function with respect to the capacity r of family , and be the cost function for expanding the capacity of family to r. The goal is to find the optimum capacity value r ^* and the corresponding expansion strategy so that the pure efficency function is the largest. Firstly, we show that this problem can be solved efficiently by figuring out a series of bottleneck capacity expansion problem defined by paper (Yang and Chen, Acta Math Sci 22:207–212, 2002) if f (r) is a piecewise linear function. Then we consider two variations and prove that these problems can be solved in polynomial time under some conditions. Finally the optimum capacity for maximum flow expansion problem is discussed. We tackle it by constructing an auxiliary network and transforming the problem into a maximum cost circulation problem on the auxiliary network.     

Relative stability of trimmed sums

Summary This paper gives extensions of Mori's strong law for ^(r) S _n =S _n –X _n ^(1)} ...–X _n ^(r) , where S _n =X₁+X₂+...+X _n ,X _i are iidrv's and (X _i ⁿ ()) is (X _i ) arranged in decreasing order of absolute magnitude. The methods differ from Mori's. Continuity of the distribution of the X _i is assumed throughout. Necessary and sufficient conditions for relative stability (^(r) S _n /B _n ±1 a.s. for some B _n ), including a generalised condition of Spitzer's and a dominated ergodic theorem, are proved. A one-sided version of the relative stability results is also given. A theorem of Kesten's is generalised to show that if (^(r) S _n –A _n )/B _n is bounded almost surely for constants A _n ,B _n + then for some _n . A corollary to this is that if ¦ ^(r) S _n ¦/B _n is bounded away from 0 and + a.s. then ^(r) S _n is relatively stable. This generalises a result of Chow and Robbins, apart from the continuity assumption.     

On the maximal number of certain subgraphs inK r -free graphs

Given two graphsH andG, letH(G) denote the number of subgraphs ofG isomorphic toH. We prove that ifH is a bipartite graph with a one-factor, then for every triangle-free graphG withn verticesH(G) H(T ₂(n)), whereT ₂(n) denotes the complete bipartite graph ofn vertices whose colour classes are as equal as possible. We also prove that ifK is a completet-partite graph ofm vertices,r > t, n max(m, r – 1), then there exists a complete (r – 1)-partite graphG* withn vertices such thatK(G) K(G*) holds for everyK _r -free graphG withn vertices. In particular, in the class of allK _r -free graphs withn vertices the complete balanced (r – 1)-partite graphT _r–1(n) has the largest number of subgraphs isomorphic toK _t (t < r),C ₄,K _2,3. These generalize some theorems of Turán, Erdös and Sauer.Dedicated to Paul Turán on his 80th Birthday     

An extreme value theory for long head runs

Summary For an infinite sequence of independent coin tosses withP(Heads)=p(0,1), the longest run of consecutive heads in the firstn tosses is a natural object of study. We show that the probabilistic behavior of the length of the longest pure head run is closely approximated by that of the greatest integer function of the maximum ofn(1-p) i.i.d. exponential random variables. These results are extended to the case of the longest head run interrupted byk tails. The mean length of this run is shown to be log(n)+klog(n)+(k+1)log(1–p)–log(k!)+k+/–1/2+ r₁(n)+ o(1) where log=log_1/p , =0.577 ... is the Euler-Mascheroni constant, =ln(1/p), andr ₁(n) is small. The variance is ₂/6²+1/12 +r ₂(n)+ o(1), wherer ₂(n) is again small. Upper and lower class results for these run lengths are also obtained and extensions discussed.This work was supported by a grant from the System Development Foundation