The Flooding Time in Random Graphs

Based on our analysis of the hopcount of the shortest path between two arbitrary nodes in the class G _p (N) of random graphs, the corresponding flooding time is investigated. The flooding time T _N (p) is the minimum time needed to reach all other nodes from one node. We show that, after scaling, the flooding time T _N (p) converges in distribution to the two-fold convolution ^(2*) of the Gumbel distribution function (z)=exp (–e ^–z ), when the link density p _N satisfies Np _N /(log N)³ if N .     

<Emphasis Type="Italic">CR</Emphasis>-Warped Products in Complex Projective Spaces with Compact Holomorphic Factor

A submanifold of a Kaehler manifold is called a CR-warped product if it is the warped product N_T ×_fN of a complex submanifold N_T and a totally real submanifold N. There exist many CR-warped products N_T ×_fN in CP^h+p, h = dim_CN_T and p = dim_RN (see [5, 6]). In contrast, we prove in this article that the situation is quite different if the holomorphic factor N_T is compact. For such CR-wraped products in CP^m (4), we prove the following: (1) The complex dimension m of the ambient space is at least h + p + hp. (2) If m = h + p + hp, then N_T is CP^h(4). We also obtain two geometric inequalities for CR-warped products in CP^m with compact N_T.     

THE NUMBER AND SIZE OF ORBITS OF SOME PERMUTATION GROUPS

Abstract

It is shown that Aut ?, the group of homeomorphisms of the rational numbers with the usual topology, has 2 ^N_o orbits on the power set P(?). We call S ? ? a moiety if S and its complement in ? are infinite. It is shown that the orbit of any moiety S under Aut ? has cardinality 2^N_o while the orbit of S under Aut(?, ≤), the group of order preserving automorphisms of ?, has cardinality N_o if and only if S is a finite union of disjoint rational intervals with rational endpoints.     

Periodic part in some Shunkov groups

A group G is saturated with groups in a set X if every finite subgroup of G is embeddable in G into a subgroup L isomorphic to some group in X. We show that a Shunkov group has a periodic part if the saturating set for it coincides with one of the following: {L₂(q)}, {Sz(q)}, {Re(q)}, or {U₃(2ⁿ)}. Translated fromAlgebra i Logika, Vol. 38, No. 1, pp. 96–125, January–February, 1999.     

Spherical 7-Designs in 2 n -Dimensional Euclidean Space

We consider a finite subgroup _n of the group O(N) of orthogonal matrices, where N = 2 ⁿ , n = 1, 2 .... This group was defined in [7]. We use it in this paper to construct spherical designs in 2 ⁿ -dimensional Euclidean space R ^N . We prove that representations of the group _n on spaces of harmonic polynomials of degrees 1, 2 and 3 are irreducible. This and the earlier results [1–3] imply that the orbit _n,2 x ^t of any initial point x on the sphere S _{N – 1} is a 7-design in the Euclidean space of dimension 2 ⁿ .     

Some properties related to torsion of a two-dimensional surface

We consider a two-dimensional surface F² of class C³ in Euclidean spaceE ⁿ ,n4. We introduce the concepts of the normal curvature vectork _N (x; t) and the Euler curvature vectorv _N (x; t) of the normalp _N (x;t) and the Euler _N (x;t) torsion of surface F² at the point x in the tangent directiont. We show that these magnitudes are characteristics of surface F² at the point x in the directiont, and derive formulas for their calculation. We establish necessary and sufficient conditions under which the directions of vectorsk _N (x; t) andp _N (x; t) are parallel displaced in the normal fiber bundle on F² from the pointx ¯F ² in the directiont. In particular, the following assertion holds: the direction of the Euler curvature vectorp _N (x; t) is parallel displaced in the normal fiber bundle on F² along any geodesic on F² if and only ifv _N (x; t) 0, x F ². t.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 45–52, 1990.     

Clean triangulations

A polyhedron on a surface is called a clean triangulation if each face is a triangle and each triangle is a face. LetS _p (resp.N _p ) be the closed orientable (resp. nonorlentable) surface of genusp. If (S) is the smallest possible number of triangles in a clean triangulation ofS, the results are: (N ₁)=20, (S ₁)=24, lim(S _p )p ^–1=4, lim(N _p )p ^–1=2 forp.     

Quasirecognizability by the Set of Element Orders for Groups 3 D 4(q), for q Even

It is proved that if G is a finite group with an element order set as in the simple group ³D₄(q), where q is even, then the commutant of G/F(G) is isomorphic to ³D₄(q) and the factor group G/G′ is a cyclic {2, 3}-group. __________ Translated from Algebra i Logika, Vol. 45, No. 1, pp. 3–19, January–February, 2006.     

Maximal Orders of Abelian Subgroups in Finite Chevalley Groups

In the present paper, for any finite group G of Lie type (except for ² F ₄(q)), the order a(G) of its large Abelian subgroup is either found or estimated from above and from below (the latter is done for the groups F ₄ (q), E ₆ (q), E ₇ (q), E ₈ (q), and ² E ₆(q ²)). In the groups for which the number a(G) has been found exactly, any large Abelian subgroup coincides with a large unipotent or a large semisimple Abelian subgroup. For the groups F ₄ (q), E ₆ (q), E ₇ (q), E ₈ (q), and ² E ₆(q ²)), it is shown that if an Abelian subgroup contains a noncentral semisimple element, then its order is less than the order of an Abelian unipotent group. Hence in these groups the large Abelian subgroups are unipotent, and in order to find the value of a(G) for them, it is necessary to find the orders of the large unipotent Abelian subgroups. Thus it is proved that in a finite group of Lie type (except for ² F ₄(q))) any large Abelian subgroup is either a large unipotent or a large semisimple Abelian subgroup.     

Moderate Deviations for the overlap parameter in the Hopfield model

We derive moderate deviation principles for the overlap parameter in the Hopfield model of spin glasses and neural networks. If the inverse temperature is different from the critical inverse temperature _c=1 and the number of patterns M(N) satisfies M(N)/N 0, the overlap parameter multiplied by N, 1/2 < < 1, obeys a moderate deviation principle with speed N^1–2 and a quadratic rate function (i.e. the Gaussian limit for = 1/2 remains visible on the moderate deviation scale). At the critical temperature we need to multiply the overlap parameter by N, 1/4 < < 1. If then M(N) satisfies (M(N)⁶ log N M(N)²N⁴ log N)/N 0, the rescaled overlap parameter obeys a moderate deviation principle with speed N^1–4 and a rate function that is basically a fourth power. The random term occurring in the Central Limit theorem for the overlap at _c = 1 is no longer present on a moderate deviation scale. If the scaling is even closer to N^1/4, e.g. if we multiply the overlap parameter by N^1/4 log log N the moderate deviation principle breaks down. The case of variable temperature converging to one is also considered. If _N converges to _c fast enough, i.e. faster than the non-Gaussian rate function persists, whereas for _N converging to one slower than the moderate deviations principle is given by the Gaussian rate. At the borderline the moderate deviation rate function is the one at criticality plus an additional Gaussian term.Research supported by the Volkswagen-Stiftung (RiP-program at Oberwolfach, Germany).Mathematics Subject Classification (2000): 60F10 (primary), 60K35, 82B44, 82D30 (secondary)     

Geometry of \mathbb{Z}^d and the Central Limit Theorem for Weakly Dependent Random Fields

We study the asymptotic distribution of where A is a subset of , A _N = A[–N, N] ^d , v(A) = lim _N card(A _N) (2N+1) ^–d (0, 1) and X is a stationary weakly dependent random field. We show that the geometry of A has a relevant influence on the problem. More specifically, S _N(A, X) is asymptotically normal for each X that satisfies certain mixting hypotheses if and only if has a limit F(n; A) as N for each . We also study the class of sets A that satisfy this condition.     

Stable complete surfaces with constant mean curvature

Letx:M ²N ³ be a stable immersion with constant mean curvatureH of a complete orientable surfaceM ² into a complete oriented three dimensional Riemannian manifoldN ³. In this paper we prove that, ifM ² is compact andH ²> –1/2 inf _M Ricc _N , thenM ² has genusg3, here Ricc _N is the Ricci curvature ofN ³. We also prove that, ifM ² is complete non compact andN ² has bounded geometry, the area ofM ² is infinite in the metric induced byx. In this case, ifH ²–1/2 inf _M Ricc _N thenx is umbilic and the equality holds.     

Continuous Linear Extension of Ultradifferentiable Functions of Beurling Type

For a weight function ω and a closed set A ? ?^N let ?_(ω)(A) denote the space of all ω-Whitney jets of Beurling type on A. It is shown that for each closed set A ? ?^N there exists an ω-extension operator EA: ?_(ω)(A) → ?_(ω)(?^N) if and only if ω is a (DN)-function (see MEISE and TAYLOR [18], 3.3). Moreover for a fixed compact set K ? ?^N there exists an ω-extension operator E_K: ?_(ω)(K) → ?_(ω)(?^N) if and only if the Fréchet space ?_(ω)(K) satisfies the property (DN) (see Vogt [29], 1.1.).     

2-Cohomologies of the groups SL(n, q)

Let G = SL(n, q), where q is odd, V be a natural module over G, and L = S²(V) be its symmetric square. We construct a 2-cohomology group H²(G, L). The group is one-dimensional over F _q if n = 2 and q ≠ 3, and also if (n, q) = (4, 3). In all other cases H²(G, L) = 0. Previously, such groups H²(G, L) were known for the cases where n = 2 or q = p is prime. We state that H²(G, L) are trivial for n ⩾ 3 and q = p^m, m ⩾ 2. In proofs, use is made of rather elementary (noncohomological) methods. __________ Translated from Algebra i Logika, Vol. 47, No. 6, pp. 687–704, November–December, 2008.     

Conditioned limit theorems for random walks with negative drift

Summary In this paper we will solve a problem posed by Iglehart. In (1975) he conjectured that if S _n is a random walk with negative mean and finite variance then there is a constant so that (S _[n.]/n ^1/2¦N>n) converges weakly to a process which he called the Brownian excursion. It will be shown that his conjecture is false or, more precisely, that if ES ₁=–a<0, ES ₁ ² <, and there is a slowly varying function L so that P(S ₁>x)x ^–q L(x) as x then (S _[n.]/n¦S _n >0) and (S _[n.]/n¦N>n) converge weakly to nondegenerate limits. The limit processes have sample paths which have a single jump (with d.f. (1–(x/a)^–q )⁺) and are otherwise linear with slope –a. The jump occurs at a uniformly distributed time in the first case and at t=0 in the second.The research for this paper was started while the author was visiting W. Vervaat at the Katholieke Universiteit in Nijmegen, Holland, and was completed while the author was at UCLA being supported by funds from NSF grant MCS 77-02121     

Quasirecognizability by the Set of Element Orders for Groups 3D4(q) and F4(q), for q Odd

It is proved that if L is one of the simple groups ³D₄(q) or F₄(q), where q is odd, and G is a finite group with the set of element orders as in L, then the derived subgroup of G/F(G) is isomorphic to L and the factor group G/G′ is a cyclic {2, 3}-group. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 517–539, September–October, 2005. Supported by RFBR grant No. 04-01-00463.     

The asymptotic number of labeled connected graphs with a given number of vertices and edges

Let c(n, q) be the number of connected labeled graphs with n vertices and q ≤ N = (2n ) edges. Let x = q/n and k = q ? n. We determine functions w_k ? 1. a(x) and φ(x) such that c(n, q) ? w_k(q^N)eⁿφ(x)+a(x) uniformly for all n and q ≥ n. If ? > 0 is fixed, n→ ∞ and 4q > (1 + ?)n log n, this formula simplifies to c(n, q) ? (N^q) exp(–ne^?2q/n). on the other hand, if k = o(n^1/2), this formula simplifies to c(n, n + k) ? 1/2 w_k (3/π)^1/2 (e/12k)^k/2nⁿ?^(3k?1)/2.     

Threshold phenomena in the transient behaviour of Markovian models of communication networks and databases

This paper accompanies a talk given at the Workshop on Mathematical Methods in Queueing Networks held at the Mathematical Sciences Institute at Cornell University in August 1988. In earlier work we had exhibited a threshold phenomenon in the transient behaviour of a closed network of ./M/1 nodes: When there areN customers circulating, and the initial state isx, letd _x ^N (t) denote the total variation distance between the distribution at timet and the stationary distribution. Let d^N(t) = max _x d _x ^N (t). We explicitly founda _N proportional toN such thatd ^N(ta_N)1 forevery t<1, andd ^N(ta_N)0 forevery t>1. Thus it appears that the network has not yet converged to stationarity uptoa _N , but has converged to stationarity aftera _N , soa _N can be naturally interpreted as the settling time of the network. Here we briefly deal with some other similar models — closed networks of ./M/m nodes, a well studied model for circuit switched networks, and a model of Mitra for studying concurrency control in databases. Similar threshold phenomena are established in the transient behaviour of these models.Research supported by the National Science Foundation, Grant No. NCR 8710840.     

A Note on Primes and Goldbach Numbers in Short Intervals

Let J(N, H) be the Selberg integral and E(x, T) the error term in Kaczorowski-Perelli's weighted form of the classical explicit formula. We prove that the estimate J(N, H) = o(H2 N) is connected with an appropriate estimate of _N ^2N| E(x,T)² dx, uniformly for H and T in some ranges. Moreover, assuming a suitable bound for _N ^2N| E(x,T)|² dx, we also obtain, for all sufficiently large N and H (log N)11/12, that every interval [N,N + H] contains H Goldbach numbers.     

Syzygies of Veronese Embeddings

We prove that the Veronese embedding _O ⁿ (d): ^{n N} with n2, d3 does not satisfy property N _p (according to Green and Lazarsfeld) if p3d–2. We make the conjecture that also the converse holds. This is true for n=2 and for n=d=3.