Complete hypersurfaces with constant mean curvature and finite L p norm curvature in Euclidean spaces

In this paper, we consider complete hypersurfaces in R ⁿ⁺¹ with constant mean curvature H and prove that M ⁿ is a hyperplane if the L ² norm curvature of M ⁿ satisfies some growth condition and M ⁿ is stable. It is an improvement of a theorem proved by H. Alencar and M. do Carmo in 1994. In addition, we obtain that M ⁿ is a hyperplane (or a round sphere) under the condition that M ⁿ is strongly stable (or weakly stable) and has some finite L ^p norm curvature. Received: 14 July 2007     

The fitting length of solvableH pn-groups

For a (finite) groupG and some prime powerp ⁿ, theH _p ⁿ -subgroupH _pn (G) is defined byH _p ⁿ (G)=〈xεG|x ^pn≠1〉. A groupH≠1 is called aH _p ⁿ -group, if there is a finite groupG such thatH is isomorphic toH _p ⁿ (G) andH _p ⁿ (G)≠G. It is known that the Fitting length of a solvableH _p ⁿ -group cannot be arbitrarily large: Hartley and Rae proved in 1973 that it is bounded by some quadratic function ofn. In the following paper, we show that it is even bounded by some linear function ofn. In view of known examples of solvableH _p ⁿ -groups having Fitting lengthn, this result is “almost” best possible.     

Compromise 4<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> plans with clear two-factor interactions

This paper gets some necessary conditions for the existence of some kinds of clear 4^m2^n compromise plans which allow estimation of all main effects and some specified two-factor interactions without assuming the remaining two-factor interactions being negligible. Some methods for constructing clear 4^m2^n compromise plans are introduced.     

Some Exponential Inequalities for Positively Associated Random Variables and Rates of Convergence of the Strong Law of Large Numbers

We present some exponential inequalities for positively associated unbounded random variables. By these inequalities, we obtain the rate of convergence n ^−1/2 β _n log ^3/2 n in which β _n can be particularly taken as (log log n)^1/σ with any σ>2 for the case of geometrically decreasing covariances, which is faster than the corresponding one n ^−1/2(log log n)^1/2log ² n obtained by Xing, Yang, and Liu in J. Inequal. Appl., doi: (2008) for the case mentioned above, and derive the convergence rate n ^−1/2 β _n log ^1/2 n for the above β _n under the given covariance function, which improves the relevant one n ^−1/2(log log n)^1/2log n obtained by Yang and Chen in Sci. China, Ser. A 49(1), 78–85 (2006) for associated uniformly bounded random variables. In addition, some moment inequalities are given to prove the main results, which extend and improve some known results.     

The bias of three pseudo-random shuffles

Three schemes for shuffling a deck ofn cards are studied, each involving a random choice from [n] ⁿ . The shuffles favor some permutations over others sincen! does not dividen ⁿ . The probabilities that the shuffles lead to some simple permutations, for instance cycles left and right and the identity, are calculated. Some inequalities are obtained which lead to information about the least and most likely permutations. Numbers of combinatorial interest occur, notably the Catalan numbers and the Bell numbers.     

On decomposition of pseudo BL-algebras

We show that under some conditions, imposed on coatoms and maximal idempotents of a pseudo BL-algebra, we can decompose a pseudo BL-algebra M as an ordinal sum and we show that then M is linearly ordered. We investigate pseudo BL-algebras with a unique coatom a and with a maximal idempotent, and analyze two main situations: either a ⁿ = a ⁿ⁺¹ holds for some n ≥ 1, or a ⁿ > a ⁿ⁺¹ hold for any n ≥ 1. We note that there exist (subdirectly irreducible) algebras with two coatoms that are not linearly ordered, so the restriction to a single coatom is natural.     

Asymptotic Behavior of Heat Kernels on Spheres of Large Dimensions

Forn2, let (μ^x_τ, n)_τ0be the distributions of the Brownian motion on the unit sphereS^{n n+1}starting in some pointxSⁿ. This paper supplements results of Saloff-Coste concerning the rate of convergence ofμ^x_τ, nto the uniform distributionU_nonSⁿforτ→∞ depending on the dimensionn. We show that,[formula]forτ_n:=(ln n+2s)/(2n), where erf denotes the error function. Our proof depends on approximations of the measuresμ^x_τ, nby measures which are known explicitly via Poisson kernels onSⁿ, and which tend, after suitable projections and dilatations, to normal distributions on forn→∞. The above result as well as some further related limit results will be derived in this paper in the slightly more general context of Jacobi-type hypergroups.     

Graphs with exactly one hamiltonian circuit

Let h(n) be the largest integer such that there exists a graph with n vertices having exactly one Hamiltonian circuit and exactly h(n) edges. We prove that h(n) = [n²/4]+1 (n ≧ 4) and discuss some related problems.     

Some Inequalities for Two Simplices

In this paper, we establish some inequalities for two n-dimensional simplices in the n-dimensional Euclidean space E ⁿ .     

Weierstrass Type Representation of Willmore Surfaces in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we reformulate the Euler-Lagrange equations of Willmore surfaces in S^n as the flatness of a family of certain loop algebra-valued 1-forms. Therefore we can give the Weierstrass type representation of conformal Willmore surfaces. We also discuss the relations between conformal Willmore surfaces in S^n and minimal surfaces in constant curvature spaces S^n, R^n, H^n, and prove that some special Willmore surfaces can be derived from minimal surfaces in S^n, R^n, H^n.     

On finitely presented,cancellative and commutative ordered monoids

We show that every finitely presented, cancellative and commutative ordered monoid is determined by a finitely generated and cancellative pseudoorder on the monoid (ℕ ⁿ ,+) for some positive integer n. Every cancellative pseudoorder on (ℕ ⁿ ,+) is determined by a submonoid of the group (ℤ ⁿ ,+), and we prove that the pseudoorder is finitely generated if and only if the submonoid is an affine monoid in ℤ ⁿ .     

Cauchy, Ferrers-Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences

In this paper some decompositions of Cauchy polynomials, Ferrers-Jackson polynomials and polynomials of the form x ²ⁿ + y ²ⁿ , n ∈ ℕ, are studied. These decompositions are used to generate the identities for powers of Fibonacci and Lucas numbers as well as for powers of the so called conjugate recurrence sequences. Also, some new identities for Chebyshev polynomials of the first kind are presented here.     

Some cyclic BIBDs with block size four

In this paper we give some results on cyclic BIBDs with block size 4. It is proved that a cyclic B(4,1;4ⁿ u) exists where u is a product of primes congruent to 1 modulo 6 and n is a positive integer and n ≥ 3. In the case of n = 2, we also give some partial results on the existence of a cyclic B(4,1;4²u) where u is a product of primes congruent to 1 modulo 6. © 2004 Wiley Periodicals, Inc.     

Rigidity of minimal submanifolds with flat normal bundle

Let M ⁿ (n ≥ 3) be an n-dimensional complete immersed $ \frac{{n - 2}} {n} $ \frac{{n - 2}} {n} -super-stable minimal submanifold in an (n + p)-dimensional Euclidean space ℝ ^n+p with flat normal bundle. We prove that if the second fundamental form of M satisfies some decay conditions, then M is an affine plane or a catenoid in some Euclidean subspace.     

Gelfand–Kirillov Dimension for Automorphism Groups of Relatively Free Algebras

Using ideas of our recent work on automorphisms of residually nilpotent relatively free groups, we introduce a new growth function for subgroups of the automorphism groups of relatively free algebras Fⁿ(V) over a field of characteristic zero and the related notion of Gelfand-Kirillov dimension, and study their behavior. We prove that, under some natural restrictions, the Gelfand-Kirillov dimension of the group of tame automorphisms of Fⁿ(V) is equal to the Gelfand-Kirillov dimension of the algebra Fⁿ(V). We show that, in some cases, the Gelfand-Kirillov dimension of the group of tame automorphisms of Fⁿ(V) is smaller than the Gelfand-Kirillov dimension of the whole automorphism group, and calculate the Gelfand-Kirillov dimension of the automorphism group of Fⁿ(V) for some important varieties V.Partially supported by Grant MM605/96 of the Bulgarian Foundation for Scientific Research.2000 Mathematics Subject Classification: primary 16R10, 16P90; secondary 16W20, 17B01, 17B30, 17B40     

Spectral properties of Jacobi matrices by asymptotic analysis

We consider two classes of Jacobi matrix operators in l² with zero diagonals and with weights of the form n^α+c_n for 0<α1 or of the form n^α+c_nn^α−1 for α>1, where {c_n} is periodic. We study spectral properties of these operators (especially for even periods), and we find asymptotics of some of their generalized eigensolutions. This analysis is based on some discrete versions of the Levinson theorem, which are also proved in the paper and may be of independent interest.     

Linear widths of Hölder-Nikol'skii classes of periodic functions of several variables

We consider the linear widths _N (W _p ^r (Tⁿ), L_q) and _N (H _p ^r (Tⁿ), L_q) of the classesW _p ^r (Tⁿ) andH _p ^r (Tⁿ) of periodic functions of one or several variables in the spaceL _q. For the Sobolev classesW _p ^r (Tⁿ) of functions of one or several variables, we state some well-known results without proof; for the Hölder-Nikol'skii classesH _p ^r (Tⁿ), we state some well-known results, prove some new results, and present some previously unpublished proofs.Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 189–199, February, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00237 and by the International Science Foundation under grant No. MP1000.     

Extremal subgraphs of random graphs

We shall prove that if L is a 3-chromatic (so called “forbidden”) graph, and —Rⁿ is a random graph on n vertices, whose edges are chosen independently, with probability p, and —Bⁿ is a bipartite subgraph of Rⁿ of maximum size, —Fⁿ is an L-free subgraph of Rⁿ of maximum size, then (in some sense) Fⁿ and Bⁿ are very near to each other: almost surely they have almost the same number of edges, and one can delete O_p(1) edges from Fⁿ to obtain a bipartite graph. Moreover, with p = 1/2 and L any odd cycle, Fⁿ is almost surely bipartite.     

Partial Ovoids in Classical Finite Polar Spaces

Ovoids in finite polar spaces are related to many other objects in finite geometries. In this article, we prove some new upper bounds for the size of partial ovoids in Q ^–(2n+1,q) and W(2n+ 1,q). Further, we give a combinatorial proof for the non-existence of ovoids of H(2n +1,q ²) for n>q ³.     

On some new type generalized difference sequence spaces

In this paper we introduce a new type of difference operator Δ _m ⁿ for fixed m, n ∈ ℕ. We define the sequence spaces ℓ_∞(Δ _m ⁿ ), c(Δ _m ⁿ ) and c ₀(Δ _m ⁿ ) and study some topological properties of these spaces. We obtain some inclusion relations involving these sequence spaces. These notions generalize many earlier existing notions on difference sequence spaces.