The Polynomial Degree of the Grassmannian {\mathcal G_{\bf 1,}{\bf n,}{\bf 2}}

For a subset ψ of PG(N, 2) a known result states that ψ has polynomial degree ≤ r, r≤ N, if and only if ψ intersects every r-flat of PG(N, 2) in an odd number of points. Certain refinements of this result are considered, and are then applied in the case when ψ is the Grassmannian , to show that for n <8 the polynomial degree of is .     

Weighted Non-Trivial Multiply Intersecting Families

Let n and r be positive integers. Suppose that a family satisfies F₁∩···∩F_r ≠∅ for all F₁, . . .,F_r ∈ and . We prove that there exists ε=ε(r) >0 such that holds for 1/2≤w≤1/2+ε if r≥13.     

On some homotopical and homological properties of monoid presentations

If a monoid S is given by some finite complete presentation ℘, we construct inductively a chain of CW-complexes

A generalization of Meshulam’s theorem on subsets of finite abelian groups with no 3-term arithmetic progression

Let r ₁, …, r _s be non-zero integers satisfying r ₁ + ⋯ + r _s = 0. Let G be a finite abelian group with k _i |k _i-1(2 ≤ i ≤ n), and suppose that (r _i , k ₁) = 1(1 ≤ i ≤ s). Let denote the maximal cardinality of a set which contains no non-trivial solution of r ₁ x ₁ + ⋯ + r _s x _s = 0 with . We prove that . We also apply this result to study problems in finite projective spaces.      

Non-commutative Clarkson Inequalities for n-Tuples of Operators

Let A ₀, ... , A _n−1 be operators on a separable complex Hilbert space , and let α₀,..., α _n−1 be positive real numbers such that 1. We prove that for every unitarily invariant norm,

Almost sure convergence of sample range

Let X ₁, X ₂, ... be i.i.d. random variables. The sample range is R _n = max {X _i , 1 ≤ i ≤ n} − min {X _i , 1 ≤ i ≤ n}. If for a non-degenerate distribution G and some sequences (α _k ), (β _k ) then we have

Addendum To Schrijver's Work On Minimum Permanents

Let denote the set of n×n binary matrices which have each row and column sum equal to k. For 2≤k≤n→∞ we show that is asymptotically equal to (k−1)^k−1k^2−k. This confirms Conjecture 23 in Minc's catalogue of open problems. * Written while the author was employed by the Department of Computer Science at the Australian National University.     

Immersions of spheres and algebraically constructible functions

Let Λ be an algebraic set and let (n is even) be a polynomial mapping such that for each there is r(λ) > 0 such that the mapping g _λ  =  g(· , λ) restricted to the sphere S ⁿ (r) is an immersion for every 0  <  r  <  r (λ), so that the intersection number I(g _λ|S ⁿ (r)) is defined. Then is an algebraically constructible function. I. Karolkiewicz and A. Nowel supported by the grant BW/5100-5-0286-7.     

Two Inequalities for Convex Functions

Let a ₀ < a ₁ < ··· < a _n be positive integers with sums $ {\sum\nolimits_{i = 0}^n {\varepsilon _{i} a_{i} {\left( {\varepsilon _{i} = 0,1} \right)}} } Let a ₀ < a ₁ < ··· < a _n be positive integers with sums distinct. P. Erd?s conjectured that The best known result along this line is that of Chen: Let f be any given convex decreasing function on [A, B] with α ₀, α ₁, ... , α _n , β ₀, β ₁, ... , β _n being real numbers in [A, B] with α ₀ ≤ α ₁ ≤ ··· ≤ α _n , Then In this paper, we obtain two generalizations of the above result; each is of special interest in itself. We prove: Theorem 1 Let f and g be two given non-negative convex decreasing functions on [A, B], and α ₀, α ₁, ... , α _n , β ₀, β ₁, ... , β _n , α' ₀, α' ₁, ... , α' _n , β' ₀ , β' ₁ , ... , β' _n be real numbers in [A, B] with α ₀ ≤ α ₁ ≤ ··· ≤ α _n , α' ₀ ≤ α' ₁ ≤ ··· ≤ α' _n , Then Theorem 2 Let f be any given convex decreasing function on [A, B] with k ₀, k ₁, ... , k _n being nonnegative real numbers and α ₀, α ₁, ... , α _n , β ₀, β ₁, ... , β _n being real numbers in [A, B] with α ₀ ≤ α ₁ ≤ ··· ≤ α _n , Then      

Topological and metric product\mathbb{Z}^d -actions and their applications-actions and their applications

We show that for a -action Ψ being the Kronecker sum of a symbolic strictly ergodic -actionT and a Chacon -actionS, the rank (covering number) of Ψ is the same as that forT. Using this result we construct, for a given natural numberr≥2 and a real numberb∈(0,1) withr\b≥1, a -action with rankr, covering numberb and a simple spectrum. On the other hand, for any positive integersr, m with 1≤m≤r≤∞ we construct a -action with rankr and spectral multiplicitym.     

Fourier and Hermite series estimates of regression functions

Summary In the paper we estimate a regressionm(x)=E {Y|X=x} from a sequence of independent observations (X ₁,Y ₁),…, (X _n, Y_n) of a pair (X, Y) of random variables. We examine an estimate of a type , whereN depends onn andϕ _N is Dirichlet kernel and the kernel associated with the hermite series. Assuming, that E|Y|<∞ and |Y|≦γ≦∞, we give condition for to converge tom(x) at almost allx, provided thatX has a density. if the regression hass derivatives, then converges tom(x) as rapidly asO(nC^{−(2s−1)/4s}) in probability andO(n ^{−(2s−1)/4s} logn) almost completely.     

Homotopy Categories for Simply Connected Torsion Spaces

For each n > 1 and each multiplicative closed set of integers S, we study closed model category structures on the pointed category of topological spaces, where the classes of weak equivalences are classes of maps inducing isomorphism on homotopy groups with coefficients in determined torsion abelian groups, in degrees higher than or equal to n. We take coefficients either on all the cyclic groups with s ∈ S, or in the abelian group where is the group of fractions of the form with s ∈ S. In the first case, for n > 1 the localized category is equivalent to the ordinary homotopy category of (n − 1)-connected CW-complexes whose homotopy groups are S-torsion. In the second case, for n > 1 we obtain that the localized category is equivalent to the ordinary homotopy category of (n − 1)-connected CW-complexes whose homotopy groups are S-torsion and the nth homotopy group is divisible. These equivalences of categories are given by colocalizations , obtained by cofibrant approximations on the model structures. These colocalization maps have nice universal properties. For instance, the map is final (in the homotopy category) among all the maps of the form Y → X with Y an (n − 1)-connected CW-complex whose homotopy groups are S-torsion and its nth homotopy group is divisible. The spaces , are constructed using the cones of Moore spaces of the form M(T, k), where T is a coefficient group of the corresponding structure of models, and homotopy colimits indexed by a suitable ordinal. If S is generated by a set P of primes and S ^p is generated by a prime p ∈ P one has that for n > 1 the category is equivalent to the product category . If the multiplicative system S is generated by a finite set of primes, then localized category is equivalent to the homotopy category of n-connected Ext-S-complete CW-complexes and a similar result is obtained for .     

Asymptotic expansion of the log-likelihood function based on stopping times defined on a Markov process

Consider the parameter space Θ which is an open subset of ℝ ^k ,k≧1, and for each θ∈Θ, let the r.v.′sY _n ,n=0, 1, ... be defined on the probability space (X,A,P _θ) and take values in a Borel setS of a Euclidean space. It is assumed that the process {Y _n },n≧0, is Markovian satisfying certain suitable regularity conditions. For eachn≧1, let υ _n be a stopping time defined on this process and have some desirable properties. For 0 < τ _n → ∞ asn→∞, set h _n →h ∈R ^k , and consider the log-likelihood function of the probability measure with respect to the probability measure . Here is the restriction ofP _θ to the σ-field induced by the r.v.′sY ₀,Y ₁, ..., . The main purpose of this paper is to obtain an asymptotic expansion of in the probability sense. The asymptotic distribution of , as well as that of another r.v. closely related to it, is obtained under both and . This research was supported by the National Science Foundation, Grant MCS77-09574. Research supported by the National Science Foundation, Grant MCS76-11620.     

Homological Connectivity Of Random 2-Complexes

Let Δ_n−1 denote the (n − 1)-dimensional simplex. Let Y be a random 2-dimensional subcomplex of Δ_n−1 obtained by starting with the full 1-dimensional skeleton of Δ_n−1 and then adding each 2−simplex independently with probability p. Let denote the first homology group of Y with mod 2 coefficients. It is shown that for any function ω(n) that tends to infinity

Dominant Families of Pointed Curves in Projective Spaces

Fix non-negative integers r, e, m, g, s such that r ≥ 3, 0 ≤ m < r, e > 0, g + s ≤ er + max{0, m − 1} + 2, g ≤ (e − 1)r + max{0,m − 1} and 0 ≤ s ≤ er + 2. Set d := er + m. Fix any such that and S is in linearly general position. Fix an ordering of the points P ₁, . . . , P _s of S. Here we prove the existence of an irreducible family Γ of smooth, non-degenerate and connected curves with degree d and genus g, all of them containing S and such that the induced map is dominant. Received: September 19, 2006.     

Interior superconvergence error estimates for mixed finite element methods for second order elliptic problem

1.IntroductionLetfibeaplanedomainwithsmoothboundaryonandWm,p(fl)betheusualSobolevspaceonnwithnormWhenp=2,pisusuallyomitted.WeshalldenotetheusualinnerproductinL'(fl)orLa(O)'by','),andinL'(ofl)by't').Weshallusethesamenotationstoindicatethedualltiesbetw...     

Stable constant mean curvature hypersurfaces in the real projective space

In this paper, we prove that the only compact two-sided hypersurfaces with constant mean curvature H which are weakly stable in and have constant scalar curvature are (i) the twofold covering of a totally geodesic projective space; (ii) the geodesic spheres in ; and (iii) the quotient to of the hypersurface obtained as the product of two spheres of dimensions k and n − k, with k = 1,..., n − 1, and radii r and , respectively, with .     

Domination inequality for martingale transforms of a Rademacher sequence

Letf _n = Σ _k=1 ⁿ v _k r _k ,n=1,…, be a martingale transform of a Rademacher sequence (r _n)and let (r _n ^′ ) be an independent copy of (r _n).The main result of this paper states that there exists an absolute constantK such that for allp, 1≤p<∞, the following inequality is true: In order to prove this result, we obtain some inequalities which may be of independent interest. In particular, we show that for every sequence of scalars (a _n)one has where is theK-interpolation norm between ℓ₁ and ℓ₂. We also derive a new exponential inequality for martingale transforms of a Rademacher sequence. This research was supported in part by an NSF grant and an FRPD grant at NCSU.     

Convergence of weighted sums of independent random variables

Let {X_k} be a sequence of i.i.d. random variables with d.f. F(x). In the first part of the paper the weak convergence of the d.f.'s F_n(x) of sums is studied, where 0<α≤2, a_nk>0, 1≤k≤m_n, and, as n→∞, bothmax _1≤k≤mn_{a nk}→0 and . It is shown that such convergence, with suitably chosen A_n's and necessarily stable limit laws, holds for all such arrays {α_nk} provided it holds for the special case α_nk=1/n, 1≤k≤n. Necessary and sufficient conditions for such convergence are classical. Conditions are given for the convergence of the moments of the sequence {F_n(x)}, as well as for its convergence in mean. The second part of the paper deals with the almost sure convergence of sums , where a_n≠0, b_n>0, andmax _1≤k≤n a_k/b_n→0. The strong law is said to hold if there are constants A_n for which S_n→0 almost surely. Let N(0)=0 and N(x) equal the number of n≥1 for which b_n/|a_n|<x if x>0. The main result is as follows. If the strong law holds,EN (|X₁|)<∞. If for some 0<p≤2, then the strong law holds with if 1≤p≤2 and A_n=0 if 0<p<1. This extends the results of Heyde and of Jamison, Orey, and Pruitt. The strong law is shown to hold under various conditions imposed on F(x), the coefficients a_n and b_n, and the function N(x). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.