Sum of sequence spaces and matrix transformations mapping in s α 0 ((Δ ? λ I ) h ) + s β ( c ) ((Δ ? μ I ) l )

We deal with the sum of sequence spaces. Then we apply these results to characterize matrix transformations mapping between s ^h,l (λ, μ) = s _α ⁰((Δ − λI) ^h ) + s _β ^(c)((Δ − μI) ^l ) and s _γ . Among other things the aim of this paper is to reduce the set (s ^h,l (λ, μ), s _γ to a set of the form S _τ,γ .      

On γ-semi-open sets

Summary We introduce the concepts of γ-semi-open set, γ_s-set, γ^s-set, generalized γ_s-set, generalized γ^s-set, semi-T₁^γspace and semi-R₀^γspace by using γ-open-sets.     

On some medial semigroups with an associate subgroup

Let S be a semigroup and s,t∈S. We say that t is an associate of s if s=sts. If S has a maximal subgroup G such that every element s of S has a unique associate in G, say s ^∗, we say that G is an associate subgroup of S and consider the mapping s→s ^∗ as a unary operation on S. In this way, semigroups with an associate subgroup may be identified with unary semigroups satisfying three simple axioms. Among them, only those satisfying the identity (st)^∗=t ^∗ s ^∗, called medial, have a structure theorem, due to Blyth and Martins.     

The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group

Let A be an Artin group with standard generating set {σ _s :s∈S}. Tits conjectured that the only relations in A amongst the squares of the generators are consequences of the obvious ones, namely that σ _s ² and σ _t ² commute whenever σ _s and σ _t commute, for s,t∈S. In this paper we prove Tits’ conjecture for all Artin groups. In fact, given a number m _s ≥2 for each s∈S, we show that the elements {T _s =σ _s ^ms :s∈S} generate a subgroup that has a finite presentation in which the only defining relations are that T _s and T _t commute if σ _s and σ _t commute. Oblatum 21-III-2000 & 1-XII-2000?Published online: 5 March 2001     

Kolmogorov and linear widths of classes of <Emphasis Type="Italic">s</Emphasis>-monotone integrable functions

Let s ∈ ℕ and let Δ ₊ ^s be the set of functions x: I ↦ ℝ on a finite interval I such that the divided differences [x; t ₀, ..., t _s ] of order s of these functions are nonnegative for all collections of s + 1 different points t ₀, ..., t _s ∈ I. For the classes Δ ₊ ^s B _p : = Δ ₊ ^s ∩ B _p , where B _p is the unit ball in L _p , we determine the orders of Kolmogorov and linear widths in the spaces _Lq for 1 ≤ q > p ≤ ∞. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 12, pp. 1633–1652, December, 2005.     

γ-Convergence of integral functionals and the variational dirichlet problem in variable domains

By using special local characteristics of domains Ω _s ⊂Ω,s=12,..., we establish necessary and sufficient conditions for the γ-convergence of sequences of integral functionalsI _λs :W ^k,m (Ω _s )→ℝ, λ⊂Ω to interal functionals defined on W ^k,m (Ω). Institute of Applied Mathematics and Mechanics, Ukrainian Academy of Sciences, Donetsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9, pp. 1236–1254, September, 1996.     

On generalized energy equality of the Navier-Stokes equations

We show that for every initial dataa εL ²(Ω) there exists a weak solutionu of the Navier-Stokes equations satisfying the generalized energy inequality introduced by Caffarelli-Kohn-Nirenberg forn=3. We also show that if a weak solutionu εL ^s (0,T;L ^q (Ω)) with 2/q + 2/s ≤ 1 and 3/q + 1/s ≤ 1 forn=3, or 2/q + 2/s ≤ 1 andq ≥ 4 forn ≥ 4, thenu satisfies both the generalized and the usual energy equalities. Moreover we show that the generalized energy equality holds only under the local hypothesis thatu εL ^s (ε, T; L ^q (K)) for all compact setsK ⊂ ⊂ Ω and all 0 <ε <T with the same (q, s) as above when 3 ≤n ≤ 10.     

A Higman inequality for regular near polygons

The inequality of Higman for generalized quadrangles of order (s,t) with s>1 states that t≤s ². We generalize this by proving that the intersection number c _i of a regular near 2d-gon of order (s,t) with s>1 satisfies the tight bound c _i ≤(s ²ⁱ −1)/(s ²−1), and we give properties in case of equality. It is known that hemisystems in generalized quadrangles meeting the Higman bound induce strongly regular subgraphs. We also generalize this by proving that a similar subset in regular near 2d-gons meeting the bounds would induce a distance-regular graph with classical parameters (d,b,α,β)=(d,−q,−(q+1)/2,−((−q) ^d +1)/2) with q an odd prime power.     

Small fractional parts of additive forms

Letf(x)=θ₁ x ₁ ^k +...+θ _s x _s ^k be an additive form with real coefficients, and ∥α∥ = min {|α-u|:uεℤ} denote the distance fromα to the nearest integer. We show that ifθ ₁,…,θ _s , are algebraic ands = 4k then there are integersx ₁,…,x _s , satisfying l ≤x ₁,≤ N and ∥f(x)∥ ≤ N ^E , withE = − 1 + 2/e. Whens = λk, 1 ≤λ ≤ 2k, the exponentE may be replaced byλE/4, and if we drop the condition thatθ ₁,…,θ _s , be algebraic then the result holds for almost all values of θεℝ ^s . Whenk ≥ 6 is small a better exponent is obtained using Heath-Brown’s version of Weyl’s estimate.     

A global volume lemma and applications

Letf ^t be aC ² Axiom A dynamical system on a compact manifold satisfying the transversality condition. We prove that ifB _x (ε,t)=[y: dist (f ^s x,f ^s y)≤ε for all 0≤s≤t], then volB _x (ε,t) has the order exp(∫ ₀ ^t φ (f ^s x)ds) in the continuous time case and exp (Σ _s ^t−1 φ (f ^s x)) in the discrete time case, whereφ is a Holder continuous extension from basic hyperbolic sets of the negative of the differential expansion coefficient in the unstable direction. An application to the theory of large deviations is given. Partially supported by US-Israel BSF. Partially supported by a Darpa grant.     

A braidlike presentation of Sp(n,p)

Forn even andp an odd prime a symplectic group Sp(n, p) is a quotient of the Artin braid groupB _n+1. Ifs ₁, …,s _n are standard generators ofB _n+1 then the kernel of the corresponding epimorphism is the normal closure of just four elements:s ₁ ^p ,(s ₁ s ₂)⁶,s ₁ ^(p+1)/2 s ₂ ⁴ s ₁ ^(p−1)/2 s ₂ ⁻² s ₁ ⁻¹ s ₂ ² and (s ₁ s ₂ s ₃)⁴ A ⁻¹ s ₁ ⁻² A, whereA=s ₂ s ₃ ⁻¹ s ₂ ^(p−1)/2 s ₄ s ₃ ² s ₄, all of them lying in the subgroupB ₅. Sp(n, p) acts on a vector space and the image of the subgroupB _n ofB _n+1 in Sp(n, p), denoted Sp(n−1,p), is a stabilizer of one vector. A sequence of inclusions …B _k+1·B _k … induces a sequence of inclusions …Sp(k,p)·Sp(k−1,p)…, which can be used to study some finite-valued invariants of knots and links in the 3-sphere via the Markov theorem. Partially supported by the Technion VPR-Fund.     

On the average hyperoscillations of planted plane trees

Assume that the leaves of a planted plane tree are enumerated from left to right by 1, 2, .... Thej-ths-turn of the tree is defined to be the root of the (unique) subtree of minimal height with leavesj, j+1, ...,j+s−1. If all trees withn nodes are regarded equally likely, the average level number of thej-ths-turn tends to a finite limitα _s (j), which is of orderj ^1/2. Thej-th ”s-hyperoscillation”α ₁(j)−α _s+1(j) is given by 1/2α ₁(s)+O(j ^−1/2) and therefore tends (forj → ∞) to a constant behaving like √8/π·s ^1/2 fors → ∞. These results are obtained by setting up appropriate generating functions, which are expanded about their (algebraic) singularities nearest to the origin, so that the asymptotic formulas are consequences of the so-called Darboux-Pólyamethod.     

Irregular wavelet frames on<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(ℝ<Superscript>n</Superscript>)

In this paper, we present the conditions on dilation parameter {s _j}_j that ensure a discrete irregular wavelet system {s _j ^n/2ψ(s _j ·−bk)} _{j∈ℤ,k∈ℤ} ⁿ to be a frame on L²(ℝⁿ), and for the wavelet frame we consider the perturbations of translation parameter b and frame function ψ respectively.     

Many partition relations below density

We force 2 ^λ to be large, and for many pairs in the interval (λ, 2 ^λ ) a strong version of the polarized partition relations holds. We apply this to problems in general topology. For example, consistently, every 2 ^λ is the successor of a singular and for every Hausdorff regular space X, hd(X) ≤ s(X)⁺³, hL(X) ≤ s(X)⁺³ and better when s(X) is regular, via a halfgraph partition relations. For the case s(X) = ℵ ₀ we get hd(X), hL(X) ≤ N ₂.     

A class of optimal linear codes of length one above the Griesmer bound

In this paper, we determine the smallest lengths of linear codes with some minimum distances. We construct a [g _q (k, d) + 1, k, d] _q code for sq ^k-1 − sq ^k-2 − q ^s  − q ² + 1 ≤ d ≤ sq ^k-1 − sq ^k-2 − q ^s with 3 ≤ s ≤ k − 2 and q ≥ s + 1. Then we get n _q (k, d) = g _q (k, d) + 1 for (k − 2)q ^k-1 − (k − 1)q ^k-2 − q ² + 1 ≤ d ≤ (k − 2)q ^k-1 − (k − 1)q ^k-2, k ≥ 6, q ≥ 2k − 3; and sq ^k-1 − sq ^k-2 − q ^s  − q + 1 ≤ d ≤ sq ^k-1 − sq ^k-2 − q ^s , s ≥ 2, k ≥ 2s + 1 and q ≥ 2s − 1. This work was partially supported by the Com²MaC-SRC/ERC program of MOST/KOSEF (grant ^# R11-1999-054) and was partially supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD)(KRF-2005-214-C00175).     

On the hot spots of a catalytic super-Brownian motion

Consider the catalytic super-Brownian motion X ^ϱ (reactant) in ℝ ^d , d≤3, which branching rates vary randomly in time and space and in fact are given by an ordinary super-Brownian motion ϱ (catalyst). Our main object of study is the collision local time L = L _[ϱ,Xϱ] (d(s,x) )of catalyst and reactant. It determines the covariance measure in themartingale problem for X ^ϱ and reflects the occurrence of “hot spots” of reactant which can be seen in simulations of X ^ϱ. In dimension 2, the collision local time is absolutely continuous in time, L(d(s,x) ) = ds K _s (dx). At fixed time s, the collision measures K _s (dx) of ϱ _s and X _s ^ϱ have carrying Hausdorff dimension 2. Spatial marginal densities of L exist, and, via self-similarity, enter in the long-term randomergodic limit of L (diffusiveness of the 2-dimensional model). We alsocompare some of our results with the case of super-Brownian motions withdeterministic time-independent catalysts. Received: 2 December 1998 / Revised version: 2 February 2001 / Published online: 9 October 2001     

Multivalued nonlinear equations on the half line: A fixed point approach

New fixed point theorems of the authors are used to establish the existence of one (or more) C[0, ∞) solutions to the nonlinear integral inclusion y(t)∈ ∫₀^∞ K(t, s) F(s, y(s))ds fort ∈ [0,∞).     

On K s -free subgraphs in K s+k -free graphs and vertex Folkman numbers

Extending the problem of determining Ramsey numbers Erdős and Rogers introduced the following function. For given integers 2 ≤ s < t let f _s,t (n) = min{max{|S|: S ⊆ V (H) and H[S] contains no K _s }}, where the minimum is taken over all K _t -free graphs H of order n. This function attracted a considerable amount of attention but despite that, the gap between the lower and upper bounds is still fairly wide. For example, when t=s+1, the best bounds have been of the form Ω(n ^1/2+o(1)) ≤ f _s,s+1(n) ≤ O(n ^1−ɛ(s)), where ɛ(s) tends to zero as s tends to infinity. In this paper we improve the upper bound by showing that f _s,s+1(n) ≤ O(n ^2/3). Moreover, we show that for every ɛ > 0 and sufficiently large integers 1 ≪ k ≪ s, Ω(n ^1/2−ɛ ) ≤ f _s,s+k (n) ≤ O(n ^1/2+ɛ . In addition, we also discuss some connections between the function f _s,t and vertex Folkman numbers.