On hilbertian subsets of finite metric spaces

The following result is proved: For everyε>0 there is aC(ε)>0 such that every finite metric space (X, d) contains a subsetY such that |Y|≧C(ε)log|X| and (Y, d _Y) embeds (1 +ε)-isomorphically into the Hilbert spacel ₂. The authors are grateful to Haim Wolfson for some discussions related to the content of this paper.     

Regular centralizers of idempotent transformations

Denote by T(X) the semigroup of full transformations on a set X. For ε∈T(X), the centralizer of ε is a subsemigroup of T(X) defined by C(ε)={α∈T(X):αε=εα}. It is well known that C(id _X )=T(X) is a regular semigroup. By a theorem proved by J.M. Howie in 1966, we know that if X is finite, then the subsemigroup generated by the idempotents of C(id _X ) contains all non-invertible transformations in C(id _X ).     

Properties of Riesz fractional derivatives for Korteweg–de Vries solitons

Riesz fractional derivatives are defined as fractional powers of the Laplacian, D ^α  = (?Δ) ^α/2 for ${\alpha \in \mathbb{R}}Riesz fractional derivatives are defined as fractional powers of the Laplacian, D ^α  = (−Δ) ^α/2 for a ? \mathbbR{\alpha \in \mathbb{R}}. For the soliton solution of the Korteweg–de Vries equation, u ₀(X) with X = x − 4t, these derivatives, u _α (X) = D ^α u ₀(X), and their Hilbert transforms, v _α (X) = −HD ^α u ₀(X), can be expressed in terms of the full range Hurwitz Zeta functions ζ₊(s, a) and ζ₋(s, a), respectively. New properties are established for u _α (X) and v _α (X). It is proved that the functions w _α (X) = u _α (X) + iv _α (X) with α > −1 are solutions of the differential equation

-\fracddX(Pa(X)\fracdwdX)+Qa(X)w = lra(X)w,       X ? \mathbbR,-\frac{\rm d}{{\rm d}X}\left(P_{\alpha}(X)\frac{{\rm d}w}{{\rm d}X}\right)+Q_{\alpha}(X)w = \lambda\rho_{\alpha}(X)w,\qquad X \in \mathbb{R},      4. Large Monotone Paths in Graphs with Bounded Degree      Raphael Yuster 《Graphs and Combinatorics》2001,17(3):579-587  We prove that for every ε>0 and positive integer r, there exists Δ0=Δ0(ε) such that if Δ>Δ0 and n>n(Δ,ε,r) then there exists a packing of K n with ⌊(n−1)/Δ⌋ graphs, each having maximum degree at most Δ and girth at least r, where at most εn 2 edges are unpacked. This result is used to prove the following: Let f be an assignment of real numbers to the edges of a graph G. Let α(G,f) denote the maximum length of a monotone simple path of G with respect to f. Let α(G) be the minimum of α(G,f), ranging over all possible assignments. Now let αΔ be the maximum of α(G) ranging over all graphs with maximum degree at most Δ. We prove that Δ+1≥αΔ≥Δ(1−o(1)). This extends some results of Graham and Kleitman [6] and of Calderbank et al. [4] who considered α(K n ). Received: March 15, 1999?Final version received: October 22, 1999      5. On the rate of convergence of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> norms in the CLT for poisson and gamma random variables      J. Sunklodas 《Lithuanian Mathematical Journal》2009,49(2):216-221 In the paper, we present upper bounds of L p norms of order ( X)-1/2 for all 1 ≤ p ≤ ∞ in the central limit theorem for a standardized random variable (X− X)/ √ X, where a random variable X is distributed by the Poisson distribution with parameter λ > 0 or by the standard gamma distribution Γ(α, 0, 1) with parameter α > 0. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-70/09.      6. Self-similar extremal processes      E. I. Pancheva 《Journal of Mathematical Sciences》1998,92(3):3911-3920 Given an extremal process X: [0,∞)→[0,∞)d with lower curve C and associated point process N={(tk, Xk):k≥0}, tk distinct and Xk independent, given a sequence ζ n =(τ n , ξ n ), n≥1, of time-space changes (max-automorphisms of [0,∞)d+1), we study the limit behavior of the sequence of extremal processes Yn(t)=ξ n -1 ○ X ○ τn(t)=Cn(t) V max {ξ n -1 ○ Xk: tk ≤ τn(t){ ⇒ Y under a regularity condition on the norming sequence ζn and asymptotic negligibility of the max-increments of Yn. The limit class consists of self-similar (with respect to a group ηα=(σα, Lα), α>0, of time-space changes) extremal processes. By self-similarity here we mean the property Lα ○ Y(t) = d Y ○ αα(t) for all α>0. The univariate marginals of Y are max-self-decomposable. If additionally the initial extremal process X is assumed to have homogeneous max-increments, then the limit process is max-stable with homogeneous max-increments. Supported by the Bulgarian Ministry of Education and Sciences (grant No. MM 234/1996). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.      7. Exponential sums over primes formed with coefficients of primitive cusp forms      Hui Xue Lao 《数学学报(英文版)》2009,25(4):687-692 Let Af（n） be the coefficient of the logarithmic derivative for the Hecke L-function. In this paper we study the cancellation of the function Ay（n） twisted with an additive character e（α√n）, α 〉0, i.e. Ef（x） = Σx〈n〈2x Af（n）e（α√n）.      8. Weak solutions for steady compressible Navier-Stokes-Fourier system in two space dimensions      Antonín Novotný  Milan Pokorný 《Applications of Mathematics》2011,56(1):137-160 We consider steady compressible Navier-Stokes-Fourier system in a bounded two-dimensional domain. We show the existence of a weak solution for arbitrarily large data for the pressure law p(ϱ, ϑ) ∼ ϱ γ + ϱϑ if γ > 1 and p(ϱ, ϑ) ∼ ϱ ln α (1 + ϱ) + ϱϑ if γ = 1, α > 0, depending on the model for the heat flux.      9. Spaces with large projection constants      Hermann König 《Israel Journal of Mathematics》1985,50(3):181-188 For every prime numberk, we give an explicit construction of a complexk-dimensional spaceX k with projection constantγ(X k ) = √k − 1/√k + 1/k. Moreover, there are realk-dimensional spacesX k withγ(x K ) ≧ √k − 1 for a subsequence of integersk. Hence in both casesγ(X k )/√k → 1 which is the maximal possible value sinceγ(X k ) ≦ √k is generally true.      10. Hilbert’s irreducibility theorem for prime degree and general polynomials      Peter Müller 《Israel Journal of Mathematics》1999,109(1):319-337 Letf (X, t)εℚ[X, t] be an irreducible polynomial. Hilbert’s irreducibility theorem asserts that there are infinitely manyt 0εℤ such thatf (X, t 0) is still irreducible. We say thatf (X, t) isgeneral if the Galois group off (X, t) over ℚ(t) is the symmetric group in its natural action. We show that if the degree off with respect toX is a prime ≠ 5 or iff is general of degree ≠ 5, thenf (X, t 0) is irreducible for all but finitely manyt 0εℤ unless the curve given byf (X, t)=0 has infinitely many points (x 0,t 0) withx 0εℚ,t 0εℤ. The proof makes use of Siegel’s theorem about integral points on algebraic curves, and classical results about finite groups, going back to Burnside, Schur, Wielandt, and others. Supported by the DFG.      11. Characterizing hyperbolic spaces and real trees      Roberto Frigerio  Alessandro Sisto 《Geometriae Dedicata》2009,142(1):139-149 Let X be a geodesic metric space. Gromov proved that there exists ε 0 > 0 such that if every sufficiently large triangle Δ satisfies the Rips condition with constant ε 0 · pr(Δ), where pr(Δ) is the perimeter of Δ, then X is hyperbolic. We give an elementary proof of this fact, also giving an estimate for ε 0. We also show that if all the triangles D í X{\Delta \subseteq X} satisfy the Rips condition with constant ε 0 · pr(Δ), then X is a real tree. Moreover, we point out how this characterization of hyperbolicity can be used to improve a result by Bonk, and to provide an easy proof of the (well-known) fact that X is hyperbolic if and only if every asymptotic cone of X is a real tree.      12. A class of risk processes with reserve-dependent premium rate: sample path large deviations and importance sampling      A. Ganesh  C. Macci  G. L. Torrisi 《Queueing Systems》2007,55(2):83-94 Let (X(t)) be a risk process with reserve-dependent premium rate, delayed claims and initial capital u. Consider a class of risk processes {(X ε (t)): ε > 0} derived from (X(t)) via scaling in a slow Markov walk sense, and let Ψ_ε(u) be the corresponding ruin probability. In this paper we prove sample path large deviations for (X ε (t)) as ε → 0. As a consequence, we give exact asymptotics for log Ψ_ε(u) and we determine a most likely path leading to ruin. Finally, using importance sampling, we find an asymptotically efficient law for the simulation of Ψ_ε(u). AMS Subject Classifications 60F10, 91B30 This work has been partially supported by Murst Project “Metodi Stocastici in Finanza Matematica”      13. 2-Summing operators on C([0, 1], lp) with values in l1      Dumitru Popa 《Proceedings Mathematical Sciences》2009,119(2):221-230 Let Ω be a compact Hausdorff space, X a Banach space, C(Ω, X) the Banach space of continuous X-valued functions on Ω under the uniform norm, U: C(Ω, X) → Y a bounded linear operator and U #, U # two natural operators associated to U. For each 1 ≤ s < ∞, let the conditions (α) U ∈ Π s (C(Ω, X), Y); (β)U # ∈ Π s (C(Ω), Π s (X, Y)); (γ) U # ε Π s (X, Π s (C(Ω), Y)). A general result, [10, 13], asserts that (α) implies (β) and (γ). In this paper, in case s = 2, we give necessary and sufficient conditions that natural operators on C([0, 1], l p ) with values in l 1 satisfies (α), (β) and (γ), which show that the above implication is the best possible result.      14. Karhunen-Loève expansions of <Emphasis Type="Italic">α</Emphasis>-Wiener bridges      Mátyás Barczy  Endre Iglói 《Central European Journal of Mathematics》2011,9(1):65-84 We study Karhunen-Loève expansions of the process(X t (α)) t∈[0,T) given by the stochastic differential equation $ dX_t^{(\alpha )} = - \frac{\alpha } {{T - t}}X_t^{(\alpha )} dt + dB_t ,t \in [0,T) $ dX_t^{(\alpha )} = - \frac{\alpha } {{T - t}}X_t^{(\alpha )} dt + dB_t ,t \in [0,T) , with the initial condition X 0(α) = 0, where α > 0, T ∈ (0, ∞), and (B t )t≥0 is a standard Wiener process. This process is called an α-Wiener bridge or a scaled Brownian bridge, and in the special case of α = 1 the usual Wiener bridge. We present weighted and unweighted Karhunen-Loève expansions of X (α). As applications, we calculate the Laplace transform and the distribution function of the L 2[0, T]-norm square of X (α) studying also its asymptotic behavior (large and small deviation).      15. Weighted approximation of random functions      YU JIARONG 《高校应用数学学报(英文版)》1995,10(4):361-366 ＷＥＩＧＨＴＥＤＡＰＰＲＯＸＩＭＡＴＩＯＮＯＦＲＡＮＤＯＭＦＵＮＣＴＩＯＮＳＹＵＪＩＡＲＯＮＧＡｂｓｔｒａｃｔ：Ｌｅｔ（Ω,Ａ,Ｐ）ｂｅａｐｒｏｂａｂｉｌｉｔｙｓｐａｃｅ,Ｘ（ｔ,ω）ａｒａｎｄｏｍｆｕｎｃｔｉｏｎｃｏｎｔｉｎｕｏｕｓｉｎｐｒｏｂａｂ...      16. On the vector space of 0-configurations      M. Deza  P. Frankl 《Combinatorica》1982,2(4):341-345 Let α be a rational-valued set-function on then-element sexX i.e. α(B) εQ for everyB ⫅X. We say that α defines a 0-configuration with respect toA⫅2 x if for everyA εA we have α(B)=0. The 0-configurations form a vector space of dimension 2 n − |A| (Theorem 1). Let 0 ≦t<k ≦n and letA={A ⫅X: |A| ≦t}. We show that in this case the 0-configurations satisfying α(B)=0 for |B|>k form a vector space of dimension , we exhibit a basis for this space (Theorem 4). Also a result of Frankl, Wilson [3] is strengthened (Theorem 6).      17. An integral strong law of large numbers for processes with independent increments      Howard G. Tucker  A. Larry Wright 《Israel Journal of Mathematics》1981,38(3):177-181 For a stochastically continuous stochastic process with independent increments overD[0,T], letN(t,ε) be the number of smaple function jumps that occur in the interval [0,t] of sizes less than −ε or greater than ε, where ε>0. LetM(t,ε)=EN(t,ε), and assumeM(t,0+)=∞ for 0<t≦T. If limε ↓0(M(t,ε)/M(T,ε)) exists and is positive for eacht∈(0,T], then limε ↓0(N(t,ε)/M(T,ε)) for allt∈(0,T] with probability one. The research of Howard G. Tucker was supported in part by the National Science Foundation, Grant No. MCS76-03591A01.      18. Coefficient Quantization in Banach Spaces      S. J. Dilworth  E. Odell  T. Schlumprecht  A. Zsák 《Foundations of Computational Mathematics》2008,8(6):703-736 Let (e i ) be a dictionary for a separable infinite-dimensional Banach space X. We consider the problem of approximation by linear combinations of dictionary elements with quantized coefficients drawn usually from a ‘finite alphabet’. We investigate several approximation properties of this type and connect them to the Banach space geometry of X. The existence of a total minimal system with one of these properties, namely the coefficient quantization property, is shown to be equivalent to X containing c 0. We also show that, for every ε>0, the unit ball of every separable infinite-dimensional Banach space X contains a dictionary (x i ) such that the additive group generated by (x i ) is (3+ε)−1-separated and 1/3-dense in X.       19. The Besov subspace consisting of most non-smooth functions      E. I. Berezhnoi 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2009,44(3):163-171 Under the assumptions that Δ(f, h)(t) = |f(t + h) − f(t)|, X is a symmetric space of functions in [0, 1], α ∈ (0, 1) and p ∈ [1, ∞) are any fixed number, by the triple (X, α, p) a Besov type space Λ X,p α is constructed, where the norm is given by the equality For any α 0 ∈ (0, 1), it is shown that there exists an infinite-dimensional, closed subspace of Λ X,p α0, such that any non-identically zero function does not belong to the subspace Λ X,p α with α > α 0. The work is done under the financial support of RFFI, Project Cod 08-01-00669      20. A collocation by spline in tension      K. Suria 《分析论及其应用》1990,6(2):101-110 The collocation method by spline in tension for the problem: −εy"+p(x)y=f(x), y(0)=α0,y(1)=α1, p(x)>0, 0<ε<<1, is derived. The method has the second order of the global uniform convergence. For the corresponding difference scheme the optimal estimate: O (himin(hi, ε) is obtained. This research was supported partly by NSF and SIZ for Science of SAP Vojvodina through funds made available to the U.S.—Yugoalav Joint Board on Scientific and Tchnological Cooperation (grants JF554, JF799).

Large Monotone Paths in Graphs with Bounded Degree

 We prove that for every ε>0 and positive integer r, there exists Δ₀=Δ₀(ε) such that if Δ>Δ₀ and n>n(Δ,ε,r) then there exists a packing of K _n with ⌊(n−1)/Δ⌋ graphs, each having maximum degree at most Δ and girth at least r, where at most εn ² edges are unpacked. This result is used to prove the following: Let f be an assignment of real numbers to the edges of a graph G. Let α(G,f) denote the maximum length of a monotone simple path of G with respect to f. Let α(G) be the minimum of α(G,f), ranging over all possible assignments. Now let α_Δ be the maximum of α(G) ranging over all graphs with maximum degree at most Δ. We prove that Δ+1≥α_Δ≥Δ(1−o(1)). This extends some results of Graham and Kleitman [6] and of Calderbank et al. [4] who considered α(K _n ). Received: March 15, 1999?Final version received: October 22, 1999     

On the rate of convergence of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> norms in the CLT for poisson and gamma random variables

In the paper, we present upper bounds of L _p norms of order ( X)^-1/2 for all 1 ≤ p ≤ ∞ in the central limit theorem for a standardized random variable (X− X)/ √ X, where a random variable X is distributed by the Poisson distribution with parameter λ > 0 or by the standard gamma distribution Γ(α, 0, 1) with parameter α > 0. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-70/09.     

Self-similar extremal processes

Given an extremal process X: [0,∞)→[0,∞)^d with lower curve C and associated point process N={(t_k, X_k):k≥0}, t_k distinct and X_k independent, given a sequence ζ _n =(τ _n , ξ _n ), n≥1, of time-space changes (max-automorphisms of [0,∞)^d+1), we study the limit behavior of the sequence of extremal processes Y_n(t)=ξ _n ^-1 ○ X ○ τ_n(t)=C_n(t) V max {ξ _n ^-1 ○ X_k: t_k ≤ τ_n(t){ ⇒ Y under a regularity condition on the norming sequence ζ_n and asymptotic negligibility of the max-increments of Y_n. The limit class consists of self-similar (with respect to a group η_α=(σ_α, L_α), α>0, of time-space changes) extremal processes. By self-similarity here we mean the property L_α ○ Y(t) ₌ ^d Y ○ α_α(t) for all α>0. The univariate marginals of Y are max-self-decomposable. If additionally the initial extremal process X is assumed to have homogeneous max-increments, then the limit process is max-stable with homogeneous max-increments. Supported by the Bulgarian Ministry of Education and Sciences (grant No. MM 234/1996). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.     

Exponential sums over primes formed with coefficients of primitive cusp forms

Let Af（n） be the coefficient of the logarithmic derivative for the Hecke L-function. In this paper we study the cancellation of the function Ay（n） twisted with an additive character e（α√n）, α 〉0, i.e. Ef（x） = Σx〈n〈2x Af（n）e（α√n）.     

Weak solutions for steady compressible Navier-Stokes-Fourier system in two space dimensions

We consider steady compressible Navier-Stokes-Fourier system in a bounded two-dimensional domain. We show the existence of a weak solution for arbitrarily large data for the pressure law p(ϱ, ϑ) ∼ ϱ ^γ + ϱϑ if γ > 1 and p(ϱ, ϑ) ∼ ϱ ln ^α (1 + ϱ) + ϱϑ if γ = 1, α > 0, depending on the model for the heat flux.     

Spaces with large projection constants

For every prime numberk, we give an explicit construction of a complexk-dimensional spaceX _k with projection constantγ(X _k ) = √k − 1/√k + 1/k. Moreover, there are realk-dimensional spacesX _k withγ(x _K ) ≧ √k − 1 for a subsequence of integersk. Hence in both casesγ(X _k )/√k → 1 which is the maximal possible value sinceγ(X _k ) ≦ √k is generally true.     

Hilbert’s irreducibility theorem for prime degree and general polynomials

Letf (X, t)εℚ[X, t] be an irreducible polynomial. Hilbert’s irreducibility theorem asserts that there are infinitely manyt ₀εℤ such thatf (X, t ₀) is still irreducible. We say thatf (X, t) isgeneral if the Galois group off (X, t) over ℚ(t) is the symmetric group in its natural action. We show that if the degree off with respect toX is a prime ≠ 5 or iff is general of degree ≠ 5, thenf (X, t ₀) is irreducible for all but finitely manyt ₀εℤ unless the curve given byf (X, t)=0 has infinitely many points (x ₀,t ₀) withx ₀εℚ,t ₀εℤ. The proof makes use of Siegel’s theorem about integral points on algebraic curves, and classical results about finite groups, going back to Burnside, Schur, Wielandt, and others. Supported by the DFG.     

Characterizing hyperbolic spaces and real trees

Let X be a geodesic metric space. Gromov proved that there exists ε ₀ > 0 such that if every sufficiently large triangle Δ satisfies the Rips condition with constant ε ₀ · pr(Δ), where pr(Δ) is the perimeter of Δ, then X is hyperbolic. We give an elementary proof of this fact, also giving an estimate for ε ₀. We also show that if all the triangles D í X{\Delta \subseteq X} satisfy the Rips condition with constant ε ₀ · pr(Δ), then X is a real tree. Moreover, we point out how this characterization of hyperbolicity can be used to improve a result by Bonk, and to provide an easy proof of the (well-known) fact that X is hyperbolic if and only if every asymptotic cone of X is a real tree.     

A class of risk processes with reserve-dependent premium rate: sample path large deviations and importance sampling

Let (X(t)) be a risk process with reserve-dependent premium rate, delayed claims and initial capital u. Consider a class of risk processes {(X ^ε (t)): ε > 0} derived from (X(t)) via scaling in a slow Markov walk sense, and let Ψ_ε(u) be the corresponding ruin probability. In this paper we prove sample path large deviations for (X ε (t)) as ε → 0. As a consequence, we give exact asymptotics for log Ψ_ε(u) and we determine a most likely path leading to ruin. Finally, using importance sampling, we find an asymptotically efficient law for the simulation of Ψ_ε(u). AMS Subject Classifications 60F10, 91B30 This work has been partially supported by Murst Project “Metodi Stocastici in Finanza Matematica”     

2-Summing operators on C([0, 1], lp) with values in l1

Let Ω be a compact Hausdorff space, X a Banach space, C(Ω, X) the Banach space of continuous X-valued functions on Ω under the uniform norm, U: C(Ω, X) → Y a bounded linear operator and U ^#, U _# two natural operators associated to U. For each 1 ≤ s < ∞, let the conditions (α) U ∈ Π _s (C(Ω, X), Y); (β)U ^# ∈ Π _s (C(Ω), Π _s (X, Y)); (γ) U _# ε Π _s (X, Π _s (C(Ω), Y)). A general result, [10, 13], asserts that (α) implies (β) and (γ). In this paper, in case s = 2, we give necessary and sufficient conditions that natural operators on C([0, 1], l _p ) with values in l ₁ satisfies (α), (β) and (γ), which show that the above implication is the best possible result.     

Karhunen-Loève expansions of <Emphasis Type="Italic">α</Emphasis>-Wiener bridges

We study Karhunen-Loève expansions of the process(X _t ^(α)) _t∈[0,T) given by the stochastic differential equation $ dX_t^{(\alpha )} = - \frac{\alpha } {{T - t}}X_t^{(\alpha )} dt + dB_t ,t \in [0,T) $ dX_t^{(\alpha )} = - \frac{\alpha } {{T - t}}X_t^{(\alpha )} dt + dB_t ,t \in [0,T) , with the initial condition X ₀^(α) = 0, where α > 0, T ∈ (0, ∞), and (B _t )t≥0 is a standard Wiener process. This process is called an α-Wiener bridge or a scaled Brownian bridge, and in the special case of α = 1 the usual Wiener bridge. We present weighted and unweighted Karhunen-Loève expansions of X ^(α). As applications, we calculate the Laplace transform and the distribution function of the L ²[0, T]-norm square of X ^(α) studying also its asymptotic behavior (large and small deviation).     

Weighted approximation of random functions

ＷＥＩＧＨＴＥＤＡＰＰＲＯＸＩＭＡＴＩＯＮＯＦＲＡＮＤＯＭＦＵＮＣＴＩＯＮＳＹＵＪＩＡＲＯＮＧＡｂｓｔｒａｃｔ：Ｌｅｔ（Ω,Ａ,Ｐ）ｂｅａｐｒｏｂａｂｉｌｉｔｙｓｐａｃｅ,Ｘ（ｔ,ω）ａｒａｎｄｏｍｆｕｎｃｔｉｏｎｃｏｎｔｉｎｕｏｕｓｉｎｐｒｏｂａｂ...     

On the vector space of 0-configurations

Let α be a rational-valued set-function on then-element sexX i.e. α(B) εQ for everyB ⫅X. We say that α defines a 0-configuration with respect toA⫅2 ^x if for everyA εA we have α(B)=0. The 0-configurations form a vector space of dimension 2 ⁿ − |A| (Theorem 1). Let 0 ≦t<k ≦n and letA={A ⫅X: |A| ≦t}. We show that in this case the 0-configurations satisfying α(B)=0 for |B|>k form a vector space of dimension , we exhibit a basis for this space (Theorem 4). Also a result of Frankl, Wilson [3] is strengthened (Theorem 6).     

An integral strong law of large numbers for processes with independent increments

For a stochastically continuous stochastic process with independent increments overD[0,T], letN(t,ε) be the number of smaple function jumps that occur in the interval [0,t] of sizes less than −ε or greater than ε, where ε>0. LetM(t,ε)=EN(t,ε), and assumeM(t,0+)=∞ for 0<t≦T. If lim_{ε ↓0}(M(t,ε)/M(T,ε)) exists and is positive for eacht∈(0,T], then lim_{ε ↓0}(N(t,ε)/M(T,ε)) for allt∈(0,T] with probability one. The research of Howard G. Tucker was supported in part by the National Science Foundation, Grant No. MCS76-03591A01.     

Coefficient Quantization in Banach Spaces

Let (e _i ) be a dictionary for a separable infinite-dimensional Banach space X. We consider the problem of approximation by linear combinations of dictionary elements with quantized coefficients drawn usually from a ‘finite alphabet’. We investigate several approximation properties of this type and connect them to the Banach space geometry of X. The existence of a total minimal system with one of these properties, namely the coefficient quantization property, is shown to be equivalent to X containing c ₀. We also show that, for every ε>0, the unit ball of every separable infinite-dimensional Banach space X contains a dictionary (x _i ) such that the additive group generated by (x _i ) is (3+ε)⁻¹-separated and 1/3-dense in X.      

The Besov subspace consisting of most non-smooth functions

Under the assumptions that Δ(f, h)(t) = |f(t + h) − f(t)|, X is a symmetric space of functions in [0, 1], α ∈ (0, 1) and p ∈ [1, ∞) are any fixed number, by the triple (X, α, p) a Besov type space Λ _X,p ^α is constructed, where the norm is given by the equality

A collocation by spline in tension

The collocation method by spline in tension for the problem: −εy"+p(x)y=f(x), y(0)=α₀,y(1)=α₁, p(x)>0, 0<ε<<1, is derived. The method has the second order of the global uniform convergence. For the corresponding difference scheme the optimal estimate: O (h_imin(h_i, ε) is obtained. This research was supported partly by NSF and SIZ for Science of SAP Vojvodina through funds made available to the U.S.—Yugoalav Joint Board on Scientific and Tchnological Cooperation (grants JF554, JF799).