Freeknot splines approximation of Sobolev-type classes of <Emphasis Type="Italic">s</Emphasis>-monotone functions

Let I be a finite interval, s ∈ ℕ₀, and r,ν,n ∈ ℕ. Given a set M, of functions defined on I, denote by M the subset of all functions y ∈ M such that the s-difference is nonnegative on I, ∀τ > 0. Further, denote by the Sobolev class of functions x on I with the seminorm . Also denote by Σ _ν,n , the manifold of all piecewise polynomials of order ν and with n – 1 knots in I. If ν ≥ max {r,s}, 1 ≤ p,q ≤ ∞, and (r,p,q) ≠ (1,1,∞), then we give exact orders of the best unconstrained approximation and of the best s-monotonicity preserving approximation . Part of this work was done while the first author visited Tel Aviv University in May 2003 and in March 2004.     

The <Emphasis Type="Italic">s</Emphasis>-energy of spherical designs on <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript>

This paper investigates the s-energy of (finite and infinite) well separated sequences of spherical designs on the unit sphere S ². A spherical n-design is a point set on S ² that gives rise to an equal weight cubature rule which is exact for all spherical polynomials of degree ≤n. The s-energy E _s (X) of a point set of m distinct points is the sum of the potential for all pairs of distinct points . A sequence Ξ = {X _m } of point sets X _m ⊂S ², where X _m has the cardinality card(X _m )=m, is well separated if for each pair of distinct points , where the constant λ is independent of m and X _m . For all s>0, we derive upper bounds in terms of orders of n and m(n) of the s-energy E _s (X _m(n)) for well separated sequences Ξ = {X _m(n)} of spherical n-designs X _m(n) with card(X _m(n))=m(n).      

Pointwise convergence for semigroups in vector-valued <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

Suppose that {T _t  : t  ≥  0} is a symmetric diffusion semigroup on L ²(X) and denote by its tensor product extension to the Bochner space , where belongs to a certain broad class of UMD spaces. We prove a vector-valued version of the Hopf–Dunford–Schwartz ergodic theorem and show that this extends to a maximal theorem for analytic continuations of on . As an application, we show that such continuations exhibit pointwise convergence.     

Normal forms for singularities of pedal curves produced by non-singular dual curve germs in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

For an n-dimensional spherical unit speed curve r and a given point P, we can define naturally the pedal curve of r relative to the pedal point P. When the dual curve germs are non-singular, singularity types of such pedal curves depend only on locations of pedal points. In this paper, we give a complete list of normal forms for singularities and locations of pedal points when the dual curve germs are non-singular. As an application of our list, we characterize C ^∞ left equivalence classes of pedal curve germs (I, s ₀) → S ⁿ produced by non-singular dual curve germ from the viewpoint of the relation between tangent space and tangent space.      

Largest Families Without an <Emphasis Type="Italic">r</Emphasis>-Fork

Let [n] = { 1,2,...,n} be a finite set, a family of its subsets, 2 ≤ r a fixed integer. Suppose that contains no r + 1 distinct members F, G ₁,..., G _r such that F ⊂ G ₁,...,F ⊂ G _r all hold. The maximum size is asymptotically determined up to the second term, improving the result of Tran. The work of the second author was supported by the Hungarian National Foundation for Scientific Research grant numbers NK0621321, AT048826, the Bulgarian National Science Fund under Grant IO-03/2005 and the projects of the European Community: INTAS 04-77-7171, COMBSTRU–HPRN-CT-2002-000278, FIST–MTKD-CT-2004-003006.     

On rectifiable curves with <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-bounds on global curvature: self-avoidance,regularity, and minimizing knots

We discuss the analytic properties of curves γ whose global curvature function ρ _G [γ]⁻¹ is p-integrable. It turns out that the L ^p -norm is an appropriate model for a self-avoidance energy interpolating between “soft” knot energies in form of singular repulsive potentials and “hard” self-obstacles, such as a lower bound on the global radius of curvature introduced by Gonzalez and Maddocks. We show in particular that for all p > 1 finite -energy is necessary and sufficient for W ^2,p -regularity and embeddedness of the curve. Moreover, compactness and lower-semicontinuity theorems lead to the existence of -minimizing curves in given isotopy classes. There are obvious extensions to other variational problems for curves and nonlinearly elastic rods, where one can introduce a bound on to preclude self-intersections.     

<Emphasis Type="Italic">K</Emphasis><Subscript>0</Subscript>-group and approximate similarity invariants of strongly irreducible operators

Let H olenote a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator T ∈ L(H) is said to be strongly irreducible if T does not commute with any nontrivial idempotent. Herrero and Jiang showed that the norm-closure of the class of all strongly irreducible operators is the class of all operators with connected spectrum. This result can be considered as an approximate inverse of the Riesz decomposition theorem. In the paper, we give a more precise charact...     

The <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> sphere of a 4-manifold

Let denote the set of even integers . We prove that when H ≥ X ^0.33, almost all integers can be represented as the sum of a prime and the square of a prime. We also prove a similar result for sums of three squares of primes.      

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-invariants of Riemannian foliations

For a Riemannian foliation on a closed manifold M, we define L ²-spectral sequence Betti numbers and spectral sequence Novikov–Shubin invariants. The spectral sequence of the lift of to the universal covering of M is used in the definitions. These invariants are natural extensions of the L ²-Betti numbers and the Novikov–Shubin invariants of differentiable manifolds. It is shown that these numbers are invariant by foliated homotopy equivalences, and they are computed for several examples.      

Riesz <Emphasis Type="Italic">s</Emphasis>-Equilibrium Measures on <Emphasis Type="Italic">d</Emphasis>-Rectifiable Sets as <Emphasis Type="Italic">s</Emphasis> Approaches <Emphasis Type="Italic">d</Emphasis>

Let A be a compact set in of Hausdorff dimension d. For s ∈ (0,d) the Riesz s-equilibrium measure μ ^s is the unique Borel probability measure with support in A that minimizes

On the <Emphasis Type="Italic">C</Emphasis>*-valued triangle equality and inequality in Hilbert <Emphasis Type="Italic">C</Emphasis>*-modules

We prove that for two elements x, y in a Hilbert C*-module V over a C*-algebra the C*-valued triangle equality |x + y| = |x| + |y| holds if and only if 〈x, y〉 = |x| |y|. In addition, if has a unit e, then for every x, y ∊ V and every ɛ > 0 there are contractions u, υ ∊ such that |x + y| ≦ u|x|u* + υ|y|υ* + ɛe.      

<Emphasis Type="Italic">r</Emphasis>-Minimal submanifolds in space forms

Let be an n-dimensional compact, possibly with boundary, submanifold in an (n + p)-dimensional space form R ^n+p (c). Assume that r is even and , in this paper we introduce rth mean curvature function S _r and (r + 1)-th mean curvature vector field . We call M to be an r-minimal submanifold if on M, we note that the concept of 0-minimal submanifold is the concept of minimal submanifold. In this paper, we define a functional of , by calculation of the first variational formula of J _r we show that x is a critical point of J _r if and only if x is r-minimal. Besides, we give many examples of r-minimal submanifolds in space forms. We calculate the second variational formula of J _r and prove that there exists no compact without boundary stable r-minimal submanifold with in the unit sphere S ^n+p . When r = 0, noting S ₀ = 1, our result reduces to Simons’ result: there exists no compact without boundary stable minimal submanifold in the unit sphere S ^n+p .      

Remarks on the <Emphasis Type="Italic">k</Emphasis>-error linear complexity of <Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>-periodic sequences

Recently the first author presented exact formulas for the number of 2 ⁿ -periodic binary sequences with given 1-error linear complexity, and an exact formula for the expected 1-error linear complexity and upper and lower bounds for the expected k-error linear complexity, k ≥ 2, of a random 2 ⁿ -periodic binary sequence. A crucial role for the analysis played the Chan–Games algorithm. We use a more sophisticated generalization of the Chan–Games algorithm by Ding et al. to obtain exact formulas for the counting function and the expected value for the 1-error linear complexity for p ⁿ -periodic sequences over prime. Additionally we discuss the calculation of lower and upper bounds on the k-error linear complexity of p ⁿ -periodic sequences over .      

<Emphasis Type="Italic">h</Emphasis>-Principles for curves and knots of constant curvature

We prove that curves of constant curvature satisfy, in the sense of Gromov, the relative -dense h-principle in the space of immersed curves in Euclidean space R ^{n ≥ 3}. In particular, in the isotopy class of any given knot f there exists a knot f͂ of constant curvature which is -close to f. More importantly, we show that if f is , then the curvature of f͂ may be set equal to any constant c which is not smaller than the maximum curvature of f. We may also require that f͂ be tangent to f along any finite set of prescribed points, and coincide with f over any compact set with an open neighborhood where f has constant curvature c. The proof involves some basic convexity theory, and a sharp estimate for the position of the average value of a parameterized curve within its convex hull. The author’s research was supported in part by NSF CAREER award DMS-0332333.     

Inequalities for derivatives of functions in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

  We obtain a new sharp inequality for the local norms of functions x ∈ L _{∞, ∞} ^r (R), namely,

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.     

<Emphasis Type="Italic">c</Emphasis>*-Supplemented subgroups and <Emphasis Type="Italic">p</Emphasis>-nilpotency of finite groups

A subgroup H of a finite group G is said to be c*-supplemented in G if there exists a subgroup K such that G = HK and H ⋂ K is permutable in G. It is proved that a finite group G that is S ₄-free is p-nilpotent if N _G (P) is p-nilpotent and, for all x ∈ G\N _G (P), every minimal subgroup of is c*-supplemented in P and (if p = 2) one of the following conditions is satisfied: (a) every cyclic subgroup of of order 4 is c*-supplemented in P, (b) , (c) P is quaternion-free, where P a Sylow p-subgroup of G and is the p-nilpotent residual of G. This extends and improves some known results. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1011–1019, August, 2007.     

Parabolic Equations and Markov Processes Over <Emphasis Type="Italic">p</Emphasis>-Adic Fields

In this paper we construct and study a fundamental solution of Cauchy’s problem for p-adic parabolic equations of the type