A Criterion for Convergence of Weak Greedy Algorithms

We find a class V of sequences such that the condition V is necessary and sufficient for convergence of weak greedy algorithm with weakness sequence for each f and all Hilbert spaces H and dictionaries D. We denote by V the class of sequences x={x _{k k=1} , x _k 0, k=1,2,..., with the following property: there exists a sequence 0=q ₀<q ₁< such that _s=1 2 ^s /q _s )< and _s=1 2^–s _k=1 ^q s ^x _k ²<, where q _s :=q _s –q _s–1.     

Positivity and Negativity of Solutions to a Schrödinger Equation in ℝ N

Weak L ² -solutions u of the Schrödinger equation, –u + q(x) u – u = f(x) in L ² , are represented by a Fourier series using spherical harmonics in order to prove the following strong maximum and anti-maximum principles in (N 2): Let ₁ denote the positive eigenfunction associated with the principal eigenvalue ₁ of the Schrödinger operator . Assume that the potential q(x) is radially symmetric and grows fast enough near infinity, and f is a `sufficiently smooth' perturbation of a radially symmetric function, f 0 and 0 f / C const a.e. in . Then u is ₁-positive for - < < ₁ (i.e., u c ₁ with c const > 0) and ₁-negative for ₁ < < ₁ + (i.e., u –c₁ with c const > 0), where > 0 is a number depending on f. The constant c > 0 depends on both and f.     

Finite extinction time for some perturbed Hamilton-Jacobi equations

In this paper we study initial value problems likeu _t–R¦u¦^m+u^q=0 in ⁿ× ₊, u(·,0⁺)=u_o(·) in ^N, whereR > 0, 0 <q < 1,m 1, andu _o is a positive uniformly continuous function verifying –R¦u _o¦^m+u ₀ ^q 0 in ^N . We show the existence of the minimum nonnegative continuous viscosity solutionu, as well as the existence of the function t(·) defined byu(x, t) > 0 if 0<t<t (x) andu(x, t)=0 ift t (x). Regularity, extinction rate, and asymptotic behavior of t(x) are also studied. Moreover, form=1 we obtain the representation formulau(x, t)=max{([(u _o(x – t))^1–q –(1–q)t]₊)^1/(1–q): ¦¦R}, (x, t) ₊ ^N+1 .Partially supported by the DGICYT No. 86/0405 project.     

Tail behavior for semigroups of measures on groups

Let (_t)_t>0 be a convolution semigroup of probability measures on a measurable group (G, ). In this paper, we provide precise information about the asymptotic behavior of _t{q>s, whereq is a measurable seminorm and (_t)_t>0 isq-continuous.     

(q, δ)-Numeration Systems with Missing Digits

We consider the (q, ) numeration system, with basis q2 and the set of digits {, +1,,q+–1} where –(q–1)0. We study properties of numbers where some digits do not occur. This is analogous to the Cantor set {0.a₁a₂a_i{0,2}}. We compute an asymptotic equivalent of the nth moment of the Cantor (q, D)-distribution which can be described as the numbers 0. w₁w₂ with w_iD{,,q+–1}, and each such letter can occur with the same probability 1/CardD. Furthermore, we consider n random strings according to the distribution and the expected minimum of them. We find a recursion which we solve asymptotically.This author was supported by the CNRS/NRF-project no 10959. Part of this work was done during the first authors visit to the John Knopfmacher Centre for Applicable Analysis and Number Theory at the University of the Witwatersrand, Johannesburg, South Africa.This author was supported by the CNRS/NRF-project no 10959.     

The probability of the triangle inequality in probabilistic metric squares

The notion of a random semi-metric space provides an alternate approach to the study of probabilistic metric spaces from the standpoint of random variables instead of distribution functions and permits a new investigation of the triangle inequality. Starting with a probability space (, , P) and an abstract setS, each pair of points,p, q, inS is assigned a random variableX _pq with the interpretation thatX _pq () is the distance betweenp andq at the instant . The probability of the eventJ _pqr = { :X _pr ()X _pq ()+X _qr ()} is studied under distribution function conditions imposed by Menger Spaces (K. Menger, Statistical Metrics, Proc. Nat. Acad. Sci., U.S.A., 28 (1942), 535–537; B. Schweizer and A. Sklar, Statistical Metric Spaces, Pacific J. Math.10 (1960), 313–334). It turns out that for > 0 there are 3 non-negative, identically-distributed random variablesX, Y andZ for whichP(X < Y + Z) < . This and other results show that distribution function triangle inequalities are very weak. Conditional probabilities are introduced to give necessary and sufficient conditions forP(J _pqr ) = 1.     

Designs on vector spaces constructed using quadratic forms

Let N be the set of nonnegative integers, let , t, v be in N and let K be a subset of N, let V be a v-dimensional vector space over the finite field GF(q), and let W _Kbe the set of subspaces of V whose dimensions belong to K. A t-[v, K, ; q]-design on V is a mapping : W _K N such that for every t-dimensional subspace, T, of V, we have (B)=. We construct t-[v, {t, t+1}, ; q-designs on the vector space GF(q ^v) over GF(q) for t2, v odd, and q ^t(q–1)² equal to the number of nondegenerate quadratic forms in t+1 variables over GF(q). Moreover, the vast majority of blocks of these designs have dimension t+1. We also construct nontrivial 2-[v, k, ; q]-designs for v odd and 3kv–3 and 3-[v, 4, q ⁶+q ⁵+q ⁴; q]-designs for v even. The distribution of subspaces in the designs is determined by the distribution of the pairs (Q, a) where Q is a nondegenerate quadratic form in k variables with coefficients in GF(q) and a is a vector with elements in GF(q ^v) such that Q(a)=0.This research was partly supported by NSA grant #MDA 904-88-H-2034.     

On the Representation of Linear Continuous Functionals in Spaces of Analytic Functions of the Hardy–Sobolev Type in a Polydisk

We describe dual spaces of classes of the Hardy–Sobolev type of functions holomorphic in a polydisk for 0 < p 1 and q (0, ) and for p (1, ) and q = 1.     

The measure of a translated ball in uniformly convex spaces

Let (E, ¦·¦) be a uniformly convex Banach space with the modulus of uniform convexity of power type. Let be the convolution of the distribution of a random series inE with independent one-dimensional components and an arbitrary probability measure onE. Under some assumptions about the components and the smoothness of the norm we show that there exists a constant such that |{·<t}–{·+r<t}|r ^q , whereq depends on the properties of the norm. We specify it in the case ofL spaces, >1.     

Асимптоический Анализ Распределения Статистики Диксона

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Largek-arcs in inversive planes of odd order

An infinite family of largek-arcs in the inversive plane over a finite field GF(q), withq 1 (mod 3),q71 orq {17,23, 27,29,41,47,49,53,59} is constructed.Research supported by G.N.S.A.G.A. of C.N.R., project Applicazioni della matematica per la tecnologia e la società, subproject Calcolo simbolico.     

Nonparametric Inference for a Class of Stochastic Partial Differential Equations II

Consider the stochastic partial differential equationdu (t,x) = (t)u (t, x)dt + dW _Q(t,x), 0 t T where = ₂/x ₂, and is a class of positive valued functions. We obtain an estimator for the linear multiplier (t) and establish the consistency, rate of convergence and asymptotic normality of this estimator as 0.     

On the nuclearity of a dual space with the convergence in probability topology

Summary Let be a probability measure on a separable locally convex Fréchet space E and let s denote the topology on E of the convergence in . Then (E, s ) is nuclear iff ((E', s ))=1.     

Two Exterior Algebras for Orthogonal and Symplectic Quantum Groups

Let be one of the N ²-dimensional bicovariant first-order differential calculi on the quantum groups O _q (N) or Sp _q (N), where q is not a root of unity. We show that the second antisymmetrizer exterior algebra _s is the quotient of the universal exterior algebra _u by the principal ideal generated by . Here denotes the unique up to scalars bi-invariant 1-form. Moreover, is central in _u and _u is an inner differential calculus.     

Note on angle-preserving mappings

Summary Let (V, K, q) be aq-regular metric vector space over a commutative field with quadratic formq and letA(V, K, q) be the corresponding affine-metric space. A metric collineation ofA(V, K, q) is a product of a translation and a semilinear bijection ( ₁, ₂) (where ₂ AutK) such that, for a K\{0}, we haveq ₁ = ₂ q. For linesA + KB, A + KC whereA, B, C V\{X Vq(X) = 0} we define an angle-measure < _q (A +KB, A +KC) f(B, C)² q(B)^–1 q(C)^–1 wheref is the bilinear form corresponding toq. For a point tripleA, B, C we define < _q ABC < _q (K(A – B),K(C – B)) whenever the right-hand side is defined. Now assume |K| > 5. In order to get minimal conditions for metric collineations we prove: If 0, 4 is an occurring angle-measure and if is a permutation of the point set such that exactly the point triples with measure are mapped to point triples with measure 0, 4, then is already a metric collineation.     

Staircase kernels for orthogonald-polytopes

LetS be a finite union of boxes inR ^d . Forx inS, defineA _x ={yx is clearly visible fromy via staircase paths inS}, and let KerS denote the staircase kernel ofS. Then KerS={A _x x is a point of local nonconvexity ofS}. A similar result holds with clearly visible replaced by visible and points of local nonconvexity ofS replaced by boundary points ofS.Supported in part by NSF grant DMS-9207019.     

Cycles and measure of bifurcation sets for two-dimensional diffeomorphisms

Summary We describe a large class of one-parameter families , {}, , of two-dimensional diffeomorphisms which arestable for <0, exhibit acycle for =0, and thereafter have a bifurcation set of positive but arbitrarily smallrelative measure for in small intervals [0, ]. A main assumption is that the basic sets involved in the cycle havelimit capacities that are not too large.The second author acknowledges hospitality and financial support from IMPA/CNPq during the period this paper was prepared     

Balanced sets andQ-polynomial association schemes

The notion of a balanced set of vectors is defined, and the classification of such sets suggested. A slightly stronger condition is considered for association schemes, with the following result. LetX be ad-class symmetric association scheme with Bose-Mesner algebraM and Krein parametersq _ij ^h , and let E = E_t (0 t d) be any primitive idempotent ofM. For eachx X letx _E denote the diagonal matrix withy, y entryE _xy(y X). Define the representation diagramD _E on the nodes 0, 1, ...,d by drawing an undirected arc between any nodesh, j for whichq _tj ^h > 0. CallD _E an augmented tree if it has a single loop, whose removal yields a tree. Writexy = i ifx, y X are ith associates (0 i d).     

Almost optimal estimates for entropy numbers of <Emphasis Type="Italic">B</Emphasis><Subscript>2,2</Subscript><Superscript>1/2</Superscript> and its consequences

Triebels conjecture on the entropy numbers for the limiting embedding case B_p,_p^1/p E, E being suitable Orlicz spaces, is almost proved for p2 – almost in the sense that a log -factor occurs in the upper estimate. Furthermore, it is shown that the breaking point, where the radii of the -balls in the covering of the B_p,_q^1/p-unit ball stop to be constant, essentially depends on the second parameter q. The methods used in the paper also give improvements of the results till now known in the case p<2, explains a result of Kashin and Temlyakov on entropy numbers of continuous functions of low smoothness, and indicates how the E-scale can be refined.Mathematics Subject Classification (2000) : 46E35, 41A46, 46E30Part of this work was done while this author was visiting the Department of Mathematics at Technical University Darmstadt in the framework of a DAAD Scholarship.     

Surjectivity of quotient maps for algebraic (ℂ,+)-actions and polynomial maps with contractible fibers

In this paper we establish two results concerning algebraic (,+)-actions on ⁿ . First, let be an algebraic (,+)-action on ³. By a result of Miyanishi, its ring of invariants is isomorphic to [t ₁,t ₂]. Iff ₁,f ₂ generate this ring, the quotient map of is the mapF:³²,x(f ₁(x), f₂(x)). By using some topological arguments we prove thatF is always surjective. Secon, we are interested in dominant polynomial mapsF: ^{n n-1} whose connected components of their generic fibers are contractible. For such maps, we prove the existence of an algebraic (,+)-action on ⁿ for whichF is invariant. Moreover we give some conditions so thatF*([t ₁,...,t _n-1 ]) is the ring of invariants of .Dedicated to all my friends and my family