Classification and geometric aspects of vector valued Fourier transforms

It is shown that for any locally compact abelian group ?? and 1 ≤ p ≤ 2, the Fourier type p norm with respect to ?? of a bounded linear operator T between Banach spaces, denoted by ‖T |???^??_p‖, satisfies ‖T |???^??_p‖ ≤ ‖T |???^??_p‖, where ?? is the direct product of ?₂, ?₃, ?₄, … It is also shown that if ?? is not of bounded order then Cⁿ_p ‖T |???^??_p‖ ≤ ‖T |???^??_p‖, where ?? is the circle group, n is a onnegative integer and C^p = . From these inequalities, for any locally compact abelian group ?? ‖T |???^??₂‖ ≤ ‖T |???^??₂‖, and moreover if ?? is not of bounded order then ‖T |???^??₂‖ = ‖T |???^??₂‖. The Hilbertian property and B‐convexity are discussed in the framework of Fourier type p norms. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

One Hundred Twenty-Seven Subsemilattices and Planarity

Let L be a finite n-element semilattice. We prove that if L has at least 127 ? 2^n??8 subsemilattices, then L is planar. For n >?8, this result is sharp since there is a non-planar semilattice with exactly 127 ? 2^n??8 ??1 subsemilattices.

     

On the Behaviour of the spectral characteristic of Feigenbaum’s map

Let T _g : [?1, 1] ?? [?1, 1] be the Feigenbaum map. It is well known that T _g has a Cantor-type attractor F and a unique invariant measure ??₀ supported on F. The corresponding unitary operator (U _g ??)(x) = ??(g(x)) has pure point spectrum consisting of eigenvalues ?? _n,r , n ?? 1, 0 ?? r ?? 2 ^n?1 ? 1 with eigenfunctions e _r ⁽ⁿ⁾ (x). Suppose that f ?? C ¹([?1, 1]), f?? is absolutely continuous on [?1, 1] and f?? ?? L ^p ([?1, 1], d??₀), p > 1. Consider the sum of the amplitudes of the spectral measure of f: $$ Sn(f): = \sum\limits_{r = 0}^{2^n - 1} {|\rho _r^{(n)} |^2 ,\rho _r^{(n)} = \int\limits_{ - 1}^1 {f(x)\overline {e_r^{(n)} (x)} d\mu _o } } (x). $$ Using the thermodynamic formalism for T _g we prove that S _n (f) ?? 2^?n q ⁿ , as n ?? ??, where the constant q ?? (0, 1) does not depend on f.     

Asymptotics of the variance of the number of real roots of random trigonometric polynomials

Let an,n 1 be a sequence of independent standard normal random variables.Consider the random trigonometric polynomial Tn(θ)=∑nj=1 aj cos(jθ),0≤θ≤2π and let Nn be the number of real roots of Tn(θ) in(0,2π).In this paper it is proved that limn →∞ Var(Nn)/n=c0,where 0相似文献   

A trichotomy for a class of equivalence relations

Let Xn, n ∈ N be a sequence of non-empty sets, ψn : Xn2 → IR+. We consider the relation E = E((Xn, ψn)n∈N) on ∏n∈N Xn by (x, y) ∈ E((Xn, ψn)n∈N) <=>Σn∈Nψn(x(n), y(n)) < +∞. If E is an equiv- alence relation and all ψn, n ∈ N, are Borel, we show a trichotomy that either IRN/e1≤B E, E1≤B E, or E≤B E0. We also prove that, for a rather general case, E((Xn, ψn)n∈N) is an equivalence relation iff it is an ep-like equivalence relation.     

Substructure lattices and almost minimal end extensions of models of Peano arithmetic

This paper concerns intermediate structure lattices Lt(??/??), where ?? is an almost minimal elementary end extension of the model ?? of Peano Arithmetic. For the purposes of this abstract only, let us say that ?? attains L if L ? Lt(??/??) for some almost minimal elementary end extension of ??. If T is a completion of PA and L is a finite lattice, then: (A) If some model of T attains L, then every countable model of T does. (B) If some rather classless, ?₁‐saturated model of T attains L, then every model of T does. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Theories of linear order

LetT be a complete theory of linear order; the language ofT may contain a finite or a countable set of unary predicates. We prove the following results. (i) The number of nonisomorphic countable models ofT is either finite or 2^ω. (ii) If the language ofT is finite then the number of nonisomorphic countable models ofT is either 1 or 2^ω. (iii) IfS ₁(T) is countable then so isS _n(T) for everyn. (iv) In caseS ₁(T) is countable we find a relation between the Cantor Bendixon rank ofS ₁(T) and the Cantor Bendixon rank ofS _n(T). (v) We define a class of modelsL, and show thatS ₁(T) is finite iff the models ofT belong toL. We conclude that ifS ₁(T) is finite thenT is finitely axiomatizable. (vi) We prove some theorems concerning the existence and the structure of saturated models. Most of the results in this paper appeared in the author’s Master of Science thesis which was prepared at the Hebrew University under the supervision of Professor H. Gaifman.     

Fractional non-Archimedean calculus in one variable

Let ?? be a natural number. A function f: ? _p ?? K into a non-Archimedeanly valued complete field K ? ? _p is ??-times continuously differentiable if and only if its Mahler coefficients (a _n ) _n??? obey |a _n |n ^?? ?? 0 as n ?? ??. For a real number r ?? 0, this suggests the ad hoc definition by [1] of a C ^r -function f: ? _p ?? K by asking its Mahler coefficients (a _n ) _n??? to satisfy |a _n |n ^r ?? 0 as n?? ??. We will present for functions f: X ?? K on subsets X ? K without isolated points a general pointwise notion of r-fold differentiability through iterated difference quotients, subsequently shown on the domain X = ? _p to coincide with the one given above. For functions on open domains, we prove this notion to admit a handier characterization by its Taylor polynomial up to degree ?r?.     

Omega- and Beta-Models of Alternative Set Theory

We present the axioms of Alternative Set Theory (AST) in the language of second-order arithmetic and study its ω- and β-models. These are expansions of the form (M, M), M ? P(M), of nonstandard models M of Peano arithmetic (PA) such that (M, M) ? AST and ω ? M. Our main results are: (1) A countable M ? PA is β-expandable iff there is a regular well-ordering for M. (2) Every countable β-model can be elementarily extended to an ω-model which is not a β-model. (3) The Ω-orderings of an ω-model (M, M) are absolute well-orderings iff the standard system SS(M) of M is a β-model of A^?_2. (4) There are ω-expandable models M such that no ω-expansion of M contains absolute Ω-orderings. (5) There are s-expandable models (i. e., their ω-expansions contain only absolute Ω-orderings) which are not β-expandable. (6) For every countable β-expansion M of M, there is a generic extension M[G] which is also a β-expansion of M. (7) If M is countable and β-expandable, then there are regular orderings <₁, <₂ such that neither <₁ belongs to the ramified analytical hierarchy of the structure (M, <₂), nor <₂ to that of (M, <₁). (8) The result (1) can be improved as follows: A countable M ? PA is β-expandable iff there is a semi-regular well-ordering for M.     

Characterization of the convergence of weighted averages of sequences and functions

The weighted averages of a sequence (c _k ), c _k ?? ?, with respect to the weights (p _k ), p _k ?? 0, with {fx135-1} are defined by {fx135-2} while the weighted average of a measurable function f: ?₊ ?? ? with respect to the weight function p(t) ?? 0 with {fx135-3}. Under mild assumptions on the weights, we give necessary and sufficient conditions under which the finite limit ?? _n ?? L as n ?? ?? or ??(t) ?? L as t ?? ?? exists, respectively. These characterizations may find applications in probability theory.     

A lower bound in Nehari’s theorem on the polydisc

By theorems of Ferguson and Lacey (d = 2) and Lacey and Terwilleger (d > 2), Nehari??s theorem (i.e., if H _?? is a bounded Hankel form on H ²(D ^d ) with analytic symbol ??, then there is a function ?? in L ^??(T ^d ) such that ?? is the Riesz projection of g4) is known to hold on the polydisc D ^d for d > 1. A method proposed in Helson??s last paper is used to show that the constant C _d in the estimate ??????_?? ?? C _d ??H _?? ?? grows at least exponentially with d; it follows that there is no analogue of Nehari??s theorem on the infinite-dimensional polydisc.     

Existence and regularity of positive solutions of elliptic equations of Schr?dinger type

We prove the existence of positive solutions with optimal local regularity of the homogeneous equation of Schr?dinger type $$ - {\rm{div}}(A\nabla u) - \sigma u = 0{\rm{ in }}\Omega $$ for an arbitrary open ?? ? ? ⁿ under only a form-boundedness assumption on ?? ?? D??(??) and ellipticity assumption on A ?? L ^??(??) ^n×n . We demonstrate that there is a two-way correspondence between form boundedness and existence of positive solutions of this equation as well as weak solutions of the equation with quadratic nonlinearity in the gradient $$ - {\rm{div}}(A\nabla u) = (A\nabla v) \cdot \nabla v + \sigma {\rm{ in }}\Omega $$ As a consequence, we obtain necessary and sufficient conditions for both formboundedness (with a sharp upper form bound) and positivity of the quadratic form of the Schr?dinger type operator H = ?div(A?·)-?? with arbitrary distributional potential ?? ?? D??(??), and give examples clarifying the relationship between these two properties.     

Positive Results and Counterexamples in Comonotone Approximation

We estimate the degree of comonotone polynomial approximation of continuous functions f, on [?1,1], that change monotonicity s??1 times in the interval, when the degree of unconstrained polynomial approximation E _n (f)??n ^??? , n??1. We ask whether the degree of comonotone approximation is necessarily ??c(??,s)n ^??? , n??1, and if not, what can be said. It turns out that for each s??1, there is an exceptional set A _s of ????s for which the above estimate cannot be achieved.     

Construction of nonseparable orthonormal compactly supported wavelet bases for L2（Rd）

Suppose M and N are two r × r and s × s dilation matrices,respectively.Let ΓM and ΓN represent the complete sets of representatives of distinct cosets of the quotient groups M-TZr/Zr and N-TZs/Zs,respe...     

Traces of Sobolev functions with one square integrable directional derivative

We consider the Sobolev spaces of square integrable functions v, from ?ⁿ or from one of its hyperquadrants Q, into a complex separable Hilbert space, with square integrable sum of derivatives ∑?_??v. In these spaces we define closed trace operators on the boundaries ?Q and on the hyperplanes {r_?? = z}, z ∈ ?\{0}, which turn out to be possibly unbounded with respect to the usual L²‐norm for the image. Therefore, we also introduce bigger trace spaces with weaker norms which allow to get bounded trace operators, and, even if these traces are not L², we prove an integration by parts formula on each hyperquadrant Q. Then we discuss surjectivity of our trace operators and we establish the relation between the regularity properties of a function on ?ⁿ and the regularity properties of its restrictions to the hyperquadrants Q. Copyright © 2005 John Wiley & Sons, Ltd.     

Triangular summability of two-dimensional Fourier transforms

We consider the triangular summability of two-dimensional Fourier transforms, and show that the maximal operator of the triangular-??-means of a tempered distribution is bounded from H _p (?²) to L _p (?²) for all 2/(2 + ??) < p ?? ??; consequently, it is of weak type (1,1), where 0 < ?? ?? 1 is depending only on ??. As a consequence, we obtain that the triangular-??-means of a function f ?? L ₁(?²) converge to f a.e. Norm convergence is also considered, and similar results are shown for the conjugate functions. Some special cases of the triangular-??-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, de la Vallée-Poussin, Rogosinski, and Riesz summations.     

On the Equations De.ning Curves in a Polynomial Algebra

Let A be a commutative Noetherian ring of dimension n (n ≥ 3). Let I be a local complete intersection ideal in A[T] of height n. Suppose I/I ² is free A[T]/I-module of rank n and (A[T]/I) is torsion in K ₀(A[T]). It is proved in this article that I is a set theoretic complete intersection ideal in A[T] if one of the following conditions holds: (1) n ≥ 5, odd; (2) n is even, and A contains the field of rational numbers; (3) n = 3, and A contains the field of rational numbers.     

Envelopes,indicators and conservativeness

A well known theorem proved (independently) by J. Paris and H. Friedman states that B Σ_{n +1} (the fragment of Arithmetic given by the collection scheme restricted to Σ_{n +1}‐formulas) is a Π_{n +2}‐conservative extension of I Σ_n (the fragment given by the induction scheme restricted to Σ_n ‐formulas). In this paper, as a continuation of our previous work on collection schemes for Δ_{n +1}(T )‐formulas (see [4]), we study a general version of this theorem and characterize theories T such that T + B Σ_{n +1} is a Π_{n +2}‐conservative extension of T . We prove that this conservativeness property is equivalent to a model‐theoretic property relating Π_n ‐envelopes and Π_n ‐indicators for T . The analysis of Σ_{n +1}‐collection we develop here is also applied to Σ_{n +1}‐induction using Parsons' conservativeness theorem instead of Friedman‐Paris' theorem. As a corollary, our work provides new model‐theoretic proofs of two theorems of R. Kaye, J. Paris and C. Dimitracopoulos (see [8]): B Σ_{n +1} and I Σ_{n +1} are Σ_{n +3}‐conservative extensions of their parameter free versions, B Σ^–_{n +1} and I Σ^–_{n +1}. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some new researches on the general connectivity index of benzenoid systems

Given H a benzenoid system, we find an expression of the general connectivity index, denoted by ?? _?? (H), in terms of inlet features and internal vertex features of H. As a consequence, the extremal n-benzenoid systems with maximal general connectivity index ?? _?? are completely characterized. Moreover, by constructing a combinatorial bijection, we prove that the linear chain L _n is the unique extremal n-benzenoid system with minimal general connectivity index ?? _?? if and only if ??>?? ^?, where ?? ^? is the positive root of the equation 2×6 ^x ?9 ^x =0.     

A Skorohod representation theorem for uniform distance

Let??? _n be a probability measure on the Borel ??-field on D[0, 1] with respect to Skorohod distance, n ?? 0. Necessary and sufficient conditions for the following statement are provided. On some probability space, there are D[0, 1]-valued random variables X _n such that X _n ~ ?? _n for all n ?? 0 and ||X _n ? X ₀|| ?? 0 in probability, where ||·|| is the sup-norm. Such conditions do not require??? ₀ separable under ||·||. Applications to exchangeable empirical processes and to pure jump processes are given as well.