Periodicity of the Spectrum of a Finite Union of Intervals

A set Ω, of Lebesgue measure 1, in the real line is called spectral if there is a set Λ of real numbers such that the exponential functions e _λ (x)=exp (2πiλx), λ∈Λ, form a complete orthonormal system on L ²(Ω). Such a set Λ is called a spectrum of Ω. In this note we present a simplified proof of the fact that any spectrum Λ of a set Ω which is finite union of intervals must be periodic. The original proof is due to Bose and Madan.     

NON-EMPTY CROSS-2-INTERSECTING FAMILIES OF SUBSETS

Let Cdenote the set of all k-subests of an n-set.Assume Alohtain in Ca,and A lohtain in (A,B) is called a cross-2-intersecting family if |A B≥2 for and A∈A,B∈B.In this paper,the best upper bounds of the cardinalities for non-empty cross-2-intersecting familles of a-and b-subsets are obtained for some a and b,A new proof for a Frankl-Tokushige theorem[6] is also given.     

A Heuristic for the Prime Number Theorem

Conclusion  Might there be a chance of proving in a simple way thatx/π(x) is asymptotic to an increasing function, thus getting another proof of PNT? This is probably wishful thinking. However, there is a natural candidate for the increasing function. LetL(x) be the upper convex hull of the full graph ofxπ(x) (precise definition to follow). The piecewise linear functionL(x) is increasing becausex/π(x) → ∞ asx → ∞. Moreover, using PNT, we can give a proof thatL(x) is indeed asymptotic tox/π(x). But the point of our work in this article is that for someone who wishes to understand why the growth of primes is governed by natural logarithms, a reasonable approach is to convince oneself via computation that the convex hull just mentioned satisfies the hypothesis of our theorem, and then use the relatively simple proof to show that this hypothesis rigorously implies the prime number theorem.     

Characters,genericity, and multiplicity one for U(3)

Let ψ:U→ℂ be a generic character of the unipotent radicalU of a Borel subgroup of a quasisplitp-adic groupG. The number (0 or 1) of ψ-Whittaker models on an admissible irreducible representation π ofG was expressed by Rodier in terms of the limit of values of the trace of π at certain measures concentrated near the origin. An analogous statement holds in the twisted case. This twisted analogue is used in [F, p. 47] to provide a local proof of the multiplicity one theorem for U(3). This assert that each discrete spectrum automorphic representation of the quasisplit unitary group U(3) associated with a quadratic extensionE/F of number fields occurs in the discrete spectrum with multiplicity one. It is pointed out in [F, p. 47] that a proof of the twisted analogue of Rodier's theorem does not appear in print. It is then given below. Detailing this proof is necessitated in particular by the fact that the attempt in [F, p. 48] at a global proof of the multiplicity one theorem for U(3), although widely quoted, is incomplete, as we point out here. Partially supported by a Lady Davis Visiting Professorship at the Hebrew University and the Max-Planck-Institut für Mathematik, Bonn.     

A proof of Selberg's orthogonality for automorphic L-functions

Let π and π′ be automorphic irreducible cuspidal representations of GL_m(Q_A) and GL_m′(Q_A), respectively. Assume that π and π′ are unitary and at least one of them is self-contragredient. In this article we will give an unconditional proof of an orthogonality for π and π′, weighted by the von Mangoldt function Λ(n) and 1−n/x. We then remove the weighting factor 1−n/x and prove the Selberg orthogonality conjecture for automorphic L-functions L(s,π) and L(s,π′), unconditionally for m≤4 and m′≤4, and under the Hypothesis H of Rudnick and Sarnak [20] in other cases. This proof of Selberg's orthogonality removes such an assumption in the computation of superposition distribution of normalized nontrivial zeros of distinct automorphic L-functions by Liu and Ye [12].     

Non-linear operators on sets of measures

Summary If M[ℬ, U(C, C)] is the collection of U(C, C)-valued (non-linear) set functions defined on the Borel subsetsℬ of the compact Hausdorff space S, one may define operators on M[ℬ, U(C, C)] which are ? of the Hammerstein type ?. We initiate a study of a concept analogous to the second dual of a space of continuous functions by inquiring as to what representation theorems one may obtain for these operators. A ? Lebesgue type ? decomposition theorem for elements of M[ℬ, U(C, C)] is obtained. A ? density ? theorem is also obtained for the space M[ℬ, U(C, C)]. Entrata in Redazione il 6 marzo 1974.     

Quasi-contractions on a nonnormal cone metric space

Ilić and Rakočević [6] proved a fixed point theorem for quasi-contractive mappings on cone metric spaces when the underlying cone is normal. Recently, Z. Kadelburg, S. Radenović, and V. Rakočević obtained a similar result without using the normality condition but only for a contractive constant λ ∈ (0, 1/2) [8]. In this note, using a new method of proof, we prove this theorem for any contractive constant λ ∈ (0, 1).     

On Birkhoff interpolation (0;2) case

P. Turán and his associates^[2] considered in detail the problem of (0,2) interpolation based on the zeros of π_n(x). Motivated by these results and an earlier result of Szabados and Varma^[9] here we consider the problem of existence, uniqueness and explicit representation of the interpolatory polynomial R_n(x) satis fying the function values at one set of nodes and the second derivative on the other set of nodes. It is important to note that this problem has a unique solution provided these two sets of nodes are chosen properly. We also promise to have an interesting convergence theorem in the second paper of this series, which will provide a solution to the related open problem of P. Turán.     

r-Qsym is free over Sym

Our main result is a proof of the Florent Hivert conjecture [F. Hivert, Local action of the symmetric group and generalizations of quasi-symmetric functions, in preparation] that the algebras of r-Quasi-Symmetric polynomials in x₁,x₂,…,xn are free modules over the ring of Symmetric polynomials. The proof rests on a theorem that reduces a wide variety of freeness results to the establishment of a single dimension bound. We are thus able to derive the Etingof-Ginzburg [P. Etingof, V. Ginzburg, On m-quasi-invariants of a Coxeter group, Mosc. Math. J. 2 (2002) 555-566] Theorem on m-Quasi-Invariants and our r-Quasi-Symmetric result as special cases of a single general principle. Another byproduct of the present treatment is a remarkably simple new proof of the freeness theorem for 1-Quasi-Symmetric polynomials given in [A.M. Garsia, N. Wallach, Qsym over Sym is free, J. Combin. Theory Ser. A 104 (2) (2003) 217-263].     

Union and symmetry preserving homomorphisms of semigroups of closed relations

The symbol β_X denotes the semigroup of all binary relations on a nonempty set X under composition which is defined by α_oβ={(x,y)} ∈ X×X: (x,z) ∈ β and (z,y) ∈ α for some z∈X} for all α,β ∈ β_x . In a recent paper [1, Theorem 3, p. 310], A. H. Clifford and D. D. Miller initiated a study of the endomorphisms of β_X when they completely determined those which preserve unions and take symmetric relations into symmetric relations. The purpose here is to place the theorem of Clifford and Miller in a topological setting and to discuss some of the problems which then arise naturally. The full results will appear in [9]. Partial financial support from Australian National University and the research foundation of State University of New York is gratefully acknowledged.     

An alternative proof of the Barker, Berman, Plemmons (BBP) result on diagonal stability and extensions

We revisit the theorem of Barker, Berman and Plemmons on the existence of a diagonal quadratic Lyapunov function for a stable linear time-invariant (LTI) dynamical system [G.P. Barker, A. Berman, R.J. Plemmons, Positive diagonal solutions to the Lyapunov equations, Linear and Multilinear Algebra 5(3) (1978) 249-256]. We use recently derived results to provide an alternative proof of this result and to derive extensions.     

On the symmetric square Unstable twisted characters

We provide a purely local computation of the (elliptic) twisted (by “transpose-inverse”) character of the representationπ=I(1) of PGL(3) over ap-adic field induced from the trivial representation of the maximal parabolic subgroup. This computation is independent of the theory of the symmetric square lifting of [IV] of automorphic and admissible representations of SL(2) to PGL(3). It leads — see [FK] — to a proof of the (unstable) fundamental lemma in the theory of the symmetric square lifting, namely that corresponding spherical functions (on PGL(2) and PGL(3)) are matching: they have matching orbital integrals. The new case in [FK] is the unstable one. A direct local proof of the fundamental lemma is given in [V].     

A 1Ψ1 Summation Theorem for Macdonald Polynomials

In this note we extend the Ramanujan's 11 summation formula to the case of a Laurent series extension of multiple q-hypergeometric series of Macdonald polynomial argument [7]. The proof relies on the elegant argument of Ismail [5] and the q-binomial theorem for Macdonald polinomials. This result implies a q-integration formula of Selberg type [3, Conjecture 3] which was proved by Aomoto [2], see also [7, Appendix 2] for another proof. We also obtain, as a limiting case, the triple product identity for Macdonald polynomials [8].     

The symmetry of the partition function of some square ice models

We consider the partition function Z(N; x ₁ , …, x_N, y ₁ , …, y_N) of the square ice model with domain wall boundary conditions. We give a simple proof that Z is symmetric with respect to all its variables when the global parameter a of the model is set to the special value a = e^iπ/3 . Our proof does not use any determinant interpretation of Z and can be adapted to other situations (e.g., to some symmetric ice models).     

A proof of Ewell’s octuple product identity

We give a new proof of Ewell’s octuple product identity in [7] by using a general theorem developed by the first author in [3].     

A note on recursive completions of the countable atomless Boolean algebras

The Baire category theorem for compact Hausdorff spaces is used to show that any recursive completion ℬ^ϕof the countable atomless Boolean algebra ℬ is ℬrecursively closed." Also, a theorem of R.D. Anderson on the simplicity of Aut(ℬ) is used to show that uncountably many automorphisms of ℬ extend to ℬ^ϕ. This revised version was published online in June 2006 with corrections to the Cover Date.     

A generalization of the Looman-Menchoff theorem

In this paper we give a generalization of the classical Looman-Menchoff theorem:If f is a complex-valued continuous function of a complex variable in a domain G, f has partial derivatives f _x and f _y everywhere in G and the Cauchy Riemann equations f _x +if _y = 0are satisfied almost everywhere, then f is holomorphic in G. From our generalization of this theorem, we deduce a theroem of Sindalovskii [9] as a corollary and also answer some of the questions raised in [9]. We note in this context that, as far as we know, Sindalovskii’s result is the best published to date in this area.     

Existence of solution of the logarithmic diffusion equation with bounded above Gauss curvature

We give a simple proof of an extension of the existence results of Ricci flow of Giesen and Topping (2010, 2011) [15], [20], on incomplete surfaces with bounded above Gauss curvature without using the difficult Shi’s existence theorem of Ricci flow on complete non-compact surfaces and the pseudolocality theorem of Perelman [7] on Ricci flow. We will also give a simple proof of a special case of the existence theorem of Topping (2010) [16] without using the existence theorem of Shi (1989) [9].     

Rationally isotropic quadratic spaces are locally isotropic

Let R be a regular local ring, K its field of fractions and (V,ϕ) a quadratic space over R. Assume that R contains a field of characteristic zero we show that if (V,ϕ)⊗ _R K is isotropic over K, then (V,ϕ) is isotropic over R. This solves the characteristic zero case of a question raised by J.-L. Colliot-Thélène in [3]. The proof is based on a variant of a moving lemma from [7]. A purity theorem for quadratic spaces is proved as well. It generalizes in the charactersitic zero case the main purity result from [9] and it is used to prove the main result in [2].