Asymptotics of variance of the lattice point count

The variance of the number of lattice points inside the dilated bounded set rD with random position in ℝ ^d has asymptotics ∼ r ^d−1 if the rotational average of the squared modulus of the Fourier transform of the set is O(ϰ ^−d−1). The asymptotics follow from Wiener’s Tauberian theorem.     

A proximal point algorithm for minimax problems

This paper presents a proximal point algorithm for solving discretel _∞ approximation problems of the form minimize ∥Ax−b∥_∞. Let ε_∞ be a preassigned positive constant and let ε _l ,l = 0,1,2,... be a sequence of positive real numbers such that 0 < ε _l < ε_∞. Then, starting from an arbitrary pointz ₀, the proposed method generates a sequence of points z _l ,l= 0,1,2,..., via the rule . One feature that characterizes this algorithm is its finite termination property. That is, a solution is reached within a finite number of iterations. The smaller are the numbers ε _l the smaller is the number of iterations. In fact, if ε ₀ is sufficiently small then z₁ solves the original minimax problem. The practical value of the proposed iteration depends on the availability of an efficient code for solving a regularized minimax problem of the form minimize where ∈ is a given positive constant. It is shown that the dual of this problem has the form maximize , and ify solves the dual thenx=A ^T y solves the primal. The simple structure of the dual enables us to apply a wide range of methods. In this paper we design and analyze a row relaxation method which is suitable for solving large sparse problems. Numerical experiments illustrate the feasibility of our ideas.     

When do random subsets decompose a finite group?

Let A, B be two random subsets of a finite group G. We consider the event that the products of elements from A and B span the whole group, i.e. [AB ∪ BA = G]. The study of this event gives rise to a group invariant we call Θ(G). Θ(G) is between 1/2 and 1, and is 1 if and only if the group is abelian. We show that a phase transition occurs as the size of A and B passes √Θ(G)|G| log |G|; i.e. for any ɛ > 0, if the size of A and B is less than (1 − ɛ)√Θ(G)|G| log |G|, then with high probability AB ∪ BA ≠ G. If A and B are larger than (1 + ɛ)√Θ(G)|G| log |G|, then AB ∪ BA = G with high probability.     

Diophantinc approximation by linear forms on manifolds

The following Khintchine-type theorem is proved for manifoldsM embedded in ℝ ^k which satisfy some mild curvature conditions. The inequality |q·x| <Ψ(|q|) whereΨ(r) → 0 asr → ∞ has finitely or infinitely many solutionsqεℤ ^k for almost all (in induced measure) points x onM according as the sum Σ _r ^{= 1/∞} Ψ(r)r ^k−2 converges or diverges (the divergent case requires a slightly stronger curvature condition than the convergent case). Also, the Hausdorff dimension is obtained for the set (of induced measure 0) of point inM satisfying the inequality infinitely often whenψ(r) =r ^−t . τ >k − 1.     

Branched two-sheeted covers

In this paper we explore the connection between Weierstrass points of subspaces of the holomorphic differentials and the geometry of the canonical curve inPC ^g−1. In particular, we consider non-hyperelliptic Riemann surfaces with involution and the Weierstrass points of the −1 eigenspace of the holomorphic differentials. The case of coverings of a torus is considered in detail. Research of the first author supported in part by the Paul and Gabriella Rosenbaum Foundation, the Landau Center for Research in Mathematical Analysis (supported by Minerva Foundation-Germany) and a US-Israel BSF grant. Research by the second author supported in part by NSF Grant DMS 9003361 and a Lady Davis Visiting Professorship at the Hebrew University.     

Finsler metrics with constant （or scalar） flag curvature

A Finsler metric on a manifold M with its flag curvature K is said to be almost isotropic flag curvature if K =3c + σ where σ and c are scalar functions on M.In this paper,we establish the intrinsic re...     

Chromatic uniqueness of certain complete tripartite graphs

Let P(G, λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P(H, λ) = P(G, λ) implies H is isomorphic to G. Liu et al. [Liu, R. Y., Zhao, H. X., Ye, C. F.: A complete solution to a conjecture on chromatic uniqueness of complete tripartite graphs. Discrete Math., 289, 175–179 (2004)], and Lau and Peng [Lau, G. C., Peng, Y. H.: Chromatic uniqueness of certain complete t-partite graphs. Ars Comb., 92, 353–376 (2009)] show that K(p − k, p − i, p) for i = 0, 1 are chromatically unique if p ≥ k + 2 ≥ 4. In this paper, we show that if 2 ≤ i ≤ 4, the complete tripartite graph K(p − k, p − i, p) is chromatically unique for integers k ≥ i and p ≥ k ²/4 + i + 1.     

A fixed point theorem for a class of Sobolev maps

We prove an analog of the Brouwer fixed point theorem for a map whose differential and adjoint are integrable with exponents n−1 and n/(n−1) respectively. Here Ω is a convex bounded open subset of Rⁿ.

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Fixed Points for Mean Non-expansive Mappings

For a Banach Space X Garcia-Falset introduced the coefficient R（X） and showed that if R（X） 〈 2 then X has a fixed point. In this paper, we define a mean non-expansive mapping T on X in the sense that ｜｜Tx - TY｜｜ ≤ a｜｜x - y｜｜ ＋ b｜｜x - Ty｜｜ for any x,y E X, where a,b ≥ 0, a ＋ b ≤ 1. We show that if R（X） 〈 1/1＋b then T has a fixed point in X.     

Solution of the spectral problem for the curl and Stokes operators with periodic boundary conditions

In this paper, we establish relations between eigenvalues and eigenfunctions of the curl operator and Stokes operator (with periodic boundary conditions). These relations show that the curl operator is the square root of the Stokes operator with ν = 1. The multiplicity of the zero eigenvalue of the curl operator is infinite. The space L ₂(Q, 2π) is decomposed into a direct sum of eigenspaces of the operator curl. For any complex number λ, the equation rot u + λu = f and the Stokes equation −ν(Δv + λ ²v) + ∇p = f, div v = 0, are solved. Bibliography: 15 titles. Dedicated to the memory of Olga Aleksandrovna Ladyzhenskaya __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 246–276.     

Gibbsianness of fermion random point fields

We consider fermion (or determinantal) random point fields on Euclidean space ℝ^d. Given a bounded, translation invariant, and positive definite integral operator J on L²(ℝ^d), we introduce a determinantal interaction for a system of particles moving on ℝ^d as follows: the n points located at x₁,· · ·,x_n ∈ ℝ^d have the potential energy given by where j(x−y) is the integral kernel function of the operator J. We show that the Gibbsian specification for this interaction is well-defined. When J is of finite range in addition, and for d≥2 if the intensity is small enough, we show that the fermion random point field corresponding to the operator J(I+J)⁻¹ is a Gibbs measure admitted to the specification.     

Small semiovals in PG(2, q)

A semioval in a projective plane is a nonempty subset S of points with the property that for every point P ∈ S there exists a unique line ℓ such that . It is known that and both bounds are sharp. We say that S is a small semioval in if . Dover [5] proved that if S has a (q − 1)-secant, then , thus S is small, and if S has more than one (q − 1)-secant, then S can be obtained from a vertexless triangle by removing some subset of points from one side. We generalize this result and prove that if there exist integers 1 ≤ t and − 1 ≤ k such that and S has a (q − t)-secant, then the tangent lines at the points of the (q − t)-secant are concurrent. Specially when t = 1 then S can be obtained from a vertexless triangle by removing some subset of points from one side. The research was supported by the Italian-Hungarian Intergovernmental Scientific and Technological Cooperation Project, Grant No. I-66/99 and by the Hungarian National Foundation for Scientific Research, Grant Nos. T 043556 and T 043758.     

On Point-Cyclic Resolutions of the 2-(63,7,15) Design Associated with PG(5,2)

 A t(v,k,λ) design is a set of v points together with a collection of its k-subsets called blocks so that t points are contained in exactly λ blocks. PG(n,q), the n-dimensional projective geometry over GF(q) is a 2(q ⁿ +q ⁿ⁻¹ +⋯+q+1,q ²+q+1, q ⁿ⁻² + q ⁿ⁻³ +⋯+q+1) design when we take its points as the points of the design and its planes as the blocks of the design. A 2(v,k,λ) design is said to be resolvable if the blocks can be partitioned as ℱ={R ₁,R ₂,…,R _s }, where s=λ(v−1)/(k−1) and each R _i consists of v/k disjoint blocks. If a resolvable design has an automorphism σ which acts as a cycle of length v on the points and ℱ^σ=ℱ, then the design is said to be point-cyclically resolvable. The design consisting of points and planes of PG(5,2) is shown to be point-cyclically resolvable by enumerating all inequivalent resolutions which are invariant under a cyclic automorphism group G=〈σ〉 where σ is a cycle of length v. These resolutions are shown to be the only resolutions which admit point-transitive automorphism group. Received: November 10, 1999 Final version received: September 18, 2000 Acknowledgments. The author would like to thank A. Munemasa for his assistance in writing computer programs on constructing projective spaces and searching for partial spreads. Moreover, she's thankful to T. Hishida and M.␣Jimbo for helpful discussions and for verifying the results of this paper. Present address: Mathematics Department, Ateneo de Manila University, Loyola Heights, Quezon City 1108, Philippines. e-mail: jumela@mathsci.math.admu.edu.ph     

A characterization theorem for the nearest point mappings of linearC-sets

Summary Suppose U is a set, F is a field of subsets of U, P_AB is the set of all bounded finitely additive functions from F into ℝ, and P _A ⁺ is the set of non-negative valued elements of P_AB. In this paper it is proved that if β is a mapping from P _A ⁺ into P _A ⁺ then β is the nearest point mapping for a linear C-set iff β commutes with a certain collection of nonlinear mappings. Entrata in Redazione il 6 ottobre 1976.     

Solutions of elliptic equations involving critical Sobolev exponents with neumann boundary conditions

We consider the problem −Δu=|u| ^p−1u+λu in Ω with on δΩ, where Ω is a bounded domain inR ^N ,p=(N+2)/(N−2) is the critical Sobolev exponent,n the outward pointing normal and λ a constant. Our main result is that if Ω is a ball inR ^N , then for every λ∈R the problem admits infinitely many solutions. Next we prove that for every bounded domain Ω inR ³, symmetric with respect to a plane, there exists a constant μ>0 such that for every λ<μ this problem has at least one non-trivial solution. This work was supported by the Paris VI-Leiden exchange program Supported by the Netherlands organisation for scientific research NWO, under number 611-306-016.     

On a Delay Reaction-Diffusion Difference Equation of Neutral Type

In this paper, we first consider a delay difference equation of neutral type of the form: Δ(y _n + py _n−k + q _n y _n−l = 0 for n∈ℤ⁺(0) (1*) and give a different condition from that of Yu and Wang (Funkcial Ekvac, 1994, 37(2): 241–248) to guarantee that every non-oscillatory solution of (1*) with p = 1 tends to zero as n→∞. Moreover, we consider a delay reaction-diffusion difference equation of neutral type of the form: Δ₁(u _n,m + pu _n−k,m ) + q _n,m u _n−l,m = a ²Δ² ₂ u _{n +1, m−1} for (n,m) ∈ℤ⁺ (0) ×Ω, (2*) study various cases of p in the neutral term and obtain that if p≥−1 then every non-oscillatory solution of (2*) tends uniformly in m∈Ω to zero as n→∞; if p = −1 then every solution of (2*) oscillates and if p < −1 then every non-oscillatory solution of (2*) goes uniformly in m∈Ω to infinity or minus infinity as n→∞ under some hypotheses. Received July 14, 1999, Revised August 10, 2000, Accepted September 30, 2000     

On weighted positivity and the Wiener regularity of a boundary point for the fractional Laplacian

A sufficient condition for the Wiener regularity of a boundary point with respect to the operator (− Δ)^μ inR ⁿ ,n≥1, is obtained, for μ∈(0,1/2n)/(1,1/2n−1). This extends some results for the polyharmonic operator obtained by Maz'ya and Maz'ya-Donchev. As in the polyharmonic case, the proof is based on a weighted positivity property of (− Δ)^μ, where the weight is a fundamental solution of this operator. It is shown that this property holds for μ as above while there is an interval [A _n , 1/2n−A _n ], whereA _n →1, asn→∞, with μ-values for which the property does not hold. This interval is non-empty forn≥8.     

The determination of convex bodies from derivatives of section functions

It is known that non-symmetric convex bodies generally cannot be characterized by the volumes of hyperplane sections through one interior point. Falconer and Gardner, however, independently proved that volumes of hyperplane sections through two different interior points determine the body uniquely. We prove that if −1 < q < n − 1 is not an integer, then the derivatives of the order q at zero of parallel section functions at one interior point completely characterize convex bodies in . If 0 ≤ q < n − 1 is an integer then one needs the derivatives of order q at two different interior points (except for the case where q = n − 2, q odd), generalizing the results of Falconer and Gardner. The first named author was partially supported by the NSF grant DMS 0455696. Received: 31 January 2006     

Maximal arcs in designs

An (α,n)-arc in a 2-design is a set ofn points of the design such that any block intersects it in at most α points. For such an arc,n is bounded by 1+(r(α−1)/λ), with equality if and only if every block meets the arc in either 0 or α points. An (α,n) arc with equality in above is said to be maximal. A maximal block arc can be dually defined. This generalizes the notion of an oval (α=2) in a symmetric design due to Asmus and van Lint. The aim of this paper is to study the infinite family of possibly extendable symmetric designs other than the Hadamard design family and their related designs using maximal arcs. It is shown that the extendability corresponds to the existence of a proper family of maximal arcs. A natural duality between point and block arcs is established, which among other things implies a result of Cameron and van Lint that extendability of a given design in this family is equivalent to extendability of its dual. Similar results are proved for other related designs.     

A 10-point circle is associated with any general point of the ellipse. New properties of Fagnano’s point

Let H be an ellipse with semiaxes a and b (a > b). Two circles concentric with H, and with radii a − b and a + b, are described, each of them being the locus of the intersections between couples of noteworthy H-related lines (Theorems 1 and 2). Tight, as well as unexpected links among such circles and Monge’s circle are shown (Theorems 4, 5, and 6). A surprising pythagorean relationship involving segments related to the ellipse is shown (Theorem 3). A set of 10 concyclic points is associated with any general point of H (Theorem 9). New properties of Fagnano’s point are described (Theorems 10 through 13). Only elementary facts from trigonometry and analytic geometry are used.