Geometric progressions in sumsets over finite fields

Given two sets , the set of d dimensional vectors over the finite field with q elements, we show that the sumset contains a geometric progression of length k of the form vΛ ^j , where j = 0,…, k − 1, with a nonzero vector and a nonsingular d × d matrix Λ whenever . We also consider some modifications of this problem including the question of the existence of elements of sumsets on algebraic varieties.     

Skew Boolean algebras derived from generalized Boolean algebras

Every skew Boolean algebra S has a maximal generalized Boolean algebra image given by S/ where is the Green’s relation defined initially on semigroups. In this paper we study skew Boolean algebras constructed from generalized Boolean algebras B by a twisted product construction for which . In particular we study the congruence lattice of with an eye to viewing as a minimal skew Boolean cover of B. This construction is the object part of a functor from the category GB of generalized Boolean algebras to the category LSB of left-handed skew Boolean algebras. Thus we also look at its left adjoint functor . This paper was written while the second author was a Visiting Professor in the Department of Education at the University of Cagliari. The facilities and assistance provided by the University and by the Department are gratefully acknowledged.     

The initial value problem for the cubic nonlinear Klein–Gordon equation

We study the initial value problem for the cubic nonlinear Klein–Gordon equation

Estimates on the weak solution of semilinear hyperbolic systems

The Cauchy problem for a semilinear hyperbolic system of the type

Large Time Asymptotics for the BBM–Burgers Equation

We study large time asymptotics of solutions to the BBM–Burgers equation

Schnorr trivial reals: a construction

A real is Martin-Löf (Schnorr) random if it does not belong to any effectively presented null ${\Sigma^0_1}A real is Martin-L?f (Schnorr) random if it does not belong to any effectively presented null (recursive) class of reals. Although these randomness notions are very closely related, the set of Turing degrees containing reals that are K-trivial has very different properties from the set of Turing degrees that are Schnorr trivial. Nies proved in (Adv Math 197(1):274–305, 2005) that all K-trivial reals are low. In this paper, we prove that if , then h contains a Schnorr trivial real. Since this concept appears to separate computational complexity from computational strength, it suggests that Schnorr trivial reals should be considered in a structure other than the Turing degrees. This material is based upon work supported under a National Science Foundation Graduate Research Fellowship and appears in the author’s Ph.D. thesis. A preliminary version of this paper appeared in Electronic Notes in Theoretical Computer Science     

Hölder continuity of surfaces with bounded mean curvature at corners where Plateau boundaries meet free boundaries

Let $P={\rm \Gamma}\cap{\cal S}Let be the point of non-tangential intersection of a closed Jordan arc and an embedded, regular support surface . Let be a conformally parametrized solution of with partially free boundaries . It is proved, that is H?lder continuous up to with , whenever meets orthogonally along its free trace. This provides a regularity result for stationary minimal surfaces and is applicable also to surfaces of prescribed bounded mean curvature. Mathematics Subject Classification (2000) 53 A 10, 35 J 65, 35 R 35, 49 Q 05     

Sharp estimates for intrinsic ultracontractivity on C 1,α-domains

Let be the first Dirichlet eigenfunction on a connected bounded C ^1,α-domain in and the corresponding Dirichlet heat kernel. It is proved that where λ₂ > λ₁ are the first two Dirichlet eigenvalues. This estimate is sharp for both short and long times. Bounded Lipschitz domains, elliptic operators on manifolds, and a general framework are also discussed. Supported in part by Creative Research Group Fund of the National Foundation of China (no. 10121101), the 973-Project in China and RFDP(20040027009).     

Existence theory of abstract approximate deconvolution models of turbulence

This report studies an abstract approach to modeling the motion of large eddies in a turbulent flow. If the Navier-Stokes equations (NSE) are averaged with a local, spatial convolution type filter, , the resulting system is not closed due to the filtered nonlinear term . An approximate deconvolution operator D is a bounded linear operator which is an approximate filter inverse

Bicomplex Quantum Mechanics: II. The Hilbert Space

Using the bicomplex numbers

Graph homology of the moduli space of pointed real curves of genus zero

The moduli space parameterizes the isomorphism classes of S-pointed stable real curves of genus zero which are invariant under relabeling by the involution σ. This moduli space is stratified according to the degeneration types of σ-invariant curves. The degeneration types of σ-invariant curves are encoded by their dual trees with additional decorations. We construct a combinatorial graph complex generated by the fundamental classes of strata of . We show that the homology of is isomorphic to the homology of our graph complex. We also give a presentation of the fundamental group of .      

Convexities of partial monounary algebras

In the present paper we prove that the collection of all convexities of partial monounary algebras is finite; namely, it has exactly 23 elements. Further, we show that for each element there exists a subset of such that is generated by and card . This work was supported by the Science and Technology Assistance Agency under the contract No. APVT-20-004104. Supported by Grant VEGA 1/3003/06.     

The Birman–Craggs–Johnson homomorphism and abelian cycles in the Torelli group

In the 1970s, Birman–Craggs–Johnson (BCJ) (Trans AMS 237: 283–309, 1978; Trans AMS 261(1):423–422, 1980) used Rochlin’s invariant for homology 3-spheres to construct a remarkable surjective homomorphism , where is the Torelli group and B ₃ is a certain -vector space of Boolean (square-free) polynomials. By pulling back cohomology classes and evaluating them on abelian cycles, we construct dimensions worth of nontrivial elements of which cannot be detected rationally. These classes in fact restrict to nontrivial classes in the cohomology of the subgroup generated by Dehn twists about separating curves. We also use the “Casson–Morita algebra” and Morita’s integral lift of the BCJ map restricted to to give the same lower bound on . The first author is partially supported by NSF grant DMS-0606882 and was also supported in part by NSF grant DMS-0504208 and by a VIGRE postdoc under NSF grant 9983660 to Cornell University. The second author is supported in part by NSF grant DMS-0244542.     

On the VC-dimension of uniform hypergraphs

Let be a k-uniform hypergraph on [n] where k−1 is a power of some prime p and n≥ n ₀(k). Our main result says that if , then there exists E ₀∊ such that {E∩ E ₀: E∊ } contains all subsets of E ₀. This improves a longstanding bound of due to Frankl and Pach [7].Research supported in part by NSF grants DMS-0400812 and an Alfred P. Sloan Research Fellowship.Research supported in part by NSA grant H98230-05-1-0079. Part of this research was done while working at University of Illinois at Chicago.     

Generalised twists,stationary loops,and the Dirichlet energy over a space of measure preserving maps

Let be a bounded Lipschitz domain and consider the Dirichlet energy functional

Convex sets with semidefinite representation

We provide a sufficient condition on a class of compact basic semialgebraic sets for their convex hull co(K) to have a semidefinite representation (SDr). This SDr is explicitly expressed in terms of the polynomials g _j that define K. Examples are provided. We also provide an approximate SDr; that is, for every fixed , there is a convex set such that (where B is the unit ball of ), and has an explicit SDr in terms of the g _j ’s. For convex and compact basic semi-algebraic sets K defined by concave polynomials, we provide a simpler explicit SDr when the nonnegative Lagrangian L _f associated with K and any linear is a sum of squares. We also provide an approximate SDr specific to the convex case.      

Asymptotic Expansions in Non-central Limit Theorems for Quadratic Forms

We consider quadratic forms of the type

Second-order Linearity of Wilcoxon Statistics

The rank statistic , with R _i (t) being the rank of and e ₁ , . . . , e _n being the random sample from a distribution with a cdf F, is considered as a random process with t in the role of parameter. Under some assumptions on c _i , x _i and on the underlying distribution, it is proved that the process converges weakly to the Gaussian process. This generalizes the existing results where the one-dimensional case was considered. We believe our method of the proof can be easily modified for the signed-rank statistics of Wilcoxon type. Finally, we use our results to find the second order asymptotic distribution of the R-estimator based on the Wilcoxon scores and also to investigate the length of the confidence interval for a single parameter β _l .     

MIP reformulations of the probabilistic set covering problem

In this paper, we address the following probabilistic version (PSC) of the set covering problem: where A is a 0-1 matrix, is a random 0-1 vector and is the threshold probability level. We introduce the concepts of p-inefficiency and polarity cuts. While the former is aimed at deriving an equivalent MIP reformulation of (PSC), the latter is used as a strengthening device to obtain a stronger formulation. Simplifications of the MIP model which result when one of the following conditions hold are briefly discussed: A is a balanced matrix, A has the circular ones property, the components of are pairwise independent, the distribution function of is a stationary distribution or has the disjunctive shattering property. We corroborate our theoretical findings by an extensive computational experiment on a test-bed consisting of almost 10,000 probabilistic instances. This test-bed was created using deterministic instances from the literature and consists of probabilistic variants of the set covering model and capacitated versions of facility location, warehouse location and k-median models. Our computational results show that our procedure is orders of magnitude faster than any of the existing approaches to solve (PSC), and in many cases can reduce hours of computing time to a fraction of a second. Anureet Saxena’s research was supported by the National Science Foundation through grant #DMI-0352885 and by the Office of Naval Research through contract N00014-03-1-0133. Vineet Goyal’s research was supported in part by NSF grant CCF-0430751 and ITR grant CCR-0122581.     

Stabilization of the damped wave equation with Cauchy–Ventcel boundary conditions

This paper is devoted to the study of uniform energy decay rates of solutions to the wave equation with Cauchy–Ventcel boundary conditions: