Spatial decay in a cross-diffusion problem

In this paper, the authors investigate the decay of end effectsfor a cross-diffusion problem defined on a semi-infinite cylindricalregion. With homogeneous Dirichlet or Neumann conditions prescribedon the lateral surface of the cylinder, it is shown that forfixed finite time and under certain restrictions on the coefficients,solutions decay point-wise as the distance d from the finiteend of the cylinder tends to infinity at least of order e^–kd2.Under less restrictive conditions, it is shown that solutionsdecay in L₂ at least as fast as e^–kd. In both cases, kis a computable function of time.     

Jacobiennes de Courbes Algebriques de Genre 2 et 3 de Grand Rang Sur Q

Infinite families of curves are constructed of genus 2 and 3over Q whose jacobians have high rank over Q. More precisely,if E is an elliptic curve with rank at least r over Q, an infinitefamily of curves are constructed of genus 2 whose jacobianshave rank at least r+4 over Q, and, under certain conditions,an infinite family of curves are constructed of genus 3 whosejacobians have rank at least 2r over Q. On specialisation, afamily of curves are obtained of genus 2 whose jacobians haverank at least 27 and a family of curves are obtained of genus3 whose jacobians have rank at least 26; one of these has rankat least 42.     

Lacunary Polynomials, Multiple Blocking Sets and Baer Subplanes

New lower bounds are given for the size of a point set in aDesarguesian projective plane over a finite field that containsat least a prescribed number s of points on every line. Thesebounds are best possible when q is square and s is small comparedwith q. In this case the smallest set is shown to be the unionof disjoint Baer subplanes. The results are based on new resultson the structure of certain lacunary polynomials, which canbe regarded as a generalization of Rédei's results inthe case when the derivative of the polynomial vanishes.     

Conditions for the Solvability of Systems of Two and Three Additive Forms Over p-Adic Fields

This paper is concerned with non-trivial solvability in p-adicintegers of systems of two and three additive forms. Assumingthat the congruence equation ax^k + by^k + cz^k d (modp) has asolution with xyz 0(modp) we have proved that any system oftwo additive forms of odd degree k with at least 6k + 1 variables,and any system of three additive forms of odd degree k withat least 14k + 1 variables always has non-trivial p-adic solutions,provided p does not divide k. The assumption of the solubilityof the congruence equation above is guaranteed for example ifp > k⁴. In the particular case of degree k = 5 we have proved the followingresults. Any system of two additive forms with at least n variablesalways has non-trivial p-adic solutions provided n 31 and p> 101 or n 36 and p > 11. Furthermore any system of threeadditive forms with at least n variables always has non-trivialp-adic solutions provided n 61 and p > 101 or n 71 andp > 11. 2000 Mathematics Subject Classification 11D72, 11D79.     

A Minimax Principle for the Eigenvalues in Spectral Gaps

A minimax principle is derived for the eigenvalues in the spectralgap of a possibly non-semibounded self-adjoint operator. Itallows the nth eigenvalue of the Dirac operator with Coulombpotential from below to be bound by the nth eigenvalue of asemibounded Hamiltonian which is of interest in the contextof stability of matter. As a second application it is shownthat the Dirac operator with suitable non-positive potentialhas at least as many discrete eigenvalues as the Schrödingeroperator with the same potential.     

Frames and the Feichtinger conjecture

We show that the conjectured generalization of the Bourgain-Tzafriri restricted-invertibility theorem is equivalent to the conjecture of Feichtinger, stating that every bounded frame can be written as a finite union of Riesz basic sequences. We prove that any bounded frame can at least be written as a finite union of linearly independent sequences. We further show that the two conjectures are implied by the paving conjecture. Finally, we show that Weyl-Heisenberg frames over rational lattices are finite unions of Riesz basic sequences.

     

Optimal policy for a general repair replacement model: average reward case

For a general repair replacement model, we study two types ofreplacement policy.Replacement policy T replaces the systemat time T since the installation or last replacement, whilereplacement policy N replaces the system at the time of Nthfailure. Let T^* and N^* be the optimal among all policies T andN respectively. Under the expected average reward criterion,then we show that the optimal policy N^* is at least as goodas the optimal policy T^*. Furthermore, for a monotone processmodel, we determine the optimal policy N^* explicitly throughtwo different approaches.     

A note on an SOR-like method for augmented systems

Golub et al. (2001, BIT, 41, 71–85) gave a generalizedsuccessive over-relaxation method for the augmented systems.In this paper, the connection between the SOR-like method andthe preconditioned conjugate gradient (PCG) method for the augmentedsystems is investigated. It is shown that the PCG method isat least as accurate (fast) as the SOR-like method. Numericalexamples demonstrate that the PCG method is much faster thanthe SOR-like method.     

Long Snakes in Powers of the Complete Graph with an Odd Number of Vertices

In [5] Abbott and Katchalski ask if there exists a constantc < 0 such that for every d 2 there is a snake (cycle withoutchords) of length at least c3^d in the product of d copies ofthe complete graph K₃. We show that the answer to the abovequestion is positive, and that in general for any odd integern there is a constant c_n such that for every d 2 there is asnake of length at least c_n n^d in the product of d copies ofthe complete graph K_n.     

Finite Subsequences of Shift Register Sequences

In a stream cipher a cryptogram is produced from a binary datastream by modulo-2-adding it to a keystream sequence. The securityof the system relies on the inability of an interceptor to determinethis keystream sequence. One obvious requirement for such asystem is that there should be sufficiently many possibilitiesfor the keystream sequence that the interceptor cannot possiblytry them all. In this paper we consider the likelihood of an interceptor beingable to decipher the cryptogram correctly even though he maybe trying the wrong keystream sequence. This possibility arisesbecause the length of any particular message is likely to beconsiderably shorter than the period of the keystream sequence,and thus only a comparatively small section of the keystreamsequence is used. Hence, if the interceptor tries a sequencewhich intersects (i.e. agrees) with the keystream sequence inthe appropriate positions, he will deduce the message correctly. A number of the standard methods for generating keystream sequencesuse shift registers as ‘building blocks’. So welook in considerable detail at the number of intersections (ofvarious lengths) for sequences generated by two different shiftregisters. We also show that if a keystream sequence has linearequivalence n, then the local linear equivalence of any subsequenceof length at least 2n is n. This means that if the message haslength at least 2n and the keystream sequence has linear equivalencen, then there is no other sequence of linear equivalence lessthan n+1 which can be used to decipher correctly.     

On the Representations of a Number as the Sum of Four Fifth Powers

It is known from Vaughan and Wooley's work on Waring's problemthat every sufficiently large natural number is the sum of atmost 17 fifth powers [13]. It is also known that at least sixfifth powers are required to be able to express every sufficientlylarge natural number as a sum of fifth powers (see, for instance,[5, Theorem 394]). The techniques of [13] allow one to showthat almost all natural numbers are the sum of nine fifth powers.A problem of related interest is to obtain an upper bound forthe number of representations of a number as a sum of a fixednumber of powers. Let R(n) denote the number of representationsof the natural number n as a sum of four fifth powers. In thispaper, we establish a non-trivial upper bound for R(n), whichis expressed in the following theorem.     

Strong Jordan Separation and Applications to Rigidity

We prove that simple, thick hyperbolic P-manifolds of dimensionat least three exhibit Mostow rigidity. We also prove a quasi-isometryrigidity result for the fundamental groups of simple, thickhyperbolic P-manifolds of dimension at least three. The keytool in the proof of these rigidity results is a strong formof the Jordan separation theorem, for maps from Sⁿ Sⁿ⁺¹ whichare not necessarily injective.     

The largest prime factor of X3+2

The largest prime factor of X³+2 was investigated in 1978 byHooley, who gave a conditional proo that it is infinitely oftenat least as large as X¹+, with a certain positive constant .It is trivial to obtain such a result with =0. One may thinkof Hooley's result as an approximation to the conjecture thatX³+2 is infinitely often prime. The condition required by Hooley,his R^* conjecture, gives a non-trivial bound for short Ramanujan–Kloostermansums. The present paper gives an unconditional proof that thelargest prime factor of X³+2 is infinitely often at least aslarge as X¹⁺, though with a much smaller constant than thatobtained by Hooley. In order to do this we prove a non-trivialbound for short Ramanujan–Kloosterman sums with smoothmodulus. It is also necessary to modify the Chebychev method,as used by Hooley, so as to ensure that the sums that occurdo indeed have a sufficiently smooth modulus. 2000 MathematicsSubject Classification: 11N32.     

The Strength of a Fabric

Consider fabrics in which every warp (and weft) strand passesover and under at least k weft (warp) strands. There is a 4kby 4k fabric of this kind that does not hang together. We provethat any n by n fabric of this kind, where n < 4k, does hangtogether. Moreover, we introduce a measure of how well a fabrichangs together, by defining the strength of a fabric, and twotheorems are proved that extend the preceding results.     

Cyclic Spaces for Grassmann Derivatives and Additive Theory

Let A be a finite subset of Z_p (where p is a prime). Erdösand Heilbronn conjectured (1964) that the set of sums of the2-subsets of A has cardinality at least min(p, 2|A| —3). We show here that the set of sums of all m-subsets of Ahas cardinality at least min {p,m(|A| — m)+ 1}. In particular,we answer affirmatively the above conjecture. We apply thisresult to the problem of finding the smallest n such that forevery subset 5 of cardinality n and every xZ_p there is a subsetof S with sum equal to x. On this last problem we improve theknown results due to Erdös and Heilbronn and to Olson. The above result will be derived from the following generalproblem on Grassmann spaces. Let F be a field and let V be afinite dimensional vector space of dimension d over F. Let pbe the characteristic of F in nonzero characteristic and otherwise. Let Df be the derivative of a linear operatorfon V, restrictedto the mth Grassmann space ^mV. We show that there is a cyclicsubspace for the derivative with dimension at least min {p,m(n–m)+ 1}, where n is the maximum dimension of the cyclic subspacesof f. This bound is sharp and is reached when f has d distincteigenvalues forming an arithmetic progression.     

Curves which Intersect Lines in Finite Sets

We study closed subsets in the plane which intersect each linein at least m points and at most n points, for which we tryto minimize the difference n – m. It is known that m cannotbe equal to n. The results in this paper show that for everyeven number n there exist closed sets in the plane for whichm = n – 2.     

Quasiisometries between negatively curved Hadamard manifolds

Let H₁, H₂ be the universal covers of two compact Riemannianmanifolds (of dimension not equal to 4) with negative sectionalcurvature. Then every quasiisometry between them lies at a finitedistance from a bilipschitz homeomorphism. As a consequence,every self-quasiconformal map of a Heisenberg group (equippedwith the Carnot metric and viewed as the ideal boundary of complexhyperbolic space) of dimension at least 5 extends to a self-quasiconformalmap of the complex hyperbolic space.     

The Berge-Martinet Constant and Slopes of Siegel Cusp Forms

We give a theoretical lower bound for the slope of a Siegelmodular cusp form that is as least as good as Eichler's lowerbound. In degrees n = 5, 6 and 7 we show that our new boundis strictly better. In the process we find the forms of smallestdyadic trace on the perfect core for ranks n 8. In degreesn = 5, 6 and 7 we settle the value of the generalized Hermiteconstant introduced by Bergé and Martinet and find all dual-critical pairs. 2000 MathematicsSubject Classification 11H55 (11F46).     

THE DISTRIBUTION OF INTEGERS WITH AT LEAST TWO DIVISORS IN A SHORT INTERVAL

We estimate the density of integers which have more than onedivisor in an interval (y, z] with z y + y/(log y)^{log 4 –1}. As a consequence, we determine the precise range of z suchthat most integers which have at least one divisor in (y, z]have exactly one such divisor.