Pattern formation in a 2D simple chemical system with general orders of autocatalysis and decay

In this paper, we investigate pattern formation in a coupledsystem of reaction–diffusion equations in two spatialdimensions. These equations arise as a model of isothermal chemicalautocatalysis with termination in which the orders of autocatalysisand termination, m and n, respectively, are such that 1 <n < m. We build on the preliminary work by Leach & Wei(2003, Physica D, 180, 185–209) for this coupled systemin one spatial dimension, by presenting rigorous stability analysisand detailed numerical simulations for the coupled system intwo spatial dimensions. We demonstrate that spotty patternsare observed over a wide parameter range.     

Long-Time Behavior of Solutions of the Fast Diffusion Equations with Critical Absorption Terms

This paper is devoted to the long-time behavior of solutionsto the Cauchy problem of the porous medium equation u_t = (u^m)– u^p in Rⁿ x (0,) with (1 – 2/n)₊ < m < 1and the critical exponent p = m + 2/n. For the strictly positiveinitial data u(x,0) = O(1 + |x|)^–k with n + mn(2 –n + nm)/(2[2 – m + mn(1 – m)]) k < 2/(1 –m), we prove that the solution of the above Cauchy problem convergesto a fundamental solution of u_t = (u^m) with an additional logarithmicanomalous decay exponent in time as t .     

On Closed Sets with Convex Projections

A shadow of a subset A of Rⁿ is the image of A under a projectiononto a hyperplane. Let C be a closed nonconvex set in Rⁿ suchthat the closures of all its shadows are convex. If, moreover,there are n independent directions such that the closures ofthe shadows of C in those directions are proper subsets of therespective hyperplanes then it is shown that C contains a copyof R^n–2. Also for every closed convex set B ‘minimalimitations’ C of B are constructed, that is, closed subsetsC of B that have the same shadows as B and that are minimalwith respect to dimension.     

A characterization of finite soluble groups

Let G be a finite soluble group of order m and let w(x₁, ...,x_n) be a group word. Then the probability that w(g₁, ..., g_n)= 1 (where (g₁, ..., g_n) is a random n-tuple in G) is at leastp^–(m–t), where p is the largest prime divisor ofm and t is the number of distinct primes dividing m. This contrastswith the case of a non-soluble group G, for which Abérthas shown that the corresponding probability can take arbitrarilysmall positive values as n .     

On the Mislin Genus of Symplectic Groups

In this paper we give lower bounds for the Mislin genus of thesymplectic groups Sp(m). This result appears to be the exactanalogue of Zabrodsky's theorem concerning the special unitarygroups SU(n). It is achieved by the determination of the stablegenus of the quasi-projective quaternionic spaces QH(m), followingthe approach of McGibbon. It leads to a symplectic version ofZabrodsky's conjecture, saying that these lower bounds are infact the exact cardinality of the genus sets. The genus of Sp(2)is well known to contain exactly two elements. We show thatthe genus of Sp(3) has exactly 32 elements and see that theconjecture is true in these two cases. Independently, we also show that any homotopy type in the genusof Sp(m) fibers over the sphere S⁴m–1 with fiber in thegenus of Sp(m–1), and that any homotopy type in the genusof SU(n) fibers over the sphere S²n–1 with fiber in thegenus of SU(n–1). Moreover, these fibrations are principalwith respect to some appropriate loop structures on the fibers.These constructions permit us to produce particular spaces realizingthe lower bounds obtained. 2000 Mathematics Subject Classification55P60 (primary), 55P15, 55R35 (secondary)     

The cyclic Barzilai--Borwein method for unconstrained optimization

^** Email: dyh{at}lsec.cc.ac.cn^*** Email: hager{at}math.ufl.edu^**** Email: klaus.schittkowski{at}uni-bayreuth.de^***** Email: hzhang{at}math.ufl.edu In the cyclic Barzilai–Borwein (CBB) method, the sameBarzilai–Borwein (BB) stepsize is reused for m consecutiveiterations. It is proved that CBB is locally linearly convergentat a local minimizer with positive definite Hessian. Numericalevidence indicates that when m > n/2 3, where n is the problemdimension, CBB is locally superlinearly convergent. In the specialcase m = 3 and n = 2, it is proved that the convergence rateis no better than linear, in general. An implementation of theCBB method, called adaptive cyclic Barzilai–Borwein (ACBB),combines a non-monotone line search and an adaptive choice forthe cycle length m. In numerical experiments using the CUTErtest problem library, ACBB performs better than the existingBB gradient algorithm, while it is competitive with the well-knownPRP+ conjugate gradient algorithm.     

Decomposition strategies for large-scale continuous location-allocation problems

^** Email: nenad.mladenovic{at}brunel.ac.uk The continuous location–allocation problem requires findingsites for m new facilities in the plane in order to serve nusers such that the total transportation costs are minimized.Several heuristics have been developed to solve this well-studiedglobal optimization problem. However, little work has been doneusing decomposition techniques. In this paper, we examine afew new ways to decompose the location–allocation problem.These strategies are then embedded in a variable neighbourhooddecomposition search heuristic and tested on large instancesof this problem.     

Shapiro's Cyclic Sum

Shapiro's cyclic sum is defined by , If K is the cone in Rⁿ of points withnon-negative coordinates, it is shown that the minimum of Ein K is a fixed point of T², where T is the non-linear operatordefined by (Tx)_i = x_n–i+1/(x_n–i+2 + x_n–i+3)²for i = 1,2,...,n. It is conjectured that Tx = S^kx, where Sis the shift operator in Rⁿ, and a proof is given under someadditional hypotheses. One of the consequences is a simple proofthat at the minimum point, a_i(x) = a_n–i+1–k(x) fori = 1,2,...,n.     

Martingales in the Ok Corral

In the model of the OK Corral formulated by Williams and McIlroy[2]: ‘Two lines of gunmen face each other, there beinginitially m on one side, n on the other. Each person involvedis a hopeless shot, but keeps firing at the enemy until eitherhe himself is killed or there is no one left on the other side.’They are interested in the number S of survivors when the shootingceases, and the surprising result, for which they give bothnumerical and heuristic evidence, is that when m=n, [formula] as m (where E denotes expectation). It is the occurrence of this curious power of m, rather thanany application to real gunfights, which makes the Williams–McIlroyprocess of interest. The purpose of this paper is to give anessentially elementary proof of the fact that if m and n arenot too different, then S/m^3/4 has a simple asymptotic distributionthat leads at once to (1.1). 1991 Mathematics Subject Classification60F05.     

Bounds For Gradient Trajectories and Geodesic Diameter of Real Algebraic Sets

Let M Rⁿ be a connected component of an algebraic set ^–1(0),where is a polynomial of degree d. Assume that M is containedin a ball of radius r. We prove that the geodesic diameter ofM is bounded by 2rv(n)d(4d–5)^n–2, where v(n) =(1/2)((n+1)/2)(n/2)^–1.This estimate is based on the bound rv(n)d(4d–5)^n–2for the length of the gradient trajectories of a linear projectionrestricted to M. 2000 Mathematics Subject Classification 32Bxx,34Cxx (primary), 32Sxx, 14P10 (secondary).     

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.     

Depth of Higher Associated Graded Rings

The depth of the associated graded ring of the powers of anideal I of a local ring R is studied. It is proved that thedepth of the associated graded ring of Iⁿ is asymptoticallyconstant when n tends to infinity, and this value is characterizedin terms of Valabrega–Valla conditions of I^m for somelarge integer m 0. As a corollary, a generalization is obtainedof the 2-dimensional algebraic version of the Grauert–Riemenschneidervanishing theorem (due to Huckaba and Huneke) to ideals satisfyingthe second Valabrega–Valla condition. The positivenessof Hilbert coefficients is also studied, and Valabrega–Vallaconditions are linked to the vanishing of the cohomology groupsof the closed fiber of the blowing up of Spec(R) along the closedsub-scheme defined by I.     

A multigrid-type method for thin plate spline interpolation on a circle

We consider the problem of thin plate spline interpolation ton equally spaced points on a circle, where the number of datapoints is sufficiently large for work of O(n³ to be unacceptable.We develop an iterative multigrid-type method, each iterationcomprising ngrid stages, and n being an integer multiple of2^ngrid–1. We let the first grid, V₁ be the full set ofdata points, V say, and each subsequent (coarser) grid, V_k,k=2, 3,...,ngrid, contain exactly half of the data points ofthe preceding (finer) grid, these data points being equallyspaced. At each stage of the iteration, we correct our current approximationto the thin plate spline interpolant by an estimate of the interpolantto the current residuals on V_k, where the correction is constructedfrom Lagrange functions of interpolation on small local subsetsof p data points in V_k. When the coarsest grid is reached, however,then the interpolation problem is solved exactly on its q=n/2^ngrid–1points. The iterative process continues until the maximum residualdoes not exceed a specified tolerance. Each iteration has the effect of premultiplying the vector ofresiduals by an n x n matrix R, and thus convergence will dependupon the spectral radius, (R), of this matrix. We investigatethe dependence of the spectral radius on the values of n, p,and q. In all the cases we have considered, we find (R) <<1, and thus rapid convergence is assured.     

On the Algorithmic Complexity of Minkowski's Reconstruction Theorem

In 1903 Minkowski showed that, given pairwise different unitvectors µ₁, ..., µ_m in Euclidean n-space Rⁿ whichspan Rⁿ, and positive reals µ₁, ..., µ_m such that^m_i=1µ_iµ_i = 0, there exists a polytope P in Rⁿ, uniqueup to translation, with outer unit facet normals µ₁, ...,µ_m and corresponding facet volumes µ₁, ..., µ_m.This paper deals with the computational complexity of the underlyingreconstruction problem, to determine a presentation of P asthe intersection of its facet halfspaces. After a natural reformulationthat reflects the fact that the binary Turing-machine modelof computation is employed, it is shown that this reconstructionproblem can be solved in polynomial time when the dimensionis fixed but is #P-hard when the dimension is part of the input. The problem of ‘Minkowski reconstruction’ has variousapplications in image processing, and the underlying data structureis relevant for other algorithmic questions in computationalconvexity.     

Low-Dimensional Representations of Special Linear Groups in Cross Characteristics

The low-dimensional projective irreducible representations incross characteristics of the projective special linear groupPSL_n(q) are investigated. If n 3 and (n, q) (3,2), (3,4), (4,2), (4,3), all such representationsof the first degree (which is (qⁿ – q)/(q – 1) – with = 0 or 1) and the second degree (which is (qⁿ –1)/(q – 1) come from Weil representations. We show thatthe gap between the second and the third degree is roughly q^2n-4.1991 Mathematics Subject Classification: 20C20, 20C33.     

Properties of the Approximations of a Matrix which Lower its Rank

Let C be a real m ? n matrix. This paper considers the generalproperties of the best approximations E to C in an arbitrarynorm by matrices of rank r. The case in which rank C = n, r= n – 1, and ||?|| is a separable norm is dealt with separately.For this particular case, the form of every best approximationE and an estimate of the error ||C – E|| are given. Thepaper also investigates when, for this particular case, thebest approximation E is unique.     

Spines and Embeddings of n-Manifolds

Every compact, connected PL manifold Mⁿ, with MⁿØ, collapsesto a codimension-one subpolyhedron Q^n–1, called a spineof Mⁿ. The purpose of this paper is to prove that, if Q^n–1is appropriately chosen, one can reconstruct Mⁿ from Q^n–1,after taking the Cartesian product with an interval I=[0, 1].     

Hypersurfaces in a Unit Sphere Sn+1(1) with Constant Scalar Curvature

The paper considers n-dimensional hypersurfaces with constantscalar curvature of a unit sphere S^n–1(1). The hypersurfaceS^k(c₁)xS^n–k(c₂) in a unit sphere Sⁿ⁺¹(1) is characterized,and it is shown that there exist many compact hypersurfaceswith constant scalar curvature in a unit sphere Sⁿ⁺¹(1) whichare not congruent to each other in it. In particular, it isproved that if M is an n-dimensional (n > 3) complete locallyconformally flat hypersurface with constant scalar curvaturen(n–1)r in a unit sphere Sⁿ⁺¹(1), then r > 1–2/n,and (1) when r (n–2)/(n–1), if then M is isometric to S¹xS^n–1(c),where S is the squared norm of the second fundamental form ofM; (2) there are no complete hypersurfaces in Sⁿ⁺¹(1) with constantscalar curvature n(n–1)r and with two distinct principalcurvatures, one of which is simple, such that r = (n–2)/(n–1)and      

Chaotic Homeomorphisms of Rn, Lifted from Torus Homeomorphisms

We establish the existence of self-homeomorphisms of Rⁿ, n 2, which are chaotic in the sense of Devaney, preserve volumeand are spatially periodic. Moreover, we show that in the spaceof volume-preserving homeomorphisms of the n-torus with meanrotation zero, those with chaotic lifts to Rⁿ are dense, withrespect to the uniform topology. An application is given forfixed points of 2-dimensional torus homeomorphisms (Conley–Zehnder–FranksTheorem). 1991 Mathematics Subject Classification 54H20.     

On Mason's conjecture concerning interpolation by polynomials in z and z-1 on an annulus

A polynomial of degree n in z^–1 and n – 1 in z isdefined by an interpolation projection from the space A(N) of functions f analytic in thecircular annulus ^–1<|z| and continuous on its boundaries|z|=^–1, . The points of interpolation are chosen to coincidewith the n roots of zⁿ=–n and the n roots of zⁿ=^–n.We prove Mason's conjecture that the corresponding Lebesguefunction attains its maximal value on the inner circle. We alsoestimate the bound of the Lebesgue constant . It is proved that the following estimate for theoperator norm holds: where _n, is the Lebesgue constant of Gronwall for equally spacedinterpolation on a circle by a polynomial of degree n.