Finite element approximation of a nonlinear cross-diffusion population model

Summary. We consider a fully discrete finite element approximation of the nonlinear cross-diffusion population model: Find u_i, the population of the i^th species, i=1 and 2, such that where ji and g_i(u₁,u₂):=(_i–_iiu_i–_iju_j)u_i. In the above, the given data is as follows: v is an environmental potential, c_i, a_i are diffusion coefficients, b_i are transport coefficients, _i are the intrinsic growth rates, and _ii are intra-specific, whereas _ij, ij, are interspecific competition coefficients. In addition to showing well-posedness of our approximation, we prove convergence in space dimensions d3. Finally some numerical experiments in one space dimension are presented.Mathematics Subject Classification (2000): 65M60, 65M12, 35K55, 92D25Acknowledgements. Part of this work was carried out while the authors participated in the 2003 programme {\it Computational Challenges in Partial Differential Equations} at the Isaac Newton Institute, Cambridge, UK.     

Discrete Spectrum of Differential Operators in Spectral Gaps under Nonnegative Perturbations of High Order

Let A be a self-adjoint elliptic second-order differential operator, let (, ) be an inner gap in the spectrum of A, and let B(t) = A + tW ^* W, where W is a differential operator of higher order. Conditions are obtained under which the spectrum of the operator B(t) in the gap (, ) is either discrete, or does not accumulate to the right-hand boundary of the spectral gap, or is finite. The quantity N(, A, W, ), (, ), > 0 (the number of eigenvalues of the operator B(t) passing the point (, ) as t increases from 0 to ) is considered. Estimates of N(, A, W, ) are obtained. For the perturbation W ^* W of a special form, the asymptotics of N(, A, W, ) as + is given. Bibliography: 5 titles.     

On Differences of Semicubical Powers

For X,Y,>0, let and define I ₈(X,Y,) to be the cardinality of the set. In this paper it is shown that, for >0, Y ²/X ³=O(), =O(Y ³/X ³) and X=O (Y ²), one has I ₈(X,Y,)=O(X ² Y ²+X min (X ^{3/2} Y ³, X ^{11/2} Y ^{–1})+X min (^{1/3} X ² Y ³, X ^{14/3} Y ^{1/3})), with the implicit constant depending only on . There is a brief report on an application of this that leads, by way of the Bombieri-Iwaniec method for exponential sums, to some improvement of results on the mean squared modulus of a Dirichlet L-function along a short interval of its critical line.     

On a Test Statistic for Linear Trend

Let {W(s)} _s 0 be a standard Wiener process. The supremum of the squared Euclidian norm Y (t)², of the R²-valued process Y(t)=(1/t W(t), {12/t ³ int_{0 t} s dW (s)– {3/t} W(t)), t [, 1], is the asymptotic, large sample distribution, of a test statistic for a change point detection problem, of appearance of linear trend. We determine the asymptotic behavior P {sup _t [, 1] Y(t)² > u as u , of this statistic, for a fixed (0,1), and for a moving = (u) 0 at a suitable rate as u . The statistical interest of our results lie in their use as approximate test levels.     

Weak Sequential Completeness of β-duals of Double Sequence Spaces

We consider dual pairs E,E ⁽⁾ of double sequence spaces E and E ⁽⁾, where E ⁽⁾ is the -dual space of E with respect to the -convergence of double sequences for = p (Pringsheim convergence), bp (bounded p-convergence) and r (regular convergence). Motivated by Boos, Fleming and Leiger [3], we introduce two oscillating properties (signed P_OSCP(k), k {1,2}) for a double sequence space E such that the signed P_OSCP(1) guarantees the (E ^(p), E)-sequential completeness of E ^(p), whereas the signed P_OSCP(2) implies the equalities E ^(r) = E ^(bp) = E ^(p) and the (E ⁽⁾, E)-sequentialcompleteness of E ⁽⁾ for = bp and r.     

Variational problems in SBV and image segmentation

We show how it is possible to prove the existence of solutions of the Mumford-Shah image segmentation functional F(u,K) = _\K [u² + (u – g)²]dx + ^{n – 1}(K), u W ^1,2(\K), K closed in .We use a weak formulation of the minimum problem in a special class SBV() of functions of bounded variation. Moreover, we also deal with the regularity of minimizers and the approximation of F by elliptic functionals defined on Sobolev spaces. In this paper, we have collected the main results of Ambrosio and others.     

On Paley-Wiener and Hardy theorems for <Emphasis Type="Italic">NA</Emphasis> groups

Let N be a H-type group and let S=NA be an one dimensional solvable extension of N. For the Helgason Fourier transform on S we prove the following analogue of Hardys theorem. Let (,Y,Z) stand for the Helgason Fourier transform of f and let h denote the heat kernel associated to the Laplace-Beltrami operator. Suppose a function f on S satisfies the conditions |f(x)| ch (x) and for all xS, where being the dimension of the centre of N. Then f=0 or f=ch depending on whether < or =. We also establish a stronger version of Hardys theorem and a Paley-Wiener theorem. These are generalisations of the corresponding results for rank one symmetric spaces of noncompact type.     

Darstellung einer Ordnung von Maßen durch Stoppzeiten

Summary Given a stochastic matrixP on the state spaceI an ordering for measures inI can be defined in the following way: iff(f)(f) for allf in a sufficiently rich subcone of the cone of positiveP-subharmonic functions. It is shown that, if, are probability measures with , then in theP-process (X _n)_n0 having as initial distribution there exists a stopping time such thatX is distributed according to. In addition, can be chosen in such a way, that for every positive subharmonicf with(f)< the submartingale (f(X _n))_n0 is uniformly integrable.     

Convex independent sets and 7-holes in restricted planar point sets

For a finite setA of points in the plane, letq(A) denote the ratio of the maximum distance of any pair of points ofA to the minimum distance of any pair of points ofA. Fork>0 letc (k) denote the largest integerc such that any setA ofk points in general position in the plane, satisfying for fixed , contains at leastc convex independent points. We determine the exact asymptotic behavior ofc (k), proving that there are two positive constants=(), such thatk ^1/3c (k)k ^1/3. To establish the upper bound ofc (k) we construct a set, which also solves (affirmatively) the problem of Alonet al. [1] about the existence of a setA ofk points in general position without a 7-hole (i.e., vertices of a convex 7-gon containing no other points fromA), satisfying . The construction uses Horton sets, which generalize sets without 7-holes constructed by Horton and which have some interesting properties.     

Supplementing Radicals and Decompositions of Near-Rings

If is a radical of near-rings and is its supplementing radical, then (N)(N) N. We address the issue when (N) (N) = N holds. In the variety F of near-rings in which the constants form an ideal, the assignment c: N N_c is a hereditary Kurosh–Amitsur radical, _c is characterized in terms of distributors and criteria are given for the decomposition N = c(N) c(N). In the subvariety A of all abstract affine near-rings, assigning the maximal torsion ideal (N) is a hereditary Kurosh–Amitsur radical. If such near-rings N A satisfy dcc on principal right ideals, then N splits into a direct sum N = (N) (N) where the additive group of (N) is torsionfree and divisible. Dropping dcc on principal right ideals, an ``essential" decomposition result is proved.     

Sharp tridiagonal pairs

Let denote a field and let V denote a vector space over with finite positive dimension. We consider a pair of -linear transformations A:V→V and A^*:V→V that satisfies the following conditions: (i) each of A,A^* is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A^*V_iV_i-1+V_i+V_i+1 for 0id, where V_-1=0 and V_d+1=0; (iii) there exists an ordering of the eigenspaces of A^* such that for 0iδ, where and ; (iv) there is no subspace W of V such that AWW, A^*WW, W≠0, W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0id the dimensions of coincide. We say the pair A,A^* is sharp whenever dimV₀=1. A conjecture of Tatsuro Ito and the second author states that if is algebraically closed then A,A^* is sharp. In order to better understand and eventually prove the conjecture, in this paper we begin a systematic study of the sharp tridiagonal pairs. Our results are summarized as follows. Assuming A,A^* is sharp and using the data we define a finite sequence of scalars called the parameter array. We display some equations that show the geometric significance of the parameter array. We show how the parameter array is affected if Φ is replaced by or or . We prove that if the isomorphism class of Φ is determined by the parameter array then there exists a nondegenerate symmetric bilinear form , on V such that Au,v=u,Av and A^*u,v=u,A^*v for all u,vV.     

Arithmetic distance on compact symmetric spaces

Let M be a compact Riemannian symmetric space. Then M=G/K, where G is the identity component of the isometry group of M and K is the isotropy subgroup of G at a point. In 1965 Nagano studied and classified the geometric transformation groups of compact symmetric spaces. Roughly speaking they are larger groups L that act on M, (i) G/L; (ii) L is a Lie transformation group acting effectively on M; (iii) L preserves the symmetric structure of M; and (iv) L is simple.Using Helgason spheres, S(), the minimal totally geodesic spheres in a compact irreducible symmetric space, we define an arithmetic distance for compact irreducible symmetric spaces and prove: THEOREM. Let M=G ^p(K ⁿ ), K=, H, or R, or M=AI(n), of rank greater that 1 and dimension greater that 3, let L be the geometric transformation group of M. Let L={: MM: is a diffeomorphism and preserves arithmetic distance}. Then L=L     

On (-<Emphasis Type="Italic">P</Emphasis> ⋅ <Emphasis Type="Italic">P</Emphasis>)-constant deformations of Gorenstein surface singularities

Let : X T be a small deformation of a normal Gorenstein surface singularity X ₀ = ^-1(0) over the complex number field . Suppose that T is a neighborhood of the origin of and that X ₀ is not log-canonical. We show that if a topological invariant -P _t P _t of X _t = ^-1(t) is constant, then, after a suitable finite base change, admits a simultaneous resolution f : M X which induces a locally trivial deformation of each maximal string of rational curves at an end of the exceptional set of M ₀ X ₀; in particular, if X ₀ has a star-shaped resolution graph, then admits a weak simultaneous resolution (in other words, is an equisingular deformation).     

Regularity and Unit-regularity of Generalized Semigroups of Linear Transformations

Let V and W be vector spaces over a division ring D and L_D (V, W) the set of all linear transformations from V into W. For L_D(W, V), let (L_D (V, W), ) denote the semigroup L_D (V, W) with the operation * defined by * = for all , L_D(V, W). By a unit-regular semigroup we mean a semigroup S with identity having the property that for each a S, a = aua for some unit u S. The main purpose of this paper is to prove the following statements. The semigroup (L_D(V, W), ) is regular if and only if V = {0}, W = {0} or is an isomorphism from W onto V. The semigroup (L_D (V, W), ) is unit-regular if and only if (i) V = {0}, (ii) W = {0} or (iii) is an isomorphism from W onto V and dim_D V .     

A relation between the zeros of an <Emphasis Type="Italic">L</Emphasis>-function belonging to the Selberg class and the zeros of an associated <Emphasis Type="Italic">L</Emphasis>-function twisted by a Dirichlet character

Let F (s) be a function belonging to the Selberg class. For a primitive Dirichlet character , we can define the -twist F(s) of F (s). If F(s) also belongs to the Selberg class and satisfies some other conditions then there is a relation between the zeros of F (s) and the zeros F(s). Further we give an operator theoretic interpretation of this relation according to A. Connes study.Received: 5 January 2004     

Quasi-conformal mapping theorem and bifurcations

LetH be a germ of holomorphic diffeomorphism at 0 . Using the existence theorem for quasi-conformal mappings, it is possible to prove that there exists a multivalued germS at 0, such thatS(ze ²ⁱ )=HS(z) (1). IfH is an unfolding of diffeomorphisms depending on (,0), withH ₀=Id, one introduces its ideal . It is the ideal generated by the germs of coefficients (a _i (), 0) at 0 ^k , whereH (z)–z=a _i ()z ⁱ . Then one can find a parameter solutionS (z) of (1) which has at each pointz ₀ belonging to the domain of definition ofS ₀, an expansion in seriesS (z)=z+b _i ()(z–z ₀) ⁱ with , for alli.This result may be applied to the bifurcation theory of vector fields of the plane. LetX be an unfolding of analytic vector fields at 0 ² such that this point is a hyperbolic saddle point for each . LetH (z) be the holonomy map ofX at the saddle point and its associated ideal of coefficients. A consequence of the above result is that one can find analytic intervals , , transversal to the separatrices of the saddle point, such that the difference between the transition mapD (z) and the identity is divisible in the ideal . Finally, suppose thatX is an unfolding of a saddle connection for a vector fieldX ₀, with a return map equal to identity. It follows from the above result that the Bautin ideal of the unfolding, defined as the ideal of coefficients of the difference between the return map and the identity at any regular pointz, can also be computed at the singular pointz=0. From this last observation it follows easily that the cyclicity of the unfoldingX , is finite and can be computed explicity in terms of the Bautin ideal.Dedicated to the memory of R. Mañé     

On the speed of convergence in the central limit theorem of log-likelihood ratio processes

Let , the parameter space, be an open subset ofR ^k ,k1. For each , let the r.v.'sX _n ,n=1, 2,... be defined on the probability space (X, P ) and take values in (S,S,L) whereS is a Borel subset of a Euclidean space andL is the -field of Borel subsets ofS. ForhR _k and a sequence of p.d. normalizing matrices _n = _n ^{k × k} (₀ set _n ^* = ^* = ₀ + _n h, where ₀ is the true value of , such that ^*, . Let _n (^*, _*)( be the log-likelihood ratio of the probability measure with respect to the probability measure , whereP _n is the restriction ofP over _n = (X ₁,X ₂,...,X _n . In this paper we, under a very general dependence setup obtain a rate of convergence of the normalized log-likelihood ratio statistic to Standard Normal Variable. Two examples are taken into account.     

A contribution to shape theory for the ising model at low temperature

Summary We consider low temperature limits of Gibbs states of the ferromagnetic nearest-neighbour Ising Hamiltonian in the positive orthant of the lattice ^d ,d=1, 2,..., under a negative boundary condition and a small positive external fieldh that decreases linearly with the temperatureT. It is shown that positive and negative spins are separated by a staircase-shaped random boundary. Its explicit distribution is computed in the case that the ratio =h/T exceeds some positive ₀. For < ₀, our results do not rule out infinite negative areas.     

Convexity of Nonlinear Image of a Small Ball with Applications to Optimization

Let f: XY be a nonlinear differentiable map, X,Y are Hilbert spaces, B(a,r) is a ball in X with a center a and radius r. Suppose f (x) is Lipschitz in B(a,r) with Lipschitz constant L and f (a) is a surjection: f (a)X=Y; this implies the existence of >0 such that f (a)^* yy, yY. Then, if r,/(2L), the image F=f(B(a,)) of the ball B(a,) is convex. This result has numerous applications in optimization and control. First, duality theory holds for nonconvex mathematical programming problems with extra constraint x–a. Special effective algorithms for such optimization problems can be constructed as well. Second, the reachability set for small power control is convex. This leads to various results in optimal control.