diffeomorphism type of the berger space so 5 so 3 we compute the eells kuiper invariant of the berger manifold so 5 so 3 and determine that it is diffeomorphic to the total space of an s 3 bundle over s 4 this answers a question raised by k grove and w ziller

on the topology of positively curved 4 manifolds with symmetry soit m une variete compacte a courbure positive orientable a 4 dimensions si m a un champ de vecteurs de killing non trivial alors m est homeomorphe a s 4 ou cp 2

a diffeomorphism classification of 7 dimensional homogeneous einstein manifolds with mathrm su 3 times mathrm su 2 times mathrm u 1 symmetry on donne des exemples qui montrent que des espaces homogenes homeomorphes ne sont pas necessairement diffeomorphes

a corrected decoupled scheme for chemotaxis models the main purpose of this paper is to present a new corrected decoupled scheme combined with a spatial finite volume method for chemotaxis models first we derive the scheme for a parabolicelliptic chemotaxis model arising in embryology we then establish the existence and uniqueness of the numerical solution and we prove that it converges to a corresponding weak solution for the studied model in the last section several numerical tests are presented by applying our approach to a number of chemotaxis systems the obtained numerical results demonstrate the efficiency of the proposed scheme and its effectiveness to capture different forms of spatial patterns

dimensions maximal growth sites and optimization in the dielectric breakdown model we study the growth of fractal clusters in the dielectric breakdown model dbm by means of iterated conformal mappings in particular we investigate the fractal dimension and the maximal growth site measured by the hoelder exponent ensuremath alpha min as a function of the growth exponent ensuremath eta of the dbm model we do not find evidence for a phase transition from fractal to nonfractal growth for a finite ensuremath eta value simultaneously we observe that the limit of nonfractal growth d ensuremath rightarrow 1 is consistent with ensuremath alpha min ensuremath rightarrow 1 2 finally using an optimization principle we give a recipe on how to estimate the effective value of ensuremath eta from temporal growth data of fractal aggregates

notes on finite difference schemes to a parabolic elliptic system modelling chemotaxis we consider numerical solutions to a nonlinear parabolic elliptic system in mathematical biology an important property of solutions to the system is the conservation of l 1 norm applying the upwind technique we make a finite difference scheme satisfying the conservation of a discrete l 1 norm we also describe a remark on time discretization from the view point of lyapunov s property which is another important feature of the system several numerical examples clarify the nature of our scheme

new examples of manifolds with strictly positive curvature berger 3 wallach 10 and berard bergery 2 have classified all simply connected smooth manifolds which allow a homogeneous riemannian metric of strictly positive curvature besides the rank one symmetric spaces there exist five exceptional manifolds and an infinite series of 7 manifolds of distinct homotopy type which have been studied by aloft and wallach 1 these are diffeomorphic to mpq su 3 upq where p q are positive integers and upq is the one parameter subgroup of diagonal matrices

kreck stolz invariants for quaternionic line bundles we generalise the kreck stolz invariants s2 and s3 by defining a new invariant the t invariant for quaternionic line bundles over closed spin manifolds m of dimension 4k 1 with h 3 m q h 4 m q 0 the t invariant classifies closed smooth oriented 2 connected rational homology 7 spheres up to almost diffeomorphism and detects exotic homeomorphisms between such manifolds the t invariant also provides information about quaternionic line bundles over a fixed manifold and we use it to give a new proof of a theorem of feder and gitler about the values of the second chern classes of quaternionic line bundles over hp k the t invariant for s 4k 1 is closely related to the adams e invariant on the 4k 5 stem

finite volume methods for a keller segel system discrete energy error estimates and numerical blow up analysis we consider the finite volume approximation for a non linear parabolic elliptic system which describes the aggregation of slime molds resulting from their chemotactic features called a simplified keller segel system first we present a linear finite volume scheme that satisfies both positivity and mass conservations which are important features of the original system we derive some inequalities on the discrete free energy then under some assumptions on the regularity of solution admissible mesh and a priori estimates of the discrete solution we establish error estimates in l p lp norm with a suitable p 2 p 2 for the two dimensional case in the last part of this paper we restrict our attention to the radially symmetric solution of chemotaxis system and we derive some inequalities concerned with the blow up phenomenon of numerical solution several numerical experiments are presented to verify the theoretical results

surgery and duality surgery as developed by browder kervaire milnor novikov sullivan wall and others is a method for comparing homotopy types of topological spaces with difieomorphism or homeomorphism types of manifolds of dimension 5 in this paper a modiflcation of this theory is presented where instead of flxing a homotopy type one considers a weaker information roughly speaking one compares n dimensional compact manifolds with topological spaces whose k skeletons are flxed where k is at least n 2 a particularly attractive example which illustrates the concept is given by complete intersections by the lefschetz hyperplane theorem a complete intersection of complex dimension n has the same n skeleton as n and one can use the modifled theory to obtain information about their difieomorphism type although the homotopy classiflcation is not known the theory reduces this classiflcation result to the determination of complete intersections in a certain bordism group this was under certain restrictions carried out in tr the restrictions are if d d1 dr is the total degree of a complete intersection x n1 dr of complex dimension n then the assumption is that for all primes p with p pi1 n 1 the total degree d is divisible by p 2n 1 2pi1 1 theorem a two complete intersections x n

multifractal structure of the harmonic measure of diffusion limited aggregates the method of iterated conformal maps allows one to study the harmonic measure of diffusion limited aggregates with unprecedented accuracy we employ this method to explore the multifractal properties of the measure including the scaling of the measure in the deepest fjords that were hitherto screened away from any numerical probing we resolve probabilities as small as 10 235 and present an accurate determination of the generalized dimensions and the spectrum of singularities we show that the generalized dimensions dq are infinite forq q where q is of the order of 20 2 in the language of f a this means that a max is finite the f a curve loses analyticity the phenomenon of phase transition at a max and a finite value of f a max we consider the geometric structure of the regions that support the lowest parts of the harmonic measure and thus offer an explanation for the phase transition rationalizing the value of q and f a max we thus offer a satisfactory physical picture of the scaling properties of this multifractal measure

stochastic particle approximation for measure valued solutions of the 2d keller segel system we construct an approximation to the measure valued global in time solutions to the patlak keller segel model in 2d based on systems of stochastic interacting particles the advantage of our approach is that it reproduces the well known dichotomy in the qualitative behavior of the system and moreover captures the solution even after the possible blow up events we present a numerical method based on this approach and show some numerical results moreover we make a first step toward the convergence analysis of our scheme by proving the convergence of the stochastic particle approximation for the keller segel model with a regularized interaction potential the proof is based on a bbgky like approach for the corresponding particle distribution function

laplacian networks growth local symmetry and shape optimization inspired by river networks and other structures formed by laplacian growth we use the loewner equation to investigate the growth of a network of thin fingers in a diffusion field we first review previous contributions to illustrate how this formalism reduces the network s expansion to three rules which respectively govern the velocity the direction and the nucleation of its growing branches this framework allows us to establish the mathematical equivalence between three formulations of the direction rule namely geodesic growth growth that maintains local symmetry and growth that maximizes flux into tips for a given amount of growth surprisingly we find that this growth rule may result in a network different from the static configuration that optimizes flux into tips

laplacian growth as one dimensional turbulence abstract a new model of laplacian stochastic growth is formulated using conformal mappings the model describes two growth regimes stable and turbulent separated by a sharp phase transition the first few fourier components of the mapping define the web an envelope of the cluster the web is used to study the transition and the dynamics of large scale features of the cluster characterized by evolution from macro to micro scales also we derive scaling laws for the cluster size

blow up dynamics for the aggregation equation with degenerate diffusion abstract we study radially symmetric finite time blow up dynamics for the aggregation equation with degenerate diffusion u t u m u k u in r d where the kernel k x is of power law form x depending on m d and the initial data the solution exhibits three kinds of blow up behavior self similar with no mass concentrated at the core imploding shock solution and near self similar blow up with a fixed amount of mass concentrated at the core computations are performed for different values of m d and using an arbitrary lagrangian eulerian method with adaptive mesh refinement

regular article finite difference schemes for u t x α δg δu that inherit energy conservation or dissipation property we propose a new procedure for designing by rote finite difference schemes that inherit energy conservation or dissipation property from nonlinear partial differential equations such as the korteweg de vries kdv equation and the cahn hilliard equation the most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts which implies that the inherited properties are satisfied exactly since the inherited properties are kept even if the time mesh size changes in the time evolution process we can use some appropriate time mesh adaptive methods to obtain numerical solutions through the derived schemes because of these properties the derived schemes are expected to be numerically stable and yield solutions converging to pde solutions and sufficiently flexible to treat the inheritance of the energy conservation and dissipation properties are verified numerically for the kdv equation and the cahn hilliard equation