from logarithmic to subdiffusive polynomial fluctuations for internal dla and related growth models we consider a cluster growth model on zd called internal diffusion limited aggregation internal dla in this model random walks start at the origin one at a time and stop moving when reaching a site not occupied by previous walks it is known that the asymptotic shape of the cluster is spherical when dimension is 2 or more we prove that fluctuations with respect to a sphere are at most a power of the logarithm of its radius in dimension d 2 in so doing we introduce a closely related cluster growth model that we call the flashing process whose fluctuations are controlled easily and accurately this process is coupled to internal dla to yield the desired bound part of our proof adapts the approach of lawler bramson and griffeath on another space scale and uses a sharp estimate written by blachere in our appendix on the expected time spent by a random walk inside an annulus

morphological instability and dynamics of fronts in bacterial growth models with nonlinear diffusion depending on the growth condition bacterial colonies can exhibit different morphologies as argued by ben jacob et al there is biological and modeling evidence that a nonlinear diffusion coefficient of the type d b d 0 b k is a basic mechanism that underlies almost all of the patterns and generates a long wavelength instability we study a reaction diffusion system with a nonlinear diffusion coefficient and find that a unique planar traveling front solution exists whose velocity is uniquely determined by k and d d 0 d n where d n is the diffusion coefficient of the nutrient due to the fact that the bacterial diffusion coefficient vanishes when stackrel ensuremath rightarrow b 0 in the front solution b vanishes in a singular way as a result the standard linear stability analysis for fronts cannot be used we introduce an extension of the stability analysis that can be applied to singular fronts and use the method to perform a linear stability analysis of the planar bacteriological growth front we show that a nonlinear diffusion coefficient generates a long wavelength instability for kg0 and dl d c k we map out the region of stability in the d ensuremath k plane and determine the onset of stability that is given by d c k both for stackrel ensuremath rightarrow d 0 and stackrel ensuremath rightarrow k ensuremath infty the dynamics of the growth zone essentially reduces to that of a sharp interface problem that is reminiscent of a so called one sided growth problem where the growth velocity is proportional to the gradient of a diffusion field ahead of the interface the moving boundary approximation that we derive in these limits is quite accurate but surprisingly does not become a proper asymptotic theory in the strict mathematical sense in the limit stackrel ensuremath rightarrow d 0 due to lack of full separation of scales on all dynamically relevant length scales our linear stability analysis and sharp interface formulation will also be applicable to other examples of interface formation due to nonlinear diffusion like in porous media or in the problem of vortex motion in superconductors

tumor growth instability and the onset of invasion motivated by experimental observations we develop a mathematical model of chemotactically directed tumor growth we present an analytical study of the model as well as a numerical one the mathematical analysis shows that i tumor cell proliferation by itself cannot generate the invasive branching behavior observed experimentally ii heterotype chemotaxis provides an instability mechanism that leads to the onset of tumor invasion and iii homotype chemotaxis does not provide such an instability mechanism but enhances the mean speed of the tumor surface the numerical results not only support the assumptions needed to perform the mathematical analysis but they also provide evidence of i ii and iii finally both the analytical study and the numerical work agree with the experimental phenomena

internal dla and the gaussian free field in previous works we showed that the internal diffusion limited aggregation dla cluster on zd with t particles is almost surely spherical up to a maximal error of o logt if d 2 and o logt if d 3 this paper addresses average error in a certain sense the average deviation of internal dla from its mean shape is of constant order when d 2 and of order r1 d 2 for a radius r cluster in general appropriately normalized the fluctuations taken over time and space scale to a variant of the gaussian free field

from algebraic cobordism to motivic cohomology let s be an essentially smooth scheme over a eld of characteristic exponent c we prove that there is a canonical equivalence of motivic spectra over s mgl a1 a2 1 c hz 1 c where hz is the motivic cohomology spectrum mgl is the algebraic cobordism spectrum and the elements an are generators of the lazard ring we discuss several applications including the computation of the slices of z 1 c local landweber exact motivic spectra and the convergence of the associated slice spectral sequences

lower bounds on fluctuations for internal dla we consider internal diffusion limited aggregation in dimension two or more this is a random cluster growth model where random walks start at the origin of the lattice one at a time and stop moving when reaching a site that is not occupied by previous walks when n random walks are sent from the origin we establish a lower bound for the inner and outer errors fluctuations of order square root of the logarithm of n when dimension is three or more this lower bound matches the upper bound recently obtained in independent works of asselah and gaudilliere ann prob arxiv 1011 4592 2010 and jerison et al internal dla and the gaussian free field arxiv 1012 3453 2010 also we produce as a corollary of our proof of asselah and gaudilliere ann prob arxiv 1011 4592 2010 an upper bound for the fluctuation of the inner error in a specified direction

logarithmic fluctuations for internal dla let each of n particles starting at the origin in z 2 perform simple random walk until reaching a site with no other particles lawler bramson and griffeath proved that the resulting random set a n of n occupied sites is with high probability close to a disk br of radius r p n we show that the discrepancy between a n and the disk is at most logarithmic in the radius i e there is an absolute constant c such that with probability 1 br c log r a r 2 br c log r for all suciently large r

comparison of cobordism theories abstract relying on results of hopkins morel we show that for x a quasi projective variety over a field of characteristic zero the canonical map n x mgl 2 n n x is an isomorphism

algebraic cobordism revisited we define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities the main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by levine and morel double point cobordism provides a simple geometric presentation of algebraic cobordism theory as a corollary the lazard ring given by products of projective spaces rationally generates all nonsingular projective varieties modulo double point degenerations

modelling the morphology of migrating bacterial colonies we present a model which aims at describing the morphology of colonies of proteus mirabilis and bacillus subtilis our model is based on a cellular automaton which is obtained by the adequate discretisation of a diffusion like equation describing the migration of the bacteria to which we have added rules simulating the consolidation process our basic assumption following the findings of the group of chuo university is that the migration and consolidation processes are controlled by the local density of the bacteria we show that it is possible within our model to reproduce the morphological diagrams of both bacteria species moreover we model some detailed experiments done by the chuo university group obtaining a fine agreement

operations and poly operations in algebraic cobordism abstract in the case of a field of characteristic zero we describe all operations including non additive ones from a theory a obtained from algebraic cobordism of m levine f morel by change of coefficients to any oriented cohomology theory b in the sense of definition 2 1 we prove that such an operation can be reconstructed out of it s action on the products of projective spaces this reduces the construction of operations to algebra and extends the additive case done in 24 as well as the topological one obtained by t kashiwabara see 6 the key new ingredients which permit us to treat the non additive operations are the use of poly operations and the discrete taylor expansion as an application we construct the only missing the 0 th non additive symmetric operation for arbitrary p see 23 which permits to sharpen results on the structure of algebraic cobordism see 22 we also prove the general riemann roch theorem for arbitrary even non additive operations over an arbitrary field this extends the case of multiplicative operations proved by i panin in 13

moving boundary approximation for curved streamer ionization fronts solvability analysis the minimal density model for negative streamer ionization fronts is investigated an earlier moving boundary approximation for this model consisted of a kinetic undercooling type boundary condition in a laplacian growth problem of hele shaw type here we derive a curvature correction to the moving boundary approximation that resembles surface tension the calculation is based on solvability analysis with unconventional features namely there are three relevant zero modes of the adjoint operator one of them diverging furthermore the inner outer matching ahead of the front must be performed on a line rather than on an extended region and the whole calculation can be performed analytically the analysis reveals a relation between the fields ahead and behind a slowly evolving curved front the curvature and the generated conductivity this relation forces us to give up the ideal conductivity approximation and we suggest to replace it by a charge neutrality approximation this implies that the electric potential in the streamer interior is no longer constant but solves a laplace equation this leads to a muskat type problem

how long does it take for internal dla to forget its initial profile internal dla is a discrete model of a moving interface on the cylinder graph mathbb z n times mathbb z a particle starts uniformly on mathbb z n times 0 and performs simple random walk on the cylinder until reaching an unoccupied site in mathbb z n times mathbb z ge 0 which it occupies forever this operation defines a markov chain on subsets of the cylinder we first show that a typical subset is rectangular with at most logarithmic fluctuations we use this to prove that two internal dla chains started from different typical subsets can be coupled with high probability by adding order n 2 log n particles for a lower bound we show that at least order n 2 particles are required to forget which of two independent typical subsets the process started from

on the structure of algebraic cobordism abstract in this paper we investigate the structure of algebraic cobordism of levine morel as a module over the lazard ring with the action of landweber novikov and symmetric operations on it we show that the associated graded groups of algebraic cobordism with respect to the topological filtration r x are unions of finitely presented l modules of very specific structure namely these submodules possess a filtration such that the corresponding factors are either free or isomorphic to cyclic modules l i p n x where deg x p n 1 p 1 as a corollary we prove the syzygies conjecture of vishik on the existence of certain free l resolutions of x and show that algebraic cobordism of a smooth surface can be described in terms of k 0 together with a topological filtration

pattern formation in diffusive advective coupled map lattices we investigate pattern formation and evolution in coupled map lattices when advection is incorporated inaddition to the usual diffusive term all patterns may be suitably grouped into ve classes three periodic supporting static patterns and traveling waves and two nonperiodic relative frequencies are determined as afunction of all model parameters diffusion advection local nonlinearity and lattice size advection plays animportant role in coupled map lattices being capable of considerably altering pattern evolution for instance advection may induce synchronization making chaotic patterns evolve periodically as a byproduct we de scribe a practical algorithm for classifying generic pattern evolutions and for measuring velocities of travelingwaves doi 10 1103 physreve 69 066206 pacs number s 89 75 kd 05 45 xt 05 45 rai introduction

a degeneration formula of gw invariants this is the sequel to the paper 23 in this paper we construct the virtual moduli cycles of the degeneration of the moduli of stable morphisms constructed in 23 we also construct the virtual moduli cycles of the moduli of relative stable morphisms of a pair of a smooth divisor in a smooth variety based on these we prove a degeneration formula of the gromov witten invariants

algebraic cobordism as a module over the lazard ring in this paper we study the structure of the algebraic cobordism ring of a variety as a module over the lazard ring and show that it has relations in positive codimensions we actually prove the stronger graded version this extends the result of levine and morel algebraic cobordism in springer monographs in mathematics 2007 claiming that this module has generators in non negative codimensions as an application we compute the algebraic cobordism ring of a curve the main tool is symmetric operations in algebraic cobordism vishik symmetric operations for all primes and steenrod operations in algebraic cobordism 2013

absorbing state phase transition for driven dissipative stochastic dynamics on ℤ we study the long time behavior of conservative interacting particle systems in the activated random walk model for reaction diffusion systems and the stochastic sandpile we prove that both systems undergo an absorbing state phase transition

a moving boundary problem motivated by electric breakdown i spectrum of linear perturbations abstract an interfacial approximation of the streamer stage in the evolution of sparks and lightning can be written as a laplacian growth model regularized by a kinetic undercooling boundary condition we study the linear stability of uniformly translating circles that solve the problem in two dimensions in a space of smooth perturbations of the circular shape the stability operator is found to have a pure point spectrum except for the eigenvalue 0 0 for infinitesimal translations all eigenvalues are shown to have negative real part therefore perturbations decay exponentially in time we calculate the spectrum through a combination of asymptotic and series evaluation in the limit of vanishing regularization parameter all eigenvalues are found to approach zero in a singular fashion and this asymptotic behavior is worked out in detail a consideration of the eigenfunctions indicates that a strong intermediate growth may occur for generic initial perturbations both the linear and the nonlinear initial value problem are considered in a second paper

sublogarithmic fluctuations for internal dla we consider internal diffusion limited aggregation in dimension larger than or equal to two this is a random cluster growth model where random walks start at the origin of the d dimensional lattice one at a time and stop moving when reaching a site that is not occupied by previous walks it is known that the asymptotic shape of the cluster is a sphere when the dimension is two or more we have shown in a previous paper that the inner resp outer fluctuations of its radius is at most of order log radius resp log2 radius using the same approach we improve the upper bound on the inner fluctuation to log radius when d is larger than or equal to three the inner fluctuation is then used to obtain a similar upper bound on the outer fluctuation