existence uniqueness and lipschitz dependence for patlak keller segel and navier stokes in mathbb r 2 with measure valued initial data we establish a new local well posedness result in the space of finite borel measures for mild solutions of the parabolic elliptic patlak keller segel pks model of chemotactic aggregation in two dimensions our result only requires that the initial measure satisfy the necessary assumption max x in mathbb r 2 mu x 8 pi this work improves the small data results of biler stud math 114 2 181 192 1995 and the existence results of senba and suzuki j funct anal 191 17 51 2002 our work is based on that of gallagher and gallay math ann 332 287 327 2005 who prove the uniqueness and log lipschitz continuity of the solution map for the 2d navier stokes equations nse with measure valued initial vorticity we refine their techniques and present an alternative version of their proof which yields existence uniqueness and lipschitz continuity of the solution maps of both pks and nse many steps are more difficult for pks than for nse particularly on the level of the linear estimates related to the self similar spreading solutions

semiparametric nonlinear mixed effects models and their applications nonlinear mixed effects models nlmms and self modeling nonlinear regression semor models are often used to fit repeated measures data they use a common function shared by all subjects to model variation within each subject and some fixed and or random parameters to model variation between subjects the parametric nlmm may be too restrictive and the semiparametric semor model ignores correlations within each subject in this article we propose a class of semiparametric nonlinear mixed effects models snmms that extend nlmms semor models and many other existing models in a natural way a snmm assumes that the mean function depends on some parameters and nonparametric functions the parameters provide an interpretable data summary the nonparametric functions provide flexibility to allow the data to decide some unknown or uncertain components such as the shape of the mean response over time a second stage model with fixed and random effects is used to model the parameters smoothing splines are us

on the subgroup structure of classical groups in this paper we establish a result on the subgroup structure of classical groups over algebraically closed elds and use this to give a new proof of a fundamental theorem of m aschbacher on subgroups of nite classical groups let v be a nite dimensional vector space over an algebraically closed eld k and let g be one of the classical algebraic groups sl v sp v or so v our result is a reduction theorem concerning the subgroups of g we de ne a certain collection c of natural proper subgroups of g and prove that any closed nite or in nite subgroup of g either lies in a member of c or is roughly speaking a simple group acting irreducibly on v aschbacher s result is an analogous reduction theorem for subgroups of nite classical groups we obtain this as a relatively easy consequence of our main result by taking xed points under the action of a frobenius morphism using a standard process involving lang s theorem the proof of the main result uses elementary linear algebra together with a few basic facts from the theory of algebraic groups various complications which arise in the nite group setting in as become much more straightforward in the algebraic group setting in particular questions involving extension elds do not occur and issues of conjugacy are easily settled when we descend to nite invent math 134 427 453 1998

singularity formation in chemotaxis a conjecture of nagai consider the initial boundary value problem for the system s ut uxx uvx x vt u av on an interval 0 1 for t 0 where a 0 with ux 0 t ux 1 t 0 suppose mu v0 are positive constants the corresponding spatially homogeneous global solution u t mu v t mu a v0 mu a exp at is stable in the sense that if mu v0 are positive constants the corresponding spatially homogeneous solution will be uniformly close to u cdot v cdot we consider in sequence space an approximate system s which is related to s in the following sense the chemotactic term uvx x is replaced by the inverse fourier transform of the finite part of the convolution integral for the fourier transform of uvx x here the finite part of the convolution on the line at a point x of two functions f g is defined as int 0 x f y g y x dy we prove the following if mu a then in every neighborhood of mu v0 there are spatially nonconstant initial data for which the solution of proble

aggregation vs global diffusive behavior in the higher dimensional keller segel model abstract we consider the classical parabolic parabolic keller segel system u t u u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r n it is proved that in space dimension n 3 for each q n 2 and p n one can find e 0 0 such that if the initial data u 0 v 0 satisfy u 0 l q e and v 0 l p e then the solution is global in time and bounded and asymptotically behaves like the solution of a discoupled system of linear parabolic equations in particular u v approaches the steady state m m as t where m is the total mass m u 0 of the population moreover we shall show that if is a ball then for arbitrary prescribed m 0 there exist unbounded solutions emanating from initial data u 0 v 0 having total mass u 0 m

subgroups of maximal rank in finite exceptional groups of lie type the purpose of this paper is to investigate a large and natural class of maximal subgroups of the finite exceptional groups of lie type which we call subgroups of maximal rank these subgroups play a prominent role both in the classification of local maximal subgroups in 9 theorem 1 and in the reduction theorem for subgroups in 28 theorem 2 in 9 it is shown that any local maximal subgroup of a finite exceptional group of lie type is either a subgroup of maximal rank or one of a small list of exceptions and in 28 using this result it is proved that any maximal subgroup is either of maximal rank or almost simple or in a known list to describe these subgroups we require the following notation let g be a simple adjoint algebraic group over an algebraically closed field k of characteristic p 0 and let o be an endomorphism of g such that l ga is a finite exceptional simple group of lie type over fq where q p let a be a group such that f x l the group aut l is generated by ga together with field and graph automorphisms see 5 38 all of which extend to morphisms of the abstract group g commuting with o thus there is a subgroup x of caut c a such that x x o and so x acts on the set of a stable subsets of g for a a stable subset y we write nx y for the stabilizer in x of y if d is a a stable closed connected reductive subgroup of g containing a maximal torus t of g and m nx d we call m a subgroup of maximal rank in x in this paper we determine the structure and conjugacy classes of those subgroups of maximal rank which are maximal in x

existence of global bounded classical solution to a quasilinear attraction repulsion chemotaxis system with logistic source abstract this paper concerns the global existence of bounded classical solution to a quasilinear attraction repulsion chemotaxis system with logistic source u t d u u u v u w u u 2 in v t v u v in w t w u w in under homogeneous neumann boundary condition with positive parameters and d s c 0 s m 1 for s 0 d 0 0 we prove that in dimension three there exists a unique global bounded classical solution provided that m 6 5 and 0 with an additional assumption d s c 0 s m 1 1 for s 0 we prove that for any m 1 6 5 there exists a constant 0 0 m such that for all 0 the above problem admits a unique global bounded classical solution

a positivity preserving finite element method for chemotaxis problems in 3d we present an implicit finite element method for a class of chemotaxis models in three spatial dimensions the proposed algorithm is designed to maintain mass conservation and to guarantee positivity of the cell density to enforce the discrete maximum principle the standard galerkin discretization is constrained using a local extremum diminishing flux limiter to demonstrate the efficiency and robustness of this approach we solve blow up problems in a 3d chemostat domain to give a flavor of more complex and realistic chemotactic applications we investigate the pattern dynamics and aggregating behavior of the bacteria escherichia coli and salmonella typhimurium the obtained numerical results are in good qualitative agreement with theoretical studies and experimental data reported in the literature

on the orders of primitive groups abstract almost all primitive permutation groups of degree n have order at most n i 0 log 2 n 1 n 2 i 1 log 2 n or have socle isomorphic to a direct power of some alternating group the mathieu groups m11 m12 m23 and m24 are the four exceptions as a corollary the sharp version of a theorem of praeger and saxl is established where m12 turns out to be the largest primitive group for an application a bound on the orders of permutation groups without large alternating composition factors is given this sharpens a lemma of babai cameron palfy and generalizes a theorem of dixon

building accurate emulators for stochastic simulations via quantile kriging computer simulation has increasingly become popular for analysis of systems that cannot be feasibly changed because of costs or scale this work proposes a method to construct an emulator for stochastic simulations by performing a designed experiment on the simulator and developing an emulative distribution existing emulators have focused on estimation of the mean of the simulation output but this work presents an emulator for the distribution of the output this construction provides both an explicit distribution and a fast sampling scheme beyond the emulator description this work demonstrates the emulator s efficiency that is its convergence rate is the asymptotically optimal among all possible emulators using the same sample size under certain conditions an example of its practical use is demonstrated using a stochastic simulation of fracture mechanics supplementary materials for this article are available online

asymptotic behavior of type i blowup solutions to a parabolic elliptic system of drift diffusion type we consider a cauchy problem for a parabolic elliptic system of drift diffusion type the problem is formally of the form u t nabla cdot nabla u u nabla delta 1 u this system describes a mass conserving aggregation phenomenon including gravitational collapse and bacterial chemotaxis our concern is the asymptotic behavior of blowup solutions when the blowup is type i in the sense that its blowup rate is the same as the corresponding ordinary differential equation y t y 2 up to a multiple constant it is shown that all type i blowup is asymptotically backward self similar provided that the solution is radial nonnegative when the blowup set is a singleton and the space dimension is greater than or equal to three

the form of blow up for nonlinear parabolic equations on considere des equations paraboliques semi lineaires de la forme u t 2 u sf u ou f est positive et 0 ds f s est finie on etudie des problemes a 1 dimension dans la region semi infinie x o et on trouve le temps t

on explosions of solutions to a system of partial differential equations modelling chemotaxis a system of partial differential equations modelling chemotactic aggregation is analysed keller segel model conditions on the system of paramaters are given implying global existence of smooth solutions in two space dimensions and radially symmetric situations explosion of the bacteria concentration in finite time is shown for a class of initial values

singular standing ring solutions of nonlinear partial differential equations abstract we present a general framework for constructing singular solutions of nonlinear evolution equations that become singular on a d dimensional sphere where d 1 the asymptotic profile and blowup rate of these solutions are the same as those of solutions of the corresponding one dimensional equation that become singular at a point we provide a detailed numerical investigation of these new singular solutions for the following equations the nonlinear schrodinger equation i t t x 2 0 with 2 the biharmonic nonlinear schrodinger equation i t t x 2 2 0 with 4 the nonlinear heat equation t t x 2 0 with 0 and the nonlinear biharmonic heat equation t t x 2 2 0 with 0

precise computations of chemotactic collapse using moving mesh methods we consider the problem of computing blow up solutions of chemotaxis systems or the so called chemotactic collapse in two spatial dimensions such solutions can have approximate self similar behaviour which can be very challenging to verify in numerical simulations cf betterton and brenner collapsing bacterial cylinders phys rev e 64 2001 061904 we analyse a dynamic scale invariant remeshing method which performs spatial mesh movement based upon equidistribution using a suitably chosen monitor function the numerical solution resolves the fine detail in the asymptotic solution structure such that the computations are seen to be fully consistent with the asymptotic description of the collapse phenomenon given by herrero and velazquez singularity patterns in a chemotaxis model math ann 306 1996 583 623 we believe that the methods we construct are ideally suited to a large number of problems in mathematical biology for which collapse phenomena are expected

blowup in infinite time of radial solutions for a parabolic elliptic system in high dimensional euclidean spaces abstract we consider radial solutions blowing up in infinite time to a parabolic elliptic system in n dimensional euclidean space the system was introduced to describe the gravitational interaction of particles in the case where n 2 we can find positive and radial solutions blowing up in finite time in the present paper in the case where n 11 we find positive and radial solutions blowing up in infinite time and investigate those blowup speeds by using the so called asymptotic matched expansion techniques and parabolic regularity

large subgroups of simple groups abstract let g be a finite group a proper subgroup h of g is said to be large if the order of h satisfies the bound h 3 g in this note we determine all the large maximal subgroups of finite simple groups and we establish an analogous result for simple algebraic groups in this context largeness is defined in terms of dimension an application to triple factorisations of simple groups both finite and algebraic is discussed

self similar solutions to a parabolic system modeling chemotaxis we study the forward self similar solutions to a parabolic system modeling chemotaxisut u u v vt v u in the whole space r2 where is a positive constant using the liouville type result and the method of moving planes it is proved that self similar solutions u v must be radially symmetric about the origin then the structure of the set of self similar solutions is investigated as a consequence it is shown that there exists a threshold in r2u for the existence of self similar solutions in particular for 0 1 2 there exists a self similar solution u v if and only if r2u 8

self similar blow up for a reaction diffusion system abstract this work is concerned with the following system which is a model to describe several phenomena in which aggregation plays a crucial role as for instance motion of bacteria by chemotaxis and equilibrium of self attracting clusters when the space dimension n is equal to three we show here that s has radial solutions with finite mass that blow up in finite time in a self similar manner when n 2 however no radial solution with finite mass may give rise to self similar blow up

on steady distributions of self attracting clusters under friction and fluctuations we consider the vlasov fokker planek equation with a newtonian attracting potential and study its stationary solutions given by the generalized lane emden equation in a two dimensional domain we obtain the existence of a critical mass beyond which the system may admit a gravitational collapse for a one dimensional model we prove some results on existence uniqueness stability and symmetry breaking of stationary solutions

boundedness vs blow up in a chemotaxis system abstract we determine the critical blow up exponent for a keller segel type chemotaxis model where the chemotactic sensitivity equals some nonlinear function of the particle density assuming some growth conditions for the chemotactic sensitivity function we establish an a priori estimate for the solution of the problem considered and conclude the global existence and boundedness of the solution furthermore we prove the existence of solutions that become unbounded in finite or infinite time in that situation where this a priori estimate fails

the maximal subgroups of the chevalley groups g2 q with q odd the ree groups 2g2 q and their automorphism groups

sublattices generated by root differences let be an irreducible root system of exceptional type in this paper we calculate the set of exponents of torsion subgroups of quotients z l where z is the root lattice and l runs through the sublattices thereof which are generated by root differences the maximum such exponent is of interest because it appears in the hypotheses of a result of liebeck and seitz concerning lifting of embeddings from finite groups of lie type to algebraic groups

local null controllability for a chemotaxis system of parabolic elliptic type abstract we consider the controllability of a chemotaxis system of parabolic elliptic type by linearizing the nonlinear system into two separate linear equations we can bypass the obstacle caused by the nonlinear drift term and establish local null controllability of the original nonlinear system this approach is different from the usual method for dealing with coupled parabolic systems

existence of solutions to a new model of biological pattern formation abstract in this paper we study the existence of classical solutions to a new model of skeletal development in the vertebrate limb the model incorporates a general term describing adhesion interaction between cells and fibronectin an extracellular matrix molecule secreted by the cells as well as two secreted diffusible regulators of fibronectin production the positively acting differentiation factor activator tgf and a negatively acting factor inhibitor together these terms constitute a pattern forming system of equations we analyze the conditions guaranteeing that smooth solutions exist globally in time we prove that these conditions can be significantly relaxed if we add a diffusion term to the equation describing the evolution of fibronectin

classification of singularities for blowing up solutions in higher dimensions consider the cauchy problem p u t u u p when x r n t 0 n 1 u x 0 u 0 x when x r n where p 1 and u 0 x is a continuous nonnegative and bounded function it is known that under fairly general assumptions on u 0 x the unique solution of p u x t blows up in a finite time by which we mean that lim sup sup u x t t t x r n in this paper we shall assume that u x t blows up at x 0 t t and derive the possible asymptotic behaviours of u x t as x t 0 t under general assumptions on the blow up rate

triple factorisations of the general linear group and their associated geometries abstract triple factorisations of finite groups g of the form g p q p are essential in the study of lie theory as well as in geometry geometrically each triple factorisation g p q p corresponds to a g flag transitive point line geometry such that each pair of points is incident with at least one line we call such a geometry collinearly complete and duality interchanging the roles of points and lines gives rise to the notion of concurrently complete geometries in this paper we study triple factorisations of the general linear group gl v as pqp where the subgroups p and q either fix a subspace or fix a decomposition of v as v 1 v 2 with dim v 1 dim v 2

high field limit for the vlasov poisson fokker planck system this paper is concerned with the analysis of the stability of the vlasov poissonfokker planck system with respect to the physical constants if the scaled thermal mean free path converges to zero and the scaled thermal velocity remains constant then a hyperbolic limit or equivalently a high field limit equation is obtained for the mass density the passage to the limit as well as the existence and uniqueness of solutions of the limit equation in l 1 global or local in time are analyzed according to the electrostatic or gravitational character of the field and to the space dimension in the one dimensional case a new concept of global solution is introduced for the gravitational field this concept is shown to be equivalent to the concept of entropy solutions of hyperbolic systems of conservation laws

singularity patterns in a chemotaxis model the authors study a chemotactic model under certain assumptions and obtain the existence of a class of solutions which blow up at the center of an open disc in finite time such a finite time blow up of solutions implies chemotactic collapse namely concentration of species to form sporae the model studied is the limiting case of a basic chemotactic model when diffusion of the chemical approaches infinity which has the form ut u u v 0 v u 1 on r2 where is an open disc with no flux homogeneous neumann boundary conditions the initial conditions are continuous functions u x 0 u0 x 0 v x 0 v0 x 0 for x under these conditions the authors prove there exists a radially symmetric solution u r t which blows up at r 0 t t a specific description of such a solution is presented the authors also discuss the strong similarity between the chemotactic model they study and the classical stefan problem

a classification of the maximal subgroups of the finite alternating and symmetric groups following the classification of finite simple groups one of the major problems in finite group theory today is the determination of the maximal subgroups of the almost simple groups that is of groups x such that x0 u x aut x for some finite non abelian simple group x0 the problem has been solved for most sporadic groups and groups of lie type of low rank see 19 sects 3 43 for discussion and references this paper is a contribution to the case where x0 is an alternating group a so that for n 6 x is a or s the maximal subgroups of a and s are known for several classes of degrees n

non existence of finite time self similar solutions of the keller segel system in the scaling invariant class let us consider the problem whether there does exist a finite time self similar solution of the backward type to the semilinear keller segel system in the case of parabolic elliptic type for n 3 we show that there is no such a solution with a finite mass in the scaling invariant class on the other hand in the case of parabolic parabolic type for n 2 non existence of finite time self similar solutions is proved in a larger class of a finite mass with some local bounds

a sufficient condition for type i blowup in a parabolic elliptic system abstract this paper is concerned with blowup of positive solutions to a cauchy problem for a parabolic elliptic system u t u u v in r n 0 t 0 v u in r n 0 t we say that a solution u v blows up at t t if lim sup t t u t with l norm in r n when a solution u v blows up at t t the blowup is called of type i if lim sup t t t t u t and of type ii otherwise it was shown in herrero and velazquez 1996 12 and mizoguchi and senba 2007 20 that there exist radial type ii blowup solutions for n 2 or n 11 when 3 n 9 type ii blowup solutions were given by formal analysis in brenner et al 1999 2 and herrero et al 1997 10 we give a sufficient condition for a solution to exhibit type i blowup in the case of 3 n 9

inference in smoothing spline analysis of variance summary smoothing spline analysis of variance decomposes a multivariate function into additive components this decomposition not only provides an efficient way to model a multivariate function but also leads to meaningful inference by testing whether a certain component equals 0 no formal procedure is yet available to test such a hypothesis we propose an asymptotic method based on the likelihood ratio to test whether a functional component is 0 this test allows us to choose an optimal model and to compare groups of curves we first develop the general theory by exploiting the connection between mixed effects models and smoothing splines we then apply this to compare two groups of curves and to select an optimal model in a two dimensional problem a small simulation is used to assess the finite sample performance of the likelihood ratio test