chemotaxis with logistic source very weak global solutions and their boundedness properties abstract we consider the chemotaxis system u t u u v g u x t 0 0 v v u x t 0 in a smooth bounded domain r n where 0 and g generalizes the logistic function g u a u b u with 1 a 0 and b 0 a concept of very weak solutions is introduced and global existence of such solutions for any nonnegative initial data u 0 l 1 is proved under the assumption that 2 1 n moreover boundedness properties of the constructed solutions are studied inter alia it is shown that if b is sufficiently large and u 0 l has small norm in l for some n 2 then the solution is globally bounded finally in the case that additionally n 2 holds a bounded set in l can be found which eventually attracts very weak solutions emanating from arbitrary l 1 initial data the paper closes with numerical experiments that illustrate some of the theoretically established results

dynamic sampling plan in shiryayev roberts procedure for detecting a change in the drift of brownian motion in this paper a dynamic sampling plan in the shiryayev roberts procedure is considered it is shown that a two rate dynamic sampling plan is optimal in the sense that it minimizes the stationary average delay time sadt analytical results as well as numerical comparisons show that it is significantly superior to the fixed sampling plan the comparison also shows that it is as powerful as the dynamic sampling procedure of assaf and ritov the generalizations to the fast initial response and to the cusum procedure are also briefly discussed

persistence of mass in a chemotaxis system with logistic source this paper studies the dynamical properties of the chemotaxis system ut u u v ru u2 x t 0 vt v v u x t 0 under homogeneous neumann boundary conditions in bounded convex domains rn n 1 with positive constants r and numerical simulations but also some rigorous evidence have shown that depending on the relative size of r and in comparison to the well understood case when 0 this problem may exhibit quite a complex solution behavior including unexpected effects such as asymptotic decay of the quantity u within large subdomains of the present work indicates that any such extinction phenomenon if occurring at all necessarily must be of spatially local nature whereas the population as a whole always persists more precisely it is shown that for any nonnegative global classical solution u v of with u 0 one can find m 0 such that u x t dx m for all t 0 the proof is based on an in this context apparently novel analysis of the functional ln u deriving a lower bound for this quantity along a suitable sequence of times by appropriately exploiting a differential inequality for a suitable linear combination of ln u u and v2

global existence and asymptotic behavior of classical solutions to a fractional logistic keller segel system abstract in this paper we consider a fractional parabolic elliptic keller segel system with a logistic source on r n first we establish the regularity of weak solutions of the fractional parabolic equation using blow up arguments combined with liouville type theorems next by the semigroup method and regularity results we prove the local existence and uniqueness of classical solutions moreover the global existence and boundedness of classical solutions for given initial data are obtained under some conditions finally we show the asymptotic behavior of the global solutions with strictly positive initial data

a new look at warning and action lines of surveillance schemes abstract some authors have suggested an efficient dynamic monitoring procedure for detection of a change in the drift of a brownian motion their procedure can be described as a sequence of extremely short sequential probability ratio tests sprt s done in zero time with infinitesimal time between consecutive sprt s the drawbacks of this procedure are that it can be described only as a limit of practical procedures and it does not take into account the cost of initiating a test in this article we suggest a procedure that takes both sampling and overhead costs into account and that can reasonably be carried out in practice in this procedure the process is monitored continuously with a constant sampling rate the accumulated data are analyzed by the standard cumulative sum cusum statistics whenever the cusum procedure raises an alarm data are accumulated as fast as possible until either the alarm is relaxed or the process is stopped

propagation in a fisher kpp equation with non local advection abstract we investigate the influence of a general non local advection term of the form k u to propagation in the one dimensional fisher kpp equation this model is a generalization of the keller segel fisher system when k l 1 r we obtain explicit upper and lower bounds on the propagation speed which are asymptotically sharp and more precise than previous works when k l p r with p 1 and is non increasing in 0 and in 0 we show that the position of the front is of order o t p if p and o e t for some 0 if p and k 0 we use a wide range of techniques in our proofs

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

fine gradings on simple classical lie algebras abstract the fine abelian group gradings on the simple classical lie algebras including d 4 over algebraically closed fields of characteristic 0 are determined up to equivalence this is achieved by assigning certain invariant to such gradings that involve central graded division algebras and suitable sesquilinear forms on free modules over them

the state of statistical process control as we proceed into the 21st century 2000 the state of statistical process control as we proceed into the 21st century journal of the american statistical association vol 95 no 451 pp 992 998

computing moments of the exit time distribution for markov processes by linear programming we provide a new approach to the numerical computation of moments of the exit time distribution of markov processes the method relies on a linear programming formulation of a process exiting from a bounded domain the lp formulation characterizes the evolution of the process through the moments of the induced occupation measure and naturally provides upper and lower bounds for the exact values of the moments the conditions the moments have to satisfy are derived directly from the generator of the markov process and are not based on some approximation of the process excellent software is readily available because the computations involve finite dimensional linear programs

blow up in a higher dimensional chemotaxis system despite logistic growth restriction abstract we study radially symmetric solutions of a class of chemotaxis systems generalizing the prototype u t u u v u u x t 0 0 v m t u x t 0 in a ball r n with parameters 0 0 0 and 1 and m t 1 u x t d x it is shown that when n 5 and 3 2 1 2 n 2 then there exist initial data such that the smooth local in time solution of blows up in finite time this indicates that even superlinear growth restrictions may be insufficient to rule out a chemotactic collapse as is known to occur in the corresponding system without any proliferation

classification of involutions on graded division simple real algebras abstract we classify up to isomorphism and up to equivalence involutions on graded division finite dimensional simple real associative algebras when the grading group is abelian

pointwise persistence in full chemotaxis models with logistic source on bounded heterogeneous environments abstract the current paper is concerned with pointwise persistence in full chemotaxis models with local as well as nonlocal time and space dependent logistic source in bounded domains we first prove the global existence and boundedness of nonnegative classical solutions under some conditions on the coefficients in the models next under the same conditions on the coefficients we show that pointwise persistence occurs that is any globally defined positive solution is bounded below by a positive constant independent of its initial condition when the time is large enough it should be pointed out that in 21 the authors established the persistence of mass for globally defined positive solutions which indicates that any extinction phenomenon if occurring at all necessarily must be spatially local in nature whereas the population as a whole always persists the pointwise persistence proved in the current paper implies that not only the population as a whole persists but also it persists at any location eventually it also implies the existence of strictly positive entire solutions

boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source abstract this paper deals with a parabolic elliptic chemotaxis system with nonlinear sensitivity and logistic source u t u u v f u x t 0 0 v v g u x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r n n 1 where 0 the function u is the chemotactic sensitivity g u is the production rate of the chemoattractant and f u is the logistic source under some suitable assumptions on the nonlinearities u g u and logistic source f u we study the global boundedness of solutions for the problem

blowup and global solutions in a chemotaxis growth system abstract we study a keller segel type of system which includes growth and death of the chemotactic species and an elliptic equation for the chemo attractant the problem is considered in bounded domains with smooth boundary as well as in the whole space in case the random motion of the chemotactic species is neglected a hyperbolic elliptic problem results for which we characterize blow up of solutions in finite time and existence of regular solutions globally in time in dependence on the systems parameters in this case convexity of the domain is needed for the parabolic elliptic problem in dimensions three and higher we establish global existence of regular solutions in a limiting case which is an extension of the results given by tello and winkler 2007

symmetrization techniques on unbounded domains application to a chemotaxis system on rn the authors study the parabolic elliptic system on rn u t u u v 0 v v u u 0 u0 a version of the mathematical model of chemotaxis proposed by keller and segel a differential inequality for the quantity s0u t d where u is the decreasing rearrangement of the solution u t with respect to the spatial variable is obtained as a consequence they obtain e g lp bounds of the solution u v on r2 and global in time existence of solutions under the condition r2u0 8 then the solution u v blows up in a finite time compared to the previous work of diaz diaz and nagai adv math sci appl 5 1995 no 2 659 680 mr1361010 96j 35246 where this problem has been considered on bounded domains of rn there are some additional technical difficulties connected with the regularity of the derivative u t

classification of group gradings on simple lie algebras of types a b c and d abstract for a given abelian group g we classify the isomorphism classes of g gradings on the simple lie algebras of types a n n 1 b n n 2 c n n 3 and d n n 4 in terms of numerical and group theoretical invariants the ground field is assumed to be algebraically closed of characteristic different from 2

a diffusion process and its applications to detecting a change in the drift of brownian motion abstract the purpose of the present paper is to make a quantitative comparison of the shiryayev roberts and page procedures we do this in the context of continuous time in order to use the machinery of diffusion processes to perform explicitly certain calculations which seem impossible in discrete time although the continuous time results are not especially good approximations to the corresponding quantities in discrete time they provide very useful comparative information on which to base selection of a stopping rule this paper is organized as follows the shiryayev roberts process is defined and shown to be a novel diffusion process with some surprising properties we also specify more precisely the basis for our comparison of the two procedures and give the results of some elementary calculations these developments contain an asymptotic evaluation we define a modification of our basic procedure and give an asymptotic evaluation of its average run length numerical comparisons and a discussion of their significance are contained next our conclusions are roughly these in simple situations where the two procedures can be directly compared neither seems dramatically superior to the other however the shiryayev roberts procedure is more easily adapted to complex circumstances and consequently warrants additional study

global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening abstract this paper deals with nonnegative solutions of the neumann initial boundary value problem for the parabolic chemotaxis system u t u u v u u 2 x t 0 v t v v u x t 0 in bounded convex domains r n n 1 with smooth boundary it is shown that if the ratio is sufficiently large then the unique nontrivial spatially homogeneous equilibrium given by u v 1 is globally asymptotically stable in the sense that for any choice of suitably regular nonnegative initial data u 0 v 0 such that u 0 0 the above problem possesses a uniquely determined global classical solution u v with u v t 0 u 0 v 0 which satisfies u t 1 l 0 and v t 1 l 0 as t

the speed of propagation for kpp type problems ii general domains this paper is devoted to nonlinear propagation phenomena in general unbounded domains of rn for reaction diffusion equations with kolmogorov petrovsky piskunov kpp type nonlinearities this article is the second in a series of two and it is the follow up of the paper 7 which dealt which the case of periodic domains this paper is concerned with general domains and we give various definitions of the spreading speeds at large times for solutions with compactly supported initial data we study the relationships between these new notions and analyze their dependence on the geometry of the domain and on the initial condition some a priori bounds are proved for large classes of domains the case of exterior domains is also discussed in detail lastly some domains which are very thin at infinity and for which the spreading speeds are infinite are exhibited the construction is based on some new heat kernel estimates in such domains

on a quasilinear parabolic elliptic chemotaxis system with logistic source abstract this paper deals with a quasilinear parabolic elliptic chemotaxis system with logistic source under homogeneous neumann boundary conditions in a smooth bounded domain for the case of positive diffusion function it is shown that the corresponding initial boundary value problem possesses a unique global classical solution which is uniformly bounded moreover if the diffusion function is zero at some point or a positive diffusion function and the logistic damping effect is rather mild we proved that the weak solutions are global existence finally it is asserted that the solutions approach constant equilibria in the large time for a specific case of the logistic source

asymptotic spreading in heterogeneous diffusive excitable media abstract we establish propagation and spreading properties for nonnegative solutions of nonhomogeneous reaction diffusion equations of the type t u a t x u q t x u f t x u with compactly supported initial conditions at t 0 here a q f have a general dependence in t r and x r n we establish properties of families of propagation sets which are defined as families of subsets s t t 0 of r n such that lim inf t inf x s t u t x 0 the aim is to characterize such families as sharply as possible in particular we give some conditions under which 1 a given path t t 0 where t r n forms a family of propagation sets or 2 one can find such a family with s t x r n x r t and lim t r t this second property is called here complete spreading furthermore in the case q 0 and inf t x r r n f u t x 0 0 as well as under some more general assumptions we show that there is a positive spreading speed that is r t can be chosen so that lim inf t r t t 0 in the general case we also show the existence of an explicit upper bound c 0 such that lim sup t r t t c on the other hand we provide explicit examples of reaction diffusion equations such that for an arbitrary e 0 any family of propagation sets s t t 0 has to satisfy s t x r n x e t for large t in connection with spreading properties we derive some new uniqueness results for the entire solutions of this type of equations lastly in the case of space time periodic media we develop a new approach to characterize the largest propagation sets in terms of eigenvalues associated with the linearized equation in the neighborhood of zero

gradings on the real forms of the albert algebra of g2 and of f4 we find all the fine group gradings on the real forms of the albert algebra and of the exceptional lie algebras g2 and f4

spatial pattern formation in a chemotaxis diffusion growth model abstract mimura and one of the authors 1996 proposed a mathematical model for the pattern dynamics of aggregating regions of biological individuals possessing the property of chemotaxis for this model tello and winkler 2007 22 obtained infinitely many local branches of nonconstant stationary solutions bifurcating from a positive constant solution while kurata et al 2008 numerically showed several spatio temporal patterns in a rectangle motivated by their work we consider some qualitative behaviors of stationary solutions from global and local bifurcation viewpoints in the present paper first we study the asymptotic behavior of stationary solutions as the chemotactic intensity grows to infinity next we construct local bifurcation branches of stripe and hexagonal stationary solutions in the special case when the habitat domain is a rectangle for this case the directions of the branches near the bifurcation points are also obtained finally we exhibit several numerical results for the stationary and oscillating patterns

on lie gradings iii gradings of the real forms of classical lie algebras abstract maximal abelian subgroups of diagonalizable automorphisms of lie algebra so called mad groups play a crucial role in the construction of fine gradings of lie algebra our aim is to give a description of mad groups for real forms of classical lie algebras we introduce four types of matrix subgroups of g l n c called out groups ad groups out groups and ad groups for each type of these subgroups we define a relation of equivalence the problem of classifying of all non conjugate mad groups on real forms of sl n c o n c or sp n c is transformed to the problem of classifying these equivalence classes the classification of these equivalence classes is presented here

a quasilinear attraction repulsion chemotaxis system of parabolic elliptic type with logistic source abstract this study considers the following quasilinear attraction repulsion chemotaxis system of parabolic elliptic type with logistic source u t d u u u v u w f u x t 0 0 v u v x t 0 0 w u w x t 0 under homogeneous neumann boundary conditions in a bounded domain r n n 2 with smooth boundary where d u c d u m 1 with m 1 0 and c d 0 and f u a b u with a 0 b 0 and 1 in the case of non degenerate diffusion i e 0 we show that the system admits a unique global bounded classical solution provided that the repulsion prevails over the attraction in the sense that 0 or the logistic dampening is sufficiently strong or the diffusion is sufficiently strong while in the case of degenerate diffusion i e 0 we show that the system admits a global bounded weak solution at least under the same assumptions finally we obtain the large time behavior of solutions for a specific logistic source

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

gradings on the lie algebra d4 revisited abstract we classify group gradings on the simple lie algebra l of type d 4 over an algebraically closed field of characteristic different from 2 fine gradings up to equivalence and g gradings with a fixed group g up to isomorphism for each g grading on l we also study graded l modules assuming characteristic 0

spreading speeds and traveling waves for abstract monostable evolution systems abstract this paper is devoted to the development of the theory of spreading speeds and traveling waves for abstract monostable evolution systems with spatial structure under appropriate assumptions we show that the spreading speeds coincide with the minimal wave speeds for monotone traveling waves in the positive and negative directions then we use this theory to study the spatial dynamics of a parabolic equation in a periodic cylinder with the dirichlet boundary condition a reaction diffusion model with a quiescent stage a porous medium equation in a tube and a lattice system in a periodic habitat

gradings on octonions abstract all the different ways of grading any cayley dickson or octonion algebra as well as the associated central simple malcev algebras will be obtained

stabilization in the logarithmic keller segel system abstract the keller segel system u t d u d u v v x t 0 v t d v v u x t 0 is considered in a bounded domain r n n 2 with smooth boundary where 0 and d 0 the main results identify a condition on the parameters 2 n and d 0 essentially reducing to the assumption that 2 d be suitably small under which for all reasonably regular and positive initial data the corresponding classical solution of an associated neumann initial boundary value problem known to exist globally according to previous findings approaches the homogeneous steady state u 0 u 0 at an exponential rate with respect to the norm in l 2 as t where u 0 1 u 0 as a particular consequence this entails a convergence statement of the above flavor in the normalized system with d 1 and fixed 2 n provided that satisfies a certain smallness condition

cusum charts with variable sampling intervals a standard cumulative sum cusum chart for controlling the process mean takes samples from the process at fixed length sampling intervals and uses a control statistic based on a cumulative sum of differences between the sample means and the target value this article proposes a modification of the standard cusum scheme that varies the time intervals between samples depending on the value of the cusum control statistic the variable sampling interval vsi cusum chart uses short sampling intervals if there is an indication that the process mean may have shifted and long sampling intervals if there is no indication of a change in the mean if the cusum statistic actually enters the signal region then the vsi cusum chart signals in the same manner as the standard cusum chart a markov chain approach is used to evaluate properties such as the average time to signal and the average number of samples to signal results show that the proposed vsi cusum chart is considerably more efficient than the standard cus

eventual smoothness and asymptotics in a three dimensional chemotaxis system with logistic source abstract we prove existence of global weak solutions to the chemotaxis system u t u u v u u 2 v t v v u under homogeneous neumann boundary conditions in a smooth bounded convex domain r n for arbitrarily small values of 0 additionally we show that in the three dimensional setting after some time these solutions become classical solutions provided that is not too large in this case we also consider their large time behaviour we prove decay if 0 and the existence of an absorbing set if 0 is sufficiently small

global existence and asymptotic behavior of classical solutions to a parabolic elliptic chemotaxis system with logistic source on rn in the current paper we consider the following parabolic elliptic semilinear keller segel model on rnrn ut u u v au bu2 x rn t 0 0 i v u x rn t 0 where 0 a 0 b 0 are constant real numbers and n is a positive integer we first prove the local existence and uniqueness of classical solutions u x t u0 v x t u0 u x t u0 v x t u0 with u x 0 u0 u0 x u x 0 u0 u0 x for various initial functions u0 x u0 x next under some conditions on the constants a b a b and the dimension n we prove the global existence and boundedness of classical solutions u x t u0 v x t u0 u x t u0 v x t u0 for given initial functions u0 x u0 x finally we investigate the asymptotic behavior of the global solutions with strictly positive initial functions or nonnegative compactly supported initial functions under some conditions on the constants a b a b and the dimension n we show that for every strictly positive initial function u0 u0 limt supx rn u x t u0 ab v x t u0 ab 0 and that for every nonnegative initial function u0 u0 with non empty and compact support supp u0 supp u0 there are 0 clow u0 cup u0 such that limt sup x ct u x t u0 ab v x t u0 ab 0 0 c clow u0 and limt sup x ct u x t u0 v x t u0 0 c cup u0

global existence and decay properties for a degenerate keller segel model with a power factor in drift term abstract the following degenerate parabolic system modelling chemotaxis is considered ks u t u m u q 1 v x r n t 0 v t v v u x r n t 0 u x 0 u 0 x v x 0 v 0 x x r n where m 1 q 2 0 or 1 and n 1 the aim of this paper is to prove the existence of a time global weak solution u v of ks with the l 0 l r n bound such a global bound is obtained in the case of i m q 2 n for large initial data and ii 1 m q 2 n for small initial data in the case of ii the decay properties of the solution u v are also discussed

boundedness of solutions to a quasilinear parabolic elliptic keller segel system with logistic source abstract we study global solutions of a class of chemotaxis systems generalizing the prototype u t u 1 m 1 u u u 1 q 1 v a u b u r x t 0 0 v v u x t 0 in a bounded domain r n n 1 with smooth boundary with parameters m 1 r 1 a 0 b q 0 it is shown that when q 1 max r m 2 n or b b 0 n r m 2 r m n 2 r 2 if q 1 r then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded the results improve the results of wang et al 2014 37 and cao and zheng 2014 6

comparison of ewma cusum and shiryayev roberts procedures for detecting a shift in the mean pollak and siegmund compared the shiryayev roberts procedure with the cusum procedure for detecting a change in the drift of a brownian motion based on the conditional average delay time in this paper the exponentially weighted moving average ewma procedure proposed by roberts is compared with the shiryayev roberts and cusum procedures the comparison is based on the stationary average delay time as advocated by shiryayev the optimal design for the ewma procedure and its asymptotic properties are studied when the average in control run length is large the results show that the ewma procedure is less efficient than the other two procedures

on a parabolic elliptic system with chemotaxis and logistic type growth we consider a nonlinear pdes system of two equations of parabolic elliptic type with chemotactic terms the system models the movement of a biological population u towards a higher concentration of a chemical agent w in a bounded and regular domain rn for arbitrary n n after normalization the system is as follows ut u div um w u 1 u in t 0 t w w u in t for some positive constants m and with positive initial datum u0 and neumann boundary conditions we study the range of parameters and constrains for which the solution exists globally in time if either m 1 or m 1 and n 22 m 1 n the solution exists globally in time moreover if m 1 and 2 and there exist positive constants u 0 and u 0 such that 0 u 0 u0 u 0 we have that u 1 l w 1 l 0 as t

gradings on the simple real lie algebras of types g2 and d4 abstract we classify group gradings on the simple lie algebras of types g 2 and d 4 over the field of real numbers or any real closed field fine gradings up to equivalence and g gradings with a fixed group g up to isomorphism