large time behavior of an attraction repulsion chemotaxis system abstract in this study we consider an attraction repulsion chemotaxis system u t u u v u w x t 0 v t v u v x t 0 w t w u w x t 0 with homogeneous neumann boundary conditions where and are positive parameters under the critical condition that 0 we prove that the system possesses a unique global solution which is uniformly bounded in the physical domain r n n 2 3 in addition we assert that we can find 0 0 such that for all of the initial data u 0 that satisfy u 0 0 the solution of the system approaches the steady state u 0 u 0 u 0 exponentially as time tends to infinity where u 0 1 u 0

special values of multiple polylogarithms historically the polylogarithm has attracted specialists and non specialists alike with its lovely evaluations much the same can be said for euler sums or multiple harmonic sums which within the past decade have arisen in combinatorics knot theory and high energy physics more recently we have been forced to consider multidimensional extensions encompassing the classical polylogarithm euler sums and the riemann zeta function here we provide a general framework within which previously isolated results can now be properly understood applying the theory developed herein we prove several previously conjectured evaluations including an intriguing conjecture of don zagier

the algebra of multiple harmonic series abstract recently there has been much interest in multiple harmonic series i 1 i 2 i k n 1 n 2 n k 1 1 n 1 i 1 n 2 i 2 n k i k which converge when the exponentsijare positive integers andi1 1 also known as multiple zeta values or euler zagier sums starting with the noncommutative polynomial algebraq x y we define a second multiplication which is commutative and associative and call the resulting structure the harmonic algebra h as a graded commutative algebra h turns out to be a free polynomial algebra with the number of generators in degreengiven by the witt formula for the numbern n of basis elements of degreenin the free lie algebra on two generators multiple harmonic series can be thought of as images under a map h 0 rwhich is a homomorphism with respect to the commutative multiplication where h 0is an appropriate subalgebra of h if we calli1 ikthe weight of the series i1 ik then forn 1 there are at mostn n series of weightnthat are irreducible in the sense that they are not sums of rational multiples of products of multiple harmonic series of lower weight in fact our approach gives an explicit set of algebraically irreducible multiple harmonic series we also show that there is a subalgebra h 0 h 1 h related to the shuffle algebra which contains the algebra s of symmetric functions in fact maps the elements of s h 0to algebraic combinations of zeta values i for integeri 2 the map is not injective we show how several results about multiple harmonic series can be recast as statements about the kernel of and propose some conjectures on the structure of the algebra h 0 ker

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

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

boundedness and finite time collapse in a chemotaxis system with volume filling effect abstract we consider the elliptic parabolic pde system u t u u u v x t 0 0 v m u x t 0 with nonnegative initial data u 0 having mean value m under homogeneous neumann boundary conditions in a smooth bounded domain r n the nonlinearities and are supposed to generalize the prototypes u u 1 p u u u 1 q 1 with p 0 and q r problems of this type arise as simplified models in the theoretical description of chemotaxis phenomena under the influence of the volume filling effect as introduced by painter and hillen k j painter t hillen volume filling and quorum sensing in models for chemosensitive movement can appl math q 10 2002 501 543 it is proved that if p q 2 n then all solutions are global in time and bounded whereas if p q 2 n q 0 and is a ball then there exist solutions that become unbounded in finite time the former result is consistent with the aggregation inhibiting effect of the volume filling mechanism the latter however is shown to imply that if the space dimension is at least three then chemotactic collapse may occur despite the presence of some nonlinearities that supposedly model a volume filling effect in the sense of painter and hillen

finite time blowup and global in time unbounded solutions to a parabolic parabolic quasilinear keller segel system in higher dimensions abstract in this paper we consider quasilinear keller segel type systems of two kinds in higher dimensions in the case of a nonlinear diffusion system we prove an optimal with respect to possible nonlinear diffusions generating explosion in finite time of solutions finite time blowup result in the case of a cross diffusion system we give results which are optimal provided one assumes some proper non decay of a nonlinear chemical sensitivity moreover we show that once we do not assume the above mentioned non decay our result cannot be as strong as in the case of nonlinear diffusion without nonlinear cross diffusion terms to this end we provide an example interesting by itself of global in time unbounded solutions to the nonlinear cross diffusion keller segel system with chemical sensitivity decaying fast enough in a range of parameters in which there is a finite time blowup result in a corresponding case without nonlinear cross diffusion

double shuffle relation for associators it is proved that drinfel 0 d s pentagon equation implies the generalized double shue relation as a corollary an embedding from the grothen

boundedness blowup and critical mass phenomenon in competing chemotaxis abstract we consider the following attraction repulsion keller segel system u t u u v u w x t 0 v t v u v x t 0 0 w u w x t 0 u x 0 u 0 x v x 0 v 0 x x with homogeneous neumann boundary conditions in a bounded domain r 2 with smooth boundary the system models the chemotactic interactions between one species denoted by u and two competing chemicals denoted by v and w which has important applications in alzheimer s disease here all parameters and are positive by constructing a lyapunov functional we establish the global existence of uniformly in time bounded classical solutions with large initial data if the repulsion dominates or cancels attraction i e if the attraction dominates i e a critical mass phenomenon is found specifically speaking we find a critical mass m 4 such that the solution exists globally with uniform in time bound if m m and blows up if m m and m 4 m m n where n denotes the set of positive integers and m u 0 d x the initial cell mass

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

eigenvalues of the moment method matrix and their effect on the convergence of the conjugate gradient algorithm em scattering a theory that relates eigenvalues of a continuous operator to those of the moment method matrix operator is discussed and confirmed by examples this theory suggests reasons for ill conditioning when certain types of basis and testing functions are used in addition the effect of eigenvalue location on the convergence of the conjugate gradient cg method is studied the convergence rate of the cg method is dependent on the eigenvalues of the iteration matrix as well as on the number of eigenvectors of the iteration matrix needed to represent the right side of the equation these findings explain the previously reported convergence behavior of the cg method when applied to electromagnetic scattering problems

global existence and boundedness in a fully parabolic attraction repulsion chemotaxis system with signal dependent sensitivities and logistic source abstract this paper deals with the fully parabolic attraction repulsion chemotaxis system with signal dependent sensitivities and logistic source u t u u v v u w w u 1 u x t 0 v t v v u x t 0 w t w w u x t 0 under homogeneous neumann boundary conditions and initial conditions where r n n 2 is a bounded domain with smooth boundary are functions satisfying some conditions and 0 is a constant when are constants it is known that the above system possesses a globally bounded classical solution in some cases however there has been no work in the case that are functions this paper develops global existence and boundedness of classical solutions to the above system in such case

algebraic differential formulas for the shuffle stuffle and duality relations of iterated integrals abstract in this paper we prove certain algebraic identities which correspond to differentiation of the shuffle relation the stuffle relation and the relations which arise from mobius transformations of iterated integrals these formulas provide fundamental and useful tools in the study of iterated integrals on a punctured projective line

repulsion effects on boundedness in the higher dimensional fully parabolic attraction repulsion chemotaxis system abstract this paper deals with an attraction repulsion chemotaxis system u t d u u u v u w x t 0 1 v t v u v x t 0 2 w t w u w x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r n n 2 where parameters i i 1 2 and are positive and diffusion coefficient d u c 2 0 satisfies d u 0 for u 0 d u d u m 1 with d 0 and m 1 for all u 0 it is proved that the corresponding initial boundary value problem possesses a unique global bounded classical solution for m 2 2 n in particular in the case 1 2 and the solution is globally bounded if m 2 2 n n 2 n 2 n 2 therefore due to the inhibition of repulsion to the attraction the range of m 2 2 n of boundedness is enlarged and the results of 21 is thus extended to the higher dimensional attraction repulsion chemotaxis system with nonlinear diffusion

quadrature over a pyramid or cube of integrands with a singularity at a vertex a simple transformation is introduced which facilitates the evaluation of integrals over certain square based pyramids or cubes in cases where the integrand has a singularity at a vertex

finite time blow up for quasilinear degenerate keller segel systems of parabolic parabolic type abstract this paper deals with the quasilinear degenerate keller segel systems of parabolic parabolic type in a ball of r n n 2 in the case of non degenerate diffusion cie lak stinner 3 4 proved that if q m 2 n where m denotes the intensity of diffusion and q denotes the nonlinearity then there exist initial data such that the corresponding solution blows up in finite time as to the case of degenerate diffusion it is known that a solution blows up if q m 2 n see ishida yokota 13 however whether the blow up time is finite or infinite has been unknown this paper gives an answer to the unsolved problem indeed the finite time blow up of energy solutions is established when q m 2 n

analysis of nearly cylindrical antennas and scattering problems using a spectrum of two dimensional solutions this paper presents a powerful method for analysing antennas which can be considered principally two dimensional 2 d or cylindrical except for some three dimensional 3 d physical or equivalent sources e g dipoles or slots it is shown by fourier transform techniques that such antennas can be analyzed as 2 d problems with harmonic longitudinal field variation the radiation pattern can often be determined directly from a finite set of such 2 d solutions each one obtained by any method e g the moment method the mutual interaction between the cylindrical scatterer and the sources must be calculated to determine the exact current distribution on the sources and their impedances or admittances this is facilitated by performing an inverse fourier transform of an infinite spectrum of the numerical 2 d solutions followed by a moment method solution in the spatial domain to satisfy the boundary conditions on the 3 d equivalent sources themselves the inverse fourier transform is simplified by the use of asymptote extraction the method is in itself a hybrid technique as one method is used to solve the harmonic 2 d problem and the other to solve for the source currents

boundedness in a quasilinear parabolic parabolic keller segel system with subcritical sensitivity abstract we consider the quasilinear parabolic parabolic keller segel system u t d u u s u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a convex smooth bounded domain r n with n 1 it is proved that if s u d u c u with 2 n and some constant c 0 for all u 1 then the classical solutions to the above system are uniformly in time bounded provided that d u satisfies some technical conditions such as algebraic upper and lower growth resp decay estimates as u this boundedness result is optimal according to a recent result by the second author winkler 2010 27 which says that if s u d u c u for u 1 with c 0 and some 2 n n 2 then for each mass m 0 there exist blow up solutions with mass u 0 m in addition this paper also proves a general boundedness result for quasilinear non uniformly parabolic equations by modifying the iterative technique of moser alikakos alikakos 1979 1

boundedness and blow up in the higher dimensional attraction repulsion chemotaxis system with nonlinear diffusion abstract this paper deals with the following quasilinear attraction repulsion chemotaxis system u t d u u u v u w x t 0 v t v u v x t 0 0 w u w x t 0 in a bounded domain r n n 2 under homogeneous neumann boundary conditions the parameters and are positive and the diffusion d u is supposed to generalize the prototype d u d 0 u with d 0 0 and r under the assumption 2 n 1 it is proved that the corresponding initial boundary problem possesses a nonnegative globally bounded solution moreover by constructing a lyapunov functional and by assuming that 2 n 1 and n 3 it is proved that there exist radially symmetric solutions which blow up in finite time in b r 0 with r 0 finally it is also shown that blow up may occur if 2 n 1 and 0 when is a ball in r n with n 3

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

analysis of a rectangular waveguide edge slot array with finite wall thickness a method to analyze a one dimensional array comprising tilted edge slots cut in the narrow wall of a rectangular waveguide is presented the fields in the slots are calculated from a hybrid finite element boundary integral fe bi equation method the boundary surface integrals on both the interior and exterior surface of the slots are computed from a spectrum of two dimensional solutions s2ds which allows the influence of both the interior and exterior structure of the waveguide to be included rigorously for initial array design the mutual coupling between slots in the exterior region is treated approximately with an infinite array assumption modeled by two slots surrounded by periodic boundary walls with an imposed linear phase progression a short finite slot array is designed based on the infinite array assumption and fabricated results from full wave analysis of the array and measurements are presented

higher order mom analysis of traveling wave waveguide antennas with matched waveports the method of mode matching mm is used to construct ideal waveports in the context of the method of moments mom the waveports can provide not only excitations as sources but also perfectly matched loads for traveling wave slotted waveguide antennas additionally the application of the higher order basis functions hobs in mom significantly reduces the number of unknowns compared to the rwg basis functions numerical results show that the proposed method is accurate and efficient for traveling wave slotted waveguide antennas finally a ka band waveguide array with 1976 narrow wall slots is solved using a high performance cluster

higher order mom analysis of the rectangular waveguide edge slot arrays a comprehensive analysis is presented for the rectangular waveguide edge slot array with finite wall thickness to analyze the waveguide structure accurately and efficiently the magnetic currents are introduced on both the inner and outer surfaces of the slot apertures and the electric currents are employed on the boundary surfaces of both slot apertures and waveguide exterior area moreover the new coupled integral equations are established according to the boundary conditions and the method of moments mom with the higher order basis functions is employed to solve for the equivalent electric and magnetic currents the proposed method results in less number of unknowns without loss of accuracy compared to the direct methods with the rigorous modeling of waveguide edge slots numerical results including the radiation patterns as well as the reflection and transmission coefficients are presented

locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion abstract we show the existence of locally bounded global solutions to the chemotaxis system u t d u u u v v in 0 v t v u v in 0 u v 0 in 0 u 0 u 0 v 0 v 0 in in smooth bounded domains r n n 2 for d u u m 1 with some 0 provided that m 1 n 4

fdtd solution of maxwell s equations for an edge slot penetrating adjacent broadwalls of a finite wall thickness waveguide the finite difference time domain fdtd method is applied to predict reflection coefficients for edge slots penetrating the adjacent broadsides of thick nailed rectangular x band waveguides measured and calculated values s sub 11 are compared to experiment for waveguides terminated by shorted loads

analysis of edge slots in rectangular waveguide with finite waveguide wall thickness a novel analysis for computing the admittance characteristics of edge slots with a salient feature in successfully considering the finite waveguide wall thickness is presented the variational equation for the electric field in the slot region between the inner and outer waveguide walls is derived by applying the variational reaction theory and solved by the finite element method the division of the slot region into small elements of triangular cylinders and the basis function for each element are described in detail in the limiting case of zero wall thickness this finite element analysis is reduced to a moment method analysis using galerkin s approach with triangular basis functions the effects of waveguide wall thickness on the characteristics of the slot are investigated by comparing the numerical results with and without considering the wall thickness negligence of the wall thickness may result in a severe underestimation of the resonant length by as large as 14 for edge slots cut in typical x band waveguides the computed results with the finite wall thickness taken into account have been checked with the measured data published in previous literature and those obtained by a carefully devised experimental setup good agreement between theoretical and experimental results is obtained to demonstrate the validity of the analysis

blow up behavior of solutions to a degenerate parabolic parabolic keller segel system qualitative properties of solutions blowing up in finite time are obtained for a degenerate parabolic parabolic keller segel system the nonlinear diffusion being of porous medium type with an exponent smaller or equal to the critical one m c 2 n 1 n in both cases it is shown that only type ii blow up is possible that is blow up at the same rate as backward self similar solutions never occurs further information on the generation of singularities induced by mass concentration are given in the critical case

iterated integrals on p1 0 1 z and a class of relations among multiple zeta values abstract in this paper we consider iterated integrals on p 1 0 1 z and define a class of q linear relations among them which arises from the differential structure of the iterated integrals with respect to z we then define a new class of q linear relations among the multiple zeta values by taking their limits of z 1 which we call confluence relations i e the relations obtained by the confluence of two punctured points one of the significance of the confluence relations is that it gives a rich family and seems to exhaust all the linear relations among the multiple zeta values as a good reason for this we show that confluence relations imply both the regularized double shuffle relations and the duality relations

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

new critical exponents in a fully parabolic quasilinear keller segel system and applications to volume filling models abstract we carry on our studies related to the fully parabolic quasilinear keller segel system started in 6 and continued in 7 in the above mentioned papers we proved finite time blowup of some radially symmetric solutions to the quasilinear keller segel system if the nonlinear chemosensitivity is strong enough and an adequate relation between nonlinear diffusion and chemosensitivity holds on the other hand we proved that once chemosensitivity is weak enough solutions exist globally in time the present paper is devoted to looking for critical exponents distinguishing between those two behaviors moreover we apply our results to the so called volume filling models with a power type jump probability function the most important consequence of our investigations of the latter is a critical mass phenomenon found in dimension 2 namely we find a value m such that when the solution to the two dimensional volume filling keller segel system starts with mass smaller than m then it is bounded while for some initial data with mass exceeding m solutions are unbounded though being defined for any time t 0

moment method analysis of sidewall inclined slots in rectangular waveguides the main feature is the use of a 90 degrees wedge green s function to model the external coupling between the parts of the slot in the narrow and broad walls of the waveguide the aperture field distributions of the inclined slots are solved directly it is found that for resonant slots the aperture field is approximately of constant phase while its amplitude is nearly sinusoidal but with a slight tilt from the field distribution various characteristics of the slot can be extracted such as the conductance susceptance and resonant length the numerical computations are in good agreement with experimental results available in the literature

higher order hierarchical legendre basis functions for electromagnetic modeling this paper presents a new hierarchical basis of arbitrary order for integral equations solved with the method of moments mom the basis is derived from orthogonal legendre polynomials which are modified to impose continuity of vector quantities between neighboring elements while maintaining most of their desirable features expressions are presented for wire surface and volume elements but emphasis is given to the surface elements in this case the new hierarchical basis leads to a near orthogonal expansion of the unknown surface current and implicitly an orthogonal expansion of the surface charge in addition all higher order terms in the expansion have two vanishing moments in contrast to existing formulations these properties allow the use of very high order basis functions without introducing ill conditioning of the resulting mom matrix numerical results confirm that the condition number of the mom matrix obtained with this new basis is much lower than existing higher order interpolatory and hierarchical basis functions as a consequence of the excellent condition numbers we demonstrate that even very high order mom systems e g tenth order can be solved efficiently with an iterative solver in relatively few iterations

boundedness in quasilinear keller segel systems of parabolic parabolic type on non convex bounded domains abstract this paper deals with the quasilinear fully parabolic keller segel system u t d u u s u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a bounded domain r n with smooth boundary n n the diffusivity d u is assumed to satisfy some further technical conditions such as algebraic growth and d 0 0 which says that the diffusion is allowed to be not only non degenerate but also degenerate the global in time existence and uniform in time boundedness of solutions are established under the subcritical condition that s u d u k u e for u 0 with 2 n k 0 and e 0 when d 0 0 this paper represents an improvement of tao and winkler 17 because the domain does not necessarily need to be convex in this paper in the case r n and d 0 0 uniform in time boundedness is an open problem left in a previous paper 7 this paper also gives an answer to it in bounded domains

motivic multiple zeta values and superstring amplitudes the structure of tree level open and closed superstring amplitudes is analyzed for the open superstring amplitude we find a striking and elegant form which allows one to disentangle its expansion into several contributions accounting for different classes of multiple zeta values this form is bolstered by the decomposition of motivic multiple zeta values i e the latter encapsulate the expansion of the superstring amplitude moreover a morphism induced by the coproduct maps the expansion onto a non commutative hopf algebra this map represents a generalization of the symbol of a transcendental function in terms of elements of this hopf algebra the expansion assumes a very simple and symmetric form which carries all the relevant information equipped with these results we can also cast the closed superstring amplitude into a very elegant form

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

mixed tate motives over z we prove that the category of mixed tate motives over z is spanned by the motivic fundamental group of p 1 minus three points we prove a conjecture by m hoffman which states that every multiple zeta value is a q linear combination of n 1 n r where n i 2 3

boundedness of the attraction repulsion keller segel system a b s t r a c t this paper considers the initial boundary value problem of the attraction repulsion keller segel model describing aggregation of microglia in the central nervous system in alzheimer s disease due to the interaction of chemoattractant and chemorepellent if repulsion dominates over attraction we show the global existence of classical solution in two dimensions and weak solution in three dimensions with large initial data u x 0 u0 x v x 0 v 0 x w x 0 w 0 x x