type cotype and twisted sums induced by complex interpolation abstract this paper deals with extensions or twisted sums of banach spaces that come induced by complex interpolation and the relation between the type and cotype of the spaces in the interpolation scale and the nontriviality and singularity of the induced extension the results are presented in the context of interpolation of families of banach spaces and are applied to the study of submodules of schatten classes we also obtain nontrivial extensions of spaces without the cap which also fail the cap

the h calculus and sums of closed operators we develop a very general operator valued functional calcu lus for operators with an h 1 calculus we then apply this to the joint functional calculus of two commuting sectorial operators when one has an h 1 calculus using this we prove theorem of dore venni type on sums of commuting sectorial operators and apply our results to the problem of lp maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of ba nach spaces are essential here in the final section we exploit the special banach space structure of l1 spaces and c k spaces to obtain some more detailed results in this setting

a decomposition theorem for polymeasures we prove that every countably additive polymeasure can be decomposed in a unique way as the sum of a discrete polymeasure plus a continuous polymeasure this generalizes a previous result of saeki

stability of rolls and hexagonal patterns in non potential systems abstract the stability of rolls hexagons and non equilateral hexagonal patterns is considered in the framework of a system of amplitude equations containing non potential resonant quadratic terms the stability analysis of roll patterns reveals a new kind of instability which can generate steady non equilateral hexagons noise sensitive transitions between roll patterns or periodic oscillations the busse balloons for equilateral and non equilateral hexagons are calculated

exactly solvable one dimensional inhomogeneous models we present a simple way of constructing one dimensional inhomogeneous models random or quasiperiodic which can be solved exactly we treat the example of an ising chain in a varying magnetic field but our procedure can easily be extended to other one dimensional inhomogeneous models for all the models we can construct the free energy and its derivatives with respect to temperature can be computed exactly at one particular temperature

r bounded approximating sequences and applications to semigroups abstract it is shown that on certain banach spaces including c 0 1 and l1 0 1 there is no strongly continuous semigroup tt 0

generalization of a theorem of bohr for bases in spaces of holomorphic functions of several complex variables abstract in the first part we generalize the classical result of bohr by proving that an analogous phenomenon occurs whenever d is an open domain in c m or more generally a complex manifold and n n 0 is a basis in the space of holomorphic functions h d such that 0 1 and n z0 0 n 1 for some z0 d namely then there exists a neighborhood u of the point z0 such that whenever a holomorphic function on d has modulus less than 1 the sum of the suprema in u of the moduli of the terms of its expansion is less than 1 too in the second part we consider some natural hilbert spaces of analytic functions and derive necessary and sufficient conditions for the occurrence of bohr s phenomenon in this setting

instability of spatially periodic patterns due to a zero mode in the phase field crystal equation abstract weakly nonlinear spatially periodic patterns coupled to a goldstone zero mode of the phase field crystal model are investigated rotationally invariant equations for the dynamics of the amplitudes of a hexagonal pattern are derived first which then allows us to determine stability regions for stripes and hexagons there are parameter regimes in which all periodic patterns become unstable as a result of long wavelength instabilities generated by the zero mode

schauder bases in quotients of subspaces of l 2 x introduction questions and problems related to schauder basis are among the most classical ones in the banach space theory and have been studied from the earliest days the fact that every infinite dimensional banach space has an infinite dimensional subspace with a basis has been observed already in banach s book ba at about the same time the existence of bases in classical banach spaces such as c 0 1 and lp 0 1 was established an importance of the concept of a basis lies in the fact that it provides a natural method of approximation of vectors and operators in the space a question related to the approximation property in banach spaces was asked by mazur in the scottish book scb this question and the problem of the existence of a basis in every banach space stayed open for about 40 years although not because of the lack of interest cf e g sin in 1972 p enflo constructed a banach space which fails the approximation property and hence does not admit a schauder basis his discovery triggered a flurry of activities resulting in constructions of subspaces failing the approximation property in all classical banach spaces not isomorphic to a hilbert space such subspaces have been also found in all banach spaces sufficiently far from the hilbert space cf e g l t l l t 2 and references therein techniques developed until that moment did not provide quantitative finitedimensional estimates for basis constants it took another 10 years before a new probabilistic argument was introduced by e d gluskin g l and consequently the existence of n dimensional banach spaces with basis constant tending to infinity was proved independently by gluskin g 2 and s j szarek sz l at last a circle got closed a method of constructing infinite dimensional examples out of finite dimensional ones was proposed by j bourgain b l and developed by szarek sz 3 to obtain an 2 sum of finite dimensional banach spaces which has no basis but obviously admits a so called finite dimensional decomposition in this paper we prove a striking isomorphic characterization of a hilbert space in terms of a schauder basis first observe that the class of banach spaces isomorphic to a hilbert space is closed under operations of passing to closed

the k function for nearly regular point processes summary we propose modeling a nearly regular point pattern by a generalized neyman scott process in which the offspring are gaussian perturbations from a regular mean configuration the mean configuration of interest is an equilateral grid but our results can be used for any stationary regular grid the case of uniformly distributed points is first studied as a benchmark by considering the square of the interpoint distances we can evaluate the first two moments of the k function these results can be used for parameter estimation and simulations are used to both verify the theory and to assess the accuracy of the estimators the methodology is applied to an investigation of regularity in plumes observed from swimming microorganisms

polynomial inequalities on the hamming cube let x vert cdot vert x be a banach space the purpose of this article is to systematically investigate dimension independent properties of vector valued functions f 1 1 n rightarrow x on the hamming cube whose spectrum is bounded above or below our proofs exploit contractivity properties of the heat flow induced by the geometry of the target space x vert cdot vert x combined with duality arguments and suitable tools from approximation theory and complex analysis we obtain a series of improvements of various well studied estimates for functions with bounded spectrum including moment comparison results for low degree walsh polynomials and bernstein markov type inequalities which constitute discrete vector valued analogues of freud s inequality in gauss space 1971 many of these inequalities are new even for scalar valued functions furthermore we provide a short proof of mendel and naor s heat smoothing theorem 2014 for functions in tail spaces with values in spaces of nontrivial type and we also prove a dual lower bound on the decay of the heat semigroup acting on functions with spectrum bounded from above finally we improve the reverse bernstein markov inequalities of meyer in seminar on probability xviii lecture notes in mathematics springer berlin 1984 https doi org 10 1007 bfb0100043 and mendel and naor publ math inst hautes etudes sci 119 1 95 2014 https doi org 10 1007 s10240 013 0053 2 for functions with narrow enough spectrum and improve the bounds of filmus et al isr j math 214 1 167 192 2016 https doi org 10 1007 s11856 016 1355 0 on the ell p sums of influences of bounded functions for p in big 1 frac 4 3 big

procrustes shape analysis of planar point subsets summary consider a set of points in the plane randomly perturbed about a mean configuration by gaussian errors in this paper a procrustes statistic based on the shapes of subsets of the points is studied and its approximate distribution is found for small variations we derive various properties of the distribution including the first two moments a central limit result and a scaled x2 approximation we concentrate on the independent isotropic gaussian error case although the results are valid for general covariance structures we investigate triangle subsets in detail and in particular the situation where the population mean is regular i e a delaunay triangulation of the mean of the process is comprised of equilateral triangles of the same size we examine the variance of the statistic for differently shaped regions and provide an asymptotic result for general shaped regions the results are applied to an investigation of regularity in human muscle fibre crosssections

a tensor norm preserving unconditionality for lp spaces we show that for each n n there is an n tensor norm a in the sense of grothendieck with the surprising property that the tensor product y 1 y n has local unconditional structure for each choice of n arbitrary cp j spaces yj in fact a is the tensor norm associated to the ideal of multiple 1 summing n linear forms on banach spaces

liapunov exponents in high dimensional symplectic dynamics the existence of a thermodynamic limit of the distribution of liapunov exponents is numerically verified in a large class of symplectic models ranging from hamiltonian flows to maps and products of random matrices in the highly chaotic regime this distribution is approximately model independent near an integrable limit only a few exponents give a relevant contribution to the kolmogorov sinai entropy

on the nontrivial projection problem abstract the nontrivial projection problem asks whether every finite dimensional normed space admits a well bounded projection of nontrivial rank and corank or equivalently whether every centrally symmetric convex body of arbitrary dimension is approximately affinely equivalent to a direct product of two bodies of nontrivial dimensions we show that this is true up to a logarithmic factor

the lyapunov exponent of products of random 2 2 matrices close to the identity we study products of arbitrary random real 2 2 matrices that are close to the identity matrix using the iwasawa decomposition of sl 2 we identify a continuum regime where the mean values and the covariances of the three iwasawa parameters are simultaneously small in this regime the lyapunov exponent of the product is shown to assume a scaling form in the general case the corresponding scaling function is expressed in terms of gauss hypergeometric function a number of particular cases are also considered where the scaling function of the lyapunov exponent involves other special functions airy bessel whittaker elliptic the general solution thus obtained allows us among other things to recover in a unified framework many results known previously from exactly solvable models of one dimensional disordered systems

holomorphic semi groups and the geometry of banach spaces according to a fundamental theorem of dvoretzky cf 24 25 any infinite dimensional banach space x contains for each integer n and each e 0 a subspace xn which is 1 e isomorphic to 12n in the classical banach spaces lp and 1p with 1 p x it is well known that this result can be improved these spaces contain uniformly complemented ln 7 s this means that we can find subspaces xn as above with the additional property that there exist projections p x xn such that supn 11 pn 11 cx moreover it is also well known that this property no longer holds for 11 or lo this motivated the question cf 13 to determine which infinite dimensional banach spaces contain uniformly complemented 1 n s we will prove below that all uniformly convex spaces have this property moreover we will show that a banach space x verifies this property locally see definition 2 10 for more precision if and only if x does not contain l n s uniformly progressively the work of previous authors cf 14 12 and particularly 5 has reduced such so called geometric questions to a purely analytic problem consisting of verifying in x a certain inequality referred to as k convexity in this paper we connect this inequality with the holomorphy of a certain semi group of operators on lp x and we show how this leads to the results already mentioned

weak disorder expansion of liapunov exponents in a degenerate case it is shown how the weak disorder expansion of the liapunov exponents of a product of random matrices can be derived when the unperturbed matrices have two degenerate eigenvalues the general expression of the liapunov exponents at the lowest nontrivial order in disorder is given

dynamics of an epidemic model with spatial diffusion mathematical models are very useful in analyzing the spread and control of infectious diseases which can be used to predict the developing tendency of the infectious disease determine the key factors and to seek the optimum strategies of disease control as a result we investigated the pattern dynamics of a spatial epidemic model with logistic growth by using amplitude equation we found that there were different types of stationary patterns including spotted mixed and stripe patterns which mean that spatial motion of individuals can form high density of diseases the obtained results can be extended in other related fields such as vegetation patterns in ecosystems

type and cotype in vector valued nakano sequence spaces abstract given a sequence of banach spaces xn n and a sequence of real numbers pn n in 1 the vector valued nakano sequence spaces l pn xn consist of elements xn n in nxn for which there is a constant 0 such that n xn pn

perturbations of random matrix products if x1 x2 are identically distributed independent random matrices with a common distribution then with the probability 1 the limit lambda mu mathop lim limits n to infty frac 1 n ln left x n x 1 right exists the paper treats the problem is it true that k if k in the weak sense

isomorphic properties of intersection bodies we study isomorphic properties of two generalizations of intersection bodies the class ikn of k intersection bodies in rn and the class bpkn of generalized k intersection bodies in rn in particular we show that all convex bodies can be in a certain sense approximated by intersection bodies namely if k is any symmetric convex body in rn and 1 k n 1 then the outer volume ratio distance from k to the class bpkn can be estimated by o v r k bpkn inf c k 1n c bpkn k c cnklogenk where c 0 is an absolute constant next we prove that if k is a symmetric convex body in rn 1 k n 1 and its k intersection body ik k exists and is convex then dbm ik k b2n c k where c k is a constant depending only on k dbm is the banach mazur distance and b2n is the unit euclidean ball in rn this generalizes a well known result of hensley and borell we conclude the paper with volumetric estimates for k intersection bodies

pattern formation by super diffusion in fitzhugh nagumo model abstract the aim of this paper is to examine the turing instability and pattern formation in the fitzhugh nagumo model with super diffusion in two dimensional numerical simulation we also studied the effects of the super diffusive exponent on pattern formation concluding that with the presence of super diffusion the stable homogenous steady state becomes unstable by using the stability analysis of local equilibrium point we procure the conditions which ensure that the turing and hopf bifurcations occur for pattern selection the weak nonlinear multi scale analysis is used to derive the amplitude equations of the stationary patterns we then apply amplitude equations and observe that this model has very rich dynamical behaviors such as stripes spots and hexagon patterns the complexity of the dynamics in this system is theoretically discussed and graphically displayed in numerical simulation the simulation helps us to show the effectiveness of theoretical analysis and patterns which appear numerically

functional calculus under the tadmor ritt condition and free interpolation by polynomials of a given degree for banach space operators t satisfying the tadmor ritt condition zi t 1 c z 1 1 z 1 we prove that the best possible constant ct n bounding the polynomial calculus for t p t ct n p deg p n behaves in the worst case as logn as n this result is based on a new free carleson type interpolation theorem for polynomials of a given degree

equivalence of kinetic theory and random matrix approaches to lyapunov spectra of hard sphere systems abstract in the study of chaotic behaviour of systems of many hard spheres lyapunov exponents of small absolute values exhibit interesting characteristics leading to speculations about connections to non equilibrium statistical mechanics analytical approaches to these exponents so far can be divided into two groups macroscopically oriented approaches using kinetic theory or hydrodynamics and more microscopically oriented random matrix approaches in quasi one dimensional systems in this paper i present an approach using random matrices and weak disorder expansion in an arbitrary number of dimensions correlations between subsequent collisions of a particle are taken into account it is shown that the results are identical to those of a previous approach based on an extended enskog equation i conclude that each approach has its merits and provides different insights into the approximations made which include the stoszahlansatz the continuum limit and the long wavelength approximation the comparison also gives insight into possible connections between lyapunov exponents and fluctuations

the distribution of lyapunov exponents exact results for random matrices simple exact expressions are derived for all the lyapunov exponents of certainn dimensional stochastic linear dynamical systems in the case of the product of independent random matrices each of which has independent gaussian entries with mean zero and variance 1 n the exponents have an exponential distribution asn in the case of the time ordered product integral of exp n 1 2dw where the entries of then n matrixw t are independent standard wiener processes the exponents are equally spaced for fixedn and thus have a uniform distribution as n

hyperreflexivity and operator ideals abstract suppose b is an operator ideal and a is a linear space of operators between banach spaces x and y modifying the classical notion of hyperreflexivity we say that a is called b hyperreflexive if there exists a constant c such that for any t b x y with sup q t i the supremum runs over all isometric embeddings i into x and all quotient maps of y satisfying q a i 0 there exists a a for which t a c in this paper we give examples of b hyperreflexive spaces as well as of spaces failing this property in the last section we apply s e hyperreflexivity of operator algebras s e is a regular symmetrically normed operator ideal to constructing operator spaces with prescribed families of completely bounded maps

lyapunov exponents for some isotropic random matrix ensembles a random matrix with rows distributed as a function of their length is said to be isotropic when these distributions are gaussian beta type i or beta type ii previous work has from the viewpoint of integral geometry obtained the explicit form of the distribution of the determinant we use these result to evaluate the sum of the lyapunov spectrum of the corresponding random matrix product and we further give explicit expressions for the largest lyapunov exponent generalisations to the case of complex or quaternion entries are also given for standard gaussian matrices x the full lyapunov spectrum for products of random matrices i n 1 over c x is computed in terms of a generalised hypergeometric function in general and in terms of a single single integral involving a modified bessel function for the largest lyapunov exponent

gaussian processes and mixed volumes on demontre la conjecture de dudley si l exposant du volume d un sous ensemble symetrique et convexe compact d un espace de hilbert est 1 alors k est un ensemble g c

perturbation theory for generalized procrustes analysis summary perturbation theory for procrustes statistics is developed when both configurations are subject to independent normal errors the total sum of squares is shown to partition into sums of squares for translation rotation reflection and scaling plus a residual these results are extended to generalized procrustes analysis this gives a theoretical justification to gower s analysis of variance summary of the results from a generalized procrustes analysis

rademacher averages on noncommutative symmetric spaces let e be a separable or the dual of a separable symmetric function space let m be a semifinite von neumann algebra and let e m be the associated noncommutative function space let ek k 1 be a rademacher sequence on some probability space for finite sequences xk k 1 of e m we consider the rademacher averages kek xk as elements of the noncommutative function space e l m and study estimates for their norms kek xk e calculated in that space we establish general khintchine type inequalities in this context then we show that if e is 2 concave kek xk e is equivalent to the infimum of yk yk 12 zkzk 12 over all yk zk in e m such that xk yk zk for any k 1 dual estimates are given when e is 2 convex and has a nontrivial upper boyd index in this case kek xk e is equivalent to xk xk 12 xkxk 12 we also study rademacher averages i jei ej xij for doubly indexed families xij i j of e m

amplitude equations for reaction diffusion systems with cross diffusion using taylor series expansion multiscaling and further expansion in powers of a small parameter we develop general amplitude equations for two variable reaction diffusion systems with cross diffusion terms in the cases of hopf and turing instabilities we apply this analysis to the oregonator and brusselator models and find that inhibitor cross diffusion induced by the activator and activator cross diffusion induced by the inhibitor have opposite effects in the two models as a result of the different structure of their community matrices our analysis facilitates finding regions of supercritical and subcritical bifurcations as well as wave and antiwave domains and domains of turbulent waves in the case of hopf instability

theory and application of shapelets to the analysis of surface self assembly imaging a method for quantitative analysis of local pattern strength and defects in surface self assembly imaging is presented and applied to images of stripe and hexagonal ordered domains the presented method uses shapelet functions which were originally developed for quantitative analysis of images of galaxies ensuremath propto 10 20 mathrm m in this work they are used instead to quantify the presence of translational order in surface self assembled films ensuremath propto 10 ensuremath 9 mathrm m through reformulation into steerable filters the resulting method is computationally efficient with respect to the number of filter evaluations robust to variation in pattern feature shape and unlike previous approaches is applicable to a wide variety of pattern types an application of the method is presented which uses a nearest neighbor analysis to distinguish between uniform defect free and nonuniform strained defect containing regions within imaged self assembled domains both with striped and hexagonal patterns

banach spaces in various positions we formulate a general theory of positions for subspaces of a banach space we define equivalent and isomorphic positions study the automorphy index a y x that measures how many non equivalent positions y admits in x and obtain estimates of a y x for x a classical banach space such as lp lp l1 c or c 0 1 then we study different aspects of the automorphic space problem posed by lindenstrauss and rosenthal namely does there exist a separable automorphic space different from c0 or l2 recall that a banach space x is said to be automorphic if every subspace y admits only one position in x i e a y x 1 for every subspace y of x we study the notion of extensible space and uniformly finitely extensible space ufo which are relevant since every automorphic space is extensible and every extensible space is ufo we obtain a dichotomy theorem every ufo must be either an l space or a weak type 2 near hilbert space with the maurey projection property we show that a banach space all of whose subspaces are ufo called hereditarily ufo spaces must be asymptotically hilbertian while a banach space for which both x and x are ufo must be weak hilbert we then refine the dichotomy theorem for banach spaces with some additional structure in particular we show that an ufo with unconditional basis must be either c0 or a superreflexive weak type 2 space that a hereditarily ufo kothe function space must be hilbert and that a rearrangement invariant space ufo must be either l or a superreflexive type 2 banach lattice

new families of nonreflexive banach spaces with the fixed point property abstract in this paper we define the concept of sequentially separating norm and we relate it to the fulfillment of the fixed point property for nonexpansive mappings we develop a technique to construct families of nonreflexive banach spaces with the fixed point property different examples will be exhibited we also study the geometry of the banach spaces that can be renormed with a sequentially separating norm

global feedback control for pattern forming systems global feedback control of pattern formation in a wide class of systems described by the swift hohenberg sh equation is investigated theoretically by means of stability analysis and numerical simulations two cases are considered i feedback control of the competition between hexagon and roll patterns described by a supercritical sh equation and ii the use of feedback control to suppress the blowup in a system described by a subcritical sh equation in case i it is shown that feedback control can change the hexagon and roll stability regions in the parameter space as well as cause a transition from up to down hexagons and stabilize a skewed mixed mode hexagonal pattern in case ii it is demonstrated that feedback control can suppress blowup and lead to the formation of spatially localized patterns in the weakly nonlinear regime the effects of a delayed feedback are also investigated for both cases and it is shown that delay can induce temporal oscillations as well as blowup

multivariate aspects of shape theory sis and thus provide a comprehensive view of shape distributions when landmark coordinates are gaussian distributed this work allows the statistical analysis of shape to be carried out using standard techniques of multivariate analysis the paper includes some new results in all dimensions and a general gaussian approximation to the size and shape distribution we also discuss some inference problems and give a numerical example

unconditional basis and gordon lewis constants for spaces of polynomials abstract no infinite dimensional banach space x is known which has the property that for m 2 the banach space of all continuous m homogeneous polynomials on x has an unconditional basis following a program originally initiated by gordon and lewis we study unconditionality in spaces of m homogeneous polynomials and symmetric tensor products of order m in banach spaces we show that for each banach space x which has a dual with an unconditional basis x i the approximable nuclear m homogeneous polynomials on x have an unconditional basis if and only if the monomial basis with respect to x i is unconditional moreover we determine an asymptotically correct estimate for the unconditional basis constant of all m homogeneous polynomials on l n p and use this result to narrow down considerably the list of natural candidates x with the above property

the ritt property of subordinated operators in the group case abstract let g be a locally compact abelian group let be a regular probability measure on g let x be a banach space let g b x be a bounded strongly continuous representation consider the average or subordinated operator s g t d t x x we show that if x is a umd banach lattice and has bounded angular ratio then s is a ritt operator with a bounded h functional calculus next we show that if is the square of a symmetric probability measure and x is k convex then s is a ritt operator we further show that this assertion is false on any non k convex space x

forced patterns near a turing hopf bifurcation we study time periodic forcing of spatially extended patterns near a turing hopf bifurcation point a symmetry based normal form analysis yields several predictions including that i weak forcing near the intrinsic hopf frequency enhances or suppresses the turing amplitude by an amount that scales quadratically with the forcing strength and ii the strongest effect is seen for forcing that is detuned from the hopf frequency to apply our results to specific models we perform a perturbation analysis on general two component reaction diffusion systems which reveals whether the forcing suppresses or enhances the spatial pattern for the suppressing case our results are consistent with features of previous experiments on the chlorine dioxide iodine malonic acid chemical reaction however we also find examples of the enhancing case which has not yet been observed in experiment numerical simulations verify the predicted dependence on the forcing parameters

reductive renormalization of the phase field crystal equation it has been known for some time that singular perturbation and reductive perturbation can be unified from the renormalization group theoretical point of view reductive extraction of space time global behavior is the essence of singular perturbation methods reductive renormalization was proposed to make this unification practically accessible actually this reductive perturbation is far simpler than most reduction methods such as the rather standard scaling expansion however a rather cryptic exposition of the method seems to have been the cause of some trouble here an explicit demonstration of the consistency of the reductive renormalization group procedure is given for partial differentiation equations of a certain type including time evolution semigroup type equations then the procedure is applied to the reduction of a phase field crystal equation to illustrate the streamlined reduction method we conjecture that if the original system is structurally stable the reductive renormalization group result and that of the original equation are diffeomorphic

pattern formation in a predator prey model with spatial effect abstract in this paper spatial dynamics in the beddington deangelis predator prey model with self diffusion and cross diffusion is investigated we analyze the linear stability and obtain the condition of turing instability of this model moreover we deduce the amplitude equations and determine the stability of different patterns numerical simulations show that this system exhibits complex dynamical behaviors in the turing space we find three types of typical patterns one is the coexistence of hexagon patterns and stripe patterns the other two are hexagon patterns of different types the obtained results well enrich the finding in predator prey models with beddington deangelis functional response

the norm of sums of independent noncommutative random variables in lp l1 we investigate the norm of sums of independent vector valued random variables in noncommutative lp spaces this allows us to obtain a uniform family of complete embeddings of the schatten class s n q in sp m q with optimal order m n 2 using these embeddings we show the surprising fact that the sharp type cotype index in the sense of operator spaces for lp 0 1 is min p p 0 max p p 0 similar techniques are used to show that the operator space notions of b convexity and k convexity are equivalent

twisted sums with c k spaces if x is a separable banach space we consider the existence of non trivial twisted sums 0 c k y x 0 where k 0 1 or for the case k 0 1 we show that there exists a twisted sum whose quotient map is strictly singular if and only if x contains no copy of 1 if k we prove an analogue of a theorem of johnson and zippin for k 0 1 by showing that all such twisted sums are trivial if x is the dual of a space with summable szlenk index e g x could be tsirelson s space a converse is established under the assumption that x has an unconditional finite dimensional decomposition we also give conditions for the existence of a twisted sum with c with strictly singular quotient map

simple and superlattice turing patterns in reaction diffusion systems bifurcation bistability and parameter collapse abstract we use equivariant bifurcation theory to investigate pattern selection at the onset of a turing instability in a general two component reaction diffusion system the analysis is restricted to patterns that periodically tile the plane in either a square or hexagonal fashion both simple periodic patterns stripes squares hexagons and rhombs and superlattice patterns are considered the latter correspond to patterns that have structure on two disparate length scales the short length scale is dictated by the critical wave number from linear theory while the periodicity of the pattern is on a larger scale analytic expressions for the coefficients of the leading nonlinear terms in the bifurcation equations are computed from the general reaction diffusion system using perturbation theory we show that no matter how complicated the reaction kinetics might be the nonlinear reaction terms enter the analysis through just four parameters moreover for hexagonal problems all patterns bifurcate unstably unless a particular degeneracy condition is satisfied and at this degeneracy we find that the number of effective system parameters drops to two allowing a complete characterization of the possible bifurcation results at this degeneracy for example we find that rhombs squares and superlattice patterns always bifurcate unstably we apply these general results to some specific model equations including the lengyel epstein cima model to investigate the relative stability of patterns as a function of system parameters and to numerically test the analytical predictions

improved bounds in the metric cotype inequality for banach spaces abstract it is shown that if x x is a banach space with rademacher cotype q then for every integer n there exists an even integer m n 1 1 q such that for every f z m n x we have 1 j 1 n e x f x m 2 e j f x x q m q e e x f x e f x x q where the expectations are with respect to uniformly chosen x z m n and e 1 0 1 n and all the implied constants may depend only on q and the rademacher cotype q constant of x this improves the bound of m n 2 1 q from mendel and naor 2008 13 the proof of 1 is based on a smoothing and approximation procedure which simplifies the proof of the metric characterization of rademacher cotype of mendel and naor 2008 13 we also show that any such smoothing and approximation approach to metric cotype inequalities must require m n 1 2 1 q

a solution to the problem of l p maximal regularity we give a negative solution to the problem of the l p maximal regularity on various classes of banach spaces including l q spaces with 1 q neq 2 infty

interpolative construction and the generalized cotype of abstract lorentz spaces the banach operator ideals generated by an interpolative construction depending on concave functions are studied these ideals are described in terms of factorization through abstract interpolation lorentz spaces the abstract notion of rademacher type and cotype for operators between banach spaces is introduced it is shown that abstract interpolation lorentz spaces that appeared in the factorization theorem are of the generalized rademacher cotype determined by orlicz sequence spaces

markov chains in smooth banach spaces and gromov hyperbolic metric spaces a metric space x has markov type 2 if for any reversible flnite state markov chain fztg with z0 chosen according to the stationary distribution and any map f from the state space to x the distance dt from f z0 to f zt satisfles e d 2 k 2 te d 2 for some k k x 2 has markov type 2 this proves a conjecture of ball we also show that trees hyperbolic groups and simply connected riemannian manifolds of pinched negative curvature have markov type 2 our results are applied to settle several conjectures on lipschitz extensions and embeddings in particular we answer a question posed by johnson and lindenstrauss in 1982 by showing that for 1 q 2 p 1 any lipschitz mapping from a subset of lp to lq has a lipschitz extension deflned on all of lp

turing pattern dynamics in an activator inhibitor system with superdiffusion the fractional operator is introduced to an activator inhibitor system to describe species anomalous superdiffusion the effects of the superdiffusive exponent on pattern formation and pattern selection are studied our linear stability analysis shows that the wave number of the turing pattern increases with the superdiffusive exponent a weakly nonlinear analysis yields a system of amplitude equations and the analysis of these amplitude equations predicts parameter regimes where hexagons stripes and their coexistence are expected numerical simulations of the activator inhibitor model near the stability boundaries confirm our analytical results since diffusion rate manifests in both diffusion constant and diffusion exponent we numerically explore their interactions on the emergence of turing patterns when the activator and inhibitor have different superdiffusive exponents we find that the critical ratio of the diffusion rate of the inhibitor to the activator required for the formation of the turing pattern increases monotonically with the superdiffusive exponent we conclude that small ratio than unity of anomalous diffusion exponent between the inhibitor and activator is more likely to promote the emergence of the turing pattern relative to the normal diffusion

volume estimates in spaces of homogeneous polynomials given m we derive upper and lower estimates for the volume of the unit ball of the banach space mathcal p m e n all m homogeneous polynomials defined on the nth section e n of a symmetric banach sequence space e with constants depending on m only we apply this result to obtain asymptotically correct orders of various important invariants of mathcal p m e n from local banach space theory as e g volume ratios cotype and type 2 constants projection constants or banach mazur distances

turing patterns with o 3 symmetry abstract we perform an explicit center manifold reduction for the general steady state bifurcation of a chemical reaction diffusion system in an arbitrary compact domain in particular we focus on those systems with two chemical species in domains with o 3 symmetry we illustrate with the brusselator on the spherical shell and produce explicit computations of the solution branches and their stabilities for the low l bifurcations because of the phenomenon of parameter collapse we can create universal bifurcation diagrams for even l this work is an extension of that done in physica d 132 1999 339 362

morphometric tools for landmark data geometry and biology 1 introduction 2 preliminaries 3 landmarks 4 distance measures 5 shape coordinates 6 principal axes of shape change for triangles 7 features of shape comparison 8 retrospect and prospect

the bohnenblust hille inequality for homogeneous polynomials is hypercontractive the bohnenblust hille inequality says that the 2m m 1 norm of the coefcients of an m homogeneous polynomial p on c n is bounded by kpk1 times a constant independent of n wherekk 1 denotes the supremum norm on the polydisc d n the main result of this paper is that this inequality is hypercontractive i e the constant can be taken to be c m for some c 1 combining this improved version of the bohnenblust hille inequality with other results we obtain the following the bohr radius for the polydisc d n behaves asymptotically as p logn n modulo a factor bounded away from 0 and innity

rigorous bounds and the replica method for products of random matrices the authors first develop a method to obtain rigorous bounds for the lyapounov exponent of products of a random matrix when applied to a class of 1d problems including localisation it reproduces the correct scaling behaviour at the band edge and gives very good approximations of the prefactors they then study analytically the successive moments of the distribution law for the trace of the random matrix product within the whole energy band the band centre anomaly is found to affect the whole statistics of the problem and the exact anomalous value of the lyapounov exponent is recovered through the replica trick

loi des grands nombres et perturbations pour des produits reductibles de matrices aleatoires independantes soit x x n n 1 une suite de matrices aleatoires reelles d d independantes de meme loi on etudie le comportement asymptotique de la suite 1 n log x n x 1 x n

pattern formation in the fitzhugh nagumo model in this paper we investigate the effect of diffusion on pattern formation in fitzhugh nagumo model through the linear stability analysis of local equilibrium we obtain the condition how the turing bifurcation hopf bifurcation and the oscillatory instability boundaries arise by using the method of the weak nonlinear multiple scales analysis and taylor series expansion we derive the amplitude equations of the stationary patterns the analysis of amplitude equations shows the occurrence of different complex phenomena including turing instability eckhaus instability and zigzag instability in addition we apply this analysis to fitzhugh nagumo model and find that this model has very rich dynamical behaviors such as spotted stripe and hexagon patterns finally the numerical simulation shows that the analytical results agree with numerical simulation

set of measures to analyze the dynamics of nonequilibrium structures we present a class of statistical measures that can be used to quantify nonequilibrium surface growth they are used to deduce information about spatiotemporal dynamics of model systems for spinodal decomposition and surface deposition pattern growth in the cahn hilliard equation used to model spinodal decomposition are shown to exhibit three distinct stages two models of surface growth namely the continuous kardar parisi zhang model and the discrete restricted solid on solid model are shown to have different saturation exponents

on the weak amenability of a x and its relation with the approximation property abstract we investigate the weak amenability of the algebra a x of approximable operators on a banach space x and its relation with the bounded approximation property in particular it will be shown that the bounded approximation property is neither necessary nor sufficient for the weak amenability of a x

weak disorder expansions for the anderson model on a one dimensional strip at the center of the band we study the asymptotic behavior of the averaged diagonal matrix elements of the greens kernel for the anderson model on a one dimensional strip and for a set of special energies close to the center of the band

cross diffusion induced turing instability for a competition model with saturation effect abstract competitive behavior like predation widely exists in the world which has influences on the dynamics of ecosystems though a good knowledge for the patten formation and selection of predator prey models with diffusion is known in the literature little is known for that of a diffused competition model as a result a competition model with saturation effect and self and cross diffusion is proposed and analyzed in this paper firstly by making use of nullcline analysis we derive the condition of existence and stability of positive equilibrium and prove global stability then using the linear stability analysis the turing bifurcation critical value and the condition of the occurrence of turing pattern are obtained when control parameter is selected finally we deduce the amplitude equations around the turing bifurcation point by using the standard multiple scale analysis meanwhile a series of numerical simulations are given to expand our theoretical analysis this work shows that cross diffusion plays a key role in the formation of spatial patterns for competitive model with saturation effect

shape analysis of paired landmark data summary this paper deals with the statistical analysis of paired landmark data using the exact distribution theory for the shape of planar correlated gaussian configurations various properties of the paired shape distribution are investigated and some special cases given this work is a natural progression of what has been done on only univariate shape analysis however one of the fundamental biological applications in shape analysis is in analysing shape change and this work offers a possible solution we illustrate our method by examining shape change in rat calvaria

malliavin calculus and decoupling inequalities in banach spaces abstract we develop a theory of malliavin calculus for banach space valued random variables using radonifying operators instead of symmetric tensor products we extend the wiener ito isometry to banach spaces in the white noise case we obtain two sided l p estimates for multiple stochastic integrals in arbitrary banach spaces it is shown that the malliavin derivative is bounded on vector valued wiener ito chaoses our main tools are decoupling inequalities for vector valued random variables in the opposite direction we use meyer s inequalities to give a new proof of a decoupling result for gaussian chaoses in umd banach spaces

zeta function for the lyapunov exponent of a product of random matrices a cycle expansion for the lyapunov exponent of a product of random matrices is derived the formula is nonperturbative and numerically effective which allows the lyapunov exponent to be computed to high accuracy in particular the free energy and heat capacity are computed for the one dimensional ising model with quenched disorder the formula is derived by using a bernoulli dynamical system to mimic the randomness

quadratic forms in random variables theory and applications textbook for a one semester graduate course for students specializing in mathematical statistics or in multivariate analysis or reference for theoretical as well as applied statisticians confines its discussion to quadratic forms and second degree polynomials in real normal random vectors and matr

type and cotype of musielak orlicz sequence spaces let n be a musielak orlicz function and l the musielak orlicz sequence space generated by it is shown that satisfies condition qiff is equivalent to aq concave function n it is also proved thatl is of cotypeq typep if and only if satisfies condition q 2and pconditions respectively as applications we obtain characterization of the type and cotype of various sequence spaces nakano weighted orlicz and orlicz sequence spaces furthermore a relation betweenq concavity p convexity lowerq estimate upperp estimate and cotype type of l is given

volume ratio and other s numbers of operators related to local properties of banach spaces abstract volume ratio numbers are studied and inequalities relating them to other s numbers of operators are established characterizations of spaces of weak cotype 2 in terms of s numbers and of operator ideal norms are obtained

weak disorder expansion of lyapunov exponents of products of random matrices a degenerate theory the weak disorder expansion of lyapunov exponents of products of random matrices is derived by a new method our treatment can be easily generalized to the problem when in the limit of zero randomness two eigenvalues of the matrices are equal for real degenerate matrices the formula for the leading term of the lyapunov exponent is derived it has the form of a continuous fraction which converges quickly to the exact value

global feedback control of a long wave morphological instability abstract nonlinear dynamics of a long wave morphological instability of a crystallization front during directional solidification in the presence of feedback control is investigated feedback control is applied by varying the pulling speed and the temperature gradient according to the measured maximum depressions of the crystal melt interface for the case of small segregation coefficient a long scale nonlinear evolution equation for the shape of the interface is derived in the frozen temperature approximation the equation contains a nonlocal global control term and describes the near threshold nonlinear dynamics of the morphological instability in the presence of the feedback control this equation is studied analytically and numerically both for 1 1 and 2 1 interfaces it is shown that the chosen global feedback control can prevent the blow up of subcritical and supercritical solutions i e the formation of deep cells it leads to the formation of stationary periodic patterns or spatially localized structures