the henselization of a valued division algebra the main result of this paper is the following if f v is a valued field fh its henselization d a division algebra with center f and d f then v extends to d iff d ffh is a division algebra it is also shown that if p char f p 1if char f 0 then d f d f d f pb for some nonnegative integer b

the defektsatz for central simple algebras let q be a central simple algebra finite dimensional over its center f and let v be a valuation ring of f the v has en extension to q i e there exists a dubrovin valuation ring b of q satisfying v f b generally the number of extensions of v to q is not finite and therefore the so called intersection property of dubrovin valuation rings b 1 b n is introduced this property is defined in terms of the prime ideals and the valuation overrings of the intersection b b n it is shown that there exists a uniquely determined natural number n depending only on v and having the following property if b 1 b k are extensions of v having the intersection property then k n and k n holds if and only if b 1 b k is integral over v

exact results for the kuramoto model with a bimodal frequency distribution we analyze a large system of globally coupled phase oscillators whose natural frequencies are bimodally distributed the dynamics of this system has been the subject of long standing interest in 1984 kuramoto proposed several conjectures about its behavior ten years later crawford obtained the first analytical results by means of a local center manifold calculation nevertheless many questions have remained open especially about the possibility of global bifurcations here we derive the system s stability diagram for the special case where the bimodal distribution consists of two equally weighted lorentzians using an ansatz recently discovered by ott and antonsen we show that in this case the infinite dimensional problem reduces exactly to a flow in four dimensions depending on the parameters and initial conditions the long term dynamics evolves to one of three states incoherence where all the oscillators are desynchronized partial synchrony where a macroscopic group of phase locked oscillators coexists with a sea of desynchronized ones and a standing wave state where two counter rotating groups of phase locked oscillators emerge analytical results are presented for the bifurcation boundaries between these states similar results are also obtained for the case in which the bimodal distribution is given by the sum of two gaussians

hole structures in nonlocally coupled noisy phase oscillators we demonstrate that a system of nonlocally coupled noisy phase oscillators can collectively exhibit a hole structure which manifests itself in the spatial phase distribution of the oscillators the phase model is described by a nonlinear fokker planck equation which can be reduced to the complex ginzburg landau equation near the hopf bifurcation point of the uniform solution by numerical simulations we show that the hole structure clearly appears in the space dependent order parameter which corresponds to the nozaki bekki hole solution of the complex ginzburg landau equation

on dubrovin valuation rings in crossed product algebras let f be a field and let v be a valuation ring in f if a is a central simple f algebra then v can be extended to a dubrovin valuation ring in a in this paper we consider the structure of dubrovin valuation rings with center v in crossed product algebras k f g f where k f is a finite galois extension with galois group g unramified over v and f is a normalized two cocycle in the case where v is indecomposed in k we introduce a family of orders naturally associated to f examine their basic properties and determine which of these orders is dubrovin in the case where v is decomposed we determine the structure in the case of certain special discrete finite rank valuations

integral semihereditary orders inside a bezout maximal order in this paper we show that over a commutative valuation ringv a semihereditaryv orderhin a finite dimensional simple artinian ring is a finite intersection of bezout maximal v orders iff eitherhis bezout orhis contained in a bezoutv order andj v is a principal ideal ofv

integral semihereditary orders extremality and henselization abstract in this paper we study integral semihereditary orders over a valuation ring in a finite dimensional simple artinian ring in the first section we prove that such orders are extremal consequently in a central division algebra admitting a total valuation ring the intersection of all the conjugates of the total valuation ring is the unique integral semihereditary order over the center of the total valuation ring in the second section we characterize up to conjugacy integral semihereditary orders over a henselian valuation ring in the last section we show that an integral order r over an arbitrary valuation ring v is semihereditary iff its henselization r v v h where v h is the henselization of v is a semihereditary v h order in this case there is an inclusion preserving bijective correspondence between semihereditary v orders inside r and semihereditary v h orders inside r v v h

maximal orders over valuation rings abstract in this paper we study maximal orders over commutative valuation rings in central simple algebras we are particularly interested in maximal orders which are either bezout or semihereditary we construct a class of bezout maximal orders and a class of semihereditary maximal orders and show that for any valuation ring v resp v with value group z m any bezout resp semihereditary maximal order over v belongs to the class constructed furthermore we classify all maximal orders in m2 f over a valuation ring with value group z m and in mn f given a mild defectless assumption

the dynamics of chimera states in heterogeneous kuramoto networks abstract we study a variety of mixed synchronous incoherent chimera states in several heterogeneous networks of coupled phase oscillators for each network the recently discovered ott antonsen ansatz is used to reduce the number of variables in the partial differential equation pde governing the evolution of the probability density function by one resulting in a time evolution pde for a variable with as many spatial dimensions as the network bifurcation analysis is performed on the steady states of these pdes the results emphasise the commonality of the dynamics of the different networks and provide stability information that was previously inferred

trees and valuation rings a subring b of a division algebra d is called a valuation ring of d if x e b or x1 e b holds for all nonzero x in d the set 13 of all valuation rings of d is a partially ordered set with respect to inclusion having d as its maximal element as a graph 13 is a rooted tree called the valuation tree of d and in contrast to the commutative case 13 may have finitely many but more than one vertices this paper is mainly concerned with the question of whether each finite rooted tree can be realized as a valuation tree of a division algebra d and one main result here is a positive answer to this question where d can be chosen as a quaternion division algebra over a commutative fielcl

bistable chimera attractors on a triangular network of oscillator populations we study a triangular network of three populations of coupled phase oscillators with identical frequencies the populations interact nonlocally in the sense that all oscillators are coupled to one another but more weakly to those in neighboring populations than to those in their own population this triangular network is the simplest discretization of a continuous ring of oscillators yet it displays an unexpectedly different behavior in contrast to the lone stable chimera observed in continuous rings of oscillators we find that this system exhibits two coexisting stable chimeras both chimeras are as usual born through a saddle node bifurcation as the coupling becomes increasingly local in nature they lose stability through a hopf bifurcation giving rise to breathing chimeras which in turn get destroyed through a homoclinic bifurcation remarkably one of the chimeras reemerges by a reversal of this scenario as we further increase the locality of the coupling until it is annihilated through another saddle node bifurcation

a note on valued division algebras let d be a finite dimensional division algebra with center f and let b be a valuation ring of d i e xeb or x 1 e b holds for all nonzero x in d then b is called an extension of v f b let s be the set of all extensions of v to d and let p b be the group of all bijections of s which are induced by the inner automorphisms of d in this note we investigate p b it is shown that p b is solvable and an upper bound for p b is given finally theorem 4 5 gives a necessary and sufficient condition for b such that p b is equal to this bound

small time two sided lil behavior for levy processes at zero we specify a function b0 t in terms of the levy triplet such that lim sup t 0xt b0 t 1 1 8 a s iff int 0 1 overline varpi b 0 t dt infty for any levy process x with unbounded variation and a brownian component 0 we show with an example that there are cases where lim sup t 0xt b t 1 a s but b t is not asymptotically equivalent to b0 t as t tends to 0 we achieve this by introducing an integral criterion which checks whether lim sup t 0xt b t is 0 infinity or a finite positive value for b t satisfying very mild conditions and any levy process

on semi hereditary orders integral over a commutative valuation ring in this paper we study semi hereditary v orders in a simple artinian ring q with finite dimension over its center where v is a commutative valuation ring let s be a minimal v overring of a semihereditary v order r in section 1 we investigate some relations between all maximal ideals of s and r and characterize the commutativities of idempotent maximal ideals of r in terms of orders of ideals in section 2 we show that there is a bijection between the set of all v overrings of r and the set of all idempotent ideals which are finitely generated as left ideals any element in the latter set is characterized by four different types of cycles in section 3 we discuss the principalness of the jacobson radical j r some results in sections 1 3 are used to derive the exact numbers of all semi hereditary maximal v orders containing r and of all v overrings of r and to study the nilpotency of j r modulo j v r some invariant properties of semi hereditary v orders are also given

division algebras over henselian fields in this chapter we focus on the tame division algebras d with center a field f with henselian valuation v as usual we approach this by first obtaining results for graded division algebras then lifting back from operatorname mathsf gr d to d this is facilitated by results in 8 1 on existence and uniqueness of lifts of tame subalgebras from operatorname mathsf gr d to d in 8 2 we describe four fundamental canonical up to conjugacy subalgebras of d that reflect its valuative structure the rest of the chapter is devoted to brauer group factorizations of d corresponding to the noncanonical direct product decomposition of operatorname mathit br t f given in cor 7 85 the factor operatorname mathit hom c operatorname mathcal g overline f mathbb t gamma f is represented by a type of division algebra n called decomposably semiramified defined in 8 3 and characterized by the property that n contains a maximal subfield inertial over f and another totally ramified over f we show in 8 4 that every tame division algebra d is brauer equivalent to some s f t where s is inertially split and t is tame and totally ramified over f we show further that every inertially split division algebra s is brauer equivalent to some i f n where i is inertial over f and n is decomposably semiramified the classes overline i for the i appearing in the i f n decompositions of s are shown to range over a single coset of mathit dec z overline s overline f in operatorname mathit br overline f called the specialization coset of s in the final subsection 8 4 6 we summarize what happens in the special case that v is discrete of rank 1 where substantial simplifications occur

stationary bumps in networks of spiking neurons we examine the existence and stability of spatially localized bumps of neuronal activity in a network of spiking neurons bumps have been proposed in mechanisms of visual orientation tuning the rat head direction system and working memory we show that a bump solution can exist in a spiking network provided the neurons fire asynchronously within the bump we consider a parameter regime where the bump solution is bistable with an all off state and can be initiated with a transient excitatory stimulus we show that the activity profile matches that of a corresponding population rate model the bump in a spiking network can lose stability through partial synchronization to either a traveling wave or the all off state this can occur if the synaptic timescale is too fast through a dynamical effect or if a transient excitatory pulse is applied to the network a bump can thus be activated and deactivated with excitatory inputs that may have physiological relevance

value functions on central simple algebras in this paper we study noncommutative valuation rings as defined by dubrovin while there is in general no valuation associated to a dubrovin valuation ring we show that there is a value function associated to any dubrovin valuation ring integral over its center by using value functions we obtain information on dubrovin valuation rings in a tensor product both generalizing and giving a much simpler proof of a result about valued division algebras by being able to work directly with central simple algebras we gain new information about division algebras over henselian fields

valuation rings in finite dimensional division algebras a subring b of a division algebra d is called a valuation ring of d if x or x 1 is contained in b for every x in d x 0 such a ring is called an extension of the valuation ring b of k the centre of d if b k b let d be a division algebra finite dimensional over its centre k d k n2 b a valuation ring of k and b bi iei the set of all extension of b in d theorem 1 b possesses at most n extensions in d i e b n theorem 2 any two extensions of b in d are conjugate in d theorems 3 and 4 the set t of elements in d which are integral over b is a subring of d if and only if b 1 in this case t bi bi in b

the small time chung wichura law for levy processes with non vanishing brownian component we give a small time functional version of chung s other law of the iterated logarithm for levy processes with non vanishing brownian component this is an analogue of the other law of the iterated logarithm at large times for levy processes and random walks with finite variance as extended to a functional version by wichura as one of many possible applications we mention a functional law for a two sided passage time process

correspondences between valued division algebras and graded division algebras abstract if d is a tame central division algebra over a henselian valued field f then the valuation on d yields an associated graded ring gd which is a graded division ring and is also central and graded simple over gf after proving some properties of graded central simple algebras over a graded field including a cohomological characterization of its graded brauer group it is proved that the map d gd g yields an index preserving isomorphism from the tame part of the brauer group of f to the graded brauer group of gf this isomorphism is shown to be functorial with respect to field extensions and corestrictions and using this it is shown that there is a correspondence between f subalgebras of d with center tame over f and graded gf subalgebras of gd

an almost sure functional limit theorem at zero for a class of levy processes normed by the square root function and applications a recent result of bertoin doney and maller ann prob 2007 gives an integral condition to characterize the class of levy processes x t for which lim sup t downarrow 0 x t sqrt t in 0 infty occurs almost surely a s for such processes we have a kind of almost sure iterated logarithm result but without the logs in the present paper we prove a functional version of this result which then opens the way to various interesting applications obtained via a continuous mapping theorem we set these out in a rigorous framework including a characterisation of the existence of an a s cluster set for the interpolated process appropriate to the continuous time situation the applications relate to functional laws for the supremum reflected and a variety of other processes including a class of stochastic differential equations where we aim to give as informative a description as we can of the functional limit sets

hereditary crossed product orders over discrete valuation rings in this paper we consider weak crossed product orders af sx with coefficients in the integral closure s of a discrete valuation ring r in a tamely ramified galois extension of the field of fractions of r in the first section we compute the jacobson radical of af when s is local and we give a characterization of the hereditarity of the order in terms of the cocycle values in the second section we prove again in the local case that every in the inertia group for s r must belong to g f 1 is a unit of s in the final section we compute the jacobson radical in the general case s is semilocal and show how the hereditarity of af can be determined locally under an additional hypothesis

stability of the overshoot for levy processes we give equivalences for conditions like x t r r rightarrow 1 and x t r allowbreak r rightarrow 1 where the convergence is in probability or almost sure both as r rightarrow 0 and r rightarrow infty where x is a l e vy process and t r and t r are the first exit times of x out of the strip t y t 0 y leq r and half plane t y t 0 y leq r respectively we also show using a result of kesten that x t r r rightarrow 1 a s as r to 0 is equivalent to x creeping across a level

value functions and associated graded rings for semisimple algebras we introduce a type of value function y called a gauge on a finite dimensional semisimple algebra a over a field f with valuation v the filtration on a induced by y yields an associated graded ring gr y a which is a graded algebra over the graded field gr v f key requirements for y to be a gauge are that gr y a be graded semisimple and that dim gr v f gr y a dim f a it is shown that gauges behave well with respect to scalar extensions and tensor products when v is henselian and a is central simple over f it is shown that gr y a is simple and graded brauer equivalent to gr w d where d is the division algebra brauer equivalent to a and w is the valuation on d extending v on f the utility of having a good notion of value function for central simple algebras not just division algebras and with good functorial properties is demonstrated by giving new and greatly simplified proofs of some difficult earlier results on valued division algebras

noise induced stabilization of bumps in systems with long range spatial coupling the position of a localized region of active neurons a bump has been proposed to encode information for working memory the head direction system and feature selectivity in the visual system stationary bumps are ordinarily stable but including spike frequency adaptation in the neural dynamics causes a stationary bump to become unstable to a moving bump through a supercritical pitchfork bifurcation in bump speed adding spatiotemporal noise to the network supporting the bump can cause the average speed of the bump to decrease to almost zero reversing the effect of the adaptation and restabilizing the bump this restabilizing occurs for noise levels lower than those required to break up the bump the restabilizing can be understood by examining the effects of noise on the normal form of the pitchfork bifurcation where the variable involved in the bifurcation is bump speed this noisy normal form can be further simplified to a persistent random walk in which the probability of changing direction is related to the noise level through an arrhenius type rate the probability density function of position for the continuous time version of this random walk satisfies the telegrapher s equation and the closed form solution of this pde allows us to find expressions for the mean and variance of the average speed of the particle the bump undergoing the random walk this noise induced stabilization is a novel example in which moderate amounts of noise have a beneficial effect on a system specifically stabilizing a spatiotemporal pattern 2001 elsevier science b v all rights reserved

extensions of valuation rings in central simple algebras certain subrings r of simple algebras q finite dimensional over their center k are studied these rings are called q valuation rings since they share many properties with commutative valuation rings let v be a valuation ring of k the center of q and let m be the set of q valuation rings r in q with r n k v then 1s 1 this extension theorem which does not hold if one considers only total valuation rings was proved by n i dubrovin here first a somewhat different proof of this result is given and then information about the set m is obtained theorem the elements in j are conjugate if v has finite rank theorem the elements in m are total valuation rings if m contains one total valuation ring in this case q is a division ring theorem i i 1 if 5 contains an invariant total valuation ring

passage times of random walks and levy processes across power law boundaries we establish an integral test involving only the distribution of the increments of a random walk s which determines whether limsup n s n n is almost surely zero finite or infinite when 1 2 1 and a typical step in the random walk has zero mean this completes the results of kesten and maller 9 concerning finiteness of one sided passage times over power law boundaries so that we now have quite explicit criteria for all values of 0 the results and those of 9 are also extended to levy processes

passage of levy processes across power law boundaries at small times we wish to characterise when a l e vy process x t crosses boundaries like t kappa kappa 0 in a one or two sided sense for small times t thus we enquire when limsup t downarrow 0 x t t kappa limsup t downarrow 0 x t t kappa and or liminf t downarrow 0 x t t kappa are almost surely a s finite or infinite necessary and sufficient conditions are given for these possibilities for all values of kappa 0 often for many values of kappa when the limsups are finite a s they are in fact zero as we show but the limsups may in some circumstances take finite nonzero values a s in general the process crosses one or two sided boundaries in quite different ways but surprisingly this is not so for the case kappa 1 2 an integral test is given to distinguish the possibilities in that case some results relating to other norming sequences for x and when x is centered at a nonstochastic function are also given

a springer theorem for hermitian forms throughout this paper we let d be a central f division algebra with involution such that f d f d d is a henselian valued field by 11 the valuation on f extends uniquely to a valuation on d we denote this valuation by v moreover we assume that the characteristic of the residue field open image in new window is not 2 if the valuation on f is discrete then any quadratic form q can be written as q q1 q2 where is a uniformizer and qi are unit forms springer s theorem states that q is isotropic if and only if at least one of the residue forms open image in new window and open image in new window is isotropic in this paper we generalize this result to hermitian forms in section 4 we use the connection between involutions on algebras and hermitian forms to prove an analog of the springer theorem for involutions

valuation rings in ore extensions abstract extensions of valuation rings v of a skew field k are considered in the skew field f k x for a monomorphism of k at least two such extensions exist if is an automorphism but no extension may exist if is a monomorphism only there exist extensions r a of v in f with xr a ar a for every non zero a in k if and only if v is invariant and is compatible with v

bezout orders and henselization abstract let b be a bezout maximal order over a valuation ring v in a central simple f algebra s if fh vh is the henselization of f v then the main result of this paper is that b vvh is a maximal vh order in s ffh furthermore we show that this maximal order is semihereditary but not bezout in general and that certain numerical invariants of b considered by wadsworth and grater readily show up in the structure of b v vh

weak crossed product orders over valuation rings abstract let f be a field let v be a valuation ring of f of arbitrary krull dimension rank let k be a finite galois extension of f with group g and let s be the integral closure of v in k let f g g k 0 be a normalized two cocycle such that f g g s 0 but we do not require that f should take values in the group of multiplicative units of s one can construct a crossed product v order a f g s x with multiplication given by x s x s f x for s s g we characterize semihereditary and dubrovin crossed product orders under mild valuation theoretic assumptions placed on the nature of the extension k f

small time compactness and convergence behavior of deterministically and self normalised levy processes the first author s research was partially supported by arc grant dp0664603 the second author s research was partially supported by nsf grant dms 0503908

structure of shocks in burgers turbulence with levy noise initial data we study the structure of the shocks for the inviscid burgers equation in dimension 1 when the initial velocity is given by levy noise or equivalently when the initial potential is a two sided levy process 0 when 0 is abrupt in the sense of vigon or has bounded variation with lim sup h 0h 2 0 h we prove that the set of points with zero velocity is regenerative and that in the latter case this set is equal to the set of lagrangian regular points which is non empty when 0 is abrupt we show that the shock structure is discrete when 0 is eroded we show that there are no rarefaction intervals

the growth of random walks and levy processes let x i be a sequence of independent identically distributed non degenerate random variables taking values in mathbb r d and s n sum n i 1 x i m n max 1 leqq i leqq n s i define for x 0 g x p x 1 x k x x 2 e x 1 2 1 x 1 leq x m x x 1 e x 1 1 x 1 leq x and h x g x k x m x then if beta sup alpha lim sup x alpha h x 0 delta sup alpha lim inf x alpha h x 0 it is proved that n 1 alpha m n rightarrow 0 for alpha delta while the lim inf is 0 and the lim sup is infty for beta alpha delta some alternative characterizations of the indices beta delta are obtained as well as the analogous results for levy processes

the lower limit of a normalized random walk soit sn une marche aleatoire et n une suite de constantes telles que n n est non decroissante on donne un test integral universel qui determine la limite inferieure de sn n jusqu a une echelle constante pour lim sup 2n n