additive rank 1 preservers between hermitian matrix spaces and applications denote the set of n n complex hermitian matrices by hn we characterize the additive maps from hn into hm that preserve rank 1 matrices when the rank of the image of in is equal to n as applications the forms of the additive preserver of rank the additive preserver of rank additivity and the additive preserver of adjoint matrix between the spaces of hermitian matrices are also given

additive mappings decreasing rank one abstract let b x be the algebra of bounded operators on a real or complex banach space x and f x a subalgebra of finite rank operators a complete description of additive mappings f x f x which map rank one operators to operators of rank at most one is given

rank one nonincreasing additive mappings on second symmetric product spaces in this note we describe those additive mappings from a second symmetric product space to another over a field of characteristic not 2 or 3 which preserve decomposable elements of the form u u where u is a vector and is a scalar this leads to the corresponding result concerning additive mappings from one vector space of symmetric matrices to another which preserve rank less than or equal to one we also discuss some consequences of this characterization theorem

additive preservers of rank additivity on matrix spaces abstract let f be a field let v denote the vector space of all m n matrices over f or the vector space of all n n symmetric matrices over f of characteristic not 2 or 3 for each fixed positive integer s 2 let q s denote the set of all matrix pairs a b in v such that rank a b rank a rank b s we characterize additive mappings on v such that a b q s whenever a b q s for a fixed s we also describe the structure of linear mappings from the space of n n matrices over f to the space of p q matrices over f that preserve rank additivity where char f 2

painleve analysis of the non linear schrodinger family of equations the authors apply the painleve test to the generalised derivative non linear schrodinger equation iut uxx iauu ux ibu2ux cu3u 2 where u denotes the complex conjugate of u and a b and c are real constants to determine under what conditions the equation might be completely integrable it is shown that apart from a trivial multiplicative factor this equation possesses the painleve property for partial differential equations as formulated by weiss tabor and carnevale 1983 only if c 1 4b 2b a when then this relation holds this is equivalent under a gauge transformation to the derivative non linear schrodinger equation dnls of kaup and newell which is known to be completely integrable or else to a linear equation

additive rank one preservers on block triangular matrix spaces in this note we address a description of the general form of additive rank one preservers on block triangular matrix spaces over an arbitrary field as an application we characterize additive mappings preserving rank additivity on block triangular matrix algebras

additive operators preserving idempotent matrices over fields and applications abstract suppose f is a field of characteristic not 2 let m n f be the algebra of all n n matrices over f and let a 2 a 3 a 1 and a be the semigroups of all additive operator on m n f that preserve idempotence preserve tripotence preserve inverses of matrices and preserve group inverses of matrices respectively the main result in this paper is that the semigroup a 2 is generated by transposition similarity the operators x x for fixed arbitrary injective endomorphisms on f and the operators x tr x for fixed arbitrary additive maps from f to m n f with 1 o as applications we determine the structures of a 3 a 1 and a when the characteristic of f is also not 3

linear preserver problems a brief introduction and some special techniques abstract linear preserver problems concern the characterization of linear operators on matrix spaces that leave certain functions subsets relations etc invariant the earliest papers on linear preserver problems date back to 1897 and a great deal of effort has been devoted to the study of this type of question since then we present a brief picture of the subject aiming at giving a gentle introduction to the reader then we describe some techniques used in our recent papers on this type of problem

integrability of kersten krasil shchik coupled kdv mkdv equations singularity analysis and lax pair the integrability of a coupled kdv mkdv system is tested by means of singularity analysis the true lax pair associated with this system is obtained by the use of prolongation technique

on integrability of the inhomogeneous heisenberg ferromagnet model examination of a new test we present an improved version of the integrability test based on application of lie point transformations acting on the bare linear problem the test is then applied to the inhomogeneous heisenberg ferromagnet model equation the integrable cases selected by this test are identical with those obtained by the standard painleve test

additive preservers of non zero decomposable tensors let t be an additive mapping from a tensor product of vector spaces over a field into itself we describe t for the following two cases i t is surjective and sends non zero decomposable elements to non zero decomposable elements and ii t a is a non zero decomposable element if and only if a is a non zero decomposable element

rank one preserving r linear maps on spaces of self adjoint operators on complex hilbert spaces abstract let h be a complex hilbert space and let s a h be the space of all self adjoint operators on h let denote the subset of s a h consisting of all rank one operators a map l from s a h to itself is called a rank one preserver on s a h if l and is said to be r linear if l a b l a l b and l ka kl a for any k r and a b s a h where r is the field of real numbers we give a complete classification of all r linear and weakly continuous rank one preservers on s a h moreover we also determine the general form of all r linear and weakly continuous rank preservers respectively rank one idempotence preservers on s a h

on nodes and modes in resting state fmri this paper examines intrinsic brain networks in light of recent developments in the characterisation of resting state fmri timeseries and simulations of neuronal fluctuations based upon the connectome its particular focus is on patterns or modes of distributed activity that underlie functional connectivity we first demonstrate that the eigenmodes of functional connectivity or covariance among regions or nodes are the same as the eigenmodes of the underlying effective connectivity provided we limit ourselves to symmetrical connections this symmetry constraint is motivated by appealing to proximity graphs based upon multidimensional scaling crucially the principal modes of functional connectivity correspond to the dynamically unstable modes of effective connectivity that decay slowly and show long term memory technically these modes have small negative lyapunov exponents that approach zero from below interestingly the superposition of modes whose exponents are sampled from a power law distribution produces classical 1 f scale free spectra we conjecture that the emergence of dynamical instability that underlies intrinsic brain networks is inevitable in any system that is separated from external states by a markov blanket this conjecture appeals to a free energy formulation of nonequilibrium steady state dynamics the common theme that emerges from these theoretical considerations is that endogenous fluctuations are dominated by a small number of dynamically unstable modes we use this as the basis of a dynamic causal model dcm of resting state fluctuations as measured in terms of their complex cross spectra in this model effective connectivity is parameterised in terms of eigenmodes and their lyapunov exponents that can also be interpreted as locations in a multidimensional scaling space model inversion provides not only estimates of edges or connectivity but also the topography and dimensionality of the underlying scaling space here we focus on conceptual issues with simulated fmri data and provide an illustrative application using an empirical multi region timeseries

comparing dynamic causal models this article describes the use of bayes factors for comparing dynamic causal models dcms dcms are used to make inferences about effective connectivity from functional magnetic resonance imaging fmri data these inferences however are contingent upon assumptions about model structure that is the connectivity pattern between the regions included in the model given the current lack of detailed knowledge on anatomical connectivity in the human brain there are often considerable degrees of freedom when defining the connectional structure of dcms in addition many plausible scientific hypotheses may exist about which connections are changed by experimental manipulation and a formal procedure for directly comparing these competing hypotheses is highly desirable in this article we show how bayes factors can be used to guide choices about model structure both concerning the intrinsic connectivity pattern and the contextual modulation of individual connections the combined use of bayes factors and dcm thus allows one to evaluate competing scientific theories about the architecture of large scale neural networks and the neuronal interactions that mediate perception and cognition

differentiable generalized synchronization of chaos we consider simple lyapunov exponent based conditions under which the response of a system to a chaotic drive is a ital smooth function of the drive state we call this ital differentiable generalized synchronization dgs when dgs does not hold we quantify the degree of nondifferentiability using the h umlt o lder exponent we also discuss the consequences of dgs and give an illustrative numerical example copyright ital 1997 ital the american physical society

on the phase reduction and response dynamics of neural oscillator populations weundertake a probabilistic analysis of the response of repetitively firing neural populations to simple pulselike stimuli recalling and extending results from the literature we compute phase response curves prcs valid near bifurcations to periodic firing for hindmarsh rose hodgkin huxley fitzhugh nagumo and morris lecar models encompassing the four generic codimension one bifurcations phase density equations are then used to analyze the role of the bifurcation and the resulting prc in responses to stimuli in particular we explore the interplay among stimulus duration baseline firing frequency and population level response patterns we interpret the results in terms of the signal processing measure of gain and discuss further applications and experimentally testable predictions

scalar second order evolution equations possessing an irreducible sl2 valued zero curvature representation we find all scalar second order evolution equations that possess an sl2 valued zero curvature representation irreducible to a proper subalgebra of sl2 none of these zero curvature representations depends on a parameter that could serve as the spectral parameter

conditional lyapunov exponents from time series a method for estimating conditional lyapunov exponents from time series of two unidirectionally coupled chaotic systems is developed it uses two scalar data sets one taken from the driving and the other from the response system and enables one to detect a generalized synchronization in an experiment without recourse to an auxiliary response system the method is illustrated on coupled maps as well as coupled chaotic flow models

variational free energy and the laplace approximation this note derives the variational free energy under the laplace approximation with a focus on accounting for additional model complexity induced by increasing the number of model parameters this is relevant when using the free energy as an approximation to the log evidence in bayesian model averaging and selection by setting restricted maximum likelihood reml in the larger context of variational learning and expectation maximisation em we show how the reml objective function can be adjusted to provide an approximation to the log evidence for a particular model this means reml can be used for model selection specifically to select or compare models with different covariance components this is useful in the context of hierarchical models because it enables a principled selection of priors that under simple hyperpriors can be used for automatic model selection and relevance determination ard deriving the reml objective function from basic variational principles discloses the simple relationships among variational bayes em and reml furthermore we show that em is formally identical to a full variational treatment when the precisions are linear in the hyperparameters finally we also consider briefly dynamic models and how these inform the regularisation of free energy ascent schemes like em and reml

additive decomposability preservers and related results on tensor products of matrices abstract in this paper we characterize additive maps between tensor spaces that send decomposable tensors to decomposable tensors as an application we classify all additive maps from tensor products of spaces of rectangular matrices to spaces of rectangular matrices which do not increase the rank of tensor product of rank one matrices

on zero curvature representations of evolution equations for zero curvature representations zcrs at bx ab ba 0 of evolution equations ut f x u ux ux x we develop a description which is invariant under gauge transformations a sas 1 sxs 1 and b sbs 1 sts 1 where a b and s are matrix functions of x u ux uxx we prove that every fixed matrix a of any dimension and order in ux x determines a continual class of evolution equations which admit zcrs with this a then we quote examples illustrating how a dependence of a on an essential parameter restricts classes of represented equations one of our examples shows that some non integrable systems can admit parametric lax pairs and infinitely many non trivial conservation laws

linear additive preservers of rank 2 on spaces of alternate matrices over fields abstract let kn f be the linear space of all n n alternate matrices over a field f and let k n 2 f be its subset consisting of all rank 2 matrices an operator kn f kn f is said to be additive if a b a b for any a b kn f linear if is additive and aa af a for every a f and a kn f and a preserver of rank 2 on kn f if k n 2 f k n 2 f when n 4 we characterize all linear respectively additive preservers of rank 2 on kn f over any field respectively any field that is not isomorphic to a proper subfield of itself

a note on additive mappings decreasing rank one kuzma characterized additive mappings on the space of all finite rank bounded linear operators on a real or complex banach space that decreases operators of rank one in this note we give a short proof of his result in a slightly more general setting of the tensor product of a right and a left vector space over a division ring

additive rank one preserving surjections on symmetric matrix spaces we characterize additive rank one preserving surjections on symmetric matrix spaces over a field of characteristic not 2 or 3 as applications the invertibility preservers the determinant preservers and characteristic polynomial preservers have been characterized

on integrability of weingarten surfaces a forgotten class rediscovered by a systematic search a forgotten class of integrable surfaces is shown to disprove the finkel wu conjecture the associated integrable nonlinear partial differential equation possesses a zero curvature representation a third order symmetry and a nonlocal transformation to the sine gordon equation sin we leave open the problem of finding a backlund autotransformation and a recursion operator that would produce a local hierarchy

additive rank one preserving mappings on triangular matrix algebras abstract we classify surjective additive maps on the space of block upper triangular matrices that preserve matrices of rank one as well as linear maps preserving matrices of rank one on fairly general subspaces of matrices

permanent polya problem for additive surjective maps abstract let m n f denote the set of square matrices of size n over a field f with characteristics different from two we say that the map f m n f m n f is additive if f a b f a f b for all a b m n f the main goal of this paper is to prove that for n 2 there are no additive surjective maps t m n f m n f such that per t a det a for all a m n f also we show that an arbitrary additive surjective map t m n f m n f which preserves permanent is linear and thus can be completely characterized

additive mappings preserving operators of rank one abstract let b x be the algebra of all bounded linear operators on a nontrivial real or complex banach space and let f x be the subalgebra of all finite rank operators a characterization of additive mappings on f x which preserve operators of rank one or projections of rank one is given in the real case such mappings are automatically linear

a classification of pseudospherical surface equations of type ut uxxx g u ux uxx the equations of type ut uxxx g u ux uxx which describe pseudospherical surfaces are characterized a new class of such equations is obtained this class together with the korteweg de vries kdv equation the modified korteweg de vries mkdv equation and the linear equation completes the classification

spectrum preserving additive maps abstract let x and y be complex banach spaces we show that a spectrum preserving surjective additive map from b x to b y is either of the form f ata 1 for a linear isomorphism a of x onto y or of the form t bt b 1 for a linear isomorphism b of x onto y

jordan isomorphisms and additive rank preserving maps on symmetric matrices over pid abstract let r be a commutative principal ideal domain pid with char r 2 n 2 denote by s n r the set of all n n symmetric matrices over r if is a jordan automorphism on s n r then is an additive rank preserving bijective map in this paper every additive rank preserving bijection on s n r is characterized thus is a jordan automorphism on s n r if and only if is of the form x tpx p where r p gln r which satisfies tpp 1i and is an automorphism of r it follows that every jordan automorphism on s n r may be extended to a ring automorphism on mn r and is a jordan automorphism on s n r if and only if is an additive rank preserving bijection on s n r which satisfies i i

additive mappings between hermitian matrix spaces preserving rank not exceeding one abstract let k be a field of characteristic not two or three with an automorphism of order two let f a k a a we characterize additive mappings t from one f vector space of hermitian matrices over k to another that preserve rank less than or equal to one for the following cases i the image of t contains a matrix of rank at least three ii every nonzero endomorphism of f is surjective

network discovery with dcm this paper is about inferring or discovering the functional architecture of distributed systems using dynamic causal modelling dcm we describe a scheme that recovers the dynamic bayesian dependency graph connections in a network using observed network activity this network discovery uses bayesian model selection to identify the sparsity structure absence of edges or connections in a graph that best explains observed time series the implicit adjacency matrix specifies the form of the network e g cyclic or acyclic and its graph theoretical attributes e g degree distribution the scheme is illustrated using functional magnetic resonance imaging fmri time series to discover functional brain networks crucially it can be applied to experimentally evoked responses activation studies or endogenous activity in task free resting state fmri studies unlike conventional approaches to network discovery dcm permits the analysis of directed and cyclic graphs furthermore it eschews implausible markovian assumptions about the serial independence of random fluctuations the scheme furnishes a network description of distributed activity in the brain that is optimal in the sense of having the greatest conditional probability relative to other networks the networks are characterised in terms of their connectivity or adjacency matrices and conditional distributions over the directed and reciprocal effective connectivity between connected nodes or regions we envisage that this approach will provide a useful complement to current analyses of functional connectivity for both activation and resting state studies

some general techniques on linear preserver problems abstract several general techniques on linear preserver problems are described the first one is based on a transfer principle in model theoretic algebra that allows one to extend linear preserver results on complex matrices to matrices over other algebraically closed fields of characteristic 0 the second one concerns the use of some simple geometric technique to reduce linear preserver problems to standard types so that known results can be applied the third one is about solving linear preserver problems on more general operator algebras by reducing the problems to idempotent preservers numerous examples will be given to demonstrate the proposed techniques