algebraic reflexivity of sets of bounded operators on vector valued lipschitz functions abstract in this paper we establish algebraic reflexivity properties of subsets of bounded linear operators acting on spaces of vector valued lipschitz functions we also derive a representation for the generalized bi circular projections on these spaces

examples of generalized bi circular idempotents on spaces of continuously differentiable functions abstract in this note we investigate the form of a class of idempotents called generalized bi circular idempotents on spaces of continuously differentiable functions defined on the interval 0 1 and with values in the complex numbers

waves and bumps in neuronal networks with axo dendritic synaptic interactions we consider a firing rate model of a neuronal network continuum that incorporates axo dendritic synaptic processing and the finite conduction velocities of action potentials the model equation is an integral one defined on a spatially extended domain apart from a spatial integral mixing the network connectivity function with space dependent delays arising from non instantaneous axonal communication the integral model also includes a temporal integration over some appropriately identified distributed delay kernel these distributed delay kernels are biologically motivated and represent the response of biological synapses to spiking inputs they are interpreted as green s functions of some linear dierential operator exploiting this green s function description we discuss formal reductions of this non local system to equivalent partial dierential equation pde models we distinguish between those spatial connectivity functions that give rise to local pde models and those that give rise to pde models with delayed non local terms for cases in which local pdes are derived we investigate traveling wave solutions in a comoving frame by numerically computing global heteroclinic connections for sigmoidal firing rate functions we also calculate exact solutions parameterized by axonal conduction velocity for the heaviside firing rate function the sigmoidal firing rate function in the limit of infinte gain the inclusion of synaptic adaptation is shown to alter traveling wave fronts to traveling pulses which we study analytically and numerically in terms of a global homoclinic orbit finally we consider the impact of dendritic interactions on waves and on static spatially localized solutions exact analysis for

generalized tricircular projections on minimal norm ideals in b h abstract a nonzero projection p on a complex banach space x is said to be a generalized tricircular projection if there exist distinct modulus one complex numbers and not equal to 1 and nonzero projections q and r on x such that p q r i and p q r is an isometry we determine the structure of generalized tricircular projections on minimal norm ideals in b h different from the hilbert schmidt class

projections in the convex hull of three surjective isometries on c ω abstract let be a compact connected hausdorff space we define generalized n circular projection on c as a natural analogue of generalized bi circular projection and show that such a projection p can always be represented as p i t t 2 t n 1 n where i is the identity operator and t is a surjective isometry on c such that t n i we next show that if convex combination of three distinct surjective isometries on c is a projection then it is a generalized 3 circular projection

stable localized solutions of arbitrary length for the quintic swift hohenberg equation abstract we show that localized solutions of arbitrary length are stable over a finite parameter interval of subcritical values for the quintic swift hohenberg equation with a destabilizing cubic term this equation is thought to model a weakly hysteretic transition to stationary patterns we argue that the stabilization of the localized states of arbitrary length can be traced back to the interaction between long wavelength modulations and spatial variations on the length scale of one unit cell these results are critically compared with other known mechanisms to stabilize localized states in various situations we also discuss for which experimental systems the states predicted here could be detected including e g the stationary onset of binary fluid convection

algebraic reflexivity of some subsets of the isometry group let x be a compact first countable space in this paper we show that the set of isometries of c x that are involutions is algebraically reflexive as a consequence of a recent work of botelho and jamison this leads to the conclusion that the set of generalized bi circular projections on c x is also algebraically reflexive we also consider these questions for the space c x e where e is a uniformly convex banach space

algebraic reflexivity of the isometry group of some spaces of lipschitz functions abstract we show that the isometry groups of lip x d and lip x d with 0 1 for a compact metric space x d are algebraically reflexive we also prove that the sets of isometric reflections and generalized bi circular projections on such spaces are algebraically reflexive in order to achieve this we characterize generalized bi circular projections on these spaces

subgroups of containing let psu n be the image of the unitary group u n under the representation x axa 1 on the space mn c of n by n complex matrices we classify all connected lie subgroups of gl n2 c containing psu n we use this result to obtain a description of all abstract overgroups of psu n in gl n2 c we apply this classification to solve the problem of describing all invertible linear transformations of mn c which preserve the set of normal matrices our results can be applied to solve many other problems of similar nature

generalized bicircular projections on jb triples abstract let a be a jb triple and let p a a be a linear projection it is proved that p id p is an isometry for some modulus one complex number 1 if and only if either 1 or p is hermitian it is also proved that every rank one bicontractive projection on a is hermitian the particular case when a is a c algebra is discussed through several examples

oscillon type structures and their interaction in a swift hohenberg model abstract motivated by the observation of localized circular excitations oscillons in vertically vibrated granular layers p umbanhowar f melo h swinney nature 382 1996 793 we numerically investigate an extension of a swift hohenberg model that exhibits a subcritical transition to square patterns for sufficiently strong damping of the basic state stable oscillon structures are found the localization mechanism is quite general and is due to non adiabatic effects bound structures of oscillons of equal and opposite polarity are found with bound states of like polarity being less robust much of the phenomena are consistent with the experimental observations and suggest that oscillons are not specific to patterns in granular media or to parametrically driven systems experimental tests are suggested that would determine whether this minimal framework is sufficient to describe the phenomena

on traveling wave solutions of the kuramoto sivashinsky equation abstract previous numerical results indicate that the kuramoto sivashinsky equation admits three classes of non periodic traveling wave solutions namely regular shocks oscillatory shocks and solitary waves however it has been shown that regular monotonic shocks cease to exist in the weak shock limit due to the radiation of oscillatory waves of exponentially small with respect to the shock strength but growing in space amplitude here oscillatory shocks and solitary waves are constructed by asymptotic analysis it thus transpires that in the weak shock limit oscillatory shocks can only be antisymmetric otherwise oscillatory and monotonic waves of exponentially small but growing in space amplitude would inevitably be excited under certain conditions however the growing waves can link a nearly antisymmetric oscillatory shock with a radiating regular shock to form a solitary wave the predictions of the asymptotic theory are supported by numerical results

exponential asymptotics and stokes lines in nonlinear ordinary differential equations a technique for calculating exponentially small terms beyond all orders in singularly perturbed ordinary differential equations is presented the approach is based on the application of a wkbj type ansatz to the late terms in the naive asymptotic expansion and the identification of stokes lines and is closely related to the well known stokes line smoothing phenomenon in linear ordinary differential equations the method is illustrated by application to examples

generalized bicircular projections via the operator equation αx2ay2 βxay a 0 abstract the operator equation x 2 ay 2 xay a 0 is used to determine the structure of a projection p such that for some complex unit 1 the mapping p i p is a uav a ua v or a ua t v for some fixed operators u v some applications of these results are also given

projections as convex combinations of surjective isometries on c ω we study properties of operators that are in the convex hull of a finite set of surjective isometries on the banach space of complex valued continuous maps defined on a compact and connected topological space we characterize those projections that are in the convex combination of two surjective isometries and we show that they are generalized bi circular projections

front dynamics in a delayed feedback system with external forcing a scalar nonlinear delay differential equation of the kind arising in optically bistable systems is reduced in some physically relevant limit to a partial differential equation first and then to a system of ordinary differential equations these simplified equations are studied analytically we describe the continuous deformation of periodic solutions from sine waves to square waves away from the oscillation threshold and determine their stability we also study the effect of a weak periodic modulation it is found that a resonant forcing can stabilize a very large number of oscillation modes taking the form of square waves with missing front pairs

generalized bi circular projections on certain hardy spaces abstract we characterize generalized bi circular projections on the hardy space of the torus and show that they can be represented as the average of the identity and an isometric reflection we also show that the average of two distinct isometries on the hardy space of the disk is a projection if and only if it is a generalized bi circular projection

some aspects of the theory of norms abstract let v be a finite dimensional vector space motivated by theory or applications one might want to consider different kinds of norms on v in this paper we discuss some results and problems involving different classes of norms on a vector space studied by this author in the past few years the paper consists of five sections section 1 concerns the conditions on two vectors x y v satisfying x y for all in a certain class of norms section 2 concerns the isometry groups of g invariant norms i e norms that satisfy g x x for all x v and for all g g where g is a group of unitary orthogonal operators on v section 3 concerns g invariant norms that satisfy some special properties section 4 concerns the best approximation s x 0 t of y where y t v with respect to different kinds of norms additional open problems topics and references are mentioned in section 5

localized states in the generalized swift hohenberg equation the swift hohenberg equation with quadratic and cubic nonlinearities exhibits a remarkable wealth of stable spatially localized states the presence of these states is related to a phenomenon called homoclinic snaking numerical computations are used to illustrate the changes in the localized solution as it grows in spatial extent and to determine the stability properties of the resulting states the evolution of the localized states once they lose stability is illustrated using direct simulations in time i introduction ever since the observation that the subcritical complex ginzburg landau equation exhibits stable spatially localized states 1 there has been considerable interest in the properties of these states the presence of these states has important consequences for other systems described by partial differential equations on the line since the ginzburg landau equation describes the behavior of such systems near bifurcation from the trivial state of the system specifically the complex ginzburg landau equation describes the evolution of a long wavelength oscillatory instability as well as oscillatory instabilities at finite wavelength in systems with broken reflection symmetry in contrast near a steady state bifurcation with finite wavelength the evolution of the instability is described by the real ginzburg landau equation and this equation possesses only unstable spatially localized states it is of interest therefore to examine what happens to these unstable states at larger amplitude where the real ginzburg landau equation no longer provides an adequate description of the system in this paper we show that the localized states can become stable at such amplitudes and indeed that there is a large multiplicity of coexisting stable localized states under very general conditions we are able to relate the existence of these states to a phenomenon sometimes called homoclinic snaking that is well known from the theory of reversible systems with 1 1 resonance and use this theory to construct a large number of such states the stability properties of these states are also determined and the evolution of nonstationary localized states is studied by numerical integration in time it is an interesting fact that closely related phenomena have already been described in several areas involving pattern formation the theory was originally developed in the context of water waves where localized states have been studied by moving into a reference frame of the waves and converting the problem into an ordinary differential equation ode the resulting localized states are called solitary waves and in some cases turn out to be solitons kirchgassner 2 has pioneered a successful approach to this type of problem that led to a number of advances in this area specifically the ode is viewed as a dynamical system in space and localized states are sought as homoclinic orbits connecting the trivial state to itself whether such orbits are possible depends in part on the stability properties of the trivial state eigenvalues with positive real part indicate that a nontrivial state can grow from x while eigenvalues with negative real part indicate that such a state may return under appropriate conditions back to the trivial state as x the spectrum of the linearization about the trivial state is influenced by spatial symmetries of the system in many cases and in particular in the case considered here the ode is reversible as a result the bifurcations that are encountered as a parameter is increased are nongeneric in the present case the spatial dynamics of the system near the trivial state turn out to be described by the reversible 1 1 resonance the unfolding of this resonance has been worked out in detail by iooss and peroueme 3 and can be used to understand the appearance of a variety of homoclinic orbits in this system and hence of localized states with different spatial structure

isometries of norm ideals of compact operators abstract let j be a symmetric norm ideal of compact operators on hilbert space h and assume that the finite rank operators are dense in j and that j is not the ideal of hilbert schmidt operators a linear transformation on j is an isometry of j onto itself if and only if there are unitary operators u and v on h such that either x uxv or x uxtv where xt denotes the transpose of x with respect to a fixed orthonormal basis of h

projections and averages of isometries on lipschitz spaces abstract we characterize projections on spaces of lipschitz functions expressed as the average of two and three linear surjective isometries generalized bi circular projections are the only projections on these spaces given as the convex combination of two surjective isometries

stokes phenomenon and matched asymptotic expansions this paper describes the use of matched asymptotic expansions to illuminate the description of functions exhibiting stokes phenomenon in particular the approach highlights the way in which the local structure and the possibility of finding stokes multipliers explicitly depend on the behaviour of the coefficients of the relevant asymptotic expansions

homoclinic snaking near a heteroclinic cycle in reversible systems snaking curves of homoclinic orbits have been found numerically in a number of ode models from water wave theory and structural mechanics along such a curve infinitely many fold bifurcation of homoclinic orbits occur thereby the corresponding solutions spread out and develop more and more bumps oscillations about their own centre a common feature of the examples is that the systems under consideration are reversible in this paper it is shown that such a homoclinic snaking can be caused by a heteroclinic cycle between two equilibria one of which is a bi focus using lin s method a snaking of 1 homoclinic orbits is proved to occur in an unfolding of such a cycle further dynamical consequences are discussed as an application a system of boussinesq equations is considered where numerically a homoclinic snaking curve is detected and it is shown that the homoclinic orbits accumulate along a heteroclinic cycle between a real saddle and a bi focus equilibrium

overgroups of some classical linear groups with applications to linear preserver problems abstract the study of linear operators on a matrix space that leave invariant certain functions subsets or relations is commonly referred to as the study of linear preserver problems and has attracted the attention of many mathematicians in the last few decades dynkin in his classic paper studied maximal subgroups of the classical groups and showed how his results may be used to study preserver problems the purpose of this paper is to further exploit this very powerful approach the subgroups g of the general linear group that we deal with are not necessarily maximal for the applications that we have in mind we obtained a description of all possible overgroups of g the results are then applied to various linear preserver problems shorter alternative proofs for various existing results are given and some open questions are answered

analysis of the t point hopf bifurcation abstract in a parameterized three dimensional system of autonomous differential equations a t point is a point of the parameter space where a special kind of codimension 2 heteroclinic cycle occurs if the parameter space is three dimensional such a bifurcation is located generically on a curve a more degenerate scenario appears when this curve reaches a surface of hopf bifurcations of one of the equilibria involved in the heteroclinic cycle we are interested in the analysis of this codimension 3 bifurcation which we call t point hopf in this work we propose a model based on the construction of a poincare map that describes the global behavior close to a t point hopf bifurcation the existence of certain kinds of homoclinic and heteroclinic connections between equilibria and or periodic orbits is proved the predictions deduced from this model strongly agree with the numerical results obtained in a modified van der pol duffing electronic oscillator

solitary wave interaction phenomena in a strut buckling model incorporating restabilisation the archetypal model of the buckling of a compressed long elastic strut resting on a nonlinear elastic foundation is studied localised buckling is investigated when the foundation has both quadratic and cubic terms which initially destabilise but subsequently restabilise the structure the primary solution can be detected by a double scale perturbation procedure and is reminiscent of a solitary wave essentially it consists of a fast periodic oscillation which is slowly modulated and decays exponentially in both directions particular interest is paid to the process of adapting the procedure to account for the post buckling behaviour of two packet or double humped solitary waves in this model we employ the methods of beyond all orders asymptotics to reveal terms formally exponentially small in the perturbation parameter which have macroscopic effects on the post buckling behaviour of the system including the interaction phenomenon of interest the analysis is reinforced by direct numerical computations which reveal the so called snaking behaviour in the subsidiary homoclinic orbits as is observed in the case of primary solutions however additional phenomena arise for these subsidiary forms including the formation of bridges linking solution paths and the appearance of a multitude of closed isolated loops disconnected from other features of the bifurcation diagram

bicircular projections on some matrix and operator spaces a projection on a complex banach space is bicircular if its linear combinations with the complementary projection having coefficients of modulus one are isometries such projections are always bicontractive in this paper we prove structure theorems for bicircular projections acting on the spaces of the full operator algebra symmetric operators and antisymmetric operators

g invariant norms and bicircular projections abstract it is shown that for many finite dimensional normed vector spaces v over c a linear projection p v v will have nice structure if p i p is an isometry for some complex unit not equal to one from these results one can readily determine the structure of bicircular projections i e those linear projections p such that p i p is an isometry for every complex unit the key ingredient in the proofs is the knowledge of the isometry group of the given norm the proof techniques also apply to real vector spaces in such cases characterizations are given to linear projections p such that p i p 2p i is an isometry

localized periodic patterns for the non symmetric generalized swift hohenberg equation abstract a new asymptotic multiple scale expansion is used to derive envelope equations for localized spatially periodic patterns in the context of the generalized swift hohenberg equation an analysis of this envelope equation results in parametric conditions for localized patterns furthermore it yields corrections for wave number selection which are an order of magnitude larger for asymmetric nonlinearities than for the symmetric case the analytical results are compared with numerical computations which demonstrate that the condition for localized patterns coincides with vanishing hamiltonian and lagrangian for periodic solutions one striking feature of the choice of scaling parameters is that the derived condition for localized patterns agrees with the numerical results for a significant range of parameters which are an o 1 distance from the bifurcation thus providing a novel approach for studying these localized patterns

front motion metastability and subcritical bifurcations in hydrodynamics metastability occurs whenever under given external conditions as defined by the reynolds number in hydrodynamics more than one linearly stable solution may exist in variational problems the preferred state is usually claimed to be the one with the lowest energy but in hydrodynamics and except for very special conditions no energy functional exists from an analogy between amplitude and reaction diffusion equations it is argued that the border separating in space two possible solutions of the flow equations as turbulent and laminar in pipe flows or boundary layers moves with a constant mean velocity depending on the control parameter the sign of this velocity allows to decide which state is stable and which is metastable possible physical implications of these ideas for various problems of hydrodynamics are discussed

generalized bi circular projections abstract recall that a projection p on a complex banach space x is a generalized bi circular projection if p i p is a surjective isometry for some such that 1 and 1 it is easy to see that every hermitian projection is generalized bi circular a generalized bi circular projection is said to be nontrivial if it is not hermitian botelho and jamison showed that a projection p on c 0 1 is a nontrivial generalized bi circular projection if and only if p i p is a surjective isometry in this article we prove that if p is a projection such that p i p is a surjective isometry for some then either p is hermitian or is an n th unit root of unity we also show that for any n th unit root of unity there are a complex banach space x and a nontrivial generalized bi circular projection p on x such that p i p is an isometry

projections as averages of isometries on minimal norm ideals abstract in this paper we consider projections on minimal norm ideals of b h that are represented as the average of two surjective isometries we describe projections of the form p t a 1 tb 1 a 2 tb 2 2 where a 1 a 2 b 1 and b 2 are unitary operators satisfying a commutativity condition we also characterize classes of projections of the form p t at t t b 2 and p t at t t t b 2 with a and b unitary operators satisfying a spectral condition

exponential asymptotics of localised patterns and snaking bifurcation diagrams abstract localised patterns emerging from a subcritical modulation instability are analysed by carrying the multiple scales analysis beyond all orders the model studied is the swift hohenberg equation of nonlinear optics which is equivalent to the classical swift hohenberg equation with a quadratic and a cubic nonlinearity applying the asymptotic technique away from the maxwell point first it is shown how exponentially small terms determine the phase of the fast spatial oscillation with respect to their slow sech type amplitude in the vicinity of the maxwell point the beyond all orders calculation yields the pinning range of parameters where stable stationary fronts connect the homogeneous and periodic states the full bifurcation diagram for localised patterns is then computed analytically including snake and ladder bifurcation curves this last step requires the matching of the periodic oscillation in the middle of a localised pattern both with an up and a down front to this end a third super slow spatial scale needs to be introduced in which fronts appear as boundary layers in addition the location of the maxwell point and the oscillation wave number of localised patterns are required to fourth order accuracy in the oscillation amplitude