the beginning of the fucik spectrum for the p laplacian has a nontrivial solution u here 1 p 2p u div u p 2 u u max u 0 u uu in 1 1 denoting by 1 2 the first two eigenvalues of 2p on w 1 p 0 0 cf the end of this introduction 7p clearly contains in particular 1 1 2 2 and the two lines 1 r and r 1 this generalized notion of spectrum was introduced for p 2 i e in the linear case in the 1970s by fuc ik 21 and dancer 15 in connection article id jdeq 1999 3645 available online at http www idealibrary com on

the closure of the set of periodic modules over a concealed canonical algebra is regular in codimension one abstract let be a concealed canonical algebra and d the dimension vector of a module which is periodic respect to the action of the auslander reiten translation in the paper we investigate the union of the closures of the orbits of the periodic modules of dimension vector d we show that this set is closed and regular in codimension one

geometry and the zero sets of semi invariants for homogeneous modules over canonical algebras abstract we characterize the canonical algebras such that for all dimension vectors of homogeneous modules the corresponding module varieties are complete intersections respectively normal we also investigate the sets of common zeros of semi invariants of non zero degree in important cases in particular we show that for sufficiently big vectors they are complete intersections and calculate the number of their irreducible components

the strong global dimension of piecewise hereditary algebras let a be a finite dimensional piecewise hereditary algebra over an algebraically closed field this text investigates the strong global dimension of a this invariant is characterised in terms of the lengths of sequences of tilting mutations relating a to a hereditary abelian category in terms of the generating hereditary abelian subcategories of the derived category of a and in terms of the auslander reiten structure of that derived category

existence of a non trivial solution for the p laplacian equation with fucik type resonance at infinity abstract we prove the existence of a non trivial weak solution for the equation p u a u p 1 b u p 1 f x u in u 0 on by variational methods we consider the case where a 1 or b 1 1 denotes the first eigenvalue of p or the case where a is close to b while the nonlinearity f belongs to a class that contains for example u q 2 u 1 q p

a flexible method to measure synchrony in neuronal firing neurons can transmit information about the characteristics of a stimulus through the spike rate of neurons and synchronization of the neurons various association measure can be used to describe how synchronous two spike trains are we propose a new measure of synchrony the conditional synchrony measure which is the probability of firing together given that at least one of the two neurons is active focus is on the specification of a flexible marginal model for multivariate correlated binary data together with a pseudolikelihood estimation approach to adequately and directly describe the measures of interest a joint model must allow different time and covariate dependent firing rates for each neuron and also must account for the association between them the association between neurons might depend on covariates as well

module varieties over canonical algebras abstract the main purpose of this paper is the study of module varieties over the class of canonical algebras providing a rich source of examples of varieties with interesting properties our main tool is a stratification of module varieties which was recently introduced by richmond this stratification does not require a precise knowledge of the module category if it is finite then it provides a method to classify irreducible components we determine the canonical algebras for which this stratification is finite in this case we describe the algorithm for calculating the dimension of the variety and the number of irreducible components of maximal dimension for an infinite family of examples we give easy combinatorial criteria for irreducibility cohen macaulay and normality

distinguishing the causes of firing with the membrane potential slope in this letter we aim to measure the relative contribution of coincidence detection and temporal integration to the firing of spikes of a simple neuron model to this end we develop a method to infer the degree of synchrony in an ensemble of neurons whose firing drives a single postsynaptic cell this is accomplished by studying the effects of synchronous inputs on the membrane potential slope of the neuron and estimating the degree of response relevant input synchrony which determines the neuron s operational mode the measure is calculated using the normalized slope of the membrane potential prior to the spikes fired by a neuron and we demonstrate that it is able to distinguish between the two operational modes by applying this measure to the membrane potential time course of a leaky integrate and fire neuron with the partial somatic reset mechanism which has been shown to be the most likely candidate to reflect the mechanism used in the brain for reproducing the highly irregular firing at high rates we show that the partial reset model operates as a temporal integrator of incoming excitatory postsynaptic potentials and that coincidence detection is not necessary for producing such high irregular firing

integrator or coincidence detector a novel measure based on the discrete reverse correlation to determine a neuron s operational mode in this letter we propose a definition of the operational mode of a neuron that is whether a neuron integrates over its input or detects coincidences we complete the range of possible operational modes by a new mode we call gap detection which means that a neuron responds to gaps in its stimulus we propose a measure consisting of two scalar values both ranging from 1 to 1 the neural drive which indicates whether its stimulus excites the neuron serves as background noise or inhibits it the neural mode which indicates whether the neuron s response is the result of integration over its input of coincidence detection or of gap detection with all three modes possible for all neural drive values this is a pure spike based measure and can be applied to measure the influence of either all or subset of a neuron s stimulus we derive the measure by decomposing the reverse correlation test it in several artificial and biological settings and compare it to other measures finding little or no correlation between them we relate the results of the measure to neural parameters and investigate the effect of time delay during spike generation our results suggest that a neuron can use several different modes simultaneously on different subsets of its stimulus to enable it to respond to its stimulus in a complex manner

sign changing and constant sign solutions for p laplacian problems with jumping nonlinearities abstract by variational methods we prove the existence of a sign changing solution for the p laplacian equation under dirichlet boundary condition with jumping nonlinearity having relation to the fucik spectrum of p laplacian we also provide the multiple existence results for the p laplacian problems

on degenerations and extensions of finite dimensional modules abstract we derive a cancellation theorem for degenerations of modules that says in particular that projective or injective common direct summands can always be neglected combining the cancellation result with the existence of almost split sequences we characterize the orbit closure of a module living on preprojective components by the fact that the dimension of the homomorphism space to any other module does not decrease for representation directed algebras whence in particular for path algebras of dynkin quivers we provide an alternative proof which shows in addition that any minimal degenerationnofmcomes from an exact sequence with middle termmwhose end terms add up ton by a careful examination the same is true for degenerations of matrix pencils having used so far the existence of certain extensions to obtain degenerations we then turn the tables and use degenerations to produce a lot of interesting short exact sequences in particular we show that any non simple indecomposable over a tame quiver is an extension of an indecomposable and a simple

sign changing semi classical solutions of the brezis nirenberg problems with jump nonlinearities in high dimensions abstract in this paper we consider the following elliptic equation e 2 u u u u p 2 u u 2 2 u in u 0 on where r n n 4 is a bounded domain u max u 0 0 are constants e 0 is a small parameter and 2 p 2 2 n n 2 we obtain a least energy sign changing solution to this problem for e 0 that is sufficiently small by means of the variational method furthermore by combining the elliptic and local energy estimates we study the concentration behavior of this least energy sign changing solution and the location of the positive and negative spikes as e 0 our results partially complete the studies in 6 2011 in the sense that due to sobolev embedding only subcritical cases are considered in that paper

invariant theory of canonical algebras abstract based on the first fundamental theorem of classical invariant theory we present a reduction technique for computing relative invariants for quivers with relations this is applied to the invariant theory of canonical algebras and yields an explicit construction of the moduli spaces together with the quotient morphisms from the corresponding representation spaces for families of modules with a fixed dimension vector belonging to the central sincere separating subcategory by means of a tilting process we extend these results to the invariant theory of concealed canonical algebras thus covering the cases of tame hereditary tame concealed and tubular algebras respectively our approach yields in particular a uniform treatment to an essential part of the invariant theory of extended dynkin quivers a topic popular over the years but stretches far beyond since also concealed canonical algebras of tubular or wild representation type are covered

concealed canonical algebras and separating tubular families we characterise those concealed canonical algebras which arise as endomorphism rings of tilting modules all of whose indecomposable summands have strictly positive rank as those artin algebras whose module categories have a separating exact subcategory that is a separating tubular family of standard tubes this paper develops further the technique of shift automorphisms which arises from the tubular structure it is related to the characterisation of hereditary noetherian categories with a tilting object as the categories of coherent sheaves on a weighted projective line

nonlinear elliptic equations with a jumping reaction abstract we study a nonlinear nonhomogeneous dirichlet problem driven by the sum of a p laplacian and a laplacian 2 p and a jumping nonlinearity under very general conditions on the reaction and without using the fucik spectrum we show that the problem has at least three nontrivial solutions and we provide sign information for all of them our approach uses critical point theory truncation and comparison techniques and morse theory

existence and multiplicity of solutions for elliptic equations with jumping nonlinearities abstract in this paper we are concerned with a semilinear elliptic dirichlet problem with jumping nonlinearity and using a completely variational method we show that the number of solutions may be arbitrarily large provided the number of jumped eigenvalues is large enough in order to prove this fact we show that for every positive integer k when a suitable parameter is large enough there exists a solution which presents k peaks under the assumptions we consider in this paper new unexpected phenomena are observed in the study of this problem and new methods are required to construct the k peaks solutions and describe their asymptotic behavior weak limits of the rescaled solutions localization of the concentration points of the peaks asymptotic profile of the rescaled peaks etc

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

singularly perturbed nonlinear dirichlet problems with a general nonlinearity let be a bounded domain in r n n 3 with a boundary c 2 we consider the following singularly perturbed nonlinear elliptic problem on e 2 u u f u 0 u 0 on u 0 on where the nonlinearity f is of subcritical growth under rather strong conditions on f it has been known that for small e 0 there exists a mountain pass solution u e of above problem which exhibits a spike layer near a maximum point of the distance function d from as e 0 in this paper we construct a solution u e of above problem which exhibits a spike layer near a maximum point of the distance function under certain conditions on f which we believe to be almost optimal

positive least energy solutions and phase separation for coupled schrodinger equations with critical exponent in this paper we study the following coupled schrodinger system which can be seen as a critically coupled perturbed brezis nirenberg problem left begin array ll delta u lambda 1 u mu 1 u 3 beta uv 2 quad x in omega delta v lambda 2 v mu 2 v 3 beta vu 2 quad x in omega u geqq 0 v geqq 0 rm in omega quad u v 0 quad rm on partial omega end array right here omega subset mathbb r 4 is a smooth bounded domain lambda 1 omega 0 and beta neq 0 where lambda 1 omega is the first eigenvalue of with the dirichlet boundary condition note that the nonlinearity and the coupling terms are both critical in dimension 4 that is frac 2n n 2 4 when n 4 we show that this critical system has a positive least energy solution for negative positive small and positive large for the case in which lambda 1 lambda 2 we obtain the uniqueness of positive least energy solutions we also study the limit behavior of the least energy solutions in the repulsive case beta to infty and phase separation is expected these seem to be the first results for this schrodinger system in the critical case

integrate and fire neurons driven by correlated stochastic input neurons are sensitive to correlations among synaptic inputs however analytical models that explicitly include correlations are hard to solve analytically so their influence on a neuron s response has been difficult to ascertain to gain some intuition on this problem we studied the firing times of two simple integrate and fire model neurons driven by a correlated binary variable that represents the total input current analytic expressions were obtained for the average firing rate and coefficient of variation a measure of spike train variability as functions of the mean variance and correlation time of the stochastic input the results of computer simulations were in excellent agreement with these expressions in these models an increase in correlation time in general produces an increase in both the average firing rate and the variability of the output spike trains however the magnitude of the changes depends differentially on the relative values of the input mean and variance the increase in firing rate is higher when the variance is large relative to the mean whereas the increase in variability is higher when the variance is relatively small in addition the firing rate always tends to a finite limit value as the correlation time increases toward infinity whereas the coefficient of variation typically diverges these results suggest that temporal correlations may play a major role in determining the variability as well as the intensity of neuronal spike trains

construction of pseudo gradient vector field and sign changing multiple solutions involving p laplacian existence of multiple and sign changing solutions for a problem involving p laplacian and jumping nonlinearities are considered via the construction of descent flow in c01 sign changing and multiple solutions are obtained under additional assumption on the nonlinearity the uniqueness of positive negative solution theorem is included too

mountain pass solutions for singularly perturbed nonlinear dirichlet problems abstract we consider the following singularly perturbed nonlinear elliptic problem 2 u u f u 0 u 0 in u 0 on where the nonlinearity f is of subcritical growth and is a bounded domain in r n under certain conditions of f there exists a mountain pass solution u for above problem and the solution u exhibits a spike layer as 0 to see the location of the spike layer for small 0 some additional assumptions certain nondegeneracy for a limiting problem or monotonicity of f t t have been assumed in this paper we characterize the location of the spike layer without such additional assumptions for n 3

estimation of time dependent input from neuronal membrane potential the set of firing rates of the presynaptic excitatory and inhibitory neurons constitutes the input signal to the postsynaptic neuron estimation of the time varying input rates from intracellularly recorded membrane potential is investigated here for that purpose the membrane potential dynamics must be specified we consider the ornstein uhlenbeck stochastic process one of the most common single neuron models with time dependent mean and variance assuming the slow variation of these two moments it is possible to formulate the estimation problem by using a state space model we develop an algorithm that estimates the paths of the mean and variance of the input current by using the empirical bayes approach then the input firing rates are directly available from the moments the proposed method is applied to three simulated data examples constant signal sinusoidally modulated signal and constant signal with a jump for the constant signal the estimation performance of the method is comparable to that of the traditionally applied maximum likelihood method further the proposed method accurately estimates both continuous and discontinuous time variable signals in the case of the signal with a jump which does not satisfy the assumption of slow variability the robustness of the method is verified it can be concluded that the method provides reliable estimates of the total input firing rates which are not experimentally measurable

moduli spaces for representations of concealed canonical algebras abstract continuing m domokos and h lenzing j algebra 228 2000 738 762 we investigate the relative invariants and moduli spaces introduced in a d king quart j math oxford ser 2 45 1994 515 530 which are related to a separating family of stable tubes in the module category of a concealed canonical algebra the class of concealed canonical algebras consists of representation infinite algebras of tame or wild representation type and contains in particular the representation theory of extended dynkin quivers we present a uniform characteristic free approach avoiding case by case analysis we introduce the notion of admissible weight for such algebras these are the weights i e elements of the dual of the grothendieck group of the module category attached to the infinite moduli spaces for families of modules from the separating subcategory by using an explicit description of the semigroup of admissible weights the corresponding notion of semistability is analyzed and is related to the formation of perpendicular categories we construct relative invariants belonging to admissible weights and prove that the constructed system is complete in the case of canonical algebras it turns out that for any concealed canonical algebra and for any admissible weight all the corresponding moduli spaces are isomorphic to some projective space moreover any such moduli space can be naturally identified with the moduli space belonging to a special weight called the rank as a consequence we show that in the case of a tame concealed algebra any infinite moduli space for families of modules is a projective space and all fields of rational invariants on irreducible components of representation spaces are purely transcendental

critical groups and multiple solutions of the p laplacian equations abstract in this paper we give some results on the critical groups of the trivial solution u 0 of the asymmetric p laplacian equation p u u p 2 u u p 2 u with dirichlet boundary value some applications to the existence of multiple solutions of the equation p u f x u are given

on the additive categories of generalized standard almost cyclic coherent auslander reiten components abstract we prove geometric and homological properties of modules in the additive categories of generalized standard almost cyclic coherent components in the auslander reiten quivers of finite dimensional algebras

precision of neural timing the small ɛ limit abstract we explore the precision of neural timing in a model neural system with n identical input neurons whose firing time in response to stimulation is chosen from a density f these input neurons stimulate a target cell which fires when it receives m hits within msec we prove that the density of the firing time of the target cell converges as 0 to the input density f raised to the mth and normalized we give conditions for convergence of the density in l 1 pointwise and uniformly as well as conditions for the convergence of the standard deviations

some remarks on the fucik spectrum of the p laplacian and critical groups let be a bounded domain in n n 1 for 1 p the fucik spectrum of the p laplacian on w 1 p 0 is defined as the set p of those points a b 2 such that pu a u p 1 b u p 1 u w 1 p 0 1 1 has a nontrivial solution here pu div u p 2 u and u max u 0 it is known that the first eigenvalue 1 of p on w 1 p 0 is positive simple and admits a positive eigenfunction 1 w 1 p 0 c1 see lindqvist 8 so p contains the two lines 1 and 1 a first nontrivial curve c2 in p through 2 2 where 2 is the second eigenvalue of p was recently constructed and variationally characterized by a

geometry of regular modules over canonical algebras we classify canonical algebras such that for every dimension vector of a regular module the corresponding module variety is normal respectively a complete intersection we also prove that for the dimension vectors of regular modules normality is equivalent to irreducibility