numerical range and functional calculus in hilbert space we prove an inequality related to polynomial functions of a square matrix involving the numerical range of the matrix we also show extensions valid for bounded and also unbounded operators in hilbert spaces which allow the development of a functional calculus

keeping track of limit cycles this article uses analytic geometry methods to bound the number of limit cycles of bounded period for a planar polynomial vector fields it introduces the notion of limit periodic sets and shows that the hilbert s 16th problem can be reduced to the existence of a uniform bound on its cyclicity it also shows the existence of a universal bound on the number of limit cycles in the perturbation of a linear focus as a consequence of the noetherian property of the bautin ideal

limit cycles of a lienard system with symmetry allowing for discontinuity abstract this paper presents new results on the limit cycles of a lienard system with symmetry allowing for discontinuity our results generalize and improve the results in 32 theorem 1 and 2 or the monograph 33 chapter 4 theorem 5 2 the results in 33 are only valid for the smooth system we emphasize that our main results are valid for discontinuous systems moreover we show the presence and an explicit upper bound for the amplitude of the two limit cycles and we estimate the position of the double limit cycle bifurcation surface in the parameter space until now there is no result to determine the amplitude of the two limit cycles the existing results on the amplitude of limit cycles guarantee that the lienard system has a unique limit cycle finally some applications and examples are provided to show the effectiveness of our results we revisit a co dimension 3 lienard oscillator see 22 34 in application 1 li and rousseau 22 studied the limit cycles of such a system when the parameters are small however for the general case of the parameters in particular the parameters are large the upper bound of the limit cycles remains open we completely provide the bifurcation diagram for the one equilibrium case moreover we determine the amplitude of the two limit cycles and estimate the position of the double limit cycle bifurcation surface for the one equilibrium case application 2 is presented to study the limit cycles of a class of the filippov system

algebraic families of analytic functions i the aim of this paper is to consider analytic functions arising in connection with algebraic differential equations from the following point of view taylor coefficients of such functions usually turn out to be polynomials or rational functions in the parameters of the problem following pioneering work of bautin 1 we relate analytic properties of the functions considered with algebraic properties of various ideals generated by their taylor coefficients bautin s approach has been successfully applied to the study of the poincare mapping of the planar polynomial systems in 4 9 27 and to the study of the period function of polynomial hamiltonian systems in 5 6 12 13 however the study of the poincare mapping of the polynomial differential equations closely related to the second part of hilbert 16th problem is probably the most difficult among a wide variety of problems to which bautin s approach can be applied in particular counting zeroes of solutions of algebraic differential equations is an important and mostly open problem to our best knowledge the first application of bautin s method to this problem was given in 26 in the present paper we systematically apply this method in the following two situations to algebraic functions and to solutions of the first order polynomial differential equations higher order linear equations are also considered in section 5 article no de963250

relative rigid objects in triangulated categories abstract let t be a krull schmidt hom finite triangulated category with suspension functor 1 let r be a basic rigid object the endomorphism algebra of r and pr r t the subcategory of objects finitely presented by r we investigate the relative rigid objects i e r 1 rigid objects of t our main results show that the r 1 rigid objects in pr r are in bijection with rigid modules and the maximal r 1 rigid objects with respect to pr r are in bijection with support tilting modules we also show that various previously known bijections involving support tilting modules are recovered under respective assumptions

rigid objects in higher cluster categories abstract we study maximal m rigid objects in the m cluster category c h m associated with a finite dimensional hereditary algebra h with n nonisomorphic simple modules we show that all maximal m rigid objects in these categories have exactly n nonisomorphic indecomposable summands and that any almost complete m rigid object in c h m has exactly m 1 nonisomorphic complements we also show that the maximal m rigid objects and the m cluster tilting objects in these categories coincide and that the class of finite dimensional algebras associated with maximal m rigid objects is closed under certain factor algebras

quotients of cluster categories higher cluster categories were recently introduced as a generalization of cluster categories this paper shows that in dynkin types a and d half of all higher cluster categories can be obtained as quotients of cluster categories the other half are quotients of 2 cluster categories the lowest type of higher cluster categories hence in dynkin types a and d all higher cluster phenomena are implicit in cluster categories and 2 cluster categories in contrast the same is not true in dynkin type e

characterization of the subdifferential of some matrix norms abstract a characterization is given of the subdifferential of matrix norms from two classes orthogonally invariant norms and operator or subordinate norms specific results are derived for some special cases

on open orbits and their complements 1 1 we fix an algebraically closed field k and a quiver q without oriented cycles whose vertex set q we identify with 1 n for a in the set q of arrows of q we denote by ta and ha the tail and the head of a respectively a representation x of q consists of a family x i in qo of finite dimensional k vector spaces and a family x a x ta x ha a e q1 of k linear maps a morphism f x y between two representations of q is a family f i x i y i in qo of k linear maps satisfying f ha ox a y a of ta for all arrows a we denote the category of representations of q by mod q it is an abelian category of global dimension at most 1 we will denote the only possibly non trivial extension group ext by ext the dimension vector of a representation x of q is the vector dim x dim x l dim x n in n and the dimension of x is the natural number dim x dim x 1 dim x n the support of x is the full subquiver of q whose vertices are i x i zo for a vector d d d e n we define r q d to be the set of representations x of q such that x i k for ie qo thus r q d is the product r q d n hom kdru kdhe

on a cluster category of infinite dynkin type and the relation to triangulations of the infinity gon let k be a field and let d be a k linear algebraic triangulated category with split idempotents let be the suspension functor of d and let s be a 2 spherical object of d that is the morphism space d s i s is k for i 0 and i 2 and vanishes otherwise assume that s classically generates d that is each object of d can be built from s using de suspensions direct sums direct summands and distinguished triangles it was proved in 15 thm 2 1 that d is uniquely determined by these properties as we will explain d is a good candidate for a cluster category of dynkin type a for instance we show that there is a bijection between the cluster tilting subcategories of d and certain triangulations of the gon we use this to give an example of a subcategory a which is weak cluster tilting that is satisfies a 1a a but fails to be functorially finite perpendicular

on the upper bound of the criticality of potential systems at the outer boundary using the roussarie ecalle compensator abstract this paper is concerned with the study of the criticality of families of planar centers more precisely we study sufficient conditions to bound the number of critical periodic orbits that bifurcate from the outer boundary of the period annulus of potential centers in the recent years the new approach of embedding the derivative of the period function into a collection of functions that form a chebyshev system near the outer boundary has shown to be fruitful in this issue in this work we tackle with a remaining case that was not taken into account in the previous studies in which the roussarie ecalle compensator plays an essential role the theoretical results we develop are applied to study the bifurcation diagram of the period function of two different families of centers the power like family x x q x p p q r with p q and the family of dehomogenized loud s centers

crouzeix s conjecture and perturbed jordan blocks abstract crouzeix conjectured in m crouzeix bounds for analytical functions of matrices integr equ oper theory 48 2004 461 477 that for any square matrix a and any polynomial p p a 2 max p z z w a where w a is the field of values of a and denotes the spectral norm in this paper we show that the conjecture holds for matrices of the form 1 1 where and are complex numbers the technique for 1 and for 1 is to show that if g is a bijective conformal mapping from the field of values of such a matrix a to the unit disk mapping to 0 then g a is similar to a contraction via a similarity transformation with condition number at most 2

quivers for silting mutation abstract we give a combinatorial mutation rule for aihara and iyama s silting mutation as an application we reprove keller yang s mutation rule for ginzburg algebras and obtain an analog of that rule for ginzburg algebras of arbitrary calabi yau dimension

global families of limit cycles of planar analytic systems the global behavior of any one parameter family of limit cycles of a planar analytic system x f x a depending on a parameter a e r is determined it is shown that any one parameter family of limit cycles belongs to a maximal one parameter family which is either open or cyclic if the family is open then it terminates as the parameter or the orbits become unbounded or it terminates at a critical point or on a compound separatrix cycle of the system this implies that the periods in a one parameter family of limit cycles can become unbounded only if the orbits become unbounded or if they approach a degenerate critical point or compound separatrix cycle of the system this is a more specific result for planar analytic systems than wintner s principle of natural termination for n dimensional systems where the periods can become unbounded in strange ways this work generalizes duff s results for one parameter families of limit cycles generated by a one parameter family of rotated vector fields in particular it is shown that the behavior at a nonsingular multiple limit cycle of any one parameter family of limit cycles is exactly the same as the behavior at a multiple limit cycle of a one parameter family of liniit cycles generated by a one parameter family of rotated vector fields

m cluster tilted algebras of euclidean type abstract we consider m cluster tilted algebras arising from quivers of euclidean type and we give necessary and sufficient conditions for those algebras to be representation finite for the case a using the geometric realization we get a description of representation finite type in terms of m 2 angulations we establish which m cluster tilted algebras arise at the same time from quivers of type a and a finally we characterize representation infinite m cluster tilted algebras arising from a quiver of type a as m relations extensions of some iterated tilted algebra of type a

partial smoothness tilt stability and generalized hessians we compare two recent variational analytic approaches to second order conditions and sensitivity analysis for nonsmooth optimization we describe a broad setting where computing the generalized hessian of mordukhovich is easy in this setting the idea of tilt stability introduced by poliquin and rockafellar is equivalent to a classical smooth second order condition

a proof of crouzeix s conjecture for a class of matrices abstract crouzeix s conjecture is that for any square matrix a and any polynomial p we have p a 2 max p z z w a where w a is the field of values of a and denotes the spectral norm in this paper we show that the conjecture holds for the matrices of the form 1 n 1 n where c and 1 n c n

from m clusters to m noncrossing partitions via exceptional sequences let w be a finite crystallographic reflection group the generalized catalan number of w coincides both with the number of clusters in the cluster algebra associated to w and with the number of noncrossing partitions for w natural bijections between these two sets are known for any positive integer m both m clusters and m noncrossing partitions have been defined and the cardinality of both these sets is the fuss catalan number cm w we give a natural bijection between these two sets by first establishing a bijection between two particular sets of exceptional sequences in the bounded derived category db h for any finite dimensional hereditary algebra h

bernstein inequalities and applications to analytic geometry and differential equations we introduce new methods of complex analysis inequalities of bernstein type to study projections of analytic sets these techniques are applied to problems of bifurcations of periodic orbits of differential equations such as the local hilbert s 16th problem

endomorphism algebras of tilting modules over m replicated algebras abstract let a be a finite dimensional hereditary algebra over an algebraically closed field k and let a m be the m replicated algebra of a in this paper we investigate the structure properties of the endomorphism algebras of tilting modules of a m and prove that all the endomorphism algebras of tilting modules of a m can be realized as the iterated endomorphism algebras of bb tilting modules that is for each pair of basic tilting a m modules t 1 and t 2 there exists a series of finite dimensional algebras 0 1 s which are the endomorphism algebras of some basic tilting a m modules and for each i there is a bb tilting i module m i such that 0 end a m t 1 i end i 1 m i 1 for 1 i s and end a m t 2 end s m s

liouvillian first integrals of differential equations liouvillian functions are functions that are built up from rational functions using exponentiation integration and algebraic functions we show that if a system of differential equations has a generic solution that satisfies a liouvillian relation that is there is a liouvillian function of several variables vanishing on the curve defined by this solution then the system has a liouvillian first integral that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set we can refine this result in special cases to show that the first integral must be of a very special form

exchange graphs and ext quivers abstract we study the oriented exchange graph eg n q of reachable hearts in the finite dimensional derived category d n q of the cy n ginzburg algebra n q associated to an acyclic quiver q we show that any such heart is induced from some heart in the bounded derived category d q via some lagrangian immersion l d q d n q we build on this to show that the quotient of eg n q by the seidel thomas braid group is the exchange graph ceg n 1 q of cluster tilting sets in the higher cluster category c n 1 q as an application we interpret buan thomas coloured quiver for a cluster tilting set in terms of the ext quiver of any corresponding heart in d n q

on the local solvability and the local integrability of systems of vector fields on etudie des systemes surdetermines de champs vectoriels dans un sous ensemble ouvert de r n 1 de type l j t j j t x x j 1 m ayant des coefficients analytiques on suppose que les champs de vecteurs l j satisfont la condition de frobenius

the algebras with discrete derived category abstract we classify the finite dimensional algebras over an algebraically closed field whose bounded derived category does not admit an infinite continuous family of pairwise non isomorphic indecomposable complexes

defining an m cluster category we show that a certain orbit category considered by keller encodes the combinatorics of the m clusters of fomin and reading in a fashion similar to the way the cluster category of buan marsh reineke reiten and todorov encodes the combinatorics of the clusters of fomin and zelevinsky this allows us to give type uniform proofs of certain results of fomin and reading in the simply laced cases

bifurcation of critical periods for plane vector fields on etudie un probleme de bifurcation dans des familles de champs de vecteurs analytiques plans qui ont un centre non degenere a l origine pour toutes les valeurs d un parametre r n on definit la fonction periode p on determine les points critiques de cette fonction pres du centre avec comme parametre de bifurcation

the criticality of centers of potential systems at the outer boundary abstract the number of critical periodic orbits that bifurcate from the outer boundary of a potential center is studied we call this number the criticality at the outer boundary our main results provide sufficient conditions in order to ensure that this number is exactly 0 and 1 we apply them to study the bifurcation diagram of the period function of x y x x 1 p x 1 q y with q p this family was previously studied for q 1 by y miyamoto and k yagasaki

entropie geometrique des feuilletages on definit et on etudie l entropie geometrique d un feuilletage generalisant directement l entropie topologique d un flot

study of the period function of a two parameter family of centers abstract in this paper we study the period function of x 1 x p 1 x q with p q r and p q we prove three independent results the first one establishes some regions in the parameter space where the corresponding center has a monotonous period function this result extends the previous ones by miyamoto and yagasaki for the case q 1 the second one deals with the bifurcation of critical periodic orbits from the center the third one is addressed to the critical periodic orbits that bifurcate from the period annulus of each one of the three isochronous centers in the family when perturbed by means of a one parameter deformation these three results together with the ones that we obtained previously on the issue lead us to propose a conjectural bifurcation diagram for the global behaviour of the period function of the family

the period function of generalized loudʼs centers abstract in this paper a three parameter family of planar differential systems with homogeneous nonlinearities of arbitrary odd degree is studied this family is an extension to higher degree of loud s systems the origin is a nondegenerate center for all values of the parameter and we are interested in the qualitative properties of its period function we study the bifurcation diagram of this function focusing our attention on the bifurcations occurring at the polycycle that bounds the period annulus of the center moreover we determine some regions in the parameter space for which the corresponding period function is monotonous or it has at least one critical period giving also its character maximum or minimum finally we propose a complete conjectural bifurcation diagram of the period function of these generalized loud s centers

the existential hilbert 16 th problem and an estimate for cyclicity of elementary polycycles the existential hilbert problem is a weak version of the part b of the hilbert 16 th problem which also asks not only about the number but also about position of limit cycles of 1 the problem about finiteness of number of limit cycles for an individual polynomial line field 1 is called dulac problem since the pioneering work of dulac du who claimed in 1923 to solve this problem but an error was found the dulac problem was solved by two independent and rather different proofs given independently

on m cluster tilted algebras and trivial extensions abstract in this work we study the connection between iterated tilted algebras and m cluster tilted algebras we show that an iterated tilted algebra induces an m cluster tilted algebra this m cluster tilted algebra can be seen as a trivial extension of another iterated tilted algebra which is derived equivalent to the original one we give a procedure to find this new iterated tilted algebra these m cluster tilted algebras are quotients of higher relation extensions

variational analysis of the crouzeix ratio let w a denote the field of values numerical range of a matrix a for any polynomial p and matrix a define the crouzeix ratio to have numerator max left p zeta zeta in w a right max p w a and denominator vert p a vert 2 p a 2 crouzeix s 2004 conjecture postulates that the globally minimal value of the crouzeix ratio is 1 2 over all polynomials p of any degree and matrices a of any order we derive the subdifferential of this ratio at pairs p a for which the largest singular value of p a is simple in particular we show that at certain candidate minimizers p a the crouzeix ratio is clarke regular and satisfies a first order nonsmooth optimality condition and hence that its directional derivative is nonnegative there in every direction in polynomial matrix space we also show that pairs p a exist at which the crouzeix ratio is not regular

genericity conditions for finite cyclicity of elementary graphics abstract in this paper we explore the khovanskii method for proving the finite cyclicity of elementary graphics and how it can be applied in practice the genericity conditions needed in that case form a proper subset of the usual methods moreover some of the conditions are non intrinsic and can be artificially created by action of the automorphism groups preserving the normal forms near the singularities and their action on regular transitions hence we introduce an extension of the method which treats the usual functional pfaffian systems together with the admissible changes of coordinates in the functional equations

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

estimating the number of limit cycles in polynomials systems abstract we describe a method based on algorithms of computational algebra for obtaining an upper bound for the number of limit cycles bifurcating from a center or a focus of polynomial vector field we apply it to a cubic system depending on six parameters and prove that in the generic case at most six limit cycles can bifurcate from any center or focus at the origin of the system

numerical investigation of crouzeix s conjecture abstract crouzeix s conjecture states that for all polynomials p and matrices a the inequality p a 2 p w a holds where the quantity on the left is the 2 norm of the matrix p a and the norm on the right is the maximum modulus of the polynomial p on w a the field of values of a we report on some extensive numerical experiments investigating the conjecture via nonsmooth minimization of the crouzeix ratio f p w a p a using chebfun to evaluate this quantity accurately and efficiently and the bfgs method to search for its minimal value which is 0 5 if crouzeix s conjecture is true almost all of our optimization searches deliver final polynomial matrix pairs that are very close to nonsmooth stationary points of f with stationary value 0 5 for which w a is a disk or smooth stationary points of f with stationary value 1 for which w a has a corner our observations have led us to some additional conjectures as well as some new theorems we hope that these give insight into crouzeix s conjecture which is strongly supported by our results

endomorphism algebras for a class of negative calabi yau categories abstract we consider an orbit category of the bounded derived category of a path algebra of type a n which can be viewed as a m 1 cluster category for m 1 in particular we give a characterisation of those maximal m rigid objects whose endomorphism algebras are connected and then use it to explicitly study these algebras specifically we give a full description of them in terms of quivers and relations and relate them with higher cluster tilted algebras of type a as a by product we introduce a larger class of algebras called tiling algebras

hilbert s 16th problem for quadratic vector fields abstract we discuss here a systematic approach towards a positive answer to hilbert s 16th problem for quadratic systems namely the existence of a uniform bound for the number of limit cycles of a quadratic system the method is the following describe the limit periodic sets surrounding the origin in a family of quadratic vector fields and prove that they have finite cyclicity in this paper we give the list of all graphics and degenerate graphics that should be considered and describe their general features we also indicate how to find or where to find concrete examples of these limit periodic sets

three kinds of mutation for a finite dimensional hereditary algebra h we consider exceptional sequences in mod h silting objects in the bounded derived category d and m cluster tilting objects in the m cluster category cm d 1 m there are mutation operations on both the set of m cluster tilting objects and the set of exceptional sequences it is also possible to define a mutation operation for silting objects we compare these three different notions of mutation