counting representations of quivers over finite fields abstract by counting the numbers of isomorphism classes of representations indecomposable or absolutely indecomposable of quivers over finite fields with fixed dimension vectors we obtain a multi variable formal identity if the quiver has no edge loops this identity turns out to be a q analogue of the kac denominator identity modulus a conjecture of kac

existence of positive solutions for first order discrete periodic boundary value problems with delay abstract let t 3 be a positive integer and t 0 1 2 t 1 we prove the existence of positive periodic solutions of the nonlinear discrete boundary value problem u t a t g u t u t b t f u t t t t u 0 u t our approach is based upon the global bifurcation techniques

frobenius morphisms and representations of algebras by introducing frobenius morphisms f on algebras a and their modules over the algebraic closure f q of the finite field f q of q elements we establish a relation between the representation theory of a over f q and that of the f fixed point algebra a f over f q more precisely we prove that the category mod a f of finite dimensional a f modules is equivalent to the subcategory of finite dimensional f stable a modules and when a is finite dimensional we establish a bijection between the isoclasses of indecomposable a f modules and the f orbits of the isoclasses of indecomposable a modules applying the theory to representations of quivers with automorphisms we show that representations of a modulated quiver or a species over f q can be interpreted as f stable representations of the corresponding quiver over f q we further prove that every finite dimensional hereditary algebra over f q is morita equivalent to some a f where a is the path algebra of a quiver q over f q and f is induced from a certain automorphism of q a close relation between the auslander reiten theories for a and a f is established in particular we prove that the auslander reiten modulated quiver of a f is obtained by folding the auslander reiten quiver of a finally by taking frobenius fixed points we are able to count the number of indecomposable representations of a modulated quiver over f q with a given dimension vector and to generalize kac s theorem for all modulated quivers and their associated kac moody algebras defined by symmetrizable generalized cartan matrices

common fixed point theorems for contractive maps two common fixed point theorems have been proved by using minimal type commutativity and contractive conditions the last theorem extends known results on compatible maps to a wider class of mappings

counting absolutely cuspidals for quivers for an arbitrary quiver q i omega and dimension vector mathbf d in mathbb n i we define the dimension of absolutely cuspidal functions on the moduli stacks of representations of dimension mathbf d of a quiver q over a finite field mathbb f q and prove that it is a polynomial in q which we conjecture to be positive and integral we obtain a closed formula for these dimensions of spaces of cuspidals for totally negative quivers

a remark on discontinuity at fixed point abstract in this note we give one more solution to the open question of the existence of contractive definitions which are strong enough to generate a fixed point but which do not force the mapping to be continuous at the fixed point r kannan 1969 9 b e rhoades 1988 14

positive rotationally symmetric solutions for a dirichlet problem involving the higher mean curvature operator in minkowski space abstract in this paper we investigate the global behavior of positive rotationally symmetric solutions for dirichlet problem involving the k th mean curvature operator in minkowski space l n 1 which is not elliptic for k 1 the proofs of our main results are based upon bifurcation techniques

drinfeld double and ringel green theory of hall algebras abstract as a continuation of the work of ringel and green on hall algebras the hopf algebra structure of a extended twisted hall algebra is presented here by introducing the ringel pairing of hall algebras the drinfeld double of a hall algebra is constructed the reduced form of this drinfeld double is applicable to the generic composition algebra which gives a complete realization of quantum group this ringel green theory makes it possible to give the formulae for the computation of commutators in quantum groups in terms of hall algebras

persistence and bifurcation of degenerate solutions abstract we consider a nonlinear equation f e u 0 where f is a differentiable mapping from r r x to y and x y are banach spaces when e varies from a fixed e e 0 bifurcation occurs to the solution curve s u s we study the degenerate solutions of the equation and we obtain several bifurcation theorems on the degenerate solutions which can be applied in many nonlinear problems to obtain precise global bifurcation diagrams

nodal solutions for nonlinear eigenvalue problems we are concerned with determining values of for which there exist nodal solutions of the boundary value problems u ra t f u 0 0 t 1 u o u 1 0 the proof of our main result is based upon bifurcation techniques

solution curves and exact multiplicity results for 2mth order boundary value problems abstract for any integer m 2 we consider the 2mth order boundary value problem 1 m u 2m x g u x u x x 1 1 u i 1 u i 1 0 i 0 m 1 where r and the function g r r is c1 and satisfies g 0 0 g 0 0 together with some further conditions as we obtain curves of nontrivial solutions of this problem bifurcating from u 0 at the eigenvalues of the linearised problem and obtain the exact number of solutions of the problem for lying in various intervals in r

global structure of radial positive solutions for a prescribed mean curvature problem in a ball abstract in this paper we are concerned with the global structure of radial positive solutions of boundary value problem div n v f x v 0 in b r v 0 on b r where n y y 1 y 2 y r n is a positive parameter b r x r n x r and denotes the euclidean norm in r n all results depending on the behavior of nonlinear term f near 0 are obtained by using global bifurcation techniques

global structure of positive solutions for nonlocal boundary value problems involving integral conditions abstract in this paper we consider the nonlinear eigenvalue problems u h t f u 0 0 t 1 u 0 0 u 1 0 1 u s d a s where 0 1 u s d a s is a stieltjes integral with a nondecreasing and a t is not a constant on 0 1 h c 0 1 0 and h t 0 on any subinterval of 0 1 f c 0 0 and f s 0 for s 0 and f 0 f 0 f 0 lim s 0 f s s f lim s f s s we investigate the global structure of positive solutions by using global bifurcation techniques

nodal solutions of second order boundary value problems with superlinear or sublinear nonlinearities abstract we consider boundary value problems for nonlinear second order differential equations of the form u a t f u 0 t 0 1 u 0 u 1 0 where a c 1 0 1 0 with a t 0 0 for some t 0 0 1 and f r r is continuous with f s s 0 for s 0 we establish existence and multiplicity results for nodal solutions of the problems if either f 0 0 f or f 0 f 0 where f s s approaches f 0 and f as s approaches 0 and respectively we use bifurcation techniques to prove our main results

a new approach to kac s theorem on representations of valued quivers by using the ringel hall algebra approach we first present a new proof of kac s theorem for species over a finite field the method used here is quite different from earlier techniques in sv sevenhant and van den bergh have discovered an interesting relation between a conjecture of kac on representations of quivers and the structure of the ringel hall algebras in some special cases we second show that this relation still holds for species

a class of nonlinear sturm liouville problems with infinitely many solutions on etudie l existence des solutions des problemes de sturm liouville non lineaires dont la partie lineaire est a la resonance on montre que de tels problemes peuvent avoir une infinite de solutions si les perturbations non lineaires sont periodiques

hall algebras hereditary algebras and quantum groups let r be an associative hereditary algebra over a finite field k and let r fin be the full subcategory of r mod whose objects are those left r modules x which are finite as sets ix oc assume also that r isfinitary in c ringel s sense i e that iext l s s c for all simple s s in r fin this condition is met for example if r is finitely generated as k algebra 4 pp 435 436 let be the set of all isomorphism classes in r fin if 2 e then u will denote an r module in class 2 the class of all zero left r modules is denoted 0 let i c be the set of all isomorphism classes of simple modules in r fin thus u i e i is a complete set of simple finite left r modules we identify the grothendieck group k0 r fin with the free abelian group e1 v i v e 7z having i as free basis so that if x e r fin then the corresponding element of k0 r fin is the dimension vector dimx y i vii where for each i e vi is the multiplicity of the simple module u in any composition series of x clearly d imx lies in the subsemigroup

uniqueness and exact multiplicity of solutions for a class of fourth order semilinear problems using techniques of bifurcation theory we give exact multiplicity and uniqueness results for the fourth order dirichlet problem which describes deflection of an elastic beam subjected to a nonlinear force and clamped at the end points the crucial part of this approach was to show positivity of non trivial solutions of the corresponding linearized problem

fixed point theorems for families of weakly non expansive maps abstract in this paper we present some fixed point theorems for families of weakly non expansive maps under some relatively weaker and more general conditions our results generalize and improve several results due to jungck g jungck fixed points via a generalized local commutativity int j math math sci 25 8 2001 497 507 jachymski j jachymski a generalization of the theorem by rhoades and watson for contractive type mappings math japon 38 6 1993 1095 1102 guo c guo an extension of fixed point theorem of krasnoselski chinese j math p o c 21 1 1993 13 20 rhoades b e rhoades a comparison of various definitions of contractive mappings trans amer math soc 226 1977 257 290 and others

coincidence and fixed points for hybrid strict contractions abstract we define the property e a for single valued and multivalued mappings and introduce the notion of t weak commutativity for a hybrid pair f t of single valued and multivalued maps we obtain some coincidence and fixed point theorems for this class of maps and derive as application an approximation theorem

on the existence of multiple solutions of the boundary value problem for nonlinear second order differential equations abstract we consider the boundary value problem for nonlinear second order differential equations of the form u a x f u 0 0 u 0 u 1 0 we establish the precise condition concerning the behavior of the ratio f s s at infinity and zero for the existence of solutions with prescribed nodal properties then we derive the existence and the multiplicity of nodal solutions to the problem our argument is based on the shooting method together with the strum s comparison theorem the results obtained here can be applied to the study of radially symmetric solutions of the dirichlet problem for semilinear elliptic equations in annular domains

spectrum and constant sign solutions for a fractional laplace problem with sign changing weight abstract in this paper we study the spectrum of the fractional laplace operator with sign changing weight and show that there exist two simple isolated principal eigenvalues 1 and 1 by use of the obtained spectrum results we study the existence multiplicity and nonexistence of constant sign solutions to corresponding nonlinear problems according to the asymptotic behavior of nonlinear term f at 0 and whether f possesses zeros in r 0 as far as we know the spectral results presented here are new and the rest theorems partially generalize the corresponding ones in the literature

global bifurcation of positive radial solutions for an elliptic system in reactor dynamics in this paper we are concerned with the existence of positive radial solutions of the elliptic system email protected email protected f x u r 1 0 is a constant m 0 is a parameter and 0 0 is continuous and f t s 0 for all t s r 1 r 2 x 0 under some appropriate conditions on the nonlinearity f we show that the above system possesses at least one positive radial solution for any email protected 0 the proof of our main results is based upon bifurcation techniques

quantum deformations of generalized kac moody algebras and their modules abstract we construct quantum groups uq g associated with generalized kac moody algebras g with admissible borcherds cartan matrices we also construct quantum deformations of highest weight modules over u g with integral highest weights we show that for generic q verma modules over u g with integral highest weights and irreducible highest weight modules over u g with dominant integral highest weights can be deformed to those over uq g in such a way that the dimensions of weight spaces are invariant under the deformation in particular for generic q the characters of irreducible highest weight modules over uq g with dominant integral highest weights are given by the weyl kac borcherds formula

generalized kac moody algebras on etudie une classe d algebres de lie qui ont une forme bilineaire contravariante qui est presque definie positive ces algebres generalisent celles de kac moody et peuvent etre considerees comme des algebres de kac moody avec des racines simples imaginaires

second order three point boundary value problems with jumping non linearities abstract we consider the non linear three point boundary value problem consisting of the equation 1 u f u h a e on 0 1 where h l 1 0 1 together with the boundary conditions 2 u 0 0 u 1 u where 0 1 the function f r r is continuous and we assume that the following limits exist and are finite f lim s f s s we allow f f such a non linearity f is said to be jumping related to 1 is the equation 3 u a u b u u on 0 1 where a b r and u x max u x 0 for x 0 1 the problem 2 and 3 is positively homogeneous and jumping values of for which 2 and 3 has a non trivial solution u will be called half eigenvalues while the corresponding solutions u will be called half eigenfunctions we show that a sequence of half eigenvalues exists the corresponding half eigenfunctions having specified nodal properties and we obtain certain spectral and degree theoretic properties of the set of half eigenvalues these properties lead to solvability and non solvability results for the problem 1 and 2 the set of half eigenvalues is closely related to the fucik spectrum of the problem and equivalent solvability and non solvability results for 1 and 2 are obtained from either the half eigenvalue or the fucik spectrum approach

positive solutions of nonlinear three point boundary value problems abstract let a c 0 1 b c 0 1 0 let 1 t be the unique solution of the linear boundary value problem u t a t u t b t u t 0 t 0 1 u 0 0 u 1 1 we study the existence of positive solutions to the nonlinear boundary value problem u t a t u t b t u t h t f u 0 t 0 1 u 0 0 u u 1 where 0 0 and f c 0 0 we show the existence of at least one positive solution if f is either superlinear or sublinear by applying the fixed point theorem in cones

spectral properties and nodal solutions for second order m point boundary value problems abstract we consider the m point boundary value problem consisting of the equation u f u on 0 1 where f r r is c 1 with f 0 0 together with the boundary conditions u 0 0 u 1 i 1 m 2 i u i where m 3 i 0 1 and i 0 for i 1 m 2 with i 1 m 2 i 1 we first show that the spectral properties of the linearisation of this problem are similar to the well known properties of the standard sturm liouville problem with separated boundary conditions with a minor modification to deal with the multi point boundary condition these spectral properties are then used to prove a rabinowitz type global bifurcation theorem for a bifurcation problem related to the above problem finally we use the global bifurcation theorem to obtain nodal solutions that is sign changing solutions with a specified number of zeros of the above problem under various conditions on the asymptotic behaviour of f

multiplicity results for second order two point boundary value problems with superlinear or sublinear nonlinearities we consider boundary value problems for nonlinear second order differential equations of the form u a t f u 0 t epsilon 0 1 u 0 u 1 0 where a epsilon c 0 1 0 infinity and f r r is continuous and satisfies f s s 0 for s not equal 0 we establish existence and multiplicity results for nodal solutions to the problems if either f 0 0 f infinity infinity or f 0 infinity f 0 0 where f s s approaches f 0 and f infinity as s approaches 0 and infinity respectively we use bifurcation techniques to prove our main results c 2004 elsevier inc all rights reserved