the stretch of a foliation and geometric superrigidity we consider compact smooth foliated manifolds with leaves isometrically covered by a fixed symmetric space of noncompact type such objects can be considered as compact models for the geometry of the symmetric space based on this we formulate and solve a geometric superrigidity problem for foliations that seeks the existence of suitable isometric totally geodesic immersions to achieve this we consider the heat flow equation along the leaves of a foliation a bochner formula on foliations and a geometric invariant for foliations with leafwise riemannian metrics called the stretch we obtain as applications a metric rigidity theorem for foliations and a rigidity type result for riemannian manifolds whose geometry is only partially symmetric

jordan all derivable points in the algebra of all upper triangular matrices abstract let tm n be the algebra of all n n upper triangular matrices we say that l tm n is a jordan derivable mapping at g if st ts s t s t t s t s for any s t tm n with st g an element g tm n is called a jordan all derivable point of tm n if every jordan derivable linear mapping at g is a derivation in this paper we show that every element in tm n is a jordan all derivable point

non constant positive steady states of a prey predator system with cross diffusions abstract in this paper we study a strongly coupled elliptic system arising from a lotka volterra prey predator system where cross diffusions are included in such a way that the prey runs away from the predator and the predator moves away from a large group of preys we establish the existence and non existence of its non constant positive solutions our results show that if m 1 b a 2 m 1 b 1 m 1 m 2 when 0 m 1 m 2 1 or a m 1 b when m 1 m 2 1 0 d 1 m 1 v u 1 d 2 0 d 3 0 and d 4 1 m 1 v u then there exists d 1 d 2 d 3 d 4 such that the stationary problem admits non constant positive solutions otherwise the stationary problem has no non constant positive solution in particular the results indicate that its non constant positive solutions are mainly created by the cross diffusion d 4

strong allee effect in a diffusive predator prey system with a protection zone article history a reaction diffusion predator prey system with strong allee effect and a protection zone for the prey is considered dynamics and steady state solutions of the system are analyzed in particular it is shown that the overexploitation phenomenon can be avoided if the allee effect threshold is low and the protection zone is large

on bilinear maps determined by rank one matrices with some applications abstract let m n be the algebra of all n n matrix over a field f a a rank one matrix in m n in this article it is shown that if a bilinear map from m n m n to m n satisfies the condition that u v i a whenever u v a then there exists a linear map from m n to m n such that x y x y x y m n if is further assumed to be symmetric then there exists a matrix b such that x y tr xy b for all x y m n applying the main result we prove that if a linear map on m n is desirable at a rank one matrix then it is a derivation and if an invertible linear map on m n is automorphisable at a rank one matrix then it is an automorphism in other words each rank one matrix in m n is an all desirable point and an all automorphisable point respectively

effect of cross diffusion in the diffusion prey predator model with a protection zone ii abstract in the current paper we continue the mathematical analysis studied in li and wu 2017 15 and oeda 2011 22 and further study the effect of cross diffusion for the predator on the stationary problem the existence of positive solutions is first established by the bifurcation theory we next discuss the limiting behavior of positive solutions when the intrinsic growth rate of the predator species tends to infinity moreover as the prevention coefficient tends to infinity we obtain two shadow systems and give the complete limiting characterization of positive solutions

all derivable points in nest algebras abstract suppose that a is an operator algebra on a hilbert space h an element v in a is called an all derivable point of a for the strong operator topology if every strong operator topology continuous derivable mapping at v is a derivation let n be a complete nest on a complex and separable hilbert space h suppose that m belongs to n with 0 m h and write m for m or m our main result is for any alg n with p m p m if m is invertible in alg n m then is an all derivable point in alg n for the strong operator topology

reflexive algebras with finite width lattices tensor products cohomology compact perturbations reflexive algebras play a central role in the study of general operator algebras for a reflexive algebra the associated invariant subspace lattice has structural importance analogous to that of the algebraic commutant in the study of algebras tomita s tensor product commutation theorem can be restated in the form alg l1 l2 alg l1 alg l2 where each li is a reflexive ortho lattice this same formula is proved for n fold tensor products in the setting when each li is a nest thus in particular a tensor product of nest algebras is again a reflexive algebra lance has shown that the hochschild cohomology of nest algebras vanishes modifications of his arguments show that cohomology vanishes for arbitrary csl algebras whose lattices are generated by finitely many independent nests this appears to be the strongest possible result in this direction the class of irreducible tridiagonal algebras with finite width commutative lattices is investigated and it is shown that these algebras have nontrivial first cohomology finally it is shown that if l is a finite width commutative subspace lattice and k is the set of compact operators then the quasitriangular algebra alg l k is closed in the norm topology this extends to arbitrary finite width csl algebras a result obtained for nest algebras by fall arveson and muhly

effect of a protection zone in the diffusive leslie predator prey model abstract in this paper we consider the diffusive leslie predator prey model with large intrinsic predator growth rate and investigate the change of behavior of the model when a simple protection zone 0 for the prey is introduced as in earlier work y du j shi a diffusive predator prey model with a protection zone j differential equations 229 2006 63 91 y du x liang a diffusive competition model with a protection zone j differential equations 244 2008 61 86 we show the existence of a critical patch size of the protection zone determined by the first dirichlet eigenvalue of the laplacian over 0 and the intrinsic growth rate of the prey so that there is fundamental change of the dynamical behavior of the model only when 0 is above the critical patch size however our research here reveals significant difference of the model s behavior from the predator prey model studied in y du j shi a diffusive predator prey model with a protection zone j differential equations 229 2006 63 91 with the same kind of protection zone we show that the asymptotic profile of the population distribution of the leslie model is governed by a standard boundary blow up problem and classical or degenerate logistic equations

multiplicative mappings at some points on matrix algebras let mn be the algebra of all n n matrices and let mn mn be a linear mapping we say that is a multiplicative mapping at g if st s t for any s t mn with st g fix g mn we say that g is an all multiplicative point if every multiplicative linear bijection at g with in in is a multiplicative mapping in mn where in is the unit matrix in mn we mainly show in this paper the following two results 1 if g mn with detg 0 then g is an all multiplicative point in mn 2 if is an multiplicative mapping at in then there exists an invertible matrix p mn such that either s psp 1 for any s mn or t pttrp 1 for any t mn

local automorphisms of nest subalgebras of factor von neumann algebras in this paper it is proved that every surjective weakly continuous linear local automorphism of nest subalgebras with non trivial nests of factor von neumann algebras is an automorphism and that every surjective linear 2 local automorphism of nest algebras with non trivial nests of factor von neumann algebras is an automorphism

characterization of generalized jordan left derivations on real nest algebras abstract let a be a real nest algebra of b h where h is a real and separable hilbert space we show that the following conditions are equivalent for a weak topology continuous linear map a b h 1 is a left preserving kernel into range mapping i e t ker t ran t for any t a 2 is a generalized left inner derivations i e t t a bt for some a b b h 3 is a generalized jordan left derivations i e t 2 t t t t t i t for any t a 4 is a left 1 preserving kernel into range mapping i e t ker t ran t for any rank one operator t a

characterizations of lie derivations of b x abstract let x be a banach space of dimension greater than 2 we prove that if b x b x is a linear map satisfying a b a b a b for any a b b x with ab 0 resp ab p where p is a fixed nontrivial idempotent then d where d is a derivation of b x and b x c i is a linear map vanishing at commutators a b with ab 0 resp ab p

local derivations of nest subalgebras of von neumann algebras in this paper it is proved that every norm continuous linear local derivation of a nest subalgebra of a factor von neumann algebra is a derivation and that every linear 2 local derivation of a nest subalgebra of a factor von neumann algebra is a derivation

additive maps derivable or jordan derivable at zero point on nest algebras abstract let alg n be a nest algebra associated with the nest n on a real or complex banach space x assume that every n n is complemented whenever n n let alg n alg n be an additive map it is shown that the following three conditions are equivalent 1 is derivable at zero point i e ab a b a b whenever ab 0 2 is jordan derivable at zero point i e ab ba a b a b b a b a whenever ab ba 0 3 has the form a a ca for some additive derivation and some scalar c it is also shown that is generalized derivable at zero point i e ab a b a b a i b whenever ab 0 if and only if is an additive generalized derivation finer characterizations of above maps are given for the case dim x

nonlinear mappings on upper triangular matrices derivable at zero point abstract let t n f be the set of all n n upper triangular matrices over a field f recently by using a technique from graph theory l wang in 8 proved that a bijective mapping on t n f with f a finite field preserves zero product i e x y 0 whenever x y 0 if and only if can be decomposed into the product of an inner automorphism an automorphism of t n f induced by an automorphism of f and a regular automorphism of the zero divisor graph of t n f algebraic group theory shows that derivations of an algebra often go parallel with automorphisms of the algebra this viewpoint encourages us to determine nonlinear mappings on t n f which are derivable at zero point i e x y x y 0 whenever x y 0 it is proved that a mapping on t n f is derivable at zero point if and only if is the sum of an inner derivation an additive derivation of t n f induced by an additive derivation of f and a regular mapping on t n f

characterizing homomorphisms derivations and multipliers in rings with idempotents in certain rings containing non central idempotents we characterize homomorphisms derivations and multipliers by their actions on elements satisfying some special conditions for example we consider the condition that an additive map h between rings mathcal a and mathcal b satisfies h x h y h z 0 whenever x y z in mathcal a are such that xy yz 0 as an application we obtain some new results on local derivations and local multipliers in particular we prove that if mathcal a is a prime ring containing a non trivial idempotent then every local derivation from mathcal a into itself is a derivation

local derivations on certain csl algebras in this paper it is shown that every norm continuous linear local derivation from an arbitrary csl algebra whose lattice is generated by finitely many independent nests into any ultraweakly closed subalgebra which contains the algebra is an inner derivation and that every norm continuous linear local derivation from an arbitrary csl algebra whose lattice is completely distributive into any ultraweakly closed subalgebra which contains the algebra is a derivation

stationary patterns of a prey predator system with a protection zone and cross diffusion of fractional type abstract this paper is concerned with the stationary patterns of a prey predator model with a protection zone and fractional type cross diffusion for the prey it is shown that the fractional type cross diffusion has negative effects on the survival of the prey when the intrinsic growth rate of the predator is positive moreover our mathematical analysis shows that compared with the results obtained in k oeda 2011 and k oeda 2012 the large cross diffusion coefficient and large growth rate of the predator species have some essentially different effects on the profiles of the solutions

maps determined by action on identity product elements abstract let m n r be the algebra of all n n matrices over a unital commutative ring r with 6 invertible for a given element z m n r a map on m n r is called preserving z product if x y z whenever xy z a map on m n r is called derivable at the given point z if x y x y z whenever xy z using elementary matrix technique we show that if an invertible linear map on m n r preserves identity product then it is a jordan automorphism and a linear map on m n r is derivable at the identity matrix if and only if it is an inner derivation

local derivations on operator algebras abstract when attempting to find sufficient conditions for a linear mapping to be a derivation an obvious candidate is the concept of a local derivation local derivations on operator algebras have been investigated in recent papers of kadison j algebra130 1990 494 509 and larson and sourour proc symp pure math 51 1990 187 194 a local derivation is a norm continuous linear map from an operator algebra a into an a bimodule m which agrees with some derivation at each point in the algebra we show that if a is the direct limit of finite dimensional csl algebras via extendable embeddings e g a triangular af algebra then a local derivation on a must be a derivation further we show that for many finite dimensional operator algebras any inner local derivation must be an inner derivation

compact kahler manifolds with nonpositive bisectional curvature in this article we prove that for any compact kahler manifold mn with real analytic metric and nonpositive bisectional curvature there exists a finite cover m of m such that m is a holomorphic and metric fiber bundle over a compact kahler manifold n with nonpositive bisectional curvature and c1 n 0 and the fiber is a flat complex torus this partially confirms a conjecture of yau

derivable maps and derivational points abstract for an algebra a and an a bimodule m let l a m be the set of all linear maps from a to m a map l a m is called derivable at c a if a b a b c for all a b a with ab c we call an element c a a derivational point of l a m if l a m the condition is derivable at c implies is a derivation we characterize derivable maps by means of peirce decompositions and determine derivational points for some general bimodules as a special case we see that for a nest algebra a on a hilbert space h every 0 c a is a derivational point of l a b h

additive lie ξ lie derivations and generalized lie ξ lie derivations on nest algebras abstract for a scalar a notion of generalized lie derivations is introduced which coincides with the notion of generalized lie derivations if 1 some characterizations of additive generalized lie derivations on the triangular algebras and the standard operator subalgebras of banach space nest algebras are given it is shown under some suitable assumption that an additive map l is an additive generalized lie derivation if and only if it is the sum of an additive generalized derivation and an additive map from the algebra into its center vanishing all commutators is an additive generalized lie derivation with 1 if and only if it is an additive generalized derivation satisfying l a l a for all a

characterizing derivations for any nest algebras on banach spaces by their behaviors at an injective operator abstract let n be a nest on a complex banach space x and let alg n be the associated nest algebra we say that an operator z alg n is an all derivable point of alg n if every linear map from alg n into itself derivable at z i e satisfies a b a b z for any a b alg n with a b z is a derivation in this paper it is shown that every injective operator and every operator with dense range in alg n are all derivable points of alg n without any additional assumption on the nest

local 2 cocycles of nest subalgebras of factor von neumann algebras in this paper it is proved that every weakly continuous local 2 cocycle of a nest subalgebra of a factor von neumann algebra with coefficients in the factor von neumann algebra is a 2 cocycle

a diffusive competition model with a protection zone this paper is concerned with a two species diffusive competition model with a protection zone for the weak competitor our mathematical results imply that when the protection zone is above a certain critical patch size determined by the birth rate of the weak competitor the weak species almost always survives but it cannot survive when the protection zone is below the critical size and its competitor is strong enough while this is the main feature of the model the actual dynamical behavior of the reaction diffusion system is more complicated the key to reveal the main feature of the system lies in a detailed analysis of the attracting regions of its steady state solutions our mathematical analysis shows that compared with the predator prey model discussed in yihong du junping shi a diffusive predator prey model with a protect zone j differential equations 226 2006 63 91 the protection zone has some essentially different effects on the fine dynamics of the competition model

generalized derivations on unital algebras determined by action on zero products abstract let a be a unital algebra having a nontrivial idempotent and let m be a unitary a bimodule we consider linear maps f g a m satisfying f x y x g y 0 whenever x y a are such that x y 0 for example when a is zero product determined algebra e g algebra generated by idempotents f and g are generalized derivations f x f 1 x d x and g x x g 1 d x for all x a where d a m is a derivation if a is not generated by idempotents then there exist also nonstandard solutions for maps f and g in the case of a being a triangular algebra under some condition on bimodule m the characterization of maps f and g is given we also consider conditions on algebra a making it a zero product determined algebra

effect of the protection zone on coexistence of the species for a ratio dependent predator prey model abstract this paper deals with the effect of the protection zone 0 on coexistence of the species for a ratio dependent predator prey model we obtain a critical value 1 n b x which is less than the well known critical value 1 d 0 obtained in the previous literatures furthermore we show that if 1 n b x then the prey persists regardless of the growth rates of the predator while if 1 n b x then there exists a real number such that the prey is ultimately extinct when as for 1 n b x and the curve of the positive steady state solutions of the model emanating from 0 c ends at a singular point 0 0 2 meantime by using the lyapunov schmidt reduction method we obtain a fine profile of its bifurcation diagrams and the uniqueness or multiplicity of its positive steady state solutions in addition as generally expected the chances of survival of the prey will increase with the size of the protection zone

hyperreflexivity of the derivation space of some group algebras let t be a continuous linear operator on a banach algebra a we address the question of whether the constant rm sup at b c a b c in a ab bc 0 a b c 1 being small implies that the distance from t to the space der a of all continuous derivations on a is small we show that this is the case for amenable group algebras as a consequence we deduce that der l 1 g is hyperreflexive for each amenable group in sin

annihilator preserving maps multipliers and derivations abstract for a commutative subspace lattice l in a von neumann algebra n and a bounded linear map f n alg l b h we show that if af b c 0 for all a b c n alg l satisfying ab bc 0 then f is a generalized derivation for a unital c algebra a a unital banach a bimodule m and a bounded linear map f a m we prove that if f a b 0 for all a b a with ab 0 then f is a left multiplier as a consequence every bounded local derivation from a c algebra to a banach a bimodule is a derivation we also show that every local derivation on a semisimple free semigroupoid algebra is a derivation and every local multiplier on a free semigroupoid algebra is a multiplier

derivable mappings at unit operator on nest algebras abstract let a be an operator subalgebra with the unit operator i in b h we say that a linear mapping from a into itself is a derivable mapping at i if st s t s t for any s t a with st i in this paper we show the following main result every strongly operator topology continuous derivable mapping at i on a nest algebra alg n is an inner derivation

maps preserving product xy yx on factor von neumann algebras abstract let a and b be two factor von neumann algebras for a b a define by a b ab ba the new product of a and b in this paper we prove that a nonlinear bijective map a b satisfies a b a b for all a b a if and only if is a ring isomorphism in particular if the von neumann algebras a and b are type i factors then is a unitary isomorphism or a conjugate unitary isomorphism

generalized derivable mappings at zero point on some reflexive operator algebras abstract let a be a subalgebra with the unit operator i in b h we say that a linear mapping from a into b h is a generalized derivable mapping at zero point if st s t s t s i t for any s t a with st 0 in this paper we show the following main result every norm continuous generalized derivable mapping at zero point on finite csl algebras is a generalized derivation

all derivable points in matrix algebras abstract let m n be the algebra of all n n matrices we say that an element g m n is an all derivable point in m n if every derivable linear mapping at g i e st s t s t for any s t m n with st g is a derivation we mainly show in this paper that a matrix g is an all derivable point in m n if and only if g 0

on nonpositively curved compact riemannian manifolds with degenerate ricci tensor in this article we prove that if the ricci tensor of a compact nonpositively curved manifold m n is nowhere negative definite then it admits local flat factors to be more precise let u c m be the open subset where the ricci tensor has maximum rank r then for any connected component u a of u its universal cover u a is isometric to r n r n r a where n r a is a nonpositively curved manifold with negative ricci curvature in particular if m n is compact nonpositively curved without euclidean de rham factor and both the manifold and the metric are real analytic then its ricci tensor must be negative definite somewhere

characterizations of lie derivations of triangular algebras abstract let t be a triangular algebra over a commutative ring r in this paper under some mild conditions on t we prove that if t t is an r linear map satisfying x y x y x y for any x y t with xy 0 resp xy p where p is the standard idempotent of t then d where d is a derivation of t and t z t where z t is the center of t is an r linear map vanishing at commutators x y with xy 0 resp xy p

characterizations of derivations on triangular rings additive maps derivable at idempotents abstract let t be a triangular ring an element z t is said to be a full derivable point of t if every additive map from t into itself derivable at z i e a b a b z for every a b t with ab z is in fact a derivation in this paper under some mild conditions on triangular ring t we show that some idempotent elements of t are full derivable points as an application we get that for any nontrivial nest n in a factor von neumann algebra r every nonzero idempotent element q satisfying pq q qp p for some projection p n is a full derivable point of the nest subalgebra alg n of r

effect of cross diffusion on the stationary problem of a prey predator model with a protection zone abstract this paper is concerned with the stationary problem of a prey predator cross diffusion system with a protection zone for the prey we discuss the existence and non existence of coexistence states of the two species by using the bifurcation theory as a result it is shown that the cross diffusion for the prey has beneficial effects on the survival of the prey when the intrinsic growth rate of the predator is positive we also study the asymptotic behavior of positive stationary solutions as the cross diffusion coefficient of the prey tends to infinity

multiple coexistence states for a prey predator system with cross diffusion abstract we study the multiple existence of positive solutions for the following strongly coupled elliptic system 1 v u u a u cv 0 in 1 u v v b du v 0 in u v 0 on where a b c d are positive constants and is a bounded domain in r n this is the steady state problem associated with a prey predator model with cross diffusion effects and u resp v denotes the population density of preys resp predators in particular the presence of represents the tendency of predators to move away from a large group of preys assuming that is small and that is large we show that the system admits a branch of positive solutions which is s or shaped with respect to a bifurcation parameter so that the system has two or three positive solutions for suitable range of parameters our method of analysis uses the idea developed by du lou j differential equations 144 1998 390 and is based on the bifurcation theory and the lyapunov schmidt procedure

effect of cross diffusion on the stationary problem of a predator prey system with a protection zone abstract this paper is concerned with the steady state problem of a predator prey cross diffusion system with herd effect allee effect and a protection zone some sufficient conditions for the existence of positive steady state solutions are given our proof is based on the local and global bifurcation theory and some a priori estimates some limiting behavior of positive steady states with respect to the allee effect constant or the diffusion coefficient d are discussed

linear mappings derivable at some nontrivial elements abstract suppose that a is an algebra and m is an a bimodule let a be any element in a a linear mapping from a into m is said to be derivable at a if st s t s t for any s t in a with st a given an algebra a such as a non abelian von neumann algebra or an irreducible cdcsl algebra on a hilbert space h with dimh 2 we show that there exists a nontrivial idempotent p in a such that for any q p a p which is invertible in p a p every linear mapping derivable at q from a into some unital a bimodule for example a or b h is derivation

existence and the dynamical behaviors of the positive solutions for a ratio dependent predator prey system with the crowing term and the weak growth abstract this paper deals with a ratio dependent predator prey system with the crowing term and the weak growth in the prey equation under the condition that the coefficient is less than a critical value 1 d 0 we obtain existence of multiple positive steady state solutions of the predator prey system and the dynamical behaviors of its positive solutions our results show that the predator and the prey possess not only the common coexistence but also the very weak coexistence which both of the predator and the prey are very low meantime the persistence of the positive solutions for the corresponding parabolic type system sometime depends strictly on the ratio of its initial data therefore our results may be used to explain some special phenomena which under some bad environment the predator and the prey may still coexist

characterizations of derivations and jordan derivations on banach algebras abstract let a be an algebra and m be an a bimodule let x be in a and a m be a linear map which satisfies ab a b a b for all a b a with ab x it is shown that is a jordan derivation if is continuous and x is left or right invertible also it is shown that is a derivation if x is idempotent such that for m m the condition x a i x m 0 implies i x m 0 and the condition mx a i x 0 implies mx 0

on the geometric rank of homogeneous spaces of nonpositive curvature in this paper we investigate differential geometric properties of homogeneous hadamard manifolds and prove a rigidity result in this class of spaces a hadamard manifold h is a simply connected complete riemannian manifold of nonpositive curvature if in addition h is homogeneous then by the work of all aw1 aw2 and hei we know that h is isometric to a solvable lie group s a n with lie algebra s of type nc see aw1 and a left invariant metric here a exp s s denotes an abelian complement to the nilpotent commutator subgroup n exp s s assuming that h has a trivial flat de rham factor the number k dim a only depends on h whereas s need not be uniquely determined and is called the aloebraic rank of h k is an upper bound for the oeometric rank r rk h dr3 i e the maximal integer such that every geodesic is contained in a r fiat a totally geodesic copy of in h the geometric meaning of rk m is well understood for a huge class of hadamard manifolds m thanks to the following remarkable result by ballmann and burns spatzier we quote a generalisation from e3 and eh

on derivable mappings abstract a linear mapping from an algebra a into an a bimodule m is called derivable at c a if a b a b c for all a b a with a b c for a norm closed unital subalgebra a of operators on a banach space x we show that if c a has a right inverse in b x and the linear span of the range of rank one operators in a is dense in x then the only derivable mappings at c from a into b x are derivations in particular the result holds for all completely distributive subspace lattice algebras j subspace lattice algebras and norm closed unital standard algebras of b x as an application every jordan derivation from such an algebra into b x is a derivation for a large class of reflexive algebras a on a banach space x we show that inner derivations from a into b x can be characterized by boundedness and derivability at any fixed c a provided c has a right inverse in b x we also show that if a is a canonical subalgebra of an af c algebra b and m is a unital banach a bimodule then every bounded local derivation from a into m is a derivation moreover every bounded linear mapping from a into b that is derivable at the unit i is a derivation

the role of protection zone on species spreading governed by a reaction diffusion model with strong allee effect abstract it is known that a species dies out in the long run for small initial data if its evolution obeys a reaction of bistable nonlinearity such a phenomenon which is termed as the strong allee effect is well supported by numerous evidence from ecosystems mainly due to the environmental pollution as well as unregulated harvesting and hunting to save an endangered species in this paper we introduce a protection zone that is governed by a fisher kpp nonlinearity and examine the dynamics of a reaction diffusion model with strong allee effect and protection zone we show the existence of two critical values 0 l l and prove that a vanishing transition spreading trichotomy result holds when the length of protection zone is smaller than l a transition spreading dichotomy result holds when the length of protection zone is between l and l only spreading happens when the length of protection zone is larger than l this suggests that the protection zone works when its length is larger than the critical value l furthermore we compare two types of protection zone with the same length a connected one and a separate one and our results reveal that the former is better for species spreading than the latter

local derivations and local automorphisms in this paper we show that if l is a completely distributive commutative subspace lattice or a j subspace lattice then the space of all bounded derivations of algl is reflexive we also study when local automorphisms on some algebras are automorphisms

stability of steady state solutions to a prey predator system with cross diffusion abstract this paper is concerned with a cross diffusion system arising in a prey predator population model the main purpose is to discuss the stability analysis for coexistence steady state solutions obtained by kuto and yamada j differential equations to appear we will give some criteria on the stability of these coexistence steady states furthermore we show that the hopf bifurcation phenomenon occurs on the steady state solution branch under some conditions

a diffusive predator prey model with a protection zone in this paper we study the effects of a protection zone 0 for the prey on a diffusive predator prey model with holling type ii response and no flux boundary condition we show the existence of a critical patch size described by the principal eigenvalue d 0 of the laplacian operator over 0 with homogeneous dirichlet boundary conditions if the protection zone is over the critical patch size i e if d 0 is less than the prey growth rate then the dynamics of the model is fundamentally changed from the usual predator prey dynamics in such a case the prey population persists regardless of the growth rate of its predator and if the predator is strong then the two populations stabilize at a unique coexistence state if the protection zone is below the critical patch size then the dynamics of the model is qualitatively similar to the case without protection zone but the chances of survival of the prey species increase with the size of the protection zone as generally expected our mathematical approach is based on bifurcation theory topological degree theory the comparison principles for elliptic and parabolic equations and various elliptic estimates

diffusion self diffusion and cross diffusion u1 x 0 u 1 0 x u2 x 0 u 2 0 x i n 0 where 2 n 1 2 x 2 is the laplace operator 0 is a bounded smooth domain of r n with n1 0 and 0 are the boundary and the closure of 0 respectively 0t 0 0 t and 0t 0 0 t for some t 0 is the outward unit normal vector on 0 di ai bi ci i 1 2 are all positive constants while ij i j 1 2 denote non negative constants the initial values u1 0 and u2 0 are non negative smooth func tions which are not identically zero the mathematical model 1 1 was proposed by shigesada et al 27 in their study of spatial segregation of interacting species where u1 and u2 represent the densities of two competing species d1 and d2 are their diffu sion rates a1 and a2 denote the intrinsic growth rates b1 and c2 account for intra specific competitions b2 and c1 are the coefficients of inter specific competitions 11 and 22 are usually referred as self diffusion pressures and 12 and 21 are cross diffusion pressures an interesting feature of 1 1 is that the movements of these two competing species are affected by the population pressures due to the mutual interference between the individuals and it was hoped that the nonlinear dispersive force due to the

uniqueness and multiplicity of positive solutions for a diffusive predator prey model in the heterogeneous environment this is the second part of our study on the spatially heterogeneous predator prey model where the interaction is governed by a crowley martin type functional response in part i we have proved that when the predator competition is strong i e k is large the model has at most one positive steady state solution for any this part is denoted to study the effect of saturation our result shows that the large saturation coefficient i e large m can not only lead to the uniqueness of positive solutions but also lead to the multiplicity of positive solutions moreover the stability of the corresponding positive solutions is also completely obtained this work can be regarded as a supplement of ref 10

all derivable points in continuous nest algebras let a be an operator algebra on a hilbert space we say that an element g a is an all derivable point of a for the strong operator topology if every strong operator topology continuous derivable linear mapping at g i e st s t s t for any s t algn with st g is a derivation let n be a continuous nest on a complex and separable hilbert space h we show in this paper that every orthogonal projection operator p m 0 m n is an all derivable point of algn for the strong operator topology

characterizations of derivations of banach space nest algebras all derivable points abstract let n be a nest on a complex banach space x with n n complemented in x whenever n n and let alg n be the associated nest algebra we say that an operator z alg n is an all derivable point of alg n if every linear map from alg n into itself derivable at z i e a b a b z for any a b a with ab z is a derivation in this paper it is shown that if z alg n is an injective operator or an operator with dense range or an idempotent operator with ran z n then z is an all derivable point of alg n particularly if n is a nest on a complex hilbert space then every idempotent operator with range in n every injective operator as well as every operator with dense range in alg n is an all derivable point of the nest algebra alg n

jordan higher all derivable points in triangular algebras abstract let t be a triangular algebra we say that d d n n n l t is a jordan higher derivable mapping at g if d n st ts i j n d i s d j t d i t d j s for any s t t with st g an element g t is called a jordan higher all derivable point of t if every jordan higher derivable linear mapping d d n n n at g is a higher derivation in this paper under some mild conditions on t we prove that some elements of t are jordan higher all derivable points this extends some results in 6 to the case of jordan higher derivations

nest subalgebras of von neumann algebras abstract the structure of a certain class of separably acting reflexive operator algebras is investigated for which the nest algebras of j ringrose can be considered prototypes to a fixed von neumann algebra and a complete nest of projections contained therein one associates the algebra of all operators in the von neumann algebra which leave every member of the nest invariant a generalization of the ringrose criterion for inclusion in the jacobson radical of a nest algebra is given for this more general class of algebras further properties of the radical are studied

jordan higher all derivable points on nontrivial nest algebras abstract let a be a banach algebra with unity i and m be a unital banach a bimodule a family of continuous additive mappings d i i n from a into m is called a higher derivable mapping at x if n ab i j n i a j b for any a b a with ab x in this paper we show that d is a jordan higher derivation if d is a higher derivable mapping at an invertible element x as an application we also get that every invertible operator in a nontrivial nest algebra is a higher all derivable point

2 local triple homomorphisms on von neumann algebras and jbw triples abstract we prove that every not necessarily linear nor continuous 2 local triple homomorphism from a jbw triple into a jb triple is linear and a triple homomorphism consequently every 2 local triple homomorphism from a von neumann algebra respectively from a jbw algebra into a c algebra respectively into a jb algebra is linear and a triple homomorphism

linear maps on c algebras which are derivations or triple derivations at a point abstract we prove new results about generalized derivations on c algebras by considering the triple product a b c 2 1 a b c c b a we introduce the study of linear maps which are triple derivations or triple homomorphisms at a point we prove that a continuous linear map t on a unital c algebra is a generalized derivation whenever it is a triple derivation at the unit element if we additionally assume t 1 0 then t is a derivation and a triple derivation furthermore a continuous linear map on a unital c algebra which is a triple derivation at the unit element is a triple derivation similar conclusions are obtained for continuous linear maps which are derivations or triple derivations at zero we also give an automatic continuity result that is we show that generalized derivations on a von neumann algebra and linear maps on a von neumann algebra which are derivations or triple derivations at zero are all continuous even if not assumed a priori to be so

additive maps derivable at some points on j subspace lattice algebras abstract let l be a j subspace lattice on a real or complex banach space dim x with x 2 and alg l be the associated j subspace lattice algebra let alg l alg l be an additive map it is shown that if is derivable at zero point i e ab a b a b whenever ab 0 then a a a a where is an additive derivation and is a scalar if is generalized derivable at zero point i e ab a b a b a i b whenever ab 0 then is a generalized derivation it is also shown that if x is complex then every linear map derivable at unit operator on alg l is a derivation

a note on 2 local automorphisms of digraph algebras abstract in this note we show that every 2 local automorphism of a digraph algebra a is an automorphism if in addition a is symmetric moreover we produce an example to clarify that the conclusion may not be true if a has not the extra condition of symmetry by this means we give a negative answer to the question about whether every 2 local automorphism of a digraph algebra is automatically an automorphism

structure of manifolds of nonpositive curvature clearly 1 rank m dim m we may regard rank m as a measurement of the flatness of m since a parallel field x along yv and perpendicular to yv is a jacobi field if and only if the sectional curvatures of the planes i t a x t are zero for all t e r in particular m is flat if rank m dim m hence a compact surface of nonpositive curvature has rank one unless it is a torus or a klein bottle it also follows that rank m1 x m2 rank ml rank m2 finally it is important that the definition of rank m above coincides with the usual one if m is locally symmetric

local lie derivations of factor von neumann algebras abstract let a be a factor von neumann algebra acting on a complex hilbert space h with dim a 2 we show that each local lie derivation from a into itself is a lie derivation

nest algebras and similarity transformations in recent years the theory of algebras of operators on hilbert space has been stimulated by developments in the theory of quasitriangularity andersen 1 has shown that up to unitary equivalence there is only one continuous quasitrianguilar algebra we use this to provide the following answer to a question posed by j r ringrose approximately 20 years ago similar continuous nests on separable hilbert space can fail to be unitarily equivalent theorem 2 2 a consequence is the existence of a nonhyperintransitive compact operator corollary 2 3 which answers a question of kadison and singer 12 and of gohberg and krein 11 we extend our initial theorem to show that arbitrary continuous nests are similar theorem 2 10 and that every maximal nest is similar to a multiplicity one nest theorem 2 11 a consequence is that every compact operator is similar to a hyperintransitive compact operator corollary 2 12 the similarity transformation can be induced by an arbitrarily small compact perturbation of a unitary operator the methods of section 2 apply only to the continuous parts of nests for general results the atomic core part must be dealt with in section 3 different methods are developed for this purpose again utilizing andersen s results these are used in section 4 to prove that a complete nest x admits an arveson factorization for every positive invertible operator if and only if x is countable as a family of subspaces theorem 4 7 a consequence is that if x is an uncountable nest with atomic core then some similarity transformation of x has a continuous part this could not be deduced from section 2 these methods also yield a weak factorization result which concludes the paper it should be noted that a negative resolution to the ringrose question was conjectured in recent years by several mathematicians including w arveson and j ringrose also many of the results presented in this paper were announced in an a m s bulletin article 16 finally we wish to thank the referee for suggesting that the original manuscript could be condensed and improved

local lie derivations of nest algebras abstract let n be a nest on a hilbert space h and alg n be the associated nest algebra we prove that every local lie derivation from alg n into b h is a lie derivation

reflexivity algebraic reflexivity and linear interpolation introduction this paper was motivated by a discovery that most finite dimensional subalgebras of b h for h an infinite dimensional hilbert space are reflexive with the only obstructions to reflexivity being finite rank considerations proofs section 2 do not depend on multiplicative structure nor on topology so extend to linear subspaces of transformations in an abstract setting abstract reflexivity has been studied in 3 4 7 primarily for singly generated algebras reflexivity properties can be interpreted as linear interpolation properties and we shall adopt this point of view the results of section 2 extend in section 3 to algebraic reflexivity counterparts for countably generated algebraically linear subspaces of bounded linear transformations acting on a banach space as consequences in section 4 we obtain generalizations of two single operator results one is a multivariate version of the result due to kaplansky 10 theorem 15 that a bounded locally algebraic operator acting on a banach space is algebraic the other is a multivariate version of the result due to douglas and foias 5 for hilbert space and to hadwin 7 for a general banach space extending work of fillmore 6 that a bounded non algebraic operator acting on a banach space is topologically algebraically reflexive in section 5 we give an application kindly suggested by d hadwin concerning joint strong similarity orbits of n tupules of operators we wish to thank e azoff k davidson j erdos d hadwin j kraus c lance and a sourour for conversations and some correspon

reflexivity of the automorphism and isometry groups of the suspension of b h the aim of this paper is to show that the automorphism and isometry groups of the suspension of b h h being a separable infinite dimensional hilbert space are algebraically reflexive this means that every local automorphism respectively every local surjective isometry ofc0 r b h is an automorphism respectively a surjective isometry

all derivable points of operator algebras abstract let a be an operator subalgebra in b h where h is a hilbert space we say that an element z a is an all derivable point of a for the norm topology strongly operator topology etc if every norm topology strongly operator topology etc continuous derivable linear mapping at z i e st s t s t for any s t a with st z is a derivation in this paper we show that every invertible operator in the nest algebra alg n is an all derivable point of the nest algebra for the strongly operator topology we also prove that every nonzero element of the algebra of all 2 2 upper triangular matrixes is an all derivable point of the algebra

idempotent elements determined matrix algebras abstract let m n r be the algebra of all n n matrices over a unital commutative ring r with 2 invertible v be an r module it is shown in this article that if a symmetric bilinear map from m n r m n r to v satisfies the condition that u u e u whenever u 2 u then there exists a linear map f from m n r to v such that x y f x y x y m n r applying the main result we prove that an invertible linear transformation on m n r preserves idempotent matrices if and only if it is a jordan automorphism and a linear transformation on m n r is a jordan derivation if and only if it is jordan derivable at all idempotent points

symmetry diffeomorphism group of a manifold of nonpositive curvature let m denote a complete simply connected manifold of nonpos itive sectional curvature for each point p s m let sp denote the diffeomor phism of m that fixes p and reverses all geodesies through p the symmetry diffeomorphism group g generated by all diffeomorphisms sp p m ex tends naturally to group of homeomorphisms of the boundary sphere m oo a subset x of m oo is called involutive if it is invariant under g theorem let x c m co be a proper closed involutive subset for each point p s m let n p denote the linear span in tpm of those vectors at p that are tangent to a geodesic 7 whose asymptotic equivalence class 7 00 belongs to x if n p is a proper subspace of tpm for some point p m then m splits as a riemannian product my x mi such that n is the distribution of m induced by my this result has several applications that include new results as well as great simplifications in the proofs of some known results in a sequel to this paper it is shown that if m is irreducible and m oo admits a proper closed involutive subset x then m is isometric to a symmetric space of noncompact type and rank k 2