growth rates and explosions in sandpiles we study the abelian sandpile growth model where n particles are added at the origin on a stable background configuration in zd any site with at least 2d particles then topples by sending one particle to each neighbor we find that with constant background height h 2d 2 the diameter of the set of sites that topple has order n1 d this was previously known only for h d our proof uses a strong form of the least action principle for sandpiles and a novel method of background modification we can extend this diameter bound to certain backgrounds in which an arbitrarily high fraction of sites have height 2d 1 on the other hand we show that if the background height 2d 2 is augmented by 1 at an arbitrarily small fraction of sites chosen independently at random then adding finitely many particles creates an explosion a sandpile that never stabilizes

additive mappings on von neumann algebras preserving absolute values abstract given hilbert spaces h and k and a von neumann algebra a b h let denote the class of all additive mappings a b k satisfying a a a a the paper shows that if a contains no nonzero abelian central projection then every preserves the operation the r linear combination and up to a commuting operator multiple i 0 the ring multiplication if a contains a nonzero abelian central projection p and if the dimension of k is at least 2 or 2 rank p according to whether or not p can be chosen to be minimal then there exists an additive mapping a b k such that i is a projection and a a for all a a but is neither multiplicative nor adjoint preserving in case a b h the result was proved by molnar bull aust math soc 53 1996 391 when a contained all finite rank operators and by radjabalipour et al linear algebra appl 327 2001 197 under the redundant restriction i i k i k

linear preservers between matrix modules over connected commutative rings abstract let r be a connected commutative ring with identity 1 r contains no idempotents except 0 and 1 and let m n r be the r module of all n n matrices over r r is said to be idempotence diagonalizable if every idempotent matrix over r is similar to a diagonal matrix for two arbitrary positive integers n and m we characterize a linear maps from m n r to m m r preserving tripotence when r is any idempotence diagonalizable ring with the units 2 and 3 and b linear maps from m n r to m m r preserving inverses respectively drazin inverses group inverses 1 inverses 2 inverses and 1 2 inverses when r is either any idempotence diagonalizable ring with the units 2 and 3 or any commutative principal ideal domain with at least one unit except for 1 and 2 these characterizations are completed by using an idempotence preserving result obtained by cao linear maps preserving idempotence on matrix modules over some rings j natur sci heilongjiang univ 16 1 1999 1 4 moreover we also give a simple proof of cao s result

linear maps preserving idempotents of tensor products of matrices abstract let f be a field of characteristic not 2 and m n the algebra of all n n matrices over f the aim of this paper is to characterize linear maps m m 1 m l m m 1 m l such that a 1 a l is an idempotent whenever a 1 a l is an idempotent

additive rank 1 preservers between hermitian matrix spaces and applications denote the set of n n complex hermitian matrices by hn we characterize the additive maps from hn into hm that preserve rank 1 matrices when the rank of the image of in is equal to n as applications the forms of the additive preserver of rank the additive preserver of rank additivity and the additive preserver of adjoint matrix between the spaces of hermitian matrices are also given

additive operators preserving idempotent matrices over fields and applications abstract suppose f is a field of characteristic not 2 let m n f be the algebra of all n n matrices over f and let a 2 a 3 a 1 and a be the semigroups of all additive operator on m n f that preserve idempotence preserve tripotence preserve inverses of matrices and preserve group inverses of matrices respectively the main result in this paper is that the semigroup a 2 is generated by transposition similarity the operators x x for fixed arbitrary injective endomorphisms on f and the operators x tr x for fixed arbitrary additive maps from f to m n f with 1 o as applications we determine the structures of a 3 a 1 and a when the characteristic of f is also not 3

linear preserver problems a brief introduction and some special techniques abstract linear preserver problems concern the characterization of linear operators on matrix spaces that leave certain functions subsets relations etc invariant the earliest papers on linear preserver problems date back to 1897 and a great deal of effort has been devoted to the study of this type of question since then we present a brief picture of the subject aiming at giving a gentle introduction to the reader then we describe some techniques used in our recent papers on this type of problem

on classical adjoint commuting mappings between matrix algebras abstract let f be a field and let m and n be integers with m n 3 let m n denote the algebra of n n matrices over f in this note we characterize mappings m n m m that satisfy one of the following conditions 1 f 2 or f n 1 and adj a b adj a b for all a b m n and f with i n 0 2 is surjective and adj a b adj a b for every a b m n here adj a denotes the classical adjoint of the matrix a and i n is the identity matrix of order n we give examples showing the indispensability of the assumption i n 0 in our results

strongly preserver problems in banach algebras and c algebras abstract let a and b be banach algebras assume that a is unital we prove that an additive map t a b strongly preserves drazin or equivalently group invertibility if and only if t is a jordan triple homomorphism when a and b are c algebras we characterize the linear maps strongly preserving generalized invertibility in the jordan systems sense and as consequence we determine the structure of selfadjoint linear maps strongly preserving moore penrose invertibility

linear maps preserving idempotence on matrix modules over principal ideal domains abstract let r be a commutative principal ideal domain t m n r m m r an r linear map which preserves idempotence we determine the forms of t when n m and r f 2 and solve some of beasley s open problems as a consequence we prove that the set l r of all r linear maps on m n r which preserve both idempotence and nonidempotence is a proper subset of f r the set of all linear maps on m n r that preserve idempotence when the characteristic of r is 2

contour surgery a topological reconnection scheme for extended integrations using contour dynamics abstract a numerical algorithm is described which it is believed can accurately model the dynamics of a two dimensional inviscid incompressible fluid with unparalled spatial resolution the fluid is assumed however to be divided into regions of uniform vorticity conservation of vorticity ensuring that this remains true for all time like contour dynamics the algorithm is concerned with following the evolution of the boundaries of vorticity discontinuity contours unlike contour dynamics the algorithm automatically removes vorticity features smaller than a predefined scale for example two contours enclosing the same uniform vorticity merge into one if they are close enough together also the curvature along a contour is not allowed to exceed the inverse of the cutoff scale at present calculations with contour surgery resolve fluid motions extending over four to five orders of magnitude of scales 13 to 20 octaves such high resolution pictures of two dimensional vortex dynamics have been facilitated by and indeed depend critically upon a nonlocal adaptive node adjustment scheme and a variety of tests quantify the accuracy of the technique

jordan isomorphisms of triangular matrix algebras over a connected commutative ring abstract let c be a 2 torsionfree commutative ring with identity 1 and let t r c r 2 be the algebra of all upper triangular r r r 2 matrices over c then c contains no idempotents except 0 and 1 if and only if every jordan isomorphism of t r c onto an arbitrary algebra over c is either an isomorphism or an anti isomorphism

linear operators preserving idempotent matrices over fields abstract suppose f is a field we show that if the characteristic of the field is not 2 then the semigroup of linear operators on the n n matrices over f that preserve idempotence is the group g f generated by transposition and similarity chan lim and tan have previously established that theorem for the real and complex fields by other methods we also show that the semigroup l f of linear operators on the n n matrices over f that preserve both idempotence and nonidempotence is g f when the characteristic of f is not 2 we determine the structure of l f when the characteristic of f is 2 and present some open problems

linear maps preserving drazin inverses of matrices over fields abstract let f be a field with at least five elements m n f the n n full matrix algebra over f in this paper we determine the forms of linear maps from m n f to m m f preserving drazin inverses of matrices under ch f 2 and m n 1 and also determine the forms of linear bijections from m n f to itself preserving drazin inverses of matrices under ch f 2 and n 1

irreversible conversion of graphs given a graph g a function f v g z and an initial 0 1 vertex labelling c 1 v g 0 1 we study an iterative 0 1 vertex labelling process on g where in each round every vertex v never changes its label from 1 to 0 and changes its label from 0 to 1 if at least f v neighbours have label 1 such processes model opinion disease spreading or fault propagation and have been studied under names such as irreversible threshold majority processes in a large variety of contexts our contributions concern computational aspects related to the minimum cardinality irr f g of sets of vertices with initial label 1 such that during the process on g all vertices eventually change their label to 1 such sets are known as irreversible conversion sets dynamic irreversible monopolies or catastrophic fault patterns answering a question posed by dreyer and roberts p a dreyer jr f s roberts irreversible k threshold processes graph theoretical threshold models of the spread of disease and of opinion discrete appl math 157 2009 1615 1627 we prove a hardness result for irr f g where f v 2 for every email protected v g furthermore we describe a general reduction principle for irr f g which leads to efficient algorithms for graphs with simply structured blocks such as trees and chordal graphs

linear operators strongly preserving r potent matrices over semirings abstract we consider the problem of characterizing those linear operators l on the matrices over a semiring such that l x is idempotent if and only if x is complete characterizations are obtained for many semirings including the nonnegative reals the nonnegative integers the two element boolean algebra and the fuzzy scalars

polynomial time algorithm for computing a minimum geodetic set in outerplanar graphs abstract given a graph g and a pair of vertices u v the interval i g u v is the set of all vertices that are in some shortest path between u and v given a subset x of vertices of g the interval i g x of x is the union of the intervals for all pairs of vertices in x and we say that x is geodetic if its interval do coincide with the set of vertices in the graph a minimum geodetic set is a minimum cardinality geodetic set of g the problem of computing a minimum geodetic set is known to be np hard for general graphs but is known to be polynomially solvable for maximal outerplanar graphs in this paper we show a polynomial time algorithm for finding a minimum geodetic set in general outerplanar graphs

classical adjoint commuting mappings on hermitian and symmetric matrices let m and n be integers such that m n 3 and let f and k be fields which possess involutions of f and of k respectively let hnf be the k linear space n n hermitian matrices over f in this note we address the general description of mappings satisfying one of the following conditions 1 hn f hm k with either k 2 or f k 3 and f and k of characteristic 2 if and are the identity maps and is surjective satisfying adj a b adj a b for every a b hn f 2 hn f hm f with either f 2 or f n 1 and adj a b adj a b for every a b hn k and f 3 hn f hm k is additive with adj a adja for every a hn f here f a f a a and k a k a a are the fixed fields with respect to the involutions of f and of k respectively and adja denotes the classical adjoint of the matrix a

convex and isometric domination of weak dominating pair graphs abstract a set d of vertices in a graph g is a dominating set if every vertex of g which is not in d has a neighbor in d a set of vertices d in g is convex respectively isometric if all vertices in all shortest paths respectively all vertices in one of the shortest paths between any two vertices in d lie in d the problem of finding a minimum convex dominating respectively isometric dominating set is considered in this paper from algorithmic point of view for the class of weak dominating pair graphs i e the graphs that contain a dominating pair which is a pair of vertices x y v g such that vertices of any path between x and y form a dominating set we present an efficient algorithm that finds a minimum isometric dominating set of such a graph on the other hand we prove that even if one restricts to weak dominating pair graphs that are also chordal graphs the problem of deciding whether there exists a convex dominating set bounded by a given arbitrary positive integer is np complete by further restricting the class of graphs to chordal dominating pair graphs i e the chordal graphs in which every connected induced subgraph has a dominating pair we are able to find a polynomial time algorithm that determines the minimum size of a convex dominating set of such a graph

some linear preserver problems on block triangular matrix algebras in this article we classify rank one nonincreasing maps on n square block triangular matrix algebras with the assumption that in is of rank n as applications we obtain complete classifications of adjugate commuting maps and compound commuting maps on block triangular matrix algebras

linear maps transforming the higher numerical ranges abstract let k 1 n the k numerical range of a m n is the set w k a tr x ax k x is n k x x i k and the k numerical radius of a is the quantity w k a max z z w k a suppose k 1 k 1 n and n c n k min k n k it is shown that there is a linear map m n m n satisfying w k a w k a for all a m n if and only if n n k k or n n k n k is a positive integer moreover if such a linear map exists then there are unitary matrix u m n and nonnegative integers p q with p q n n such that has the form a u a a p a t a t q u or a u a a p a t a t q u where m n m n has the form a tr a i n n k a k linear maps m n m n satisfying w k a w k a for all a m n are also studied furthermore results are extended to triangular matrices

bootstrap percolation in three dimensions by bootstrap percolation we mean the following deterministic process on a graph g given a set a of vertices infected at time 0 new vertices are subsequently infected at each time step if they have at least n previously infected neighbors when the set a is chosen at random the main aim is to determine the critical probability p c g r at which percolation infection of the entire graph becomes likely to occur this bootstrap process has been extensively studied on the d dimensional grid n d with 2 r d fixed it was proved by cerf and cirillo for d r 3 and by cerf and manzo in general that p c n d r 1 log r 1 n d r 1 where log r is an r times iterated logarithm however the exact threshold function is only known in the case d r 2 where it was shown by holroyd to be 1 o 1 2 18 log n in this paper we shall determine the exact threshold in the crucial case d r 3 and lay the groundwork for solving the problem for all fixed d and r

linear preservers of zeros of matrix polynomials abstract we classify linear operators on matrices with semiring entries that preserve the zeros of multivariable matrix polynomials these matrix polynomials are defined via the adjoint operator x y xy yx as if the matrices were being considered as if over a field also we compare the results over fields with the results over semirings

linear preservers on triangular matrices abstract let t n f denote the vector space of all n n upper triangular matrices over a field f we characterize those linear transformations t on t n f that satisfy one of the following properties 1 t preserves rank one matrices 2 t preserves both singular and nonsingular matrices 3 t is nonsingular and t adj a adj t a for all a t n f

linear operators preserving adjoint matrix between matrix spaces abstract we characterize the linear operators from the linear space of n n matrices into the linear space of m m matrices over any field f that preserve adjoint matrix where n 3

linear transformations which preserve or decrease rank abstract we describe some results concerning a linear transformation on a space v of matrices which is rank preserving or rank nonincreasing on a certain subset of v

on the caratheodory number of interval and graph convexities inspired by a result of caratheodory uber den variabilitatsbereich der fourierschen konstanten von positiven harmonischen funktionen rend circ mat palermo 32 1911 193 217 the caratheodory number of a convexity space is defined as the smallest integer k such that for every subset u of the ground set v and every element u in the convex hull of u there is a subset f of u with at most k elements such that u in the convex hull of f we study the caratheodory number for generalized interval convexities and for convexity spaces derived from finite graphs we establish structural properties bounds and hardness results

on the hull number of some graph classes in this paper we study the geodetic convexity of graphs focusing on the problem of the complexity of computing a minimum hull set of a graph in several graph classes for any two vertices u email protected v of a connected graph g v e the closed interval i u v of u and v is the set of vertices that belong to some shortest u v path for any email protected v let i s email protected u v si u v a subset email protected v is geodesically convex or convex if i s s in other words a subset s is convex if for any u email protected s and for any shortest u v path p v p s given a subset email protected v the convex hull i h s of s is the smallest convex set that contains s we say that s is a hull set of g if i h s v the size of a minimum hull set of g is the hull number of g denoted by hn g the hull number problem is to decide whether hn g k for a given graph g and an integer k dourado et al showed that this problem is np complete in general graphs in this paper we answer an open question of dourado et al 2009 1 by showing that the hull number problem is np hard even when restricted to the class of bipartite graphs then we design polynomial time algorithms to solve the hull number problem in several graph classes first we deal with the class of complements of bipartite graphs then we generalize some results in araujo et al 2011 2 to the class of q q 4 graphs and to cacti finally we prove tight upper bounds on the hull numbers in particular we show that the hull number of an n node graph g without simplicial vertices is at most email protected 3 n 1 email protected in general at most email protected email protected if g is regular or has no triangle and at most email protected email protected if g has girth at least 6

short note on the simulation of nearly inviscid two dimensional turbulence 2009 elsevier inc all rights reserved we here compare two numerical algorithms in a simple test case involving the simulation of two dimensional freelydecaying turbulence the first is a standard pseudo spectral algorithm as has been widely used in studies of this problem the second is a less well known hybrid eulerian lagrangian algorithm based on contour dynamics the comparison verifies the near equivalence in detail of the two algorithms at early times at later times significant advantages of the second algorithm become clear in particular the ability to achieve a much higher effective reynolds number at a negligible fraction of the computational effort specifically a comparable simulation using the pseudo spectral algorithmwould take o 10 times more computer power we solve the two dimensional navier stokes equation in vorticity form xt j w x 1 4 mrx x 1 4 rw 1 in a doubly periodic domain and using a simple random initial condition as specified below here x is the vorticity w is the streamfunction j is the jacobean and m is the viscosity coefficient the initial conditions for 1 are constructed by specifying an initial energy spectrum of the form e k 1 4 ak exp 1 2 8p k 8 where k 1 4 jkj is the wavenumber with random phases for all fourier coefficients this e k is peaked at k 1 4 ffiffiffiffiffiffiffiffiffiffiffi 32 p p 3 2 and is more than 10 times smaller by k 1 4 32 the coefficient a is chosen to give unit r m s velocity or energy per unit area 2 hjrwj i 1 4 2 the maximum initial velocity is about 2 3 while the peak vorticity jxjmax 1 4 11 942 and the initial r m s vorticity xrms 1 4 3 541 first we solve 1 using a standard pseudo spectral algorithm and a fourth order runga kutta time scheme approximate de aliasing is achieved using a spectral filter following 6 with parameters as specified in 5 hereafter dts the viscous term is treated exactly by means of an integrating factor we perform a series of integrations at resolutions n 1 4 512 1024 2048 4096 and 8192 where n is the number of grid points in x and y in each case the time step is chosen to be 0 0025 512 n this time step is less than half the cfl time step required for stability at the initial time and is verified a posteriori to never exceed 75 of the cfl time step during the flow evolution in tests using resolutions up to n 1 4 2048 halv all rights reserved ax 44 1334 463748

linear additive preservers of rank 2 on spaces of alternate matrices over fields abstract let kn f be the linear space of all n n alternate matrices over a field f and let k n 2 f be its subset consisting of all rank 2 matrices an operator kn f kn f is said to be additive if a b a b for any a b kn f linear if is additive and aa af a for every a f and a kn f and a preserver of rank 2 on kn f if k n 2 f k n 2 f when n 4 we characterize all linear respectively additive preservers of rank 2 on kn f over any field respectively any field that is not isomorphic to a proper subfield of itself

the hypercasl algorithm a new approach to the numerical simulation of geophysical flows we describe a major extension to the contour advective semi lagrangian casl algorithm d g dritschel m h p ambaum a contour advective semi lagrangian numerical algorithm for simulating fine scale conservative dynamical fields quart j roy meteorol soc 123 1997 1097 1130 d g dritschel m h p ambaum the diabatic contour advective semi lagrangian algorithm mon weather rev 134 9 2006 2503 2514 the extension called hypercasl hcasl uses lagrangian advection of material potential vorticity contours like casl but a vortex in cell vic method for the treatment of diabatic forcing or damping in this way hypercasl is fully lagrangian regarding advection a grid is used as in casl to deal with inversion computing the velocity field from the potential vorticity field first the novel aspects of the algorithm are described including several improvements to the underlying casl algorithm all numerical parameters are chosen so as to minimise the computational cost while improving conservation properties finally a thorough inter code comparison is conducted using a two dimensional inviscid unforced turbulence test case this enables us to point out the advantages of this new algorithm in terms of resolution computational cost and numerical diffusion compared to other existing methods namely casl vic and pseudo spectral ps methods

linear transformations between matrix spaces that map one rank specific set into another we characterize in several instances those linear transformations from the linear space of m n matrices into the linear space of p q matrices that map the set of matrices having a fixed rank into the set of matrices having a fixed rank examples are given showing that in contrast with the case of linear transformations on the linear space of m n matrices mapping a rank specific set into itself in the more general case of linear transformations between two full matrix spaces often one cannot expect neat and predictable results

the sharp threshold for bootstrap percolation in all dimensions in r neighbour bootstrap percolation on a graph g a typically random set a of initially infected vertices spreads by infecting at each time step vertices with at least r already infected neighbours this process may be viewed as a monotone version of the glauber dynamics of the ising model and has been extensively studied on the d dimensional grid n d the elements of the set a are usually chosen independently with some density p and the main question is to determine p c n d r the density at which percolation infection of the entire vertex set becomes likely in this paper we prove for every pair d ge r ge 2 that there is a constant l d r such that p c n d r l d r o 1 log r 1 n d r 1 as n o infty where log r denotes an r times iterated logarithm we thus prove the existence of a sharp threshold for percolation in any fixed number of dimensions moreover we determine l d r for every pair d r

rank k preservers and preservers of sets of ranks abstract let t be a linear transformation on the set of m n matrices with entries in an algebraically closed field if t maps the set of all matrices whose rank is k into itself and if n 3 k 2 then the rank of a is the rank of t a for every m n matrix

linear preservers on powers of matrices abstract this paper is concerned with linear maps l on n x n matrices such that i l ak l a k for all a where k is a fixed integer 2 or ii l preserves idempotent or tripotent matrices respectively

linear preservers on matrices abstract let u denote either the vector space of n n matrices or the vector space of n n symmetric matrices over an infinite field f in this paper we characterize linear mappings l on u that satisfy one of the following properties i l adja adjl a for all a in u ii l preserves idempotent matrices and l in in where f is the real field r or the complex field c iii l ea el a for all a in u where f r or c

on the geodetic hull number of pk free graphs partially answering a question posed by araujo morel sampaio soares and weber we show that computing the geodetic hull number of a given p 9 free graph is np hard similarly we show that computing the geodetic interval number of a given p 5 free graph is np hard furthermore we show that the geodetic hull number can be computed in polynomial time for paw p 5 free graphs triangle free graphs in which every six vertices induce at most one p 5 and c 3 c k 2 p k free graphs for every integer k

on contour crossings in contour advective simulations part 2 analysis of crossing errors and methods for their prevention this is the second of two papers devoted to the analysis of contour crossing errors that occur in contour advective simulations of fluid motion where either vorticity or potential vorticity is represented by contours we begin with a detailed discussion on some of the potential mechanisms for contour crossing past work has suggested that the formation of contour crossings is due to inadequate spatial resolution of contours 1 the implementation of two schemes for preventing contour crossings within the framework of the contour advective semi lagrangian casl algorithm is detailed here we then present an analysis of contour crossing errors in simulations of quasigeostrophic turbulence on the f plane and the quasigeostrophic motion of an initially circular vortex patch on the b plane using the algorithm detailed in part 1 we find that in general individual crossings occur at scales smaller than the inversion grid scale on which velocity is calculated but at scales larger than that of the surgical scale that defines the smallest resolved features vorticity of a flow if the resolution of a quasigeostrophic turbulence simulation on the f plane is increased by doubling the number of grid points in each coordinate direction used in the calculation of the velocity field then the total area in error due to contour crossings remains unchanged a smaller number of crossings introducing larger scale area errors is replaced by a greater number of smaller local errors uniformly increasing the density of nodes along all contours and placement of nodes at points of close approach on contours are both effective methods for limiting contour crossings

linear preservers and representations with a 1 dimensional ring of invariants we determine the group of linear transformations on a vector space v that preserve a polynomial function f on v for several interesting pairs v f using the theory of semisimple algebraic groups in an 1897 paper fro97 frobenius proved that every linear transformation of the n by n real matrices that preserves the determinant is of the form x 7 axb or x 7 ax t b for some a b gln r such that det ab 1 this is the basic example of a linear preserver problem lpp one is given a finite dimensional vector space v over a field k and a polynomial function f v k and one wants to determine the linear transformations of v that preserve f since frobenius many such prob lems have been solved see for example the surveys lp01 pll 92 lt92 and mar62 we develop here a general method that solves several new problems see examples 3 14 and 3 15 and corollaries 7 10 8 3 8 7 8 5 8 8 and 8 11 our method is to introduce an auxiliary group g gl v that is semisimple and such that v is an irreducible representation or weyl module of g in section 2 we determine the normalizer ngl v g of g in gl v we prove in theorem 3 2 that this subgroup equals the stabilizer of a closed g orbit o in the projective space p v i e stabgl v o ngl v g using this result in sections 7 and 8 we solve two families of lpps by reducing the problems in each family to determining this stabilizer these two families consist of representations v with a 1 dimensional ring of g invariant functions generated by f and are examples of prehomogenous vector spaces of parabolic type the two families correspond to the cases where the unipotent radical u of the parabolic subgroup is abelian i e u u 0 or u u is 1 dimensional respectively and we use the general results on representations in these families from rrs92 roh93 and hel12 besides obtaining new results we also recover many known solutions to linear preserver problems the generality of our method is in contrast to many of the proofs in the literature which typically are highly dependent on the particular choice of v and f the arguments in pd95 and gur97 are notable exceptions further we require only very weak assumptions about the field k at most we require that the characteristic is 6 2 3 and determine the preserver precisely and not just its identity component or lie algebra

sharp metastability threshold for two dimensional bootstrap percolation in the bootstrap percolation model sites in an l by l square are initially independently declared active with probability p at each time step an inactive site becomes active if at least two of its four neighbours are active we study the behaviour as p 0 and l simultaneously of the probability i l p that the entire square is eventually active we prove that i l p 1 if and i l p 0 if where 2 18 we prove the same behaviour with the same threshold for the probability j l p that a site is active by time l in the process on the infinite lattice the same results hold for the so called modified bootstrap percolation model but with threshold 2 6 the existence of the thresholds settles a conjecture of aizenman and lebowitz 3 while the determination of their values corrects numerical predictions of adler stauffer and aharony 2

modular automorphisms preserving idempotence and jordan isomorphisms of triangular matrices over commutative rings abstract suppose r is a commutative ring with 1 and 2 being the units of r let t n r be the n n upper triangular matrix modular over r and let l r be the set of all r module automorphisms on t n r that preserve idempotence the main result in this paper is that f l r if and only if there exist an invertible matrix u t n r and an idempotence e r such that f x u ex 1 e x u 1 for any x x ij t n r where x x n 1 j n 1 i as applications we determine all jordan isomomorphisms of t n r over the ring r

linear operators that preserve pairs of matrices which satisfy extreme rank properties a supplementary version abstract a pair of m n matrices a b is said to be rank sum maximal if a b a b and rank sum minimal if a b a b we characterize the linear operators preserving the set of rank sum maximal pairs over any field and the linear operators preserving the set of rank sum minimal pairs over any field except for 0 1 the linear preservers of the set of rank sum maximal pairs are characterized by using a result about rank preservers proposed by li and pierce amer math monthly 108 2001 591 605 and thereby the linear preservers of the set of rank sum minimal pairs are characterized the paper can be viewed as a supplementary version of several related results

linear maps preserving r potents of tensor products of matrices abstract let m n be the algebra of all n n complex matrices and r 2 a fixed integer the aim of this paper is to characterize linear maps m m 1 m l m m 1 m l such that a 1 a l is r potent whenever a 1 a l is r potent

the characteristic polynomial of an algebra and representations we pose two new linear preserver problems first given a field k and a finite dimensional k algebra b we show that the only linear maps k d b k d b which preserve the unit and the n th roots of unity for some n 2n 2 coprime to char k char k are the algebra homomorphisms second we consider linear maps a b a b between finite dimensional k algebras which preserve the cayley hamilton relations we show that if preservation of the cayley hamilton property is understood in a certain non commutative sense and a k da k d then the only such linear mappings are the algebra homomorphisms

on maps preserving square zero matrices the problem of characterizing linear operators on matrix algebras that leave invariant certain functions subsets or relations has attracted the attention of many mathematicians see survey papers 20 22 27 for details for instance linear operators preserving zeroproduct of matrices are studied in 15 16 29 31 linear operators preserving idempotent matrices are studied in 1 11 13 14 linear operators preserving matrices annihilated by a fixed polynomial are studied in 12 13 18 linear operators preserving nilpotent matrices are studied in 9 and linear operators preserving square zero matrices are studied in 29 most linear preserver problems were investigated in the case of matrix algebras over fields in contrast not too much is known about matrix algebras over commutative rings to the best of our knowledge beside the papers by mcdonald 25 and waterhouse 30 only bresar and semrl 11 used elementary calculations to describe linear maps preserving idempotent matrices over commutative rings the reason why people seldom study matrix algebras over commutative rings is probably that one might encounter some difficulties that are not easy to overcome for example unlike matrix algebras over fields the automorphisms on matrix algebras over commutative rings can fail to be inner see 19 28

additive mappings preserving operators of rank one abstract let b x be the algebra of all bounded linear operators on a nontrivial real or complex banach space and let f x be the subalgebra of all finite rank operators a characterization of additive mappings on f x which preserve operators of rank one or projections of rank one is given in the real case such mappings are automatically linear

linear k power k potent preservers between matrix spaces abstract suppose f is a field let mn f be the linear space of all n n matrices over f and let sn f be its subspace consisting of all symmetric matrices let m n k be positive integers with k 2 and let v m n f s n f and w m m f s m f a linear map f v w is called a k power preserver if f a k f ak for every a v and a k potent preserver if f a k f a for any a v with ak a we characterize i linear k power preservers from mn f to mm f when ch f k or ch f 0 ii linear k potent preservers from mn f to mm f when f is an algebraically closed field with ch f 0 iii linear k power preservers from sn f to mm f respectively sm f when ch f k 5 or ch f 0 and iv linear k potent preservers from sn f to mm f respectively sm f when f is an algebraically closed field with ch f 0

vortex design problem in this investigation we propose a computational approach for the solution of optimal control problems for vortex systems with compactly supported vorticity the problem is formulated as a pde constrained optimization in which the solutions are found using a gradient based descent method recognizing such euler flows as free boundary problems the proposed approach relies on shape differentiation combined with adjoint analysis to determine cost functional gradients in explicit tracking of interfaces vortex boundaries this method offers an alternative to grid based techniques such as the level set methods and represents a natural optimization formulation for vortex problems computed using the contour dynamics technique we develop and validate this approach using the design of 2d equilibrium euler flows with finite area vortices as a model problem it is also discussed how the proposed methodology can be applied to euler flows featuring other vorticity distributions such as vortex sheets and to time dependent phenomena

the maximum infection time in the geodesic and monophonic convexities recent papers investigated the maximum infection time t p 3 g of the p 3 convexity also called maximum time of 2 neighbor bootstrap percolation and the maximum infection time t m o g of the monophonic convexity in 2014 it was proved that for bipartite graphs deciding whether t p 3 g k is polynomial time solvable for k 4 but is np complete for k 5 23 in 2015 it was proved that deciding whether t m o g 2 is np complete even for bipartite graphs 12 in this paper we investigate the maximum infection time t g d g of the geodesic convexity we prove that deciding whether t g d g k is polynomial time solvable for k 1 but is np complete for k 2 even for bipartite graphs we also present an o n 3 m time algorithm to determine t g d g and t m o g in distance hereditary graphs for this we characterize all minimal hull sets of a general graph in the monophonic convexity moreover we improve the complexity of the fastest known algorithm for finding a minimum hull set of a general graph in the monophonic convexity

on the parameterized complexity of the geodesic hull number abstract several recent papers obtained complexity results regarding the geodesic hull number hn gd g of a graph g in this paper we prove that determining whether hn gd g k is w 2 hard parameterized by k in diameter two graphs and is w 1 hard parameterized by tw k where tw is the treewidth of g despite this for graphs with bounded treewidth tw we prove that hn gd g is computable in polynomial time o tw 2 tw 5 n 2 tw 5

linear additive preservers of ranks 2 and 4 on alternate matrix spaces over fields abstract let kn f be the linear space of all n n alternate matrices over a field f an operator f kn f kn f is said to be additive if f a b f a f b for any a b kn f linear if f is additive and f aa af a for every a f and a kn f and a preserver of ranks 2 and 4 on kn f if rank f x rank x for every x kn f with rank x 2 or 4 when n 4 we characterize all linear respectively additive preservers of ranks 2 and 4 on kn f over any field respectively any field that is not isomorphic to a proper subfield of itself furthermore it is also shown that the condition f is not isomorphic to a proper subfield of itself is necessary for the obtained conclusion on additive preservers of ranks 2 and 4 on kn f

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

maps on spaces of symmetric matrices preserving idempotence abstract suppose f is an arbitrary field let f be the number of the elements of f let m n f be the space of all n n matrices over f and let s n f be the subset of m n f consisting of all symmetric matrices let v s n f m n f a map v v is said to preserve idempotence if a b is idempotent if and only if a b is idempotent for any a b v and f it is shown that when the characteristic of f is not 2 f 3 and n 4 s n f s n f is a map preserving idempotence if and only if there exists an invertible matrix p m n f such that a pap 1 for every a s n f and p t p ai n for some nonzero scalar a in f

idempotence preserving maps without the linearity and surjectivity assumptions abstract let m n f be the space of all n n matrices over a field f of characteristic not 2 and let p n f be the subset of m n f consisting of all n n idempotent matrices we denote by n f the set of all maps from m n f to itself satisfying a b p n f if and only if a b p n f for every a b m n f and f it was shown that n f if and only if there exists an invertible matrix p m n f such that either a pap 1 for every a m n f or a patp 1 for every a m n f this improved dolinar s result by omitting the surjectivity assumption and extending the complex field to any field of characteristic not 2

inapproximability results for graph convexity parameters in this paper we prove several inapproximability results on the p 3 convexity and the geodesic convexity in graphs we prove that determining the p 3 hull number and the geodetic hull number are apx hard problems we prove that the caratheodory number the radon number and the convexity number of both convexities are o n 1 e inapproximable in polynomial time for every e 0 unless p np we also prove that the interval numbers of both convexities are w 2 hard and o log n inapproximable in polynomial time unless p np moreover these results hold for bipartite graphs in the p 3 convexity