on the depth of the tangent cone and the growth of the hilbert function for a d dimensional cohen macaulay local ring r mn we study the depth of the associated graded ring of r with respect to an rm primary ideal i in terms of the vallabrega valla conditions and the length of it jit where j is a j minimal reduction of i and t 1 as a corollary we generalize sally s conjecture on the depth of the associated graded ring with respect to a maximal ideal to rm primary ideals we also study the growth of the hilbert function

existence of continuous solutions and stability for a general iterative functional equation in this paper c 0 solutions for a general form of iterative functional equation are considered when the coefficient of the first order iterate can be zero which is called the leading coefficient problem for polynomial like iterative equations moreover the stability of those c 0 solutions is also discussed finally as an application we give an algorithm to compute the piecewise linear solutions of polynomial like iterative equations

convex solutions of polynomial like iterative equations in this paper convex solutions and concave solutions of polynomial like iterative equations are investigated a result for non monotonic solutions is given first and applied then to prove the existence of convex continuous solutions and concave ones furthermore another condition for convex solutions which is weaker in some aspects is also given the uniqueness and stability of those solutions are also discussed

an algebraic equation for linear operators let t be a linear operator on a vector space v possibly of infinite dimension over a general field k we solve the functional equation p t f where p k x and f an algebraic operator on v are given for nilpotent f we give an explicit linear system which determines the solutions by their similarity classes the method is based on a canonical decomposition theorem

analytic solutions of iterative functional equations abstract let r be a given positive number denote by d dr the closed disc in the complex plane c whose center is the origin and radius is r for any subset k of c and any integer m 1 write a d m k f f d m k is a continuous map and f dm is analytic suppose g a d n 1 c and h k a d k c k 2 n in this paper we study the iterative functional equation g z f z f2 h2 z f z fn hn z f z fn 1 z 0 and give some conditions for the equation to have a solution and a unique solution in a d d

local c 1 stability versus global c 1 unstability for iterative roots abstract stability of iterative roots is important in the numerical computation of iterative roots known results show that under some conditions iterative roots of strictly monotonic self mappings are c 0 stable in both the local sense and the global sense in this paper we discuss the c 1 stability for iterative roots of strictly increasing self mappings on a compact interval between two fixed points we prove that those iterative roots are locally c 1 stable but globally c 1 unstable

some np hard polygon decomposition problems the inherent computational complexity of polygon decomposition problems is of theoretical interest to researchers in the field of computational geometry and of practical interest to those working in syntactic pattern recognition three polygon decomposition problems are shown to be np hard and thus unlikely to admit efficient algorithms the problems are to find minimum decompositions of a polygonal region into perhaps overlapping convex star shaped or spiral subsets we permit the polygonal region to contain holes the proofs are by transformation from boolean three satisfiability a known np complete problem several open problems are discussed

planar formulae and their uses we define the set of planar boolean formulae and then show that the set of true quantified planar formulae is polynomial space complete and that the set of satisfiable planar formulae is np complete using these results we are able to provide simple and nearly uniform proofs of np completeness for planar node cover planar hamiltonian circuit and line geometric connected dominating set and of polynomial space completeness for planar generalized geography the np completeness of planar node cover and planar hamiltonian circuit and line were first proved elsewhere m r garey and d s johnson the rectilinear steiner tree is np complete siam j appl math 32 1977 pp 826 834 and m r garey d s johnson and r e tarjan the planar hamilton circuit problem is np complete siam j comp 5 1976 pp 704 714

lower bounds on systolic gossip gossiping is an extensively investigated information dissemination process in which each processor has a distinct item of information and has to collect all the items possessed by the other processors in this paper we provide an innovative and general lower bound technique relying on the novel notion of delay digraph of a gossiping protocol and on the use of matrix norm methods such a technique is very powerful and allows the determination of new and significantly improved lower bounds in many cases in fact we derive the first general lower bound on the gossiping time of systolic protocols i e constituted by a periodic repetition of simple communication steps in particular given any network of n processors and any systolic period s in the directed and the undirected half duplex cases every s systolic gossip protocol takes at least log n log 1 o log log n time steps where is the unique solution between 0 and 1 of p s 2 p s 2 1 with pi 1 2 2i 2 for any integer i 0 we then provide improved lower bounds in the directed and half duplex cases for many well known network topologies such as butterfly de bruijn and kautz graphs all the results are extended also to the full duplex case our technique is very general as for s it allows the determination of improved results even for non systolic protocols in fact for general networks as a simple corollary it yields a lower bound only an o log log n additive factor far from the general one independently proved in proc 1st acm symposium on parallel algorithms and architectures spaa 1989 p 318 topics in combinatorics and graph theory 1990 451 siam journal on computing 21 1 1992 111 discrete applied mathematics 42 1993 75 for all graphs and any non systolic gossip protocol moreover for specific networks it significantly improves with respect to the previously known results even in the full duplex case correspondingly better lower bounds on the gossiping time of non systolic protocols are determined in the directed half duplex and full duplex cases for butterfly de bruijn and kautz graphs even if in this paper we give only a limited number of examples our technique has wide applicability and gives a general framework that often allows to get improved lower bounds on the gossiping time of systolic and non systolic protocols in the directed half duplex and full duplex cases

iterative equations in banach spaces abstract let x be a banach space let k x f x x lipschitz f id sup and let p k x k x f k x by applying the banach contraction principle we prove that if p is sufficiently close in a certain sense to the identity then the equation p f f has a unique solution f as a corollary we obtain results on iterative equations of the types i a i f i x f x or i a i f i x f x with operator coefficients in banach spaces

gossiping in minimal time the gossip problem involves communicating a unique item from each node in a graph to every other node this paper studies the minimum time required to do this under the weakest model of parallel communication which allows each node to participate in just one communication at a time as either sender or receiver a number of topologies are studied including the omplete graph grids hypercubes and rings definitive new optimal time algorithms are derived for complete graphs rings regular grids and toroidal grids that significantly extend existing results in particular an open problem about minimum time gossiping in complete graphs is settled specifically for a graph with n nodes at least log rho n communication steps where the logarithm is in the base of the golden ratio rho are required by any algorithm under the weakest model of communication this bound which is approximately 1 44 log 2 n can be realized for some networks and so the result is optimal

leading coefficient problem for polynomial like iterative equations abstract because of difficulties in applying fixed point theorems most of known results on the polynomial like iterative equation j 1 n j f j x f x were given by assuming 1 0 and the existence of solutions under the most natural assumption n 0 is an interesting problem called leading coefficient problem for this problem locally expansive invertible c 1 solutions are obtained in the expansive case and the non hyperbolic case in w zhang on existence for polynomial like iterative equations results math 45 2004 185 194 and c 0 increasing solutions are constructed in b xu w zhang construction of continuous solutions and stability for the polynomial like iterative equation j math anal appl 325 2007 1160 1170 in this paper we discuss c 1 solutions for more combinations between expansive mappings and contractive ones and combinations between increasing mappings and decreasing ones

transversal homoclinic points of a class of conservative diffeomorphisms abstract we develop a global graph transformation to obtain estimates for certain invariant manifolds of a class of area preserving diffeomorphisms with symmetries the existence of homoclinic or heteroclinic points and the analyticity of the angle between the invariant manifolds at them is proved for the henon and the standard maps a lower bound of this angle is given for a large set of values of the parameters

deterministic broadcasting time with partial knowledge of the network we consider the time of deterministic broadcasting in networks whose nodes have limited knowledge of network topology each node v knows only the part of the network within knowledge radiusr from it i e it knows the graph induced by all nodes at distance at most r from v apart from that each node knows the maximum degree of the network one node of the network called the source has a message which has to reach all other nodes we adopt the widely studied communication model called the one way model in which in every round each node can communicate with at most one neighbor and in each pair of nodes communicating in a given round one can only send a message while the other can only receive it this is the weakest of all store and forward models for point to point networks and hence our algorithms work for other models as well in at most the same time we show trade offs between knowledge radius and time of deterministic broadcasting when the knowledge radius is small i e when nodes are only aware of their close vicinity while for knowledge radius 0 minimum broadcasting time is e where e is the number of edges in the network broadcasting can be usually completed faster for positive knowledge radius our main results concern knowledge radius 1 we develop fast broadcasting algorithms and analyze their execution time we also prove lower bounds on broadcasting time showing that our algorithms are close to optimal

fast gossiping by short messages gossiping is the process of information diffusion in which each node of a network holds a packet that must be communicated to all other nodes in the network we consider the problem of gossiping in communication networks under the restriction that communicating nodes can exchange up to a fixed number p of packets at each round in the first part of the paper we study the extremal case p 1 and we exactly determine the optimal number of communication rounds to perform gossiping for several classes of graphs including hamiltonian graphs and complete k ary trees for arbitrary graphs we give asymptotically matching upper and lower bounds we also study the case of arbitrary p and we exactly determine the optimal number of communication rounds to perform gossiping under this hypothesis for complete graphs hypercubes rings and paths finally we investigate the problem of determining sparse networks in which gossiping can be performed in the minimum possible number of rounds

how violently do smooth iterative roots oscillate abstract a crisis in the smoothness of iterative roots appears at fixed points the local c r smoothness of iterative roots near a hyperbolic fixed point is well known but its global c r extension to both fixed points is generically impossible because of their violent oscillation near the next hyperbolic fixed point in this study we consider how violently these smooth iterative roots oscillate we prove that their first order derivatives are bounded but their r th r 2 order derivatives are generically unbounded in addition we prove that the global c 1 smoothness is of vital significance for the global higher order derivatives as a corollary we show that iterative roots on the maximal interval oscillate with c 1 boundedness but c r r 2 unboundedness

small divisor problem for an analytic q difference equation abstract similar to the problem of linearization the small divisor problem also arises in the discussion of invertible analytic solutions of a class of q difference equations in this paper we give the existence of such solutions under the brjuno condition and prove that the equation may not have a nontrivial analytic solution when the brjuno condition is violated these results are applied to discussing a nonlinear iterative equation

solutions of equivariance for a polynomial like iterative equation using fixed point theorems we discuss continuous solutions of equivariance for a polynomial like iterative equation on the real line where is a closed subgroup of the general linear group gl r our main results guarantee the existence of solutions with certain kinds of symmetry we show that under restrictive hypotheses similar results can be proved in a higher dimensional case where the symmetry group is a topologically finitely generated subgroup of the group generated by rotations and dilations in n dimensional euclidean space

polynomial algorithms for open plane graph and subgraph isomorphisms graphs are used as models in a variety of situations in some cases e g to model images or maps the graphs will be drawn in the plane and this feature can be used to obtain new algorithmic results in this work we introduce a special class of graphs called open plane graphs which can be used to represent images or maps for robots they are planar graphs embedded in the plane in which certain faces can be removed are absent or unreachable we give a normal form for such graphs and prove that one can check in polynomial time if two normalised graphs are isomorphic or if two open plane graphs are equivalent their normal forms are isomorphic then we consider a new kind of subgraphs built from subsets of faces and called patterns we show that searching for a pattern in an open plane graph is tractable if and only if the faces are contiguous that is we prove that the problem is np complete otherwise

analytic solutions of a nonlinear iterative equation near neutral fixed points and poles in this paper existence of analytic solutions of a nonlinear iterative equations is studied when given functions are all analytic and when given functions have poles as well as in many previous works we reduce this problem to finding analytic solutions of a functional equation without iteration of the unknown function f for technical reasons in previous works an indeterminate constant related to the eigenvalue of the linearized f at its fixed point o is required to fulfill the diophantine condition that o is an irrationally neutral fixed point of f in this paper the case of rationally neutral fixed points is also discussed where the diophantine condition is not required

decreasing solutions and convex solutions of the polynomial like iterative equation abstract in this paper we prove the existence and uniqueness of decreasing solutions for the polynomial like iterative equation so as to answer problem 2 in j zhang l yang w zhang some advances on functional equations adv math china 24 1995 385 405 or problem 3 in w zhang j a baker continuous solutions of a polynomial like iterative equation with variable coefficients ann polon math 73 2000 29 36 furthermore we completely investigate increasing convex or concave solutions and decreasing convex or concave solutions of this equation so that the results obtained in w zhang k nikodem b xu convex solutions of polynomial like iterative equations j math anal appl 315 2006 29 40 are improved

the general exact bijective continuous solution of feigenbaum s functional equation solutionsf of feigenbaum s functional equationf f x 1 f x where 0 is a fixed real number account for many of the fascinating properties of the behaviour of successive iterates of one parameter families of nonlinear maps in connection with the phenomenon of intermittency interesting families of exact solutions have recently been found for 0 these solutions can all be derived from continuous bijective solutions which are topologically equivalent to translations in this paper thegeneral exact continuous bijective solution is found for any 0 positive or negative in particular it is shown that forany 0 there are solutions which areinequivalent to translations and it is shown that bijective solutions equivalent to translations exist only when 0 1 these results considerably enlarge the stock of available exact solutions of feigenbaum s equation

optimal algorithms for broadcast and gossip in the edge disjoint path modes the communication power of the one way and two way edge disjoint path modes for broadcast and gossip is investigated the complexity of communication algorithms is measured by the number of communication steps rounds the main results achieved are the following 1 for each connected graphgnofnnodes the complexity of broadcast ingn bmin gn satisfies log2n bmin gn log2n 1 the complete binary trees meet the upper bound and all graphs containing a hamiltonian path meet the lower bound 2 for each connected graphgnofnnodes the one way two way gossip complexityr gn r2 gn satisfies log2n r2 gn 2 log2n 1 1 44 log2n r gn 2 log2n 2 all these lower and upper bounds are shown to be sharp up to 1 3 all planar graphs ofnnodes and degreehhave a two way gossip complexity of at least 1 5log2n log2log2n 0 5log2h 8 and the two dimensional grid ofnnodes has the gossip complexity 1 5log2n log2log2n o 1 i e two dimensional grids are optimal gossip structures among planar graphs of bounded degree some upper bounds are also obtained for the one way mode 4 thed dimensional grid d 3 ofnnodes has the two way gossip complexity 1 1 d log2n log2nlog2n o d

construction of continuous solutions and stability for the polynomial like iterative equation abstract most of known results such as existence uniqueness and stability for polynomial like iterative equations were given under the assumption that the coefficient of the first order iteration term does not vanish the existence with a non zero leading coefficient was therefore raised as an open problem it was positively answered for local c 1 solutions later in this paper this problem is answered further by constructing c 0 solutions moreover we discuss the stability of those c 0 solutions which consequently implies a result of the stability for iterative roots

hyersulam stability of the iterative equation with a general boundary restriction hyersulam stability has played an important role not only in the theory of functional equations but also in a variety of branches of mathematics such as differential equations integral equations and linear operators in the present paper we will discuss the hyersulam stability of the iterative equation with a general boundary restriction by the construction of a uniformly convergent sequence of functions we prove that for every approximate solution of such an equation there exists an exact solution near it

the minimum broadcast time problem for several processor networks abstract broadcasting is the information dissemination process in a communication network a subset of processors v 0 v called originators knows an unique message which has to be transferred by calls between adjacent processors each call requires one time unit and each processor can participate in at most one call per time unit the problem is to find a schedule such that the time needed to inform all processors is less than or equal to a deadline k n we present np completeness results for this problem restricted to several communication networks bipartite planar graphs grid graphs complete grid graphs split graphs and chordal graphs with constant deadline k 2 or one originator v 0

on the hilbert function of one dimensional local complete intersections abstract the sequences that occur as hilbert functions of standard graded algebras a are well understood by macaulay s theorem those that occur for graded complete intersections are elementary and were known classically however much less is known in the local case once the dimension of a is greater than zero or the embedding dimension is three or more using an extension to the power series ring r of grobner bases with respect to local degree orderings we characterize the hilbert functions h of one dimensional quadratic complete intersections a r i i f g of type 2 2 that is that are quotients of the power series ring r in three variables by a regular sequence f g whose initial forms are linearly independent and of degree 2 we also give a structure theorem up to analytic isomorphism of a for the minimal system of generators of i given the hilbert function more generally when the type of i is 2 b we are able to give some restrictions on the hilbert function in this case we can also prove that the associated graded algebra of a is cohen macaulay if and only if the hilbert function of a is strictly increasing

finding a minimum medial axis of a discrete shape is np hard the medial axis is a classical representation of digital objects widely used in many applications however such a set of balls may not be optimal subsets of the medial axis may exist without changing the reversivility of the input shape representation in this article we first prove that finding a minimum medial axis is an np hard problem for the euclidean distance then we compare two algorithms which compute an approximation of the minimum medial axis one of them providing bounded approximation results

non monotonic solutions and continuously differentiable solutions of conjugacy equations in this paper solutions of conjugacy equation f f x g f x for a strictly decreasing continuous given function f and a continuous given function g maybe non monotonic are constructed by piecewise defining we determine the conditions for piecewise continuously differentiable solutions of conjugacy equations with a strictly decreasing continuously differentiable given function f and a continuously differentiable given function g finally the recursive algorithm is implemented in matlab software and two examples are respectively presented for a non monotonic solution and a continuously differentiable one

analytic solutions of a general nonlinear functional equations near resonance existence of analytic solutions of a general class of nonlinear functional equations is discussed this general class includes some specific functional equations studied recently moreover we can generalize this problem to finding analytic solutions of a general class of iterative equations

continuity of iteration and approximation of iterative roots motivated by computing iterative roots for general continuous functions in this paper we prove the continuity of the iteration operators t n defined by t nf f n we apply the continuity and introduce the concept of continuity degree to answer positively the approximation question if lim m f m f can we find an iterative root f m of f m of order n for each email protected n such that the sequence f m tends to the iterative root of f of order n associated with a given initial function we not only give the construction of such an approximating sequence f m but also illustrate the approximation of iterative roots with an example some remarks are presented in order to compare our approximation with the hyers ulam stability

communication complexity of fault tolerant information diffusion this paper considers problems of fault tolerant information diffusion in a network with cost function we show that the problem of determining the minimum cost necessary to perform fault tolerant gossiping among a given set of participants is np hard and give approximate with respect to the cost fault tolerant gossiping algorithms we also analyze the communication time and communication complexity of fault tolerant gossiping algorithms finally we give an optimal cost fault tolerant broadcasting algorithm and apply our results to the atomic commitment problem

iterative functional equation x x t f t with f t piecewise linear the equation x x t f t is discussed in many papers purely from the theoretical point of view existence properties we consider the properties of the solution and corresponding algorithms for computing the exact solution of such equations in the case of f t being a line or polygon

on ptak s derivation of the jordan normal form some readers of 1 might appreciate the following comments that make more explicit how ptak s beautiful insight there leads to a trivial proof of the basics of the jordan normal form the proof of theorem 1 of 1 can also be based on the observation that x being finite dimensional the sequence 0 ker a ker a2 must eventually be stationary i e ker a ker aq p for some q and all p 0 for such q let xr and xs be the range and the kernel respectively of a hence dim x dim xr dim xs further for any x xr xs x az for some z and so z ker a2q ker a hence x 0 therefore x is the direct sum of the two a invariant subspaces xs and xr and a is regular on xr since a is and nilpotent on xs in the setup and notation of theorem 2 of 1 there must be by duality some y0 in y for which x0aq 1 y0 0 hence the q order matrix x0aj 1 y0a q i i j 1 q is triangular with nonzero diagonal entries therefore invertible and this guarantees that x is the direct sum x0 x with x0 the linear span of x0aj 1 j 1 q and x the annihilator of y0a q i i 1 q both of which are ainvariant moreover it shows x0aj 1 j 1 q to be a basis for x0 and the matrix representation with respect to this basis of a restricted to x0 has the familiar form of a jordan block for the eigenvalue 0 now x being finite dimensional there are a invariant direct sum decompositions x x1 xm that are minimal in the sense that none of its summands is the direct sum of two nontrivial a invariant subspaces take any one such then the matrix representation for a with respect to any basis made up from bases for the summands xi is block diagonal with the ith block the matrix representation of the restriction ai of a to xi with respect to the chosen basis for xi assuming the underlying field to be algebraically closed the restriction ai of a to xi has some eigenvalue i and in view of the minimality of xi theorem 1 ensures

neighbourhood information dissemination in the star graph this paper studies two neighborhood information dissemination problems in the star graph n star the neighborhood broadcast problem nbp is the problem of disseminating a message from the source vertex to all the vertices adjacent to the source vertex and the neighborhood gossiping problem ngp is the problem of exchanging messages among all neighboring vertices in the given graph we propose two neighborhood information dissemination schemes in n star under the single port half duplex model the first scheme completes a neighborhood broadcast in 1 5 log sub 2 n o 1 time units and the second scheme completes a neighborhood gossiping in 2 88 log sub 2 n o logn time units we also derive a lower bound on the neighborhood gossiping of 2 88 log sub 2 n o 1 time units which implies that the latter scheme is optimal with respect to the leading term

stability of functional equations in single variable abstract this paper discusses hyers ulam stability for functional equations in single variable including the forms of linear functional equation nonlinear functional equation and iterative equation surveying many known and related results we clarify the relations between hyers ulam stability and other senses of stability such as iterative stability continuous dependence and robust stability which are used for functional equations applying results of nonlinear functional equations we give the hyers ulam stability of bottcher s equation we also prove a general result of hyers ulam stability for iterative equations

applications of the duality method to generalizations of the jordan canonical form abstract the jordan normal form for a matrix over an arbitrary field and the canonical form for a pair of matrices under contragredient equivalence are derived using ptak s duality method

note on optimal gossiping in some weak connected graphs abstract the problems of gossiping and broadcasting in one way communication mode are investigated optimal algorithms for gossip problem are known only for the complete graphs paths some simple trees and cycles in this paper some lower bounds on gossiping in graphs with bridges or with edge disjoint cycles are proved a direct consequence of these lower bounds is the construction of optimal gossip algorithms for some families of weak connected graphs another question considered here is the relationship between the gossip problem and the broadcast problem gossiping cannot be more than twice as hard as broadcasting and only for trees it is known that gossiping is exactly two times harder than broadcasting using the above mentioned l lower bounds some other graphs having cycles with this property are found

analytic solutions of a functional equation for invariant curves abstract an iterative functional equation is deduced by c t ng and w zhang 1997 j differ equations appl 3 147 168 from the problem of invariant curves in this paper its analytic solutions are discussed by locally reducing the equation to another functional equation without iteration and by constructing solutions in uniformly convergent power series for the latter equation

dynamics of a quadratic map in two complex variables abstract we study the map c 2 c 2 defined by w z z z w2 and the associated collection of sequences zj satisfying the recurrence zj 2 zj 1 z2j iteration of is equivalent to the study of such sequences we analyze growth rates of the sequences positive sequences periodic and asymptotically periodic sequences and establish the existence of doubly infinite homoclinic sequences non zero sequences whose limits as j are 0 we investigate some associated functional equations f x 2 f x 1 f x 2 and l l x l x x2