the sets of divergence points of self similar measures are residual abstract let be a self similar measure supported on a self similar set k with the open set condition for x k let a d x be the set of accumulation points of d r x log b x r log r as r 0 in this paper we show that for any closed non singleton subinterval i r the set of points x for which the set a d x equals i is either empty or residual

ergodic limits on the conformal repellers abstract let j be the repeller of an expanding c1 conformal topological mixing map g let j r d be a continuous function and let x limn 1 n n 1j 0 gjx when the limit exists be the ergodic limit it is known that the possible x are just the values d for all g invariant measures for any in the range of the ergodic limits we prove the following variational formula dim x j x max h g log d x g d x d where is a g invariant borel probability measure on j hg is the entropy of dxg is the operator norm of the differential dxg and dim is the hausdorff dimension or the packing dimension this result gives a substantial extension of the well known case that is holder continuous we also prove that unless the same ergodic limit exists everywhere the set of points whose ergodic limit does not exist has the same hausdorff dimension as the whole space j

normalization of modules we introduce techniques to derive estimates for the degrees of the generators of the integral closure of several classes of rees algebras of modules and to bound the length of normalization processes in the case of regular base rings the bounds are expressed in terms of buchsbaum rim multiplicities and a module version of briancon skoda numbers

on the arithmetic and homology of algebras of linear type three modifications of the symmetric algebra of a module are introduced and their arithmetical and homological properties studied emphasis is placed on converting syzygetic properties of the modules into ideal theoretic properties of the algebras e g cohen macaulayness factoriality the main tools are certain fitting ideals of the module and an extension to modules of a complex of not necessarily free modules that we have used in studying blowing up rings

hilbert coefficients and the gorenstein property of the associated graded ring in this article we investigate the gorenstein property of the associated graded ring ga i of an ideal i in a gorenstein local ring a m of positive dimension we especially concentrate on the case where i is m primary assuming that the rees algebra of i is cohen macaulay we then give necessary and sufficient conditions for the gorensteiness of ga i in terms of the hilbert coefficients of i

the structure of the core of ideals the core of an r ideal i is the intersection of all reductions of i this object was introduced by d rees and j sally and later studied by c huneke and i swanson who showed in particular its connection to j lipman s notion of adjoint of an ideal being an a priori infinite intersection of ideals the core is difficult to describe explicitly we prove in a broad setting that core i is a finite intersection of minimal reductions core i is a finite intersection of general minimal reductions core i is the contraction to r of a universal ideal core i behaves well under flat extensions the proofs are based on general multiplicity estimates for certain modules

cohen macaulay rees algebras and degrees of polynomial relations e mail aron sunrnp vfba br 2 department of mathematics michigan state university east lansing mi 48824 usa e mail 21144bfu msu edu 3 department of mathematics rutgers university new brunswick nj 08903 usa e mail vasconce rings rutgers edu received 5 october 1993

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

on the longest block in luroth expansion abstract in this paper for a finite subset a 2 3 we introduce the notion of longest block function l n x a for the luroth expansion of x 0 1 with respect to a and consider the asymptotic behavior of l n x a as n tends to we also obtain the hausdorff dimensions of the level sets and exceptional set arising from the longest block function

towards an algorithmic guide to spiral galaxies we study the one player game spiral galaxies from an algorithmic viewpoint spiral galaxies has been shown to be np hard friedman 2002 more than a decade ago but so far it seems that no one has dared exploring its algorithmic universe we take this trip and visit some of its corners

the fractional dimensions of intersections of the besicovitch sets and the erdos renyi sets abstract in this paper we completely determine the hausdorff dimension and packing dimension of the intersections of two kinds of typical level sets in which one is the besicovitch sets and the other is the erdos renyi sets with relation to the well known erdos renyi limit theorem our research extends the work of besicovitch in another direction and simultaneously complements the erdos renyi limit theorem from the point of view of dimension theory

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

the core of zero dimensional monomial ideals the core of an ideal is the intersection of all its reductions we describe the core of a zero dimensional monomial ideal i as the largest monomial ideal contained in a general reduction of i this provides a new interpretation of the core in the monomial case as well as an efficient algorithm for computing it we relate the core to adjoints and first coefficient ideals and in dimension two and three we give explicit formulas

rees algebras of modules we study rees algebras of modules within a fairly general framework we introduce an approach through the notion of bourbaki ideals that allows the use of deformation theory one can talk about the essentially unique generic bourbaki ideal i e of a module e which in many situations allows one to reduce the nature of the rees algebra of e to that of its bourbaki ideal i e properties such as cohen macaulayness normality and being of linear type are viewed from this perspective the known numerical invariants such as the analytic spread the reduction number and the analytic deviation of an ideal and its associated algebras are considered in the case of modules corresponding notions of complete intersection almost complete intersection and equimultiple modules are examined in some detail special consideration is given to certain modules which are fairly ubiquitous because interesting vector bundles appear in this way for these modules one is able to estimate the reduction number and other invariants in terms of the buchsbaum rim multiplicity

the largest non vanishing degree of graded local cohomology modules abstract this paper gives characterizations for the largest non vanishing degree of the local cohomology modules of a graded ring s in terms of a reduction of s and of the associated graded ring of an ideal i in terms of any reduction of i as a consequence this invariant can be computed explicitly for the associated graded ring of any ideal generated by a d sequence and of ideals having small analytic deviation

minimum weight triangulation is np hard a triangulation of a planar point set s is a maximal plane straight line graph with vertex set s in the minimum weight triangulation mwt problem we are looking for a triangulation of a given point set that minimizes the sum of the edge lengths we prove that the decision version of this problem is np hard using a reduction from planar 1 in 3 sat the correct working of the gadgets is established with computer assistance using dynamic programming on polygonal faces as well as the skeleton heuristic to certify that certain edges belong to the minimum weight triangulation

rees algebras of truncations of complete intersections abstract in this paper we describe the defining equations of the rees algebra and the special fiber ring of a truncation i of a complete intersection ideal in a polynomial ring over a field with homogeneous maximal ideal m to describe explicitly the rees algebra r i in terms of generators and relations we map another rees ring r m onto it where m is the direct sum of powers of m we compute a grobner basis of the ideal defining r m it turns out that the normal domain r m is a koszul algebra and from this we deduce that in many instances r i is a koszul algebra as well

asymptotic behaviour of arithmetically cohen macaulay blow ups this paper addresses problems related to the existence of arithmetic macaulayfications of projective schemes let y be the blow up of a projective scheme x projr along the ideal sheaf of i r it is known that there are embeddings y projk i e c for c d i e 1 where d i denotes the maximal generating degree of i and that there exists a cohen macaulay ring of the form k i e c if and only if h 0 y oy k h i y oy 0 for i 1 dimy 1 y is equidimensional and cohen macaulay cutkosky and herzog asked when there is a linear bound on c and e ensuring that k i e c is a cohen macaulay ring we obtain a surprising compelte answer to this question namely that under the above conditions there are well determined invariants and e0 such that k i e c is cohen macaulay for all c d i e and e e0 our approach is based on recent results on the asymptotic linearity of the castelnuovo mumford regularity of ideal powers we also investigate the existence of a cohen macaulay rees algebra of the form r i e ct which provides an arithmetic macaulayfication for x if r has negative a invariant we prove that such a cohen macaulay rees algebra exists if and only if oy ox r i oy 0 for i 0 y is equidimensional and cohen macaulay especially these conditions imply the cohen macaulayness of r i e ct for all c d i e and e e0 the above results can be applied to obtain several new classes of cohen macaulay algebras

ergodic properties of systems with asymptotic average shadowing property abstract in this paper we explore a topological system f m m with asymptotic average shadowing property and extend sigmund s results from bowen s specification case we show that every non empty compact and connected subset v m inv f coincides with some v f y moreover we show that the set m v y m v f y v is dense in v v supp in particular if max m inv f supp coincides with m then m max y v f y m inv f is residual in m as consequences we have several corollaries one is that every invariant measure has generic points another is that the set consisting of those points for which the birkhoff ergodic average does not exist called irregular set is either dense in max residual provided that max m or empty in particular we give an uncountable division of irregular set and obtain a refined characterization which can be as a substantial generalization of 18 with a different method remark that for systems with asymptotic average shadowing property this article studies birkhoff ergodic average from topological viewpoint but it is still unknown how about the theory from the perspective of dimension theory

on the depth of the rees algebra of an ideal module we study the rees algebra r e s e r s e of an ideal module e g re we use the technique of bourbaki ideals introduced by simis ulrich and vasconcelos 2003 22 to relate the rees algebra of a module e to the rees algebra of an ideal i i e r where r is a nagata extension of r we shall prove that depthr e depthr i e 1 and use it to deduce formulae for the depth of r e for ideal modules having small reduction number

the dimension and components of symmetric algebras let r be a commutative noetherian ring and let a4 be any finitely generated r module in this paper we compute the dimension of s m the symmetric algebra of m and identify many of the components of s m while the computation of the dimension of s m is not difficult the answer is intriguing and in view of the recent interest in symmetric algebras we hope the computation will be useful the dimension is given by the for mula dim s m max dim r p p m

the tractability of segmentation and scene analysis one of the fundamental problems in computer vision is the segmentation of an image into semantically meaningful regions based only on image characteristics a single segmentation can be determined using a linear number of evaluations of a uniformity predicate however minimising the number of regions is shown to be an np complete problem we also show that the variational approach to segmentation based on minimising a criterion combining the overall variance of regions and the number of regions also gives rise to an np complete problem when a library of object models is available segmenting the image becomes a problem of scene analysis a sufficient condition for the reconstruction of a 3d scene from a 2d image to be solvable in polynomial time is that the scene contains no cycles of mutually occluding objects and that no range information can be deduced from the image it is known that relaxing the no cycles condition renders the problem np complete we show that relaxing the no range information condition also produces an np complete problem

linkage and the koszul homology of ideals 0 introduction let r be a local ring and i an ideal of r associated to i are several graded algebras the symmetric algebra of the module i sym i the rees algebra of i defined to be the algebra r i gn o inf and the associated graded algebra of i gr1 r rii g ii 2 in in 1 i i i our purpose in this paper is to study the behavior of these graded algebras when we change i through linkage recall the definition of linkage found in the paper of peskine and szpiro 22

the complexity of the residual node connectedness reliability problem this paper considers a probabilistic network in which the edges are perfectly reliable but the nodes fail with some known probabilities the network is in an operational state if the surviving nodes induce a connected graph the residual node connectedness reliability r g of a network g is the probability that the graph induced by the surviving nodes is connected this reliability measure is very different from the widely studied k terminal network reliability measure it is proven that the problem of computing the residual connectedness reliability is np hard by showing that the problem of counting the number of node induced connected subgraphs of a given graph is bf p complete the problem remains bf p complete for split graphs as well as planar and bipartite graphs

on exceptional sets in erdős renyi limit theorem abstract for x 0 1 the run length function r n x is defined as the length of the longest run of 1 s amongst the first n dyadic digits in the dyadic expansion of x erd s and renyi proved that lim n r n x log 2 n 1 for lebesgue almost all x 0 1 in this paper we study the hausdorff dimensions of the exceptional sets in erd s renyi limit theorem let n 0 be a monotonically increasing function satisfying lim n n n 1 with some 0 1 we prove that the set e max x 0 1 lim inf n r n x n 0 lim sup n r n x n has hausdorff dimension one and is residual in 0 1

on the depth of the associated graded ring of an ideal abstract let r be a local cohen macaulay ring let i be an r ideal and let g be the associated graded ring of i we give an estimate for the depth of g when g fails to be cohen macaulay we assume that i has a small reduction number and sufficiently good residual intersection properties and satisfies local conditions on the depth of some powers the main theorem unifies and generalizes several known results we also give conditions that imply the serre properties of the blow up rings

divergence points of self similar measures satisfying the osc recently barreira and schmeling 2000 1 and chen and xiong 1999 2 have shown that for self similar measures satisfying the ssc the set of divergence points typically has the same hausdorff dimension as the support k it is natural to ask whether we obtain a similar result for self similar measures satisfying the osc however with only the osc satisfied we cannot do most of the work on a symbolic space and then transfer the results to the subsets of rd which makes things more difficult in this paper by the box counting principle we show that the set of divergence points has still the same hausdorff dimension as the support k for self similar measures satisfying the osc

radiocoloring in planar graphs complexity and approximations the frequency assignment problem fap in radio networks is the problem of assigning frequencies to transmitters by exploiting frequency reuse while keeping signal interference to acceptable levels the fap is usually modelled by variations of the graph coloring problem a radiocoloring rc of a graph g v e is an assignment function v n such that u v 2 when u v are neighbors in g and u v 1 when the distance of u v in g is two the number of discrete frequencies and the range of frequencies used are called order and span respectively the optimization versions of the radiocoloring problem rcp are to minimize the span or the order in this paper we prove that the radiocoloring problem for general graphs is hard to approximate unless np zpp within a factor of n1 2 e for any e 0 where n is the number of vertices of the graph however when restricted to some special cases of graphs the problem becomes easier we prove that the min span rcp is np complete for planar graphs next we provide an o n time algorithm v n which obtains a radiocoloring of a planar graph g that approximates the minimum order within a ratio which tends to 2 where the maximum degree of g finally we provide a fully polynomial randomized approximation scheme fpras for the number of valid radiocolorings of a planar graph g with colors in the case where 4 50

on the canonical module of the rees algebra and the associated graded ring of an ideal the investigations of this paper originate in the following question suppose i is an ideal of a local gorenstein ring r m whose associated graded ring gr r is cohenmacaulay under which extra conditions is gr r actually a gorenstein ring hochster shows in 151 that gr r is a gorenstein ring if r is factorial and gr r is a domain hochster s arguments work as well if one only requires r to be a gorenstein ring however the condition that gr r is a domain cannot be weakened much in fact we given an example of a local complete intersection r m whose associated graded ring gr r is a reduced cohen macaulay ring but not a gorenstein ring the question when gr r is a gorenstein ring has a more satisfying answer for ideals i generated by a d sequence for which r i is a cohen macaulay ring d sequences were introduced and studied by huneke their relevance in the study of powers of ideals was shown by huneke in 161 for further investigations on d sequences the reader is referred to 111 now given an ideal i generated by a d sequence and such that r i is cohenmacaulay we show that gr r is a gorenstein ring if and only if i is strongly cohen macaulay again the notion strongly cohen macaulay was introduced by huneke it means that the koszul homology of a system of generators of i is cohen macaulay huneke showed in 191 that any ideal in the linkage class of a complete intersection is strongly cohen macaulay this result provides us with plenty of interesting examples the main subject of the paper however is the study of the canonical module of gr r and of the rees algebra r if r zr 12t2 sr t in 14 the canonical class os of the rees algebra s r zr was determined under the assumption that i is

rees algebras of ideals having small analytic deviation in this article we identify two large families of ideals of a cohen macaulay sometimes gorenstein local ring whose rees algebras are cohen macaulay our main results imply for example that if r m is a regular local ring and p is a prime ideal of r such that p n is unmixed for all n 1 then the rees algebra r pt is cohen macaulay if either dim r p 2 o dim r p 3 r p is cohen macaulay and r p is integrally closed

ideals having the expected reduction number 1 introduction let r be a noetherian local ring with infinite residue field k and let be an r ideal when studying algebraic properties of the blow up of spec along v l it is often important to have good upper bounds for the reduction number of the ideal recall that an ideal c is a reduction of if the extension of rees algebras r jt c r it is module finite or equivalently if ir l j r for some r 0 32 the least such r is denoted by rj i a reduction is minimal if it is minimal with respect to inclusion and the reduction number r l of is defined as min o j a minimal reduction of finally the analytic spread i of is the krull dimension of the fiber ring r lt r k or equivalently the minimal number of generators jbi j of any minimal reduction j of 1 32 philosophically speaking j is a simplification of with the reduction number r l measuring how closely the two ideals are related of the numerous recent results about rees algebras and associated graded rings of ideals having small reduction number 18 19 38 35 12 13 14 4 3

rees algebras and almost linearly presented ideals abstract consider a grade 2 perfect ideal i in r k x 1 x d which is generated by forms of the same degree assume that the presentation matrix is almost linear that is all but the last column of consist of entries which are linear for such ideals we find explicit forms of the defining ideal of the rees algebra r i we also introduce the notion of iterated jacobian duals

an erdos renyi law with shifts abstract motivated by the comparison of dna sequences a generalization is given of the result of erdos and renyi on the length r n of the longest run of heads in the first n tosses of a coin consider two sequences x 1 x 2 x n and y 1 y 2 y n the length of the longest matching consecutive subsequence allowing shifts is m n max m x i k y j k for k 1 to m for some 0 i j n m suppose that all the letters are independent and identically distributed the length of the longest match without shifts has the same distribution as r n the length of the longest head run for a biased coin with p p x i y i described by the erdos renyi law p lim n r n log 1 p n 1 1 for matching with shifts our result is p lim n m n log 1 p n 2 1 loosely speaking allowing shifts doubles the length of the longest match the case of markov chains is also handled

rational normal scrolls and the defining equations of rees algebras consider a height two ideal i which is minimally generated by m homogeneous forms of degree d in the polynomial ring r k x y suppose that one column in the homogeneous presenting matrix of i has entries of degree n and all of the other entries of are linear we identify an explicit generating set for the ideal a which defines the rees ring r r it so r s a for the polynomial ring s r t1 tm we resolve r as an s module and is as an r module for all powers s the proof uses a rational normal scroll ring a s h with aa isomorphic to the nth symbolic power of a height one prime ideal k of a the ideal k n is generated by monomials whenever possible we study a k n in place of a aa because the generators of k n are much less complicated then the generators of aa we obtain a filtration of k n in which the factors are polynomial rings hypersurface rings or modules resolved by eagon northcott complexes the generators of i give rise to an algebraic curve c in projective m 1 space the defining equations of the fiber ring r x y r yield a solution of the implicitization problem for c introduction let i be a height two ideal in the polynomial ring r k x y with i minimally generated by m forms of degree d we consider the following questions what is the defining ideal a of the rees ring r r it r it it what is the hilbert function of r the rees ring r is equal to r t1 tm a what is the minimal resolution of r what are the defining equations of the fiber 1supported in part by the national security agency 2supported in part by the national science foundation and the national security agency 3supported in part by the national science foundation 2000 mathematics subject classification primary 14q05 secondary 13c20 13d02 14e05 14m12 68w30

on a invariant formulas abstract let g i be the form ring of an ideal i of positive height in a local ring a in this work we will provide formulas for the a invariant of g i our main result will only need the assumption that a is cohen macaulay and that g i fulfills serre s condition sl where l is the analytic spread of i as a consequence of our formula we will prove upper bounds for the reduction exponent of i in the case that a is a regular local ring if g i fulfills serre s condition sl then this upper bound is l 1 and in the case that g i is even gorenstein it is l 2

fast mixed searching and related problems on graphs abstract in this paper we introduce the fast mixed search model which is a combination of the fast search model and the mixed search model given a graph g in which a fugitive hides on vertices or along edges the fast mixed search problem is to find the minimum number of searchers to capture the fugitive in the fast mixed search model we establish a relation between the fast mixed search problem and the induced path cover problem we also consider relations between the fast mixed search problem and other graph search problems we prove that fast mixed searching from leaves is np complete and it remains np complete for planar graphs with maximum degree 3 we also prove that fast mixed searching between leaves is np complete and it remains np complete for graphs with maximum degree 4 we present linear time algorithms for computing the fast mixed search number and optimal search strategies of some classes of graphs including trees cacti and proper interval graphs

artin nagata properties minimal multiplicities and depth of fiber cones abstract minimal values of multiplicities of ideals have a strong relation with the depth of blowup algebras in this paper we introduce the notion of goto minimal j multiplicity for ideals of maximal analytic spread in a cohen macaulay ring inspired by the work of s goto a jayanthan t puthenpurakal and j verma we study the interplay among this new notion the notion of minimal j multiplicity introduced by c polini and y xie and the cohen macaulayness of the fiber cone of ideals satisfying certain residual assumptions we are also able to provide a bound on the reduction number of ideals of goto minimal j multiplicity having either cohen macaulay associated graded algebra or linear decay in the depth of their powers

computing coverage kernels under restricted settings abstract given a set b of d dimensional boxes i e axis aligned hyperrectangles a minimum coverage kernel is a subset of b of minimum size covering the same region as b computing it is np hard but as for many similar np hard problems e g box cover and orthogonal polygon covering the problem becomes solvable in polynomial time under restrictions on b we show that computing minimum coverage kernels remains np hard even when restricting the graph induced by the input to a highly constrained class of graphs alternatively we present two polynomial time approximation algorithms for this problem one deterministic with an approximation ratio within o log n and one randomized with an improved approximation ratio within o lg opt with high probability

gorenstein associated graded rings of analytic deviation two ideals abstract this paper studies the question of when the associated graded ring g i n 0in in 1 of a certain ideal i in a local ring is gorenstein the main result implies for example that if a is a regular local ring 708 is a prime ideal in a with dim a p 2 and a p is a complete intersection in codimension one then the associated graded ring g p is gorenstein if and only if the reduction number of p is at most 1

on the complexity of clustering with relaxed size constraints in fixed dimension abstract we study the computational complexity of the problem of computing an optimal clustering a 1 a 2 a k of a set of points assuming that every cluster size a i belongs to a given set m of positive integers we present a polynomial time algorithm for solving the problem in dimension 1 i e when the points are simply rational values for an arbitrary set m of size constraints which extends to the l 1 norm an analogous procedure known for the euclidean norm moreover we prove that in dimension 2 assuming euclidean norm the problem is strongly np hard with size constraints m 2 4 this result is extended also to the size constraints m 2 3 both in the case of euclidean and l 1 norm