test ideals in local rings it is shown that certain aspects of the theory of tight closure are well behaved under localization let j be the parameter test ideal for r a complete local cohen macaulay ring of positive prime characteristic for any multiplicative system u c r it is shown that ju 1r is the parameter test ideal for u 1 r this is proved by proving more general localization results for the here introduced classes of f ideals of r and f submodules of the canonical module of r which are annihilators of r modules with an action of frobenius it also follows that the parameter test ideal cannot be contained in any parameter ideal of r tight closure has produced surprising new results for instance that the absolute integral closure of a complete local domain r of prime characteristic is a cohen macaulay algebra for r as well as tremendously simple proofs for otherwise difficult theorems such as the fact that the ring of invariants of a linearly reductive group acting on a regular ring is cohen macaulay the definition of tight closure is recalled in section 1 but we refer the reader to the papers of hochster and huneke listed in the bibliography for more about its applications while primarily a prime characteristic notion tight closure offers insight into arbitrary commutative rings containing q by fairly standard reduction to characteristic p techniques despite its successes the tight closure operation which in its principal setting is a closure operation performed on ideals in a commutative noetherian ring of prime characteristic remains poorly understood for example tight closure is not known to behave well under localization given an ideal i of a ring r letting i denote the tight closure of i it is not known whether or not

statistical analysis of airport network of china through the study of airport network of china anc composed of 128 airports nodes and 1165 flights edges we show the topological structure of anc conveys two characteristics of small worlds a short average path length 2 067 and a high degree of clustering 0 733 the cumulative degree distributions of both directed and undirected anc obey two regime power laws with different exponents i e the so called double pareto law in degrees and out degrees of each airport have positive correlations whereas the undirected degrees of adjacent airports have significant linear anticorrelations it is demonstrated both weekly and daily cumulative distributions of flight weights frequencies of anc have power law tails besides the weight of any given flight is proportional to the degrees of both airports at the two ends of that flight it is also shown the diameter of each subcluster consisting of an airport and all those airports to which it is linked is inversely proportional to its density of connectivity efficiency of anc and of its subclusters is measured through a simple definition in terms of that the efficiency of anc s subclusters increases as the density of connectivity does anc is found to have an efficiency of 0 484

typing correspondence assertions for communication protocols woo and lam propose correspondence assertions for specifying authenticity properties of security protocols prior work on checking correspondence assertions depends on model checking and is limited to finite state systems we propose a dependent type and effect system for checking correspondence assertions since it is based on type checking our method is not limited to finite state systems this paper presents our system in the simple and general setting of the calculus we show how to type check correctness properties of example communication protocols based on secure channels in a related paper we extend our system to the more complex and specific setting of checking cryptographic protocols based on encrypted messages sent over insecure channels

extension of weakly and strongly f regular rings by flat maps this is a preprint of an article published in the journal of algebra vol 241 2001 no 2 799 807

statistical analysis of 22 public transport networks in poland public transport systems in 22 polish cities have been analyzed the sizes of these networks range from n 152 to 2881 depending on the assumed definition of network topology the degree distribution can follow a power law or can be described by an exponential function distributions of path lengths in all considered networks are given by asymmetric unimodal functions clustering assortativity and betweenness are studied all considered networks exhibit small world behavior and are hierarchically organized a transition between dissortative small networks n approximately or 500 is observed

behavior of test ideals under smooth and etale homomorphisms we investigate the behavior of the test ideal of an excellent reduced ring of prime characteristic under base change it is shown that if h a d is a smooth homomorphism then ad d assuming that all residue fields of a at maximal ideals are perfect and that formation of the test ideal commutes with localization it is also shown that if h a m d is a finite flat homomorphism of gorenstein normal rings etale in codimension 1 then ad d more generally this last result holds under the assumption that the closed fiber of h a m d is gorenstein provided one knows that the tight closure of zero and the finitistic tight closure of zero in the injective hulls of the residue fields of a and s are equal

empirical analysis of a scale free railway network in china abstract we present a detailed empirical analysis of the statistical properties of the china railway network crn consisting of 3915 nodes train stations and 22 259 edges railways based on this crn displays two explicit features already observed in numerous real world and artificial networks one feature the small world property has the fingerprint of a small characteristic shortest path length 3 5 accompanied by a high degree of clustering 0 835 another feature is characterized by the scale free distributions of both degrees and weighted degrees namely strengths correlations between strength and degree degree and degree and clustering coefficient and degree have been studied and the forms of such behaviors have been identified in addition we investigate distributions of clustering coefficients topological distances and spatial distances

a characterization of rational singularities in terms of injectivity of frobenius maps the notions of f rational and f regular rings are defined via tight closure which is a closure operation for ideals in a commutative ring of positive characteristic the geometric significance of these notions has persisted and k e smith proved that f rational rings have rational singularities we now ask about the converse implication the answer to this question is yes and no for a fixed positive characteristic there is a rational singularity which is not f rational so the answer is no in this paper however we aim to show that the answer is yes in the following sense if a ring of characteristic zero has rational singularity then its modulo p reduction is f rational for almost all characteristic p this result leads us to the correspondence of f regular rings and log terminal singularities

tight closure plus closure and frobenius closure in cubical cones we consider tight closure plus closure and frobenius closure in the rings r k x y z x3 y3 z3 where k is a field of characteristic p and p 74 3 we use a z3 grading of these rings to reduce questions about ideals in the quotient rings to questions about ideals in the regular ring k x y we show that frobenius closure is the same as tight closure in certain classes of ideals when p 2 mod 3 since if c ir n r c 1 we conclude that ir n r 1 for these ideals using injective modules over the ring r the union of all peth roots of elements of r we reduce the question of whether if i for 23 graded ideals to the case of 23 graded irreducible modules we classify the irreducible m primary z3 graded ideals we then show that jf 1 for most irreducible rn primary z3 graded ideals in k x y z x3 y3 z3 where k is a field of characteristic p and p 2 mod 3 hence 1 irmf n r for these ideals in this paper we discuss the conjecture that i ir n r where r denotes the integral closure of a domain r of characteristic p in an algebraic closure of its fraction field and i denotes the tight closure of i the ring r is characterized by the property that it is a domain integral over r and every monic polynomial with coefficients in r factors into monic linear factors this characterization can be used to prove the following property of r if w is a multiplicatively closed set of r then w lr w 1r aside from providing a much more concrete description of tight closure proving that i ir n r would solve the localization problem for tight closure it is known that i ir n r for parameter ideals sml and for rings in which every ideal of the normalization is tightly closed also for those ideals i of an excellent local domain r such that r i has finite phantom projective dimension it is known that i ir n r ab however the conjecture is open even for two dimensional normal gorenstein domains in particular the conjecture is open for the cubical cone k x y z x3 y3 z3 where k is a field of characteristic p and p 5l 3 and more generally for rings of the form k x y z f x y z where f is a homogeneous cubic polynomial we consider tight closure plus closure and frobenius closure in the rings r k x y z x3 y3 z3 where k is a field of characteristic p and p 5l 3 in section 1 we use a z3 grading of these rings to reduce questions about ideals in the quotient rings to questions about ideals in the regular rings k x y in section 2 we show that the frobenius closure of an ideal i denoted if is the same as the tight closure in certain classes of ideals when p 2 mod 3 since if c ir n r c i received by the editors august 27 1997 1991 mathematics subject classification primary 13a35 13a02 13hi10

strong and weak f regularity are equivalent for graded rings it is shown that the tight closure of a submodule in a artinian module is the same as its finitistic tight closure when the modules are graded over a finitely generated inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i graded ring over a perfect field as a corollary it is deduced that for such a graded ring strong and weak f regularity are equivalent as another application the following conjecture of hochster and huneke is proved let r m be a finitely generated inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i graded ring over a field with unique homogeneous maximal ideal m then r is weakly f regular if and only if r m is weakly f regular

a calculus for cryptographic protocols we introduce the spi calculus an extension of the pi calculus designed for describing and analyzing cryptographic protocols we show how to use the spi calculus particularly for studying authentication protocols the pi calculus without extension suffices for some abstract protocols the spi calculus enables us to consider cryptographic issues in more detail we represent protocols as processes in the spi calculus and state their security properties in terms of coarse grained notions of protocol equivalence

f rational rings have rational singularities it is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational a key point in the proof is a characterization of f rational local rings as those cohen macaulay local rings r m in which the local cohomology module h d m r where d is the dimension of r have no submodules stable under the natural action of the frobenius map an analog for finitely generated algebras over a field of characteristic zero is developed which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i without reference to a desingularization

empirical analysis of the worldwide maritime transportation network abstract in this paper we present an empirical study of the worldwide maritime transportation network wmn in which the nodes are ports and links are container liners connecting the ports using the different representations of network topology the spaces l and p we study the statistical properties of wmn including degree distribution degree correlations weight distribution strength distribution average shortest path length line length distribution and centrality measures we find that wmn is a small world network with power law behavior important nodes are identified based on different centrality measures through analyzing weighted clustering coefficient and weighted average nearest neighbors degree we reveal the hierarchy structure and rich club phenomenon in the network

test ideals and base change problems in tight closure theory test ideals are an important concept in tight closure theory and their behavior via flat base change can be very difficult to understand our paper presents results regarding this behavior under flat maps with reasonably nice but far from smooth fibers this involves analyzing in depth a special type of ideal of test elements called the cs test ideal besides providing new results the paper also contains extensions of a theorem by g lyubeznik and k e smith on the completely stable test ideal and of theorems by f enescu and independently m hashimoto on the behavior of f rationality under flat base change

a generic type system for the pi calculus we propose a general powerful framework of type systems for the calculus and show that we can obtain as its instances a variety of type systems guaranteeing non trivial properties like deadlock freedom and race freedom a key idea is to express types and type environments as abstract processes we can check various properties of a process by checking the corresponding properties of its type environment the framework clarifies the essence of recent complex type systems and it also enables sharing of a large amount of work such as a proof of type preservation making it easy to develop new type systems

nilpotence of frobenius action and the hodge filtration on local cohomology abstract an f nilpotent local ring is a local ring r m of prime characteristic defined by the nilpotence of the frobenius action on its local cohomology modules h m i r a singularity in characteristic zero is said to be of f nilpotent type if its modulo p reduction is f nilpotent for almost all p in this paper we give a hodge theoretic interpretation of three dimensional normal isolated singularities of f nilpotent type in the graded case this yields a characterization of these singularities in terms of divisor class groups and brauer groups

scaling and correlations in three bus transport networks of china abstract we report the statistical properties of three bus transport networks btn in three different cities of china these networks are composed of a set of bus lines and stations serviced by these network properties including the degree distribution clustering and average path length are studied in different definitions of network topology we explore scaling laws and correlations that may govern intrinsic features of such networks besides we create a weighted network representation for btn with lines mapped to nodes and number of common stations to weights between lines in such a representation the distributions of degree strength and weight are investigated a linear behavior between strength and degree s k k is also observed