a complete classification of bifurcation diagrams of a dirichlet problem with concave convex nonlinearities abstract we study the bifurcation diagrams of positive solutions of the two point boundary value problem u x f u x 0 1 u 1 u 1 0 where f u g u h u g h c 0 c2 0 and 0 is a bifurcation parameter we assume that functions g and h satisfy hypotheses h1 h3 under hypotheses h1 h3 we give a complete classification of bifurcation diagrams and we prove that on the u plane each bifurcation diagram consists of exactly one curve which is either a monotone curve or has exactly one turning point where the curve turns to the left hence the problem has at most two positive solutions for each 0 more precisely we prove the exact multiplicity of positive solutions

simultaneous prime specializations of polynomials over finite fields recently the author proposed a uniform analogue of the bateman horn conjectures for polynomials with coefficients from a finite field that is for polynomials in f q t rather than z t here we use an explicit form of the chebotarev density theorem over function fields to prove this conjecture in particular ranges of the parameters we give some applications including the solution of a problem posed by hall

large gaps between consecutive prime numbers let g x denote the size of the largest gap between consecutive primes below x answering a question of erd os we show that g x f x logx log logx log log log logx log log logx 2 where f x is a function tending to innity with x our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions as such we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes

central sets and a non commutative roth theorem in 1952 k roth showed that any subset of n having positive upper density contains arithmetic progressions of length 3 a result e szemer edi extended to progressions of arbitrary length in 1975 an approach involving multiple recurrence for measure preserving systems developed by h furstenberg led to extensions to linear configurations in more general groups while demonstrating families of such configurations to be large in various senses using ultrafilter techniques for doing ergodic theory without averaging we prove a version of roth s theorem in its ergodic theoretic formulation for general countable groups g a combinatorial consequence valid in amenable groups is the presence of many configurations of the form a b ag b ag bg in positive density subsets of g g here many implies in particular central in the variable g

an equivalence classes model of fuzzy relational databases abstract the technique of employing sets of values for tuple components to express imprecision in relational databases was proposed by buckles and petry in their classic works on fuzzy relational databases in addition to providing an intuitively appealing scheme for representing fuzzy information the model of buckles and petry possesses several key properties of the classical relational model by employing finite scalar domains with similarity relations and special fuzzy number domains buckles and petry have demonstrated that the classical properties of uniqueness of tuple interpretations and well definedness of the relational algebra can be retained in the fuzzy relational database model the key to the preservation of these properties is the fact that scalar domains with similarity relations and the fuzzy number domains can be partitioned into equivalence classes however since equivalence classes can be constructed without assuming the existence of similarity relations or special fuzzy number domains it is desirable to generalize the fuzzy relational database model to one based only on equivalence classes in this work we show that the important properties of classical relational databases and of fuzzy relational databases are preserved in a generalized model built on equivalence relations on finite database domains further we generalize the notion of a functional dependency to the fuzzy relational model

roth s theorem in the primes we show that any set containing a positive proportion of the primes contains a 3 term arithmetic progression an important ingredient is a proof that the primes enjoy the so called hardy littlewood majorant property we derive this by giving a new proof of a rather more general result of bourgain which because of a close analogy with a classical argument of tomas and stein from euclidean harmonic analysis might be called a restriction theorem for the primes

on s shaped bifurcation curves where sz c r n 2 1 is bounded with smooth boundary aq readers interested in the derivation of the problem are referred to the work by bebernen l particularly section 2 relevant work on the analysis of this problem has been done by brown et al 2 shivaji 3 61 hastings and mcleod 7 sattinger 8 parks 9 and tam lo clearly u 0 is a solution of 1 l when 2 0 it follows from the contraction mapping theorem that l l always has a solution for a sufficiently small moreover let

s shaped bifurcation curves for nonlinear two parameter problems abstract we consider the nonlinear eigenvalue problem u t 1 u t u t 2 u t p u t 0 t i 1 2 1 2 u 1 2 u 1 2 0 where p 2 is a constant 0 and 0 are parameters this equation was introduced by crandall and rabinowitz 1973 as the typical model which develops the s shaped bifurcation curve when p 3 and 0 1 it is known that if p 2 and 0 1 then there exists a unique solution u r c 2 i with u where 0 is a given constant furthermore is s shaped in this paper we establish the asymptotic formulas for as 0 1 2 when 0 1 where 1 and 2 are two turning points of and is the unique positive solution to the algebraic equation 1 x x 2 x p 0 we also characterize the asymptotic shape of u t as by establishing the asymptotic behavior of u q as when 0 1 is a fixed constant

papr problem for walsh systems and related problems high peak values of transmission signals in wireless communication systems lead to wasteful energy consumption and degradation of several transmission performances we continue the theoretical contributions made by boche and farell toward the understanding of peak value reduction using the strategy known as tone reservation for orthogonal transmission schemes there it was shown that for orthogonal frequency division multiplexing ofdm systems the combinatorial object called arithmetic progression plays an important role in setting limitations for the applicability of the tone reservation method in this paper we show that the combinatorial object introduced as perfect walsh sum pws plays a similar role for code division multiple access cdma systems as arithmetic progression for ofdm systems by specific construction we show that for a chosen numbers m and n all subsets mathcal i of the set n of the first n 2 n natural numbers which has the density in n larger than a given delta in 0 1 i e left mathcal i right n geq delta and which is sufficiently large enough in the sense that left i right geq 2 2 delta 2 m 1 contains a pws of size 2 m by means of this result and motivated by the previously mentioned connection between arithmetic progression and pws we show results for the pws which are analogous to the famous szemeredi theorem on arithmetic progressions conlon gower s theorem on probabilistic construction of sparse sets containing an arithmetic progression and even a solution of an analogon to the erd s conjecture on arithmetic progressions those results give in particular an insight into the asymptotic limitations of tone reservation method for the cdma systems besides we show that a subset i of n is a pws if and only if the embedding inequality of the subspace of l 1 0 1 containing linear combinations of walsh functions indexed by elements of mathcal i holds with the minimum possible embedding constant sqrt left mathcal i right the corresponding approach based in particular by the fact that the pwss are the only walsh sums having unit l 1 norm proven in this paper by means of that results we show that the minimum possible threshold constant for which the tone reservation method is applicable yields sqrt left mathcal i right if and only if the information set mathcal i is a pws

rigorous analysis and estimates of s shaped bifurcation curves in a combustion problem with general arrhenius reaction rate laws we study the bifurcation of solutions of the combustion problem with general arrhenius reaction rate laws d 2 u d x 2 1 e u m e u 1 e u 0 1 x 1 u 1 u 1 0 for this two point boundary value problem we give a bifurcation analysis in the physically important ranges 0 m 1 and m 0 it is proved that the bifurcation curve is s shaped in certain parameter ranges and a lower bound is given for the frank kamenetskii transition value

from harmonic analysis to arithmetic combinatorics arithmetic combinatorics or additive combinatorics is a fast developing area of research combining elements of number theory combinatorics harmonic analysis and ergodic theory its arguably best known result and the one that brought it to global prominence is the proof by ben green and terence tao of the long standing conjecture that primes contain arbitrarily long arithmetic progressions there are many accounts and expositions of the green tao theorem including the articles by kra 119 and tao 182 in the bulletin the purpose of the present article is to survey a broader highly interconnected network of questions and results built over the decades and spanning several areas of mathematics of which the green tao theorem is a famous descendant an old geometric problem lies at the heart of key conjectures in harmonic analysis a major result in partial differential equations invokes combinatorial theorems on intersecting lines and circles an unexpected argument points harmonic analysts towards additive number theory with consequences that could have hardly been anticipated we will not try to give a comprehensive survey of harmonic analysis combinatorics or additive number theory we will not even be able to do full justice to our specific areas of focus instead referring the reader to the more complete expositions and surveys listed in section 7 our goal here is to emphasize the connections between these areas we will thus concentrate on relatively few problems chosen as much for their importance to their fields as for their links to each other the article is written from the point of view of an analyst who in the course of her work was gradually introduced to the questions discussed here and found them fascinating we hope that the reader will enjoy a taste of this experience

extending the relational model to deal with fuzzy values abstract in this paper the relational model with fuzzy attribute values is discussed based on the concepts of the semantic distance the definitions of the fuzzy relational operations and the fuzzy functional dependency are given making use of fuzzy functional dependencies we can compress the domain of the fuzzy data and make the fuzzy data clearer furthermore an algorithm for testing the satisfiability of a fuzzy instance to the set of fuzzy functional dependencies is given

extremal results in sparse pseudorandom graphs abstract szemeredi s regularity lemma is a fundamental tool in extremal combinatorics however the original version is only helpful in studying dense graphs in the 1990s kohayakawa and rodl proved an analogue of szemeredi s regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs many of the key applications of szemeredi s regularity lemma use an associated counting lemma in order to prove extensions of these results which also apply to sparse graphs it remained a well known open problem to prove a counting lemma in sparse graphs the main advance of this paper lies in a new counting lemma proved following the functional approach of gowers which complements the sparse regularity lemma of kohayakawa and rodl allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph we use this to prove sparse extensions of several well known combinatorial theorems including the removal lemmas for graphs and groups the erd s stone simonovits theorem and ramsey s theorem these results extend and improve upon a substantial body of previous work

polynomial extensions of van der waerden s and szemeredi s theorems an extension of the classical van der waerden and szemeredi the orems is proved for commuting operators whose exponents are polynomials as a consequence for example one obtains the following result let s zl be a set of positive upper banach density let p1 n pk n be polynomials with rational coefficients taking integer values on the integers and satisfying pi 0 0 i 1 k then for any v1 vk zl there exist an integer n and a vector u zl such that u pi n vi s for each i k department of mathematics ohio state university columbus ohio 43210 e mail address vitaly math ohio state edu department of mathematics technion haifa 23000 israel e mail address sashal techunix technion ac il current address department of mathematics stanford university stanford california 94305 e mail address leibman math stanford edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

classification of type 2 fuzzy sets represented as sequences of vertical slices granulation of information by using type 2 fuzzy sets is receiving more attention nowadays this is due to the superior capability of type 2 fuzzy sets in handling the data uncertainty from a theoretical perspective a set containing type 2 fuzzy sets has no trivial geometric structure therefore a proper metric cannot be easily defined as a consequence common pattern recognition systems which in one way or another rely on some geo metric structure of the input space are not easily applicable to a space of type 2 fuzzy sets in this paper we study the problem of designing a classifier in the input space of type 2 fuzzy sets type 2 fuzzy sets are hence interpreted as granular patterns forming a given input dataset by decomposing a type 2 fuzzy set into a sequence of simpler lower type fuzzy sets we explore the possibility of defining and building dissimilarity and kernel based classification systems on input spaces of type 2 fuzzy sets such an interpretation provided in terms of sequences allows us to conceive an effective sequence matching strategy which can be suitably embedded into well established pattern recognition systems we support the methodological developments by performing experiments on synthetically generated classification problems for datasets composed of type 2 fuzzy sets with adjustable and controlled level of difficulty results are promising and suggest to further investigate on the possibility of interpreting type 2 fuzzy sets as input patterns of a given data driven inference system

random unconditional convergence of vector valued dirichlet series abstract we study random unconditionality of dirichlet series in vector valued hardy spaces h p x it is shown that a banach space x has type 2 respectively cotype 2 if and only if for every choice x n n x it follows that x n n s n is random unconditionally convergent respectively divergent in h 2 x the analogous question on h p x spaces for p 2 is also explored we also provide explicit examples exhibiting the differences between the unconditionality of x n n s n in h p x and that of x n z n n in h p x

linear equations in primes consider a system of nonconstant affine linear forms 1 t d no two of which are linearly dependent let n be a large integer and let k n n d be convex a generalisation of a famous and difficult open conjecture of hardy and littlewood predicts an asymptotic as n for the number of integer points n d k for which the integers 1 n t n are simultaneously prime this implies many other well known conjectures such as the twin prime conjecture and the weak goldbach conjecture it also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations in which all unknowns are required to be prime in this paper we conditionally verify this asymptotic under the assumption that no two of the affine linear forms 1 t are affinely related this excludes the important binary cases such as the twin prime or goldbach conjectures but does allow one to count nondegenerate configurations such as arithmetic progressions our result assumes two families of conjectures which we term the inverse gowers norm conjecture gi s and the mobius and nilsequences conjecture mn s where s 1 2 is the complexity of the system and measures the extent to which the forms i depend on each other the case s 0 is somewhat degenerate and follows from the prime number theorem in aps roughly speaking the inverse gowers norm conjecture gi s asserts the gowers u s 1 norm of a function f n 1 1 is large if and only if f correlates with an s step nilsequence while the mobius and nilsequences conjecture mn s asserts that the mobius function is strongly asymptotically orthogonal to s step nilsequences of a fixed complexity these conjectures have long been known to be true for s 1 essentially by work of hardy littlewood and vinogradov and were established for s 2 in two papers of the authors thus our results in the case of complexity s 2 are unconditional in particular we can obtain the expected asymptotics for the number of 4 term progressions p 1 p 2 p 3 p 4 n of primes and more generally for any nondegenerate problem involving two linear equations in four prime unknowns

a sound and sometimes complete query evaluation algorithm for relational databases with null values a sound and in certain cases complete method is described for evaluating queries in relational databases with null values where these nulls represent existing but unknown individuals the soundness and completeness results are proved relative to a formalization of such databases as suitable theories of first order logic because the algorithm conforms to the relational algebra it may easily be incorporated into existing relational systems

the ergodic theoretical proof of szemeredi s theorem partial results were obtained previously by k f roth 1952 who established the existence of arithmetic progressions of length three in subsets of z of positive upper density and by e szemeredi 1969 who proved the existence of progressions of length four in 1976 furstenberg noticed that the statement of theorem i is equivalent to a statement about multiple recurrence of measure preserving transformations namely

the structure factor of primes although the prime numbers are deterministic they can be viewed by some measures as pseudo random numbers in this article we numerically study the pair statistics of the primes using statistical mechanical methods particularly the structure factor in an interval with m large and l m smaller than unity we show that the structure factor of the prime number configurations in such intervals exhibits well defined bragg like peaks along with a small diffuse contribution this indicates that primes are appreciably more correlated and ordered than previously thought our numerical results definitively suggest an explicit formula for the locations and heights of the peaks this formula predicts infinitely many peaks in any non zero interval similar to the behavior of quasicrystals however primes differ from quasicrystals in that the ratio between the location of any two predicted peaks is rational we also show numerically that the diffuse part decays slowly as m and l increases this suggests that the diffuse part vanishes in an appropriate infinite system size limit

negation as failure careful closure procedure abstract the aim of this paper is a modification of minker s generalized closed world assumption that would allow application of the negation as failure rule with respect to a set p of not necessarily all predicates of a database db a careful closure procedure is introduced which when applied to a database db produces a new database db that is used to answer queries about predicates from db it is shown that db is consistent iff db is consistent if p is the set of all predicates from db and db does not contain functional symbols then db coincides with minker s gcwa the soundness and completeness of the careful closure procedure with respect to a minimal model style semantic is shown as an inference engine associated with db we propose a query evaluation procedure qep which is a combination of a method of splitting an indefinite database db into a disjunction of horn databases and clark s query evaluation procedure qep soundness of qep with respect to db is shown for a broad class of databases

the study of joins in fuzzy relational databases abstract it is often necessary to join naturally two or more fuzzy relations to answer a user s query however the result of a join may not be what one expects in this paper we study the decomposition problem of fuzzy relation schemes we show that the algorithm proposed by aho et al 1 to test lossless join decomposition for classical relations can be applied in the fuzzy domain when particularizations are induced by functional dependencies

generalizing database relational algebra for the treatment of incomplete or uncertain information and vague queries abstract this paper deals with relational databases which are extended in the sense that fuzzily known values are allowed for attributes precise as well as partial imprecise uncertain knowledge concerning the value of the attributes are represented by means of 0 1 valued possibility distributions in zadeh s sense thus we have to manipulate ordinary relations on cartesian products of sets of fuzzy subsets rather than fuzzy relations besides vague queries whose contents are also represented by possibility distributions can be taken into account the basic operations of relational algebra union intersection cartesian product projection and selection are extended in order to deal with partial information and vague queries approximate equalities and inequalities modeled by fuzzy relations can also be taken into account in the selection operation then the main features of a query language based on the extended relational algebra are presented an illustrative example is provided this approach which enables a very general treatment of relational databases with fuzzy attribute values makes an extensive use of dual possibility and necessity measures

random graphs and the parity quantifier the classical zero one law for first order logic on random graphs says that for every first order property p in the theory of graphs and every p 0 1 the probability that the random graph g n p satisfies p approaches either 0 or 1 as n approaches infinity it is well known that this law fails to hold for any formalism that can express the parity quantifier for certain properties the probability that g n p satisfies the property need not converge and for others the limit may be strictly between 0 and 1 in this work we capture the limiting behavior of properties definable in first order logic augmented with the parity quantifier foltr over g n p thus eluding the above hurdles specifically we establish the following modular convergence law for every foltr sentence p there are two explicitly computable rational numbers a0 a1 such that for i 0 1 as n approaches infinity the probability that the random graph g 2npi p satisfies p approaches ai our results also extend appropriately to fo equipped with modq quantifiers for prime q in the process of deriving this theorem we explore a new question that may be of interest in its own right specifically we study the joint distribution of the subgraph statistics modulo 2 of g n p namely the number of copies mod 2 of a fixed number of graphs f1 fe of bounded size in g n p we first show that every foltr property p is almost surely determined by subgraph statistics modulo 2 of the above type next we show that the limiting joint distribution of the subgraph statistics modulo 2 depends only on n mod 2 and we determine this limiting distribution completely interestingly both these steps are based on a common technique using multivariate polynomials over finite fields and in particular on a new generalization of the gowers norm the first step is analogous to the razborov smolensky method for lower bounds for ac0 with parity gates yet stronger in certain ways for instance it allows us to obtain examples of simple graph properties that are exponentially uncorrelated with every foltr sentence which is something that is not known for ac0ltr

the primes contain arbitrarily long arithmetic progressions we prove that there are arbitrarily long arithmetic progressions of primes there are three major ingredients the first is szemeredi s theorem which asserts that any subset of the integers of positive density contains progressions of arbitrary length the second which is the main new ingredient of this paper is a certain transference principle this allows us to deduce from szemeredi s theorem that any subset of a suficiently pseudorandom set or measure of positive relative density contains progressions of arbitrary length the third ingredient is a recent result of goldston and yildirim which we reproduce here using this one may place a large fraction of the primes inside a pseudorandom set of almost primes or more precisely a pseudorandom measure concentrated on almost primes with positive relative density

poincare recurrence and number theory thirty years later hillel furstenberg s 1981 article in the bulletin gives an elegant introduction to the interplay between dynamics and number theory summarizing the major developments that occurred in the few years after his landmark paper 21 the field has evolved over the past thirty years with major advances on the structural analysis of dynamical systems and new results in combinatorics and number theory furstenberg s article continues to be a beautiful introduction to the subject drawing together ideas from seemingly distant fields furstenberg s article 21 gave a general correspondence between regularity properties of subsets of the integers and recurrence properties in dynamical systems now dubbed the furstenberg correspondence principle he then showed that such recurrence properties always hold proving what is now referred to as the multiple recurrence theorem combined these results gave a new proof of szemeredi s theorem 45 if s z has positive upper density then s contains arbitrarily long arithmetic progressions this proof lead to an explosion of activity in ergodic theory and topological dynamics beginning with new proofs of classic results of ramsey theory and ultimately leading to significant new combinatorial and number theoretic results the full implications of these connections have yet to be understood the approach harks back to the earliest results on recurrence in the measurable setting and in the topological setting a measure preserving system is a quadruple x b t where x denotes a set b is a algebra on x is a probability measure on x b and t x x is a measurable transformation such that t 1 a a for all a b poincare recurrence states that if x b t is a measure preserving system and a b with a 0 there exists n n such that a t na 0 a topological dynamical system is a pair x t where x is a compact metric space and t x x is a continuous map one can show that any such topological space admits a borel probability measure that preserves t in particular poincare recurrence implies recurrence in the topological setting birkhoff 13 gave a direct proof of

ell p mathbb z boundedness of discrete maximal functions along thin subsets of primes and pointwise ergodic theorems we establish the first pointwise ergodic theorems along thin sets of prime numbers a set with zero density with respect to the primes for instance we will be able to achieve this with the piatetski shapiro primes our methods will be robust enough to solve the ternary goldbach problem for some thin sets of primes

a hardy field extension of szemeredi s theorem abstract in 1975 szemeredi proved that a set of integers of positive upper density contains arbitrarily long arithmetic progressions bergelson and leibman showed in 1996 that the common difference of the arithmetic progression can be a square a cube or more generally of the form p n where p n is any integer polynomial with zero constant term we produce a variety of new results of this type related to sequences that are not polynomial we show that the common difference of the progression in szemeredi s theorem can be of the form n where is any positive real number and x denotes the integer part of x more generally the common difference can be of the form a n where a x is any function that is a member of a hardy field and satisfies a x x k and a x x k 1 0 for some non negative integer k the proof combines a new structural result for hardy sequences techniques from ergodic theory and some recent equidistribution results of sequences on nilmanifolds

an extended version of the fuzzy relational database model abstract the fuzzy relational database model proposed by buckles and petry is a formal method for organizing and using fuzzy information in relational databases the model possesses two key properties that hold for classical relational databases no two tuples have identical interpretations and each relational algebra operation has a unique result the original fuzzy relational database model was based on similarity relations which were defined on finite scalar domains buckles and petry later extended the model to incorporate fuzzy number domains this extension was done without loss of consistency with the representation or the relational algebra in a recent paper we have extended the original model to proximity relations which generalize similarity relations the key properties of buckles and petry s model are preserved in the extension in this work we demonstrate that the existence of partitions on finite scalar domains in the fuzzy relational model is the key to preserving the important properties of the classical relational model we explain how proximity relations defined on finite scalar domains are used to partition the domains further we show that normal fuzzy sets can be employed as domain values these fuzzy sets become members of fi nite scalar domains and their characteristic functions are employed in generating the proximity relations of the fuzzy relational database model

a theorem on reversed s shaped bifurcation curves for a class of boundary value problems and its application abstract we study the bifurcation curves of positive solutions of the boundary value problem u x f e u x 0 1 x 1 u 1 u 1 0 where f e u g u e h u e r is a bifurcation parameter and functions g h c 0 c 2 0 satisfy five hypotheses presented herein assuming these hypotheses on fixed g and h we prove that the bifurcation curve is reverse s shaped on the e u plane that is the bifurcation curve has exactly two turning points at some points e u e and e u e such that e e and u e u e in addition we prove that e 0 thus the exact number of positive solutions can be precisely determined by the values of e and e we give an application to the two parameter bifurcation problem u x 1 u 2 e u 3 0 1 x 1 u 1 u 1 0 where e are two positive bifurcation parameters some new results are obtained

fuzzy functional dependencies and independencies in extended fuzzy relational database models abstract in this paper we have considered data dependencies more specifically restricted fuzzy functional dependencies in the framework of extended fuzzy relational database model a formal system is presented for the derivation of fuzzy functional dependencies which is both sound and complete the problem of lossless join decomposition is discussed in the framework of extended fuzzy relational model then we introduce a formal statement of fuzzy functional independency which expresses the fact that the corresponding restricted fuzzy functional dependency is not a legitimate constraint of the real world a formal system is presented for the derivation of restricted fuzzy functional dependencies and independencies the various problems such as the membership problem the contradiction problem and the problem of canonical represents are discussed a cover called minimal cover of restricted fuzzy functional independencies is shown to be unique

fuzzy information in extended fuzzy relational databases abstract the queries on type 2 fuzzy relational databases may get non reasonable answers and as pointed out by buckles a tuple s membership value is not static but is a measure of the appropriateness of that tuple to a given query here we use two supplementary attributes r and i e to model the concepts of fuzziness and uncertainty respectively we also use these two attribute values to compute the appropriateness of a tuple to a given query in this paper the proposed fuzzy relational model is defined by first order logical in our proposed model the classical properties of the database which are 1 database interpretation being redundant free being uniquely determined and 2 the well definedness of relational algebra are preserved for the fuzzy relational algebra therefore the proposed fuzzy relational model can capture more meaning than the conventional approaches and is more suitable to handle the real life application

the first digit frequencies of prime numbers and riemann zeta zeros prime numbers seem to be distributed among the natural numbers with no law other than that of chance however their global distribution presents a quite remarkable smoothness such interplay between randomness and regularity has motivated scientists across the ages to search for local and global patterns in this distribution that could eventually shed light on the ultimate nature of primes in this paper we show that a generalization of the well known first digit benford s law which addresses the rate of appearance of a given leading digit d in datasets describes with astonishing precision the statistical distribution of leading digits in the prime number sequence moreover a reciprocal version of this pattern also takes place in the sequence of the non trivial riemann zeta zeros we prove that the prime number theorem is in the final analysis responsible for these patterns

global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications we study the global bifurcation and exact multiplicity of positive solutions of u x lambda f epsilon u 0 1 f epsilon u epsilon u 3 sigma u 2 tau u rho where lambda epsilon 0 are two bifurcation parameters and sigma rho 0 tau 0 are constants by developing some new time map techniques we prove the global bifurcation of bifurcation curves for varying epsilon 0 more precisely we prove that for any sigma rho 0 tau 0 there exists epsilon 0 such that on the lambda parallel to u parallel to infinity plane the bifurcation curve is s shaped for 0 epsilon we also prove the global bifurcation of bifurcation curves for varying lambda 0 thus we are able to determine the exact number of positive solutions by the values of epsilon and lambda we give an application to prove a long standing conjecture for global bifurcation of positive solutions for the problem u x lambda epsilon u 3 u 2 u 1 0 1 u 1 u 1 0 which was studied by crandall and rabinowitz arch rational mech anal 52 1973 p 177 in addition we give an application to prove a conjecture of smoller and wasserman j differential equations 39 1981 p 283 lines 2 3 on the maximum number of positive solutions of a positone problem

incomplete information in relational databases abstract this paper concerns the semantics of codd s relational model of data formulated are precise conditions that should be satisfied in a semantically meaningful extension of the usual relational operators such as projection selection union and join from operators on relations to operators on tables with null values of various kinds allowed these conditions require that the system be safe in the sense that no incorrect conclusion is derivable by using a specified subset of the relational operators and that it be complete in the sense that all valid conclusions expressible by relational expressions using operators in are in fact derivable in this system two such systems of practical interest are shown the first based on the usual codd s null values supports projection and selection the second based on many different marked null values or variables allowed to appear in a table is shown to correctly support projection positive selection with no negation occurring in the selection condition union and renaming of attributes which allows for processing arbitrary conjunctive queries a very desirable property enjoyed by this system is that all relational operators on tables are performed in exactly the same way as in the case of the usual relations a third system mainly of theoretical interest supporting projection selection union join and renaming is also discussed under a so called closed world assumption it can also handle the operator of difference it is based on a device called a conditional table and is crucial to the proof of the correctness of the second system all systems considered allow for relational expressions containing arbitrarily many different relation symbols and no form of the universal relation assumption is required categories and subject descriptors h 2 3 database management languages query languages h 2 4 database management systems query processing general terms theory

gefred a generalized model of fuzzy relational databases abstract in this paper we present a fuzzy relational databases model whose main characteristics are the integration of previous models in the same framework representation capabilities for a wide series of fuzzy information and a coherent and flexible handling of it this model aims to solve each problem of representation and handling of fuzzy information taking into account its specific nature and hence it allows the user to choose the comparison operator and the fuzzy compatibility measure to be used in a query besides it permits the user to specify the precision with which the conditions involved in a query are satisfied

a refinement of the cameron erdős conjecture in this paper we study sum free subsets of the set 1 n that is subsets of the first n positive integers which contain no solution to the equation x y z and erdos conjectured in 1990 that the lumber of such sets is o 2 n 2 this conjecture was confirmed by green and independently by sapozhenko here we prove a refined version of their theorem by showing that the number of sum free subsets of n of size in is 2 o n m n 2 for every 1 to 17 21 for m root t this result is sharp up to the constant implicit in the oh our proof uses a general bound on the number of independent sets of size to in 3 uniform hypergraphs proved recently by the authors and new bounds on the number of integer partitions with small sumset