random polynomials having few or no real zeros consider a polynomial of large degree n whose coefficients are independent identically distributed nondegenerate random variables having zero mean and finite moments of all orders we show that such a polynomial has exactly k real zeros with probability n b o 1 as n through integers of the same parity as the fixed integer k 0 in particular the probability that a random polynomial of large even degree n has no real zeros is n b o 1 the finite positive constant b is characterized via the centered stationary gaussian process of correlation function sech t 2 the value of b depends neither on k nor upon the specific law of the coefficients under an extra smoothness assumption about the law of the coefficients with probability n b o 1 one may specify also the approximate locations of the k zeros on the real line the constant b is replaced by b 2 in case the i i d coefficients have a nonzero mean

order sorted algebra i equational deduction for multiple inheritance overloading exceptions and partial operations abstract this paper generalizes many sorted algebra msa to order sorted algebra osa by allowing a partial ordering relation on the set of sorts this supports abstract data types with multiple inheritance in roughly the sense of object oriented programming several forms of polymorphism and overloading partial operations as total on equationally defined subsorts exception handling and an operational semantics based on term rewriting we give the basic algebraic constructions for osa including quotient image product and term algebra and we prove their basic properties including quotient homomorphism and initiality theorems the paper s major mathematical results include a notion of osa deduction a completeness theorem for it and an osa birkhoff variety theorem we also develop conditional osa including initiality completeness and mckinsey malcev quasivariety theorems and we reduce osa to conditional msa which allows lifting many known msa results to osa retracts which intuitively are left inverses to subsort inclusions provide relatively inexpensive run time error handling we show that it is safe to add retracts to any osa signature in the sense that it gives rise to a conservative extension a final section compares and contrasts many different approaches to osa this paper also includes several examples demonstrating the flexibility and applicability of osa including some standard benchmarks like stack and list as well as a much more substantial example the number hierarchy from the naturals up to the quaternions

a climate based malaria transmission model with structured vector population in this paper we present a malaria transmission model with periodic birth rate and age structure for the vector population we first introduce the basic reproduction ratio for this model and then show that there exists at least one positive periodic state and that the disease persists when mathcal r 0 1 it is also shown that the disease will die out if mathcal r 0 1 provided that the invasion intensity is not strong we further use these analytic results to study the malaria transmission cases in kwazulu natal province south africa some sensitivity analysis of mathcal r 0 is performed and in particular the potential impact of climate change on seasonal transmission and populations at risk of the disease is analyzed

fractional model for malaria transmission under control strategies we study a fractional model for malaria transmission under control strategies we consider the integer order model proposed by chiyaka et al 2008 in 15 and modify it to become a fractional order model we study numerically the model for variation of the values of the fractional derivative and of the parameter that models personal protection b from observation of the figures we conclude that as b is increased from 0 to 1 there is a corresponding decrease in the number of infectious humans and infectious mosquitoes for all values of a this means that this result is invariant for variation of fractional derivative in the values tested these results are in agreement with those obtained in chiyaka et al 2008 15 for a 1 0 and suggest that our fractional model is epidemiologically well posed

diagonal cycles and euler systems ii the birch and swinnerton dyer conjecture for hasse weil artin l functions this article establishes new cases of the birch and swinnerton dyer conjecture in analytic rank 0 for elliptic curves over mathbb q viewed over the fields cut out by certain self dual artin representations of dimension at most 4 when the associated l function vanishes to even order ge 2 at its central point two canonical classes in the corresponding selmer group are constructed and shown to be linearly independent assuming the non vanishing of a garrett hida p adic l function at a point lying outside its range of classical interpolation the key tool for both results is the study of certain p adic families of global galois cohomology classes arising from gross kudla schoen diagonal cycles in a tower of triple products of modular curves

the cassels tate pairing on polarized abelian varieties let a e be a principally polarized abelian variety defined over a global field k and let a be its shafarevich tate group let a nd denote the quotient of a by its maximal divisible subgroup cassels and tate constructed a nondegenerate pairing a nd a nd q z if a is an elliptic curve then by a result of cassels the pairing is alternating but in general it is only antisymmetric using some new but equivalent definitions of the pairing we derive general criteria deciding whether it is alternating and whether there exists some alternating nondegenerate pairing on a nd these criteria are expressed in terms of an element c a nd that is canonically associated to the polarization e in the case that a is the jacobian of some curve a down to earth version of the result allows us to determine effectively whether a if finite is a square or twice a square we then apply this to prove that a positive proportion in some precise sense of all hyperelliptic curves of even genus g 2 over q have a jacobian with nonsquare if finite for example it appears that this density is about 13 for curves of genus 2 the proof makes use of a general result relating global and local densities this result can be applied in other situations

on fuzzy type theory this paper is a generalization of classical simple type theory we have developed a formal system of fuzzy type theory which differs from the classical one essentially in extension of truth values from two to infinitely many the structure of truth values is assumed to be an imtl algebra residuated lattice with prelinearity and double negation extended by the baaz delta operation various properties of fuzzy type theory are proved including its completeness

derived p adic heights and p adic l functions if e is an elliptic curve defined over a number field and p is a prime of good ordinary reduction for e a theorem of rubin relates the p adic height pairing on the p power selmer group of e to the first derivative of a cohomologically defined p adic l function attached to e bertolini and darmon have defined a sequence of derived p adic heights in this paper we give an alternative definition of the p adic height pairing and prove a generalization of rubin s result relating the derived heights to higher derivatives of p adic l functions we also relate degeneracies in the derived heights to the failure of the selmer group of e over a z p extension to be semi simple as an iwasawa module generalizing results of perrin riou

neron models lie algebras and reduction of curves of genus one let k be a discrete valuation field let ok denote the ring of integers of k and let k be the residue field of ok of characteristic p 0 let s specok let x k be a smooth geometrically connected projective curve of genus 1 over k denote by ek the jacobian of x k let x s and e s be the minimal regular models of x k and ek respectively in this article we investigate the possible relationships between the special fibers xk and ek in doing so we are led to study the geometry of the picard functor pic x s when x s is not necessarily cohomologically flat as an application of this study we are able to prove in full generality a theorem of gordon on the equivalence between the artin tate and birch swinnertondyer conjectures recall that when k is algebraically closed the special fibers of elliptic curves are classified according to their kodaira type which is denoted by a symbol t in i n n z 0 ii ii iii iii iv iv given a type t and a positive integer m we denote by mt the new type obtained from t by multiplying all the multiplicities of t by m when k is algebraically closed the relationships between the type of a curve of genus 1 and the type of its jacobian can be summarized as follows

local points of twisted mumford quotients and shimura curves we determine over which fields twisted mumford quotients have rational points using the p adic uniformization we apply these results to shimura curves and show some new cases for which the jacobians are even in the sense of ps

pairings and functional equations over the gl2 extension we construct a pairing on the dual selmer group over the gl2 extension q e p of an elliptic curve without complex multiplication and with good ordinary reduction at a prime p 5 whenever it satisfies certain conjectured torsion properties this gives a functional equation of the characteristic element which is compatible with the conjectural functional equation of the p adic l function as an application we reduce the parity conjecture for the p selmer rank and the analytic root number for the twists of elliptic curves with self dual artin representation to the case when the artin representation factors through the quotient of gal q e p q b y its maximal pro p normal subgroup this gives a new proof of the parity conjecture whenever the elliptic curve e has a p isogeny over the rationals

a mathematical analysis of the effects of control strategies on the transmission dynamics of malaria we formulate a deterministic model with two latent periods in the non constant host and vector populations in order to theoretically assess the potential impact of personal protection treatment and possible vaccination strategies on the transmission dynamics of malaria the thresholds and equilibria for the model are determined the model is analysed qualitatively to determine criteria for control of a malaria epidemic and is used to compute the threshold vaccination and treatment rates necessary for community wide control of malaria in addition to having a disease free equilibrium which is locally asymptotically stable when the basic reproductive number is less than unity the model exhibits the phenomenon of backward bifurcation where a stable disease free equilibrium coexists with a stable endemic equilibrium for a certain range of associated reproductive number less than one from the analysis we deduce that personal protection has a positive impact on disease control but to eradicate the disease in the absence of any other control measures the efficacy and compliance should be very high our results show that vaccination and personal protection can suppress the transmission rates of the parasite from human to vector and vice versa if the treated populations are infectious then certain conditions should be satisfied for treatment to reduce the spread of malaria in a community among the interesting dynamical behaviours of the model numerical simulations show a backward bifurcation which gives a challenge to the designing of effective control measures

explicit arithmetic intersection theory and computation of neron tate heights we describe a general algorithm for computing intersection pairings on arithmetic surfaces we have implemented our algorithm for curves over q and we show how to use it to compute regulators for a number of jacobians of smooth plane quartics and to numerically verify the conjecture of birch and swinnerton dyer for the jacobian of the split cartan curve of level 13 up to squares

a survey on the categorical term construction with applications this paper gives a survey on the categorical term construction based on the free algebra algorithm in the framework of monoidal biclosed and cocomplete categories a possible concept of signature for finitary theories is introduced applications of these constructions are given in goguen s category and in the category of complete lattices and join preserving maps

period and index of genus one curves over global fields the period of a curve is the smallest positive degree of galois invariant divisor classes the index is the smallest positive degree of rational divisors we construct examples of genus one curves with prescribed period and index over a given global field as long as the characteristic of the field does not divide the period

the mathematics of infectious diseases many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases threshold theorems involving the basic reproduction number r 0 the contact number sigma and the replacement number r are reviewed for the classic sir epidemic and endemic models similar results with new expressions for r 0 are obtained for mseir and seir endemic models with either continuous age or age groups values of r 0 and sigma are estimated for various diseases including measles in niger and pertussis in the united states previous models with age structure heterogeneity and spatial structure are surveyed

the order of higher brauer groups let x be a smooth projective variety of dimension 2r over a finite field mathbb f we prove that for every prime ell ne mathrm char mathbb f the order of the non divisible quotient of the ell primary torsion group of the higher brauer group mathrm br r x ell mathrm nd is a square number

monadic extensions of institutions in this paper we show how partially ordered monads can be used to provide a non classical extension of institutions and entailment systems as appearing in the framework of general logics general logics is an axiomatization of a general framework for logics this framework builds upon traditional power sets of sentences as appearing in satisfaction relations and entailment the underlying power set monad is implicit and the utility of a monadic machinery was therefore not explored making the use of the power set monad more explicit in the satisfaction and entailment relations opens up possibilities to experiment with various non classical representations of sentences as appearing in inference these representations are enabled by partially ordered monads where the partial order attached to the underlying set functor is essential for extending the axiomatization

on the birch swinnerton dyer quotients modulo squares let a be an abelian variety over a number field k an identity between the l functions l a k i s for extensions k i of k induces a conjectural relation between the birch swinnerton dyer quotients we prove these relations modulo finiteness of lii and give an analogous statement for selmer groups based on this we develop a method for determining the parity of various combinations of ranks of a over extensions of k as one of the applications we establish the parity conjecture for elliptic curves assuming finiteness of iii e k e 2 6 and some restrictions on the reduction at primes above 2 and 3 the parity of the mordell weil rank of e k agrees with the parity of the analytic rank as determined by the root number we also prove the p parity conjecture for all elliptic curves over q and all primes p the parities of the p selmer rank and the analytic rank agree

random maximal isotropic subspaces and selmer groups under suitable hypotheses we construct a probability measure on the set of closed maximal isotropic subspaces of a locally compact quadratic space over f p a random subspace chosen with respect to this measure is discrete with probability 1 and the dimension of its intersection with a fixed compact open maximal isotropic subspace is a certain nonnegative integer valued random variable we then prove that the p selmer group of an elliptic curve is naturally the intersection of a discrete maximal isotropic subspace with a compact open maximal isotropic subspace in a locally compact quadratic space over f p by modeling the first subspace as being random we can explain the known phenomena regarding distribution of selmer ranks such as the theorems of heath brown swinnerton dyer and kane for 2 selmer groups in certain families of quadratic twists and the average size of 2 and 3 selmer groups as computed by bhargava and shankar our model is compatible with delaunay s heuristics for p torsion in shafarevich tate groups and predicts that the average rank of elliptic curves over a fixed number field is at most 1 2 many of our results generalize to abelian varieties over global fields

annihilating selmer modules we prove that for a wide class of motives the local and global non commutative tamagawa number conjectures of fukaya and kato imply explicit restrictions on the galois structure of selmer modules in the sense of bloch and kato we thereby obtain a variety of new concrete and general conjectures concerning the structures of selmer modules in several important cases we prove these conjectures

nontrivial elements of sha explained through k3 surfaces we present a new method to show that a principal homogeneous space of the jacobian of a curve of genus two is nontrivial the idea is to exhibit a brauer manin obstruction to the existence of rational points on a quotient of this principal homogeneous space in an explicit example we apply the method to show that a specific curve has infinitely many quadratic twists whose jacobians have nontrivial tate shafarevich group

tate shafarevich groups and k3 surfaces this paper explores a topic taken up recently by logan and van luijk finding nontrivial 2 torsion elements of the tate shafarevich group of the jacobian of a genus 2 curve by exhibiting brauer manin obstructions to rational points on certain quotients of principal homogeneous spaces of the jacobian whose desingularizations are explicit k3 surfaces the main difference between the methods used in this paper and those of logan and van luijk is that the obstructions are obtained here from explicitly constructed quaternion algebras rather than elliptic fibrations

empirical evidence for the birch and swinnerton dyer conjectures for modular jacobians of genus 2 curves this paper provides empirical evidence for the birch and swinnerton dyer conjectures for modular jacobians of genus 2 curves the second of these conjectures relates six quantities associated to a jacobian over the rational numbers one of these six quantities is the size of the shafarevich tate group unable to compute that we computed the five other quantities and solved for the last one in all 32 cases the result is very close to an integer that is a power of 2 in addition this power of 2 agrees with the size of the 2 torsion of the shafarevich tate group which we could compute

galois sections for abelianized fundamental groups given a smooth projective curve x of genus at least 2 over a number field k grothendieck s section conjecture predicts that the canonical projection from the etale fundamental group of x onto the absolute galois group of k has a section if and only if the curve has a rational point we show that there exist curves where the above map has a section over each completion of k but not over k in the appendix victor flynn gives explicit examples in genus 2 our result is a consequence of a more general investigation of the existence of sections for the projection of the etale fundamental group with abelianized geometric part onto the galois group we also point out the relation to the elementary obstruction of colliot thelene and sansuc

a generalization of the coleman map for hida deformations in this paper we give a coleman perrin riou type map for an ordinary type deformation and construct a two variable p adic l function for a hida family from the beilinson kato elements 1 introduction fix a prime p 3 we denote by q jlp the extension of the rational number field q obtained by adjoining all power roots of unity we fix a complex embedding q c and a p adic embedding q qp of an algebraic closure q of q throughout the paper where c is the field of complex numbers and qp is an algebraic closure of the field qp of p adic numbers let goo be the galois group of q jjlpo q and let dqo be the group of diamond operators for the tower of modular curves fi i see 2 we have the

fuzzy terms in this paper we will show how purely categorical constructions of terms are advantageous when investigating situations concerning uncertainty more specifically where uncertainty comes from and how uncertainty is integrated when dealing with terms over selected signatures there are basically two ways of invoking uncertainty for terms on one hand we may proceed by building composed monads where uncertainty is provided by some suitable monad composed with the traditional term monad on the other hand we can provide a strictly formal basis for term monads being created over categories themselves carrying uncertainty this is the distinction between computing with fuzzy and fuzzy computing and the fundamental question raised by these constructions is where uncertainty resides in language constructions for logic this paper also shows how the notion of signature often needs to be expanded to levels of signatures in particular when dealing with type constructors such levels allow us to strictly delineate e g primitive operations type terms and value level terms levels of signature will in this paper be exemplified by the construction of the signature of simply typed lambda calculus

visible evidence for the birch and swinnerton dyer conjecture for modular abelian varieties of analytic rank zero this paper provides evidence for the birch and swinnerton dyer conjecture for analytic rank 0 abelian varieties a f that are optimal quotients of j 0 n attached to newforms we prove theorems about the ratio l a f 1 af develop tools for computing with af and gather data about certain arithmetic invariants of the nearly 20 000 abelian varieties a f of level 2333 over half of these a f have analytic rank 0 and for these we compute upper and lower bounds on the conjectural order of iii a f we find that there are at least 168 such af for which the birch and swinnerton dyer conjecture implies that iii a f is divisible by an odd prime and we prove for 37 of these that the odd part of the conjectural order of iii a f really divides iii a f by constructing nontrivial elements of iii a f using visibility theory we also give other evidence for the conjecture the appendix by cremona and mazur fills in some gaps in the theoretical discussion in their paper on visibility of shafarevich tate groups of elliptic curves

two cover descent on hyperelliptic curves we describe an algorithm that determines a set of unramified covers of a given hyperelliptic curve with the property that any rational point will lift to one of the covers in particular if the algorithm returns an empty set then the hyperelliptic curve has no rational points this provides a relatively efficiently tested criterion for solvability of hyperelliptic curves we also discuss applications of this algorithm to curves of genus 1 and to curves with rational points

attributive concept descriptions with complements abstract we investigate the consequences of adding unions and complements to attributive concept descriptions employed in terminological knowledge representation languages it is shown that deciding coherence and subsumption of such descriptions are pspace complete problems that can be decided with linear space

on the brauer group of a surface artin and tate 15 5 1 have shown in 1966 the existence of a canonical skew symmetric pairing on the non p part of br x whose kernel is exactly the set of divisible elements it follows from this fact that if the non p part of br x is finite then its order is a square or twice a square a few years later manin published examples of rational surfaces that is surfaces birational over k to the projective plane with brauer groups equal to z 2z it is only in 1996 that the examples of manin were revisited by urabe who found a mistake in them for rational surfaces the brauer group is relatively easy to understand and urabe 16 remark after 1 17 showed improving on a result of milne 9 that the brauer group of a rational surface has order a square in 17 0 3 urabe then proves in full generality that when p 2 the 2 part of br x modulo its divisible subgroup has order a square thus to complete the proof of theorem 1 it remains to treat the case where p 2 and x k is not rational as we remarked above it was wrongly assumed for about 30 years that br x was not always a square our method of proof provides for any p that the 2 part of br x has order a square via the knowledge of the 2 part

powersets of terms and composite monads composing various powerset functors with the term monad gives rise to the concept of generalized terms this in turn provides a technique for handling many valued sets of terms in a framework of variable substitutions thus being the prerequisite for categorical unification in many valued logic programming using an extended notion of terms as constructions of monads involve complicated calculations with natural transformations proofs are supported by a graphical approach that provides a useful tool for handling various conditions such as those well known for distributive laws

unobstructed modular deformation problems let be a newform of weight k 3 with fourier coefficients in a number field k we show that the universal deformation ring of the mod a galois representation associated to is unobstructed and thus isomorphic to a power series ring in three variables over the witt vectors for all but finitely many primes a of k we give an explicit bound on such a for the 6 known cusp forms of level 1 trivial character and rational fourier coefficients we also prove a somewhat weaker result for weight 2

topological systems as a framework for institutions recently j t denniston a melton and s e rodabaugh introduced a lattice valued analogue of the concept of institution of j a goguen and r m burstall comparing it moreover with the lattice valued version of the notion of topological system of s vickers in this paper we show that a suitable generalization of topological systems provides a convenient framework for certain kinds of lattice valued institutions

on the bloch kato conjecture for elliptic modular forms of square free level let kappa ge 6 be an even integer m an odd square free integer and f in s 2 kappa 2 gamma 0 m a newform we prove that under some reasonable assumptions that half of the lambda part of the bloch kato conjecture for the near central critical value l kappa f is true we do this by bounding the ell valuation of the order of the appropriate bloch kato selmer group below by the ell valuation of algebraic part of l kappa f we prove this by constructing a congruence between the saito kurokawa lift of f and a cuspidal siegel modular form

how many zeros of a random polynomial are real we provide an elementary geometric derivation of the kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients we show that the expected number of real zeros is simply the length of the moment curve 1 t tn projected onto the surface of the unit sphere divided by the probability density of the real zeros is proportional to how fast this curve is traced out we then relax kac s assumptions by considering a variety of random sums series and distributions and we also illustrate such ideas as integral geometry and the fubini study metric

finding large selmer rank via an arithmetic theory of local constants we obtain lower bounds for selmer ranks of elliptic curves over dihedral extensions of number fields suppose k k is a quadratic extension of number fields e is an elliptic curve defined over k and p is an odd prime let denote the maximal abelian p extension of k that is unramified at all primes where e has bad reduction and that is galois over k with dihedral galois group i e the generator c of gal k k acts on gal k by inversion we prove under mild hypotheses on p that if the zp rank of the pro p selmer group sp e k is odd then rank zp s p e f f k for every finite extension f of k in

the coleman map for hida families of gsp4 in this paper we construct a coleman map for ordinary hida deformations associated to the symplectic group rm gsp 4 which interpolates the dual exponential maps when arithmetic specializations of this family vary the main result theorem 3 2 will play an important role in a forthcoming work by the authors discussing a conjecture on the existence of an euler system which would give rise via our map to a three variable p adic l function

on the geometry of principal homogeneous spaces let b be a curve defined over an algebraically closed field k and let pi colon x to b be an elliptic surface with base curve b we investigate the geometry of everywhere locally trivial principal homogeneous spaces for x i e elements of the tate shafarevich group of the generic fiber of pi if y is such a principal homogeneous space of order n we find strong restrictions on the bbb p n 1 bundle over b into which y embeds examples for small values of n show that at least in some cases these restrictions are sharp finally we determine these bundles in case k has characteristic zero b bbb p 1 and x is generic in a suitable sense

a non commutative and non idempotent theory of quantale sets in fuzzy set theory non idempotency arises when the conjunction is interpreted by arbitrary t norms there are many instances in mathematics where set theory ought to be non commutative and or non idempotent the purpose of this paper is to combine both ideas and to present a theory of non commutative and non idempotent quantale sets among other things standard concepts like fuzzy preorders and fuzzy equivalence relations will be exhibited as special cases more specifically the category q set is investigated where q is an arbitrary involutive quantale objects quantale sets are pairs consisting of a set and a q valued equality with a suitable symmetry axiom three important properties of q set are shown it is complete cocomplete and has the epi extremal mono factorization property its subcategory s q set of separated quantale sets is reflective in q set and shares the same fundamental properties with q set in particular s q set is also a complete and cocomplete epi extremal mono category the objects of the category q set are interesting categories in their own right two categorical frameworks for objects of q set are exhibited first it is shown that q valued equalities arise from q valued preorders with self adjoint extents by symmetrization which leaves q valued equalities invariant here sets with quantale preorders are shown to be b categories with base b being a specific quantaloid the second approach is based on involutive quantaloids a combination of two well known things quantaloids and ordered categories with involution in this context quantale sets are precisely symmetric b categories w r t appropriately chosen quantaloid b with involution further the cauchy completion preserves the symmetry axiom for a large class of involutive quantales which include quantic frames our non commutative generalization of frames and all left continuous t norms there exist at least two monads on q set the singleton monad and quasi singleton monad which are of special interest for fuzzy set theory the kleisli category associated with the singleton monad is the non commutative and non idempotent analogue of higgs topos while the eilenberg moore category of the quasi singleton monad permits the internalization of lukasiewicz negation as truth arrow finally an application of quantale sets to c algebras is given and the change of base is treated

computational verification of the birch and swinnerton dyer conjecture for individual elliptic curves we describe theorems and computational methods for verifying the birch and swinnerton dyer conjectural formula for specific elliptic curves over q of analytic ranks 0 and 1 we apply our techniques to show that if e is a non cm elliptic curve over q of conductor 1000 and rank 0 or 1 then the birch and swinnerton dyer conjectural formula for the leading coefficient of the l series is true for e up to odd primes that divide either tamagawa numbers of e or the degree of some rational cyclic isogeny with domain e since the rank part of the birch and swinnerton dyer conjecture is a theorem for curves of analytic rank 0 or 1 this completely verifies the full conjecture for these curves up to the primes excluded above

on the characteristic polynomial of the frobenius on etale cohomology let x be a smooth projective variety of even dimension d over a finite field then we establish a restriction on the value at 1 of the characteristic polynomial of the frobenius on the middle dimensional etale cohomology of x with coefficients in ql d 2

foundations of fuzzy sets abstract this paper gives an overview of the origins of fuzzy set theory and the problems for the foundations of fuzzy sets arising from those origins and current practice it then gives detailed accounts of categorical approaches using a closed structure to capture the fuzzy and connective within these categories weak forms of subobject representations provide an internal second order logic an approach to fuzzy real numbers and fuzzy topology is included to illustrate the use of this internal second order theory