on the modular curves y e 7 let e denote an elliptic curve over q and y e 7 the modular curve classifying the elliptic curves e over q such that the representations of gal q q in the 7 torsion points of e and of e are symplectically isomorphic in case e is given by a weierstrass equation such that the c 4 invariant is a square we exhibit here nontrivial points of y e 7 q from this we deduce an infinite family of curves e for which y e 7 q has at least four nontrivial points

arithmetic representations of fundamental groups i let x be a normal algebraic variety over a finitely generated field k of characteristic zero and let ell be a prime say that a continuous ell adic representation rho of pi 1 acute mathrm e text t x bar k is arithmetic if there exists a finite extension k of k and a representation tilde rho of pi 1 acute mathrm e text t x k with rho a subquotient of tilde rho pi 1 x bar k we show that there exists an integer n n x ell such that every nontrivial semisimple arithmetic representation of pi 1 acute mathrm e text t x bar k is nontrivial mod ell n as a corollary we prove that any nontrivial ell adic representation of pi 1 acute mathrm e text t x bar k which arises from geometry is nontrivial mod ell n

a relational perspective of attribute reduction in rough set based data analysis attribute reduction is very important in rough set based data analysis rsda because it can be used to simplify the induced decision rules without reducing the classification accuracy the notion of reduct plays a key role in rough set based attribute reduction in rough set theory a reduct is generally defined as a minimal subset of attributes that can classify the same domain of objects as unambiguously as the original set of attributes nevertheless from a relational perspective rsda relies on a kind of dependency principle that is the relationship between the class labels of a pair of objects depends on component wise comparison of their condition attributes the larger the number of condition attributes compared the greater the probability that the dependency will hold thus elimination of condition attributes may cause more object pairs to violate the dependency principle based on this observation a reduct can be defined alternatively as a minimal subset of attributes that does not increase the number of objects violating the dependency principle while the alternative definition coincides with the original one in ordinary rsda it is more easily generalized to cases of fuzzy rsda and relational data analysis

p torsion monodromy representations of elliptic curves over geometric function fields given a complex quasiprojective curve b and a nonisotrivial familye of elliptic curves over b the p torsione p yields a monodromy representation e p 1 b gl2 fp we prove that if e p e0 p thene ande 0 are isogenous provided p is larger than a constant depending only on the gonality of b this can be viewed as a function eld analog of the frey mazur conjecture which states that an elliptic curve over q is determined up to isogeny by its p torsion galois representation for p 17 the proof relies on hyperbolic geometry and is therefore only applicable in characteristic 0

applications of soft sets in ideal theory of bck bci algebras the notion of soft ideals and idealistic soft bck bci algebras is introduced and several examples are given relations between soft bck bci algebras and idealistic soft bck bci algebras are provided the intersection union and operation and or operation of soft ideals and idealistic soft bck bci algebras are established

soft sets and soft groups molodtsov introduced the concept of soft set theory which can be used as a generic mathematical tool for dealing with uncertainty in this paper we introduce the basic properties of soft sets and compare soft sets to the related concepts of fuzzy sets and rough sets we then give a definition of soft groups and derive their basic properties using molodtsov s definition of the soft sets

the hessian of a genus one curve we continue our development of the invariant theory of genus one curves with the aim of computing certain twists of the universal family of elliptic curves parametrised by the modular curve x n for n 2 3 4 5 our construction makes use of a covariant we call the hessian generalising the classical hessian that exists in degrees 2 and 3 in particular we give explicit formulae and algorithms for computing the hessian in degrees 4 and 5 this leads to a practical algorithm for computing equations for visible elements of order n in the tate shafarevich group of an elliptic curve taking jacobians we also recover the formulae of rubin and silverberg for families of n congruent elliptic curves

there exist multilinear bohnenblust hille constants cn n 1 with limn cn 1 cn 0 abstract the n linear bohnenblust hille inequality asserts that there is a constant c n 1 such that the l 2 n n 1 norm of u e i 1 e i n i 1 i n 1 n is bounded above by c n times the supremum norm of u for any n linear form u c n c n c and n n the same holds for real scalars we prove what we call fundamental lemma which brings new information on the optimal constants k n n 1 for both real and complex scalars for instance k n 1 k n 0 87 n 0 473 for infinitely many n s for complex scalars we give a formula of surprisingly low growth in which e and the famous euler mascheroni constant appear k n 1 4 1 e 2 1 2 j 1 n 1 j log 2 e 2 1 2 1 n 2 we study the interplay between the kahane salem zygmund and the bohnenblust hille polynomial and multilinear inequalities and provide estimates for bohnenblust hille type inequality constants for any exponent q 2 n n 1

volumes of complex analytic subvarieties of hermitian symmetric spaces we give lower bounds of volumes of k dimensional complex analytic subvarieties of certain naturally defined domains in n dimensional complex space forms of constant positive zero or negative holomorphic sectional curvature for each 1 k n the lower bounds are sharp in the sense that these bounds are attained by k dimensional complete totally geodesic complex submanifolds such lower bounds are obtained by constructing singular potential functions corresponding to blow ups of the kahler metrics involved similar lower bounds are also obtained in the case of hermitian symmetric spaces of noncompact type in this case the lower bounds are sharp for those values of k at which the hermitian symmetric space contains k dimensional complete totally geodesic complex submanifolds which are complex hyperbolic spaces of minimum holomorphic sectional curvature

coordinatewise multiple summing operators in banach spaces we invent the new notion of coordinatewise multiple summing operators in banach spaces and use it to study various vector valued extensions of the well know bohnenblust hille inequality which originally extended littlewood s 4 3 inequality our results have application on the summability of monomial coefficients of m homogeneous polynomials p l lp as well as for the convergence theory of products of vector valued dirichlet series

selmer companion curves we say that two elliptic curves e1 e2 over a number field k are n selmer companions for a positive integer n if for every quadratic character of k there is an isomorphism seln e 1 k seln e 2 k between the nselmer groups of the quadratic twists e 1 e 2 we give sufficient conditions for two elliptic curves to be n selmer companions and give a number of examples of non isogenous pairs of companions

billiards in rectangles with barriers in this paper we consider billiards in a square with a barrier we use ratner s theorem to compute the asymptotics for the number of closed orbits

on the period matrix of a riemann surface of large genus with an appendix by j h conway and n j a sloane riemann showed that a period matrix of a compact riemann surface of genusg 1 satisfies certain relations we give a further simple combinatorial property related to the length of the shortest non zero lattice vector satisfied by such a period matrix see 1 13 in particular it is shown that for large genus the entire locus of jacobians lies in a very small neighborhood of the boundary of the space of principally polarized abelian varieties

on the growth of the optimal constants of the multilinear bohnenblust hille inequality let kn n 1 be the optimal constants satisfying the multilinear real or complex bohnenblust hille inequality the exact values of the constants kn are still waiting to be discovered since eighty years ago with the publication of bohnenblust and hille paper in the annals of mathematics recently it was proved that kn n 1 has a subpolynomial growth moreover it is now known that if there is an l such that limn kn kn 1 l then l 0 in this note we show that if there is an l 0 such that limn k2nkn l then l 1 d with d e12 12 for real scalars and d e1 12 2 for complex scalars here is the famous euler mascheroni constant we show that this result generalizes the former

modular diagonal quotient surfaces the main objective of this paper is to analyze the geometry of the modular diagonal quotient surface zn e e x n x n which classifies pairs of elliptic curves e1 and e2 together with an isomorphism of determinant e between their associated modular representations mod n in particular we calculate some of the numerical invariants of its minimal desingularization zn e such as its betti and chern numbers and determine its place in the enriques kodaira classification table

generalised fuzzy soft sets in this paper we define generalised fuzzy soft sets and study some of their properties application of generalised fuzzy soft sets in decision making problem and medical diagnosis problem has been shown

improving the closed formula for subpolynomial constants in the multilinear bohnenblust hille inequalities abstract for k r or c the bohnenblust hille inequality asserts that there exists a sequence of scalars c k m m 1 such that i 1 i m 1 n u e i 1 e i m 2 m m 1 m 1 2 m c k m sup z 1 z m d n tu z 1 z m for all m linear forms u k n k n k and every positive integer n where e i i 1 n denotes the canonical basis of k n and d n represents the open unit polydisc in k n very recently 2012 it was shown that there exist constants c k m m 1 with subpolynomial growth satisfying this inequality however these constants were obtained via a complicated recursive formula we improve the best known closed non recursive approximation for these constants

new upper bounds for the constants in the bohnenblust hille inequality a classical inequality due to bohnenblust and hille states that for every positive integer m there is a constant c m 0 so that sigma n i1 im 1 vertical bar u e i1 e im vertical bar 2m m 1 m 1 2m c where c m m m 1 2m 2 m 1 2 the value of c m was improved to c m 2 m 1 2 by s kaijser and more recently h queffelec and a defant and p sevilla peris remarked that c m 2 root pi m 1 also works the bohnenblust hille inequality also holds for real banach spaces with the constants c m 2 m 1 2 in this note we show that a recent new proof of the bohnenblust hille inequality due to defant popa and schwarting provides in fact quite better estimates for c m for all values of m is an element of n in particular we will also show that for real scalars if m is even with 2 m 24 then c r m 2 1 2 c r m 2 we will mainly work on a paper by defant popa and schwarting giving some remarks about their work and explaining how to numerically improve the previously mentioned constants

the normal parameter reduction of soft sets and its algorithm this paper is concerned with the reduction of soft sets and fuzzy soft sets firstly the problems of suboptimal choice and added parameter set of soft sets are analyzed then we introduce the definition of normal parameter reduction in soft sets to overcome these problems in addition a heuristic algorithm of normal parameter reduction is presented two new definitions parameter important degree and decision partition are proposed for analyzing the algorithm of normal parameter reduction furthermore the normal parameter reduction is also investigated in fuzzy soft sets

dynamics of sl2 ℝ over moduli space in genus two this paper classifies orbit closures and invariant measures for the natural action of sl2 r on um2 the bundle of holomorphic 1 forms over the moduli space of riemann surfaces of genus two

connected components of the moduli spaces of abelian differentials with prescribed singularities consider the moduli space of pairs c where c is a smooth compact complex curve of a given genus and is a holomorphic 1 form on c with a given list of multiplicities of zeroes we describe connected components of this space this classification is important in the study of dynamics of interval exchange transformations and billiards in rational polygons and in the study of geometry of translation surfaces

a new multilinear insight on littlewood s 4 3 inequality abstract we unify littlewood s classical 4 3 inequality a forerunner of grothendieck s inequality together with its m linear extension due to bohnenblust and hille which originally settled bohr s absolute convergence problem for dirichlet series with a scale of inequalties of bennett and carl in l p spaces which are of fundamental importance in the theory of eigenvalue distribution of power compact operators as an application we give estimates for the monomial coefficients of homogeneous l p valued polynomials on c 0

invariant theory for the elliptic normal quintic i twists of x 5 a genus one curve of degree 5 is defined by the 4 times 4 pfaffians of a 5 times 5 alternating matrix of linear forms on mathbb p 4 we describe a general method for investigating the invariant theory of such models we use it to explain how we found our algorithm for computing the invariants and to extend our method for computing equations for visible elements of order 5 in the tate shafarevich group of an elliptic curve as a special case of the latter we find a formula for the family of elliptic curves 5 congruent to a given elliptic curve in the case the 5 congruence does not respect the weil pairing we also give an algorithm for doubling elements in the 5 selmer group of an elliptic curve and make a conjecture about the matrices representing the invariant differential on a genus one normal curve of arbitrary degree

soft matrix theory and its decision making in this work we define soft matrices and their operations which are more functional to make theoretical studies in the soft set theory we then define products of soft matrices and their properties we finally construct a soft max min decision making method which can be successfully applied to the problems that contain uncertainties

the asymptotic growth of the constants in the bohnenblust hille inequality is optimal the search of sharp estimates for the constants in the bohnenblust hille inequality besides its challenging nature has quite important applications in different fields of mathematics and physics for homogeneous polynomials it was recently shown that the bohnenblust hille inequality for complex scalars is hypercontractive this result interesting by itself has found direct striking applications in the solution of several important problems for multilinear mappings precise information on the asymptotic behavior of the constants of the bohnenblust hille inequality is of particular importance for applications in quantum information theory and multipartite bell inequalities in this paper using elementary tools we prove a quite surprising result the asymptotic growth of the constants in the multilinear bohnenblust hille inequality is optimal besides its intrinsic mathematical interest and potential applications to different areas the mathematical importance of this result also lies in the fact that all previous estimates and related results for the last 80 years such as for instance the multilinear version of the famous grothendieck theorem for absolutely summing operators always present constants c m s growing at an exponential rate of certain power of m

rough set based logics for multicriteria decision analysis in this paper we propose some decision logic languages for rule representation in rough set based multicriteria analysis the semantic models of these logics are data tables each of which is comprised of a finite set of objects described by a finite set of criteria attributes the domains of the criteria may have ordinal properties expressing preference scales while the domains of the attributes may not the validity support and confidence of a rule are defined via its satisfaction in the data table

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

hodge teichmuller planes and finiteness results for teichmuller curves we prove that there are only finitely many algebraically primitive teichmuller curves in the minimal stratum in each prime genus at least 3 the proof is based on the study of certain special planes in the first cohomology of a translation surface which we call hodge teichmuller planes we also show that algebraically primitive teichmuller curves are not dense in any connected component of any stratum in genus at least 3 the closure of the union of all such curves in a fixed stratum is equal to a finite union of affine invariant submanifolds with unlikely properties results of this type hold even without the assumption of algebraic primitivity combined with work of nguyen and the second author a corollary of our results is that there are at most finitely many nonarithmetic teichmuller curves in h 4

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

ranks of twists of elliptic curves and hilbert s tenth problem in this paper we investigate the 2 selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields we give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2 selmer rank and we give lower bounds for the number of twists with bounded conductor that have a given 2 selmer rank as a consequence under appropriate hypotheses we can find many twists with trivial mordell weil group and assuming the shafarevich tate conjecture many others with infinite cyclic mordell weil group using work of poonen and shlapentokh it follows from our results that if the shafarevich tate conjecture holds then hilbert s tenth problem has a negative answer over the ring of integers of every number field

a combined forecasting approach based on fuzzy soft sets forecasting the export and import volume in international trade is the prerequisite of a government s policy making and guidance for a healthier international trade development however an individual forecast may not always perform satisfactorily while combination of forecasts may result in a better forecast than component forecasts we believe the component forecasts employed in combined forecasts are a description of the actual time series which is fuzzy this paper attempts to use forecasting accuracy as the criterion of fuzzy membership function and proposes a combined forecasting approach based on fuzzy soft sets this paper also examines the method with data of international trade from 1993 to 2006 in the chongqing municipality of china and compares it with a combined forecasting approach based on rough sets and each individual forecast the experimental results show that the combined approach provided in this paper improves the forecasting performance of each individual forecast and is free from a rough sets approach s restrictions as well it is a promising forecasting approach and a new application of soft sets theory

soft semirings molodtsov introduced the concept of soft sets which can be seen as a new mathematical tool for dealing with uncertainty in this paper we initiate the study of soft semirings by using the soft set theory the notions of soft semirings soft subsemirings soft ideals idealistic soft semirings and soft semiring homomorphisms are introduced and several related properties are investigated

on the bohnenblust hille inequality and a variant of littlewoodʼs 4 3 inequality the search for sharp constants for inequalities of the type littlewood s 4 3 and bohnenblust hille has lately shown unexpected applications in many fields such as analytic number theory quantum information theory or in results on n dimensional bohr radii recent estimates obtained for the multilinear bohnenblust hille inequality for real scalars have been used as a crucial tool by a montanaro in order to solve problems in quantum xor games here among other results we obtain new upper bounds for the bohnenblust hille constants for complex scalars for bilinear forms we provide optimal constants of variants of littlewood s 4 3 inequality for real scalars when the exponent 4 3 is replaced by any r 4 3 we also prove that the optimal constants in real case are always strictly greater than those from the complex case

combination of interval valued fuzzy set and soft set the soft set theory proposed by molodtsov can be used as a general mathematical tool for dealing with uncertainty by combining the interval valued fuzzy set and soft set models the purpose of this paper is to introduce the concept of the interval valued fuzzy soft set the complement and and or operations are defined on the interval valued fuzzy soft sets the demorgan s associative and distribution laws of the interval valued fuzzy soft sets are then proved finally a decision problem is analyzed by the interval valued fuzzy soft set some numerical examples are employed to substantiate the conceptual arguments

on some new operations in soft set theory molodtsov introduced the theory of soft sets which can be seen as a new mathematical approach to vagueness in this paper we first point out that several assertions proposition 2 3 iv vi proposition 2 4 and proposition 2 6 iii iv in a previous paper by maji et al p k maji r biswas a r roy soft set theory comput math appl 45 2003 555 562 are not true in general by counterexamples furthermore based on the analysis of several operations on soft sets introduced in the same paper we give some new notions such as the restricted intersection the restricted union the restricted difference and the extended intersection of two soft sets moreover we improve the notion of complement of a soft set and prove that certain de morgan s laws hold in soft set theory with respect to these new definitions

some applications of hypercontractive inequalities in quantum information theory hypercontractive inequalities have become important tools in theoretical computer science and have recently found applications in quantum computation in this note we discuss how hypercontractive inequalities in various settings can be used to obtain fairly concise proofs of several results in quantum information theory a recent lower bound of lancien and winter on the bias achievable by local measurements which are 4 designs spectral concentration bounds for k local hamiltonians and a recent result of pellegrino and seoane sepulveda giving general lower bounds on the classical bias obtainable in multiplayer xor games

the size of selmer groups for the congruent number problem where np is the number of solutions of the congruence y x d2x mod p then ld s has an analytic continuation as an entire function on the complex plane the conjecture of birch and swinnerton dyer then states in particular that the rank r d of ed is equal to the order r d of ld s at s 1 this being the so called analytic rank while we cannot at present find r d in all cases we can at least determine whether or not ld 1 0 and hence conjecturally whether or not r d 0 moreover one has a functional equation for ld s which relates its values at s and 2 s via a sign change ed 1 one may deduce that 1 r d ed it would then follow from the conjecture of birch and swinnerton dyer that the rank is positive whenever ed 1 according to the calculations of birch and stephens 2 one has

ergodicity of billiard flows and quadratic differentials on considere un systeme billard de 2 objets de masses m 1 et m 2 on montre que pour un ensemble dense de paires m 1 m 2 ce systeme est ergodique

application of level soft sets in decision making based on interval valued fuzzy soft sets molodtsov s soft set theory was originally proposed as a general mathematical tool for dealing with uncertainty research on fuzzy soft set based decision making has received much attention in recent years this paper aims to give deeper insights into decision making involving interval valued fuzzy soft sets a hybrid model combining soft sets with interval valued fuzzy sets the concept called reduct fuzzy soft sets of interval valued fuzzy soft sets is introduced using reduct fuzzy soft sets and level soft sets flexible schemes for decision making based on weighted interval valued fuzzy soft sets are proposed and some illustrative examples are employed to show that the proposals presented here are not only more reasonable but more efficient in practical applications

explicit moduli spaces for congruences of elliptic curves we determine explicit birational models over mathbb q for the modular surfaces parametrising pairs of n congruent elliptic curves in all cases where this surface is an elliptic surface in each case we also determine the rank of the mordell weil lattice and the geometric picard number

multicriteria decision support using rules that represent rough graded preference relations the approach described in this paper aims to support multicriteria choice and ranking of actions when the input preference information acquired from the decision maker is a graded comprehensive pairwise comparison or ranking of reference actions it is based on decision rule preference model induced from a rough approximation of the graded comprehensive preference relation among the reference actions the set of decision rules applied to a new set of actions provides a graded fuzzy preference relation which can be exploited by weighted fuzzy net flow score or lexicographic fuzzy net flow score procedure to obtain a final recommendation in terms of the best choice or of the ranking

arithmetic of elliptic curves upon quadratic extension this paper is a study of variations in the rank of the mordell weil group of an elliptic curve e defined over a number field f as one passes to quadratic extensions k of f let s k be the selmer group for multiplication by 2 on e k in analogy with genus theory we describe s k in terms of various objects defined over f and the local norm indices dimf2 ft norm ah for each completion fv of f in particular we show that dim s k dim e k 2 has the same parity as zi we compute i when e has good or multiplicative reduction modulo v assuming that the 2 primary component of the tate shafare vitch group u1 k is finite as conjectured we obtain the parity of rank e k for semistable elliptic curves defined over q and parametrized by modular functions our parity results agree with those predicted analytically by the conjectures of birch and swinnerton dyer 1 introduction let e be an elliptic curve defined over a number field f our motivating question is this what can be said about variations in the rank of the mordell weil group e k over quadratic extensions k f dl 2 1 let e d denote the twist of e which becomes isomorphic to e over k but not over f concretely if we choose for e a model over f of the form y2 f x then a model for d is given by dy2 f x if a denotes the generator of gal k f then e f can be identified with the l eigenspace and e d f with the l eigenspace of a acting on e k it follows that rank e k rank e f rank e d f an equiv alent question therefore is to describe changes in the rank of e d f as d varies for certain specific curves defined over q this question has been discussed for

the square free sieve and the rank of elliptic curves let e be an elliptic curve over q a celebrated theorem of mordell asserts that e q the abelian group of rational points of e is finitely generated by the rank of e we mean the rank of e q thus the rank of e is positive if and only if e possesses an infinity of rational points relatively few general qualitative assertions can be made about the rank as e varies how large can the rank get although we expect that there are elliptic curves over q with arbitrarily high ranks this is presently unknown what is the average size of the rank recently brumer and mcguinness have reported on their study of 310 716 elliptic curves of prime conductor less than 108 where they have found that 20 06 of those curves have even rank 2 cf bm also see forthcoming work of brumer where subject to a number of standard conjectures he shows that 2 3 is an upper bound for the average rank for all elliptic curves over q ordered in terms of their faltings height what is the behavior of the rank over the family of twists of a given elliptic curve here one has three natural kinds of families of twists 1 quadratic twists one can take any elliptic curve and systematically twist it by all quadratic characters this is the type of family of elliptic curves we are concerned with in this paper specifically if e is an elliptic curve over q given by the weierstrass equation y2 x a x b and d is any squarefree integer the quadratic twist of e by d ed is given by the equation

fuzzy soft set theory applied to bck bci algebras abstract in the paper p k maji r biswas a r roy fuzzy soft sets j fuzzy math 9 3 2001 589 602 maji et al introduced the concept of fuzzy soft sets as a generalization of the standard soft sets and presented an application of fuzzy soft sets in a decision making problem the aim of this manuscript is to apply fuzzy soft set for dealing with several kinds of theories in bck bci algebras the notions of fuzzy soft bck bci algebras closed fuzzy soft ideals and fuzzy soft p ideals are introduced and related properties are investigated

pruning method for optimal solutions of intm intn decision making scheme abstract this paper aims to find a faster method for optimal solutions of feng et al s int m int n decision making scheme we first give theoretical characterizations of optimal decision sets then we develop a pruning method which filters out those objects that cannot be elements of any optimal decision sets in the beginning experimental results have shown that our method has higher efficiency in computing the optimal solutions of this scheme particularly when we are processing soft sets with a great quantity of data

teichmuller curves in moduli space eisenstein series and an application to triangular billiards there exists a teichmuller disc n containing the riemann surface ofy2 x n 1 in the genus n 1 2 teichmuller space such that the stabilizer of n in the mapping class group has a fundamental domain of finite poincare volume in n application is given to an asymptotic formula for the length spectrum of the billiard in isosceles triangles with angles n n n 2 n and to the uniform distribution of infinite billiard trajectories in the same triangles

injectivity radius and gonality of a compact riemann surface we obtain a sharp lower bound for the volumes of purely 1 dimensional complex analytic subvarieties in a geodesic tubular neighborhood of the diagonal of the cartesian product of a compact riemann surface with itself this leads to a lower bound of the seshadri number of the canonical line bundle of the cartesian product with respect to the diagonal as a consequence we obtain an upper bound for the hyperbolic injectivity radii of compact riemann surfaces of a fixed gonality in particular we obtain the limiting behavior of the gonalities of a tower of compact riemann surfaces we also give an application of our results to an invariant related to the ample cone of the symmetric product of a riemann surface

knowledge acquisition in incomplete information systems a rough set approach abstract this paper deals with knowledge acquisition in incomplete information systems using rough set theory the concept of similarity classes in incomplete information systems is first proposed two kinds of partitions lower and upper approximations are then formed for the mining of certain and association rules in incomplete decision tables one type of optimal certain and two types of optimal association decision rules are generated two new quantitative measures random certainty factor and random coverage factor associated with each decision rule are further proposed to explain relationships between the condition and decision parts of a rule in incomplete decision tables the reduction of descriptors and induction of optimal rules in such tables are also examined

billiards and teichmuller curves on hilbert modular surfaces this paper exhibits an infinite collection of algebraic curves isometrically embedded in the moduli space of riemann surfaces of genus two these teichmuller curves lie on hilbert modular surfaces parameterizing abelian varieties with real multiplication explicit examples constructed from l shaped polygons give billiard tables with optimal dynamical properties