characterization of log del pezzo pairs via anticanonical models we give a characterization of log del pezzo pairs in terms of their anticanonical models as a first application we show that every non rational del pezzo surface with log canonical singularities is of picard number one and contains a unique singularity which is simple elliptic as a second application we verify a conjecture of smith and schwede in dimension two which says that every surface of globally f regular type is of fano type

on fuzzy linear representations of finite groups abstract a linear representation of a finite group g in a finite dimensional vector space v induces through zadeh s extension principle a function 9 from igto into igl v where gl v is the group of all linear automorphisms of v if w is a fuzzy subspace of v the group of all fuzzy linear automorphisms of w gl w is a subgroup of gl v w is said to be stable under the action of a fuzzy subgroup a of g if 9 a is a subset of gl w i e 9 a is zero at every f inside gl v and outside gl q if w is stable under the action of a then its support subspace is stable under the action of the support subgroup of a in the crisp sense finally we show that if there are two stable fuzzy subspaces one of them is contained in the other then the smaller one has a fuzzy direct summand in the bigger which is also a stable fuzzy subspace

relative canonical sheaves of a family of curves abstract in this paper we study the relative canonical sheaf of a relatively minimal fibration of curves of genus g 2 over a one dimensional regular scheme using the configurations of 2 chains we show that its m tensored product is base point free for any m 2 we utilize koszul cohomology to prove that the relative canonical algebra of the fibration is generated in degree up to five it is a generalization of k konno s work on the 1 2 3 conjecture of m reid

k fuzzy ideals in semirings abstract we introduce the k fuzzy ideal of a semiring also we construct an extension of a fuzzy ideal in a k semiring and study the quotient structure of a k semiring by a fuzzy ideal

a note on fuzzy relations and fuzzy groups abstract it is shown that for a group g if is a fuzzy subgroup of g for all fuzzy subgroups of g g then either g is a cyclic group of order p m for some m 1 or g z p where p is a prime conversely we prove that if either g is a cyclic group of order p m for some m 1 or g z p then is a fuzzy subgroup of g for any fuzzy subgroup of g g where p is a prime

fuzzy homomorphisms of rings abstract in this paper we introduce the concept of fuzzy homomorphisms of rings we prove for fuzzy homomorphisms analogues of the homomorphism theorems of ordinary ring homomorphisms

fuzzy relations and fuzzy groups abstract given a group s we consider fuzzy relations on s that is maps from s s into 0 1 of particular interest is to investigate conditions under which the fuzzy relation becomes a fuzzy subgroup on s s we prove that if is a fuzzy subset of s and is the strongest fuzzy relation on s that is a fuzzy relation on then is a fuzzy subgroup if and only if is a fuzzy subgroup a number of other results are obtained about the interrelationships between fuzzy relations on s including the weakest fuzzy relation and fuzzy subgroups on s s

fuzzy normal subgroups and fuzzy cosets abstract we introduce here the notion of 1 i a fuzzy normal subgroup and 2 ii a fuzzy coset we prove that for a group g a fuzzy subgroup is fuzzy normal if and only if it is constant on the conjugate classes of g further we show that the level subgroups of a fuzzy normal subgroup are all normal we obtain several results which are analogs of some basic theorems of group theory in particular a fuzzy version of lagrange s theorem is established

fuzzy groups some group theoretic analogs abstract we introduce the concepts of 1 order of a fuzzy subgroup of a finite group 2 a fuzzy abelian group and 3 a fuzzy solvable group we obtain some results involving these notions which are analogs of results from group theory also we prove analogs of two basic theorems of group theory namely cayley s theorem and lagrange s theorem

the cone of type a level 1 conformal blocks divisors abstract we prove that the type a level 1 conformal blocks divisors on m 0 n span a finitely generated full dimensional subcone of the nef cone each such divisor induces a morphism from m 0 n and we identify its image as a git quotient parameterizing configurations of points supported on a flat limit of veronese curves we show how scaling git linearizations gives geometric meaning to certain identities among conformal blocks divisor classes this also gives modular interpretations in the form of git constructions to the images of the hyperelliptic and cyclic trigonal loci in m g under an extended torelli map

m 0 n is not a mori dream space building on the work of goto nishida and watanabe on symbolic rees algebras of monomial primes we prove that the moduli space of stable rational curves with n punctures is not a mori dream space for n 133 this answers a question posed by hu and keel

congruence for rational points over finite fields and coniveau over local fields if the l adic cohomology of a projective smooth variety defined over a local field k with finite residue field k is supported in codimension 1 then every model over the ring of integers of k has a k rational point for k a p adic field this is proved in esnault 2007 theorem 1 1 if the model x is regular one has a congruence x k 1 modulo k for the number of k rational points esnault 2006 theorem 1 1 the congruence is violated if one drops the regularity assumption

fuzzy direct sums of fuzzy rings abstract in this paper we introduce the concepts of the fuzzy weak direct sum and the fuzzy complete direct sum of fuzzy subrings of commutative rings we prove some basic properties analogous to those of ordinary rings

tl subrings and tl ideals part 1 basic concepts abstract in 25 27 we put forward a rudimentary frame of the theory of tl subalgebras in 26 we established a theory of tl subgroups and tl subgroups in this paper we will continue our work in concern with basic ideas provided in 25 27 introduce the notions of tl subrings and tl ideals of a ring and study some of their properties where t is an arbitrary infinitely distributive t norm on a given complete brouwerian lattice l

on the effective cone of m g n abstract for every g 2 and n g 1 we exhibit infinitely many extremal effective divisors in m g n coming from the strata of abelian differentials

fuzzy relations on rings and groups abstract let s be any nonempty set a fuzzy relation on s is a fuzzy subset of s s in this paper we study fuzzy relations on rings and groups in particular we show that if and are fuzzy left right ideals of a ring r then is a fuzzy left right ideal of r r and conversely if is a fuzzy left right ideal of r r then either or is a fuzzy left right ideal of r an example is given to show that if is a fuzzy left right ideal of r r then and both need not be fuzzy left right ideals of r an example is also given to show that if is a fuzzy left right ideal of r r then the weakest fuzzy subset of r on which is a fuzzy relation need not be a fuzzy left right ideal of r we obtain similar results for groups we also show that certain results of bhattacharya and mukherjee 1985 are not true

basepoint freeness for nef and big line bundles in positive characteristic a necessary and sufficient condition is given for semi ampleness of a numerically effective nef and big line bundle in positive characteristic one application is to the geometry of the universal stable curve over mg specifically the semi ampleness of the relative dualizing sheaf in positive characteristic an example is given which shows this and the semi ampleness criterion fail in characteristic zero a second application is to mori s program for minimal models of 3 folds in positive characteristic namely to the existence of birational extremal contractions

fuzzy ideals and fuzzy subalgebras of lie algebras abstract a fuzzy idea l of a lie algebra l factors it into a lie algebra l l called a fuzzy quotient lie algebra this fuzzy algebra lie algebra is always a fuzzy partition of l and a fuzzy partition of l will be a fuzzy quotient lie algebra if and only if its associated fuzzy similarity relation is compatible with the lie algebra operations relationship between the operations of the lie algebras l i t and zadeh s extension principle are studied we also study homomorphisms of lie algebras and the relationship between them and fuzzy ideals and fuzzy subalgebras of the domains and the co domains of these homomorphisms finally solvability of fuzzy ideals and studied in defining these solvable fuzzy ideals we use zadeh s extension principle

fuzzy partitions and fuzzy quotient rings abstract the cosets of a fuzzy ideal m in a ring a form another ring a m called the fuzzy quotient ring of a and m we show that a m is always a fuzzy partition of a and a fuzzy partition of a will be a fuzzy quotient ring with respect to some fuzzy ideal if and only if its associated fuzzy similarity relation is compatible with the ring operations we also discuss relationships between the two operations of the fuzzy quotient ring a m and the extensions over ia of the two operations of a by zadeh s extension principle we study relationships between the level ring structures of a fuzzy quotient ring and the level relations of its associated fuzzy similarity relations we introduce and study a new notion of the factorization of a ring homomorphism through fuzzy quotients finally a relationship between fuzzy prime ideals and fuzzy similarity relation has been established

extremal effective divisors on overline mathcal m _ 1 n for every n ge 3 we exhibit infinitely many extremal effective divisors on overline mathcal m 1 n the deligne mumford moduli space parameterizing stable genus one curves with n ordered marked points

towards the ample cone of mg n in this paper we study the ample cone of the moduli space m g n of stable n pointed curves of genus g our motivating conjecture is that a divisor on m g n is ample iff it has positive intersection with all 1 dimensional strata the components of the locus of curves with at least 3g n 2 nodes this translates into a simple conjectural description of the cone by linear inequalities and as all the 1 strata are rational includes the conjecture that the mori cone is polyhedral and generated by rational curves our main result is that the conjecture holds iff it holds for g 0 more precisely there is a natural finite map r m0 2g n m g n whose image is the locus rg n of curves with all components rational any 1 strata either lies in rg n or is numerically equivalent to a family e of elliptic tails and we show that a divisor d is nef iff d e 0 and r d is nef we also give results on contractions i e morphisms with connected fibers to projective varieties of m g n for g 1 showing that any fibration factors through a tautological one given by forgetting points and that the exceptional locus of any birational contraction is contained in the boundary

on probabilistic n ary hypergroups abstract the t fuzzy n ary subhypergroups of an n ary hypergroup are defined by using triangular norms and some related properties are hence obtained in particular we consider the probabilistic version of n ary hypergroups by using random sets and show that the fuzzy n ary hypergroups defined by triangular norms are consequences of some probabilistic n ary hypergroups under certain conditions some results on n ary hypergroups recently given by davvaz and corsini are extended

fuzzy ideals and fuzzy bi ideals in fuzzy semigroups abstract the purpose of this paper is to introduce some basic concepts of fuzzy algebra such as fuzzy left right ideal and fuzzy bi ideal in fuzzy semigroup through the new approach of fuzzy space and fuzzy group introduced by dib 1994 our notion of fuzzy ideal and fuzzy bi ideal includes the classical concepts of fuzzy ideal and fuzzy bi ideal of ordinary semigroup many counterexamples are also given

effective cones of quotients of moduli spaces of stable n pointed curves of genus zero let x n m 0 n the moduli space of n pointed stable genus zero curves and let x n m be the quotient of x n by the action of s n m on the last n m marked points the cones of effective divisors ne 1 x n m m 0 1 2 are calculated using this upper bounds for the cones mov x n m generated by divisors with moving linear systems are calculated m 0 1 along with the induced bounds on the cones of ample divisors of m g and m g 1 as an application the cone ne 1 m 2 1 is analyzed in detail

fuzzy cosets and fuzzy normal subgroups abstract in this paper besides generalizing some of the results of mukherjee and bhattacharya 9 we investigate conditions under which a homomorphic image of a fuzzy normal subgroup is fuzzy normal we also prove that a fuzzy subgroup of order p n 1 of a group of order p n is a fuzzy normal subgroup if p is a prime

naive blowups and canonical birationally commutative factors in 2008 rogalski and zhang math z 259 2 433 455 2008 showed that if r is a strongly noetherian connected graded algebra over an algebraically closed field bbbk then r has a canonical birationally commutative factor this factor is up to finite dimension a twisted homogeneous coordinate ring b x mathcal l sigma here x is the projective parameter scheme for point modules over r as well as tails of points in mathrm qgr r as usual sigma is an automorphism of x and mathcal l is a sigma ample invertible sheaf on x we extend this result to a large class of noetherian but not strongly noetherian algebras specifically let r be a noetherian connected graded bbbk algebra where bbbk is an uncountable algebraically closed field let y infty denote the parameter space or stack or proscheme parameterizing r point modules and suppose there is a projective variety x that corepresents tails of points there is a canonical map p y infty rightarrow x if the indeterminacy locus of p 1 is 0 dimensional and x satisfies a mild technical assumption we show that there is a homomorphism g r rightarrow b x mathcal l sigma and that g r is up to finite dimension a naive blowup on x in the sense of keeler et al duke math j 126 3 491 546 2005 rogalski and stafford j algebra 318 2 794 833 2007 and satisfies a universal property we further show that the point space y infty is noetherian

invariance of plurigenera in this paper we give a proof of the following long conjectured result on the invariance of the plurigenera main theorem let p x d be a smooth projective family of compact complex manifolds parametrized by the open unit 1 disk d assume that the bers xt py1 t t 2 d are of general type then for every positive integer m the plurigenus dimc c xt mkxt is independent of t 2 d where kxt is the canonical line bundle of xt

spectrum of prime fuzzy ideals abstract a topological space called a spectrum of prime fuzzy ideals and denoted by fspec r is obtained by introducing a topology on the set of prime fuzzy ideals of a ring r the prime fuzzy ideals used here introduced by the author elsewhere are not necessarily two valued the topology introduced is compact and the usual nonfuzzy prime spectrum spec r is a subspace of fspec r and dense in it situations under which fuzzy spectrums of different rings are homeomorphic are discussed in particular fspec r is proved to be homeomorphic to fspec r frad r where frad r is the fuzzy prime radical of r the correspondence associating a ring r to the topological space fspec r is shown to define a contravarient functor from the category of rings with unity into the category of compact topological spaces

fuzzy implicative ideals in bck algebras abstract in this paper we introduce the concept of fuzzy implicative ideals in bck algebras and study some of their properties

on the projectivity of the moduli spaces of curves 1 there are basically three algebraic proofs of the projectivity of the moduli spaces of stable curves the original one by knudsen 8 9 the proof by mumford 12 and gieseker 5 which makes heavy use of the machinery of geometric invariant theory and the more recent one by viehweg 15 and kollar 10 which relies instead on the semipositivity of the direct images of powers of the relative dualizing sheaf and at least in kollar s version does not use geometric invariant theory at all in this note combining ideas from the above papers and from 4 we shall outline a proof of projectivity which we believe is simpler and more direct than any of the existing ones however while the proofs by mumford gieseker viehweg and kollar are applicable at least in principle to a wide variety of moduli problems ours uses in an essential way the peculiarities of the problem at hand and it is hard to see how it could be extended to other situations mumford s idea is to use the machinery of geometric invariant theory to directly construct the moduli space of stable curves as a projective quotient of the chow or hilbert scheme of pluricanonically embedded stable curves by the action of the projective linear group to be able to do so one needs to show that the k canonical images of stable curves are stable in the invariant theoretic sense for high k this is quite well understood although not really easy for smooth curves one notices that k canonically embedded stable curves are linearly stable and then proves that for smooth curves linear stability implies invariant theoretic stability in the singular case by contrast both mumford and gieseker have to rely on indirect arguments which are quite long and involved our main point is that one can avoid proving the invariant theoretic stability of singular stable curves provided one is willing to grant that moduli space exists as a complete algebraic space that this is the case incidentally is relatively easy to prove as in 10 we prove projectivity by applying to a suitable invertible sheaf one of the standard numerical ampleness criteria seshadri s criterion in our case the necessary numerical estimates are obtained using the techniques of 4 more precisely the invariant theoretic stability of linearly stable smooth curves is used to prove an inequality for families of stable curves which suitably interpreted says that the hypothesis of seshadri s criterion is satisfied for curves in moduli which are not contained in the boundary the case of curves lying in the boundary is then reduced to the previous one by standard elementary techniques in order to properly carry out this procedure which can be viewed as a sort of induction on the genus it is convenient to deal simultaneously with the moduli spaces of stable n pointed curves as well in a sense then our approach is to use invariant theoretic stability as a means of obtaining numerical inequalities rather than as a step in the construction of quotients by group actions

some results on l fuzzy modules abstract an l fuzzy quotient module of a module is redefined and a modular complete pseudo complemented lattice structure is given to the set of all l fuzzy modules of a given module some isomorphisms between the lattices or posets of l fuzzy modules are obtained finally a galois connection is established between two lattices of l fuzzy modules of homomorphic modules

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

hypersurfaces with defect abstract a projective hypersurface x p n has defect if h i x h i p n for some i n 2 n 2 in a suitable cohomology theory this occurs for example when x p 4 is not q factorial we show that hypersurfaces with defect tend to be very singular in characteristic 0 we present a lower bound on the tjurina number where x is allowed to have arbitrary isolated singularities for x with mild singularities we prove a similar result in positive characteristic as an application we obtain an estimate on the asymptotic density of hypersurfaces without defect over a finite field

correspondence theorem for fuzzy ideals abstract the ring r j of cosets of a fuzzy ideal j of a ring r is defined and shown to be isomorphic to a factor ring of r in a natural way if f r r is an epimorphism of rings a one to one order preserving correspondence is established between the fuzzy ideals of r and those of r which are constant on the kernel of f this generalizes the correspondence theorem of non fuzzy ideals

length functions and fuzzy groups fuzzy sets were introduced by zadeh 7 in 1965 and fuzzy groups by rosenfeld 4 in 1971 fuzzy groups were also considered by anthony and sherwood i in 1979 and in 1981 the theory was developed by das 2 as in the paper of das 2 we also show how a fuzzy subgroup of a group g is determined by a chain of subgroups of g however we wish to point out that fuzzy subgroups are constructed in a way which is analogous to that of constructing certain length functions of g wilkens s 61 thus in section 3 we consider the theory of length functions and we show the connection between certain length functions and fuzzy subgroups of a group in section 4 we apply the structural results of section 3 on finite solvable groups and we gain a generalization of a theorem of das 2 our notation is standard however we use the symbol e to denote the identity element of a group g and i denotes a real number we start with a preliminary section where we briefly analyse the definition of a fuzzy subgroup given by rosenfeld 4 naturally we use this definition of a fuzzy subgroup throughout the paper

on fuzzy h ideals in hemirings the fuzzy setting of a left h ideal in a hemiring is constructed and basic properties are investigated using a collection of left h ideals of a hemiring s fuzzy left h ideals of s are established the notion of a finite valued fuzzy left h ideal is introduced and its characterization is given fuzzy relations on a hemiring s are discussed

existence of mori fibre spaces for 3 folds in char p abstract we prove the following results for projective klt pairs of dimension 3 over an algebraically closed field of characteristic p 5 the cone theorem the base point free theorem the contraction theorem finiteness of minimal models termination with scaling existence of mori fibre spaces etc

moving codimension one subvarieties over finite fields we give the first examples of nef line bundles on smooth projective varieties over finite fields which are not semi ample more concretely we find smooth curves on smooth projective surfaces over finite fields such that the normal bundle has degree zero but no positive multiple of the curve moves in a family of disjoint curves this answers questions by keel and mumford the proof uses an obstruction theory in the spirit of homotopy theory which links the infinitely many obstructions to moving higher and higher multiples of a given codimension one subvariety on 3 folds over a finite field we find nef and big line bundles which are not semi ample finally we reprove some of the known positive results about semi ampleness over finite fields in topology the normal bundle of a submanifold determines a neighborhood of the submanifold up to isomorphism in particular the normal bundle of a codimension one submanifold is trivial if and only if the submanifold can be moved in a family of disjoint submanifolds in algebraic geometry however there are higher order obstructions to moving a given subvariety in this paper we develop an obstruction theory in the spirit of homotopy theory which gives some control over when a codimension one subvariety moves in a family of disjoint subvarieties even if a subvariety does not move in a family some positive multiple of it may we find a pattern linking the infinitely many obstructions to moving higher and higher multiples of a given subvariety as an application we find the first examples of line bundles l on smooth projective varieties over finite fields which are nef l has nonnegative degree on every curve but not semi ample no positive power of l is spanned by its global sections this answers questions by keel and mumford determining which line bundles are spanned by their global sections or more generally are semi ample is a fundamental issue in algebraic geometry if a line bundle l is semi ample then the powers of l determine a morphism from the given variety onto some projective variety one of the main problems of the minimal model program the abundance conjecture predicts that a variety with nef canonical bundle should have semi ample canonical bundle 15 conjecture

a characterization of fuzzy subgroups abstract an ordinary subgroup of a group g is 1 a subset of g 2 closed under the group operation in a fuzzy subgroup it is precisely these two notions that lose their deterministic character a fuzzy subgroup of a group g associates with each group element a number the larger the number the more certainly that element belongs to the fuzzy subgroup the closure property is captured by the inequality x y t x y in a rosenfeld s original definition t was the function minimum however any t norm t provides a meaningful generalization of the closure property two classes of fuzzy subgroups are investigated the fuzzy subgroups in one class are subgroup generated those in the other are function generated each fuzzy subgroup in these classes satisfies the above inequality with t given by t a b max a b 1 0 while the two classes look different each fuzzy subgroup in either is isomorphic to one in the other it is shown that a fuzzy subgroup satisfies the above inequality with t minimum if and only if it is subgroup generated of a very special type finally these notions are applied to some abstract pattern recognition problems

projectivity of the moduli space of stable log varieties and subadditivity of log kodaira dimension reference epfl article 231100doi 10 1090 jams 871 url http dx doi org 10 1090 jams 871 record created on 2017 09 19 modified on 2017 10 02

fuzzy sets fuzzy relations and fuzzy groups some interrelations abstract we define closure operator on the set of fuzzy relations on s and show that the closed hull of a fuzzy relation s is given by s s we define strong fuzzy relation and derive some results to investigate the relationship between closed strong and symmetric fuzzy relations we give a counterexample to show that the composition of two closed fuzzy relations need not be a closed fuzzy relation we also give a much simpler proof of theorem 4 2 of bhattacharya and mukherjee and provide counterexamples to show that their lemma 3 4 and theorem 3 1 of das do not always remain true

isomorphism theorems and fuzzy k ideals of k semirings we define and study the fuzzy maximal k ideal of a k semiring also we characterize the quotient k semiring r a of a k semiring r by a fuzzy maximal k ideal a and we have some isomorphism theorems in commutative k semirings and further we have some properties of the quotient k semiring r a of all cosets of a fuzzy semiprimary semiprime primary prime k ideal a in a commutative k semiring r

surfaces of globally f regular and f split type we prove that normal projective surfaces of dense globally f split type respectively globally f regular type are of calabi yau type respectively fano type

intersection theory of moduli space of stable n pointed curves of genus zero we give a new construction of the moduli space via a composition of smooth codimension two blowups and use our construction to determine the chow ring

projectivity of the witt vector affine grassmannian we prove that the witt vector affine grassmannian which parametrizes w k lattices in w k frac 1 p n for a perfect field k of characteristic p is representable by an ind perfect scheme over k this improves on previous results of zhu by constructing a natural ample line bundle along the way we establish various foundational results on perfect schemes notably h descent results for vector bundles

on base point free theorem and mori dream spaces for log canonical threefolds over the algebraic closure of a finite field the authors and d martinelli proved in algebra number theory 9 3 725 747 2015 the base point free theorem for big line bundles on a three dimensional log canonical projective pair defined over the algebraic closure of a finite field in this paper we drop the bigness condition when the characteristic is larger than five additionally we discuss mori dream spaces defined over the algebraic closure of a finite field

cone theorem via deligne mumford stacks we prove the cone theorem for varieties with lciq singularities using deformation theory of stable maps into deligne mumford stacks we also obtain a sharper bound on kx d degree of kx d negative extremal rays for projective mathbb q factorial log terminal threefold pairs x d

a characterization of l fuzzy prime ideals abstract in this paper the concepts of a l fuzzy prime and a l fuzzy completely prime ideal are given and some fundamental lemmas are proved in particular bijections among some of the l fuzzy prime ideals of two homomorphic rings are constructed also a characterization of a l fuzzy prime ideal of a ring is given

boundedness of pluricanonical maps of varieties of general type using the techniques of 20 and 10 we prove that certain log forms may be lifted from a divisor to the ambient variety as a consequence of this result following 22 we show that for any positive integer n there exists an integer rn such that if x is a smooth projective variety of general type and dimension n then phi rk x colon x dasharrow mathbb p h 0 mathcal o x rk x is birational for all r rn

generalized fuzzy quotient subgroups abstract in this paper we consider the fuzzy subgroup of a given group defined by generalized triangular norms and conorms introduced by yager 1986 the definitions of a anti fuzzy subgroup and a anti pseudo fuzzy subgroup are reviewed then a generalized fuzzy cayley theorem as well as a partial generalization of fuzzy lagrange theorem are given also a unified approach for studying fuzzy quotient subgroups is discussed some correspondence theorems for several kinds of fuzzy subgroups are proved and examples of generalized triangular norms conorms are given

on prime and primary fuzzy ideals and their radicals abstract the merits and demerits of various definitions of prime and primary fuzzy ideals are discussed and new definitions are suggested it is shown that the new definitions are more satisfactory the nil radical is defined as against the prime radical of a fuzzy ideal some theorems regarding the nil radical are proved it is shown that the two radicals coincide when the grade membership lattice is totally ordered

a canonical bundle formula a higher dimensional analogue of kodaira s canonical bundle formula is obtained as applications we prove that the log canonical ring of a klt pair with 3is finitely generated and that there exists an effectively computable natural number m such that mk x induces the iitaka fibering for every algebraic threefold x with kodaira dimension 1

the augmented base locus of real divisors over arbitrary fields we show that the augmented base locus coincides with the exceptional locus i e null locus for any nef mathbb r cartier divisor on any scheme projective over a field of any characteristic next we prove a semi ampleness criterion in terms of the exceptional locus generalizing a result of keel we also discuss some problems related to augmented base loci of log divisors

not so fuzzy fuzzy ideals abstract a characterization of rings which are both artinian and noetherian is obtained they are proved to be the rings whose fuzzy ideals assume only finitely many distinct values prime fuzzy ideals have been defined and conditions have been obtained under which they have only two distinct values

a special cayley octad first published in transactions of the american mathematical society 370 pp 5359 5379 published by the american mathematical society 2017 by a prendergast smith

normality and congruence in fuzzy subgroups abstract it is proved that normal fuzzy subgroups and fuzzy congruence relations determine each other in a group theoretic situation this is analogous to the crisp set theoretic case of groups

closure systems and l subalgebras abstract let x be a set and l a class of l subsets of x l is a closure system if it is closed with respect to greatest lower bounds if is an l subset then the g l b tf of g l g is said to be generated by we examine closure systems of l subalgebras and propose formulas for the l subalgebra of a required type generated by a given l subset

fuzzy congruences and fuzzy normal subgroups abstract a complete description of the congruences on a group in terms of its normal subgroups can be seen in papers by a rosenfeld and w j howie in this paper we shall introduce the concept of fuzzy congruences on a groupoid and a group and characterize fuzzy congruences on a group in terms of fuzzy normal subgroups

factoring 3 fold flips and divisorial contractions to curves flips flops and divisorial contractions are the elementary birational maps of the minimal model program divisorial contractions are the higher dimensional analog of the inverse of blowing up a smooth point on a surface flips are instead a new operation which consists of a surgery in codimension 2 which replaces certain kx negative curves by some kx positive curves flops are also surgeries in codimension 2 but they replace certain kx trivial curves by some new kx trivial curves not surprisingly the geometry of divisorial contractions is better understood than that of flips and flops in fact the proof of the existence of flips and flops cf 1 7 is very abstract and gives little insight to their geometry in dimension 3 these operations are reasonably well understood kollar has shown 23 that if x 99k x is a flop of terminal 3 folds then x and x have analytically isomorphic singularities in 34 mori gives a precise understanding of the geometry of all flips with irreducible flipping curve we would like to remark that 3 fold divisorial contractions may be subdivided into two classes those that map the divisor to a curve and those that map the divisor to a point divisorial contractions to a point are classified in 4 14 18 19 20 21 and 31 there are also several partial results on divisorial contractions to a curve cf 5 37 38 39 40 and 41 the purpose of this talk is try to understand what happened to the singularities under these elementary birational map in dimension three and how can we decompose these maps into simplers maps that explicit description of singularities is possible in fact we can factor flips and divisorial contractions to curves via the simpler operations

fuzzy subgroup some characterizations ii abstract we consider an analysis of fuzzy subgroups in terms of the corresponding family of level subgroups given a finite chain of subgroups of a group g we prove that there exists a fuzzy subgroup of g whose level subgroups are exactly the subgroups of this chain as a corollary we obtain an interpretation of the number of chains of subgroups of a group g in which a subgroup h is a member when the group g is a supersolvable group some further interpretation of this number is obtained