ramsey type results for semi algebraic relations a k ary semi algebraic relation e on r d is a subset of r kd the set of k tuples of points in r d which is determined by a finite number of polynomial inequalities in kd real variables the description complexity of such a relation is at most t if d k t and the number of polynomials and their degrees are all bounded by t a set a subset of r d is called homogeneous if all or none of the k tuples from a satisfy e a large number of geometric ramseytype problems and results can be formulated as questions about finding large homogeneous subsets of sets in r d equipped with semi algebraic relations in this paper we study ramsey numbers for k ary semi algebraic relations of bounded complexity and give matching upper and lower bounds showing that they grow as a tower of height k 1 this improves upon a direct application of ramsey s theorem by one exponential and extends a result of alon pach pinchasi radoicic and sharir who proved this for k 2 we apply our results to obtain new estimates for some geometric ramsey type problems relating to order types and one sided sets of hyperplanes we also study the off diagonal case achieving some partial results

proof of the bandwidth conjecture of bollobas and komlos in this paper we prove the following conjecture by bollobas and komlos for every 0 and integers r 1 and there exists 0 with the following property if g is a sufficiently large graph with n vertices and minimum degree at least r 1 r n and h is an r chromatic graph with n vertices bandwidth at most n and maximum degree at most then g contains a copy of h

ramsey numbers of degenerate graphs a graph is d degenerate if all its subgraphs have a vertex of degree at most d we prove that there exists a constant c such that for all natural numbers d and r every d degenerate graph g of chromatic number r has ramsey number at most 2 d2 cr v g this solves a conjecture of burr and erd os from 1973

dissipative non self adjoint sturm liouville operators and completeness of their eigenfunctions abstract in this paper non self adjoint sturm liouville operators in weyl s limit circle case are studied we first determine all the non self adjoint boundary conditions yielding dissipative operators for each allowed sturm liouville differential expression then using the characteristic determinant the completeness of the system of eigenfunctions and associated functions for these dissipative operators is proved

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

on fuzzy congruence of a near ring module the aim of this paper is to introduce fuzzy submodule and fuzzy congruence of an r module near ring module to obtain the correspondence between fuzzy congruences and fuzzy submodules of an r module to define quotient r module of an r module over a fuzzy submodule and to obtain correspondence between fuzzy congruences of an r module and fuzzy congruences of quotient r module over a fuzzy submodule of an r module

a new view of fuzzy hypernear rings in this survey article we first introduce the concept of quasi coincidence of a fuzzy interval value with an interval valued fuzzy set this concept is a generalized concept of quasi coincidence of a fuzzy point within a fuzzy set by using this new idea we consider the interval valued q fuzzy sub hypernear rings hyperideals of a hypernear ring and hence a generalization of a fuzzy sub near ring ideal is given some related properties of fuzzy hypernear rings are described finally we consider the concept of implication based interval valued fuzzy sub hypernear rings hyperideals in a hypernear ring in particular the implication operators in lukasiewicz system of continuous valued logic are discussed

j self adjoint extensions of a class of hamiltonian differential systems abstract this paper is concerned with j self adjoint extensions for a class of hamiltonian differential systems the domains of the corresponding minimal and maximal operators are described a classification of limit types of the systems at each of endpoints is given a property of the regularity field of the corresponding minimal operator is derived and an expression of the j defect index of the minimal operator by those of the left and right minimal operators is presented based on them a characterization of all j self adjoint extensions for the hamiltonian system is obtained in terms of square integrable solutions of a certain equation which is generated by this hamiltonian system and its formal adjoint as a consequence characterizations of all j self adjoint extensions are given for the systems in some special cases in particular all j self adjoint extensions are characterized in terms of square integrable solutions of the system itself in certain special cases

the lower and upper approximations in a hypergroup this paper presents a relationship between rough sets and hypergroup theory we analyze the lower and upper approximations of a subset with respect to an invertible subhypergroup and we consider some particular situations moreover the notion of a rough subhypergroup is introduced finally fuzzy rough subhypergroups are introduced and characterized

two extensions of ramsey s theorem ramsey s theorem in the version of erd s and szekeres states that every 2 coloring of the edges of the complete graph on 1 2 n contains a monochromatic clique of order 12 log n in this paper we consider two well studied extensions of ramsey s theorem improving a result of rodl we show that there is a constant c 0 such that every 2 coloring of the edges of the complete graph on 2 3 n contains a monochromatic clique s for which the sum of 1 log i over all vertices i s is at least c log log log n this is tight up to the constant factor c and answers a question of erd s from 1981 motivated by a problem in model theory vaananen asked whether for every k there is an n such that the following holds for every permutation of 1 k 1 every 2 coloring of the edges of the complete graph on 1 2 n contains a monochromatic clique a1 a 2 1 a 2 a k 1 1 a k 1 that is not only do we want a monochromatic clique but the differences between consecutive vertices must satisfy a prescribed order alon and independently erd s hajnal and pach answered this question affirmatively alon further conjectured that the true growth rate should be exponential in k we make progress towards this conjecture obtaining an upper bound on n which is exponential in a power of k this improves a result of shelah who showed that n is at most double exponential in k

fuzzy γ hypernear rings the concept of c hypernear rings which is a generalization of c near rings and hypernear rings is introduced some related properties of c hypernear rings are described in particular three isomorphism theorems of c hypernear rings are established by using this new idea we consider the fuzzy hyperideals of c hypernear rings finally the fundamental relations between c hypernear rings and c near rings are discussed

on the strong limit point and dirichlet properties of second order differential expressions second order differential expressions of the form w 1 pf qf are considered at a singular end point some of the known relationships between properties such as dirichlet weak dirichlet and strong limit point are extended to incorporate an arbitrary positive weight function and complexvalued coefficients

pawlak s approximations in γ semihypergroups the c semihypergroup is a generalization of a semigroup a generalization of a semihypergroup and a generalization of a c semigroup in this paper we discuss the roughness of c subsemihypergroups and c hyperideals bi c hyperideals in c semihypergroups in addition we consider rough sets with respect to the idempotent regular relations in a quotient c semihypergroup the upper approximations with respect to green s equivalence relations are derived in the context of c subsemihypergroups and c hyperideals

on the boundary conditions characterizing j selfadjoint extensions of j symmetric operators the study of boundary value problems involving linear differentiai equations with real valued coefficients is by now a well established area of analysis on the other hand much less is known about the solvability of such problems when the coefficients or boundary conditions are known to be complex examples of this latter type arise naturally for example in nuclear physics e g the so called optical mode1 for low energy scattering 1 p ijo electromagnetic field theory dielectric waveguides with heat loss c f 19 i or the propagation of radio waves through inhomogeneous media s and elsewhere in cases like these the relevant differential expressions are no longer formally symmetric and hence the powerful methods associated with the spectral theory of selfadjoint operators are not available to facilitate the study of such problems glazman introduced in 4 the concept of a j symmetric operator in a complex hilbert space r let j be a given conjugation operator on 2 i e j is a conjugate linear involution with jx 3 u x for all x and y in 2 a closed densely defined linear operator t in 3 is said to be j symmetric if

topological hypergroupoids hypergroups are generalizations of groups if this binary operation is taken to be multivalued then we arrive at a hypergroup the motivation for generalization of the notion of group resulted naturally from various problems in non commutative algebra another motivation for such an investigation came from geometry in various branches of mathematics we encounter important examples of topologico algebraical structures like topological groupoids groups rings fields etc in this contribution various kinds of continuity of hyperoperations will be introduced namely pseudocontinuous strongly pseudocontinuous and continuous hyperoperations further the relationship between them is studied our aim is to generalize the concept of topological groupoid to topological hypergroupoid

on fuzzy subnear rings and ideals abstract the aim of this paper is to introduce the notion of a fuzzy subnear ring to study fuzzy ideals of a near ring and to give some properties of fuzzy prime ideals of a near ring

transposition hypergroups noncommutative join spaces the study of various kinds of algebraic hypergroups is unified in the theory of transposition hypergroups the hyperoperation in any hypergroup has two inverses given bya b x a xb andb a x a bx a transposition hypergroup is a hypergroup where transposition b a c d impliesad bc holds the algebra of transposition hypergroups is developed closed subhypergroupsnthat are reflexive a n n a have distinguished structural significance the quotient space of a transposition hypergroup modulo a reflexive closed subhypergroup forms a transposition hypergroup that is a polygroup then generalizations of the isomorphism theorems and the jordan holder theorem of group theory are obtained

normal fuzzy r subgroups in near rings in this paper we introduce the notion of a normal fuzzy r subgroup in a near ring and investigate some related properties

a conjecture of erdős on graph ramsey numbers abstract the ramsey number r g of a graph g is the minimum n such that every red blue coloring of the edges of the complete graph on n vertices contains a monochromatic copy of g determining or estimating these numbers is one of the central problems in combinatorics one of the oldest results in ramsey theory proved by erd s and szekeres in 1935 asserts that the ramsey number of the complete graph with m edges is at most 2 o m motivated by this estimate erd s conjectured more than a quarter century ago that there is an absolute constant c such that r g 2 c m for any graph g with m edges and no isolated vertices in this short note we prove this conjecture

maximal and minimal eigenvalues and their associated nonlinear equations the spectral theory of uniformly elliptic operators a under perturbations v giving rise to operators of the form hv a v x on a bounded or unbounded region such as schrodinger operators are considered suppose that v p is constrained but v is otherwise unspecified the theory of the potentials v that maximize or minimize the eigenvalues of hv is presented the optimizing potentials are typically determined by equations of the form u w x u cu u the optimization of eigenvalues also turns out to be related to the determination of the best constants in sobolev s inequality and in its one dimensional simplification to a classical oscillator problem with instanton properties

a new upper bound for diagonal ramsey numbers we prove a new upper bound for diagonal two colour ramsey numbers showing that there exists a constant c such that r k 1 k 1 k clog k log log k 2k k

hypergraph ramsey numbers the ramsey number rk s n is the minimum n such that every red blue coloring of the k tuples of an n element set contains a red set of size s or a blue set of size n where a set is called red blue if all k tuples from this set are red blue in this paper we obtain new estimates for several basic hypergraph ramsey problems we give a new upper bound for rk s n for k 3 and s fixed in particular we show that r3 s n 2 s 2 log which improves by a factor of ns 2 polylogn the exponent of the previous upper bound of erd s and rado from 1952 we also obtain a new lower bound for these numbers showing that there is a constant c 0 such that r3 s n 2 sn log n s 1 for all 4 s n for constant s it gives the first superexponential lower bound for r3 s n answering an open question posed by erd s and hajnal in 1972 next we consider the 3 color ramsey number r3 n n n which is the minimum n such that every 3 coloring of the triples of an n element set contains a monochromatic set of size n improving another old result of erd s and hajnal we show that r3 n n n 2 c log n finally we make some progress on related hypergraph ramsey type problems

fully simple semihypergroups abstract in this paper we consider the class of semihypergroups h such that all subsemihypergroups k h are simple and when k 3 the fundamental relation k is not transitive for these semihypergroups we prove that hyperproducts of elements in h have size 2 and the quotient semigroup h is trivial this last result allows us to completely characterize these semihypergroups in terms of a small set of simple semihypergroups of size 3 finally we solve a problem on strongly simple semihypergroups introduced in 11

a study on fuzzy interior ideals of γ semigroups in this paper we fuzzify the notions of interior ideals and characteristic interior ideals of a c semigroup and investigate some of their basic properties finally a characterization of a simple c semigroup in terms of fuzzy interior ideals is obtained

induced ramsey type theorems abstract we present a unified approach to proving ramsey type theorems for graphs with a forbidden induced subgraph which can be used to extend and improve the earlier results of rodl erd s hajnal promel rodl nikiforov chung graham and uczak rodl the proofs are based on a simple lemma generalizing one by graham rodl and rucinski that can be used as a replacement for szemeredi s regularity lemma thereby giving much better bounds the same approach can be also used to show that pseudo random graphs have strong induced ramsey properties this leads to explicit constructions for upper bounds on various induced ramsey numbers

polygroups derived from cogroups abstract a polygroup is a completely regular reversible in itself multigroup in the sense of dresher and ore 4 these systems occur naturally in the study of algebraic logic 1 2 some algebraic and combinatorial properties were developed in 3 the question of whether or not q 2 group the class of double quotients of groups is the same as the class of all chromatic polygroups is important for many reasons in this paper double quotients of groups are constructed in a natural way from cogroups cogroups were introduced by eaton 5 in an attempt to axiomatize d hypergroups i e systems obtainable from groups by right coset decompositions with respect to not necessarily normal subgroups eaton s axioms apply only to finite systems utumi 7 formulated a general notion and gave an example of a cogroup not isomorphic to a d hypergroup utumi s scalar partition hypergroups appear in section 2 the main results in section 3 show that polygroups derivable from cogroups are chromatic and they include all double quotients of groups

fuzzy hyperlattices in this paper we introduce the notion of fuzzy hyperlattices and establish connections between fuzzy hyperlattices and hyperlattices then we discuss the relations between the weak homomorphism of fuzzy hyperlattices and the weak homomorphism of corresponding hyperlattices moreover we define and analyze the hypercongruence on hyperlattices and the fuzzy strong hypercongruence on fuzzy hyperlattices and we study the interrelation between the fuzzy hypercongruence on fuzzy hyperlattices and the homomorphism of fuzzy hyperlattices also a relation between the fuzzy hypercongruence on fuzzy hyperlattices and the hypercongruence on corresponding hyperlattices is obtained finally we apply the hypercongruence and the fuzzy strong hypercongruence to construct quotient structures of hyperlattices and fuzzy hyperlattices respectively

on fuzzy isomorphism theorems of γ hyperrings in this paper we first derive three isomorphism theorems and a jordan holder theorem of c hyperrings by fuzzy c hyperideals we establish three fuzzy isomorphism theorems of c hyperrings