on the arithmetical rank of certain segre embeddings we study the number of set theoretically defining equations of segre products of projective spaces times certain projective hypersurfaces extending results by singh and walther meanwhile we prove some results about the cohomological dimension of certain schemes in particular we solve a conjecture of lyubeznik about an inequality involving the cohomological dimension and the etale cohomological dimension of a scheme in the characteristiczero case and under a smoothness assumption furthermore we show that a relationship between depth and cohomological dimension discovered by peskine and szpiro in positive characteristic also holds true in characteristic zero up to dimension three

on the ideal theory of graphs we study algebras defined by finite sets g m1 mq of monomials of a polynomial ring r there are two basic algebras i k g k m1 mq the k subalgebra of r spanned by the mi and ii the quotient ring r i g where i g m1 mq they come together in the construction of the rees algebra r i g of the ideal i g the emphasis is almost entirely on sets of squarefree monomials of degree two and their attached graphs the main results are assertions about the cohen macaulay behaviour of the koszul homology of i g and how normality or cohen macaulayness of one of the algebras can be read off the properties of the graph or in the other algebra

symbolic powers of sums of ideals let i and j be nonzero ideals in two noetherian algebras a and b over a field k let i j denote the ideal generated by i and j in a otimes k b we prove the following expansion for the symbolic powers begin aligned i j n sum i j n i i j j end aligned if a and b are polynomial rings and if mathrm char k 0 or if i and j are monomial ideals we give exact formulas for the depth and the castelnuovo mumford regularity of i j n which depend on the interplay between the symbolic powers of i and j the proof involves a result of independent interest which states that the induced map mathrm tor i a k i n rightarrow mathrm tor i r k i n 1 is zero for any homogeneous ideal i and i ge 0 n ge 0 we also investigate other properties and invariants of i j n such as the equality between ordinary and symbolic powers the waldschmidt constant and the cohen macaulayness

ordinary and symbolic powers are golod abstract let s be a positively graded polynomial ring over a field of characteristic 0 and i s a proper graded ideal in this note it is shown that s i is golod if i 2 i here i denotes the ideal generated by all the partial derivatives of elements of i we apply this result to find large classes of golod ideals including powers symbolic powers and saturations of ideals

the integral closure of subrings associated to graphs letgbe a graph on the vertex setv andr k v a polynomial ring over a fieldk we give a very explicit description of the integral closure of the monomial subringk g r generated by the squarefree monomials of degree two defining the edges of the graphg

complete classification of shape functions of self similar solutions abstract in this paper we study some asymptotic profiles of shape functions of self similar solutions to the initial boundary value problem with neumann boundary condition for the generalized kpz equation u t u x x u x q where q is positive number the shapes of solutions of the corresponding nonlinear ordinary differential equation are of very different nature the properties depend on the critical value q 1 3 2 2 and initial data as usual

analytic spread modulo an element and symbolic rees algebras let t n be a d dimensional regular local ring p a prime ideal of t having height equal d 1 and f p within this setting we are interested in computing the analytic spread of p f t we have two reasons for seeking this analytic spread the first is to identify prime ideals in local hypersurface rings whose analytic spreads and heights are the same the second is to aid in a better understanding of the symbolic rees algebra p t of p the ideal pen is the nth symbolic power of p our interest in hypersurface rings essentially originated in hug there we were interested in determining when powers and symbolic powers are equal for a prime ideal in a local hypersurface ring for a prime ideal q adjacent to the maximal ideal i e height m q l the only time such an equality can occur for all large powers is when the analytic spread of q equals the height of q equality does not however necessarily occur under this analytic spread condition cf hu2 example 2 lo in order to force equality for all powers one must make additional assumptions on the associated graded ring of p f t q cf hu2 theorem 2 11 section 2 of the present article contains some general methods for recognizing the analytic spread of p f t and section 3 contains some illustrative examples where the analytic spread of p f t equals the height of p f t the noetherian property of the symbolic rees algebra has created recent interest due to a question of cowsik in connection with his result in c that p t noetherian implies p is a set theoretic complete intersection when t is a regular local ring and dim t p 1 cowsik asked whether p t is noetherian for every prime ideal in a regular local ring an example due to p roberts ro showed the answer is no in general but the question has continued to draw attention hl h2 kr 0 e the

average twin prime conjecture for elliptic curves let e be an elliptic curve over bbb q in 1988 n koblitz conjectured a precise asymptotic for the number of primes p up to x such that the order of the group of points of e over bbb f p is prime this is an analogue of the hardy littlewood twin prime conjecture in the case of elliptic curves koblitz s conjecture is still widely open in this paper we prove that koblitz s conjecture is true on average over a two parameter family of elliptic curves one of the key ingredients in the proof is a short average distribution result of primes in the style of barban davenport halberstam where the average is taken over prime differences and over arithmetic progressions

subvarieties of moduli space determined by finite groups acting on surfaces suppose the finite group g acts as orientation preserving homeomorphisms of the oriented surface s of genus g this determines an irreducible subvariety m g g of the moduli space m g of riemann surfaces of genus g consisting of all surfaces with a group g 1 of holomorphic homeomorphisms of the same topological type as g this family is determined by an equivalence class of epimorphisms from a fuchsian group to g whose kernel is isomorphic to the fundamental group of s to examine the singularity of m g along this family one needs to know the full automorphism group of a generic element of m g g

cohen macaulayness of monomial ideals and symbolic powers of stanley reisner ideals abstract we present criteria for the cohen macaulayness of a monomial ideal in terms of its primary decomposition these criteria allow us to use tools of graph theory and of linear programming to study the cohen macaulayness of monomial ideals which are intersections of prime ideal powers we can characterize the cohen macaulayness of the second symbolic power or of all symbolic powers of a stanley reisner ideal in terms of the simplicial complex these characterizations show that the simplicial complex must be very compact if some symbolic power is cohen macaulay in particular all symbolic powers are cohen macaulay if and only if the simplicial complex is a matroid complex we also prove that the cohen macaulayness can pass from a symbolic power to another symbolic powers in different ways

equality of ordinary and symbolic powers of stanley reisner ideals abstract this paper studies properties of simplicial complexes with the equality i m i m for a given m 2 the main results are combinatorial characterizations of such complexes in the two dimensional case it turns out that there exists only a finite number of complexes with this property and that these complexes can be described completely as a consequence we are able to determine all complexes for which i m is cohen macaulay for some m 2 in particular there are complexes with i 2 i 2 or i 3 i 3 but i m i m for all m 4 and that if i m i m for some m 4 then i m i m for all m 1 similarly there are complexes for which i 2 is cohen macaulay but i m is not cohen macaulay for all m 3 and if i m is cohen macaulay for some m 3 then i is a complete intersection

hyperelliptic curves and their invariants geometric arithmetic and algorithmic aspects we apply classical invariant theory of binary forms to explicitly characterize isomorphism classes of hyperelliptic curves of small genus and conversely propose algorithms for reconstructing hyperelliptic models from given invariants we focus on genus 3 hyperelliptic curves both geometric and arithmetic aspects are considered

depth functions of symbolic powers of homogeneous ideals this paper addresses the problem of comparing minimal free resolutions of symbolic powers of an ideal our investigation is focused on the behavior of the function mathrm depth r i t dim r mathrm pd i t 1 where i t denotes the t th symbolic power of a homogeneous ideal i in a noetherian polynomial ring r and mathrm pd denotes the projective dimension it has been an open question whether the function mathrm depth r i t is non increasing if i is a squarefree monomial ideal we show that mathrm depth r i t is almost non increasing in the sense that mathrm depth r i s ge mathrm depth r i t for all s ge 1 and t in e s where begin aligned e s bigcup i ge 1 t in mathbb n i s 1 1 le t le is end aligned which contains all integers t ge s 1 2 1 the range e s is the best possible since we can find squarefree monomial ideals i such that mathrm depth r i s mathrm depth r i t for t not in e s which gives a negative answer to the above question another open question asks whether the function mathrm depth r i t is always constant for t gg 0 we are able to construct counter examples to this question by monomial ideals on the other hand we show that if i is a monomial ideal such that i t is integrally closed for t gg 0 e g if i is a squarefree monomial ideal then mathrm depth r i t is constant for t gg 0 with begin aligned lim t rightarrow infty mathrm depth r i t dim r dim oplus t ge 0 i t mathfrak m i t end aligned our last result which is the main contribution of this paper shows that for any positive numerical function phi t which is periodic for t gg 0 there exist a polynomial ring r and a homogeneous ideal i such that mathrm depth r i t phi t for all t ge 1 as a consequence for any non negative numerical function psi t which is periodic for t gg 0 there is a homogeneous ideal i and a number c such that mathrm pd i t psi t c for all t ge 1

the cohen lenstra heuristic methodology and results in number theory great efforts have been undertaken to study the cohen lenstra probability measure on the set of all finite abelian p groups on the other hand group theorists have studied a probability measure on the set of all partitions induced by the probability that a randomly chosen n n matrix over fp is contained in a conjugacy class associated with this partitions for n this paper shows that both probability measures are identical as a consequence a multitude of results can be transferred from each theory to the other one the paper contains a survey about the known methods to study the probability measure and about the results that have been obtained so far from both communities

on the finite generation of symbolic blow ups symbolic blow up p n s a noetherian ring if s is noetherian and dim n 0 rip l cowsik observed that p must be a set theoretic complete intersection if r is not regular there is no hope in general of s being noetherian in fact if r is any normal two dimensional local ring and p is a height one prime ideal of r which is not torsion in the class group of r then s is never noetherian this idea is behind the rees counterexample 6 to the generalized zariski question as far as the author knows there are no known counter examples to the question of cowsik the most trivial case is the following if r is nagata and

cohen macaulayness of large powers of stanley reisner ideals abstract we prove that for m 3 the symbolic power i m of a stanley reisner ideal is cohen macaulay if and only if the simplicial complex is a matroid similarly the ordinary power i m is cohen macaulay for some m 3 if and only if i is a complete intersection these results solve several open questions on the cohen macaulayness of ordinary and symbolic powers of stanley reisner ideals moreover they have interesting consequences on the cohen macaulayness of symbolic powers of facet ideals and cover ideals

on fields of moduli of curves the field of moduli k of a curve x a priori defined over the separable closure k5 of k need not be a field of definition this paper shows that the obstruction is essentially the same as the obstruction to k being a field of definition of the cover x x aut x using previous results of debes douai we then obtain a cohomological measure of the obstruction this yields concrete criteria for the field of moduli to be a field of definition an interesting application is the following local global principle if a curve x together with all of its automorphisms is defined over p for all primes p then it is defined over

a probabilistic approach toward conjugacy classes in the finite general linear and unitary groups abstract the conjugacy classes of the finite general linear and unitary groups are used to define probability measures on the set of all partitions of all natural numbers probabilistic algorithms for growing random partitions according to these measures are obtained these algorithms are applied to prove group theoretic results which are typically proved by techniques such as character theory and moebius inversion among the theorems studied are steinberg s count of unipotent elements rudvalis and shinoda s work on the fixed space of a random matrix and lusztig s count of nilpotent matrices of a given rank generalizations of these algorithms based on macdonald s symmetric functions are given

combinatorial symbolic powers abstract symbolic powers are studied in the combinatorial context of monomial ideals when the ideals are generated by quadratic squarefree monomials the generators of the symbolic powers are obstructions to vertex covering in the associated graph and its blowups as a result perfect graphs play an important role in the theory dual to the role played by perfect graphs in the theory of secants of monomial ideals we use grobner degenerations as a tool to reduce questions about symbolic powers of arbitrary ideals to the monomial case among the applications are a new unified approach to the grobner bases of symbolic powers of determinantal and pfaffian ideals

how to compute the stanley depth of a monomial ideal abstract let j i be monomial ideals we show that the stanley depth of i j can be computed in a finite number of steps we also introduce the fdepth of a monomial ideal which is defined in terms of prime filtrations and show that it can also be computed in a finite number of steps in both cases it is shown that these invariants can be determined by considering partitions of suitable finite posets into intervals

on a conjecture of murthy abstract this article concerns murthy s conjecture on complete intersections made in 1975 the sole breakthrough on this conjecture has still been the result proved by mohan kumar in 1978 the conjecture is open in general in this article we improve mohan kumar s bound when the base field is f p as an application we prove that any local complete intersection surface in the affine space a f p d is a set theoretic complete intersection generalizing a result of bloch murthy szpiro

zero cycles and projective modules let a be a reduced affine k algebra of dimension n over an algebraically closed field k let fnko a denote the sub group of ko a generated by the images of all the residue fields of all smooth maximal ideals of height n for a module of finite projective dimension m let m denote the image of m in ko a for a projective a module p of rank n we define the nth chern class of p to be cn p e l i aip where p is the dual of p suppose fnko a has no n 1 torsion then our main result th 3 8 is that p has a free direct summand of rank one if and only if cn p 0 when the characteristic of k is zero or a is normal and n 3 it is known that fnko a is torsion free le sr hence our theorem is applicable in these cases also when a is regular fnko a coincides with chn x the chow group of zero cycles of spec a and cn p coincides with the usual nth chern class as defined by grothendieck see fu when n 3 and a is regular this result was proved in mkm when n 3 and the characteristic of k is not equal to two this result in mkm was extended to the singular case by m levine in this paper we extend the results of mkm to all dimensions with the assumption stated above on torsion in fnko a we first show that cor 3 4 if i c a is a local complete intersection of height n then a i is zero in ko a if and only if i is a complete intersection corollary 3 4 together with a result of mohan kumar mk cor 1 9 here at once gives theorem 3 8 in section 1 we give some preliminaries and generalize results in mk2 these results are crucially used in the rest of the paper the basic ideas in section 1 are all taken from mk2 in section 2 we prove theorem 2 2 which strengthens a result of boratynski bo for local complete intersections with trivial co normal bundle theorem 2 2 is one of the crucial ingredients in the proofs of the results in section 3 as an amusing application of theorem 2 2 we give a new proof of a theorem of srinivas sr about torsion in zero cycles

the golod property for products and high symbolic powers of monomial ideals abstract we show that for any two proper monomial ideals i and j in the polynomial ring s k x 1 x n the ring s i j is golod we also show that if i is squarefree then for large enough k the quotient s i k of s by the kth symbolic power of i is golod as an application we prove that the multiplication on the cohomology algebra of some classes of moment angle complexes is trivial

on triviality of the euler class group of a deleted neighbourhood of a smooth local scheme let k be an infinite perfect field and let x specr where r is the localization of a k algebra b of finite type with respect to a regular prime ideal p specb letd be a divisor onx with a decompositiond i 1di into its irreducible components di such that each di is nonsingular and they intersect transversally consider y x d a deleted neighbourhood of the closed point x0 of x let e be a vector bundle on y in this set up e is trivial in the following cases i dim x 3 gabber ga1 ii t 1 bhatwadekar rao br iii ranke min dimr 2 t rao ra1 the above exposition is borrowed from a paper of nisnevich ni which contains some beautiful applications of these purity theorems in the same set up now let y y be a closed point and let my be the corresponding maximal ideal of oy y it is natural to ask whether my is a complete intersection observing that the conormal bundle my my is trivial we consider a more general situation let i be the defining ideal of a closed subscheme z of y such that the conormal bundle i i2 defined on z is trivial is i a complete intersection the questions we are addressing in this paper are a little stronger in nature can a given basis of i i2 be lifted to a minimal set of generators of i we remark that with the hypothesis on x each component di is defined by a single equation fi which vanishes at the closed point x0 of x let m be the maximal ideal at x0 the condition of transversality of intersection of di s is equivalent to saying that the images of f1 ft in the a m vector space m m2 are linearly independent keeping this discussion in mind we now give an excerpt of our main results in commutative algebraic terms see 4 2 4 3 4 4

resurgences for ideals of special point configurations in bf p n coming from hyperplane arrangements abstract symbolic powers of ideals have attracted interest in commutative algebra and algebraic geometry for many years with a notable recent focus on containment relations between symbolic powers and ordinary powers see for example 3 7 13 16 18 19 20 to cite just a few several invariants have been introduced and studied in the latter context including the resurgence and asymptotic resurgence 3 15 there have been exciting new developments in this area recently it had been expected for several years that i n r n 1 i r should hold for the ideal i of any finite set of points in p n for all r 0 but in the last year various counterexamples have now been constructed see 11 17 8 all involving point sets coming from hyperplane arrangements in the present work we compute their resurgences and obtain in particular the first examples where the resurgence and the asymptotic resurgence are not equal

stability of depths of symbolic powers of stanley reisner ideals abstract we give a bound for n 0 i from which the depth function depth r i n of the quotient ring by symbolic powers of a squarefree monomial ideal i stabilizes in the unmixed codimension two case we show that depth r i n is a non increasing function of n and use this property to provide a sharp bound for n 0 i

asymptotic linear bounds for the castelnuovo mumford regularity we prove asymptotic linear bounds for the castelnuovo mumford regularity of certain filtrations of homogeneous ideals whose rees algebras need not be noetherian

the theory of d sequences and powers of ideals in this paper we study the question of finding depth r i given an ideal i in a commutative ring this problem is difficult in general even for simple examples depth r z can be difficult to compute brodmann has shown that for any commutative noetherian ring depth r z become stable for large n finding this value however is quite hard this problem originally arose through consideration of the following example suppose k is a field and x xij is an n

computing the stanley depth in the first chapter some definitions and necessary results from commutative algebra are given also some detail are given which leads us towards the stanley decompositions of multigraded s modules where s is a polynomi ring in n varables over a field f in the end of the chapter we give some important results about the stanledecompositions and stanley conjecture in chapter 2 for the given monomial primary ideals and of s we gave an upper bound for the stanley depth of s which is reached if are irreducible also we showed that stanley conjecture holds for i being some irreducible monomial ideals of s these results are published in our paper 23 for integers 1 n consider the ideal i n s in chapter 3 we gave an upper bound for the stanley depth of the ideal we gave similar upper bounds for the stanley depth of the ideal where in 2 is the square free veronese ideal of degree 2 in n variables these results are from our paper 11

factoring integers with elliptic curves this paper is devoted to the description and analysis of a new algorithm to factor positive integers it depends on the use of elliptic curves the new method is obtained from pollard s p 1 method proc cambridge philos soc 76 1974 521 528 by replacing the multiplicative group by the group of points on a random elliptic curve it is conjectured that the algorithm determines a non trivial divisor of a composite number n in expected time at most k p log n 2 where p is the least prime dividing n and k is a function for which log k x 2 o 1 log x log log x for x x o in the worst case when n is the product of two primes of the same order of magnitude this is

a probabilistic approach to conjugacy classes in the finite symplectic and orthogonal groups abstract markov chains are used to give a purely probabilistic way of understanding the conjugacy classes of the finite symplectic and orthogonal groups in odd characteristic as a corollary of these methods one obtains a probabilistic proof of steinberg s count of unipotent matrices and generalizations of formulas of rudvalis and shinoda

symbolic powers of monomial ideals and vertex cover algebras abstract we introduce and study vertex cover algebras of weighted simplicial complexes these algebras are special classes of symbolic rees algebras we show that symbolic rees algebras of monomial ideals are finitely generated and that such an algebra is normal and cohen macaulay if the monomial ideal is squarefree for a simple graph the vertex cover algebra is generated by elements of degree 2 and it is standard graded if and only if the graph is bipartite we also give a general upper bound for the maximal degree of the generators of vertex cover algebras

non uniqueness of solutions to the cauchy problem for semilinear heat equations with singular initial data the cauchy problem for semilinear heat equations with singular initial data is studied where n 2 0 is a parameter and a 0 a 0 we show that when p n 2 n and n 2 p n 2 there exists a positive constant such that the problem has two positive self similar solutions and with if and no positive self similar solutions if furthermore for each fixed and in l rn as 0 where w0 is a non unique solution to the problem with zero initial data which is constructed by haraux and weissler

symbolic powers of monomial curves the answer is positive for ideal theoretic local complete intersection e g smooth curves s mk and for arbitrary curves if char k is non zero c n problem 1 remains open for most singular curves over say c in fact we look at an algebraic problem related to problem 1 let p be a prime ideal in a commutative noetherian ring a denote by s p the direct sum ora0 p of the symbolic powers of p see section 0 for definitions which is canonically a graded a algebra it is called the symbolic rees algebra of p or of x if x is the variety defined by p in spec a

standard graded vertex cover algebras cycles and leaves the aim of this paper is to characterize simplicial complexes which have standard graded vertex cover algebras this property has several nice consequences for the squarefree monomial ideals defining these algebras it turns out that such simplicial complexes are closely related to a range of hypergraphs which generalize bipartite graphs and trees these relationships allow us to obtain very general results on standard graded vertex cover algebras which cover previous major results on rees algebras of squarefree monomial ideals

on the arithmetical rank of monomial ideals the arithmetical rank of monomial ideals has been studied by several authors in 12 131 for example the result of the above theorem is proved in some special cases on the other hand in 8 we stated the conjecture to the effect that every algebraic set l c p of pure codimension t can be set theoretically cut out by n n t 1 hypersurfaces supporting evidence for this conjecture is as follows

a very singular solution for the slow diffusion equation with nonlinear convection abstract we here investigate an existence and uniqueness of the nontrivial nonnegative solution of a nonlinear ordinary differential equation f m r f f f q 0 satisfying a specific decay rate lim r r f r 0 with 1 2 q m 1 and q m 2 q m 1 here m 1 m q m 1 and 1 or 1 such a solution arises naturally when we study a very singular solution for a slow diffusion equation with nonlinear convection u t u m x x u q x defined either on the whole real line or on the half line 0

cohen macaulayness of powers of two dimensional squarefree monomial ideals abstract two dimensional squarefree monomial ideals can be seen as the stanley reisner ideals of graphs the main results of this paper are combinatorial characterizations for the cohen macaulayness of ordinary and symbolic powers of such an ideal in terms of the associated graph

counting hyperelliptic curves abstract we find a closed formula for the number hyp g of hyperelliptic curves of genus g over a finite field k f q of odd characteristic these numbers hyp g are expressed as a polynomial in q with integer coefficients that depend on g and the set of divisors of q 1 and q 1 as a by product we obtain a closed formula for the number of self dual curves of genus g a hyperelliptic curve is defined to be self dual if it is k isomorphic to its own hyperelliptic twist

quasi finite modules and asymptotic prime divisors abstract let a be a noetherian ring j a an ideal and c a finitely generated a module in this note we would like to prove the following statement let i n n 0 be a collection of ideals satisfying i i n j n for all n ii j s i s i r s for all r s 0 and iii i n i m whenever m n then ass a i n c j n c is independent of n for n sufficiently large note that the set of prime ideals n 1 ass a i n c j n c is finite so the issue at hand is the realization that the primes in ass a i n c j n c do not behave periodically as one might have expected say if n 0 i n were a noetherian a algebra generated in degrees greater than one we also give a multigraded version of our results

a note on complete intersections let r be a regular local ring and let r t be a polynomial algebra in one variable over r in this paper the author proves that every maximal ideal of r t is complete intersection in each of the following cases 1 r is a local ring of an affine algebra over an infinite perfect field 2 r is a power series ring over a field introduction let f be a regular local ring let r t be a polynomial algebra in one variable over r in d g the following question has been asked question is every maximal ideal of r t complete intersection in this paper we prove that the answer to the above question is affirmative in each of the following cases 1 f is a local ring of an affine algebra over an infinite perfect field 2 f is a power series ring over a field this paper is divided into three sections in 1 we fix notations and state a theorem without proof which is used in 2 and 3 in 2 we prove some lemmas and propositions which are used in proving the result when f is a local ring of an affine algebra 3 deals with the power series case 1 throughout this paper we consider commutative noetherian rings with 1 for a ring r dim r denotes its krull dimension which we always assume to be finite if r is a local ring then 911 f will always denote its unique maximal ideal if m is a finitely generated f module then ii m will denote the minimal number of genera tors of m for an ideal i of r i i denotes the height of definition let be an unmixed ideal of r of height r then i is said to be complete intersection in r if i 2lri xrai where ax a2 ar is a regular r sequence remark if r is cohen macaulay then i is complete intersection if and only if p ht 7 let r and s be two local rings definition r is said to be a local extension of s if s is a subring of r and 91t s 911 f d s r is said to be unramified over s if gjt s r 911 f and f 91t f is separable over 5 91l s let l k be a finite separable extension of k then l is a simple extension of k by a minimal polynomial of l over k we always mean an irreducible monic polynomial over k satisfied by a generator of l over k now we state a theorem which has been proved in d g theorem 3

conjugacy class properties of the extension of gl n q generated by the inverse transpose involution abstract letting denote the inverse transpose automorphism of gl n q a formula is obtained for the number of g in gl n q so that gg is equal to a given element h this generalizes a result of gow and macdonald for the special case that h is the identity we conclude that for g random gg behaves like a hybrid of symplectic and orthogonal groups it is shown that our formula works well with both cycle index generating functions and asymptotics and is related to the theory of random partitions the derivation makes use of models of representation theory of gl n q and of symmetric function theory including a new identity for hall littlewood polynomials we obtain information about random elements of finite symplectic groups in even characteristic and explicit bounds for the number of conjugacy classes and centralizer sizes in the extension of gl n q generated by the inverse transpose automorphism we give a second approach to these results using the theory of bilinear forms over a field the results in this paper are key tools in forthcoming work of the authors on derangements in actions of almost simple groups and we give a few examples in this direction

symbolic powers of edge ideals abstract let i r k x k x 1 x n be an ideal in a polynomial ring over the field k we define the essential symbolic module of i to be the r i module f i r 2 i r r i where r i i 1 r 1 i i i r i and i m stands for the mth symbolic power of i we will mainly focus on the case where i is generated by square free monomials of degree two among our main results are optimal bounds for the degrees of the minimal generators of f i several criteria for a monomial to be such a generator and an upper bound for the generation type of the symbolic rees algebra of i as a byproduct we recapture the result of simis vasconcelos and villarreal on when such an ideal is normally torsionfree

stanley depths of certain stanley reisner rings abstract let be a simplicial complex on n s k x 1 x n the polynomial ring in n variables over a field k and k s i the stanley reisner ring of with respect to k it is proved that the stanley conjecture holds for k i e sdepth s k depth s k when i has four associated prime ideals it is also obtained that sdepth s k size s i

variational problems related to self similar solutions of the heat equation on demontre la compacite de certains plongements de sobolev sous de bonnes conditions sur les poids k x exp x c 2 r n r on considere des methodes variationnelles pour la resolution d equations elliptiques

the gorensteinness of the symbolic blow ups for certain space monomial curves let p p n 1 n 2 n 3 denote the prime ideal in the formal power series ring a k x y z over a field k defining the space monomial curve x t n1 y t n2 and z t n3 with gcd n 1 n 2 n 3 1 then the symbolic rees algebras r s p n 0 p n are gorenstein rings for the prime ideals p p n 1 n 2 n 3 with min n 1 n 2 n 3 4 and p p m m 1 m 4 with m 9 13 the rings r s p for p p 9 10 13 and p p 13 14 17 are noetherian but non cohen macaulay if ch k 3

classification of singular solutions of a nonlinear heat equation on considere toutes les solutions non negatives de l equation de la chaleur non lineaire avec absorption u t u m u p dans q r n 0 t pour n 1 m p 1 avec des conditions initiales u x 0 0 pour x 0

linear equivalence of ideal topologies it is proved that whenever p is a prime ideal in a commutative noetherian ring such that the p adic and thep symbolic topologies are equivalent then the two topologies are equivalent linearly several explicit examples are calculated in particular for all prime ideals corresponding to non torsion points on nonsingular elliptic cubic curves

the probability that the number of points on the jacobian of a genus 2 curve is prime in 2000 galbraith and mckee heuristically derived a formula that estimates the probability that a randomly chosen elliptic curve over a fixed finite prime field has a prime number of rational points we show how their heuristics can be generalized to jacobians of curves of higher genus we then elaborate this in genus g 2 and study various related issues such as the probability of cyclicity and the probability of primality of the number of points on the curve itself finally we discuss the asymptotic behavior for g

colorings of hypergraphs perfect graphs and associated primes of powers of monomial ideals abstract there is a natural one to one correspondence between squarefree monomial ideals and finite simple hypergraphs via the cover ideal construction let h be a finite simple hypergraph and let j j h be its cover ideal in a polynomial ring r we give an explicit description of all associated primes of r j s for any power j s of j in terms of the coloring properties of hypergraphs arising from h we also give an algebraic method for determining the chromatic number of h proving that it is equivalent to a monomial ideal membership problem involving powers of j our work yields two new purely algebraic characterizations of perfect graphs independent of the strong perfect graph theorem the first characterization is in terms of the sets ass r j s while the second characterization is in terms of the saturated chain condition for associated primes

a semilinear heat equation with singular initial data we first prove existence and uniqueness of non negative solutions of the equation in in the range 1 p n when initial data u x 0 a x 2 p 1 x 0 for a 0 it is proved that the maximal and minimal solutions are self similar with the form where g g a satisfies after uniqueness is proved the asymptotic behaviour of solutions of is studied in particular we show that the case for a 0 is also considered and a sharp decay rate of the above equation is derived in the final we reveal existence of solutions of the first and third equations above which change sign