gaussian interference channel capacity to within one bit the capacity of the two user gaussian interference channel has been open for 30 years the understanding on this problem has been limited the best known achievable region is due to han and kobayashi but its characterization is very complicated it is also not known how tight the existing outer bounds are in this work we show that the existing outer bounds can in fact be arbitrarily loose in some parameter ranges and by deriving new outer bounds we show that a very simple and explicit han kobayashi type scheme can achieve to within a single bit per second per hertz bit s hz of the capacity for all values of the channel parameters we also show that the scheme is asymptotically optimal at certain high signal to noise ratio snr regimes using our results we provide a natural generalization of the point to point classical notion of degrees of freedom to interference limited scenarios

combinatorial decompositions of a class of rings in this paper we introduce some combinatorial methods for the study of graded rings and use these methods to analyze a class of rings that are of importance in combinatorics our methods are based on a unique representation theorem which is shown section 2 to hold for any graded algebra and which in a sense is an extension of the direct sum decomposition for cohen macaulay algebras this unique representation is reminiscent of a similar result obtained by rees 20 both results yield a decomposition of the algebra however our result differs from that of rees in that it carries more information about the algebra our main results are concerned with a class of rings see section 3 for the definition associated to simplicial complexes there are several natural operations that can be performed on simplicial complexes to obtain new complexes which in turn have an interpretation for the corresponding rings we show sections 4 5 and 6 that our basic decompositions can be transferred during these operations from the original rings to the newly constructed ones the operations studied here are the chain transform or barycentric subdivision section 4 rank selection both for partially ordered sets and for simplicial complexes section s and localization section 6 as we develop these tools we give some applications in section 5 we give a new topological characterization of the cohen macaulay property for partially ordered sets an immediate consequence of which is the rankselection theorem baclawski 2 this characterization is closely related to those obtained by garsia in 14

alexander duality and stanley depth of multigraded modules abstract we apply miller s theory on multigraded modules over a polynomial ring to the study of the stanley depth of these modules several tools for stanley s conjecture are developed and a few partial answers are given for example we show that taking the alexander duality twice but with different centers is useful for this subject generalizing a result of apel we prove that stanley s conjecture holds for the quotient by a cogeneric monomial ideal

grobner bases of syzygies and stanley depth abstract let f be any free resolution of a finitely generated z n graded module over the polynomial ring k x 1 x n we show that for a suitable term order on f then for 0 p n the initial module of the p 1 th syzygy module z p f p is generated by terms m i e i where the m i are monomials in k x p 1 x n also for a large class of free resolutions f encompassing eliahou kervaire resolutions we show that a grobner basis for z p is given by the boundaries of generators of f p 1 we apply the above to give lower bounds for the stanley depth of the p 1 th syzygy module z p in particular showing it is greater or equal to p 1 we also show that if i is any squarefree ideal in k x 1 x n the stanley depth of i is at least of order 2 n

cohen macaulayness of semi invariants for tori in this paper we give a new method in terms of one parameter subgroups to study semi invariants for algebraic tori in some cases we obtain extensions to results in 7 in other cases we obtain different proofs

the worst additive noise under a covariance constraint the maximum entropy noise under a lag p autocorrelation constraint is known by burg s theorem to be the pth order gauss markov process satisfying these constraints the question is what is the worst additive noise for a communication channel given these constraints is it the maximum entropy noise the problem becomes one of extremizing the mutual information over all noise processes with covariances satisfying the correlation constraints r sub 0 r sub p for high signal powers the worst additive noise is gauss markov of order p as expected but for low powers the worst additive noise is gaussian with a covariance matrix in a convex set which depends on the signal power

a non partitionable cohen macaulay simplicial complex abstract a long standing conjecture of stanley states that every cohen macaulay simplicial complex is partitionable we disprove the conjecture by constructing an explicit counterexample due to a result of herzog jahan and yassemi our construction also disproves the conjecture that the stanley depth of a monomial ideal is always at least its depth

saturation and castelnuovo mumford regularity abstract let i be a homogeneous ideal of the polynomial ring k x 0 x n where k is an arbitrary field avoiding the construction of a minimal graded free resolution of i we provide effective methods for computing the castelnuovo mumford regularity of i that also compute other cohomological invariants of k x 0 x n i

stanley depth of complete intersection monomial ideals and upper discrete partitions abstract let i be an m generated complete intersection monomial ideal in s k x 1 x n we show that the stanley depth of i is n m 2 we also study the upper discrete structure for monomial ideals and prove that if i is a squarefree monomial ideal minimally generated by 3 elements then the stanley depth of i is n 1

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

a new entropy power inequality a strengthened version of shannon s entropy power inequality for the case where one of the random vectors involved is gaussian is proved in particular it is shown that if independent gaussian noise is added to an arbitrary multivariate random variable the entropy power of the resulting random variable is a concave function of the variance power of the added noise the strengthened inequality is shown to hold for the class of stable distributions

on the rational homotopy type of function spaces the main result of this paper is the construction of a minimal model for the function space f x y of continuous functions from a finite type finite dimensional space x to a finite type nilpotent space y in terms of minimal models for x and y for the component containing the constant map 7r f x y 0 q 7r y 0 h x q in positive dimensions when x is formal there is a simple formula for the differential of the minimal model in terms of the differential of the minimal model for y and the coproduct of h x q we also give a version of the main result for the space of cross sections of a fibration

the number of solutions to linear diophantine equations and multivariate splines etude de la variation du nombre des solutions entieres non negatives de s equations diophantiennes lineaires en fonction des termes non homogenes

dualizing complexes of affine semigroup rings for an affine semigroup ring we construct the dualizing complex in terms of the semigroup and the homology of the face lattice of the polyhedral cone spanned by the semigroup as a consequence there are characterizations of locally cohen macaulay rings buchsbaum rings and cohen macaulay rings as well as serre s condition 9 introduction let s denote a finitely generated submonoid of the additive monoid n n being a positive integer by the affine semigroup ring i s of s over a field x let us denote the subring of x xl xn generated by all monomials xs s e s so one has up to isomorphisms a one to one correspondence between affine semigroup rings and affine varieties given parametrically by monomials now it is of some interest to characterize ring theoretic properties as cohenmacaulay gorenstein etc in terms of the semigroup s a first breakthrough in this direction was done by kempf et al kkms and hochster hol who showed that x s is a cohen macaulay ring provided it is normal moreover the normality of x s is described in terms of s see hol another particular case if s is a simplicial semigroup is treated by goto suzuki and watanabe gsw resp stanley st2 who showed that x s is a cohen macaulay ring if it satisfies the 22 condition of serre note that normality is equivalent to 5y2 and r1 moreover there is a description of the 5 condition in terms of s see theorem 6 3 in general the 52 condition is not sufficient for i s to be a cohen macaulay ring as follows by examples in hol resp th 1 2 on the other side grobner g posed the problem to classify cohen macaulay affine semigroup rings the solution of this problem was done by hoa and trung th related to structural properties of graded rings is a technical tool of the duality theory this is the dualizing complex introduced by grothendieck see ha in recent years it became a helpful tool in commutative algebra in particular its knowledge allows characterizations of cohen macaulay rings locally cohenmacaulay rings buchsbaum rings gorenstein rings and the 5 conditions of received by the editors december 18 1988 1980 mathematics subject classification 1985 revision primary 13h10 secondary 14m05 06a99 1990 american mathematical society 0002 9947 90 1 q0 25 per page

shellable nonpure complexes and posets ii this is a direct continuation of shellable nonpure complexes and posets i which appeared in transactions of the american mathematical society 348 1996 1299 1327 8 interval generated lattices and dominance order in this section and the following one we will continue exemplifying the applicability of lexicographic shellability to nonpure posets let f i1 i2 in be a family of intervals of integers by which is meant sets of the form a b a a 1 b a b we assume that there are no containments among these intervals and that they are ordered so that their left and right endpoints are increasing let l f be the lattice of all sets that are unions of subfamilies of f ordered by inclusion such interval generated lattices l f were introduced and studied by greene g define an edge labeling of l f as follows if a b is a covering and a max b a then a b a if a 1 a and a is the left endpoint of some i f

a new achievable rate region for the interference channel a new achievable rate region for the general interference channel which extends previous results is presented and evaluated the technique used is a generalization of superposition coding to the multivariable case a detailed computation for the gaussian channel case clarifies to what extent the new region improves previous ones the capacity of a class of gaussian interference channels is also established

outer bounds on the capacity of gaussian interference channels two outer bounds on the capacity region of the two user gaussian interference channel ifc are derived the idea of the first bound is to let a genie give each receiver just enough information to decode both messages this bound unifies and improves the best known outer bounds of sato and carleial furthermore the bound extends to discrete memoryless ifcs and is shown to be equivalent to another bound of carleial the second bound follows directly from existing results of costa and sato and possesses certain optimality properties for weak interference

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

stanley depth of multigraded modules abstract the stanley s conjecture on cohen macaulay multigraded modules is studied especially in dimension 2 in codimension 2 similar results were obtained by herzog soleyman jahan and yassemi as a consequence of our results stanley s conjecture holds in 5 variables

depth and stanley depth of symbolic powers of cover ideals of graphs abstract let g be a graph with n vertices and let s k x 1 x n be the polynomial ring in n variables over a field k assume that j g is the cover ideal of g and j g k is its k th symbolic power we prove that the sequences sdepth s j g k k 1 and sdepth j g k k 1 are non increasing and hence convergent suppose that o g denotes the ordered matching number of g we show that for every integer k 2 o g 1 the modules j g k and s j g k satisfy the stanley s inequality we also provide an alternative proof for 9 theorem 3 4 which states that depth s j g k n o g 1 for every integer k 2 o g 1

stanley depth of squarefree monomial ideals in this paper we answer a question posed by y h shen we prove that if i is an m generated squarefree monomial ideal in the polynomial ring s k x1 xn with k a field then sdepthi n m 2 the proof is inductive and uses the correspondence between a stanley decomposition of a monomial ideal and a partition of a particular poset into intervals established by herzog vladoiu and zheng

on the han kobayashi region for theinterference channel in this correspondence we derive a simplified description of the han kobayashi rate region for the general interference channel using this result we establish that the recently discovered chong motani garg rate region is a new representation of the han kobayashi region moreover a tighter bound for the cardinality of the time sharing auxiliary random variable emerges from our simplified description

lie models for the components of sections of a nilpotent fibration we give an explicit lie model for any component of the space of free and pointed sections of a nilpotent fibration and in particular of the free and pointed mapping spaces among the applications presented we obtain a lie model of the exponential law and prove that in many cases the rank of the homotopy groups of the mapping space grows at the same rate as the rank of the homotopy groups of the target space

local cohomology of modules of covariants abstract let g be a connected reductive algebraic group over an algebraically closed field of characteristic zero and let w u be two finite dimensional representations of g in this paper we compute the local cohomology of u sw g provided a certain relatively weak technical condition is true

capacity and rate regions of a class of broadcast interference channels in this paper a class of broadcast interference channels bic is investigated where one of the two broadcast receivers is subject to interference coming from a point to point transmission for a general discrete memoryless broadcast interference channel dm bic an achievable scheme based on message splitting superposition and binning is proposed and a concise representation of the corresponding achievable rate region mathcal r is obtained two partial order broadcast conditions interference oblivious less noisy and interference cognizant less noisy are defined thereby extending the usual less noisy condition for a regular broadcast channel by taking interference into account under these conditions a reduced form of mathcal r is shown to be equivalent to a rate region based on a simpler scheme where the broadcast transmitter uses only superposition furthermore if interference is strong for the interference oblivious less noisy dm bic the capacity region is given by the aforementioned two equivalent rate regions for the interference cognizant less noisy dm bic it is argued that the strong but not very strong interference condition does not exist and in this case the capacity region for the very strong interference is obtained for a gaussian broadcast interference channel channel parameters are categorized into three regimes for the first two regimes which are closely related to the two partial order broadcast conditions achievable rate regions are derived by specializing the corresponding achievable schemes of dm bics with gaussian input distributions the entropy power inequality based outer bounds are obtained by combining bounding techniques for a gaussian broadcast channel and a gaussian interference channel these inner and outer bounds lead to either exact or approximate characterizations of capacity regions and sum capacity under various conditions for the remaining complementing regime inner and outer bounds are also provided

on the stanley depth of powers of edge ideals abstract let k be a field and s k x 1 x n be the polynomial ring in n variables over k let g be a graph with n vertices assume that i i g is the edge ideal of g and p is the number of its bipartite connected components we prove that for every positive integer k the inequalities sdepth i k i k 1 p and sdepth s i k p hold as a consequence we conclude that s i k satisfies stanley s inequality for every integer k n 1 also it follows that i k i k 1 satisfies stanley s inequality for every integer k 0 furthermore we prove that if i g is a non bipartite graph or ii at least one of the connected components of g is a tree with at least one edge then i k satisfies stanley s inequality for every integer k n 1 moreover we verify a conjecture of the author in special cases

rational homotopy of the space of sections of a nilpotent bundle we show that an algebraic construction proposed by sullivan is indeed a model for the rational homotopy type of the space of sections of a nilpotent bundle in his paper uhomologie des espaces fonctionnels r thom studied the homotopy type of the space f of continuous maps of x into f homotopic to a given map starting from a postnikov decomposition of f he built the functional space f step by step he also indicated how one could construct a differential graded algebra describing the rational homotopy type of f later on sullivan gave an algebraic model which mirrors this construction in terms of a dg algebra representing x and the minimal model of f the aim of this paper is to show following the method of thom that the model of sullivan is indeed a model for the functional space under suitable restrictions as in 3 we consider the slightly more general problem of the determination of the rational homotopy type of the space of sections ts of a nilpotent fiber space p y x homotopic to a given section s in 1 we explain thorn s geometric construction in 2 we describe an algebraic model for an abelian galois covering of a nilpotent space in 3 we show how the model of sullivan fits with the geometry i thank the referee for many improvements of this paper 1 a postnikov factorization of the space of sections 1 1 let g be a finitely generated abehan group and let x be a path connected space whose cohomology groups hk x g are finitely generated for each k to avoid difficulties with the topologies cf 4 we can work in the category of simplicial sets proposition thom 4 the space k g m x of continuous maps of x in the eilenberg mac lane complex k g m is homotopically equivalent to the product nr ra fthe eilenberg mac lane spaces ki k hm x g i more precisely let x g hm k g m g be the fundamental class of k g m if e k g m x xx k g m

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

prime filtrations of monomial ideals and polarizations abstract we show that an arbitrary monomial ideal i is pretty clean if and only if its polarization i p is clean this yields a new characterization of pretty clean monomial ideals in terms of the arithmetic degree and it also implies that a multicomplex is shellable if and only the simplicial complex corresponding to its polarization is non pure shellable we also discuss stanley decompositions in relation to prime filtrations

rational homotopy of the homotopy fixed point sets of circle actions in this paper when g is the circle s 1 and m is a g space we study the rational homotopy type of the fixed point set m g the homotopy fixed point set m hg and the natural injection m g m hg c 2009 elsevier inc all rights reserved

the ubiquity of smooth hilbert schemes we investigate the geography of hilbert schemes parametrizing closed subschemes of projective space with specified hilbert polynomials we classify hilbert schemes with unique borel fixed points via combinatorial expressions for their hilbert polynomials these expressions naturally lead to an arrangement of nonempty hilbert schemes as the vertices of an infinite full binary tree here we discover regularities in the geometry of hilbert schemes specifically under natural probability distributions on the tree we prove that hilbert schemes are irreducible and nonsingular with probability greater than 0 5

non commutative resolutions of toric varieties abstract let r be the coordinate ring of an affine toric variety we prove using direct elementary methods that the endomorphism ring end r a where a is the finite direct sum of all isomorphism classes of conic r modules has finite global dimension equal to the dimension of r this gives a precise version and an elementary proof of a theorem of spenko and van den bergh implying that end r a has finite global dimension furthermore we show that end r a is a non commutative crepant resolution if and only if the toric variety is simplicial for toric varieties over a perfect field k of prime characteristic we show that the ring of differential operators d k r has finite global dimension

the action of the special orthogonal group on planar vectors integrity bases via a generalization of the symbolic interpretation of molien functions the present article completes the mathematical description initiated in the paper by dhont and zhilinskii 2013 the action of the orthogonal group on planar vectors invariants covariants and syzygies j phys a math theor 46 455202 of the algebraic structures that emerge from the symmetry adapted polynomials in the coordinates of n planar vectors under the action of the so 2 group the set of covariant polynomials contains all the polynomials that transform according to the weight of so 2 and is a free module for but a non free module for the sum of the rational functions of the molien function for covariants describes the decomposition of the ring of invariants or the module of covariants as a direct sum of submodules a method for extracting the generating function for covariants from the comprehensive generating function for all polynomials is introduced the approach allows the direct construction of the integrity basis for the module of covariants decomposed as a direct sum of submodules and gives insight into the expressions for the molien functions found in our earlier paper in particular a generalized symbolic interpretation in terms of the integrity basis of a rational function is discussed where the requirement of associating the different terms in the numerator of one rational function with the same subring of invariants is relaxed

stanley conjecture in small embedding dimension we show that stanley s conjecture holds for a polynomial ring over a field in four variables in the case of polynomial ring in five variables we prove that the monomial ideals with all associated primes of height two are stanley ideals

a gaussian input is not too bad we consider the problem of choosing a robust input for communicating over an input constrained additive noise channel where the noise distribution is arbitrary we show that the mutual information rate achievable using a white gaussian input never incurs a loss of more than half a bit per sample with respect to the power constrained capacity for comparison for the family of colored gaussian noise channels a white gaussian input loses at most log e 2e spl ap 0 265 bit per sample with respect to the optimum water pouring solution for general input constraints we derive a formula for choosing the best input in the min max capacity loss bound sense the bound on the capacity loss is tight for pulse position modulation ppm in the presence of a bursty jammer

simplicity of rings of differential operators in prime characteristic let w be a finite dimensional representation of a linearly reductive group g over a field k motivated by their work on classical rings of invariants levasseur and stafford asked whether the ring of invariants under g of the symmetric algebra of w has a simple ring of differential operators in this paper we show that this is true in prime characteristic indeed if r is a graded subring of a polynomial ring over a perfect field of characteristic p 0 and if the inclusion r hookrightarrow s splits then d k r is a simple ring in the last section of the paper we discuss how one might try to deduce the characteristic zero case from this result as yet however this is a subtle problem and the answer to the question of levasseur and stafford remains open in characteristic zero http www luc ac be research algebra 1991 mathematics subject classification 16s32 16g60 13a35

local cohomology with support in ideals of maximal minors and sub maximal pfaffians abstract we compute the gl equivariant description of the local cohomology modules with support in the ideal of maximal minors of a generic matrix as well as of those with support in the ideal of 2 n 2 n pfaffians of a 2 n 1 2 n 1 generic skew symmetric matrix as an application we characterize the cohen macaulay modules of covariants for the action of the special linear group sl g on g m the main tool we develop is a method for computing certain ext modules based on the geometric technique for computing syzygies and on matlis duality