linear free resolutions and minimal multiplicity let s k x x be a polynomial ring over a field and let a4 a m be a finitely generated graded module in the most interesting case a4 is an ideal of s for a given natural number p there is a great interest in the question can m be generated by homogeneous elements of degree p no simple answer say in terms of the local cohomology of m is known but somewhat surprisingly the stronger question can the jth syzygy of m be generated by elements of degree qp j for all j 0 l n does admit a simple response the following is neither new nor difficult to prove though it seems not well known

direction finding in partly calibrated sensor arrays composed of multiple subarrays we consider the direction finding problem in partly calibrated arrays composed of several calibrated and identically oriented but possibly nonidentical subarrays that are displaced by unknown and possibly time varying vector translations a new search free eigenstructure based direction finding approach is proposed for such class of sensor arrays it is referred to as the rank reduction rare estimator and enjoys simple implementation that entails computing the eigendecomposition of the sample array covariance matrix and polynomial rooting closed form expressions for the deterministic cramer rao bounds crbs on direction of arrival doa estimation for the considered class of sensor arrays are derived comparison of these expressions with simulation results show that the finite sample performance of rare algorithms in both time invariant and time varying array cases is close to the corresponding bounds moreover comparisons of the derived crbs with the well known bounds for the fully calibrated time invariant array case help to discover several interesting properties of doa estimation in partly calibrated and time varying arrays

castelnuovo mumford regularity bounds for singular surfaces we prove the regularity conjecture namely eisenbud goto conjecture for a normal surface with rational gorenstein elliptic and log canonical singularities along the way we bound the regularity for a dimension zero scheme by its loewy length and for a curve allowing embedded or isolated point components by its arithmetic degree

multigraded castelnuovo mumford regularity we develop a multigraded variant of castelnuovo mumford regularity motivated by toric geometry we work with modules over a polynomial ring graded by a finitely generated abelian group as in the standard graded case our definition of multigraded regularity involves the vanishing of graded components of local cohomology we establish the key properties of regularity its connection with the minimal generators of a module and its behavior in exact sequences for an ideal sheaf on a simplicial toric variety x we prove that its multigraded regularity bounds the equations that cut out the associated subvariety we also provide a criterion for testing if an ample line bundle on x gives a projectively normal embedding

normalization of modules we introduce techniques to derive estimates for the degrees of the generators of the integral closure of several classes of rees algebras of modules and to bound the length of normalization processes in the case of regular base rings the bounds are expressed in terms of buchsbaum rim multiplicities and a module version of briancon skoda numbers

almost blind steering vector estimation using second order moments the problem of separation and reconstruction of superimposed signals using an array of sensors attracted considerable interest the estimation of the steering vectors of an uncalibrated array is considered we identify a cost function whose minimizer is a statistically consistent estimate of the unknown parameters next we present an iterative algorithm for finding a local minimum of that cost function the proposed algorithm is guaranteed to converge the performance of the algorithm is compared with the cramer rao bound crb

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

algebraic theories for name passing calculi in a theory of processes the names are atomic data items which can be exchanged and tested for identify a well known example of a calculus for name passing is the calculus where names are additionally used as communication ports we provide complete axiomatisations of late and early bisimulation equivalences in such calculi since neither of the equivalences is a congruence we also axiomatise the corresponding largest congruences we consider a few variations of the signature of the language among these a calculus of deterministic processes which is reminiscent of sequential functional programs with a conditional construct most of our axioms are shown to be independent the axiom systems differ only by a few simple axioms and reveal the similarities and the symmetries of the calculi and the equivalences

cohen macaulay rees algebras and degrees of polynomial relations e mail aron sunrnp vfba br 2 department of mathematics michigan state university east lansing mi 48824 usa e mail 21144bfu msu edu 3 department of mathematics rutgers university new brunswick nj 08903 usa e mail vasconce rings rutgers edu received 5 october 1993

projective resolutions of cohen macaulay algebras the problem of explicitly finding a free resolution minimal in some suitable sense of a module over a polynomial ring is solved in principle by the algorithm of hilbert h however this algorithm is of enormous computational difficulty if the module happens to be finite dimensional over the ground field and if the module structure is given by specifying the commuting linear transformations induced by the indeterminates then a little known result of scheja and storch resumed in sect 1 allows one to write down an explicit free resolution of the right length without computation the same idea can be used for many cohenmacaulay modules example 1 1 but although the scheja storch resolution is minimal in some cases it is not minimal in the main case of interest where the module is a factor ring of the polynomial ring and not the ground field itself

joint 2 d doa estimation and phase calibration for uniform rectangular arrays a precise model of the array response is required to maintain the performance of direction of arrival doa estimation when modeling errors are present or the sensor environment is time varying autocalibration becomes necessary in this paper the problem of phase autocalibration for uniform rectangular array ura geometries is considered for the case with a single source a simple and robust least squares algorithm for joint 2 d doa estimation and phase calibration is presented when performing phase autocalibration with a ura the phase and doa parameters cannot be identified together without ambiguity this problem is discussed and a suitable remedy is suggested an approximate cramer rao bound and analytical expressions for the mean squared error performance of the proposed estimator are presented the proposed algorithm for phase autocalibration is extended for the case with multiple sources the results are evaluated using a representative body of simulations

a unified framework and sparse bayesian perspective for direction of arrival estimation in the presence of array imperfections self calibration methods play an important role in reducing the negative effects of array imperfections during direction of arrival doa estimation however the dependence of most such methods on the eigenstructure techniques greatly degrades their adaptation to demanding scenarios such as low signal to noise ratio snr and limited snapshots this paper aims at formulating a unified framework and sparse bayesian perspective for array calibration and doa estimation a comprehensive model of the array output is presented first when a single type of array imperfection is considered with mutual coupling gain phase uncertainty and sensor location error treated as typical examples the spatial sparsity of the incident signals is then exploited and a bayesian method is proposed to realize array calibration and source doa estimation the array perturbation magnitudes are assumed to be small according to most application scenarios and the geometries of mutually coupled arrays are assumed to be uniform linear and those of arrays with sensor location errors are assumed to be linear cramer rao lower bounds crlbs for the array calibration and doa estimation precisions are also obtained the sparse bayesian method is finally extended to deal with the doa estimation problem when more than one type of array perturbation coexists

a combinatorial proof of the eisenbud goto conjecture for monomial curves and some simplicial semigroup rings abstract we will give a pure combinatorial proof of the eisenbud goto conjecture for arbitrary monomial curves moreover we will show that the conjecture holds for certain simplicial affine semigroup rings

small schemes and varieties of minimal degree we prove that if x p r is any 2 regular scheme in the sense of castelnuovo mumford then x is small this means that if l is a linear space and y l x is finite then y is linearly independent in the sense that the dimension of the linear span of y is deg y 1 the converse is true and well known for finite schemes but false in general the main result of this paper is that the converse small implies 2 regular is also true for reduced schemes algebraic sets this is proven by means of a delicate geometric analysis leading to a complete classification we show that the components of a small algebraic set are varieties of minimal degree meeting in a particularly simple way from the classification one can show that if x p r is 2 regular then so is xred and so also is the projection of x from any point of x our results extend the del pezzo bertini classification of varieties of minimal degree the charac terization of these as the varieties of regularity 2 by eisenbud goto and the construction of 2 regular square free monomial ideals by froberg 0 introduction throughout this paper we will work with projective schemes x p r over an algebraically closed field k the castelnuovo mumford regularity of x p r is a basic homological measure of the complexity of x and its embedding in p r that gives a bound for the degrees of the generators of the defining ideal ix of x and for many other invariants the only schemes of regular ity 1 are the linear spaces but no classification is known for projective schemes of regularity 2 in this paper we prove a structure theorem for reduced 2 regular schemes showing that their irreducible components are varieties of minimal degree and characterizing how these components can meet we also show that the reduced structure on any 2 regular scheme is 2 regular and thus we obtain a complete description of the reduced structures on 2 regular schemes since a high veronese re embedding of any zero dimensional scheme is 2 regular one cannot hope to characterize the isomorphism types of all 2 regular nonreduced schemes before stating our results we review some basic notions for any subscheme x p r we write span x for the smallest linear subspace of p r containing x recall that every variety reduced irreducible scheme x p r satisfies the condition deg x 1 codim x span x

liaison and castelnuovo mumford regularity in this article we establish bounds for the castelnuovo mumford regularity of projective schemes in terms of the degrees of their defining equations the main new ingredient in our proofs is to show that generic residual intersections of complete intersection rational singularities again have rational singularities when applied to the theory of residual intersections this circle of ideas also sheds new light on some known classes of free resolutions of residual ideals

theoretical analyses of gain and phase error calibration with optimal implementation for linear equispaced array in array signal processing high resolution direction of arrival doa estimation algorithms are known to be sensitive to system errors in practice the system should be properly calibrated before doa estimation in this paper a calibration method of gain and phase errors of linear equispaced arrays leas is considered a class of simplified calibration algorithms based on different diagonal lines of the covariance matrix is proposed the statistical performance analyses of the calibration algorithms due to finite data perturbations are presented expressions for average bias abias and square root of average variance srav of the calibration algorithms are derived using first order approximation these statistical and computer simulation results reveal and explain why the more diagonal lines that are used for gain and phase error estimations the more inferior performance that may be obtained based on this conclusion the simple and optimal gain and phase error calibration algorithms for such model are obtained

ccs expressions finite state processes and three problems of equivalence abstract we examine the computational complexity of testing finite state processes for equivalence in milner s calculus of communicating systems ccs the equivalence problems in ccs are presented as refinements of the familiar problem of testing whether two nondeterministic finite automata nfa are equivalent i e accept the same language three notions of equivalence proposed for ccs are investigated namely observational equivalence strong observational equivalence and failure equivalence we show that observational equivalence can be tested in polynomial time as defined in ccs observational equivalence is the limit of a sequence of successively finer equivalence relations k where 1 is nondeterministic finite automaton equivalence we prove that for each fixed k deciding k is pspace complete we show that strong observational equivalence can be decided in polynomial time by reducing it to generalized partitioning a new combinatorial problem of independent interest finally we demonstrate that testing for failure equivalence is pspace complete even for a very restricted type of process

esprit estimation of signal parameters via rotational invariance techniques an approach to the general problem of signal parameter estimation is described the algorithm differs from its predecessor in that a total least squares rather than a standard least squares criterion is used although discussed in the context of direction of arrival estimation esprit can be applied to a wide variety of problems including accurate detection and estimation of sinusoids in noise it exploits an underlying rotational invariance among signal subspaces induced by an array of sensors with a translational invariance structure the technique when applicable manifests significant performance and computational advantages over previous algorithms such as mem capon s mlm and music

on the depth of the symmetric algebra let r m be a local ring assume that r a i where a n is a regular local ring and c n2 is an ideal the depth of the symmetric algebra s m of m over r is computed in terms of the depth of the associated graded module gr 7 and the so called strong socle condition explicit results are obtained for instance if is generated by a super regular sequence if has a linear resolution or if has projective dimension one introduction in recent years the symmetric algebra s i of an ideal and some related algebras such as the rees algebra a i of or the associated graded ring gr r of have been studied very extensively one central problem always has been to give reasonable conditions for one of these algebras to be cohen macaulay as it turned out the symmetric algebra has a strong tendency not to be cohen macaulay for instance if we consider the symmetric algebra s m of the maximal ideal of a cohen macaulay local ring r m of positive dimension rossi 11 has shown that s m is cohen macaulay if and only if r is an abstract hypersurface ring in other words s m is almost never cohen macaulay one reason for this behavior of the symmetric algebra seems to be the fact that its dimension compared with the one of r may be much higher in their paper 7 huneke and rossi succeeded in giving a very explicit formula for the dimension dims m of the symmetric algebra of a module m applied to the above situation one finds that dim 5 111 embedding dimension of r provided r is not regular if r is regular then dims rrt dim 1 it follows from a result in the same paper 7 that depth 5 m dim 1 here the gap between depth and dimension becomes apparent unfortunately the bound depth s m depth r 1 is not true in general if r is not cohen macaulay see the remark after 2 12 the main theme of the paper actually will be to compute or to estimate the depth of 5 m for a noetherian local ring r m in the range up to depth r 1 we shall see that depth s rrt is a much more sensitive invariant of s rrt than its dimension before going into a detailed description of our paper we just quote two of our results to give an impression of how delicate depth s m behaves

on linear complete intersections or some of its localizations becomes a complete intersection is a major factor in section 1 we describe a jacobian formulation of serre s normality criterion for symmetric algebras it is used to provide several classes of examples of normal and non normal rees algebras of prime ideals in regular domains adding to the catalog of cohen macaulay domains that are symmetric algebras we examine almost complete intersections these bear a close relationship to certain height three ideals of

on syzygies of non complete embedding of projective varieties let x be a non degenerate not necessarily linearly normal projective variety in mathbb p r recently the generalization of property np to non linearly normal projective varieties have been considered and its algebraic and geometric properties are studied extensively one of the generalizations is the property nd p for the saturated ideal ix eisenbud et al in compos math 141 1460 1478 2005 and the other is the property n p s for the graded module of the twisted global sections of mathcal o x 1 kwak and park in j reine angew math 582 87 105 2005 in this paper we are interested in the algebraic and geometric meaning of properties n p s for every p 0 and the syzygetic behaviors of isomorphic projections and hyperplane sections of a given variety with property n p s

an eigenstructure method for estimating doa and sensor gain phase errors in this paper we consider the problem of direction of arrival doa estimation in the presence of sensor gain phase errors under some mild assumptions we propose a new doa estimation method based on the eigendecomposition of a covariance matrix which is constructed by the dot product of the array output vector and its conjugate by combining the new doa estimation with the conventional gain phase error estimation a method is proposed to simultaneously estimate the doa and gain phase errors without joint iteration theoretical analysis shows that the proposed method performs independently of phase errors and thus behaves well regardless of phase errors however the resolution capability of the proposed method is lower than that of the method in a j weiss and b friedlander eigenstructure methods for direction finding with sensor gain and phase uncertainties circuits systems signal process vol 9 no 3 pp 271 300 1990 named as the wf method in order to improve the resolution capability and maintain phase error independence a combined strategy is developed using the proposed and wf methods the advantage of the proposed methods is that they are independent of phase errors leading to the cancellation of phase error calibration during the operation life of an array moreover the proposed methods avoid the problem of suboptimal convergence which occurs in the wf method the drawbacks of the proposed methods are their high computational complexity and their requirement for the condition that at least two signals are spatially far from each other and they are not applicable to a linear array simulation results verify the effectiveness of the proposed methods

the dimension and components of symmetric algebras let r be a commutative noetherian ring and let a4 be any finitely generated r module in this paper we compute the dimension of s m the symmetric algebra of m and identify many of the components of s m while the computation of the dimension of s m is not difficult the answer is intriguing and in view of the recent interest in symmetric algebras we hope the computation will be useful the dimension is given by the for mula dim s m max dim r p p m

on a theorem of castelnuovo and the equations defining space curves introduction 491 w notat ion and conventions 493 w a regularity theorem 494 w a rationality theorem 498 w the existence of secant lines 501 w open problems 505 references 506

castelnuovo mumford regularity of canonical and deficiency modules abstract we give two kinds of bounds for the castelnuovo mumford regularity of the canonical module and the deficiency modules of a ring respectively in terms of the homological degree and the castelnuovo mumford regularity of the original ring

hypersurfaces cutting out a projective variety let x be a nondegenerate projective variety of degree d and codimension e in a projective space p n defined over an algebraically closed field we study the following two problems is the length of the intersection of x and a line l in p n at most d e 1 if l g x is the scheme theoretic intersection of all hypersurfaces of degree at most d e 1 containing x equal to x to study the second problem we look at the locus of points from which x is projected nonbirationally

identifiability analysis of local oscillator phase self calibration based on hybrid cramer rao bound in mimo radar the identifiability of self calibration of local oscillator phase perturbation in multiple input multiple output mimo radar is analyzed in this paper four cases are considered the absolute phase perturbation app and phase perturbation differences ppds in mimo radar with widely separated antennas smimor and the app and ppds in co located mimo radar cmimor the tightness of the hybrid cramer rao bound hcrb is proved in all four cases it is interesting that the problem in this paper is a special case in which hcrb is asymptotically tight for nonlinear observation models containing unknown deterministic and random parameters for each case a signal model for multiple stationary targets is constructed the closed form hcrb of phase perturbation is derived and used to analyze the identifiability of phase perturbation it is proven that only the ppds in cmimor in the above mentioned four cases are identifiable numerical simulations verify the conclusion

deciding bisimulation equivalences for a class of non finite state programs traditionally many automatic program verification techniques have been applicable only to finite state programs in this paper we extend some of these techniques to a class of infinite state programs that in addition to having a finite state control component may read store and write but not perform any other computations on data such programs are data independent in the sense that their behavior does not depend on the actual data values supplied we consider the problems of deciding strong equivalence and observation equivalence defined by bisimulations as in ccs between such programs these equivalences have major applications in verification of communication protocols we present reductions of these problems to the problem of deciding strong equivalence and observation equivalence between finite state programs for which polynomial time algorithms exist the equivalence problems on data independent programs are shown to be np hard in the size of the programs

the bi graded structure of symmetric algebras with applications to rees rings abstract consider a rational projective plane curve c parameterized by three homogeneous forms of the same degree in the polynomial ring r k x y over a field k the ideal i generated by these forms is presented by a homogeneous 3 2 matrix with column degrees d 1 d 2 the rees algebra r r i t of i is the bi homogeneous coordinate ring of the graph of the parameterization of c and accordingly there is a dictionary that translates between the singularities of c and algebraic properties of the ring r and its defining ideal finding the defining equations of rees rings is a classical problem in elimination theory that amounts to determining the kernel a of the natural map from the symmetric algebra sym i onto r the ideal a d 2 1 which is an approximation of a can be obtained using linkage we exploit the bi graded structure of sym i in order to describe the structure of an improved approximation a d 1 1 when d 1 d 2 and has a generalized zero in its first column the latter condition is equivalent to assuming that c has a singularity of multiplicity d 2 in particular we give the bi degrees of a minimal bi homogeneous generating set for this ideal when 2 d 1 d 2 and has a generalized zero in its first column then we record explicit generators for a when d 1 d 2 we provide a translation between the bi degrees of a bi homogeneous minimal generating set for a d 1 2 and the number of singularities of multiplicity d 1 that are on or infinitely near c we conclude with a table that translates between the bi degrees of a bi homogeneous minimal generating set for a and the configuration of singularities of c when the curve c has degree six

non commutative castelnuovo mumford regularity and as regular algebras abstract let a be a connected graded k algebra with a balanced dualizing complex we prove that a is a koszul as regular algebra if and only if that the castelnuovo mumford regularity and the ext regularity coincide for all finitely generated a modules this can be viewed as a non commutative version of t romer on the regularity over positively graded algebras j algebra 319 2008 1 15 theorem 1 3 by using castelnuovo mumford regularity we prove that any koszul standard as gorenstein algebra is as regular as a preparation to prove the main result we also prove that the following statements are equivalent 1 a is as gorenstein 2 a has finite left injective dimension 3 the dualizing complex has finite left projective dimension this generalizes i mori homological properties of balanced cohen macaulay algebras trans amer math soc 355 2002 1025 1042 corollary 5 9

order ideals and a generalized krull height theorem let n be a finitely generated module over a noetherian local ring r m we give criteria for the height of the order ideal n x of an element x n to be bounded by the rank of n the generalized principal ideal theorem of bruns eisenbud and evans says that this inequality always holds if x mn we show that the inequality even holds if the hypothesis becomes true after first extending scalars to some local domain and then factoring out torsion we give other conditions in terms of residual intersections and integral closures of modules we derive information about order ideals that leads to bounds on the heights of trace ideals of modules even in circumstances where we do not have the expected bounds for the heights of the order ideals

three partition refinement algorithms we present improved partition refinement algorithms for three problems lexicographic sorting relational coarsest partition and double lexical ordering our double lexical ordering algorithm uses a new efficient method for unmerging two sorted sets

asymptotically optimal blind calibration of uniform linear sensor arrays for narrowband gaussian signals an asymptotically optimal blind calibration scheme of uniform linear arrays for narrowband gaussian signals is proposed rather than taking the direct maximum likelihood ml approach for joint estimation of all the unknown model parameters which leads to a multi dimensional optimization problem with no closed form solution we revisit paulraj and kailath s p k s classical approach in exploiting the special toeplitz structure of the observations covariance however we offer a substantial improvement over p k s ordinary least squares ls estimates by using asymptotic approximations in order to obtain simple non iterative quasi linear optimally weighted ls owls estimates of the sensors gains and phases offsets with asymptotically optimal weighting based only on the empirical covariance matrix of the measurements moreover we prove that our resulting estimates are also asymptotically optimal w r t the raw data and can therefore be deemed equivalent to the ml estimates mle which are otherwise obtained by joint ml estimation of all the unknown model parameters after deriving computationally convenient expressions of the respective cramer rao lower bounds we also show that our estimates offer improved performance when applied to non gaussian signals and or noise as quasi mle in a similar setting the optimal performance of our estimates is demonstrated in simulation experiments with a considerable improvement reaching an order of magnitude and more in the resulting mean squared errors w r t p k s ordinary ls estimates we also demonstrate the improved accuracy in a multiple sources directions of arrivals estimation task

a complexity analysis of bisimilarity for value passing processes abstract we study the complexity of deciding bisimilarity between non deterministic processes with explicit primitives for manipulating data values in particular we consider a language with value passing input output of data and parametric definitions of processes we distinguish the case in which data cannot be tested and the case in which a simple equality test over data is permitted in the first case our main result shows that the problem is pspace hard for the full calculus in the second case we first show that the problem is conp complete in the fragment with value passing and no parametric definitions we then define a compositional polynomial time translation of the full calculus to the fragment with parametric definitions but no value passing the translation preserves bisimilarity this fact establishes the decidability of the full calculus and shows that the fragment without value passing is computationally equivalent to the full calculus for the latter bisimilarity is then proved to be exp complete finally we add to our language a parallel composition operator and show that for a certain restricted syntactic format the bisimilarity problem is still decidable and exp complete

cohen macaulay rees algebras and their specialization the rees algebra of an ideal j in a commutative ring r is by definition the graded algebra where t is an indeterminate in this paper we are concerned with proving that certain rees algebras are cohen macaulay and with answering under strong hypotheses the question if r s n is a factor ring of s and i jr when is i r a specialization of j s that is when is the natural epimorphism r a j s r i r an isomorphism these questions have interest partly because if s and s j s are cohen macaulay then so is gr s s j j j 16 1 and under these hypotheses if n is perfect and r s j s i r then r and 7 1 r are cohen macaulay too thus gr r is cohen macaulay and its torsion freeness and normality for exmple can be characterized in terms of analytic spreads as in 161 see section 3 for details we deal with the specialization question in section 1 it is answered for the case where as above s and 2 j s are cohen macaulay and n is 202

minimal free resolution of a finitely generated module over a regular local ring abstract numerical invariants of a minimal free resolution of a module m over a regular local ring r n can be studied by taking advantage of the rich literature on the graded case the key is to fix suitable n stable filtrations m of m and to compare the betti numbers of m with those of the associated graded module gr m m this approach has the advantage that the same module m can be detected by using different filtrations on it it provides interesting upper bounds for the betti numbers and we study the modules for which the extremal values are attained among others the koszul modules have this behavior as a consequence of the main result we extend some results by aramova conca herzog and hibi on the rigidity of the resolution of standard graded algebras to the local setting

a subspace method for estimating sensor gains and phases the problem of array signal processing under model errors is studied a signal subspace constraint is used to obtain a simple way for computing a set of sensor gains and phases consistent with a given set of doa angles then the issue of uniqueness of the set of sensor gains and phases and doa angles is addressed

rees algebras and almost linearly presented ideals abstract consider a grade 2 perfect ideal i in r k x 1 x d which is generated by forms of the same degree assume that the presentation matrix is almost linear that is all but the last column of consist of entries which are linear for such ideals we find explicit forms of the defining ideal of the rees algebra r i we also introduce the notion of iterated jacobian duals

rees algebras of linearly presented ideals abstract let i be a height two perfect ideal with a linear presentation matrix in a polynomial ring r k x 1 x d assume that i d 1 and i satisfies the artin nagata condition g d 1 we determine the defining ideal of the rees algebra r i explicitly and we show that r i is cohen macaulay

music maximum likelihood and cramer rao bound the performance of the music and ml methods is studied and their statistical efficiency is analyzed the cramer rao bound crb for the estimation problems is derived and some useful properties of the crb covariance matrix are established the relationship between the music and ml estimators is investigated as well a numerical study is reported of the statistical efficiency of the music estimator for the problem of finding the directions of two plane waves using a uniform linear array an exact description of the results is included

on rings with linear resolutions let r denote a positively graded noetherian ring over a field k r eori with r k and m the irrelevant maximal ideal generated by forms of degree 1 or let r be the m adic completion of such a ring if we consider x as a homomorphic image of a polynomial ring resp of a power series ring over k with a homogeneous defining ideal one may ask whether r has a minimal linear resolution as an s module cohen macaulay rings r of maximal embedding dimension e g have a linear resolution as sally showed in 9 she also proved formulas for the hilbert series hr and the poincari series pi pz schenzel lo later characterized cm rings r with a linear resolution and called these rings extremal he also gave formulas for hr and pi s goto see 3 1 below proved besides other facts the linearity of the resolution for buchsbaum rings of maximal embedding dimension i e edim r dim r e r i r 1 i r denotes the invariant of the buchsbaum ring r and a formula for hh in section 1 we generalize these results and characterize rings with a linear resolution which have a parameter system that forms a d sequence in the buchsbaum case we then show that the h pi and pf of such rings are determined by the embedding dimension the krull dimension the dimensions of the local cohomology modules as k vector spaces and what we call the type of r in section 2 we determine all buchsbaum rings r k r of maximal embedding dimension and multiplicity 2 sally proved in 8 that for a local cm ring r m of maximal embedding dimension the associated graded ring is also cm of maximal embedding dimension goto showed a similar result for buchsbaum rings of

a critical study of a self calibrating direction finding method for arrays the self calibrating direction finding method was developed by friedlander and weiss 1988 it refines the estimates of the array element gains and phases while it estimates the signal directions this study shows that the outputs of the method could be inaccurate even if noise is absent

on the dimension and integrality of symmetric algebras the symmetric algebra of an r module e sym e has a relatively simple description since it is given as a quotient of a polynomial ring over r modulo an ideal j generated by linear homogeneous polynomials our first result relates the grade of j to certain determinantal ideals associated to such presentation in particular at least when r is cohen macaulay the krull dimension of sym e can be easily estimated from such formulation the expression of the grade of j is also useful in setting up conditions for the integrality and or factoriality of sym e it turns out that the integrality of sym e for r regular is related to phenomena of cohen macaulayness of the symmetric algebra there are however considerable difficulties in translating the details of a projective resolution of e into properties of the ideal 3 this analysis is carried out for some modules finally we consider for an ideal i the symmetric algebra over the ring

sums of squares and varieties of minimal degree the study of nonnegativity and its relation with sums of squares is a basic challenge in real algebraic geometry the classification of varieties of minimal degree is one of the milestones of classical complex algebraic geometry the goal of this paper is to establish the deep connection between these apparently separate topics to achieve this let x p be an embedded real projective variety with homogeneous coordinate ring r the variety x has minimal degree if it is nondegenerate not contained in a hyperplane and deg x 1 codim x a homogeneous element f r of even degree is nonnegative if its evaluation at each real point of x is at least zero our main theorem is a broad generalization of hilbert s 1888 classification of nonnegative forms and provides a tight connection between real and complex algebraic geometry

the jacobian module of a lie algebra there is a natural way to associate to the commuting variety c a of an algebra a a module over a polynomial ring it serves as a vehicle to study the arithmetical properties of c a particularly cohen macaulayness the focus here is on lie algebras and some of their representations

approximation complexes of blowing up rings ii this paper is a sequel to 111 where we studied the relationships that hold between the arithmetical properties of the rees ring 9 i and the symmetric algebra sym l of an ideal i and some of their fibers and the depth properties of the koszul homology modules h i r on a set of generators of i the connection between these objects is realized by certain differential graded algebras the so called approximation complexes that are built out of ordinary koszul complexes these compiexes show however different sensitivities being acyclic in situations much broader than the usual context of regular sequences it has been found that certain extensions thereof d sequences and proper sequences play here that role of acyclic sequence the interplay between sequences and approximation complexes will be a basic point of view here where we give

smooth varieties of almost minimal degree abstract in this article we study non linearly normal smooth projective varieties x p r of deg x codim x p r 2 we first give geometric characterizations for x theorem 1 1 indeed x is the image of an isomorphic projection of smooth varieties x p r 1 of minimal degree also if x is not the veronese surface then there exists a smooth rational normal scroll y p r which contains x as a divisor linearly equivalent to h 2 f where h is the hyperplane section of y and f is a fiber of the projection morphism y p 1 by using these characterizations 1 we determine all the possible types of y from the type of x theorem 1 2 and 2 we investigate the relation between the betti diagram of x and the type of y theorem 1 3 in particular we clarify the relation between the number of generators of the homogeneous ideal of x and the type of y as an application we construct non linearly normal examples where the converse to theorem 1 1 in d eisenbud m green k hulek s popescu restriction linear syzygies algebra and geometry compos math 141 2005 1460 1478 fails to hold remark 2

schur functors and schur complexes the study of schur functors has a relatively long history its main impetus derived from representation theory originally in characteristic zero over the years however with the development of modular representations and algebraic geometry over fields of positive characteristic the need for a theory of universal polynomial functors increased and since the mid 1960s approaches to a characteristic free treatment of schur functors have been developing see for instance the recent book of green 1 l in which the treatments by carter and lusztig 6 higman 12 and towber 20 among others are discussed our own interest in such a treatment was awakened by the work of lascoux 141 on resolutions of determinantal ideals although his thesis treated only the characteristic zero case it suggested that a general and elementary theory of schur functors could be developed using only the rudiments of multilinear algebra involving the hopf algebra structures of the symmetric exterior and divided power algebras moreover this elementary development admitted of a natural generalization to the idea of schur complexes whose usefulness was demonstrated for instance in our construction of a universal minimal