from canonical tensor operators of su 3 and 3 to bi orthogonal coupling coefficients explicit expansion expansion of the matrix elements of su 3 and canonical tensor operators in terms of the bi orthogonal coupling coefficients and overlaps of the draayer akiyama construction are considered special bi orthogonal extremal isoscalar factors with subscripts as multiplicity labels and proportional to q racah q 6j coefficients or the generalized wilson rahman rational bi orthogonal functions in terms of balanced or hypergeometric series are used as a natural basis for extremal matrix elements of the highest weight component of the canonical tensor operators of su 3 in the generating function approach of biedenharn lohe and louck the expansion that is obtained triple sum together with previously derived asymmetric seed isofactors and elementary overlaps gives the explicit overlap coefficients and can be used to derive su 3 and canonical tensor operators as well as new explicit normalized seed isofactors with regge type symmetry specified for the minimal null space case

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

space complexity in propositional calculus we study space complexity in the framework of propositional proofs we consider a natural model analogous to turing machines with a read only input tape and such popular propositional proof systems as resolution polynomial calculus and frege systems we propose two different space measures corresponding to the maximal number of bits and clauses monomials that need to be kept in the memory simultaneously we prove a number of lower and upper bounds in these models as well as some structural results concerning the clause space for resolution and frege systems

design of stable parallel feedforward compensator and its application to synchronization problem this paper addresses the design problem of a stable parallel feedforward compensator v for a given siso lti plant p possibly being of non minimum phase and or having relative degree greater than one the objective of the problem is that their parallel interconnection p v becomes minimum phase having relative degree one based on the classical results of simultaneous stabilization a necessary and sufficient condition for solving the problem is presented as well as a design procedure for constructing such a compensator the proposed feedforward compensator allows the control system to have the three useful features 1 the ability that assigns the zeros of p v to a region of complex numbers having arbitrary negative real parts 2 infinite gain margin property of p v controlled by a static output feedback and 3 block diagonal structure of p v these features are extensively exploited in the synchronization problem of multi agent systems to achieve arbitrary fast convergence rate and to have the synchronized trajectory independent of the initial conditions and parameters of the involved dynamic controllers

symmetries of clebsch gordan coefficients of the quantum group uq n permutation symmetries for clebsch gordan coefficients of the quantum group uq n and for uq n 1 reduced clebsch gordan coefficients of uq n are derived in particular the formula relating clebsch gordan coefficients for the tensor product tm1 tm2 with clebsch gordan coefficients for the tensor product with permuted representations tm1 and tm is obtained where tm is one of the irreducible representations in the decomposition of tm1 tm2 contrary to the classical case coefficients connecting these clebsch gordan coefficients depend on gel fand tsetlin patterns of one of irreducible representations this dependence is explicitly determined

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

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

optimal simultaneous stabilization of linear single input systems via linear state feedback control simultaneous stabilization of a collection of linear time invariant systems is considered the aim is to develop a practical method for the design of a single feedback controller such that all the systems involved are stabilized with good transient responses hurwitz s necessary and sufficient conditions are used as the required set of constraints for simultaneous stability for transient behaviour we introduce the usual quadratic objective function as performance index for each system their sum is the objective function for the collection of systems and the problem is to minimize it by choosing a feedback gain vector subject to boundedness and the hurwitz stability constraints a computational technique is proposed for solving this problem the results obtained give good transient behaviour for illustration two numerical examples are solved

matrix elements and wigner coefficients for uq gl n in this series of papers the fundamental wigner coefficients and the matrix elements of the generators for the quantum group uq gl n are derived the uq gl n uq gl n 1 reduced wigner coefficients and reduced matrix elements are determined algebraically as eigenvalues of certain uq gl n 1 invariants in the quantum group uq gl n the matrix elements of the elementary quantum group generators in the gel fand tsetlin basis are derived below while all fundamental wigner coefficients are derived in the second paper of the series

algebraic design techniques for reliable stabilization in this paper we study two problems in feedback stabilization the first is the simultaneous stabilization problem which can be stated as follows given plants g 0 g 1 g l does there exist a single compensator c that stabilizes all of them the second is that of stabilization by a stable compensator or more generally a least unstable compensator given a plant g we would like to know whether or not there exists a stable compensator c that stabilizes g if not what is the smallest number of right half place poles counted according to their mcmillan degree that any stabilizing compensator must have we show that the two problems are equivalent in the following sense the problem of simultaneously stabilizing l 1 plants can be reduced to the problem of simultaneously stabilizing l plants using a stable compensator which in turn can be stated as the following purely algebraic problem given 2l matrices a 1 a l b 1 b l where a i b i are right coprime for all i does there exist a matrix m such that a i mb i is unimodular for all i conversely the problem of simultaneously stabilizing l plants using a stable compensator can be formulated as one of simultaneously stabilizing l 1 plants the problem of determining whether or not there exists an m such that a bm is unimodular given a right coprime pair a b turns out to be a special case of a question concerning a matrix division algorithm in a proper euclidean domain we give an answer to this question and we believe this result might be of some independent interest we show that given two n times m plants g 0 and g 1 we can generically stabilize them simultaneously provided either n or m is greater than one in contrast simultaneous stabilizability of two single input single output plants g 0 and g 1 is not generic

pp is as hard as the polynomial time hierarchy in this paper two interesting complexity classes pp and oplus text p are compared with ph the polynomial time hierarchy it is shown that every set in ph is polynomial time turing reducible to a set in pp and ph is included in text bp cdot oplus text p as a consequence of the results it follows that text pp subseteq text ph or oplus text p subseteq text ph implies a collapse of ph a stronger result is also shown every set in pp ph is polynomial time turing reducible to a set in pp

tensor operators and projection techniques in infinite dimensional representations of semi simple lie algebras polynomial identities satisfied by the infinitesimal generators of a semi simple lie group are employed to construct projection operators which project a tensor product representation v lambda x v v lambda finite dimensional and irreducible v an infinite dimensional representation admitting an infinitesimal character onto a primary subrepresentation such projection operators project out the generalised shift components of a tensor operator acting on an infinite dimensional representation v in direct analogy with the finite dimensional analysis of bracken and green 1971 applications of these methods to evaluating the matrix elements of the generators of the locally compact groups u p q and o p q in unitary discrete irreducible representations are discussed

the complexity of partial derivatives abstract let l denote the nonscalar complexity in k x 1 x n we prove l x 1 x n 3l using this we determine the complexity of single power sums single elementary symmetric functions the resultant and the discriminant as root functions up to order of magnitude also we linearly reduce matrix inversion to computing the determinant

random self reducibility of complete sets this paper generalizes the previous formal definitions of random self reducibility it is shown that even under a very general definition sets that are complete for any level of the polynomial hierarchy are not nonadaptively random self reducible unless the hierarchy collapses in particular np complete sets are not nonadaptively random self reducible unless the hierarchy collapses at the third level by contrast we show that sets complete for the classes pp and text mod m text p are random self reducible

a diagonal bound for cohomological postulation numbers of projective schemes abstract let x be a projective scheme over a field k and let f be a coherent sheaf of o x modules we show that the cohomological postulation numbers f i of f e g the ultimate places at which the cohomological hilbert functions n dim k h i x f n h f i n start to be polynomial for n 0 are polynomially bounded in terms of the cohomology diagonal h f i i i 0 dim f of f as a consequence we obtain that there are only finitely many different cohomological hilbert functions h f i if f runs through all coherent sheaves of o x modules with fixed cohomology diagonal in order to prove these results we extend the regularity bound of bayer and mumford computational algebraic geometry and commutative algebra proc cortona 1991 cambridge univ press 1993 pp 1 48 from graded ideals to graded modules moreover we prove that the castelnuovo mumford regularity of the dual f of a coherent sheaf of o p k r modules f is polynomially bounded in terms of the cohomology diagonal of f

simultaneous stabilization of single input single output discrete time systems the problem of finding one compensator which simultaneously stabilizes a family of single input single output siso discrete time plants is considered the family of plants is described by the transfer functions pz q q in q which are generated via a z transformation a number of assumptions describing the set of allowable plants are then given these assumptions include some regularity conditions on the plants and a minimum phase requirement the satisfaction of these assumptions guarantees the existence of a strictly proper stable compensator c z for simultaneous stabilization an iterative computation method is provided for control design

on the effective nullstellensatz let mathbb k be an algebraically closed field and let x subset mathbb k m be an n dimensional affine variety assume that f1 fk are polynomials which have no common zeros on x we estimate the degrees of polynomials a i in mathbb k x such that 1 ki 1aifi on x our estimate is sharp for k n and nearly sharp for k n now assume that f1 fk are polynomials on x let i f 1 dots f k subset mathbb k x be the ideal generated by fi it is well known that there is a number e i the noether exponent such that ie i i we give a sharp estimate of e i in terms of n deg x and deg fi we also give similar estimates in the projective case finally we obtain a result from the elimination theory if f 1 dots f n in mathbb k x 1 dots x n is a system of polynomials with a finite number of common zeros then we have the following optimal elimination phi i x i sum n j 1 f jg ij quad i 1 dots n where deg f jg ij le prod n i 1 deg f i

recent improvements in the complexity of the effective nullstellensatz we bring up to date the estimates on the complexity of the effective nullstellensatz and the membership problem

biorthogonal coupling coefficients of uq n the coupling wigner clebsch gordan coefficients and their isofactors for the unitary quantum algebras uq n with the repeating irreducible representations in the coproduct decomposition are considered generalizing the u n case the biorthogonal systems of the uq n isofactors with the dual multiplicity labels are constructed by means of the recoupling technique in terms of the isofactors with simpler multiplicity structure a first construction correlated with the inverted littlewood richardson rules gives the bilinear combinations of isofactors after applying the proportionality of the q recoupling racah coefficients to the boundary q isofactors an alternative recursive construction gives the nonorthogonal q isofactors satisfying the most elementary boundary conditions and proportional to the uq n 1 recoupling coefficients for some less restricted values of parameters some multiplicity free and more general uq n recoupling coefficients are found the blocks bilinear combinations of which equal to the resubducing coefficients of the complementary chains of q algebras are proposed to use for the orthonormalization of some uq n biorthogonal isofactors including the general uq 3 case

the information encoded in initial ideals we consider homogeneous ideals i and the initial ideal in i for the revlex order first we give a sequence of invariants computed from i giving better and better approximations to the initial ideal and ending in an equivalent description then we apply this to different settings in algebraic geometry to understand what information is encoded in the generic initial ideal of the ideal of a projective scheme we also consider the higher initial ideals as defined in a paper by fl0ystad in particular we show that giving the generic higher initial ideal of a space curve is equivalent to giving the generic initial ideal of a linked curve

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

multiplicity free uq n coupling coefficients the coupling wigner clebsch gordan coefficients of the unitary quantum algebras uq n are considered the tensorial properties of the generator powers and their ordered products used in the explicit projectors and weight lowering procedures are established different expressions for the multiplicity free isofactors of uq n coupling coefficients those coupling an arbitrary and symmetric representations are derived explicit expression of the arbitrary isofactors in terms of their boundary values are proposed proportionality of the semistretched isofactors to the stretched q recoupling coefficients of uq n 1 q analogues of 9j symbols is demonstrated the stretched isofactors of uq 3 are expressed in terms of clebsch gordan coefficients of uq 2

strong stabilization over polytopes the problem considered consists of finding a stable controller assigning a closed loop characteristic polynomial over a stability domain specified as a polytopic region of the space of characteristic polynomial coefficients the controller is required to be stable with respect to the same polytopic stability region specified for the closed loop system

a geometric effective nullstellensatz the purpose of this paper is to present a geometric theorem which clarifies and extends in several directions work of brownawell kollar and others on the effective nullstellensatz specifically we work on an arbitrary smooth complex projective variety x with the previous classical results corresponding to the case when x is projective space in this setting we prove a local effective nullstellensatz for ideal sheaves and a corresponding global division theorem for adjoint type bundles we also make explicit the connection with the intersection theory of fulton and macpherson finally constructions involving products of prime ideals that appear in earlier work are replaced by geometrically more natural conditions involving orders of vanishing along subvarieties much of the previous activity in this area has been algebraic in nature and seems perhaps not well known in detail among geometers therefore we have felt it worthwhile to include here a rather extended introduction we start with an overview of the questions and earlier work on them then we present the set up and statement of our main theorem we conclude with a series of examples which can be read before the general result of what it yields in special cases

cohomology of projective schemes from annihilators to vanishing we provide bounds on the castelnuovo mumford regularity in terms of defining equations by using elements that annihilates some cohomology modules inspired by works of miyazaki nagel schenzel and vogel the elements in these annihilators are provided either by liaison or by tight closure theories our results hold in any characteristic

discrete time pole placement with stable controller the classical polynomial approach to pole placement involves the computation of a particular solution of a given diophantine equation the solution corresponds to a proper controller whose order is fixed by the number of nonminimum phase zeros of the plant transfer function by its order and by the desired closed loop characteristic polynomial this controller might be unstable for some desired closed loop characteristic polynomials even though the plant can be stabilized by a stable controller now unstable controllers are known to deteriorate closed loop system performances compared with stable ones hence we propose a procedure to build for a specific class of plants a stable pole placement controller when the classical pole placement controller is unstable the properties of both stable and unstable controllers are compared via simulations it appears that for the considered plants the stable controller improves at least one of the following properties compared with its unstable counterpart step response transient gain margin measurement noise rejection the results obtained with our procedure are also compared with the results obtained via middleton s approach which consists in modifying the desired closed loop poles so as to obtain a stable and minimum phase controller with the classical pole placement method finally our pole placement design with stable controller is applied to solve a particular simultaneous partial pole placement problem

a state space interpretation of simultaneous stabilization a state space interpretation of some previously known results on simultaneous stabilization is given the results reviewed were obtained using the yuola lemma

circuit complexity proof complexity and polynomial identity testing the ideal proof system we introduce a new and natural algebraic proof system whose complexity measure is essentially the algebraic circuit size of nullstellensatz certificates this enables us to exhibit close connections between effective nullstellensatze proof complexity and algebraic circuit complexity in particular we show that any super polynomial lower bound on any boolean tautology in our proof system implies that the permanent does not have polynomial size algebraic circuits vnp vp we also show that super polynomial lower bounds on the number of lines in polynomial calculus proofs imply the permanent versus determinant conjecture note that there was no proof system prior to ours for which lower bounds on an arbitrary tautology implied any complexity class lower bound our proof system helps clarify the relationships between previous algebraic proof systems in doing so we highlight the importance of polynomial identity testing pit in proof complexity in particular we use pit to illuminate ac0 p frege lower bounds which have been open for nearly 30 years with no satisfactory explanation as to their apparent difficulty finally we explain the obstacles that must be overcome in any attempt to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity using the algebraic structure of our proof system we propose a novel route to such lower bounds although such lower bounds remain elusive this proposal should be contrasted with the difficulty of extending ac0 p circuit lower bounds to ac0 p frege lower bounds

sharp effective nullstellensatz the usual proofs of this result however give no information about the g1 s for instance they give no bound on their degrees this question was first considered by g hermann h she used elimination theory to get a bound on the degree of the gi s which was doubly exponential in the number of variables her results were later improved in mw and in th all these produce bounds that are doubly exponential in the number of variables a major breakthrough was achieved by brownawell b1 who proved the following result

multiplicity free wigner coefficients for semisimple lie groups i the u n pattern calculus this is the first paper in a series of two dedicated to a new algebraic determination of the multiplicity free reduced wigner coefficients for the lie groups u n and o n the approach employed enables a direct nonrecursive derivation of reduced wigner coefficients the absolute value squared of the reduced wigner coefficients is expressed as a rational polynomial function numerator polynomial divided by denominator polynomial whose weyl group symmetries are seen to fall out naturally in our approach from the transformation properties of polynomial functions determined by casimir invariants a unified treatment of the composition laws of reduced wigner coefficients and the evaluation of their associated denominator polynomials is presented which applies to both u n and o n an explicit formula for the numerator polynomials of u n is also derived the numerator polynomials for the orthogonal groups will be given in the second paper of the series

the pattern calculus for tensor operators in quantum groups an explicit algebraic evaluation is given for all q tensor operators with unit norm belonging to the quantum group uq u n and having extremal operator shift patterns acting on arbitrary uq u n irreps these rather complicated results are shown to be easily comprehensible in terms of a diagrammatic calculus of patterns a more conceptual derivation of these results is discussed using fundamental properties of the q 6j coefficients

np is as easy as detecting unique solutions abstract for every known np complete problem the number of solutions of its instances varies over a large range from zero to exponentially many it is therefore natural to ask if the inherent intractability of np complete problem is caused by this wide variation we give a negative answer to this question using the notion of randomized polynomial time reducibility we show that the problems of distinguishing between instances of sat having zero or one solution or of finding solutions to instances of sat having a unique solution are as hard as sat under randomized reductions several corollaries about the difficulty of specific problems follow for example computing the parity of the number of solutions of a sat formula is shown to be np hard and deciding if a sat formula has a unique solution is shown to be d p hard under randomized reduction central to the study of cryptography is the question as to whether there exist np problems whose instances have solutions that are unique but are hard to find our result can be interpreted as strengthening the belief that such problems exist

lower bounds in a parallel model without bit operations we define a natural and realistic model of parallel computation called the pram model without bit operations it is like the usual pram model the main difference being that no bit operations are provided it encompasses virtually all known parallel algorithms for weighted combinatorial optimization and algebraic problems in this model we prove that for some large enough constant b the mincost flow problem for graphs with n vertices cannot be solved deterministically or with randomization in sqrt n b expected time using 2 sqrt n b processors this is so even if we restrict every cost and capacity to be an integer nonnegative if it is a capacity of bitlength at most an for some large enough constant a a similar lower bound is also proved for the max flow problem it follows that these problems cannot be solved in our model deterministically or with randomization in omega n c expected time with 2 omega n c processors where c is an appropriate positive constant and n is the total bitlength of the input since these problems were known to be p complete this provides concrete support for the belief that p completeness implies high parallel complexity and for the p not nc conjecture our lower bounds also extend to the pram model with limited bit operations which provides instructions for parity and left or right shift by one bit our proof is based on basic algebraic geometry so we investigate if the algebrogeometric approach could also work for the p versus nc problem our results support this possibility and a close analysis of the limitation of our technique in this context suggests that such a proof of p not nc should somehow use geometric invariant theory in a deep way

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

relations for clebsch gordan and racah coefficients in suq 2 and yang baxter equations relations are exploited among clebsch gordan cg and racah coefficients in the algebra suq 2 known as a deformation of su 2 these are used to show that the yang baxter yb relation for the irf interaction round a face model results from one of the symmetry relations for the 9 j symbol specific to suq 2 and that in an asymptotic limit this yb relation becomes the yb relation for the two dimensional vertex model the racah coefficient which has a particularly simple dependence on q is efficiently used such that an asymptotic limit of the racah coefficient is the cg coefficient and another limit gives the factorized s matrix of the vertex model

towards the canonical tensor operators of uq 3 i the maximal null space case generalizing the su 3 canonical tensor operator concept biedenharn and louck to the quantum algebra uq 3 the wigner clebsch gordan coefficients of uq 3 with repeating irreducible representations are considered extremal projectors of the quantum algebra uq 3 in terms of the ordered generator polynomials are used for evaluation of the bilinear combinations of the uq 3 canonical isoscalar factors explicit expressions of the uq 3 isofactors corresponding to the maximal null space case of the uq 3 unit canonical tensor operators and their normalization factors denominator functions are presented the transposition and conjugation phase factors for the su 3 and uq 3 canonical isofactors are correlated with phases and zeros of boundary isofactors invariance of the canonical isofactors or absence of such invariance under interchange of the tensor operator and the initial or final state parameters is correlated with the existence and invariance or numerical degeneracy of the usual splitting

new perspective on the u n wigner racah calculus ii elementary reduced wigner coefficients for u n for pt i see ibid vol 20 no 9 p 2241 50 1987 exploiting the powerful formalism of the vector coherent state theory expounded in part i the group theoretical content of the complementarity principle and a novel interpretation of the operator pattern of biedenharn and louck the authors rederive in a simple fashion all u n u n 1 reduced wigner coefficients for the elementary u n tensors

cook s versus valiant s hypothesis we first investigate the dependency of our complexity concepts on the field k we prove that for algebraically closed fields k the truth of valiant s hypothesis vp vnp depends at most on the characteristic of k

geometric complexity theory i an approach to the p vs np and related problems we suggest an approach based on geometric invariant theory to the fundamental lower bound problems in complexity theory concerning formula and circuit size specifically we introduce the notion of a partially stable point in a reductive group representation which generalizes the notion of stability in geometric invariant theory due to mumford geometric invariant theory springer verlag berlin 1965 then we reduce fundamental lower bound problems in complexity theory to problems concerning infinitesimal neighborhoods of the orbits of partially stable points we also suggest an approach to tackle the latter class of problems via construction of explicit obstructions

on the q symplecton realization of the quantum group suq 2 a q symplecton realization of the quantum group suq 2 is constructed giving explicitly the product law for the weyl ordered characteristic polynomials a new form of the q racah coefficients is obtained which is especially adapted to proving a new symmetry for these functions a surprising result is that unlike the su 2 case there are two distinct invariant q triangle functions in the q symplecton extension

conditions for generic initial ideals to be almost reverse lexicographic abstract let i be a homogeneous artinian ideal in a polynomial ring r k x 1 x n over a field k of characteristic 0 we study an equivalent condition for the generic initial ideal gin i with respect to reverse lexicographic order to be almost reverse lexicographic as a result we show that the moreno socias conjecture implies the froberg conjecture and the pardue conjecture and for the case codim i 3 we show that r i has the strong lefschetz property if and only if gin i is almost reverse lexicographic finally we give a positive partial answer to the moreno socias conjecture and hence to the pardue conjecture

a stratification of the hilbert scheme of points in the projective plane this result is analogous to a theorem of iarrobino and brian on on the stratification of the punctual hilbert scheme where a different hilbert function defined by initial forms instead is used however cp i l thm 2 9 p 26 the proof essentially makes use of the smoothness of h a cp f p 517 and a general result of bialynicki birula on the action of the multiplicative group on a smooth projective scheme as already was done in a study of the homology of h a by ellingsrud and stromme this method is applied here to a study of h a around points which are invariant under the borel group in the appendix a method for computing the dimension of h is represented which rests on iarrobino s description of the tangent space to certain points of h a and the theorem of bialynicki birula finally i would like to mention that hirschowitz has outlined a general program for studying stratifications including those defined by hilbert functions cp h p 175 but the result obtained here can at best be regarded only as the most simple perhaps non trivial example

quantum group invariants and link polynomials a general method is developed for constructing quantum group invariants and determining their eigenvalues applied to the universal matrix this method leads to the construction of a closed formula for link polynomials to illustrate the application of this formula the quantum groups uq es uq so 2m 1 and uq gl m are considered as examples and corresponding link polynomials are obtained

lefschetz properties and hyperplane arrangements abstract in this article we study the weak and strong lefschetz properties and the related notion of almost revlex ideal in the non artinian case proving that several results known in the artinian case hold also in this more general setting we then apply the obtained results to the study of the jacobian algebra of hyperplane arrangements

multiplicity free wigner coefficients for semisimple lie groups ii a pattern calculus for o n in this paper a direct algebraic derivation of a large class of multiplicity free reduced wigner coefficients is presented for the orthogonal groups in particular all elementary reduced wigner coefficients including those for the fundamental spinor and tensor representations are obtained the results are presented in a form directly analogous to the corresponding results obtained for the unitary groups

direct methods for primary decomposition leti be an ideal in a polynomial ring over a perfect field we given new methods for computing the equidimensional parts and radical ofi for localizingi with respect to another ideal and thus for finding the primary decomposition ofi our methods rest on modern ideas from commutative algebra and are direct in the sense that they avoid the generic projections used by hermann 1926 and all others until now

sharp estimates for the arithmetic nullstellensatz we present sharp estimates for the degree and the height of the polynomials in the nullstellensatz over the integer ring zz the result improves previous work of philippon berenstein yger and krick pardo we also present degree and height estimates of intrinsic type which depend mainly on the degree and the height of the input polynomial system as an application we derive an effective arithmetic nullstellensatz for sparse polynomial systems the proof of these results relies heavily on the notion of local height of an affine variety defined over a number field we introduce this notion and study its basic properties

borel fixed ideals and reduction number abstract the aim of this paper is to study the relationship between the reduction number and borel fixed ideals in all characteristics especially it is shown that r r i r r i lex where i lex denotes the unique lex segment ideal whose hilbert function is equal to that of i this solves a recent question by conca

a sufficient condition for simultaneous stabilization the condition under which it is possible to find a single controller that stabilizes k single input single output linear time invariant systems p sub i s i 1 k is investigated the concept of avoidance in the complex plane is introduced and used to derive a sufficient condition for k systems to be simultaneously stabilizable a method for constructing a simultaneous stabilizing controller is also provided and is illustrated by an example

overlaps of the biorthogonal 3 coupling coefficients and related basic hypergeometric and other q factorial series the intermediate expansion technique is developed for the overlap coefficients of different biorthogonal coupled states and coupling coefficients of the quantum algebra and group su 3 with repeating irreducible representations and the role of separate basic and classical hypergeometric functions is demonstrated some expansion coefficients such as new prime overlap functions equivalent to the bilinear combinations of the different boundary isofactors are expressed in terms of the balanced saalschutzian basic or classical hypergeometric series these new overlap functions their analytical inversion some known triangular expansion matrices transition matrices related to definite racah coefficients the well poised or series and compositions of q factorial series resembling the well poised and series or equivalent to and series provide themselves as generators for mutual expansion of different non orthonormal systems of and su 3 isofactors which may be orthonormalized to the paracanonical or with a definite caution canonical version

the intractability of resolution abstract we prove that for infinitely many disjunctive normal form propositional calculus tautologies the length of the shortest resolution proof of cannot be bounded by any polynomial of the length of the tautologies we use were introduced by cook and reckhow 1979 and encode the pigeonhole principle extended resolution can furnish polynomial length proofs of these formulas

irreducible tensor operators in the regular coaction formalisms of compact quantum group algebras the defining conditions for the irreducible tensor operators associated with the unitary irreducible corepresentations of compact quantum group algebras are deduced first in both the right and left regular coaction formalisms in each case it is shown that there are two types of irreducible tensor operator which may be called ordinary and twisted the consistency of the definitions is demonstrated and various consequences are deduced including generalizations of the wigner eckart theorem for both the ordinary and twisted operators also included are discussions within the regular coaction formalisms for compact quantum group algebras of inner products basis functions projection operators clebsch gordan coefficients and two types of tensor product of corepresentations the formulation of quantum homogeneous spaces for compact quantum group algebras is discussed and the defining conditions for the irreducible tensor operators associated with such quantum homogeneous spaces and with the uni

some results on simultaneous stabilization with multiple domains of stability abstract in this paper the problem of simultaneous stabilization in the case where each plant has a different domain of stability associated with it is studied specifically it is supposed that one is given plants p0 p1 pr together with domains of stability d0 dr and the objective is to find a common controller c such that the poles of the closed loop transfer matrix h pi c are all inside the corresponding domain of stability di for i 0 r this formulation of the simultaneous stabilization problem generalizes earlier problem formulations wherein it was assumed that all the domains of stability were the same and is believed to be a more realistic formulation of the problem of preserving system stability in the event of sensor or actuator failure in the paper necessary and sufficient conditions for the existence of a common controller are derived in the case where one of the domains say d0 is a subset of the rest the case of two plants p0 p1 with d0 a subset of d1 is studied in great detail and the following result is established the two plants can be simultaneously stabilized in the above generalized sense if and only if they can be simultaneously stabilized with respect to the larger of the two regions more precisely there exists a controller c such that the poles of h pi c lie inside di for i 0 1 if and only if there exists a controller such that the poles of bothh p0 c and of h p1 c lie inside the larger region d1

geometric complexity theory ii towards explicit obstructions for embeddings among class varieties in k d mulmuley and m sohoni siam j comput 31 2001 pp 496 526 henceforth referred to as part i we suggested an approach to the p vs np and related lower bound problems in complexity theory through geometric invariant theory in particular it reduces the arithmetic characteristic zero version of the np not subseteq p conjecture to the problem of showing that a variety associated with the complexity class np cannot be embedded in a variety associated with the complexity class p we shall call these class varieties associated with the complexity classes p and np this paper develops this approach further reducing these lower bound problems which are all nonexistence problems to some existence problems specifically to proving the existence of obstructions to such embeddings among class varieties it gives two results towards explicit construction of such obstructions the first result is a generalization of the borel weil theorem to a class of orbit closures which include class varieties the second result is a weaker form of a conjectured analogue of the second fundamental theorem of invariant theory for the class variety associated with the complexity class nc these results indicate that the fundamental lower bound problems in complexity theory are in turn intimately linked with explicit construction problems in algebraic geometry and representation theory the results here were announced in k d mulmuley and m sohoni in advances in algebra and geometry hyderabad 2001 hindustan book agency new delhi india 2003 pp 239 261

on interpolation and automatization for frege systems the interpolation method has been one of the main tools for proving lower bounds for propositional proof systems loosely speaking if one can prove that a particular proof system has the feasible interpolation property then a generic reduction can usually be applied to prove lower bounds for the proof system sometimes assuming a usually modest complexity theoretic assumption in this paper we show that this method cannot be used to obtain lower bounds for frege systems or even for tc0 frege systems more specifically we show that unless factoring of blum integers is feasible neither frege nor tc0 frege has the feasible interpolation property in order to carry out our argument we show how to carry out proofs of many elementary axioms theorems of arithmetic in polynomial sized tc0 frege as a corollary we obtain that tc0 frege as well as any proof system that polynomially simulates it is not automatizable under the assumption that factoring of blum integers is hard we also show under the same hardness assumption that the k provability problem for frege systems is hard

arithmetic circuits the chasm at depth four gets wider in their paper on the chasm at depth four agrawal and vinay have shown that polynomials in m variables of degree o m which admit arithmetic circuits of size 2 o m also admit arithmetic circuits of depth four and size 2 o m this theorem shows that for problems such as arithmetic circuit lower bounds or black box derandomization of identity testing the case of depth four circuits is in a certain sense the general case in this paper we show that smaller depth four circuits can be obtained if we start from polynomial size arithmetic circuits for instance we show that if the permanent of nxn matrices has circuits of size polynomial in n then it also has depth 4 circuits of size n o n l o g n if the original circuit uses only integer constants of polynomial size then the same is true for the resulting depth four circuit these results have potential applications to lower bounds and deterministic identity testing in particular for sums of products of sparse univariate polynomials we also use our techniques to reprove two results on the existence of nontrivial boolean circuits of constant depth for languages in logcfl reduction to polylogarithmic depth for arithmetic circuits of polynomial size and polynomially bounded degree

fast parallel computation of polynomials using few processors it is shown that any multivariate polynomial of degree d that can be computed sequentially in c steps can be computed in parallel in o log d log c log d steps using only cd o 1 processors

fractional representation algebraic geometry and the simultaneous stabilization problem an explicit relationship between the fractional representation approach feedback system design and the algebro geometric approach to system theory is formulated and used to derive a global solution to the feedback system problem these techniques are then applied to the simultaneous stabilization problem yielding a natural geometric criterion for a set of plants to be simultaneously stabilized by a single compensator

tensor operators for quantum groups and applications tensor operators are defined for hopf algebras if the hopf algebra is quasitriangular the properly defined tensor product of tensor operators is a tensor operator in one application the problem of defining invariant correlation functions for the uq sl 2 symmetric spin 1 2 heisenberg chain is considered in another application it is shown that suitably modified bosonic and fermionic q creation and annihilation operators are tensor operators of uq sl n

the complexity of the word problems for commutative semigroups and polynomial ideals abstract any decision procedure for the word problems for commutative semigroups and polynomial deals inherently requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied this bound is achieved by a simple procedure for the semigroup problem