arithmetic of normal rees algebras two reasons drive this study of the arithmetical properties of blow up like algebras first one still lacks effective criteria for the rees algebra of an ideal to be normal and second there are several new phenomena related to torsionfree symmetric algebras of modules notably bundlesthat need clarification as yet another related question there is the description of the canonical modules of such algebras a necessary step in determining their gorenstein loci the main thread for these questions is the understanding of the divisor class group essentially defined by a pair made up of a morphism and a free subgroup of finite rank the behaviour of the mapping has puzzling

asymptotic stability of two dimensional digital filters under quantization the stability of 2 d digital filters implemented with two s complement quantization is investigated the difference equation model is considered for the following cases i quantization after each product and ii quantization after a complete sum of products for both cases sufficient conditions are established for asymptotic stability the 2 d state space roesser s 1975 model is also considered under two s complement quantization and a sufficient condition is derived for its stability some examples are given to illustrate the results

m sequences graph ideals and ladder ideals of linear type in this paper we study monomial ideals and ladder determinantal ideals of linear type and their blow up algebras our main tools are grobner bases and sagbi bases deformations and the notion of m sequence of monomials w x let r s k x be a polynomial ring over a field k equipped with a monomial order t let i be an ideal of r generated by polynomials f f consider the presentation 1 s w x w x w x c r t s r t t a r i s r f t f t 1 s i s

the abundance of unknots in various models of polymer loops a veritable zoo of different knots is seen in the ensemble of looped polymer chains whether created computationally or observed in vitro at short loop lengths the spectrum of knots is dominated by the trivial knot unknot the fractional abundance of this topological state in the ensemble of all conformations of the loop of n segments follows a decaying exponential form exp n n0 where n0 marks the crossover from a mostly unknotted i e topologically simple to a mostly knotted i e topologically complex ensemble in the present work we use computational simulation to look closer into the variation of n0 for a variety of polymer models among models examined n0 is smallest about 240 for the model with all segments of the same length it is somewhat larger 305 for gaussian distributed segments and can be very large up to many thousands when the segment length distribution has a fat power law tail

two dimensional dissipative control and filtering for roesser model this paper investigates the problems of two dimensional 2 d dissipative control and filtering for a linear discrete time roesser model first a novel sufficient condition is proposed such that the discrete time roesser system is asymptotically stable and 2 d q s r hbox alpha dissipative special cases such as 2 d passivity performance and 2 d h infty performance and feedback interconnected systems are also discussed based on this condition new 2 d q s r hbox alpha dissipative state feedback and output feedback control problems are defined and solved for a discrete time roesser model the design problems of 2 d q s r hbox alpha dissipative filters of observer form and general form are also considered using a linear matrix inequality lmi approach two examples are given to illustrate the effectiveness and potential of the proposed design techniques

a computer aided test for the absence of limit cycles in fixed point digital filters the problem of determining the existence of limit cycles in direct form digital filters with quantization nonlinearities is considered a method based on an exhaustive search is introduced it is applicable to all direct form digital filters regardless of the type of quantization arithmetic the number of quantizers and the order of the filter the equivalence between the concept of global asymptotic stability and the exclusion of limit cycles is proved for linearly stable finite wordlength digital filters the robustness of this test with respect to coefficient perturbation in the digital filter is presented the computational aspects of the algorithm are also discussed the algorithm is shown to be capable of always determining the global asymptotic stability of the tested realization

l gain performance analysis for two dimensional roesser systems with persistent bounded disturbance and saturation nonlinearity the l gain approach has been an essential tool in one dimensional system theory however limited results have been presented in the literature for the two dimensional 2 d l gain approach this paper investigates the l gain performance for 2 d systems in the roesser model with persistent bounded disturbance input and saturation nonlinearity a linear matrix inequality lmi based condition is established to reduce the effect of persistent bounded disturbance input on 2 d systems within a given disturbance attenuation level based on the discrete jensen inequality lower bounds lemma and diagonally dominant matrices we apply the obtained results to the l gain performance analysis for 2 d digital filters with saturation arithmetic

bibo stability of multidimensional md shift invariant discrete systems bibo stability of 1 d and multidimensional md shift varying discrete systems is analyzed the test for one particular shift varying md system involves one md linear shift invariant stability test only the stability conditions derived also provides insight into robustness and margin of stability of the shift varying system two classes of systems are investigated 1 shift varying discrete direct realization systems and 2 shift varying discrete state space realization systems the results extend an implication of the perron frobenius theorem from 1 d shift invariant systems to shift invariant and or multidimensional systems

on the flatness of models of certain shimura varieties of pel type consider a pel shimura variety associated to a unitary group that splits over an unramified extension of q p rapoport and zink have defined a model of the shimura variety over the ring of integers of the completion of the reflex field at a place lying over p with parahoric level structures at p we show that this model is flat as conjectured by rapoport and zink and that its special fibre is reduced

a stability analysis of two dimensional nonlinear digital state space filters two different methods for a stability analysis of 2 d nonlinear discrete state space systems under zero input conditions are provided the first method reduces the task of testing a 2 d discrete nonlinear or shift varying system to a single 2 d linear stability test of a system matrix with nonnegative system matrix entries the second method is based on a number of norm tests for products of extreme matrices and can be considered the 2 d counterpart of the method for 1 d systems described by k t erickson and a n michel 1985 both of the introduced methods are based on the sector description of the nonlinearity and can be used to analyze digital filter stability under finite word length effects

some remarks on the robust schur stability analysis of interval polynomials the schur stability analysis of an interval polynomial family can be quickly performed through a unique suitably defined extreme polynomial the purpose of this article is to provide some improvements with respect to the actually existing methods based on this approach

algebras defined by powers of determinantal ideals let x be an m x n matrix of indeterminates over a field k a k x and i z x the ideal generated by the t minors of x the main objects of this article are the rees algebra 2 y these characteristics will be called non exceptional we show how this result generalizes to an arbitrary integral domain of coefficients the intersection of primary ideals which gives i in non exceptional characteristics always is the integral closure of i it follows immediately from the primary decomposition that the powers i are integrally closed in non exceptional characteristics therefore is a normal domain and the primary decomposition of i turns out easy giving some insight into the structure of y an interesting observation the primary decomposition of the ideals i can be computed very quickly if one knows in advance that all these ideals are integrally closed the best results are obtained in characteristic 0 since one has a multiplicity free action of the linearly reductive group gl m k x gl n k on k x under which z is stable cf dep or bv sect 111 applying the theory of u invariants kraft kr one shows that g has rational singularities in particular and hence 9 are cohen macaulay rings we have no doubt that and 3 are cohen macaulay in arbitrary non exceptional characteristic it seems however that in exceptional characteristic they are as far as possible from this property the case t 2

on support varieties for lie superalgebras and finite supergroup schemes abstract we study the spectrum of the cohomology rings of cocommutative hopf superalgebras restricted and non restricted lie superalgebras and finite supergroup schemes we also investigate support varieties in these settings and demonstrate that they have the desirable properties of such a theory we completely characterize support varieties for finite supergroup schemes over algebraically closed fields of characteristic zero while for non restricted lie superalgebras in positive characteristic we obtain descriptions that are suggestive of varieties previously defined by duflo and serganova in characteristic zero but which have no known cohomological interpretation our computations for restricted lie superalgebras and infinitesimal supergroup schemes provide natural generalizations of foundational results of friedlander and parshall and of bendel friedlander and suslin in the classical setting

average size of random polygons with fixed knot topology we have evaluated by numerical simulation the average size r k of random polygons of fixed knot topology k ensuremath emptyset 3 1 3 1 ensuremath sharp 4 1 and we have confirmed the scaling law r k 2 ensuremath sim n 2 ensuremath nu k for the number n of polygonal nodes in a wide range n 100 char21 2200 the best fit gives 2 ensuremath nu k ensuremath simeq 1 11 char21 1 16 with good fitting curves in the whole range of n the estimate of 2 ensuremath nu k is consistent with the exponent of self avoiding polygons in a limited range of n n ensuremath gtrsim 600 however we have another fit with 2 ensuremath nu k ensuremath simeq 1 01 char21 1 07 which is close to the exponent of random polygons

limit cycles due to roundoff in state space digital filters a deterministic analysis is presented for limit cycles due to roundoff in state space digital filters two new sufficient conditions are established for the roundoff stability i e suppression of limit cycles in digital filters using these results restrictions on the location of poles are found for various realizations to be free from roundoff limit cycles

two s complement quantization in two dimensional state space digital filters two dimensional 2 d state space digital filters are investigated for stability with and without quantization a sufficient condition is obtained for the stability of the linear fornasini marchesini 1976 state space model the same model is then studied for stability when implemented using two s complement truncation quantization and a sufficient condition for asymptotic stability of the nonlinear filter is obtained in the process a theorem is proved which gives a sufficient condition for the stability of a one dimensional 1 d state space digital filter under the same type of quantization

a solvable lie algebra condition for stability of linear multidimensional systems this note analyzes exponential stability of a class of linear discrete multidimensional systems using a multidimensional comparison principle for estimating the system componentwise exponential convergence and a solvable lie algebra condition a sufficient condition for exponential stability of linear multidimensional systems is presented the stability condition can be easily examined by computing the system matrices in finite steps this is demonstrated by an example

on the variety of complexes publisher summary this chapter focuses on the characteristic free study of the buchsbaum eisenbud varieties of complexes in detail with the use of young diagrams and young tableaux to give an explicit basis for buchsbaum and eisenbud have studied the varieties of complexes from the point of view of resolutions and have proven the results for the variety w k 1 a young tableau is a filling of the boxes of a young diagram with integers out of 1 2 and is called the shape of the young tableau the chapter presents a proposition that states that ach element in can be written as a linear combination of standard multitableaux

on categories of equivariant d modules abstract let x be a variety with an action by an algebraic group g in this paper we discuss various properties of g equivariant d modules on x such as the decompositions of their global sections as representations of g when g is reductive and descriptions of the categories that they form when g acts on x with finitely many orbits the category of equivariant d modules is equivalent to the category of finite dimensional representations of a finite quiver with relations we describe explicitly these categories for irreducible g modules x that are spherical varieties and show that in such cases the quivers are almost always representation finite i e with finitely many indecomposable representations

stability of distributed 3 d systems implemented on grid sensor networks using the givone roesser and the fornasini marchesini state space models for m d systems a method for distributed information processing in rectangular grid based wireless sensor networks is presented it can be used for distributed implementation of any general linear system on a grid sensor network nonlinearities caused by finite precision number representation schemes used for in node computations and inter node communication are modeled stability of the system is analyzed with special consideration given to the influence of inter node communication on system dynamics a necessary and sufficient condition for global asymptotic stability under finite precision arithmetic is established simulation results are presented to illustrate the theoretical results

intersection theory of linear embeddings we study intersection theoretic properties of subschemes defined by ideal sheaves of linear type in particular their behavior with respect to blowups and segre classes

delay dependent stability for 2d systems with time varying delay subject to state saturation in the roesser model this paper addresses the problem of delay dependent stability of 2d systems with time varying delay subject to state saturation in the roesser model by introducing diagonally dominant matrices new delay dependent conditions are obtained in terms of linear matrix inequalities lmis where the lower and upper delay bounds along horizontal and vertical directions respectively are known numerical examples are provided to demonstrate the proposed results

limits of analogy between self avoidance and topology driven swelling of polymer loops the work addresses the analogy between trivial knotting and excluded volume in looped polymer chains of moderate length where the effects of knotting are small a simple expression for the swelling seen in trivially knotted loops is described and shown to agree with simulation data contrast between this expression and the well known expression for excluded volume polymers leads to a graphical mapping of excluded volume to trivial knots which may be useful for understanding where the analogy between the two physical forms is valid the work also includes description of a new method for the computational generation of polymer loops via conditional probability although computationally intensive this method generates loops without statistical bias and thus is preferable to other loop generation routines in the region of interest

scaling behavior of random knots using numerical simulations we investigate how overall dimensions of random knots scale with their length we demonstrate that when closed non self avoiding random trajectories are divided into groups consisting of individual knot types then each such group shows the scaling exponent of 0 588 that is typical for self avoiding walks however when all generated knots are grouped together their scaling exponent becomes equal to 0 5 as in non self avoiding random walks we explain here this apparent paradox we introduce the notion of the equilibrium length of individual types of knots and show its correlation with the length of ideal geometric representations of knots we also demonstrate that overall dimensions of random knots with a given chain length follow the same order as dimensions of ideal geometric representations of knots

probability theory of random polygons from the quaternionic viewpoint we build a new probability measure on closed space and plane polygons the key construction is a map given by hausmann and knutson using the hopf map on quaternions from the complex stiefel manifold of 2 frames in n space to the space of closed n gons in 3 space of total length 2 our probability measure on polygon space is defined by pushing forward haar measure on the stiefel manifold by this map a similar construction yields a probability measure on plane polygons that comes from a real stiefel manifold the edgelengths of polygons sampled according to our measures obey beta distributions this makes our polygon measures different from those usually studied which have gaussian or fixed edgelengths one advantage of our measures is that we can explicitly compute expectations and moments for chord lengths and radii of gyration another is that direct sampling according to our measures is fast linear in the number of edges and easy to code some of our methods will be of independent interest in studying other probability measures on polygon spaces we define an edge set ensemble ese to be the set of polygons created by rearranging a given set of n edges a key theorem gives a formula for the average over an ese of the squared lengths of chords skipping k vertices in terms of k n and the edgelengths of the ensemble this allows one to easily compute expected values of squared chord lengths and radii of gyration for any probability measure on polygon space invariant under rearrangements of edges 2014 wiley periodicals inc

a fast direct sampling algorithm for equilateral closed polygons sampling equilateral closed polygons is of interest in the statistical study of ring polymers over the past 30 years previous authors have proposed a variety of simple markov chain algorithms but have not been able to show that they converge to the correct probability distribution and complicated direct samplers which require extended precision arithmetic to evaluate numerically unstable polynomials we present a simple direct sampler which is fast and numerically stable and analyze its runtime using a new formula for the volume of equilateral polygon space as a dirichlet type integral

the robustness properties of univariate and multivariate reciprocal polynomials the author investigates the robustness properties of univariate and multivariate reciprocal polynomials that are nonzero on the unit circle and the unit polycircle respectively he shows that any polytope of univariate reciprocal polynomials are nonzero on the unit circle if and only if a set of real valued rationals corresponding to its vertices are entirely either positive or negative on the unit circle ensuring that these vertex rationals are entirely either positive or negative on the unit circle can be carried out by the tests described by lakshmanan 1992 when these existing tests are combined with the results contained in this paper it provides a complete procedure for testing the nonzeroness of polytopes of univariate reciprocal polynomials over the unit circle he shows that this result generalizes to the case of multivariate polynomials for any polytope of multivariate polynomials to be nonzero on the unit polycircle it is necessary and sufficient that a set of real valued multivariate rationals corresponding to its vertices are entirely either positive or negative on the unit polycircle again by using the test the positivity or the negativity of the vertex rationals can be ensured as well thereby resulting in a complete procedure for testing the nonzeroness of an entire polytope of multivariate reciprocal polynomials over the unit polycircle although he develops the results for polytopic families he then extends those results to the case of non polytopic reciprocal polynomial families

new criteria for asymptotic stability of one and multidimensional state space digital filters in fixed point arithmetic this paper addresses the problem of global asymptotic stability of one dimensional 1 d and multidimensional m d digital filters with any combination of overflow and quantization nonlinearities the stability analysis is carried out using 1 d and m d state space representations the approach introduced allows one to determine the stability behavior of single input single output systems with overflow and quantization nonlinearities the new criteria based on previous stability results of digital filters with quantization schemes are applicable to all arithmetic schemes for the first time results concerning general state variable representations of 1 d and m d digital filters with the naturally occurring combination of two s complement truncation quantization and overflow are reported furthermore significantly improved stability regions are obtained for digital filters with roundoff nonlinearities

on the conormal bundle of the determinantal variety call w such variety if we choose bases b v a a b u v i b for each module we can identify w with the variety of pairs of matrices y x with entries in r x being an m x n matrix y an ii x m matrix such that x y omx y x onxm now let a be the coordinate ring of the affine space a of dimension 2 m x n i e the polynomial ring r x yji i l m j i n and

criteria for the absence of limit cycles in two dimensional discrete systems sufficient conditions for the nonexistence of zero input limit cycles in two dimensional systems which contain a finite number of memoryless nonlinearities are presented two different conditions are given the first one is based on the frequency domain representation of the linear part of the discrete system and it is shown to be more general than the one already presented in 1 the second condition is formulated using the state space representation and is based on the properties of quasidominant matrices and on partial results on the 2 d lyapunov equation

stability and absence of overflow oscillations for 2 d discrete time systems it is shown that a 2 d linear time invariant discrete time system could be bounded input bounded output bibo unstable even if the 2 d system which is implemented by finite word length is free of overflow oscillations with zero input according to a previous criterion for the absence of overflow oscillations in 2 d discrete time systems an improved criterion is presented to ensure both stability properties

stability analysis of multidimensional m d direct realization digital filters under the influence of nonlinearities a stability test of m d direct realization digital filters with nonlinearities such as quantization overflow saturation etc is presented the method presented reduces a given nonlinear m d system to a nonlinear 1 d system such that the stability of the latter is sufficient for the stability of the former to check stability of the nonlinear 1 d system some new results on 1 d recursive systems with positive coefficients are applied the case of multiple nonlinearities in multidimensions is also addressed examples are given illustrating the application of the method

robust stability of discrete time state delayed systems employing generalized overflow nonlinearities abstract this paper considers the problem of global asymptotic stability of a class of nonlinear uncertain discrete time state delayed systems the class of systems under investigation involves multiple state delays norm bounded parameter uncertainties and generalized overflow nonlinearities which cover usual types of overflow arithmetic used in practice a new criterion for the global asymptotic stability of such systems is presented a numerical example is given to illustrate the applicability of the criterion presented

stability and overflow oscillations in 2 d state space digital filters an important theorem relating to the lyapunov stability of two dimensional discrete systems is proven using this theorem it is shown that for any 2 d digital filter satisfying shanks criterion there exists a realization that cannot support overflow oscillations in the process of proving the theorem some interesting results on the multi dimensional bilinear transformation are developed one of these results yields a simple test that can be used to check the stability of a 2 d discrete transfer function that has been obtained from the bilinear transform of a 2 d continuous transfer function with a 2 d hurwitzian denominator polynomial a technique is given for determining whether a normal realization exists for a given 2 d discrete system also a theorem is presented that allows the determination of the norm of the minimum norm realization of a given transfer function a noniterative technique for obtaining a low norm realization and an iterative technique for obtaining a minimum norm realization are developed

finite word length effects in m d digital filters with singularities on the stability boundary the author addresses stability robustness with respect to finite word length conditions for multidimensional m d digital filters with singularities on the distinguished boundary in particular it is shown that all purely first order digital filters with nonessential singularities of the second kind nssks are asymptotically stable under certain types of nonlinearities even bounded input bounded output bibo unstable m d digital filters with singularities on the distinguished boundary proved to be asymptotically stable in the first order case the results are then extended to certain classes of higher order m d digital filters it is demonstrated that although linear m d digital filters with nssks on the distinguished boundary have no margin of stability in the l sub 1 l sub 2 or asymptotic sense they might exhibit a rather robust asymptotic behavior with respect to nonideal effects such as finite word length conditions

stability of digital filters implemented with two s complement truncation quantization the stability of digital filters implemented with two s complement truncation quantization is investigated both the direct form and normal form realizations are considered and regions in the parameter plane are obtained where the filters are globally asymptotically stable regions within the stability triangle of a direct form digital filter are also studied where limit cycles of period one and two are possible the results are illustrated with several examples

stability analysis of 1 d and 2 d fixed point state space digital filters using any combination of overflow and quantization nonlinearities a new criterion together with its frequency domain interpretation for the global asymptotic stability of zero input one dimensional 1 d state space digital filters under various combinations of overflow and quantization nonlinearities and for the situation where quantization occurs after summation only is presented a condition in closed form involving solely the parameters of the state transition matrix for the nonexistence of limit cycles in second order digital filters is derived improved versions of some of the stability results due to leclerc and bauer 1994 are established finally the approach is extended to two dimensional 2 d digital filters described by the roesser and the fornasini marchesini second local state space models

the generation of random equilateral polygons freely jointed random equilateral polygons serve as a common model for polymer rings reflecting their statistical properties under theta conditions to generate equilateral polygons researchers employ many procedures that have been proved or at least are believed to be random with respect to the natural measure on the space of polygonal knots as a result the random selection of equilateral polygons as well as the statistical robustness of this selection is of particular interest in this research we study the key features of four popular methods the polygonal folding the crankshaft rotation the hedgehog and the triangle methods in particular we compare the implementation and efficacy of these procedures especially in regards to the population distribution of polygons in the space of polygonal knots the distribution of edge vectors the local curvature and the local torsion in addition we give a rigorous proof that the crankshaft rotation method is ergodic

finite size and asymptotic behaviors of the gyration radius of knotted cylindrical self avoiding polygons several nontrivial properties are shown for the mean square radius of gyration r k 2 of ring polymers with a fixed knot type k through computer simulation we discuss both finite size and asymptotic behaviors of the gyration radius under the topological constraint for self avoiding polygons consisting of n cylindrical segments with radius r we find that the average size of ring polymers with the knot k can be much larger than that of no topological constraint the effective expansion due to the topological constraint depends strongly on the parameter r that is related to the excluded volume the topological expansion is particularly significant for the small r case where the simulation result is associated with that of random polygons with the knot k

asymptotics of knotted lattice polygons we use monte carlo methods to investigate the asymptotic behaviour of the number and mean square radius of gyration of polygons in the simple cubic lattice with fixed knot type let be the number of n edge polygons of a fixed knot type in the cubic lattice and let be the mean square radius of gyration of all the polygons counted by if we assume that where is the growth constant of polygons of knot type and is the entropic exponent of polygons of knot type then our numerical data are consistent with the relation where is the unknot and is the number of prime factors of the knot if we assume that then our data are consistent with both and being independent of these results support the claims made in janse van rensburg and whittington 1991a 24 3935 and orlandini et al 1996 29 l299 1998 topology and geometry in polymer science ima volumes in mathematics and its applications berlin springer