shannon entropy fisher information and uncertainty relations for log periodic oscillators abstract we calculate the time dependent shannon s x and s p entropy and fisher f x and f p information of three log periodic oscillators we obtain a general expression for s x p and f x p in the state n 0 in terms of a c number quantity satisfying a nonlinear differential equation for two out of three oscillators s x p and f x p depend on time but s x s p and f x f p do not the other oscillator behaves as the time independent harmonic oscillator where s x p and f x p are all constants relations among the fisher information and the stam and cramer rao inequalities are also discussed

lattice polytopes cut out by root systems and the koszul property we show that lattice polytopes cut out by root systems of classical type are normal and koszul generalizing a well known result of bruns gubeladze and trung in type a we prove similar results for cayley sums of collections of polytopes whose minkowski sums are cut out by root systems the proofs are based on a combinatorial characterization of diagonally split toric varieties

on the relation between classical and quantum thermodynamic entropy we discuss the unresolved problem of proving rigorously that in the classical limit 0 the quantum thermodynamic entropy functional tends to the classical entropy functional we state rather restrictive conditions that define the general problem of finding a complete classical phase space representation of quantum kinematics whether the problem admits of solutions remains an unresolved question we discuss a physically interesting attempt to relate the structure in the classical limit 0 of the well known blokhintzev wigner and wehrl phase space functions to the spectral expansion of the quantum state operator

frequency weighted model reduction via quasi kalman decomposition a frequency weighted model reduction method based on quasi kalman decomposition is proposed for discrete systems the method is essentially an extension of the technique presented by lu and lee 1985 to the frequency weighted case and can handle both single sided as well as double sided weights two illustrative examples show the effectiveness of the method and results are compared with the method developed by enns 1984

singular loci of ladder determinantal varieties and schubert varieties abstract we relate certain ladder determinantal varieties associated to one sided ladders to certain schubert varieties in sl n q for a suitable n and a suitable parabolic subgroup q and we determine the singular loci of these varieties we state a conjecture on the irreducible components of the singular locus of a schubert variety in the flag variety which is a refinement of the conjecture of lakshmibai and sandhya proc indian acad sci math sci 100 1990 45 52 we prove the conjecture for a certain class of schubert varieties

hilbert functions of points on schubert varieties in the symplectic grassmannian we give an explicit combinatorial description of the multiplicity as well as the hilbert function of the tangent cone at any point on a schubert variety in the symplectic grassmannian

on quantum statistical inference interest in problems of statistical inference connected to measurements of quantum systems has recently increased substantially in step with dramatic new developments in experimental techniques for studying small quantum systems furthermore developments in the theory of quantum measurements have brought the basic mathematical framework for the probability calculations much closer to that of classical probability theory the present paper reviews this field and proposes and interrelates some new concepts for an extension of classical statistical inference to the quantum context

pattern avoidance and rational smoothness of schubert varieties abstract letwbe an element of the weyl groupsn and letxwbe the schubert variety associated towin the flag manifoldsln c b lakshmibai and sandhya showed thatxwis smooth if and only ifwavoids the patterns 4231 and 3412 using two tests for rational smoothness from carrell and peterson we show that rational smoothness ofxwis characterized by pattern avoidance for typesbandcas well a key step in the proof of this result is a sequence of rules for factoring the poincare polynomials for the cohomology ring ofxw generalizing the work of gasharov patterns can also be used to characterize actual smoothness of schubert varieties

fisher information and shannon entropy of position dependent mass oscillators abstract we calculate the fisher information and the shannon entropy for three position dependent mass oscillators these systems can be seen as deformed harmonic oscillators in the sense that when the deformation parameter goes to zero they are identical to the constant mass harmonic oscillator for two out of the three oscillators we observe that as increases the position fisher information f x increases while the momentum fisher information f p decreases on the other hand the shannon entropy always increases for the three systems with increasing discussion about squeezing effect in either position or momentum due to the variation and a relation between the product of fisher information and the shannon entropy are also presented

a unifying theorem for three subspace system identification algorithms the aim of this paper is to indicate and explore the similarities between three different subspace algorithms for the identification of combined deterministic stochastic systems the similarities between these algorithms have been obscured due to different notations and backgrounds it is shown that all three algorithms are special cases of one unifying theorem the comparison also reveals that the three algorithms use exactly the same subspace to determine the order and the extended observability matrix but that the weighting matrix used to calculate a basis for the column space of the observability matrix is different in the three cases

the arc space of the grassmannian abstract we study the arc space of the grassmannian from the point of view of the singularities of schubert varieties our main tool is a decomposition of the arc space of the grassmannian that resembles the schubert cell decomposition of the grassmannian itself just as the combinatorics of schubert cells is controlled by partitions the combinatorics in the arc space is controlled by plane partitions sometimes also called 3d partitions a combination of a geometric analysis of the pieces in the decomposition and a combinatorial analysis of plane partitions leads to invariants of the singularities as an application we reduce the computation of log canonical thresholds of pairs involving schubert varieties to an easy linear programming problem we also study the nash problem for schubert varieties showing that the nash map is always bijective in this case

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

normality of torus orbit closures in g p abstract the purpose of this note is to classify the torus orbit closures in an arbitrary algebraic homogeneous space g p that are toric varieties

a classical bound on quantum entropy a classical upper bound for quantum entropy is identified and illustrated 0 sq ln e 2 2 involving the variance 2 in phase space of the classical limit distribution of a given system a fortiori this further bounds the corresponding information theoretical generalizations of the quantum entropy proposed by renyi

rigidity of smooth schubert varieties in hermitian symmetric spaces in this paper we study the space z k g p r x w of effective k cycles x in g p with the homology class equal to an integral multiple of the homology class of schubert variety x w of type w when x w is a proper linear subspace p k k n of a linear space p n in g p p v we know that z k p n r p k is already complicated we will show that for a smooth schubert variety x w in a hermitian symmetric space any irreducible subvariety x with the homology class x r x w r z is again a schubert variety of type w unless x w is a non maximal linear space in particular any local deformation of such a smooth schubert variety in hermitian symmetric space g p is obtained by the action of the lie group g

a lookup conjecture for rational smoothness the purpose of this paper is to present a new combinatorial criterion for rational smooth ness of a point in a schubert variety the main result proves the criterion for schubert varieties associated to partial flag varieties of the form g p where g is semisimple of classical type and p is a maximal parabolic subgroup in the case where g p is of cominuscule type not necessarily clas sical we prove a stronger result we conjecture that the criterion is valid for all schubert varieties the paper also contains some general results related to rational smoothness in schubert varieties 1 introduction the purpose of this paper is to present a new combinato rial criterion for rational smoothness of a point in a schubert variety the main result proves the criterion for schubert varieties associated to certain partial flag varieties we conjecture that it is true in general the paper also contains some general results related to rational smoothness in schubert varieties we now explain our criterion in more detail let g be a complex semisimple

billey postnikov decompositions and the fibre bundle structure of schubert varieties a theorem of ryan and wolper states that a type a schubert variety is smooth if and only if it is an iterated fibre bundle of grassmannians we extend this theorem to arbitrary finite type showing that a schubert variety in a generalized flag variety is rationally smooth if and only if it is an iterated fibre bundle of rationally smooth grassmannian schubert varieties the proof depends on deep combinatorial results of billey postnikov on weyl groups we determine all smooth and rationally smooth grassmannian schubert varieties and give a new proof of peterson s theorem that all simply laced rationally smooth schubert varieties are smooth taken together our results give a fairly complete geometric description of smooth and rationally smooth schubert varieties using primarily combinatorial methods

classical information entropy and phase distributions of optical fields the wehrl phase distribution is defined as a phase density of the wehrl classical information entropy the new measure is applied to describe the quantum phase properties of some optical fields including fock states coherent and squeezed states and superposition of chaotic and coherent fields the wehrl phase distribution is compared with both the conventional wehrl entropy and husimi phase distribution the marginal husimi q function it is shown that the wehrl phase distribution is a good measure of the phase space uncertainty noise phase locking and phase bifurcation effects it is also demonstrated that the wehrl phase distribution properly describes phase randomization processes and thus can be used in a description of the quantum optical phase

on the shape of bruhat intervals let w s be a crystallographic coxeter group this includes all finite and affine weyl groups and let j subset of s let w j denote the set of minimal coset representatives modulo the parabolic

a combinatorial approach to the singularities of schubert varieties schubert subvarieties of grassmannians and more generally varieties g p with g a linear algebraic group and p a parabolic subgroup have long been a rich source for algebraic geometers and topologists recent interest has focussed on their singularities which are of interest due to the relationship between the intersection homological properties of a singularity of a schubert variety in g p and invariants of composition series of verma modules kl gm this work presents a simple combinatorial algorithm for deciding whether a schubert variety in g p where g sl is singular this leads to a geometric characterization of the nonsingular schubert varieties as sequences of grassmannian bundles over grassmannians the combinatorial techniques are those of p of whose existence the author was unaware a more general result of this nature has been proved by seshadri and lakshmibai laks who use standard monomial theory to find the ideal of the singular locus of a schubert variety in g b where g is an algebraic group with a standard monomial theory and b is a bore1 subgroup of g ryan r has obtained results similar to ours using general techniques for understanding determinantal varieties the present work has the advantage of being quite elementary while sharing aspects of both of the above or special interest here are the combinatorial techniques which appear to be a basic connection between the singularities of schubert varieties the geometry of the tibres of springer s resolution of the singularities of the nilpotent scheme see wl and the kazhdan lusztig polynomials a related note w2 discusses some applications of these techniques to the computation of the kazhdan lusztig polynomials

multiplicities of points on a schubert variety in a minuscule gp in this paper we prove the results announced in 13 let g be a semi simple simply connected algebraic group defined over an algebraically closed field k let t be a maximal torus b a bore1 subgroup b 3 t let w be the weyl group of g let r resp r be the set of roots resp positive roots relative to t resp b let s be the set of simple roots in r let p be a maximal parabolic subgroup in g with associated fundamental weight w let w be the weyl group of p and wp be the set of minimal representatives of w w for w e wp let e w be the point and x w the schubert variety in g p associated to w in this paper we deter mine the multiplicity m w of x w at e z where e z e x w for all minuscule p s and also for p pgn g being of type c here pun denotes the maximal parabolic subgroup obtained by omitting a the determina tion of m w is done as follows let l be the ample generator of pic g p a basis has been constructed for x w l in terms of standard monomials on x w cf 16 11 i let u be the unipotent subgroup of g generated by u et r rp here r denotes the set of roots of p and u denotes the unipotent subgroup of g associated to tl e r then u e r gives an affme neighborhood of e z in g p let a be the affine algebra of u e z and a a where is the ideal of elements of a that vanish on x w n u e z let m be the maximal ideal in a h corresponding to e r then using the results of 16 111 we obtain a basis of m f this enables us to obtain an inductive formula for f the hilbert polynomial of x w at e t cf corollaries 3 8 and 4 11 and also express m w in terms of m w s x w s being the schubert divisors in x w such that e t e x w cf theorems 3 7 and 4 10 using this we

investigations of information quantifiers for the tavis cummings model abstract in this article a system of two two level atoms interacting with a single mode quantized electromagnetic field in a lossless resonant cavity via a multi photon transition is considered the quantum fisher information negativity classical fisher information and reduced von neumann entropy for the two atoms are investigated we found that the number of photon transitions plays an important role in the dynamics of different information quantifiers in the cases of two symmetric and two asymmetric atoms our results show that there is a close relationship between the different quantifiers also the quantum and classical fisher information can be useful for studying the properties of quantum states which are important in quantum optics and information

generic singularities of certain schubert varieties let g be a connected semisimple algebraic group b a borel subgroup t a maximal torus in b with weyl group w and q a subgroup containing b for w in w let x wq denote the schubert variety overline bwq q for y in w such that x yq subseteq x wq one knows that byq q admits a t stable transversal in x wq which we denote by cal n yq wq we prove that under certain hypotheses cal n yq wq is isomorphic to the orbit closure of a highest weight vector in a certain weyl module we also obtain a generalisation of this result under slightly weaker hypotheses further we prove that our hypotheses are satisfied when q is a maximal parabolic subgroup corresponding to a minuscule or cominuscule fundamental weight and x yq is an irreducible component of the boundary of x wq that is the complement of the open orbit of the stabiliser in g of x wq as a consequence we describe the singularity of x wq along byq q and obtain that the boundary of x wq equals its singular locus

model reduction via balanced realizations an extension and frequency weighting techniques two model reduction methods for discrete systems related to balanced realizations are described the first is a technique which utilizes the least controllable and observable subsystem in deriving a balanced discrete reduced order model for this technique as l sup infinity norm bound on the reduction error is given the second method is a frequency weighting technique for discrete and continuous time systems where the input normal or output normal realizations are modified to include a simple frequency weighting for this technique l sup infinity norm bounds on the weighted reduction errors are obtained

the gorenstein locus of minuscule schubert varieties in this article we explicitly describe the gorenstein locus of all minuscule schubert varieties this proves a special case of the conjecture of a woo and a yong alexander woo alexander yong governing singularities of schubert varieties j algebra 320 2 2008 495 520 regarding the gorenstein locus of schubert varieties

pattern avoidance and smoothness of closures for orbits of a symmetric subgroup in the flag variety abstract we give a pattern avoidance criterion to classify the orbits of sp p c sp q c resp gl n c on the flag variety of type c p q resp d n with rationally smooth closure we show that all such orbit closures fiber with smooth fiber over a smaller flag variety and hence are in fact smooth in addition we prove that the classification is insensitive to isogeny

wehrl information entropy and phase distributions of schrodinger cat and cat like states the wehrl information entropy and its phase density the so called wehrl phase distribution are applied to describe schrcat and cat like kitten states the advantages of the wehrl phase distribution over the wehrl entropy in a description of the superposition principle are presented the entropic measures are compared with a conventional phase distribution from the husimi q function compact form formulae for the entropic measures are found for superpositions of well separated states examples of schr odinger cats including even odd and yurke stoler coherent states as well as the cat like states generated in the kerr medium are analysed in detail it is shown that in contrast to the wehrl entropy the wehrl phase distribution properly distinguishes between different superpositions of unequally weighted states with respect to their number and phase space configuration

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

the hodge theory of soergel bimodules we prove soergel s conjecture on the characters of indecomposable soergel bimodules we deduce that kazhdan lusztig polynomials have positive coecients for arbitrary coxeter systems using results of soergel one may deduce an algebraic proof of the kazhdan lusztig conjecture

rational curves on minuscule schubert varieties introduction let us denote by c the variety of lines in p3 meeting a fixed line it is a grassmannian and hence minuscule schubert variety in 16 we described the irreducible components of the scheme of morphisms from p1 to c and the general morphism in each irreducible component in this text we study the scheme of morphisms from p1 to any minuscule schubert variety x let us recall that we studied in 15 the scheme of morphisms from p1 to any homogeneous variety the main idea in the case of a minuscule schubert variety x is to restrict ourselves to the dense orbit under the stabilizer stab x of x and apply the results of 15 more precisely let u be the dense orbit under stab x in x and let y be the complement because x is a minuscule schubert variety the closed subset y of x is of codimension at least 2 see section 2 2 this fact and the stratification of x by schubert subvarieties gives us a surjective morphism see section 1 s pic u a1 x

the nil hecke ring and singularity of schubert varieties let g be a semi simple simply connected complex algebraic group and t b a maximal torus and a borel subgroup respectively let lie t be the cartan subalgebra of the lie algebra lie g and w n t t the weyl group associated to the pair g t where n t is the normalizer of t in g we can view any element omega overline omega mod t i w as the element denoted by the corresponding german character xo of g b denned as m overline omega b for any w i w there is associated the schubert variety rm x omega overline b omega b b subset g b and the t fixed points of x w under the canonical left action are precisely rm i omega left o upsilon in wand upsilon leqslant omega right

model reduction via a quasi kalman decomposition a system theoretic model reduction algorithm is proposed which appears to be simple to work with the algorithm is based on a quasi kalman decomposition

excited young diagrams and equivariant schubert calculus we describe the torus equivariant cohomology ring of isotropic grassmannians by using a localization map to the torus fixed points we present two types of formulas for equivariant schubert classes of these homogeneous spaces the first formula involves combinatorial objects which we call excited young diagrams and the second one is written in terms of factorial schur q or p functions as an application we give a giambelli type formula for the equivariant schubert classes we also give combinatorial and pfaffian formulas for the multiplicity of a singular point in a schubert variety

pattern characterization of rationally smooth affine schubert varieties of type a abstract schubert varieties in finite dimensional flag manifolds g p are a well studied family of projective varieties indexed by elements of the corresponding weyl group w in particular there are many tests for smoothness and rational smoothness of these varieties one key result due to lakshmibai and sandhya is that in type a the smooth schubert varieties are precisely those that are indexed by permutations that avoid the patterns 4231 and 3412 recently there has been a flurry of research related to the infinite dimensional analogs of flag manifolds corresponding with g being a kac moody group and w being an affine weyl group or parabolic quotient in this paper we study the case when w is the affine weyl group of type a or the affine permutations we develop the notion of pattern avoidance for affine permutations our main result is a characterization of the rationally smooth schubert varieties corresponding to affine permutations in terms of the patterns 4231 and 3412 and the twisted spiral permutations

a grobner basis for kazhdan lusztig ideals it kazhdan lusztig ideals a family of generalized determinantal ideals investigated in woo yong 08 provide an explicit choice of coordinates and equations encoding a neighborhood of a torus fixed point of a schubert variety on a type a flag variety our main result is a gr o bner basis for these ideals this provides a single geometric setting to transparently explain the naturality of pipe dreams on the it rothe diagram of a permutation and their appearance in begin itemize item combinatorial formulas fomin kirillov 94 for schubert and grothendieck polynomialsof lascoux sch u tzenberger 82 item the equivariant k theory specialization formula of buch rim a nyi 04 and item a positive combinatorial formula for multiplicities of schubert varieties in good cases including those for which the associated kazhdan lusztig ideal is homogeneous under the standard grading end itemize our results generalize with alternate proofs knutson miller 05 s gr o bner basis theorem for schubert determinantal ideals and their geometric interpretation of the monomial positivity of schubert polynomials we also complement recent work of knutson 08 09 on degenerations of kazhdan lusztig varieties in general lie type as well as work of goldin 01 on equivariant localization and of lakshmibai weyman 90 rosenthal zelevinsky 01 and krattenthaler 01 on grassmannian multiplicity formulas

schubert classes in the equivariant cohomology of the lagrangian grassmannian abstract let lg n denote the lagrangian grassmannian parametrizing maximal isotropic lagrangian subspaces of a fixed symplectic vector space of dimension 2n for each strict partition 1 k with 1 n there is a schubert variety x let t denote a maximal torus of the symplectic group acting on lg n consider the t equivariant cohomology of lg n and the t equivariant fundamental class of x the main result of the present paper is an explicit formula for the restriction of the class to any torus fixed point the formula is written in terms of factorial analogue of the schur q function introduced by ivanov as a corollary to the restriction formula we obtain an equivariant version of the giambelli type formula for lg n as another consequence of the main result we obtained a presentation of the ring h t lg n

some degenerations of kazhdan lusztig ideals and multiplicities of schubert varieties abstract we study hilbert samuel multiplicity for points of schubert varieties in the complete flag variety by grobner degenerations of the kazhdan lusztig ideal in the covexillary case we give a manifestly positive combinatorial rule for multiplicity by establishing with a grobner basis a reduced limit whose stanley reisner simplicial complex is homeomorphic to a shellable ball or sphere we show that multiplicity counts the number of facets of this complex we also obtain a formula for the hilbert series of the local ring in particular our work gives a multiplicity rule for grassmannian schubert varieties providing alternative statements and proofs to formulae of lakshmibai and weyman 1990 26 rosenthal and zelevinsky 2001 37 krattenthaler 2001 22 kodiyalam and raghavan 2003 21 kreiman and lakshmibai 2004 24 ikeda and naruse 2009 13 and woo and yong 2009 40 we suggest extensions of our methodology to the general case

smooth points of t stable varieties in g b and the peterson map let g be a connected semi simple algebraic group defined over an algebraically closed field k and let t b p be respectively a maximal torus a borel subgroup and a parabolic subgroup of g inspired by a beautiful result of dale peterson describing the singular locus of a schubert variety in g b we characterize the t fixed points in the singular locus of an arbitrary irreducible t stable subvariety of g p a t variety for short peterson s result cf the deformation theorem 1 says that if k then a schubert variety x g b is smooth at a t fixed point x if and only if it is smooth at every t fixed point y x in the bruhat chevalley order on the fixed point set x t and all the limits c x x lim z x t z x z c c t of the zariski tangent spaces t z x of x coincide as c varies over the set of all t stable curves in x with c t x y where y x using this peterson showed that if g is simply laced and defined over then every rationally smooth point of a schubert variety in g b is smooth more generally the deformation c x x is defined for any k variety x with a t action provided c is what we call good i e c is a curve of the form c overline tz where z is a smooth point of x x t and x c t our first main result theorem 1 4 says that if x x is an attractive fixed point then x is smooth at x if and only if there exist at least two good c containing x such that c x x te x x where te x x denotes the span of the tangent lines of the t stable curves in x containing x in addition if x is cohen macaulay at x and c x x te x x for even one good c then x is smooth at x our second main result theorem 1 6 says that if x is a t variety in g p where g is simply laced then c x x te x x for each good c this is not true for general g but when g has no g2 factors then c x x is always contained in the linear span of the reduced tangent cone to x at x these results lead to several descriptions of the smooth fixed points of a t variety in g p and in particular they give simple proofs of peterson s results valid for any algebraically closed field we also show cf example 7 1 that there can exist t stable subvarieties in g b where g is simply laced which have rationally smooth t fixed points in their singular loci

the nash blow up of a cominuscule schubert variety abstract we compute the nash blow up of a cominuscule schubert variety in particular we show that the nash blow up is algebraically isomorphic to another schubert variety of the same lie type as a consequence we give a new characterization of the smooth locus and for grassmannian schubert varieties determine when the nash blow up is a resolution of singularities we also study the induced torus action on the nash blow up and give a bijection between its torus fixed points and peterson translates on the schubert variety