k double schur functions and equivariant co homology of the affine grassmannian the schubert bases of the torus equivariant homology and cohomology rings of the affine grassmannian of sl k 1 are realized by new families of symmetric functions called k double schur functions and affine double schur functions

the nil hecke ring and cohomology of g p for a kac moody group g abstract let g be the group with borel subgroup b associated to a kac moody lie algebra unk with weyl group w and cartan subalgebra unk then h g b has among others four distinguished structures i an algebra structure ii a distinguished basis given by the schubert cells iii a module for w and iv a module for hecke type operators aw for w unk w we construct a ring r which we refer to as the nil hecke ring which is very simply and explicitly defined as a functor of w together with the w module unk alone and such that all these four structures on h g b arise naturally from the ring r

forced vibration analysis of a multidegree impact vibrator abstract the dynamics of a multidegree impact vibrator subject to harmonic loading is investigated the system is represented by a lumped mass model which hits and rebounds from a rigid wall during vibration the periodic solution to the equations of motion with n forcing cycles and p impacts is formulated the variational equations and the resulting transition matrix for investigating local stability of the periodic solutions are derived a two degree of freedom example is analysed and a variety of motion types are found chaotic windows are present between regions of periodic response and at these boundaries n p motions are prevalent low velocity impacts are evident at exciting frequencies away from the natural frequencies two basins of attraction are computed and the sensitivity to initial conditions is noted the quality of the n p motion is discussed from an engineering application perspective

schubert polynomials for the affine grassmannian of the symplectic group we study the schubert calculus of the affine grassmannian gr of the symplectic group the integral homology and cohomology rings of gr are identified with dual hopf algebras of symmetric functions defined in terms of schur s p and q functions an explicit combinatorial description is obtained for the schubert basis of the cohomology of gr and this is extended to a definition of the affine type c stanley symmetric functions a homology pieri rule is also given for the product of a special schubert class with an arbitrary one

the hilbert phenomenon in chaotic motions abstract it is proved that chaotic dynamical systems have d 1 singularity due to such singularity and its periodicity in hausdorff metric spaces a chaotic dynamical system induces a hilbert iterated system which was discovered at the end of the 19th century this implies that the most puzzling discovery in nature in the 20th century can be put into deep correspondence with that in thoughts in the 19th century all this is verified by the numerical experiments of chaotic vibrations of a beam and a tension slack oscillator

construction of central elements in the affine hecke algebra via nearby cycles 0 1 overview let g be a connected reductive group over a finite field fq and let g k be the corresponding group over the local field k fq t let g o g k be a maximal compact subgroup of g k here o fq t and let hsph denote the hecke algebra of g k with respect to g o in other words hsph as a vector space consists of compactly supported bi g o invariant functions g k q and the product is defined by

convex polytopes for the central degeneration of the affine grassmannian abstract we study the algebraic geometry and combinatorics of the central degeneration the degeneration that shows up in local models of shimura varieties and gaitsgory s central sheaves in type a more specifically we elucidate the central degeneration of semi infinite orbits and explain its relations with levi restriction also we discuss the central degeneration of mirkovic vilonen cycles in the affine grassmannian and the corresponding transformations of mirkovic vilonen polytopes in addition we shed some light on the geometry of iwahori mv cycles in the affine grassmannian and generalized mv cycles in the affine flag variety which are closely related to demazure modules and affine deligne lusztig varieties respectively

quantum and affine schubert calculus and macdonald polynomials abstract we definitively establish that the theory of symmetric macdonald polynomials aligns with quantum and affine schubert calculus using a discovery that distinguished weak chains can be identified by chains in the strong bruhat order poset on the type a affine weyl group we construct two one parameter families of functions that respectively transition positively with hall littlewood and macdonald s p functions and specialize to the representatives for schubert classes of homology and cohomology of the affine grassmannian our approach leads us to conjecture that all elements in a defining set of 3 point genus 0 gromov witten invariants for flag manifolds can be formulated as strong covers

schubert polynomials and the nilcoxeter algebra abstract schubert polynomials were introduced and extensively developed by lascoux and schutzenberger after an earlier less combinatorial version had been considered by bernstein gelfand and gelfand and demazure we give a new development of the theory of schubert polynomials based on formal computations in the algebra of operators u1 u2 satisfying the relations u2i 0 uiuj ujui if i j 2 and uiui 1ui ui 1uiui 1 we call this algebra the nilcoxeter algebra of the symmetric group s n our development leads to simple proofs of many standard results in particular a symmetry of the stable schubert polynomials fw b an explicit combinatorial formula for schubert polynomials due to billey jockusch and stanley c the cauchy formula for schubert polynomials and d a formula of macdonald for s w 1 1 our main new result is a proof of a conjectured q analogue of d due to macdonald which gives a formula for s w 1 q q2

affine approach to quantum schubert calculus this paper presents a formula for products of schubert classes in the quantum cohomology ring of the grassmannian we introduce a generalization of schur symmetric polynomials for shapes that are naturally embedded in a torus then we show that the coefficients in the expansion of these toric schur polynomials in terms of the regular schur polynomials are exactly the 3 point gromov witten invariants which are the structure constants of the quantum cohomology ring this construction implies three symmetries of the gromov witten invariants of the grassmannian with respect to the groups s3 z nz 2 and z 2z the last symmetry is a certain curious duality of the quantum cohomology which inverts the quantum parameter q our construction gives a solution to a problem posed by fulton and woodward about the characterization of the powers of the quantum parameter q which occur with nonzero coefficients in the quantum product of two schubert classes the curious duality switches the smallest such power of q with the highest power we also discuss the affine nil temperley lieb algebra that gives a model for the quantum cohomology

k schur expansions of catalan functions abstract we make a broad conjecture about the k schur positivity of catalan functions symmetric functions which generalize the parabolic hall littlewood polynomials we resolve the conjecture with positive combinatorial formulas in cases which address the k schur expansion of 1 hall littlewood polynomials proving the q 0 case of the strengthened macdonald positivity conjecture from 24 2 the product of a schur function and a k schur function when the indexing partitions concatenate to a partition describing a class of gromov witten invariants for the quantum cohomology of complete flag varieties 3 k split polynomials solving a substantial special case of a problem of broer and shimozono weyman on parabolic hall littlewood polynomials 37 in addition we prove the conjecture that the k schur functions defined via k split polynomials 25 agree with those defined in terms of strong tableaux 21

two views of belief belief as generalized probability and belief as evidence abstract belief functions are mathematical objects defined to satisfy three axioms that look somewhat similar to the kolmogorov axioms defining probability functions we argue that there are at least two useful and quite different ways of understanding belief functions the first is as a generalized probability function which technically corresponds to the inner measure induced by a probability function the second is as a way of representing evidence evidence in turn can be understood as a mapping from probability functions to probability functions it makes sense to think of updating a belief if we think of it as a generalized probability on the other hand it makes sense to combine two beliefs using say dempster s rule of combination only if we think of the belief functions as representing evidence many previous papers have pointed out problems with the belief function approach the claim of this paper is that these problems can be explained as a consequence of confounding these two views of belief functions

affine stanley symmetric functions we define a new family inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of generating functions for w inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i which are affine analogues of stanley symmetric functions we establish basic properties of these functions including symmetry dominance and conjugation we conjecture certain positivity properties in terms of a subfamily of symmetric functions called affine schur functions as applications we show how affine stanley symmetric functions generalize the dual of the k schur functions of lapointe lascoux and morse as well as the cylindric schur functions of postnikov conjecturally affine stanley symmetric functions should be related to the cohomology of the affine flag variety

equivariant k theory of affine flag manifolds and affine grothendieck polynomials we study the equivariant k group of the affine flag manifold with respect to the borel group action we prove that the structure sheaf of the infinite dimensional schubert variety in the k group is represented by a unique polynomial which we call the affine grothendieck polynomial

pieri rule for the affine flag variety abstract we prove the affine pieri rule for the cohomology of the affine flag variety conjectured by lam lapointe morse and shimozono we study the cap operator on the affine nilhecke ring that is motivated by kostant and kumar s work on the equivariant cohomology of the affine flag variety we show that the cap operators for pieri elements are the same as pieri operators defined by berg saliola and serrano this establishes the affine pieri rule we also discuss properties of cap operators which are not necessarily affine a type

the center of the affine niltemperley lieb algebra we give a description of the center of the affine niltemperley lieb algebra based on a certain grading of the algebra and on a faithful representation of it on fermionic particle configurations we present a normal form for monomials hence construct a basis of the algebra and use this basis to show that the affine niltemperley lieb algebra is finitely generated over its center as an application we obtain a natural embedding of the affine niltemperley lieb algebra on n generators into the affine niltemperley lieb algebra on n 1 generators

quantum cohomology of g p and homology of affine grassmannian let g be a simple and simply connected complex algebraic group p g a parabolic subgroup we prove an unpublished result of d peterson which states that the quantum cohomology qh g p of a flag variety is up to localization a quotient of the homology h gr g of the affine grassmannian gr g of g as a consequence all three point genus zero gromov witten invariants of g p are identified with homology schubert structure constants of h gr g establishing the equivalence of the quantum and homology affine schubert calculi

belief function representations of statistical evidence cet article a pour objet d apporter des reponses aux questions soulevees par shafer 1976 1982 et dempster 1966 1968

gromov witten invariants for and pontryagin product for ω we give an explicit formula for t equivariant 3 pointed genus zero gromov witten invariants for g b we derive it by finding an explicit formula for the pontryagin product on the equivariant homology of the based loop group k

schubert polynomials for the affine grassmannian let g be a complex simply connected simple group and k a maximal compact subgroup let f c t denote the field of formal laurent series and o c t the ring of formal power series the quotient qr g f g 0 is known as the affine grassmannian of g it is known that qr is homotopy equivalent to the space of based loops k into the compact group k see 9 30 the multiplication on qk endows the homology h qr and cohomology h qr with structures of dual hopf algebras these hopf algebras were first identified by bott 4 another descrip tion is given by ginzburg 10 which we shall not use the bruhat decomposition of g f induces a stratification of qr by schubert cells ilw indexed by grassman nian elements w of the affine weyl group waffwe denote by aw g h qr and aw g h qr the schubert classes in homology and cohomology when g sl n c bott identifies h qr and h qr with a subring an and a quotient an of the ring of symmetric functions our main theorem theorem 7 1 identifies the schubert classes aw g qr and aw g h qr as explicit symmet ric functions under this isomorphism these symmetric functions are combinatori ally defined in homology the aw g h qr are represented by lascoux lapointe morse s fc schur functions sw x g an and in cohomology the aw g h qr are represented by the dual fc schur functions or affine schur functions fw x g an see 21 24 17 our theorem was originally conjectured by mark shimozono the conjecture was made explicit in the cohomology case by jennifer morse kostant and kumar 14 studied the topology of homogeneous spaces for arbi trary kac moody groups and in particular calculated the structure constants of h qr in the schubert basis using the algebraic construction of the nilhecke ring our connection between the topology and the combinatorics proceeds via the study of three subalgebras of the nilhecke ring the first subalgebra is the nilcoxeter al gebra ao which is the algebra generated by the divided difference operators the second algebra is a certain centralizer algebra aaff s which we call the peterson subalgebra peterson 28 constructs an isomorphism j from the t equivariant ho mology hr qr of the affine grassmannian to z ff s the third algebra b is a combinatorially defined subalgebra of a0 which we call the affine fomin stanley

quantum cohomology and the k schur basis we prove that structure constants related to hecke algebras at roots of unity are special cases of k littlewood richardson coefficients associated to a product of k schur functions as a consequence both the 3 point gromov witten invariants appearing in the quantum cohomology of the grassmannian and the fusion coefficients for the wzw conformal field theories associated to su l are shown to be k littlewood richardson coefficients from this mark shimozono conjectured that the k schur functions form the schubert basis for the homology of the loop grassmannian whereas k schur coproducts correspond to the integral cohomology of the loop grassmannian we introduce dual k schur functions defined on weights of k tableaux that given shimozono s conjecture form the schubert basis for the cohomology of the loop grassmannian we derive several properties of these functions that extend those of skew schur functions

the periodicity of chaotic impact oscillators in hausdorff phase spaces it is well known that non periodic behavior is one of the most puzzling characteristics of chaotic oscillators so far chaotic dynamical systems have been investigated in euclidean spaces in this paper the concept of non autonomous dynamical systems and that of hausdorff phase spaces are proposed the behavior of chaotic impact oscillators is investigated in hausdorff phase spaces it is discovered that although the non autonomous dynamical systems described by chaotic impact oscillators are non periodic in euclidean phase spaces they are periodic in hausdorff phase spaces this shows that euclidean spaces in which we stayed for hundreds of years may no longer be suitable for the investigation into chaotic phenomena in addition the periodicity of chaotic dynamical systems in hausdorff metric spaces induces a new class of strange invariant sets in euclidean spaces such strange invariant sets may be an ideal symbol of chaotic dynamical systems

combinatorics of the k theory of affine grassmannians abstract we introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize grothendieck polynomials and k schur functions we prove that the polynomials drawn from these tableaux are the affine grothendieck polynomials and k k schur functions schubert representatives for the k theory of affine grassmannians and their dual in the nil hecke ring we prove a number of combinatorial properties including pieri rules

priors and likelihood ratios as evidence abstract arguments based on diagnostic data concerning a particular case and ones based on prior experience with like cases can be represented by belief functions and combined by dempster s rule in limiting cases the belief functions depend on likelihood ratio or on prior odds and when these limiting cases occur together bayes theorem is applicable as is dempster s rule the resulting functional equations lead to a single parameter family of functions relating belief strength to probability and to likelihood ratio the parameter can be measured by studying the belief strength produced by conceptually independent dissonant arguments that point to a common conclusion this provides one possible solution to the problem of fundamental measurement for shafer s theory of belief