scholarly journals A dual approach to structure constants for K-theory of Grassmannians

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Huilan Li ◽  
Jennifer Morse ◽  
Pat Shields
Keyword(s):  
Generating Functions ◽  
Cohomology Ring ◽  
Point Of View ◽  
Schur Polynomials ◽  
Dual Approach ◽  
Schur Polynomial ◽  
International Audience ◽  
Structure Constants ◽  
Schubert Classes ◽  
A Chain

International audience The problem of computing products of Schubert classes in the cohomology ring can be formulated as theproblem of expanding skew Schur polynomial into the basis of ordinary Schur polynomials. We reformulate theproblem of computing the structure constants of the Grothendieck ring of a Grassmannian variety with respect to itsbasis of Schubert structure sheaves in a similar way; we address the problem of expanding the generating functions forskew reverse-plane partitions into the basis of polynomials which are Hall-dual to stable Grothendieck polynomials. From this point of view, we produce a chain of bijections leading to Buch’s K-theoretic Littlewood-Richardson rule.

scholarly journals Interval positroid varieties and a deformation of the ring of symmetric functions

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Allen Knutson ◽  
Mathias Lederer
Keyword(s):  
Symmetric Functions ◽  
Schur Functions ◽  
Dimensional Subspace ◽  
Schubert Varieties ◽  
International Audience ◽  
Structure Constants ◽  
Schubert Classes ◽  
K Theory ◽  
Two Parameter ◽  
Parameter Deformation

International audience Define the <b>interval rank</b> $r_[i,j] : Gr_k(\mathbb C^n) →\mathbb{N}$ of a k-plane V as the dimension of the orthogonal projection $π _[i,j](V)$ of V to the $(j-i+1)$-dimensional subspace that uses the coordinates $i,i+1,\ldots,j$. By measuring all these ranks, we define the <b>interval rank stratification</b> of the Grassmannian $Gr_k(\mathbb C^n)$. It is finer than the Schubert and Richardson stratifications, and coarser than the positroid stratification studied by Lusztig, Postnikov, and others, so we call the closures of these strata <b>interval positroid varieties</b>. We connect Vakil's "geometric Littlewood-Richardson rule", in which he computed the homology classes of Richardson varieties (Schubert varieties intersected with opposite Schubert varieties), to Erd&odblac;s-Ko-Rado shifting, and show that all of Vakil's varieties are interval positroid varieties. We build on his work in three ways: (1) we extend it to arbitrary interval positroid varieties, (2) we use it to compute in equivariant K-theory, not just homology, and (3) we simplify Vakil's (2+1)-dimensional "checker games" to 2-dimensional diagrams we call "IP pipe dreams". The ring Symm of symmetric functions and its basis of Schur functions is well-known to be very closely related to the ring $\bigoplus_a,b H_*(Gr_a(\mathbb{C}^{(a+b)})$ and its basis of Schubert classes. We extend the latter ring to equivariant K-theory (with respect to a circle action on each $\mathbb{C}^{(a+b)}$, and compute the structure constants of this two-parameter deformation of Symm using the interval positroid technology above.

scholarly journals Chevalley-Monk and Giambelli formulas for Peterson Varieties

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Elizabeth Drellich
Keyword(s):  
Cohomology Ring ◽  
Quantum Cohomology ◽  
Type A ◽  
Flag Variety ◽  
Flag Varieties ◽  
Dimensional Torus ◽  
One Dimensional ◽  
International Audience ◽  
Schubert Classes ◽  
Partial Flag Varieties

International audience A Peterson variety is a subvariety of the flag variety $G/B$ defined by certain linear conditions. Peterson varieties appear in the construction of the quantum cohomology of partial flag varieties and in applications to the Toda flows. Each Peterson variety has a one-dimensional torus $S^1$ acting on it. We give a basis of Peterson Schubert classes for $H_{S^1}^*(Pet)$ and identify the ring generators. In type A Harada-Tymoczko gave a positive Monk formula, and Bayegan-Harada gave Giambelli's formula for multiplication in the cohomology ring. This paper gives a Chevalley-Monk rule and Giambelli's formula for all Lie types.

Do Event-Related Brain Potentials Reflect Mental (Cognitive) Operations?

2002 ◽  
Vol 16 (3) ◽  
pp. 129-149 ◽  
Author(s):  
Boris Kotchoubey
Keyword(s):  
Cognitive Processes ◽  
Point Of View ◽  
Internal Variables ◽  
Brain Potentials ◽  
Cognitive Operations ◽  
External Variables ◽  
Series Of Experiments ◽  
Mental Operations ◽  
A Chain ◽  
Processing Mechanisms

Abstract Most cognitive psychophysiological studies assume (1) that there is a chain of (partially overlapping) cognitive processes (processing stages, mechanisms, operators) leading from stimulus to response, and (2) that components of event-related brain potentials (ERPs) may be regarded as manifestations of these processing stages. What is usually discussed is which particular processing mechanisms are related to some particular component, but not whether such a relationship exists at all. Alternatively, from the point of view of noncognitive (e. g., “naturalistic”) theories of perception ERP components might be conceived of as correlates of extraction of the information from the experimental environment. In a series of experiments, the author attempted to separate these two accounts, i. e., internal variables like mental operations or cognitive parameters versus external variables like information content of stimulation. Whenever this separation could be performed, the latter factor proved to significantly affect ERP amplitudes, whereas the former did not. These data indicate that ERPs cannot be unequivocally linked to processing mechanisms postulated by cognitive models of perception. Therefore, they cannot be regarded as support for these models.

scholarly journals Area of Brownian Motion with Generatingfunctionology

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Michel Nguyên Thê
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Generating Functions ◽  
Limit Distributions ◽  
Dyck Paths ◽  
Different Types ◽  
International Audience

International audience This paper gives a survey of the limit distributions of the areas of different types of random walks, namely Dyck paths, bilateral Dyck paths, meanders, and Bernoulli random walks, using the technology of generating functions only.

scholarly journals Finite Eulerian posets which are binomial or Sheffer

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Hoda Bidkhori
Keyword(s):  
Generating Functions ◽  
Complete Classification ◽  
International Audience ◽  
Original Question ◽  
Eulerian Posets

International audience In this paper we study finite Eulerian posets which are binomial or Sheffer. These important classes of posets are related to the theory of generating functions and to geometry. The results of this paper are organized as follows: (1) We completely determine the structure of Eulerian binomial posets and, as a conclusion, we are able to classify factorial functions of Eulerian binomial posets; (2) We give an almost complete classification of factorial functions of Eulerian Sheffer posets by dividing the original question into several cases; (3) In most cases above, we completely determine the structure of Eulerian Sheffer posets, a result stronger than just classifying factorial functions of these Eulerian Sheffer posets. We also study Eulerian triangular posets. This paper answers questions posed by R. Ehrenborg and M. Readdy. This research is also motivated by the work of R. Stanley about recognizing the \emphboolean lattice by looking at smaller intervals. Nous étudions les ensembles partiellement ordonnés finis (EPO) qui sont soit binomiaux soit de type Sheffer (deux notions reliées aux séries génératrices et à la géométrie). Nos résultats sont les suivants: (1) nous déterminons la structure des EPO Euleriens et binomiaux; nous classifions ainsi les fonctions factorielles de tous ces EPO; (2) nous donnons une classification presque complète des fonctions factorielles des EPO Euleriens de type Sheffer; (3) dans la plupart de ces cas, nous déterminons complètement la structure des EPO Euleriens et Sheffer, ce qui est plus fort que classifier leurs fonctions factorielles. Nous étudions aussi les EPO Euleriens triangulaires. Cet article répond à des questions de R. Ehrenborg and M. Readdy. Il est aussi motivé par le travail de R. Stanley sur la reconnaissance du treillis booléen via l'étude des petits intervalles.

scholarly journals Enzyme catalysis as a chain reaction

1991 ◽  
Vol 279 (3) ◽  
pp. 855-861 ◽  
Author(s):  
S E Szedlacsek ◽  
R G Duggleby ◽  
M O Vlad
Keyword(s):  
Reaction Mechanism ◽  
Kinetic Mechanism ◽  
Point Of View ◽  
Simple Type ◽  
Chain Reaction ◽  
Proposed Model ◽  
New Type ◽  
A Chain ◽  
Theoretical Considerations ◽  
Classical Mechanism

A new type of enzyme kinetic mechanism is suggested by which catalysis may be viewed as a chain reaction. A simple type of one-substrate/one-product reaction mechanism has been analysed from this point of view, and the kinetics, in both the transient and the steady-state phases, has been reconsidered. This analysis, as well as literature data and theoretical considerations, shows that the proposed model is a generalization of the classical ones. As a consequence of the suggested mechanism, the expressions, and in some cases even the significance of classical constants (Km and Vmax.), are altered. Moreover, this mechanism suggests that, between two successive enzyme-binding steps, more than one catalytic act could be accomplished. The reaction catalysed by alcohol dehydrogenase was analysed, and it was shown that this chain-reaction mechanism has a real contribution to the catalytic process, which could become exclusive under particular conditions. Similarly, the mechanism of glycogen phosphorylase is considered, and two partly modified versions of the classical mechanism are proposed. They account for both the existing experimental facts and suggest the possibility of chain-reaction pathways for any polymerase.

scholarly journals El currículum de Educación Musical en los Programas Renovados y el Real Decreto 1006/1991: un estudio comparado

2019 ◽  
pp. 182
Author(s):  
Alba María López Melgarejo ◽  
Gregorio Vicente Nicolas ◽  
Eva María González Barea
Keyword(s):  
Music Education ◽  
Legal Framework ◽  
Point Of View ◽  
Teaching Staff ◽  
Royal Decree ◽  
Perceptual Dimension ◽  
Dual Approach ◽  
Musical Education ◽  
Documentary Analysis ◽  
High Degree

The aim of this work has been to detect the differences and similarities from the point of view of Music Education among the Programas Renovados (Renewed Programs), the last document published in Spain before the emergence of the curriculum concept and how we conceive it today, and the Real Decree 1006/1991, that established the curricula for Primary School. For this, a comparative analysis of the aforementioned texts has been carried out through a documentary analysis that has allowed to contrast the legal framework, the character, the curricular functions, the configuration of the music area, structure, elements, musical areas, as well as the degree of concretion of both documents. The results reveal a high degree of difference between the Renewed Programs and Royal Decree 1006/1991 regarding the curricular functions, because the former served as a guide for the teaching staff, while the latter, in addition to the previous function, made explicit the intentions of the educational system for the stage. Likewise, a high degree of difference between the structure of the Musical Education elements of both documents has been verified. However, it has been observed a common presence of contents related to different areas of Music Education (singing, instrumentation, listening, movement and dance...), as well as a dual approach that includes the perceptual dimension and expressive of them.

scholarly journals Double theta polynomials and equivariant Giambelli formulas

2015 ◽  
Vol 160 (2) ◽  
pp. 353-377 ◽  
Author(s):  
HARRY TAMVAKIS ◽  
ELIZABETH WILSON
Keyword(s):  
Equivariant Cohomology ◽  
Cohomology Ring ◽  
Generators And Relations ◽  
Raising Operators ◽  
Schubert Classes ◽  
Symplectic Grassmannians

AbstractWe use Young's raising operators to introduce and study double theta polynomials, which specialize to both the theta polynomials of Buch, Kresch, and Tamvakis, and to double (or factorial) Schur S-polynomials and Q-polynomials. These double theta polynomials give Giambelli formulas which represent the equivariant Schubert classes in the torus-equivariant cohomology ring of symplectic Grassmannians, and we employ them to obtain a new presentation of this ring in terms of intrinsic generators and relations.

On fluctuation problems in the theory of queues

1976 ◽  
Vol 8 (03) ◽  
pp. 548-583 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Markov Chains ◽  
Markov Processes ◽  
Generating Functions ◽  
Finite Differences ◽  
Historical Development ◽  
Banach Algebras ◽  
Point Of View ◽  
Mathematical Methods ◽  
Operator Calculus ◽  
Complex Functions

This paper gives a survey of the historical development of the solutions of various fluctuation problems in the theory of queues from the point of view of the mathematical methods used. These methods include Markov chains, Markov processes, integral equations, complex functions, generating functions, Laplace transforms, factorization of functions, operator calculus, Banach algebras and some particular methods, such as calculus of finite differences and combinatorics. In addition, the paper contains several recent results of the author for semi-Markov queuing processes.

scholarly journals PARABOLIC KAZHDAN–LUSZTIG BASIS, SCHUBERT CLASSES, AND EQUIVARIANT ORIENTED COHOMOLOGY

2019 ◽  
Vol 19 (6) ◽  
pp. 1889-1929
Author(s):  
Cristian Lenart ◽  
Kirill Zainoulline ◽  
Changlong Zhong
Keyword(s):  
Cohomology Ring ◽  
Flag Varieties ◽  
Special Cases ◽  
Oriented Cohomology ◽  
Schubert Classes ◽  
New Interpretation ◽  
The Moment ◽  
The Right ◽  
Relationship Of ◽  
The Relationship

We study the equivariant oriented cohomology ring $\mathtt{h}_{T}(G/P)$ of partial flag varieties using the moment map approach. We define the right Hecke action on this cohomology ring, and then prove that the respective Bott–Samelson classes in $\mathtt{h}_{T}(G/P)$ can be obtained by applying this action to the fundamental class of the identity point, hence generalizing previously known results of Chow groups by Brion, Knutson, Peterson, Tymoczko and others. Our main result concerns the equivariant oriented cohomology theory $\mathfrak{h}$ corresponding to the 2-parameter Todd genus. We give a new interpretation of Deodhar’s parabolic Kazhdan–Lusztig basis, i.e., we realize it as some cohomology classes (the parabolic Kazhdan–Lusztig (KL) Schubert classes) in $\mathfrak{h}_{T}(G/P)$. We make a positivity conjecture, and a conjecture about the relationship of such classes with smoothness of Schubert varieties. We also prove the latter in several special cases.

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