79,410 research outputs found
Free integro-differential algebras and Groebner-Shirshov bases
No full textThe notion of commutative integro-differential algebra was introduced for the algebraic study of boundary problems for linear ordinary differential equations. Its noncommutative analog achieves a similar purpose for linear systems of such equations. In both cases, free objects are crucial for analyzing the underlying algebraic structures, e.g. of the (matrix) functions.
In this paper we apply the method of Groebner-Shirshov bases to construct the free (noncommutative) integro-differential algebra on a set. The construction is from the free Rota-Baxter algebra on the free differential algebra on the set modulo the differential Rota-Baxter ideal generated by the noncommutative integration by parts formula. In order to obtain a canonical basis for this quotient, we first reduce to the case when the set is finite. Then in order to obtain the monomial order needed for the Composition-Diamond Lemma, we consider the free Rota-Baxter algebra on the truncated free differential algebra. A Composition-Diamond Lemma is proved in this context, and a Groebner-Shirshov basis is found for the corresponding differential Rota-Baxter ideal
Rota–Baxter Operators on Skew Braces
No full textIn this paper, we introduce the concept of Rota–Baxter skew braces, and provide classifications of Rota–Baxter operators on various skew braces, such as (Z,+,∘) and (Z/(4),+,∘). We also present a necessary and sufficient condition for a skew brace to be a co-inverse skew brace. Additionally, we describe some constructions of Rota–Baxter quasiskew braces, and demonstrate that every Rota–Baxter skew brace can induce a quasigroup and a Rota–Baxter quasiskew brace
Pattern avoidance in partial permutations
No full textMotivated by the concept of partial words, we introduce an analogous concept of partial permutations. A partial permutation of length n with k holes is a sequence of symbols in which each of the symbols from the set {1,2,...,n-k} appears exactly once, while the remaining k symbols of are "holes". We introduce pattern-avoidance in partial permutations and prove that most of the previous results on Wilf equivalence of permutation patterns can be extended to partial permutations with an arbitrary number of holes. We also show that Baxter permutations of a given length k correspond to a Wilf-type equivalence class with respect to partial permutations with (k-2) holes. Lastly, we enumerate the partial permutations of length n with k holes avoiding a given pattern of length at most four, for each n >= k >= 1
Lax pairs for new -symmetric coset -models and their Yang-Baxter deformations
No full textTwo-dimensional -models with -symmetric homogeneous
target spaces have been shown to be classically integrable when introducing
WZ-terms in a particular way. This article continues the search for new models
of this type now allowing some kinetic terms to be absent, analogously to the
Green-Schwarz superstring -model on -symmetric
homogeneous spaces. A list of such integrable -symmetric
(super)coset -models for and their Lax pairs is presented.
For arbitrary , a big class of integrable models is constructed that
includes both the known pure spinor and Green-Schwarz superstring on
-symmetric cosets.
Integrable Yang-Baxter deformations of this class of -symmetric
(super)coset -models can be constructed in same way as in the known
- or -cases. Deformations based on solutions of the
modified classical Yang-Baxter equation, the so-called -deformation,
require deformation of the constants defining the Lagrangian and the
corresponding Lax pair. Homogeneous Yang-Baxter deformations (i.e. those based
on solutions to the classical Yang-Baxter equation) leave the equations of
motion and consequently the Lax pair invariant and are expected to be
classically equivalent to the undeformed model.
As an example, the relationship between -symmetric homogeneous
spaces and nearly (para-)K\"ahler geometries is revisited. Confirming existing
literature it is shown that the integrable choice of WZ-term in the
-symmetric coset -model associated to a nearly K\"ahler
background gives an imaginary contribution to the action.Comment: 23 page
YPFS Lessons Learned Oral History Project: An Interview with Thomas Baxter
No full textSuggested Citation Form: Baxter, Thomas, 2018. “Lessons Learned Interview. Interview by Rosalind Wiggins and Alec Buchholtz. Yale Program on Financial Stability Lessons Learned Oral History Project. November 20, 2018. Transcript. https://ypfs.som.yale.edu/library/ypfs-lesson-learned-oral-history-project-interview-thomas-baxte
Rota-Baxter operators and Bernoulli polynomials
No full textsummary:We develop the connection between Rota-Baxter operators arisen from algebra and mathematical physics and Bernoulli polynomials. We state that a trivial property of Rota-Baxter operators implies the symmetry of the power sum polynomials and Bernoulli polynomials. We show how Rota-Baxter operators equalities rewritten in terms of Bernoulli polynomials generate identities for the latter
Lessons Learned: Thomas C. Baxter, Jr., Esq.
No full textBaxter, who was General Counsel of the Federal Reserve Bank of New York during the crisis, gives us his take on how best to prepare for future crises
Short-time critical dynamics of the Baxter-Wu model
No full textWe study the early time behavior of the Baxter-Wu model, an Ising model with three-spin interactions on a triangular lattice. Our estimates for the dynamic exponent z are compatible with results recently obtained for two models which belong to the same universality class of the Baxter-Wu model: the two-dimensional four-state Potts model and the Ising model with three-spin interactions in one direction. However, our estimates for the dynamic exponent theta of the Baxter-Wu model are completely different from the values obtained for those models. This discrepancy could be related to the absence of a marginal operator in the Baxter-Wu model
On the algebraic structure of factorized S-matrices
No full textThis thesis investigates the algebraic structure of certain quantum field theories in one space and one time dimension. These theories are integrable - essentially, highly constrained and therefore soluble. Thus, instead of having to use perturbative techniques, it is possible to conjecture their exact 5-matrices, which have the property that they are factorized into two-particle 5-matrices. In particular, there are two types of such theory: in one, scattering is purely elastic, whilst in the other, there is additional structure dictated by the Yang-Baxter equation. This thesis explores the algebraic structure of the latter and its links with the former. We begin, in chapter one, with an informal summary of the development of the subject, followed by a more mathematical exposition in chapter two. Chapter three constructs explicitly some exact factorized 5-matrices with Yang-Baxter structure, and comments on their features, both intrinsic and in relation to purely elastic 5-matrices. In particular, there is an unexplained close correspondence between the mass spectra and particle fusings in the two types of theory. The next three chapters attempt to shed some light on these features. Chapter four constructs similar 5-matrices, but based on quantum-deformed algebras rather than classical algebras. In chapter five we describe the structure of the 5-matrices when the particles they describe transform in irreducible representations of classical algebras. This leads us to consider the Yangian algebra, the representation theory of which underlies Yang-Baxter dependent 5-matrices, and which we therefore review briefly. We begin chapter six by reviewing the work which shows that the Yangian is also the charge algebra of the integrable quantum field theory, and subsequently show that the Yangian is also to a great extent present in the corresponding classical theory. We conclude with a brief seventh chapter describing the outlook for further research, followed by appendices containing respectively details of the Lagrangians of some integrable quantum field theories, a continuum formulation of the quantum inverse problem, explicit expressions for some of the R-matrices computed in the text, and a summary of known solutions of the Yang-Baxter equation
The Benefits of Being Economics Professor A (and not Z)
No full textAlphabetic name ordering on multi-authored academic papers, which is the convention in the economics discipline and various other disciplines, is to the advantage of people whose last name initials are placed early in the alphabet. As it turns out, Professor A, who has been a first author more often than Professor Z, will have published more articles and experienced afaster growth rate over the course of her career as a result of reputation and visibility. Moreover, authors know that name ordering matters and indeed take ordering seriously: Several characteristics of an author group composition determine the decision to deviate from the default alphabetic name order to a significant extent.performance measurement, incentives, economists, name ordering
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