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In the first part of the paper, we propose a different viewpoint on the theory of higher Airy structures (or Airy ideals), which may shed light on its origin. We define Airy ideals in the ħ -adic completion of the Rees Weyl algebra and show that Airy ideals are defined exactly such that they are always related to the canonical left ideal generated by derivatives by automorphisms of the Rees Weyl algebra of a simple type, which we call transvections. The standard existence and uniqueness result in the theory of Airy structures then follow immediately. In the second part of the paper, we construct Airy ideals generated by the nonnegative modes of the strong generators of the principal W -algebra of sp 2 N at level - N - 1 / 2 , following the approach developed in Borot et al. (Mem Am Math Soc, 2021). This provides an example of an Airy ideal in the Heisenberg algebra that requires realizing the zero modes as derivatives instead of variables, which leads to an interesting interpretation for the resulting partition function. [ABSTRACT FROM AUTHOR]

It is well-known that a formal deformation of a commutative algebra A leads to a Poisson bracket on A and that the classical limit of a derivation on the deformation leads to a derivation on A , which is Hamiltonian with respect to the Poisson bracket. In this paper we present a generalization of it for formal deformations of an arbitrary noncommutative algebra A . The deformation leads in this case to a Poisson algebra structure on Π (A) : = Z (A) × (A / Z (A)) and to the structure of a Π (A) -Poisson module on A . The limiting derivations are then still derivations of A , but with the Hamiltonian belong to Π (A) , rather than to A . We illustrate our construction with several cases of formal deformations, coming from known quantum algebras, such as the ones associated with the nonabelian Volterra chains, Kontsevich integrable map, the quantum plane and the quantized Grassmann algebra. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study a generalization of pre-Malcev algebras and pre-Lie superalgebras, which is called pre-Malcev superalgebras. We first introduce the definition of pre-Malcev superalgebras with representations, and obtain some important properties and constructions of the super Kupershmidt operators on pre-Malcev superalgebras. Then we give the relationship among pre-Malcev superalgebras, alternative superalgebras and M-dendriform superalgebras. Furthermore, we give infinitesimal deformations of super Kupershmidt operators. Finally, we define bimodules of pre-Malcev superalgebras and study their relationships with super Kupershmidt operators. [ABSTRACT FROM AUTHOR]

In this paper, we give the notion of (lax or weak) Hom-corings, which leads to the introduction of partial coaction of a Hom-Hopf algebra on a Hom-algebra. We construct several pairs of adjoint functors, and define Galois Hom-corings, as a special case of which Galois Hom-coalgebras are introduced. [ABSTRACT FROM AUTHOR]

As a quantum affinization, the quantum toroidal algebra U q , q ¯ (gl ¨ n) is defined in terms of its "left" and "right" halves, which both admit shuffle algebra presentations (Enriquez in Transform Groups 5(2):111–120, 2000; Feigin and Odesskii in Am Math Soc Transl Ser 2:185, 1998). In the present paper, we take an orthogonal viewpoint, and give shuffle algebra presentations for the "top" and "bottom" halves instead, starting from the evaluation representation U q (gl ˙ n) ↷ C n (z) and its usual R-matrix R (z) ∈ End (C n ⊗ C n) (z) (see Faddeev et al. in Leningrad Math J 1:193–226, 1990). An upshot of this construction is a new topological coproduct on U q , q ¯ (gl ¨ n) which extends the Drinfeld–Jimbo coproduct on the horizontal subalgebra U q (gl ˙ n) ⊂ U q , q ¯ (gl ¨ n) . [ABSTRACT FROM AUTHOR]

Let A be the Steenrod algebra over the finite field k : = F 2 and G(q) be the general linear group of rank q over k. A well-known open problem in algebraic topology is the explicit determination of the cohomology groups of the Steenrod algebra, Ext A q , ∗ (k , k) , for all homological degrees q ⩾ 0. The Singer algebraic transfer of rank q, formulated by William Singer in 1989, serves as a valuable method for describing that Ext groups. This transfer maps from the coinvariants of a certain representation of G(q) to Ext A q , ∗ (k , k). Singer predicted that the algebraic transfer is always injective, but this has gone unanswered for all q ⩾ 4. This paper establishes Singer's conjecture for rank four in the generic degrees n = 2 s + t + 1 + 2 s + 1 - 3 whenever t ≠ 3 and s ⩾ 1 , and n = 2 s + t + 2 s - 2 whenever t ≠ 2 , 3 , 4 and s ⩾ 1. In conjunction with our previous results, this completes the proof of the Singer conjecture for rank four. [ABSTRACT FROM AUTHOR]

This paper presents variants of the topological Jacobson radical of algebras in quasi-regular and modular versions of the definition, which use the ideas of the construction of the Brown–McCoy radical and the descriptions of the Jacobson radical of alternative algebras. [ABSTRACT FROM AUTHOR]

In this survey paper, we collect properties of one-relator algebras in varieties of linear algebras. The main attention is paid to Schreier varieties of algebras. A variety of linear algebras is said to be Schreier if any subalgebra of a free algebra of this variety is free. [ABSTRACT FROM AUTHOR]

In this paper, numerical characteristics of identities with involution of nonassociative algebras are studied. We construct a series of examples of algebras with overexponential growth of codimensions with involution. [ABSTRACT FROM AUTHOR]

In this paper, we consider a metric graph of the simplest structure, which is a three-ray star with edges of pairwise different lengths. For it, all steps of the procedure which reduces the algebra of the eikonals of the graph to a canonical form, are described in detail. Using this form, the spectrum of the eikonal algebra is endowed with a graph structure. [ABSTRACT FROM AUTHOR]

The rook monoid, also known as the symmetric inverse monoid, is the archetypal structure when it comes to extend the principle of symmetry. In this paper, we establish a Schur–Weyl duality between this monoid and an extension of the classical Schur algebra, which we name the extended Schur algebra. We also explain how this relates to Solomon's Schur–Weyl duality between the rook monoid and the general linear group and mention some advantages of our approach. [ABSTRACT FROM AUTHOR]

Bhargava and the first-named author of this paper introduced a functorial Galois closure operation for finite-rank ring extensions, generalizing constructions of Grothendieck and Katz–Mazur. In this paper, we generalize Galois closures and apply them to construct a new infinite family of irreducible components of Hilbert schemes of points. We show that these components are elementary, in the sense that they parametrize algebras supported at a point. Furthermore, we produce secondary families of elementary components obtained from Galois closures by modding out by suitable socle elements. [ABSTRACT FROM AUTHOR]

This paper considers theoretical foundations and mathematical methods of data analysis under the conditions of the Rice statistical distribution. The problem involves joint estimation of the signal and noise parameters. It is shown that this estimation requires the solution of a complex system of essentially nonlinear equations with two unknown variables, which implies significant computational costs. This study is aimed at mathematical optimization of computer algebra methods for numerical solution of the problem of Rician data analysis. As a result of the optimization, the solution of the system of two nonlinear equations is reduced to the solution of one equation with one unknown variable, which significantly simplifies algorithms for the numerical solution of the problem, reduces the amount of necessary computational resources, and enables the use of advanced methods for parameter estimation in information systems with priority of real-time operation. Results of numerical experiments carried out using Wolfram Mathematica confirm the effectiveness of the developed methods for two-parameter analysis of Rician data. The data analysis methods considered in this paper are useful for solving many scientific and applied problems that involve analysis of data described by the Rice statistical model. [ABSTRACT FROM AUTHOR]

In this paper, by considering the notion of L-algebras and CL-algebras, we construct a topology on a bounded CL-algebra. Then, we investigate some of its topological properties, such as Hausdorff space, T 0 -space and T 1 -space and connectedness. Moreover, we express the relation between closed and compact sets in this topology. Finally, by considering the notion of semi-topological algebra, we prove that any commutative bounded CL-algebra is a right semi-topological algebra. [ABSTRACT FROM AUTHOR]

In this paper, first, we consider an algebra that has a binary operation and a join of arbitrary nonempty subset. A lattice implication algebra is a lattice with a binary operation, which has a join and a meet of finite nonempty subsets. In this work, the notion of join-complete implication algebras L is defined as a join-complete lattice with a binary operation, and some properties of this algebra L are searched. Moreover, we prove that the interval [a, 1] in L is a lattice implication algebra and show that L satisfies the completely distributive law when it has the smallest element 0. Finally, we state the concept of filter and multipliers of L and provide finite and infinite examples of them. In addition, we research some properties of these concepts in detail. [ABSTRACT FROM AUTHOR]

Studies of issues related to computability and computational complexity involve the use of a model of computation. Central in such a model are computational processes. Processes of this kind can be described using an imperative process algebra based on ACP (Algebra of Communicating Processes). In this paper, it is investigated whether the imperative process algebra concerned can play a role in the field of models of computation. It is demonstrated that the process algebra is suitable to describe in a mathematically precise way models of computation corresponding to existing models based on sequential, asynchronous parallel, and synchronous parallel random access machines as well as time and work complexity measures for those models. [ABSTRACT FROM AUTHOR]

This paper introduces a novel method for solving the resection problem in two and three dimensions based on conformal geometric algebra (CGA). Advantage is taken because of the characteristics of CGA, which enables the representation of points, lines, planes, and volumes in a unified mathematical framework and offers a more intuitive and geometric understanding of the problem, in contrast to existing purely algebraic methods. Several numerical examples are presented to demonstrate the efficacy of the proposed method and to compare its validity with established techniques in the field. Numerical simulations indicate that our vector geometric algebra implementation is faster than the best-known algorithms to date, suggesting that the proposed GA-based methods can provide a more efficient and comprehensible solution to the two- and three-dimensional resection problem, paving the way for further applications and advances in geodesy research. Furthermore, the method's emphasis on graphical and geometric representation makes it particularly suitable for educational purposes, allowing the reader to grasp the concepts and principles of resection more effectively. The proposed method has potential applications in a wide range of other fields, including surveying, robotics, computer vision, or navigation. [ABSTRACT FROM AUTHOR]

In this paper we provide an account of the Todd–Coxeter algorithm for computing congruences on semigroups and monoids. We also give a novel description of an analogue for semigroups of the so-called Felsch strategy from the Todd–Coxeter algorithm for groups. [ABSTRACT FROM AUTHOR]

Let H be a real or complex Hilbert space with the dimension greater than one and B (H) the algebra of all bounded linear operators on H . Assume that δ is a linear mapping from B (H) into itself which is Jordan derivable at a given element Ω ∈ B (H) , in the sense that δ (A ∘ B) = δ (A) ∘ B + A ∘ δ (B) holds for all A , B ∈ B (H) with A ∘ B = Ω , where ∘ denotes the Jordan product A ∘ B = A B + B A . In this paper, we show that if Ω is an arbitrary but fixed nonzero operator, then δ is a derivation; if Ω is a zero operator, then δ is a generalized derivation. [ABSTRACT FROM AUTHOR]

A permutation statistic st is said to be shuffle-compatible if the distribution of st over the set of shuffles of two disjoint permutations π and σ depends only on st π , st σ , and the lengths of π and σ . Shuffle-compatibility is implicit in Stanley's early work on P-partitions, and was first explicitly studied by Gessel and Zhuang, who developed an algebraic framework for shuffle-compatibility centered around their notion of the shuffle algebra of a shuffle-compatible statistic. For a family of statistics called descent statistics, these shuffle algebras are isomorphic to quotients of the algebra of quasisymmetric functions. Recently, Domagalski, Liang, Minnich, Sagan, Schmidt, and Sietsema defined a version of shuffle-compatibility for statistics on cyclic permutations, and studied cyclic shuffle-compatibility through purely combinatorial means. In this paper, we define the cyclic shuffle algebra of a cyclic shuffle-compatible statistic, and develop an algebraic framework for cyclic shuffle-compatibility in which the role of quasisymmetric functions is replaced by the cyclic quasisymmetric functions recently introduced by Adin, Gessel, Reiner, and Roichman. We use our theory to provide explicit descriptions for the cyclic shuffle algebras of various cyclic permutation statistics, which in turn gives algebraic proofs for their cyclic shuffle-compatibility. [ABSTRACT FROM AUTHOR]

This paper is to look for bi-Frobenius algebra structures on quantum complete intersections over field k. We find a class of comultiplications, such that if − 1 ∈ k , then a quantum complete intersection becomes a bi-Frobenius algebra with comultiplication of this form if and only if all the parameters qij = ±1. Also, it is proved that if − 1 ∈ k then a quantum exterior algebra in two variables admits a bi-Frobenius algebra structure if and only if the parameter q = ±1. While if − 1 ∉ k , then the exterior algebra with two variables admits no bi-Frobenius algebra structures. We prove that the quantum complete intersections admit a bialgebra structure if and only if it admits a Hopf algebra structure, if and only if it is commutative, the characteristic of k is a prime p, and every ai a power of p. This also provides a large class of examples of bi-Frobenius algebras which are not bialgebras (and hence not Hopf algebras). In commutative case, other two comultiplications on complete intersection rings are given, such that they admit non-isomorphic bi-Frobenius algebra structures. [ABSTRACT FROM AUTHOR]

In this paper, we discuss an important and nontrivial theorem on evaluation homomorphisms. We state this theorem as a canonical duality between the family of all smooth mappings f ∈ Hom(M,M′) of a smooth real finite-dimensional manifold M into a similar manifold M′ and the family of homomorphisms φ of the algebra C∞ (M′) of smooth scalar-valued functions on M′ into the analogous algebra C∞ (M) on M, φ ∈ Hom (C∞ (M′), C∞ (M)). This formulation possesses the maximum natural generality and, at the same time, allows it to be used in applications in the standard canonical form. [ABSTRACT FROM AUTHOR]

The paper is devoted to the problem of determining the number of subalgebras in the matrix algebra that are generated by nonderogatory matrices and are distinct up to similarity. A similarity criterion in terms of isomorphism of quotient algebras of the algebra of polynomials for matrix algebras of the type indicated is established. The existence of infinite fields for which the number of distinct algebras is infinite for all values of the matrix order is proved. For algebraically closed fields, the field of reals, and finite fields of sufficiently large cardinalities, the number of distinct algebras generated by nonderogatory matrices is determined as a function of the matrix order. [ABSTRACT FROM AUTHOR]

It is known that determinantal point processes have an intimate relation to operator algebras. In the paper, we extend this relationship to encompass dynamical aspects. Especially, we delve into two types of determinantal point processes. One is related to discrete orthogonal polynomials of hypergeometric type, and another is z-measures, which arise in the asymptotic representation theory of the symmetric groups. Within the framework of operator algebras, we investigate both unitary and stochastic dynamics on these point processes. [ABSTRACT FROM AUTHOR]

This paper explores the nonconvex second-order cone as a nonconvex conic extension of the known convex second-order cone in optimization, as well as a higher-dimensional conic extension of the known causality cone in relativity. The nonconvex second-order cone can be used to reformulate nonconvex quadratic programming and nonconvex quadratically constrained quadratic program in conic format. The cone can also arise in real-world applications, such as facility location problems in optimization when some existing facilities are more likely to be closer to new facilities than other existing facilities. We define notions of the algebraic structure of the nonconvex second-order cone and show that its ambient space is commutative and power-associative, wherein elements always have real eigenvalues; this is remarkable because it is not the case for arbitrary Jordan algebras. We will also find that the ambient space of this nonconvex cone is rank-independent of its dimension; this is also notable because it is not the case for algebras of arbitrary convex cones. What is more noteworthy is that we prove that the nonconvex second-order cone equals the cone of squares of its ambient space; this is not the case for all non-Euclidean Jordan algebras. Finally, numerous algebraic properties that already exist in the framework of the convex second-order cone are generalized to the framework of the nonconvex second-order cone. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the notion of an HLieDer triple, which includes a Hom-Lie algebra and a derivation. We define a cohomology theory for HLieDer triples with coefficients in a representation. We study central extensions of an HLieDer triple. Finally, we consider homotopy derivations on HLieb∞ algebras and 2-derivations on Hom-Lie 2-algebras, and we prove that the category of 2-term HLieb∞ algebras with homotopy derivations and the category of Hom-Lie 2-algebras with 2-derivations are equivalent. [ABSTRACT FROM AUTHOR]

In this paper we show how the Functional Separation of Variables (FSoV) method can be applied to the problem of computing overlaps with integrable boundary states in integrable systems. We demonstrate our general method on the example of a particular boundary state, a singlet of the symmetry group, in an su 3 rational spin chain in an alternating fundamental-anti-fundamental representation. The FSoV formalism allows us to compute in determinant form not only the overlaps of the boundary state with the eigenstates of the transfer matrix, but in fact with any factorisable state. This includes off-shell Bethe states, whose overlaps with the boundary state have been out of reach with other methods. Furthermore, we also found determinant representations for insertions of so-called Principal Operators (forming a complete algebra of all observables) between the boundary and the factorisable state as well as certain types of multiple insertions of Principal Operators. Concise formulas for the matrix elements of the boundary state in the SoV basis and su N generalisations are presented. Finally, we managed to construct a complete basis of integrable boundary states by repeated action of conserved charges on the singlet state. As a result, we are also able to compute the overlaps of all of these states with integral of motion eigenstates. [ABSTRACT FROM AUTHOR]

In 1993, I. P. Shestakov posed the question of the existence of a central simple finite-dimensional algebra over a field of characteristic zero whose identities are not given by a finite set of identities. In 2012, I. M. Isaev and the author of the present paper constructed an example of a seven-dimensional central simple algebra over any field that does not have a finite basis of identities. In the present paper, we construct an example of a six-dimensional central simple algebra over a field of characteristic zero which has no finite basis of identities. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity. For positive integers m and n, Anderson and Badawi (Journal of Algebra and Its Applications 16(1):1750013 (21 pp), 2017) defined an ideal I of a ring R to be an (m,n)-closed if whenever x m ∈ I , then x n ∈ I . In this paper we define and study a new generalization of the class of (m,n)-closed ideals which is the class of quasi (m,n)-closed ideals. A proper ideal I is called quasi (m,n)-closed in R if for x ∈ R , x m ∈ I implies either x n ∈ I or x m - n ∈ I . That is, I is quasi (m,n)-closed in R if and only if I is either (m, n)-closed or ( m , m - n )-closed in R. We justify several properties and characterizations of quasi (m,n)-closed ideals with many supporting examples. Furthermore, we investigate quasi (m,n)-closed ideals under various contexts of constructions such as direct products, localizations and homomorphic images. Finally, we discuss the behavior of this class of ideals in idealization rings. [ABSTRACT FROM AUTHOR]

Let A and A ′ be two alternative ∗ -algebras with identities 1 A and 1 A ′ , respectively, and let e 1 and e 2 = 1 A - e 1 be nontrivial symmetric idempotents in A . In this paper, we study the characterization of multiplicative ∗ -Jordan-type maps on alternative algebras. [ABSTRACT FROM AUTHOR]

In enumerative geometry, Virasoro constraints were first conjectured in Gromov-Witten theory with many new recent developments in the sheaf theoretic context. In this paper, we rephrase the sheaf theoretic Virasoro constraints in terms of primary states coming from a natural conformal vector in Joyce's vertex algebra. This shows that Virasoro constraints are preserved under wall-crossing. As an application, we prove the conjectural Virasoro constraints for moduli spaces of torsion-free sheaves on any curve and on surfaces with only (p , p) cohomology classes by reducing the statements to the rank 1 case. [ABSTRACT FROM AUTHOR]

It has been shown that by using a Lagrange multiplier field to ensure that the classical equations of motion are satisfied, radiative effects beyond one-loop order are eliminated. It has also been shown that through the contribution of some additional ghost fields, the effective action becomes form invariant under a redefinition of field variables, and furthermore, the usual one-loop results coincide with the quantum corrections obtained from this effective action. In this paper, we consider the consequences of a gauge invariance being present in the classical action. The resulting gauge transformations for the Lagrange multiplier field as well as for the additional ghost fields are found. These gauge transformations result in a set of Faddeev–Popov ghost fields arising in the effective action. If the gauge algebra is closed, we find the Becci–Rouet–Stora–Tyutin (BRST) transformations that leave the effective action invariant. [ABSTRACT FROM AUTHOR]

A (k, m)-Furstenberg set is a subset S ⊂ F q n with the property that each k-dimensional subspace of F q n can be translated so that it intersects S in at least m points. Ellenberg and Erman (Algebra Number Theory 10(7), 1415–1436 (2016)) proved that (k, m)-Furstenberg sets must have size at least C n , k m n / k , where C n , k is a constant depending only n and k. In this paper, we adopt the same proof strategy as Ellenberg and Erman, but use more elementary techniques than their scheme-theoretic method. By modifying certain parts of the argument, we obtain an improved bound on C n , k , and our improved bound is nearly optimal for an algebraic generalization the main combinatorial result. We also extend our analysis to give lower bounds for sets that have large intersection with shifts of a specific family of higher-degree co-dimension n - k varieties, instead of just co-dimension n - k subspaces. [ABSTRACT FROM AUTHOR]

The secure transmission of information is receiving more and more attention nowadays. Aeolus is a novel platform designed to enhance the development of distributed applications by preventing unauthorized disclosure of information. And one of the most representative systems for information transmission is the file system, therefore it is of great significance to formally analyze the Aeolus-based file system. In this paper, we use Communicating Sequential Processes (CSP) to model and formalize the file system based on Aeolus. Moreover, we utilize the Process Analysis Toolkit (PAT) to simulate and verify the CSP description of our established model. We specifically verify the validity of five properties: Deadlock Freedom, Divergence Freedom, Reachability, Secrecy, and Integrity. The verification results demonstrate that the model successfully satisfies these properties, affirming the effectiveness of the framework in ensuring file operations and guaranteeing the secure transmission of information. [ABSTRACT FROM AUTHOR]

In this paper we introduce a notion of duality for matrix valued orthogonal polynomials with respect to a measure supported on the nonnegative integers. We show that the dual families are closely related to certain difference operators acting on the matrix orthogonal polynomials. These operators belong to the so called Fourier algebras, which play a key role in the construction of the families. In order to illustrate duality, we describe a family of Charlier type matrix orthogonal polynomials with explicit shift operators which allow us to find explicit formulas for three term recurrences, difference operators and squared norms. These are the essential ingredients for the construction of different dual families. [ABSTRACT FROM AUTHOR]

As is well-known, the subalgebras of any universal algebra form an algebraic closure system. Conversely, every algebraic closure system arises as the family of subalgebras of some universal algebra, but this algebra is far from uniquely determined. This paper investigates the realization of algebraic closure systems by algebras given either by a single operation or by operations of the lowest arity. In particular, it is shown that an algebraic closure system with arity n in which the empty set is closed and every finitely generated closed set is countable can be realized by a single (n + 1) -ary operation. The algebraic closure system of cosets on any group is realized by the single ternary Mal'cev term x y - 1 z . It is shown that the closure system of cosets on an Abelian group A can be realized by a single binary operation if and only if A has at most one element of order 2. Similar results are obtained for modules over an arbitrary ring. [ABSTRACT FROM AUTHOR]

Gel'fand-Dorfman bialgebra, which is both a Lie algebra and a Novikov algebra with some compatibility condition, appeared in the study of Hamiltonian pairs in completely integrable systems. They also emerged in the description of a class special Lie conformal algebras called quadratic Lie conformal algebras. In this paper, we investigate the extending structures problem for Gel'fand-Dorfman bialgebras, which is equivalent to some extending structures problem of quadratic Lie conformal algebras. Explicitly, given a Gel'fand-Dorfman bialgebra (A, ○, [·, ·]), this problem is how to describe and classify all Gel'fand-Dorfman bialgebra structures on a vector space E (A ⊂ E) such that (A, ○, [·, ·]) is a subalgebra of E up to an isomorphism whose restriction on A is the identity map. Motivated by the theories of extending structures for Lie algebras and Novikov algebras, we construct an object G ℋ 2 (V , A) to answer the extending structures problem by introducing the notion of a unified product for Gel'fand-Dorfman bialgebras, where V is a complement of A in E. In particular, we investigate the special case when dim(V) = 1 in detail. [ABSTRACT FROM AUTHOR]

Let H be a complex infinite dimensional Hilbert space and B (H) be the algebra of all bounded linear operators on H . In this paper, we mainly study the operators that satisfy both a-Weyl's theorem and property (R). Also, the operators whose functional calculus satisfies the two properties are also explored. We give the features for the operator or its functional calculus for which both a-Weyl's theorem and property (R) hold. [ABSTRACT FROM AUTHOR]

The words "Programming is the second literacy" were coined more than 40 years ago [13], but never came to life. The paper develops and details that old slogan by proposing that the mainstream mathematics education in schools should merge with education in computer science/programming. Of course, this means a deep structural reform of school mathematics education. We are not talking about adapting the 20th century mathematics to the 21st century—as it outlined in [10, 19], we mean the 21st century mathematics education for the 21st century mathematics. To the best of our knowledge, this paper is perhaps the first known attempt to start a proper feasibility study for this reform. The scope of the paper does not allow us to touch the delicate socio-political (and financial) sides of the reform, we are looking only at general curricular and didactic aspects and possible directions of the reform. In particular, we indicate approaches to development of a Domain Specifiic Language (DSL) as a basis for all programming aspects of a new course. [ABSTRACT FROM AUTHOR]

Let H be a complex Hilbert space and let B (H) be the algebra of all bounded linear operators on H. In this paper, for T ∈ B (H) and a unit vector x ∈ H , we introduce a local version of the reduced minimum modulus of T at x, noted by γ (T , x) . Properties of this quantity are investigated. We study the relations between γ (T , x) and the Moore–Penrose inverse, spectrum of | T | and the local spectrum of | T | at x. At the end of this paper we will be interested in several problems around this quantity (preserving, continuity, local spectral theory). [ABSTRACT FROM AUTHOR]

In this paper, we study conformal points among the class of E -models. The latter are σ-models formulated in terms of a current Poisson algebra, whose Lie-theoretic definition allows for a purely algebraic description of their dynamics and their 1-loop RG-flow. We use these results to formulate a simple algebraic condition on the defining data of such a model which ensures its 1-loop conformal invariance and the decoupling of its observables into two chiral Poisson algebras, describing the classical left- and right-moving fields of the theory. In the case of so-called non-degenerate E -models, these chiral sectors form two current algebras and the model takes the form of a WZW theory once realised as a σ-model. The case of degenerate E -models, in which a subalgebra of the current algebra is gauged, is more involved: the conformal condition yields a wider class of theories, which includes gauged WZW models but also other examples, seemingly different, which however sometimes turn out to be related to gauged WZW models based on other Lie algebras. For this class, we build non-local chiral fields of parafermionic-type as well as higher-spin local ones, forming classical W -algebras. In particular, we find an explicit and efficient algorithm to build these local chiral fields. These results (and their potential generalisations discussed at the end of the paper) open the way for the quantisation of a large class of conformal E -models using the standard operator formalism of two-dimensional CFT. [ABSTRACT FROM AUTHOR]

The goal of this paper is to make first steps towards the quantisation of integrable nonlinear sigma models using the formalism of affine Gaudin models, by approaching these theories through their conformal limits. We focus mostly on the example of the Klimčík model, which is a two-parameter deformation of the principal chiral model on a Lie group G. We show that the UV fixed point of this theory is described classically by two decoupled chiral affine Gaudin models, encoding its left- and right-moving degrees of freedom, and give a detailed analysis of the chiral and integrable structures of these models. Their quantisation is then explored within the framework of Feigin and Frenkel. We study the quantum local integrals of motion using the formalism of quantised affine Gaudin models and show agreement of the first two integrals with known results in the literature for G = SU (2) . Evidence is given for the existence of a monodromy matrix satisfying the Yang–Baxter algebra for this model, thus paving the way for the quantisation of the non-local integrals of motion. We conclude with various perspectives, including on generalisations of this programme to a larger class of integrable sigma models and applications of the ODE/IQFT correspondence to the description of their quantum spectrum. [ABSTRACT FROM AUTHOR]

We present a generalized version of a quantum oscillator described by means of a ternary Heisenberg algebra. The model leads to a sixth-order Hamiltonian whose energy levels can be discretized using the Bohr–Sommerfeld quantization procedure. We note the similarity with the -extended version of Dirac's equation applied to quark color dynamics, which also leads to sixth-order field equations. The paper also contains a comprehensive guide to -graded structures, including ternary algebras, which form a mathematical basis for the proposed generalization. The symmetry properties of the model are also discussed. [ABSTRACT FROM AUTHOR]

In this paper we present the Hietarinta Chern–Simons supergravity theory in three space-time dimensions which extends the simplest Poincaré supergravity theory. After approaching the construction of the action using the Chern–Simons formalism, the analysis of the corresponding asymptotic symmetry algebra is considered. For this purpose, we first propose a consistent set of asymptotic boundary conditions for the aforementioned supergravity theory whose underlying symmetry corresponds to the supersymmetric extension of the Hietarinta algebra. We then show that the corresponding charge algebra contains the super- bms 3 algebra as subalgebra, and has three independent central charges. We also show that the obtained asymptotic symmetry algebra can alternatively be recovered as a vanishing cosmological constant limit of three copies of the Virasoro algebra, one of which is augmented by supersymmetry. [ABSTRACT FROM AUTHOR]

Let be an embedded projective variety. The Lie algebra defined by the tangent directions to at smooth points is an interesting algebraic invariant of . In some cases, this algebra is isomorphic to the symbol algebra of a filtered system of distributions on a Fano manifold, which plays an important role in the theory of these manifolds. In addition, algebras defined by tangent directions are interesting on their own right. In this paper, we study the Lie algebra corresponding to a variety that is the projectivization of the orbit of the lowest weight vector of an irreducible representation of a complex semisimple Lie group. We describe these algebras in terms of generators and relations. In many cases, we can describe their structure completely. [ABSTRACT FROM AUTHOR]

In the TQFT formalism of Moore–Tachikawa for describing Higgs branches of theories of class S , the space associated to the unpunctured sphere in type g is the universal centraliser Z G , where g = L i e (G) . In more physical terms, this space arises as the Coulomb branch of pure N = 4 gauge theory in three dimensions with gauge group G ˇ , the Langlands dual. In the analogous formalism for describing chiral algebras of class S , the vertex algebra associated to the sphere has been dubbed the chiral universal centraliser. In this paper, we construct an open, symplectic embedding from a cover of the Kostant–Toda lattice of type g to the universal centraliser of G—extending a classic result of Kostant. Using this embedding and some observations on the Poisson algebraic structure of Z G , we propose a free field realisation of the chiral universal centraliser for any simple group G. We exploit this realisation to develop free field realisations of chiral algebras of class S of type a 1 for theories of genus zero with n = 1 , ... , 6 punctures. These realisations make generalised S-duality completely manifest, and the generalisation to n ⩾ 7 punctures is conceptually clear, though technically burdensome. [ABSTRACT FROM AUTHOR]

Similar to gravity, an infinite tower of symmetries generated by higher-spin charges has been identified in Yang-Mills theory by studying collinear limits or celestial operator products of gluons. This work aims to recover this loop symmetry in terms of charge aspects constructed on the gluonic Fock space. We propose an explicit construction for these higher spin charge aspects as operators which are polynomial in the gluonic annihilation and creation operators. The core of the paper consists of a proof that the charges we propose form a closed loop algebra to quadratic order. This closure involves using the commutator of the cubic order expansion of the charges with the linear (soft) charge. Quite remarkably, this shows that this infinite-dimensional symmetry constrains the non-linear structure of Yang-Mills theory. We provide a similar all spin proof in gravity for the so-called global quadratic (hard) charges which form the loop wedge subalgebra of w1+∞. [ABSTRACT FROM AUTHOR]

In this paper we analyze the relativistic corrections to the leading order three-nucleon (3N) contact interactions. These boost corrections are derived first from the nonrelativistic reduction of covariant Lagrangians and later from the Poincaré algebra constraints on nonrelativistic theories. We show that in order to describe the 3N potential in reference frames other than the center-of-mass frame, the inclusion of five additional terms with fixed coefficients is required. These terms will be relevant in systems with mass number A > 3 . How they will affect EFT calculations of binding energies and scattering observables in these systems should be investigated. [ABSTRACT FROM AUTHOR]

In this paper, a generalization of composition algebras is given, in which the quadratic form is replaced by a form of arbitrary degree. Two series of generalized composition algebras are found: cyclic algebras of arbitrary order and algebras that are obtained by the generalized doubling procedure applied to cyclic algebras. [ABSTRACT FROM AUTHOR]