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This is a review article on the quantum toroidal algebras, focusing on their roles in various solvable structures of 2d conformal field theory, supersymmetric gauge theory, and string theory. Using W -algebras as our starting point, we elucidate the interconnection of affine Yangians, quantum toroidal algebras, and double affine Hecke algebras. Our exploration delves into the representation theory of the quantum toroidal algebra of gl 1 in full detail, highlighting its connections to partitions, W -algebras, Macdonald functions, and the notion of intertwiners. Further, we also discuss integrable models constructed on Fock spaces and associated R -matrices, both for the affine Yangian and the quantum toroidal algebra of gl 1. The article then demonstrates how quantum toroidal algebras serve as a unifying algebraic framework that bridges different areas in physics. Notably, we cover topological string theory and supersymmetric gauge theories with eight supercharges, incorporating the AGT duality. Drawing upon the representation theory of the quantum toroidal algebra of gl 1 , we provide a rather detailed review of its role in the algebraic formulations of topological vertex and q q -characters. Additionally, we briefly touch upon the corner vertex operator algebras and quiver quantum toroidal algebras. [ABSTRACT FROM AUTHOR]

This work deals with the stability analysis of nonlinear sampled-data systems under nonuniform sampling. It establishes novel relationships between specific stability properties of the exact discrete-time model and stability properties of the continuous-time system that is obtained when the maximum admissible sampling period tends to zero. These results can be used to infer stability properties for the sampled-data system by direct inspection of the stability of the mentioned continuous-time system, a task which is typically easier than the analysis of the closed-loop sampled-data system. Compared to the literature, our results allow to prove stronger (asymptotic) sampled-data stability properties for nonlinear systems in cases for which existing results only guarantee practical stability. • Novel links between continuous- and discrete-time models' stability properties. • Simpler stability proofs for nonlinear sampled-data systems under nonuniform sampling. • Present results guarantee stronger (asymptotic) stability properties. • Mild conditions allow for non-globally Lipschitz and non-input-affine systems. [ABSTRACT FROM AUTHOR]

Learning and synthesizing stabilizing controllers for unknown nonlinear control systems is a challenging problem for real-world and industrial applications. Koopman operator theory allows one to analyze nonlinear systems through the lens of linear systems and nonlinear control systems through the lens of bilinear control systems. The key idea of these methods lies in the transformation of the coordinates of the nonlinear system into the Koopman observables, which are coordinates that allow the representation of the original system (control system) as a higher dimensional linear (bilinear control) system. However, for nonlinear control systems, the bilinear control model obtained by applying Koopman operator based learning methods is not necessarily stabilizable. Simultaneous identification of stabilizable lifted bilinear control systems as well as the associated Koopman observables is still an open problem. In this paper, we propose a framework to construct such stabilizable bilinear models and identify their associated observables from data by simultaneously learning a bilinear Koopman embedding for the underlying unknown control affine nonlinear system as well as a Control Lyapunov Function (CLF) for the Koopman based bilinear model using a learner and falsifier. Our proposed approach thereby provides provable guarantees of asymptotic stability for the Koopman based representation of the unknown control affine nonlinear control system as a bilinear system. Numerical simulations are provided to validate the efficacy of our proposed class of stabilizing feedback controllers for unknown control-affine nonlinear systems. [ABSTRACT FROM AUTHOR]

We prove that the Kazhdan-Lusztig category of sl ˆ m at level k , KL k (sl m) , is a semi-simple, rigid braided tensor category for all even m ≥ 4 , and k = − m + 1 2. Moreover, all modules in KL k (sl m) are simple-currents and they appear in the decomposition of conformal embeddings gl m ↪ sl m + 1 at level k = − m + 1 2. For this we inductively identify minimal affine W -algebra W k − 1 (sl m + 2 , θ) as simple current extension of L k (sl m) ⊗ H ⊗ M , where H is the rank one Heisenberg vertex algebra, and M the singlet vertex algebra for c = − 2. The proof uses previously obtained results for the tensor categories of singlet algebra. We also classify all irreducible ordinary modules for W k − 1 (sl m + 2 , θ). The semi-simple part of the category of W k − 1 (sl m + 2 , θ) –modules comes from KL k − 1 (sl m + 2) , using quantum Hamiltonian reduction, but this W -algebra also contains indecomposable ordinary modules. [ABSTRACT FROM AUTHOR]

It is essential to guarantee product quality by detecting defects in industrial printed labels. Defect detection based on reference comparison is a common method to achieve this task. However, this method yields poor performance under large deformations of printed labels, due to the lack of accurate registration ability between testing images and reference images. Therefore, it is an urgent task to achieve accurate image registration for printed label defect detection. In this paper, a coarse-to-fine end-to-end registration network, named APPR-Net, is proposed to deal with the large deformation. First, we adopt a strategy of image patch splitting and stitching to improve the scalability of image resolution. Second, we design a four-stream affine transformation module followed by a multi-scale pyramid registration network, where a coarse registration is obtained by the former module and then gradually refined by the later network in a coarse-to-fine manner. Third, we introduce a distortion loss function to constrain the text distortion of the warped image after image registration. Finally, to simulate printed labels with defects and large deformation, we build a synthetic database based on real-world industrial printed labels for performance comparison. The results demonstrate that the proposed APPR-Net significantly outperforms other compared methods. • Propose a coarse-to-fine registration network to deal with large deformation. • Take advantages of image patch splitting and stitching strategy. • Design a four-stream affine transformation module followed by a pyramid network. • Introduce a distortion loss function to constrain text distortions. • Build a synthetic database based on real-world printed labels. [ABSTRACT FROM AUTHOR]

We compute the Hausdorff dimension of any random statistically self-affine Sierpinski sponge K ⊂ R k (k ≥ 2) obtained by using some percolation process in [ 0 , 1 ] k. To do so, we first exhibit a Ledrappier-Young type formula for the Hausdorff dimensions of statistically self-affine measures supported on K. This formula presents a new feature compared to its deterministic or random dynamical version. Then, we establish a variational principle expressing dim H ⁡ K as the supremum of the Hausdorff dimensions of statistically self-affine measures supported on K , and show that the supremum is uniquely attained. The value of dim H ⁡ K is also expressed in terms of the weighted pressure function of some deterministic potential. As a by-product, when k = 2 , we give an alternative approach to the Hausdorff dimension of K , which was first obtained by Gatzouras and Lalley [27]. The value of the box counting dimension of K and its equality with dim H ⁡ K are also studied. We also obtain a variational formula for the Hausdorff dimensions of some orthogonal projections of K , and for statistically self-affine measures supported on K , we establish a dimension conservation property through these projections. [ABSTRACT FROM AUTHOR]

Let Δ be the set of all quadruples (ν , μ , d , e) of integers that satisfy 0 ≤ D − d 2 ≤ ν ≤ μ ≤ D − d ≤ D , e + d + D is even , | e | ≤ 2 ν − D + d , d ∈ { e + D − 2 ν , min ⁡ { D − μ , e + D − 2 ν + 2 (N − 2 D) } }. In [6] , it is shown for the Terwilliger algebra T of the Grassmann scheme J q (N , D) , N ≥ 2 D , that the isomorphism classes of irreducible T -modules W are determined by their endpoint ν , dual endpoint μ , diameter d , and auxiliary parameter e , that come from the Leonard system attached to W , and it is claimed without proof that the quadruples (ν , μ , d , e) belong to Δ, if d ≥ 1. Let Λ be the set of triples (α , β , ρ) of non-negative integers that satisfy 0 ≤ α ≤ D − ρ 2 , 0 ≤ β ≤ N − D − ρ 2 , 0 ≤ α + β ≤ D − ρ. We construct a mapping from Λ to Δ which is bijective if N > 2 D and 2 : 1 if N = 2 D. We show that the set Λ naturally parameterizes the isomorphism classes of irreducible T -modules, by embedding the standard module of J q (N , D) in a bigger space that allows a U q ( s l ˆ 2) -module structure [9]. As a byproduct we have the following: for a fixed ρ , 0 ≤ ρ ≤ D , set N ′ = N − 2 ρ , D ′ = D − ρ , and Λ ρ = { (α , β) | (α , β , ρ) ∈ Λ }. Then Λ ρ is precisely the set that parameterizes the isomorphism classes of irreducible T -modules for the Johnson scheme J (N ′ , D ′) [3]. [ABSTRACT FROM AUTHOR]

• We propose a novel algorithm that uses local affine/convex hulls for high-dimensional data clustering. • The proposed clustering method circumvents "hole artifacts" problem and improves accuracy. • Experimental results show that the proposed method is very efficient. • The proposed method outperforms most of the other tested subspace clustering methods on wide range of datasets. In this paper we propose a novel clustering algorithm that uses local affine/convex hulls for high-dimensional data clustering. In high-dimensional spaces, the sparse and irregular distributions make the nearest-neighbor distances unreliable (hole artifact), and this deteriorates the clustering performance. Therefore, there is a need to fill in these gaps between the nearest samples. To this end, we use local affine/convex hulls of the nearest neighbors of a given sample to fill in the holes, and this greatly improves the Euclidean distance metric and the clustering accuracy. The proposed method can also be seen as the local extension of the well-known iterative subspace clustering algorithms in which the entire cluster is approximated with a single linear/affine subspace. Experimental results show that the proposed method is efficient and it outperforms other subspace clustering algorithms on a wide range of datasets. [ABSTRACT FROM AUTHOR]

Let R be an affine algebra of dimension d ≥ 4 over a perfect field k of char ≠2 and I be an ideal of R. Then (1) M S d + 1 (R) is uniquely divisible prime to char k if R is reduced and k is infinite with c. d. (k) ≤ 1. (2) Um d + 1 (R , I) / E d + 1 (R , I) has nice group structure if c. d. 2 (k) ≤ 2. (3) Um d (R , I) / E d (R , I) has nice group structure if k is algebraically closed of char k ≠ 2 , 3 and either (i) k = F ‾ p or (ii) R is normal. [ABSTRACT FROM AUTHOR]

A finitely presented group Γ is called flawed if Hom (Γ , G) / / G deformation retracts onto its subspace Hom (Γ , K) / K for all reductive affine algebraic groups G and maximal compact subgroups K ⊂ G. After discussing generalities concerning flawed groups, we show that all finitely generated groups isomorphic to a free product of nilpotent groups are flawed. This unifies and generalizes all previously known classes of flawed groups. We also provide further evidence for the authors' conjecture that RAAGs are flawed. Lastly, we show direct products between finite groups and some flawed group are also flawed. These latter two theorems enlarge the known class of flawed groups. [ABSTRACT FROM AUTHOR]

In this paper, we will study a fourth order fully nonlinear PDE u i j D i j w = 0 , w ≡ [ det D 2 u ] − θ , θ > 0 of affine maximal type. By a formula of quasi-norm of third derivatives derived from pure PDE tricks, we will prove that any entire convex solution must be a quadratic function, provided θ ∈ (0 , ( 1 2 − N − 1 2 (N + 8) ) + ) ⋃ ( 1 2 + N − 1 2 (N + 8) , + ∞) and the minimal convex constant λ (R) decays slower than R − α (θ , N) for some positive constant α (θ , N) given explicitly below. [ABSTRACT FROM AUTHOR]

In this paper we propose and investigate in full generality new notions of (continuous, non-isometric) symmetry on hyperkähler spaces. These can be grouped into two categories, corresponding to the two basic types of continuous hyperkähler isometries which they deform: tri-Hamiltonian isometries, on one hand, and rotational isometries, on the other. The first category of deformations gives rise to Killing spinors and generate what are known as hidden hyperkähler symmetries. The second category gives rise to hyperholomorphic line bundles over the hyperkähler manifolds on which they are defined and, by way of the Atiyah–Ward correspondence, to holomorphic line bundles over their twistor spaces endowed with meromorphic connections, generalizing similar structures found in the purely rotational case by Haydys and Hitchin. Examples of hyperkähler metrics with this type of symmetry include the c-map metrics on cotangent bundles of affine special Kähler manifolds with generic prepotential function, and the hyperkähler constructions on the total spaces of certain integrable systems proposed by Gaiotto, Moore and Neitzke in connection with the wall-crossing formulas of Kontsevich and Soibelman, to which our investigations add a new layer of geometric understanding. [ABSTRACT FROM AUTHOR]

We present a systematic study of involutions on the moduli space M (G) of principal G -Higgs bundles over an elliptic curve X , where G is complex reductive affine algebraic group. The fixed point loci of these involutions are described, providing a large class of examples of (B , B , B) , (B , A , A) , (A , B , A) and (A , A , B) -branes in M (G). Our study leads to an explicit description of the moduli spaces of pseudo-real G -Higgs bundles over the elliptic curve X. [ABSTRACT FROM AUTHOR]

False match removal is a crucial and fundamental task in photogrammetry and computer vision. This paper proposes a robust and efficient mismatch-removal algorithm based on the concepts of local barycentric coordinate (LBC) and matching coordinate matrices (MCMs), called locality affine-invariant matching (LAM). LAM is suitable for both rigid and nonrigid image matching problems. We define a novel LBC system based on area ratios, which is invariant to local affine transformations. We also present the MCMs based on the coordinates of matches, whose degeneracy is able to indicate the correctness of correspondences. Our LAM method first builds a mathematical model based on the LBCs to extract good matches that preserve local neighborhood structures. Then, LAM constructs local MCMs using the extracted reliable correspondences and identifies the correctness for the remaining matches via minimizing the rank of the MCMs. LAM has linear space and linearithmic time complexities. Extensive experiments on both rigid and nonrigid real datasets demonstrate the power of the proposed method; i.e., LAM is more robust to complex transformations compared to other methods and is two orders of magnitude faster than RANSAC under low inlier rates. The source code of the proposed LAM method will be publicly available in http://www.escience.cn/people/lijiayuan/index.html. [ABSTRACT FROM AUTHOR]

Abstract In this note, with a purely geometric approach, the covering radius of the group PGL (3 , q) is determined. Also, a new proof establishing the covering radii of PGL (2 , q) and AGL (1 , q) is provided. [ABSTRACT FROM AUTHOR]

Abstract We introduce a new model of congestion games that captures several extensions of the classical congestion game introduced by Rosenthal in 1973. The idea here is to parameterize both the perceived cost of each player and the social cost function of the system designer. Intuitively, each player perceives the load induced by the other players by an extent of ρ ≥ 0 , while the system designer estimates that each player perceives the load of all others by an extent of σ ≥ 0. For specific choices of ρ and σ , we obtain extensions such as altruistic player behavior, risk sensitive players and the imposition of taxes on the resources. We derive tight bounds on the price of anarchy and the price of stability for a large range of parameters. Our bounds provide a complete picture of the inefficiency of equilibria for these games. As a result, we obtain tight bounds on the price of anarchy and the price of stability for the above mentioned extensions. Our results also reveal how one should "design" the cost functions of the players in order to reduce the price of anarchy. Somewhat counterintuitively, if each player cares about all other players to the extent of ρ = 0.625 (instead of 1 in the standard setting) the price of anarchy reduces from 2.5 to 2.155 and this is best possible. [ABSTRACT FROM AUTHOR]

We provide a class of commutative Noetherian domains R of dimension d such that every finitely generated projective R -module P of rank d splits off a free summand of rank one. On this class, we also show that P is cancellative. At the end we give some applications to the number of generators of a module over the Rees algebras. [ABSTRACT FROM AUTHOR]

We establish the Schur-Weyl type duality between double affine Hecke algebras and quantum toroidal superalgebras, generalizing the well known result of Vasserot-Varagnolo [26] to the super case. [ABSTRACT FROM AUTHOR]

We show a diagonalisation variant of Lindström calculation method. As an application of this result we calculate the dimension of the affine space of Negami's splitting matrices. We do so by writing down an expression for the Möbius function in term of Möbius algebra identities. As a corollary we get Lindström's result in a self contained way. Finally, we calculate the partition lattice characteristic polynomial via the Negami's polynomial. [ABSTRACT FROM AUTHOR]

In this paper, we give a complete description of when the image of a monomial preserving derivation (or E -derivation) of the polynomial algebra over a field of characteristic zero is a Mathieu–Zhao subspace. In particular we show that the LFED and LNED conjectures hold for these derivations. The problem of whether the image of a derivation is a Mathieu–Zhao subspace arose from the study of the Jacobian conjecture. [ABSTRACT FROM AUTHOR]

Wavelet analysis has been extended to the p -adic line Q p . The p -adic wavelets are complex valued functions with compact support. As in the case of real wavelets, the construction of the basis functions is recursive, employing scaling and translation. Consequently, wavelets form a representation of the affine group generated by scaling and translation. In addition, p -adic wavelets are eigenfunctions of a pseudo-differential operator, as a result of which they turn out to have a larger symmetry group. The enhanced symmetry of the p -adic wavelets is demonstrated. [ABSTRACT FROM AUTHOR]

The q-deformed Hamiltonian for the S O ( 6 ) ↔ U ( 5 ) transitional case in s, d interaction boson model (IBM) can be constructed by using affine S U q ( 1 , 1 ) Lie algebra in the both IBM-1 and 2 versions and IBFM. In this research paper, we have studied the energy spectra of Xe 120 − 128 isotopes and Xe 123 − 131 isotopes and B(E2) transition probabilities of Xe 120 − 128 isotopes in the shape phase transition region between the spherical and gamma unstable deformed shapes of the theory of quantum deformation. The theoretical results agree with the experimental data fairly well. It is shown that the q-deformed S O ( 6 ) ↔ U ( 5 ) transitional dynamical symmetry remains after deformation. [ABSTRACT FROM AUTHOR]

We show that the problem of constructing an affine real rational knot of a reasonably low degree can be reduced to an algebraic problem involving the pure braid group: expressing an associated element of the pure braid group in terms of the standard generators of the pure braid group. We also predict the existence of a real rational knot in a degree that is expressed in terms of the edge number of its polygonal representation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a rewriting theory for linear monoidal categories. Those categories are a particular case of linear ( n , p ) -categories that we define in this paper. We also define linear ( n , p ) -polygraphs, a linear adaptation of n-polygraphs, to present linear ( n − 1 , p ) -categories. We focus then on linear ( 3 , 2 ) -polygraphs to give presentations of linear monoidal categories. We finally give an application of this theory to prove a basis theorem on the category A O B . Our method uses decreasingness, a property introduced by van Ostroom. [ABSTRACT FROM AUTHOR]

Let U ( gl ˆ n) be the universal enveloping algebra of gl ˆ n. A certain Z -form U Z ( gl ˆ n) of U ( gl ˆ n) was introduced in [15] such that the affine Schur algebra S ▵ (n , r) Z is a homomorphic image of U Z ( gl ˆ n). Let k be a field and let U k ( gl ˆ n) = U Z ( gl ˆ n) ⊗ k. Let S ▵ (n , r) k = S ▵ (n , r) Z ⊗ k. By tensoring with k , we have a surjective algebra homomorphism ξ r , k : = ξ r ⊗ i d : U k ( gl ˆ n) ↠ S ▵ (n , r) k. For h ⩾ 1 and r ∈ N let s ▵ (n , r) h = ξ r , k (s ▵ (n) h) , where s ▵ (n) h is a certain subalgebra of U k ( gl ˆ n). The algebra s ▵ (n , r) h is the affine analogue of infinitesimal Schur algebras introduced in [7]. In this paper we will give a presentation of s ▵ (n , r) 1 by generators and defining relations. [ABSTRACT FROM AUTHOR]

(1) Let R be an affine algebra over an algebraically closed field of characteristic 0 with dim ( R ) = n . Let P be a projective A = R [ T 1 , ⋯ , T k ] -module of rank n with determinant L . Suppose I is an ideal of A of height n such that there are two surjections α : P ↠ I and ϕ : L ⊕ A n − 1 ↠ I . Assume that either (a) k = 1 and n ≥ 3 or (b) k is arbitrary but n ≥ 4 is even. Then P has a unimodular element (see 4.1 , 4.3 ). (2) Let R be a ring containing Q of even dimension n with height of the Jacobson radical of R ≥ 2 . Let P be a projective R [ T , T − 1 ] -module of rank n with trivial determinant. Assume that there exists a surjection α : P ↠ I , where I ⊂ R [ T , T − 1 ] is an ideal of height n such that I is generated by n elements. Then P has a unimodular element (see 3.4 ). [ABSTRACT FROM AUTHOR]

We obtain an explicit upper bound on the torsion of the Picard group of the forms of A k 1 and their regular completions. We also obtain a sufficient condition for the Picard group of the forms of A k 1 to be nontrivial and we give examples of nontrivial forms of A k 1 with trivial Picard groups. [ABSTRACT FROM AUTHOR]

Bell and Zhang have shown that if A and B are two connected graded algebras finitely generated in degree one that are isomorphic as ungraded algebras, then they are isomorphic as graded algebras. We exploit this result to solve the isomorphism problem in the cases of quantum affine spaces, quantum matrix algebras, and homogenized multiparameter quantized Weyl algebras. Our result involves determining the degree one normal elements, factoring out, and then repeating. This creates an iterative process that allows one to determine relationships between relative parameters. [ABSTRACT FROM AUTHOR]

Infinite-dimensional Galilean conformal algebras can be constructed by contracting pairs of symmetry algebras in conformal field theory, such as W -algebras. Known examples include contractions of pairs of the Virasoro algebra, its N = 1 superconformal extension, or the W 3 algebra. Here, we introduce a contraction prescription of the corresponding operator-product algebras, or equivalently, a prescription for contracting tensor products of vertex algebras. With this, we work out the Galilean conformal algebras arising from contractions of N = 2 and N = 4 superconformal algebras as well as of the W -algebras W ( 2 , 4 ) , W ( 2 , 6 ) , W 4 , and W 5 . The latter results provide evidence for the existence of a whole new class of W -algebras which we call Galilean W -algebras. We also apply the contraction prescription to affine Lie algebras and find that the ensuing Galilean affine algebras admit a Sugawara construction. The corresponding central charge is level-independent and given by twice the dimension of the underlying finite-dimensional Lie algebra. Finally, applications of our results to the characterisation of structure constants in W -algebras are proposed. [ABSTRACT FROM AUTHOR]

Let P be a parabolic subgroup of a semisimple affine algebraic group G defined over C and X a compact Kähler manifold. L. Álvarez-Cónsul and O. García-Prada associated to these a quiver Q and representations of Q into holomorphic vector bundles on X (Álvarez-Cónsul and García-Prada, 2003) [1, 2]. Our aim here is to investigate the differential geometric properties of the moduli spaces of representations of Q into vector bundles on X . In particular, we construct a Hermitian form on these moduli spaces. A fiber integral formula is proved for this Hermitian form; this fiber integral formula implies that the Hermitian form is Kähler. We compute the curvature of this Kähler form. Under an assumption which says that X is a complex projective manifold, this Kähler form is realized as the curvature of a certain determinant line bundle equipped with a Quillen metric. [ABSTRACT FROM AUTHOR]

We propose a construction of affine space (or “polynomial rings”) over a triangulated category, in the context of stable derivators. [ABSTRACT FROM AUTHOR]

With respect to the two-sided cells, we give a decomposition formula for the Kazhdan–Lusztig basis elements of the affine Hecke algebra of type A ˜ n − 1 . The main tool is the star operators introduced by Kazhdan and Lusztig. In the appendix, we prove that the left star operator commutes with the right star operator even in the unequal parameter case. [ABSTRACT FROM AUTHOR]

Given a matrix polytope we consider the existence problem of a stable member in it. We suggest an algorithm in which part of uncertainty parameters is chosen randomly. Applications to affine families and 3 × 3 interval families are considered. A necessary and sufficient condition for the existence of a stable member is given for general interval families with nonnegative off-diagonal intervals. [ABSTRACT FROM AUTHOR]

In this paper, we study flag-transitive 2-designs. Let D be a 2- (v , k , λ) design with (v − 1 , k − 1) ≤ (v − 1) 1 2 admitting a flag-transitive automorphism group G. We prove that G is of affine, almost simple type, or product type with r a n k (G) = 3. In particular, if (v − 1 , k − 1) = 3 or 4, then G is of affine, almost simple type except the case that S o c (G) = A 5 × A 5 and D is a 2- (25 , 4 , 12) or 2- (25 , 4 , 18) design. Finally, an application to the case where k is prime is also discussed. [ABSTRACT FROM AUTHOR]

This paper presents a new framework for stability assessment of discrete-time piecewise affine systems. An implicit representation for piecewise functions based on ramp nonlinearities is proposed. Instead of the usual sector inequalities adopted in the study of Lurie systems, we directly exploit properties of the ramp function, which are given by a particular set of identities and inequalities. These properties are then used to derive stability conditions associated to piecewise quadratic Lyapunov functions. These functions are implicitly defined from quadratic forms involving ramp functions. These conditions can be cast as LMIs and their numerical solution is illustrated by examples highlighting the advantages of the proposed method over existing stability analysis methods for PWA systems. [ABSTRACT FROM AUTHOR]

The algebraic engineering technique is applied to a class of 3D N = 2 gauge theories on the omega-deformed background R ε 2 × S 1. The vortex partition function and the fundamental qq-character are obtained from a network of intertwiners between representations of the shifted (or asymptotic) quantum affine sl (2) algebra. This network involves two types of representations, the prefundamental representation of Hernandez-Jimbo, and a new vertex representation acting on a bosonic Fock space. The brane system associated to this network is identified: D3 branes carry the prefundamental module while NS5-branes (+D5) support the Fock module. In the process, we highlight the role of shifted quantum algebras in implementing the Higgsing procedure. [ABSTRACT FROM AUTHOR]

Three aspects of the SU(3) fusion coefficients are revisited: the generating polynomials of fusion coefficients are written explicitly; some curious identities generalizing the classical Freudenthal–de Vries formula are derived; and the properties of the fusion coefficients under conjugation of one of the factors, previously analyzed in the classical case, are extended to the affine algebra s u ˆ ( 3 ) at finite level. [ABSTRACT FROM AUTHOR]

Bivariate Gončarov polynomials are a basis of the solutions of the bivariate Gončarov Interpolation Problem in numerical analysis. A sequence of bivariate Gončarov polynomials is determined by a set of nodes Z = { ( x i , j , y i , j ) ∈ R 2 } and is an affine sequence if Z is an affine transformation of the lattice grid N 2 , i.e., ( x i , j , y i , j ) T = A ( i , j ) T + ( c 1 , c 2 ) T for some 2×2 matrix A and constants c 1 , c 2 . In this paper we prove that a sequence of bivariate Gončarov polynomials is of binomial type if and only if it is an affine sequence with c 1 = c 2 = 0 . Such polynomials form a higher-dimensional analog of the Abel polynomial A n ( x ; a ) = x ( x − a n ) n − 1 . We present explicit formulas for a general sequence of bivariate affine Gončarov polynomials and its exponential generating function, and use the algebraic properties of Gončarov polynomials to give some new two-dimensional generalizations of Abel identities. [ABSTRACT FROM AUTHOR]

This paper deals with output feedback switching function control design for continuous-time switched affine systems, assuring global asymptotic stability of a desired equilibrium point. The set of all attainable equilibrium points is provided. More specifically, the main purpose is to design a full order switched affine filter together with a switching rule assuring not only global asymptotic stability, but also a guaranteed H ∞ performance level. The design conditions do not require that the open-loop subsystems be stable and outperform other techniques available in the literature to date. The results are compared with recent ones based on a quadratic Lyapunov function and on a max-type Lyapunov function. Numerical examples illustrate the theoretical results and are used for comparisons. [ABSTRACT FROM AUTHOR]

Inspired by a profound observation on the Racah–Wigner coefficients of U q ( sl 2 ) , the Askey–Wilson algebras were introduced in the early 1990s. A universal analog △ q of the Askey–Wilson algebras was recently studied. For q not a root of unity, it is known that Z ( △ q ) is isomorphic to the polynomial ring of four variables. A presentation for Z ( △ q ) at q a root of unity is displayed in this paper. As an application, a presentation for the center of the double affine Hecke algebra of type ( C 1 ∨ , C 1 ) at roots of unity is obtained. [ABSTRACT FROM AUTHOR]

We present a proof of Kemer's representability theorem for affine PI algebras over a field of characteristic zero. [ABSTRACT FROM AUTHOR]

Since unity step size could guarantee the fastest convergence and more detailed analysis for the affine projection (AP) algorithm, a statistical tracking behavior of AP algorithm is discussed in this paper. Deterministic recursive equations are derived for the mean weight error and mean-square error. All the possible correlations between the adaptive filtering coefficients and the past measurement noise are considered as well. [ABSTRACT FROM AUTHOR]

We classify the irreducible zero-level integrable modules with finite dimensional weight spaces for the untwisted affine superalgebras A ˆ ( m , n ) ( m ≠ n ) and C ˆ ( m ) . Such modules must be of highest weight type, but are not necessarily loop modules. This is in sharp contrast to the case of ordinary affine algebras. Combined with previous results obtained by other people (Rao and Zhao in particular), our work completes the program of classifying the irreducible integrable modules with finite dimensional weight spaces for all the untwisted affine superalgebras. [ABSTRACT FROM AUTHOR]

The Maxwell extension of the affine algebra in four dimensions with additional tensor generator is given. Using the methods of nonlinear realizations, we find the transformation rules for the group parameters and the corresponding generators. Gauging the Maxwell-affine algebra we present two possible invariant actions for gravity: one is first order and the other one is second order in the affine curvature. We notice that equations of motion for the action, second order in the affine curvature, lead to the generalized Bianchi identities on the choice of appropriate coefficients for a particular solution of the constraint equation. [ABSTRACT FROM AUTHOR]

In this article we extend a cancellation theorem of D. Wright to the case of affine normal domains. We shall show that if A is an algebra over a Noetherian normal domain R containing a field k and if A [ T ] = R [ 3 ] , then A = R [ 2 ] if and only if A [ T ] has a variable of the form b T n − a for some a , b ∈ A with n ≥ 2 and ch ( k ) ∤ n . [ABSTRACT FROM AUTHOR]

We introduce the notion of standard (Kleshchev) multipartitions and establish a one-to-one correspondence between standard multipartitions and irreducible representations with integral weights for the affine Hecke algebra of type A with a parameter q ∈ C × which is not a root of unity. We then extend the correspondence to all Kleshchev multipartitions for Ariki-Koike algebras of integral type. By the affine Schur–Weyl duality, we further extend this to a correspondence between standard multipartitions and Drinfeld multipolynomials of integral type whose associated irreducible polynomial representations completely determine all irreducible polynomial representations for the quantum loop algebra U q ( gl ˆ n). We will see, in particular, the notion of standard multipartitions gives rise to a combinatorial description of the affine Schur–Weyl duality in terms of a column-reading vs. row-reading of residues of a multipartition. [ABSTRACT FROM AUTHOR]

So far, two sensor models for the geometric modeling of satellite imagery have been studied and compared: a rigorous sensor model (RSM) and a rational function model (RFM). Even though it was concluded that the RFM could replace the RSM, this paper points out that the previous conclusions were drawn only for a strong geometry because of the conventional use of single-sensor stereo and that they may not apply to the weak geometry of dual-sensor stereo pairs. This work highlights that dual-sensor stereo often creates a weak geometry and that for such weak geometry, accuracy differences may occur between the RSM and the RFM, and also between the RFMs with different bias correction methods. The positioning accuracy of the three sensor models, RSM, RFM using second-order polynomials model, and RFM using an affine model, were compared on various geometries, using pairs from every conceivable combination of two QuickBird and IKONOS as well as four KOMPSAT-2 images covering the same area. Our results showed that the three sensor models differed slightly owing to the strong geometry. However, for the weak geometry, the RSM or second-order RFM performed better than RFM with an affine model, resulting in increase in the difference between the accuracies of the sensor models. This implies that the physically weak geometry of a satellite stereo may require a rigorous or high-order model for a more accurate geo-positioning. [ABSTRACT FROM AUTHOR]

We develop an invariant deformation theory, in a form accessible to practice, for affine schemes W equipped with an action of a reductive algebraic group G . Given the defining equations of a G -invariant subscheme X ⊂ W , we device an algorithm to compute the universal deformation of X in terms of generators and relations up to a given order. In many situations, our algorithm even computes an algebraization of the universal deformation. As an application, we determine new families of examples of the invariant Hilbert scheme of Alexeev and Brion, where G is a classical group acting on a classical representation, and we describe their singularities. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of noise reduction and speech enhancement by adaptive filtering using two-sensor configuration. The well-known Gauss–Seidel fast affine projection (GSFAP) algorithm is used in various applications such as acoustic echo cancelation (AEC), and adaptive noise control. In this paper, we use the GSFAP algorithm in noise reduction and speech enhancement application. Moreover, we derive a new two-sensor GSFAP (TS-GSFAP) algorithm for speech enhancement and acoustic noise reduction application. The application of the proposed TS-GSFAP algorithm to the forward blind source separation (FBSS) structure leads to a significant improvement of the output speech signals quality even in noisy conditions. A fair comparison of the proposed TS-GSFAP algorithm with other standard two-sensor type algorithms is presented. This comparison is based on the evaluation of several objective and subjective criteria such as: the segmental signal-to-noise ratio (SegSNR), the Cepstral distance (CD) and the system mismatch (SM), the perceptual evaluation of speech quality (PESQ), and the mean opinion score (MOS) criteria. The obtained results show the good performances of the proposed TS-GSFAP algorithm. [ABSTRACT FROM AUTHOR]