Context Of Algebras Research Articles

Application of complex extreme learning machine to multiclass classification problems with high dimensionality: A THz spectra classification problem

We extend extreme learning machine (ELM) classifiers to complex Reproducing Kernel Hilbert Spaces (RKHS) where the input/output variables as well as the optimization variables are complex-valued. A new family of classifiers, called complex-valued ELM (CELM) suitable for complex-valued multiple-input–multiple-output processing is introduced. In the proposed method, the associated Lagrangian is computed using induced RKHS kernels, adopting a Wirtinger calculus approach formulated as a constrained optimization problem similarly to the conventional ELM classifier formulation. When training the CELM, the Karush–Khun–Tuker (KKT) theorem is used to solve the dual optimization problem that consists of satisfying simultaneously smallest training error as well as smallest norm of output weights criteria. The proposed formulation also addresses aspects of quaternary classification within a Clifford algebra context. For 2D complex-valued inputs, user-defined complex-coupled hyper-planes divide the classifier input space into four partitions. For 3D complex-valued inputs, the formulation generates three pairs of complex-coupled hyper-planes through orthogonal projections. The six hyper-planes then divide the 3D space into eight partitions. It is shown that the CELM problem formulation is equivalent to solving six real-valued ELM tasks, which are induced by projecting the chosen complex kernel across the different user-defined coordinate planes. A classification example of powdered samples on the basis of their terahertz spectral signatures is used to demonstrate the advantages of the CELM classifiers compared to their SVM counterparts. The proposed classifiers retain the advantages of their ELM counterparts, in that they can perform multiclass classification with lower computational complexity than SVM classifiers. Furthermore, because of their ability to perform classification tasks fast, the proposed formulations are of interest to real-time applications.

Discrimination in a General Algebraic Setting.

Discriminating groups were introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups became the main method of attack on the solution of the celebrated Tarski conjectures. In this paper we explore the notion of discrimination in a general universal algebra context. As an application we provide a different proof of a theorem of Malcev on axiomatic classes of Ω-algebras.

Formal Pseudodifferential Operators in One and Several Variables, Central Extensions, and Integrable Systems

We review some aspects of the theory of Lie algebras of (twisted and untwisted) formal pseudodifferential operators in one and several variables in a general algebraic context. We focus mainly on the construction and classification of nontrivial central extensions. As applications, we construct hierarchies of centrally extended Lie algebras of formal differential operators in one and several variables, Manin triples and hierarchies of nonlinear equations in Lax and zero curvature form.

N =2 Superconformal Nets

We provide an Operator Algebraic approach to N=2 chiral Conformal Field Theory and set up the Noncommutative Geometric framework. Compared to the N=1 case, the structure here is much richer. There are naturally associated nets of spectral triples and the JLO cocycles separate the Ramond sectors. We construct the N=2 superconformal nets of von Neumann algebras in general, classify them in the discrete series c<3, and we define and study an operator algebraic version of the N=2 spectral flow. We prove the coset identification for the N=2 super-Virasoro nets with c<3, a key result whose equivalent in the vertex algebra context has seemingly not been completely proved so far. Finally, the chiral ring is discussed in terms of net representations.

Variants of the Kakeya problem over an algebraically closed field

First, we study constructible subsets of $${\mathbb{A}^n_k}$$ which contain a line in any direction. We classify the smallest such subsets in $${\mathbb{A}^3}$$ of the type $${R \cup \{g \neq 0\},}$$ where $${g \in k[x_1,\ldots, x_n]}$$ is irreducible of degree d and $${R \subset V(g)}$$ is closed. Next, we study subvarieties $${X \subset \mathbb{A}^N}$$ for which the set of directions of lines contained in X has the maximal possible dimension. These are variants of the Kakeya problem in an algebraic geometry context.

Abelian pattern avoidance in partial words

Pattern avoidance is an important topic in combinatorics on words which dates back to the beginning of the twentieth century when Thue constructed an infinite word over a ternary alphabet that avoids squares, i.e. , a word with no two adjacent identical factors. This result finds applications in various algebraic contexts where more general patterns than squares are considered. On the other hand, Erdős raised the question as to whether there exists an infinite word that avoids abelian squares, i.e. , a word with no two adjacent factors being permutations of one another. Although this question was answered affirmately years later, knowledge of abelian pattern avoidance is rather limited. Recently, (abelian) pattern avoidance was initiated in the more general framework of partial words, which allow for undefined positions called holes. In this paper, we show that any pattern p with n > 3 distinct variables of length at least 2 n is abelian avoidable by a partial word with infinitely many holes, the bound on the length of p being tight. We complete the classification of all the binary and ternary patterns with respect to non-trivial abelian avoidability, in which no variable can be substituted by only one hole. We also investigate the abelian avoidability indices of the binary and ternary patterns.

Fuzzy Weak Regular, Strong and Preassociative Filters in Residuated Lattices

In this paper, the notions of fuzzy weak regular, strong and preassociative filters are introduced with some properties of them investigated. In particular, under the context of Glivenko algebras, fuzzy weak regular filters and regular ones are equivalent and characterizations of Glivenko algebras are concluded by characterizations of fuzzy strong filters. At last the notion of fuzzy preassociative filters are defined, which are proved to coincide with fuzzy Boolean filters, and then some new alternative definitions of Boolean algebras are obtained by this type of filters.

Invariants, exponents and formal group laws

Let W be the Weyl group of a crystallographic root system acting on the associated weight lattice by means of reflections. In the present note we extend the notion of exponent of the W-action to the context of an arbitrary algebraic oriented cohomology theory of Levine–Morel and Panin–Smirnov and the associated formal group law. From this point of view the classical Dynkin index of the associated Lie algebra will be the second exponent of the deformation map from the multiplicative to the additive formal group law. We apply this generalized exponent to study the torsion part of an arbitrary oriented cohomology theory of a twisted flag variety.

Formula omitted]-torsion points in finite abelian groups and combinatorial identities

The main aim of this article is to compute all the moments of the number of pℓ-torsion elements in some type of finite abelian groups. The averages involved in these moments are those defined for the Cohen–Lenstra heuristics for class groups and their adaptation for Tate–Shafarevich groups. In particular, we prove that the heuristic model for Tate–Shafarevich groups is compatible with the recent conjecture of Poonen and Rains about the moments of the orders of p-Selmer groups of elliptic curves. For our purpose, we are led to define certain polynomials indexed by integer partitions and to study them in a combinatorial way. Moreover, from our probabilistic model, we derive combinatorial identities, some of which appearing to be new, the others being related to the theory of symmetric functions. In some sense, our method therefore gives for these identities a somehow natural algebraic context.

Dixmier traces and extrapolation description of noncommutative Lorentz spaces

We study the connections between Dixmier traces, ζ-functions and traces of heat semigroups beyond the dual of the Macaev ideal and in the general context of semifinite von Neumann algebras. We show that the correct framework for this investigation is that of operator Lorentz spaces possessing an extrapolation description. We demonstrate the applicability of our results to Hörmander–Weyl pseudo-differential calculus on Rn. In this context, we prove that the Dixmier trace of a pseudo-differential operator coincides with the ‘Dixmier integral’ of its symbol.

Zariski Closed Algebras in Varieties of Universal Algebra

The Zariski closure of an arbitrary representable (not necessarily associative) algebra is studied in the general context of universal algebra, with an application being that the codimension sequence is exponentially bounded.

Kähler-Ricci flow with degenerate initial class

In the earlier joint work [3], we introduced the weak Kahler-Ricci flow for various geometric motivations. In this current work, we take further consideration on setting up the weak flow by allowing the initial class to be not necessary Kahler. It’s shown that the construction is compatible with the earlier construction in Kahler case. We also discuss the convergence as t→ 0+ which is of great interest in this topic, and provide related motivation. 1. Motivation and set-up of weak flow The Kahler-Ricci flow, the complex version of Ricci flow, has been under intensive study over the last twenty some years. In [18] and more recently [17], G. Tian proposed the intriguing program of constructing globally existing (weak) KahlerRicci flow with canonical (singular) limit at time infinity and applying it to the study of algebraic manifolds (and even Kahler manifolds in general). It can be viewed as the analytic version of the famous Minimal Model Program in algebraic geometry. In general, people should expect the classic smooth Kahler-Ricci flow to encounter singularity at some finite time which is completely decided by cohomology information according to the optimal existence result in [20]. Just as what people have been doing and had successes in cases for Ricci flow, surgeries for the underlying manifold should be expected. For the Kahler-Ricci flow, we naturally expect the surgery to have flavours from algebraic geometry. A simple example would be surfaces of general type. One only needs the blowingdown of (−1)-curves when applying the general construction at the end of [3] to push the flow through the finite time of singularities, where the measure restriction is actually not so involved as explained later in Example 4.4. The degenerate class at the singularity time would become Kahler for the new manifold because those (−1)-curves causing the cohomology degeneration would eventually been crushed to points. Things can get significantly more complicated for higher dimension. For example, a standard procedure called flip is introduced in the algebraic geometry context, which is of great importance for the business about the algebraic Minimal Model Program. Simply speaking, one needs to blow up the manifold and then perform a different blowing-down process. Naturally, we should expect the transformation of the degenerate class is still not Kahler. In this note, we confirm that this is not a problem under the assumption that formally the Kahler-Ricci flow is instantly Date: received on January 6, 2012 and accepted on August 22, 2012. 2010 Mathematics Subject Classification. Primary 53C44; Secondary 14E30, 58J35.

Modules as Learning Tools in Linear Algebra

This paper reports on the experience of STEM and mathematics faculty at four different institutions working collaboratively to integrate learning theory with curriculum development in a core undergraduate linear algebra context. The faculty formed a Professional Learning Community (PLC) with a focus on learning theories in mathematics and curriculum development. Beginning with an online reading seminar, faculty paired with at least one other STEM faculty member at each institution to develop modules collaboratively with each other and with the group as a whole through regular project meetings. Two representative modules from this collaboration are presented with student and faculty feedback. A link is provided to other modules. We consider this a proof-of-concept paper with broader applicability beyond linear algebra to other mathematics and even STEM courses. We focus on linear algebra because it is widely taught in undergraduate mathematics programs and lends itself easily to a variety of applications and geometric analyses. We intend to continue to apply this approach to other content domains. We chose a particular theoretical framework, Action-Process-Object-Schema (APOS), because the mathematics education faculty in this group had considerable experience in the development and use of the framework.

Homotopy theory of non-symmetric operads, II: Change of base category and left properness

In this paper we study spaces of algebras over an operad (non-symmetric) in symmetric monoidal model categories. We first compute the homotopy fiber of the forgetful functor sending an algebra to its underlying object, extending a result of Rezk. We then apply this computation to the construction of geometric moduli stacks of algebras over an operad in a homotopical algebraic geometry context in the sense of To\"en and Vezzosi. We show under mild hypotheses that the moduli stack of unital associative algebras is a Zariski open substack of the moduli stack of non-necessarily unital associative algebras. The classical analogue for finite-dimensional vector spaces was noticed by Gabriel.

The construction of left-continuous t-norms: a geometric approach in two dimensions

Each t-norm can be identified with its Cayley tomonoid, which consists of pairwise commuting order-preserving functions from the real unit interval to itself. Cayley tomonoids provide an easily manageable, yet versatile tool for the construction of t-norms. To give evidence to this claim, we review and reformulate several construction methods that are known in the literature. We adopt, on the one hand, a geometric point of view. Manipulations with t-norms have often been inspired by their three-dimensional graphs; by means of Cayley tomonoids, the process gains the character of putting together the pieces of a jigsaw puzzle. We consider, on the other hand, construction methods in their algebraic context. We show that those constructions that correspond to certain tomonoid extensions can be described on the basis of Cayley tomonoids in a particularly transparent way.

Isomorphism conjectures with proper coefficients

Let G be a group and F a nonempty family of subgroups of G, closed under conjugation and under subgroups. Also let E be a functor from small Z-linear categories to spectra, and let A be a ring with a G-action. Under mild conditions on E and A one can define an equivariant homology theory HG(−,E(A)) of G-simplicial sets such that H⁎G(G/H,E(A))=E(A⋊H). The strong isomorphism conjecture for the quadruple (G,F,E,A) asserts that if X→Y is an equivariant map such that XH→YH is an equivalence for all H∈F, thenHG(X,E(A))→HG(Y,E(A)) is an equivalence. In this paper we introduce an algebraic notion of (G,F)-properness for G-rings, modeled on the analogous notion for G-C⁎-algebras, and show that the strong (G,F,E,P) isomorphism conjecture for (G,F)-proper P is true in several cases of interest in the algebraic K-theory context.

Motivic bivariant characteristic classes

Let K0(V/X) be the relative Grothendieck group of varieties over X∈Obj(V), with V=Vk(qp) (resp. V=Vcan) the category of (quasi-projective) algebraic (resp. compact complex analytic) varieties over a base field k. Then we constructed the motivic Hirzebruch class transformation Ty⁎:K0(V/X)→H⁎(X)⊗Q[y] in the algebraic context for k of characteristic zero, with H⁎(X)=CH⁎(X) (resp. in the complex algebraic or analytic context, with H⁎(X)=H2⁎BM(X)). It “unifies” the well-known three characteristic class transformations of singular varieties: MacPhersonʼs Chern class, Baum–Fulton–MacPhersonʼs Todd class and the L-class of Goresky–MacPherson and Cappell–Shaneson. In this paper we construct a bivariant relative Grothendieck group K0(V/X→Y) for V=Vk(qp) (resp., Vcan) so that K0(V/X→pt)=K0(V/X) in the algebraic context with k of characteristic zero (resp., complex analytic context).We also construct in the algebraic context (in any characteristic) two Grothendieck transformations mCy=Λymot:K0(Vqp/X→Y)→Kalg(X→Y)⊗Z[y] and Ty:K0(Vqp/X→Y)→H(X→Y)⊗Q[y] with Kalg(f) the bivariant algebraic K-theory of f-perfect complexes and H the bivariant operational Chow groups (or the even degree bivariant homology in the case k=C). Evaluating at y=0, we get a “motivic” lift T0 of Fulton–MacPhersonʼs bivariant Riemann–Roch transformation τ:Kalg→H⊗Q. The covariant transformations mCy:K0(Vqp/X→pt)→G0(X)⊗Z[y] and Ty⁎:K0(Vqp/X→pt)→H⁎(X)⊗Q[y] agree for k of characteristic zero with our motivic Chern and Hirzebruch class transformations defined on K0(Vqp/X). Finally, evaluating at y=−1, for k of characteristic zero we get a “motivic” lift T−1 of Ernström–Yokuraʼs bivariant Chern class transformation γ:F˜→CH.

Equivalent notions of normal quantum subgroups, compact quantum groups with properties F and FD, and other applications

The notion of normal quantum subgroup introduced in algebraic context by Parshall and Wang when applied to compact quantum groups is shown to be equivalent to the notion of normal quantum subgroup introduced by the author. As applications, a quantum analog of the third fundamental isomorphism theorem for groups is obtained, which is used along with the equivalence theorem to obtain results on structure of quantum groups with property F and quantum groups with property FD. Other results on normal quantum subgroups for tensor products, free products and crossed products are also proved.

The coarse moduli space of a flat analytic groupoid

Let \(\mathcal X \) be a flat analytic groupoid \(R_X\stackrel{s}{\underset{t}{\rightrightarrows }}X\) such that the holomorphic map \(j=(s,t):R_X\rightarrow X\times X\) is finite. In this paper, we prove that there exist a (unique up to isomorphism) complex space \(Q(\mathcal X )\) and a holomorphic map \(q:X\rightarrow Q(\mathcal X )\) which is a GC quotient (see Definition 3.1). This extends to analytic groupoids the Main Theorem proved by Keel and Mori in the algebraic context (Keel and Mori in Ann Math 145(1):193–213, 1997, 1.1 Theorem).

Homogenization of degenerate porous medium type equations in ergodic algebras

We consider the homogenization problem for general porous medium type equations of the form ut=Δf(x,xε,u). The pressure function f(x,y,⋅) may be of two different types. In the type 1 case, f(x,y,⋅) is a general strictly increasing function; this is a mildly degenerate case. In the type 2 case, f(x,y,⋅) has the form h(x,y)F(u)+S(x,y), where F(u) is just a nondecreasing function; this is a strongly degenerate case. We address the initial–boundary value problem for a general, bounded or unbounded, domain Ω, with null (or, more generally, steady) pressure condition on the boundary. The homogenization is carried out in the general context of ergodic algebras. As far as the authors know, homogenization of such degenerate quasilinear parabolic equations is addressed here for the first time. We also review the existence and stability theory for such equations and establish new results needed for the homogenization analysis. Further, we include some new results on algebras with mean value, specially a new criterion establishing the null measure of level sets of elements of the algebra, which is useful in connection with the homogenization of porous medium type equations in the type 2 case.