Symmetric Quotients Research Articles

EEG-Based Emotion Recognition for Hearing Impaired and Normal Individuals With Residual Feature Pyramids Network Based on Time–Frequency–Spatial Features

With the development of affective computing, discriminative feature selection is critical for electroencephalography (EEG) emotion recognition. In this article, we fused four EEG feature matrices constructed by the preprocessed signal, the differential entropy (DE), the symmetric difference, and the symmetric quotient based on the International 10–20 system, which integrates time-, frequency-, and spatial-domain information of EEG signals. For the feature classification model, we used the space-to-depth (S2D) layer instead of the convolutional neural network (CNN) as the backbone to reduce the calculation of the model without affecting the classification performance. The residual feature pyramid network (RFPN) was proposed to obtain the correlation of channels, and then, the deep multiscale semantic information of EEG feature maps is captured. The emotion classification strategy was evaluated by DEAP, SEED, SEED-IV, and our hearing-impaired EEG dataset (HIED). The classification accuracies were 93.56% (four-class, DEAP), 96.84% (three-class, SEED), 91.62% (four-class, SEED-IV), and 87.74% (six-class, HIED). Furthermore, we also found that the difference in emotional response between the left and right brain regions of hearing-impaired subjects is more obvious than that of normal subjects.

Symmetric quotients of knot groups and a filtration of the Gordian graph

AbstractWe define a metric filtration of the Gordian graph by an infinite family of 1-dense subgraphs. The nth subgraph of this family is generated by all knots whose fundamental groups surject to a symmetric group with parameter at least n, where all meridians are mapped to transpositions. Incidentally, we verify the Meridional Rank Conjecture for a family of knots with unknotting number one yet arbitrarily high bridge number.

Highly Symmetric Quintic Quotients

AbstractThe quintic family must be the most studied family of Calabi‐Yau threefolds. Particularly symmetric members of this family are known to admit quotients by freely acting symmetries isomorphic to . The corresponding quotient manifolds may themselves be symmetric. That is, they may admit symmetries that descend from the symmetries that the manifold enjoys before the quotient is taken. The formalism for identifying these symmetries was given a long time ago by Witten and instances of these symmetric quotients were given also, for the family , by Goodman and Witten. We rework this calculation here, with the benefit of computer assistance, and provide a complete classification. Our motivation is largely to develop methods that apply also to the analysis of quotients of other CICY manifolds, whose symmetries have been classified recently. For the quotients of the quintic family, our list contains families of smooth manifolds with symmetry , and , families of singular manifolds with four conifold points, with symmetry and , and rigid manifolds, each with at least a curve of singularities, and symmetry . We intend to return to the computation of the symmetries of the quotients of other CICYs elsewhere.

Congruence subgroups of braid groups

In this paper, we give a description of the generators of the prime level congruence subgroups of braid groups. Also, we give a new presentation of the symplectic group over a finite field, and we calculate the symmetric quotients of the prime level congruence subgroups of braid groups. Finally, we find a finite generating set for the level-3 congruence subgroup of the braid group on three strands.

Symmetric quotient stacks and Heisenberg actions

For every smooth projective variety X, we construct an action of the Heisenberg algebra on the direct sum of the Grothendieck groups of all the symmetric quotient stacks \([X^n/{\mathfrak {S}}_n]\) which contains the Fock space as a subrepresentation. The action is induced by functors on the level of the derived categories which form a weak categorification of the action.

Equivalences of equivariant derived categories

We investigate conditions for a Fourier-Mukai transform between derived categories of coherent sheaves on smooth projective stacks endowed with actions by finite groups to lift to the associated equivariant derived categories. As an application we give a condition under which a global quotient stack cannot be derived equivalent to a variety. We also apply our techniques to generalised Kummer stacks and symmetric quotients.

On symmetric quotients of symmetric algebras

We investigate symmetric quotient algebras of symmetric algebras, with an emphasis on finite group algebras over a complete discrete valuation ring O. Using elementary methods, we show that if an ordinary irreducible character χ of a finite group G gives rise to a symmetric quotient over O which is not a matrix algebra, then the decomposition numbers of the row labelled by χ are all divisible by the characteristic p of the residue field of O.

Modifications of Newton’s method for even-grade palindromic polynomials and other twined polynomials

The paper describes some modifications of Newton's method for refining the zeros of even-grade f(x)-twined (f(x)-egt) polynomials, defined as polynomials whose roots appear in pairs {x i ,f(x i )}. Particular attention is given to even-grade palindromic (egp) polynomials. The algorithms are derived from certain symmetric division processes for computing a symmetric quotient and a symmetric remainder of two given f(x)-egt polynomials. Numerical results indicate that the presented algorithms can be more accurate than other methods which do not take into consideration the symmetry of the coefficients.

Alternating quotients of free groups

We strengthen Marshall Hall's theorem to show that free groups are locally extended residually alternating. Let F be any free group of rank at least two, let H be a finitely generated subgroup of infinite index in F and let \{\gamma_1,\ldots,\gamma_n\}\subseteq F\smallsetminus H be a finite subset. Then there is a surjection f from F to a finite alternating group such that f(\gamma_i)\notin f(H) for any i . The techniques of this paper can also provide symmetric quotients.

On Commutativity of Semiprime Right Goldie C<i><sub>k</sub></i>-Rings

This short exposition is about some commutativity conditions on a semiprime right Goldie Ck-ring. In particular, it is observed here that a semiprime right Goldie Ck-ring with symmetric quotient is commutative. The statement holds if the symmetric ring is replaced by reduced, 2-primal, left duo, right duo, abelian, NI, NCI, IFP, or Armendariz ring.

“On Invertible Contractions of Quotients Generated by a Differential Expression and by a Nonnegative Operator Function”

In the present paper, we describe invertible contractions of the maximal quotient generated by a differential expression with bounded operator coefficients and by a nonnegative operator function. We show that the operators inverse to such contractions are integral operators and prove a criterion for such operators to be holomorphic. Using the results obtained, we describe the generalized resolvents of symmetric quotients.

Maximal algebras of Martindale-like quotients of strongly prime linear Jordan algebras

In this paper we prove the existence and give precise descriptions of maximal algebras of Martindale like quotients for arbitrary strongly prime linear Jordan algebras. As a consequence, we show that Zelmanov's classification of strongly prime Jordan algebras can be viewed exactly as the description of their maximal algebras of Martindale-like quotients. As a side result, we show that the Martindale associative algebra of symmetric quotients can be expressed in terms of the symmetrized product, i.e., in purely Jordan terms.

Tits–Kantor–Koecher algebras of strongly prime hermitian Jordan pairs

In this paper we describe the Tits–Kantor–Koecher algebras of strongly prime hermitian Jordan pairs in terms of ∗-prime associative pairs and their Martindale pairs of symmetric quotients.

Positive symmetric quotients and their selfadjoint extensions

Positive symmetric quotients and their selfadjoint extensions

Local Borcherds products

The local Picard group at a generic point of the one-dimensional Baily-Borel boundary of a Hermitean symmetric quotient of type O(2,n) is computed. The main ingredient is a local version of Borcherds’ automorphic products. The local obstructions for a Heegner divisor to be principal are given by certain theta series with harmonic coefficients. Sometimes they generate Borcherds’ space of global obstructions. In these particular cases we obtain a simple proof of a result due to the first author: Suppose that Γ⊂O(2,n) is the orthogonal group attached to an even unimodular lattice. Then every meromorphic modular form for Γ, whose zeros and poles lie on Heegner divisors, is given by a Borcherds product.

Geodesic rays in locally symmetric spaces

Let X be a Riemannian symmetric space of noncompact type. Let V be a locally symmetric quotient of X by an (arithmetic) lattice such that V has finite volume but is not compact. We investigate geodesic rays in the ends (or cusps) of V. Technically our approach is based on a differential geometric interpretation of well-known facts from the (algebraic) reduction theory of arithmetic groups. As an application we obtain a description of the Tits geometry of locally symmetric spaces V.

Quadratic Jordan systems of hermitian type

In this paper, we extend Zel'manov's classification of linear Jordan systems (triple systems and pairs) of hermitian type to quadratic Jordan systems over an arbitrary ring of scalars. Using associative triple systems (of the first kind) for what we believe to be a more natural tool to describe special Jordan triples, we show that an i-special prime quadratic Jordan triple system (the ideals of which remain semiprime) with a nonzero hermitian part lies between an ample subspace of hermitian elements in a ∗-prime associative triple system and those in its Martindale system of symmetric quotients. Modulo some extra definitions, the structure of strongly prime Jordan pairs of hermitian type follows almost immediately from the Jordan triple result.

Relation algebraic domain constructions

Aiming at a constructive approach to domain theory, the definition of domains with deflations is presented. This class of domains is closed with respect to the common domain constructions. Another concern of this paper is to provide a formal calculus for a uniform algebraic treatment of order theoretic and functional aspects of domain theory. The abstract relation algebra turns out to be an appropriate technical means for the characterization and construction of domains. As partial functions present no problems in relation algebra, domains need not contain an additional ⊥-element and functions between domains are generally not total. Using symmetric quotients the relation algebraic approach is extended to cope with higher order functions.

Symmetric quotients and domain constructions

We introduce the symmetric quotient of two relations as a new construct in abstract relational algebra generalizing the notion of a “noyau” of Riguet. After exhibiting the main properties of symmetric quotients we study applications in domain theory. In particular, we give a monomorphic characterization of powersets and function domains.

On the symmetric difference quotient and its application to the correction of orbits

In this paper we consider the analytical foundations of numerical applications of the symmetric difference quotient for orbits correction. From these considerations it follows that the better results obtained in numerical calculations of values of derivatives at point, obtained by replacing the ordinary difference quotient with the symmetric difference quotient, have not been obtained fortuitandy.