Equivalency Research Articles

Drinfeld–Gaitsgory–Vinberg interpolation Grassmannian and geometric Satake equivalence

Let $G$ be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups and consider the corresponding semiinfinite orbits in the affine Grassmannian $Gr_G$. We prove Simon Schieder's conjecture identifying his bialgebra formed by the top compactly supported cohomology of the intersections of opposite semiinfinite orbits with $U(\check{\mathfrak n})$ (the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra $\check{\mathfrak g}$). To this end we construct an action of Schieder bialgebra on the geometric Satake fiber functor. We propose a conjectural construction of Schieder bialgebra for an arbitrary symmetric Kac-Moody Lie algebra in terms of Coulomb branch of the corresponding quiver gauge theory.

Global Topological Configurations of Singularities for the Whole Family of Quadratic Differential Systems

In Artes et al. (Geometric configurations of singularities of planar polynomial differential systems. A global classification in the quadratic case. Birkhauser, Basel, 2019) the authors proved that there are 1765 different global geometrical configurations of singularities of quadratic differential systems in the plane. There are other 8 configurations conjectured impossible, all of them related with a single configuration of finite singularities. This classification is completely algebraic and done in terms of invariant polynomials and it is finer than the classification of quadratic systems according to the topological classification of the global configurations of singularities, the goal of this article. The long term project is the classification of phase portraits of all quadratic systems under topological equivalence. A first step in this direction is to obtain the classification of quadratic systems under topological equivalence of local phase portraits around singularities. In this paper we extract the local topological information around all singularities from the 1765 geometric equivalence classes. We prove that there are exactly 208 topologically distinct global topological configurations of singularities for the whole quadratic class. The 8 global geometrical configurations conjectured impossible do not affect this number of 208. From here the next goal would be to obtain a bound for the number of possible different phase portraits, modulo limit cycles.

Three Types Generalized Zn-Heisenberg Ferromagnet Models

By taking values in a commutative subalgebra gln,C, we construct a new generalized Zn-Heisenberg ferromagnet model in (1+1)-dimensions. The corresponding geometrical equivalence between the generalized Zn-Heisenberg ferromagnet model and Zn-mixed derivative nonlinear Schrödinger equation has been investigated. The Lax pairs associated with the generalized systems have been derived. In addition, we construct the generalized Zn-inhomogeneous Heisenberg ferromagnet model and Zn-Ishimori equation in (2+1)-dimensions. We also discuss the integrable properties of the multi-component systems. Meanwhile, the generalized Zn-nonlinear Schrödinger equation, Zn-Davey–Stewartson equation and their Lax representation have been well studied.

Geometrical equivalence and action type geometrical equivalence of group representations

In this paper we construct an example of two representations (V1,G1) and (V2,G2) which are action type geometrically equivalent and groups G1 and G2 are geometrically equivalent, but the representations (V1,G1) and (V2,G2) are not geometrically equivalent.

The extended Zn-Heisenberg ferromagnet model

By virtue of taking values in a commutative subalgebra [Formula: see text] of Lie algebra [Formula: see text], we construct the [Formula: see text]-Heisenberg ferromagnet model which contains many Heisenberg ferromagnet-type equations. Moreover, we investigate the integrable properties of the [Formula: see text]-Heisenberg ferromagnet model. In terms of the gauge transformation, the gauge equivalent counterpart of the [Formula: see text]-Heisenberg ferromagnet model has been presented. Based on the differential geometry of curves and surfaces, the corresponding geometrical equivalence between the [Formula: see text]-Heisenberg ferromagnet model and [Formula: see text]-nonlinear Schrödinger equation has also been established. Furthermore, we also discuss the [Formula: see text]-generalized inhomogeneous Heisenberg ferromagnet model.

Plotkin’s Geometric Equivalence, Mal’cev’s Closure, and Incompressible Nilpotent Groups

In 1997, B. I. Plotkin introduced a concept of geometric equivalence of algebraic structures and posed a question: is it true that every nilpotent torsion-free group is geometrically equivalent to its Mal’cev’s closure? A negative answer in the form of three counterexamples was given by V. V. Bludov and B. V. Gusev in 2007. In the present paper, an infinite series of counterexamples of unbounded Hirsch rank and nilpotency degree is constructed.

Automorphic equivalence in the varieties of representations of Lie algebras

In this paper, we consider the wide class of subvarieties of the variety of all representation of Lie algebras over a field k of characteristic 0. We study the relation between the geometric equivalence and automorphic equivalence of the representations from these subvarieties.

Formas sociais, derivação e conformação

Uma rigorosa compreensão da sociabilidade a partir das formas sociais contrasta tanto com velhas leituras marxistas, mais dadas a algum grau de historicismo, quanto com o institucionalismo liberal do pensamento universitário atual. Há determinação social pela mercadoria e pelo valor, a partir do trabalho abstrato: o fundamento da determinação é a produção. A equivalência entre mercadorias e entre seus possuidores erige formas por cujos constructos articulam-se as relações sociais. Das formas nucleares da sociabilidade capitalista derivam uma forma política estatal e uma forma de subjetividade jurídica, específicas de tal modo de produção e inexoráveis a ele. A derivação de formas sociais se faz atravessada por contradições e antagonismos factuais. Tais formas derivadas conformam-se em derivação secundária.

Matter growth in extended ΛCDM cosmology

On the basis of a previously established scalar–tensor extension of the [Formula: see text]CDM model, we develop an effective fluid approach for the matter growth function. This extended [Formula: see text]CDM (henceforth [Formula: see text]CDM) cosmology takes into account deviations from the Standard Model both via a modified background expansion and by the inclusion of geometric anisotropic stresses as well as of perturbations of the geometric dark-energy equivalent. The background dynamics is governed by an explicit analytic expression for the Hubble rate in which modifications of the Standard Model are given in terms of a single constant parameter [W. C. Algoner, H. E. S. Velten and W. Zimdahl, J. Cosmol. Astropart. Phys. 1611 (2016) 034]. To close the system of fluid-dynamical perturbation equations, we introduce two phenomenological parameters through which the anisotropic stress is related both to the total energy density perturbation of the cosmic substratum and to relative perturbations in the effective two-component system. We quantify the impact of deviations from the standard background, of anisotropic stresses and of nonvanishing perturbations of the effective dark-energy component on the matter growth rate function [Formula: see text] and confront the results with recent redshift-space distortion (RSD) measurements.

On Banach space properties that are not invariant under the Banach–Mazur distance 1

We consider which properties of Banach spaces are not invariant under the Banach–Mazur distance 1; we will say that a given Property P is not invariant under the Banach–Mazur distance 1 if there exist two Banach spaces X and Y such that X has Property P, Y fails Property P, and the Banach–Mazur distance d(X,Y)=1. The main results of this paper concern the case of separable Lindenstrauss spaces. First, in the setting of preduals of ℓ1, we give some geometric equivalences for polyhedral properties. Next, we show that, even in the restricted framework of preduals of ℓ1, most of the notions of polyhedral Banach spaces (and their geometric equivalences) fail to be invariant under the Banach–Mazur distance 1. Then, in the general case of Banach spaces, we indicate some other geometrical properties that are invariant or not invariant under the Banach–Mazur distance 1.

Using synthetic CT for partial brain radiation therapy: Impact on image guidance

PurposeRecent advancements in synthetic computed tomography (synCT) from magnetic resonance (MR) imaging data have made MRI-only treatment planning feasible in the brain, although synCT performance for image guided radiation therapy (IGRT) is not well understood. This work compares geometric equivalence of digitally reconstructed radiographs (DRRs) from CTs and synCTs for brain cancer patients and quantifies performance for partial brain IGRT. Methods and materialsTen brain cancer patients (12 lesions, 7 postsurgical) underwent MR-SIM and CT-SIM. SynCTs were generated by combining ultra-short echo time, T1, T2, and fluid attenuation inversion recovery datasets using voxel-based weighted summation. SynCT and CT DRRs were compared using patient-specific thresholding and assessed via overlap index, Dice similarity coefficient, and Jaccard index. Planar IGRT images for 22 fractions were evaluated to quantify differences between CT-generated DRRs and synCT-generated DRRs in 6 quadrants. Previously validated software was implemented to perform 2-dimensional (2D)–2D rigid registrations using normalized mutual information. Absolute (planar image/DRR registration) and relative (differences between synCT and CT DRR registrations) shifts were calculated for each axis and 3-dimensional vector difference. A total of 1490 rigid registrations were assessed. ResultsDRR agreements in anteroposterior and lateral views for overlap index, Dice similarity coefficient, and Jaccard index were >0.95. Normalized mutual information results were equivalent in 75% of quadrants. Rotational registration results were negligible (<0.07°). Statistically significant differences between CT and synCT registrations were observed in 9/18 matched quadrants/axes (P < .05). The population average absolute shifts were 0.77 ± 0.58 and 0.76 ± 0.59 mm for CT and synCT, respectively, for all axes/quadrants. Three-dimensional vectors were <2 mm in 77.7 ± 10.8% and 76.5 ± 7.2% of CT and synCT registrations, respectively. SynCT DRRs were sensitive in postsurgical cases (vector displacements >2 mm in affected quadrants). ConclusionsDRR synCT geometry was robust. Although statistically significant differences were observed between CT and synCT registrations, results were not clinically significant. Future work will address synCT generation in postsurgical settings.

Model geometricheskogo ekvivalenta tela cheloveka

In determining of geometric parameters of bodies of complex shape, their geometric equivalents are often used in the form of bodies of regular shape having the same values of linear dimensions, volume or surface area. The aim of the study is to obtain formulas for finding the semi-axes of an equivalent cylinder in values of growth, body weight and density or volume, as well as the surface area of the human body. It is shown that the geometric equivalent of a human body is an elliptical cylinder whose height, volume and surface area are equal to the corresponding body parameters. Such a geometric equivalent can be used as a generalized human anthropometric parameter, as well as a phantom for experimental studies on human and environmental heat exchange, in evaluating the thermal insulation properties of tissues and for other experiments in applied anthropology.

Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale

For any reductive groupGGover a global function field, we use the cohomology ofGG-shtukas with multiple modifications and the geometric Satake equivalence to prove the global Langlands correspondence forGGin the “automorphic to Galois” direction. Moreover we obtain a canonical decomposition of the spaces of cuspidal automorphic forms indexed by global Langlands parameters. The proof does not rely at all on the Arthur-Selberg trace formula.

Convexity in Elementary Calculus: Some Geometric Equivalences

What is convexity? For the beginning student of calculus, it is a concept usually introduced to explain the geometric significance of the second derivative. It is easy enough to illustrate graphically or describe verbally (holds water), but what sort of definition is appropriate in this setting? There are several equivalent ways to formulate the definition and relate it to the sign of the second derivative, but many widely used textbooks do not reveal the variety of alternatives or, in the case of some more advanced ones, give proofs of equivalence which are unnecessarily technical or nonintuitive. For texts which contain a more thorough discussion of convexity, see [1], [2]. Here we will look at six well-known formulations of convexity which have easily visualized geometric interpretations and prove their equivalence in an elementary way. We hope that the presentation of some of this (admittedly optional) material in the classroom may hint at the fundamental character of convexity and help to dispel the possible suspicion that the second derivative test for convexity is little more than a disguised definition. Though infrequently seen in calculus texts, perhaps the most common definition of convexity for an arbitrary function is as follows:

How can the neutrino interact with the electromagnetic field?**Supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), FAPERJ (Fundação do Amparo Pesquisa do Rio de Janeiro, FINEP (Financiadora de Estudos e Projetos) and Coordenação do Aperfeiçoamento do Pessoal do Ensino Superior (CAPES)

Maxwell electrodynamics in the fixed Minkowski space-time background can be described in an equivalent way in a curved Riemannian geometry that depends on the electromagnetic field and that we call the electromagnetic metric (e-metric for short). After showing such geometric equivalence we investigate the possibility that new processes dependent on the e-metric are allowed. In particular, for very high values of the field, a direct coupling of uncharged particles to the electromagnetic field may appear.

Automorphic equivalence in the classical varieties of linear algebras

This research is a continuation of [Tsurkov, Automorphic equivalence of linear algebras, J. Algebra Appl. 13(7) (2014), doi:10.1142/S0219498814500261]. In this paper, we consider some classical varieties of linear algebras over the field [Formula: see text] such that [Formula: see text]. We study the relation between the geometric equivalence and automorphic equivalence of the algebras of these varieties. If we denote by [Formula: see text] one of these varieties, then [Formula: see text] is a category of the finite generated free algebras of the variety [Formula: see text]. In this paper, we calculate for the considered varieties the quotient group [Formula: see text], where [Formula: see text] is a group of all the automorphisms of the category [Formula: see text] and [Formula: see text] is a subgroup of all inner automorphisms of this category. The quotient group [Formula: see text] measures the possible difference between the geometric equivalence and automorphic equivalence of algebras from the variety [Formula: see text]. The results of this paper and [Tsurkov, Automorphic equivalence of linear algebras, J. Algebra Appl. 13(7) (2014), doi: 10.1142/S0219498814500261] are summarized in the table at the end of Sec. 5. We can see from this table that in all considered varieties of the linear algebras the group [Formula: see text] is generated by cosets which are presented by no more than two types of the strongly stable automorphisms of the category [Formula: see text]. The first type of automorphisms is connected with the changing of the multiplication by scalar and a the second type is connected with the changing of the multiplication of the elements of the algebras. In Sec. 6, we present some examples of the pairs of linear algebras such that the considered strongly stable automorphisms provide the automorphic equivalence of these algebras, but these algebras are not geometrically equivalent.

Magnetic Field Model for Permanent Magnet Spherical Motor With Double Polyhedron Structure

This paper presents an analytical method for calculating the magnetic field for a permanent magnet spherical motor (PMSM) with double polyhedron structure. For the PMSM, stator coils and cylindrical permanent magnet (PM) poles locate at the fixed point of octahedron and hexahedron, respectively, and provide the uniform magnetic field. To analyze the magnetic field in the spherical coordinate, the cylindrical PM pole is approximated as a dihedral cone pole using geometrical equivalence principle. Moreover, considering the characteristics of hexahedron structure, a virtual equatorial layer is emphasized, and the corresponding periodic boundary condition can be built for solving the Laplace equation related to the virtual equator. Then, the Tait–Bryan rotation transformation is applied three times; thereby the overall magnetic field of the PM poles is obtained by the superposition principle. Both simulations and experiments are conducted to demonstrate the effectiveness of the proposed method. The results indicate that the analytical modeling can represent the magnetic field of the PMSM with a high precision.

Algebraic Geometry Over Algebraic Structures. VI. Geometrical Equivalence

The present paper is one in our series of works on algebraic geometry over arbitrary algebraic structures, which focuses on the concept of geometrical equivalence. This concept signifies that for two geometrically equivalent algebraic structures $$ \mathcal{A} $$ and ℬ of a language L, the classification problems for algebraic sets over $$ \mathcal{A} $$ and ℬ are equivalent. We establish a connection between geometrical equivalence and quasiequational equivalence.

B-11Evaluation of the Equivalency of the Stanford-Binet-5 Comprehensive and Abbreviated Versions for Measuring Intelligence in Autism

B-11Evaluation of the Equivalency of the Stanford-Binet-5 Comprehensive and Abbreviated Versions for Measuring Intelligence in Autism

Universal geometrical equivalence of the algebraic structures of common signature

This article is a part of our effort to explain the foundations of algebraic geometry over arbitrary algebraic structures [1–8]. We introduce the concept of universal geometrical equivalence of two algebraic structures A and B of a common language L which strengthens the available concept of geometrical equivalence and expresses the maximal affinity between A and B from the viewpoint of their algebraic geometries. We establish a connection between universal geometrical equivalence and universal equivalence in the sense of equality of universal theories.