On the state independency and log-linearity of error propagation for discrete group affine systems with application to attitude estimation

Purpose This paper aims to propose a general and rigorous study on the propagation property of invariant errors for the model conversion of state estimation problems with discrete group affine systems. Design/methodology/approach The evolution and operation properties of error propagation model of discrete group affine physical systems are investigated in detail. The general expressions of the propagation properties are proposed together with the rigorous proof and analysis which provide a deeper insight and are beneficial to the control and estimation of discrete group affine systems. Findings The investigation on the state independency and log-linearity of invariant errors for discrete group affine systems are presented in this work, and it is pivotal for the convergence and stability of estimation and control of physical systems in engineering practice. The general expressions of the propagation properties are proposed together with the rigorous proof and analysis. Practical implications An example application to the attitude dynamics of a rigid body together with the attitude estimation problem is used to illustrate the theoretical results. Originality/value The mathematical proof and analysis of the state independency and log-linearity property are the unique and original contributions of this work.

Hecke operators in Bredon (co)homology, K-(co)homology and Bianchi groups

In this paper, we provide a framework for the study of Hecke operators acting on the Bredon (co)homology of an arithmetic discrete group. Our main interest lies in the study of Hecke operators for Bianchi groups. Using the Baum–Connes conjecture, we can transfer computations in Bredon homology to obtain a Hecke action on the [Formula: see text]-theory of the reduced [Formula: see text]-algebra of the group. We show the power of this method giving explicit computations for the group [Formula: see text]. In order to carry out these computations we use an Atiyah–Segal type spectral sequence together with the Bredon homology of the classifying space for proper actions.

Examples of Calabi-Yau threefolds with small Hodge numbers

We observe some suitable examples of Calabi-Yau threefolds for heterotic superstring compactifications. It is reasonable to seek CY threefolds with Euler characteristic equals ±6 because of generation’s number. Hosotani mechanism for violations of the gauge group by the Wilson loops requires such CY space has a non-trivial fundamental group. These spaces can be obtained by factoring the complete intersection Calabi-Yau spaces by the free action of some discrete group. Also we shortly discuss cases when discrete groups act with fixed point sets.

Non-binary Nonlinear Error-correcting Codes from Finite Upper Half-planes

Current work builds new families of non-binary nonlinear error-correcting codes from Finite Upper Half-Plane and p a prime number. A fundamental domain is defined to a discrete group acting over Hq. We establish some concepts and results on Hq, such that the geometric properties allow us to get codification and decodification.

Wrapped NS5-branes, consistent truncations and Inönü-Wigner contractions

We construct consistent Kaluza-Klein truncations of type IIA supergravity on (i) Σ2 × S3 and (ii) Σ3 × S3, where Σ2 = S2/Γ, ℝ2/Γ, or ℍ2/Γ, and Σ3 = S3/Γ, ℝ3/Γ, or ℍ3/Γ, with Γ a discrete group of symmetries, corresponding to NS5-branes wrapped on Σ2 and Σ3. The resulting theories are a D = 5, $$ \mathcal{N} $$ N = 4 gauged supergravity coupled to three vector multiplets with scalar manifold SO(1, 1) × SO(5, 3)/(SO(5) × SO(3)) and gauge group SO(2) × (SO(2) $$ {\ltimes}_{\Sigma_2} $$ ⋉ Σ 2 ℝ4) which depends on the curvature of Σ2, and a D = 4, $$ \mathcal{N} $$ N = 2 gauged supergravity coupled to one vector multiplet and two hypermultiplets with scalar manifold SU(1, 1)/U(1) × G2(2)/SO(4) and gauge group ℝ+ × ℝ+ for truncations (i) and (ii) respectively. Instead of carrying out the truncations at the 10-dimensional level, we show that they can be obtained directly by performing Inönü-Wigner contractions on the 5 and 4-dimensional gauged supergravity theories that come from consistent truncations of 11-dimensional supergravity associated with M5-branes wrapping Σ2 and Σ3. This suggests the existence of a broader class of lower-dimensional gauged supergravity theories related by group contractions that have a 10 or 11-dimensional origin.

Strong Morita equivalence for inclusions of $C^*$-algebras induced by twisted actions of a countable discrete group

We consider two twisted actions of a countable discrete group on $\sigma$-unital $C^*$-algebras. Then by taking the reduced crossed products, we get two inclusions of $C^*$-algebras. We suppose that they are strongly Morita equivalent as inclusions of $C^*$-algebras. Also, we suppose that one of the inclusions of $C^*$-algebras is irreducible, that is, the relative commutant of one of the $\sigma$-unital $C^*$-algebra in the multiplier $C^*$-algebra of the reduced twisted crossed product is trivial. We show that the two actions are then strongly Morita equivalent up to some automorphism of the group.

Locally finite groups and configurations

Let 𝐺 be a discrete group. In 2001, Rosenblatt and Willis proved that 𝐺 is amenable if and only if every possible system of configuration equations admits a normalized solution. In this paper, we show independently that 𝐺 is locally finite if and only if every possible system of configuration equations admits a strictly positive solution. Also, we give a procedure to get equidecomposable subsets 𝐴 and 𝐵 of an infinite finitely generated or a locally finite group 𝐺 such that A ⊊ B A\subsetneq B , directly from a system of configuration equations not having a strictly positive solution.

HIGHER GENERATION BY ABELIAN SUBGROUPS IN LIE GROUPS

To a compact Lie group G one can associate a space E(2;G) akin to the poset of cosets of abelian subgroups of a discrete group. The space E(2;G) was introduced by Adem, F. Cohen and Torres-Giese, and subsequently studied by Adem and Gómez, and other authors. In this short note, we prove that G is abelian if and only if πi(E(2;G)) = 0 for i = 1; 2; 4. This is a Lie group analogue of the fact that the poset of cosets of abelian subgroups of a discrete group is simply connected if and only if the group is abelian.

The reduced Dijkgraaf–Witten invariant of twist knots in the Bloch group of a finite field

Let [Formula: see text] be a closed oriented 3-manifold and let [Formula: see text] be a discrete group. We consider a representation [Formula: see text]. For a 3-cocycle [Formula: see text], the Dijkgraaf–Witten invariant is given by [Formula: see text], where [Formula: see text] is the map induced by [Formula: see text], and [Formula: see text] denotes the fundamental class of [Formula: see text]. Note that [Formula: see text], where [Formula: see text] is the map induced by [Formula: see text], we consider an equivalent invariant [Formula: see text], and we also regard it as the Dijkgraaf–Witten invariant. In 2004, Neumann described the complex hyperbolic volume of [Formula: see text] in terms of the image of the Dijkgraaf–Witten invariant for [Formula: see text] by the Bloch–Wigner map from [Formula: see text] to the Bloch group of [Formula: see text]. In this paper, by replacing [Formula: see text] with a finite field [Formula: see text], we calculate the reduced Dijkgraaf–Witten invariants of the complements of twist knots, where the reduced Dijkgraaf–Witten invariant is the image of the Dijkgraaf–Witten invariant for SL[Formula: see text] by the Bloch–Wigner map from [Formula: see text] to the Bloch group of [Formula: see text].

HYPERBOLIC GROUPS AND NON-COMPACT REAL ALGEBRAIC CURVES

In this paper we study the spaces of non-compact real algebraic curves, i.e. pairs (P, τ), where P is a compact Riemann surface with a finite number of holes and punctures and τ: P → P is an anti-holomorphic involution. We describe the uniformisation of non-compact real algebraic curves by Fuchsian groups. We construct the spaces of non-compact real algebraic curves and describe their connected components. We prove that any connected component is homeomorphic to a quotient of a finite-dimensional real vector space by a discrete group and determine the dimensions of these vector spaces.