Homogeneous Spaces Research Articles

Equivariant Filters are Equivariant

Observers for systems with Lie group symmetries are an active area of research that is seeing significant impact in a number of practical domains, including aerospace, robotics, and mechatronics. This paper builds on the theory of the recently proposed Equivariant Filter (EqF), which is a general observer design for systems on homogeneous spaces that takes advantage of symmetries to yield significant performance advantages. It is shown that the EqF error dynamics are invariant to transformation of the input signal and equivariant as a parametrised vector field. The main theorem shows that two EqF's with different choices of local coordinates and origins and with equivalent noise modelling yield identical performance. In other words, the EqF is intrinsic to the system equations and symmetry. This is verified in a simulation of a 2D robot localisation problem, which also shows how the ability to choose an origin for the EqF can yield practical performance advantages by mitigating floating point precision errors.

Role of media coverage in a SVEIR-I epidemic model with nonlinear incidence and spatial heterogeneous environment.

In this paper, we propose a SVEIR-I epidemic model with media coverage in a spatially heterogeneous environment, and study the role of media coverage in the spread of diseases in a spatially heterogeneous environment. In a spatially heterogeneous environment, we first set up the well-posedness of the model. Then, we define the basic reproduction number $ R_0 $ of the model and establish the global dynamic threshold criteria: when $ R_0 < 1 $, disease-free steady state is globally asymptotically stable, while when $ R_0 > 1 $, the model is uniformly persistent. In addition, the existence and uniqueness of the equilibrium state of endemic diseases were obtained when $ R_0 > 1 $ in homogeneous space and heterogeneous diffusion environment. Further, by constructing appropriate Lyapunov functions, the global asymptotic stability of disease-free and positive steady states was established. Finally, through numerical simulations, it is shown that spatial heterogeneity can increase the risk of disease transmission, and can even change the threshold for disease transmission; media coverage can make people more widely understand disease information, and then reduce the effective contact rate to control the spread of disease.

Review of Making Livable Worlds: Afro-Puerto Rican Women Building Environmental Justice by Hilda Lloréns (University of Washington Press)

This review examines Hilda Lloréns’s research into the role that Afro-Puerto Rican women play in advocating for environmental justice and building a sustainable environment in the Puerto Rican archipelago, particularly after the devastation left behind by Hurricanes Irma and Maria in 2017, and the subsequent catastrophic effects of COVID-19 in 2020. Lloréns shows that Afro-Puerto Rican women are able to survive in the face of racial and ecological discriminations and marginalizations, and their survival is emblematic of Puerto Rico’s own survival. The author devotes the entirety of her book to show that as an "ethnographer of home," as she calls herself, it is essential for people to create livable worlds within which they can survive. Survival in the midst of catastrophic climate change is difficult, Lloréns argues, primarily because Puerto Ricans are often on the receiving end of austerity measures that make their existence tenuous, at best. These austerity measures typically come after a climatic event, and result in limited access to clean water, food, electricity, healthcare, housing, and education, which only serve to exacerbate the desperation that many on the island feel. While this desperation was widespread across the island after the hurricanes in 2017, residents in the southeastern region of the island (predominantly Afro-Puerto Ricans) were even more affected. Lloréns shows how these people used their limited resources to cull an existence out of a seemingly hostile land and create a community that sustained them. Lloréns draws on personal experiences, the experiences of her family, ethnography, anthropology, and interviews to show how vital it is to examine Puerto Rico not as a homogenous space but rather as a heterogeneous one with its unique complexities. And by centering the work and experiences of Black Puerto Ricans, Lloréns gives voice to a group that is largely left on the margins of society, but who demonstrates the importance of building community as a sustaining entity.

The homotopy solvability of compact Lie groups and homogenous topological spaces

The homotopy solvability of compact Lie groups and homogenous topological spaces

A new regularity criterion for the 3D incompressible Boussinesq equations in terms of the middle eigenvalue of the strain tensor in the homogeneous Besov spaces with negative indices

A new regularity criterion for the 3D incompressible Boussinesq equations in terms of the middle eigenvalue of the strain tensor in the homogeneous Besov spaces with negative indices

Ozu in 2.5D

“Ozu’s films are comparable to deep, clear water, perfectly still.” (Maurice Pinguet). The simplicity of Ozu’s images is perfectly apparent. It is in the transparency of these lines that the true depth of his films is played out: Everything happens as if we were dealing with the most natural, most common world, and yet from the outset we notice a lapse—the characters move in an artificial way, their expression is always missing, they seem to float, between presence and absence. This pan.able is an invitation to meditate at the borders between visual arts and the psychology of perception. To think this in-between, between the 2D of the image and the 3D of reality, the 2.5D can be understood both as a new approach to perspective construction and as a specific relationship to the world. In the sense of perspective, the Ozuian image does not seem to be constructed linearly according to the rules of the vanishing point that unify a homogeneous and structured space. Rather, it is constructed according to a principle of superimposed layers, like cartoons. The pan.able shows the passage from an image as a section of the real to an image as a recomposition of the real, from a conception of unified space to a conception of stratified space. Metaphorically, this in-between time of 2.5D reveals that behind the purity and serenity of Ozu’s images a form of drama is at play: the character haunts the world more than he inhabits it. He is in the lining. He is no longer a singular character in a fiction, but a prototype, an idea: There Was a Father, in reference to the title of another Ozu film. The procedure of the pan.able enables us to appreciate this disconnect of the real space and its substitution by an artificial and stratified image. It gives the impression of both looking at a fun flip book and facing a ghost, escaped from its body, lost in its own reverie. Between an amused tenderness and a sensory reflexivity, this is precisely where Ozu’s films take place, between the surface and the depth, between presence and absence, between the singular and the typical—in this in-between space of the 2.5D.

On the fibre structure of compact homogeneous spaces

The paper considers properties of several bundles (natural, structural and Borel ones) related to compact homogeneous spaces. Some results on elements of these bundles are proved, illustrative examples are given, counterexamples to some naturally arising assumptions are put forward.

Equidistribution of Hodge loci II

Let $\mathbb {V}$ be a polarized variation of Hodge structure over a smooth complex quasi-projective variety $S$. In this paper, we give a complete description of the typical Hodge locus for such variations. We prove that it is either empty or equidistributed with respect to a natural differential form, the pull–push form. In particular, it is always analytically dense when the pull–push form does not vanish. When the weight is two, the Hodge numbers are $(q,p,q)$ and the dimension of $S$ is least $rq$, we prove that the typical locus where the Picard rank is at least $r$ is equidistributed in $S$ with respect to the volume form $c_q^r$, where $c_q$ is the $q$th Chern form of the Hodge bundle. We obtain also several equidistribution results of the typical locus in Shimura varieties: a criterion for the density of the typical Hodge loci of a variety in $\mathcal {A}_g$, equidistribution of certain families of CM points and equidistribution of Hecke translates of curves and surfaces in $\mathcal {A}_g$. These results are proved in the much broader context of dynamics on homogeneous spaces of Lie groups which are of independent interest. The pull–push form appears in this greater generality, we provide several tools to determine it, and we compute it in many examples.

Fractal dimensions of crumpled paper

Fractals are infinitely complex shapes that can be split into two types: mathematical fractals and statistical fractals. Mathematical fractals are infinitely complex and self-similar across multiple scales, whereas statistical fractals are not self-similar but can be applied to real life. Crumpling paper is a mathematical fractal process whereby a sheet of paper undergoes deformation via compression. This yields a three- dimensional structure that introduces isotropic and homogenous spaces of varying sizes inside the crumpled paper structure. There is a hierarchy of spaces: a few large, a moderate number of medium, and many small. Consequently, a crumpled paper ball is a fractal object. Fractal dimension is a non- integer measure of an object’s “roughness” and is useful in studying natural objects consisting of many irregular shapes like crumpled paper. In this study, we determined the fractal dimension of a sheet of paper crumpled into a series of spheres of decreasing radii using continuous Χ2 analysis. The result is a tested model relating mass to radius via fractal dimension for crumpled paper, which in this study, yielded a fractal dimension of 2.40. Using Chi-Squared, we were able to test and validate our model and compare it to other experiments’ results to verify consistency, offering advantages over more conventional least squares.

AN INVERSE PROBLEM FOR ELECTRODYNAMIC EQUATIONS WITH A NONLINEAR CONDUCTIVITY

An inverse problem of determination of a variable coefficient in electrodynamic equations with a nonlinear conductivity is considered. It is supposed that the unknown coefficient is a smooth function of space variables and finite in \({{\mathbb{R}}^{3}}\). From a homogeneous space a plane wave going in a direction fall down on a heterogeneousness. The direction is a parameter of the problem. The module of the electrical strength vector for some diapason of directions and for moments of the time close to arriving the wave at points of a surface of a ball, inside of which the heterogeneousness is contained, is given as the information for solution of the inverse problem. It is shown that this information reduces the inverse problem to the well known X-ray tomography. Algorithms of the numerical solution of the later problem is well developed.

Threshold dynamics of an age-space structure vector-borne disease model with multiple transmission pathways

This paper aims to study threshold dynamics of an age-space structured vector-borne epidemic model with multiple transmission pathways. We develop a model in a bounded space, and demonstrate the well-posedness by proving the global existence of the solution and the model admits a global attractor with using the theory of fixed point problem, Picard sequences and iteration. In the special case of homogenous space, the explicit formula of basic reproduction number $ R_{0} $ is established, which can be used to discuss whether a disease is persistent or extinct. The local and global stability of steady states are achieved by analyzing the distribution of characteristic roots of characteristic equations and designing Lyapunov functions. These theoretical results are illustrated by numerical simulations. Besides, sensitive analysis on $ R_{0} $ reveals that certain parameters are more effective to impact the $ R_{0} $. We find that the PRCC indexes of solutions can be time-dependent, and the relative importance of parameters can also depend on time. Relying on this analysis and exhaustive numerical simulations, we suggest that insecticides and mosquito nets should be used in a timely manner to disinfect vectors and reduce the contact between vectors and hosts, and that ignoring the horizontal transmission in the host underestimates the risk of disease transmission.

Homogeneous geodesics in sub-Riemannian geometry

We study homogeneous geodesics of sub-Riemannian manifolds, i.e., normal geodesics that are orbits of one-parametric subgroups of isometries. We obtain a criterion for a geodesic to be homogeneous in terms of its initial momentum. We prove that any weakly commutative sub-Riemannian homogeneous space is geodesic orbit, that means all geodesics are homogeneous. We discuss some examples of geodesic orbit sub-Riemannian manifolds. In particular, we show that geodesic orbit Carnot groups are only groups of step 1 and 2. Finally, we get a broad condition for existence of at least one homogeneous geodesic.

On the fibre structure of compact homogeneous spaces

В статье рассматриваются свойства нескольких расслоений, связанных с компактными однородными пространствами, - натурального, структурного и борелевского. Доказаны некоторые утверждения об элементах этих расслоений, приведены иллюстрирующие их примеры, а также и контрпримеры к некоторым естественно возникающим предположениям. Библиография: 29 наименований.

Fast Recognition on Shallow Groundwater and Anomaly Analysis Using Frequency Selection Sounding Method

The validity of the frequency selection method (FSM) in shallow (&lt;150 m) groundwater exploration was illustrated by practical applications, and the relationship between potential electrode spacing MN and groundwater depth in FSM sounding method was analyzed and preliminary theoretical research was carried out by a simple geologic-geophysical model of sphere. Firstly, under the combined action of horizontal alternating electric field and alternating magnetic field, a simplified geophysical model of low resistivity conductive sphere in homogeneous half space was established, and the forward calculation was performed on the FSM sounding curve. Then, the water yield of 131 wells in the application of FSM in the Rural Drinking Water Safety Project of 12th Five-Year Plan in Guangxi Province was counted. In addition, detailed tabular statistical analysis was carried out on the drilling results of 98 drilling wells, and the relationship between potential electrode spacing MN at abnormal sounding curve and actual drilling water depth was compared and studied. Theoretical analysis and practical application show that FSM has obvious effectiveness in shallow groundwater exploration, and it is an effective method to determine shallow groundwater well locations in the future. The cause of FSM anomaly is the comprehensive effect of the natural 3D alternating electromagnetic signal underground. At the same time, the practical statistics show that there is 1:1 approximation between the size of potential electrode spacing MN at the anomaly curve of the frequency selection method and the actual drilling water depth, which verifies the correctness of the theoretical simulation results. FSM could be widely used in the shallow groundwater exploration in the future, and it is an effective, non-destructive, fast, and low-cost geophysical method.

Frobenius–Ehresmann structures and Cartan geometries in positive characteristic

The aim of the present paper is to lay the foundation for a theory of Ehresmann structures in positive characteristic, generalizing the Frobenius-projective and Frobenius-affine structures defined in the previous work. This theory deals with atlases of étale coordinate charts on varieties modeled on homogeneous spaces of algebraic groups, which we call Frobenius–Ehresmann structures. These structures are compared with Cartan geometries in positive characteristic, as well as with higher-dimensional generalizations of dormant indigenous bundles. In particular, we investigate the conditions under which these geometric structures are equivalent to each other. Also, we consider the classification problem of Frobenius–Ehresmann structures on algebraic curves. The latter half of the present paper discusses the deformation theory of indigenous bundles in the algebraic setting. The tangent and obstruction spaces of various deformation functors are computed in terms of the hypercohomology groups of certain complexes. As a consequence, we formulate and prove the Ehresmann–Weil–Thurston principle for Frobenius–Ehresmann structures. This fact asserts that deformations of a variety equipped with a Frobenius–Ehresmann structure are completely determined by their monodromy crystals.

Global Small Solutions to a Special $$2\frac{1}{2}$$-D Compressible Viscous Non-resistive MHD System

This paper solves the global well-posedness and stability problem on a special \(2\frac{1}{2}\)-D compressible viscous non-resistive MHD system near a steady-state solution. The steady-state here consists of a positive constant density and a background magnetic field. The global solution is constructed in \(L^p\)-based homogeneous Besov spaces, which allow general and highly oscillating initial velocity. The well-posedness problem studied here is extremely challenging due to the lack of the magnetic diffusion and remains open for the corresponding 3D MHD equations. Our approach exploits the enhanced dissipation and stabilizing effect resulting from the background magnetic field, a phenomenon observed in physical experiments. In addition, we obtain the solution’s optimal decay rate when the initial data is further assumed to be in a Besov space of negative index.

Modeling of a homogeneous isotropic half space in the context of multi-phase lag coupled thermoelasticity

A two-dimensional multi-phase lag model in the context of generalized thermoelasticity is established for an isotropic half-space medium. A vector-matrix differential equation is obtained from the governing equations using normal mode analysis. The eigenvalue approach is applied to obtain the solutions. The temperature-dependent displacements, stresses, strains are calculated numerically and represented graphically to show the accuracy of the solution under mechanical and thermal loads.

Об одной контракции дискретной группы D3

The groups of motions of two-dimensional space-time models&#x0D; from a geometric point of view are groups of motions of&#x0D; two-dimensional homogeneous spaces of a constant curvature.&#x0D; When quantizing these models, discrete subgroups of&#x0D; these groups play an important role. The standard model is&#x0D; based on the gauge groups U(1), SU(2) and SU(3). Discrete&#x0D; subgroups of these groups are now also in the center&#x0D; of attention in the elementary particle physics. In this paper&#x0D; we present contractions of the discrete dihedron group&#x0D; D3 induced by limit transitions in the continuous groupO(2)&#x0D; and O(3). The resulting groups are D4 and infinite groups&#x0D; D∞, generating discrete lattices of two-dimensional spacetime&#x0D; models. We also discuss transitions of discrete symmetry&#x0D; groups of the Platonic solids.

Tame Sets in Homogeneous Spaces

AbstractWe prove the existence of strongly tame sets in affine algebraic homogeneous spaces of linear algebraic Lie groups. We also show that $(\mathbb {C}^{n},A)$ ( ℂ n , A ) enjoys the strong relative density property when A is a tame set, and we provide examples of Stein manifolds admitting non-equivalent tame sets.

The Action of the Thompson Group $F$ on Infinite Trees

We construct an action of the Thompson group $F$ on a compact space built from pairs of infinite, binary rooted trees. The action arises as an $F$-equivariant compactification of the action of $F$ by translations on one of its homogeneous spaces, $F/H_2$, corresponding to a certain subgroup $H_2$ of $F$. The representation of $F$ on the Hilbert space $\ell^2(F/H_2)$ is faithful on the complex group algebra $\mathbb{C}[F]$.