Topological Sense Research Articles

A Koopman–Takens Theorem: Linear Least Squares Prediction of Nonlinear Time Series

The least squares linear filter, also called the Wiener filter, is a popular tool to predict the next element(s) of time series by linear combination of time-delayed observations. We consider observation sequences of deterministic dynamics, and ask: Which pairs of observation function and dynamics are predictable? If one allows for nonlinear mappings of time-delayed observations, then Takens’ well-known theorem implies that a set of pairs, large in a specific topological sense, exists for which an exact prediction is possible. We show that a similar statement applies for the linear least squares filter in the infinite-delay limit, by considering the forecast problem for invertible measure-preserving maps and the Koopman operator on square-integrable functions.

Canonical decompositions and algorithmic recognition of spatial graphs

AbstractWe prove that there exists an algorithm for determining whether two piecewise-linear spatial graphs are isomorphic. In its most general form, our theorem applies to spatial graphs furnished with vertex colourings, edge colourings and/or edge orientations.We first show that spatial graphs admit canonical decompositions into blocks, that is, spatial graphs that are non-split and have no cut vertices, in a suitable topological sense. Then, we apply a result of Haken and Matveev in order to algorithmically distinguish these blocks.

Existence and Properties of the Solution of Nonlinear Differential Equations with Impulses at Variable Times

In this paper, a class of nonlinear ordinary differential equations with impulses at variable times is considered. The existence and uniqueness of the solution are given. At the same time, modifying the classical definitions of continuous dependence and Gâteaux differentiability, some results on the continuous dependence and Gâteaux differentiable of the solution relative to the initial value are also presented in a new topology sense. For the autonomous impulsive system, the periodicity of the solution is given. As an application, the properties of the solution for a type of controlled nonlinear ordinary differential equation with impulses at variable times is obtained. These results are a foundation to study optimal control problems of systems governed by differential equations with impulses at variable times.

Sequence entropy for amenable group actions* *C Liu is partially supported by NNSF of China (12090012, 12090010). K Yan is supported by NNSF of China (11861010, 12171175), NSF of Guangxi Province (2018GXNSFFA281008)and Project of First Class Disciplines of Statistics and Guangxi Key Laboratory of Quantity Economics.

We study the sequence entropy for amenable group actions and investigate systematically spectrum and several mixing concepts via sequence entropy both in measure-theoretic dynamical systems and topological dynamical systems. Moreover, we use sequence entropy pairs to characterize weakly mixing and null systems in the topological sense.

Computer‐assisted analysis of chaos in a three‐species food chain model

AbstractIn this paper, a three‐species food chain model with Holling type IV and Beddington–DeAngelis functional responses is formulated. Numerical simulations show that this system can generate chaos for some parameter values. But the mechanism behind chaos is still unclear only through numerical simulations. Then, using the topological horseshoe theories and Conley–Moser conditions, we present a computer‐assisted analysis to show the chaoticity of this system in the topological sense, that is, it has positive topological entropy. We prove that the Poincaré map of this model possesses a closed uniformly hyperbolic chaotic invariant set, and it is topologically conjugate to a 2‐shift map. At last, we consider the impact of fear on this three‐species model. It is an important factor in controlling chaos in biological models, which has been validated in other models.

Topological Approach for Material Structure Analyses in Terms of R2 Orientation Distribution Function

The application of solid mechanics theory for material behavior faces the discrete nature of modern or biological material. Despite the developed methods of homogenization, there are deviations between simulated and experiments results. The reason is homogenization, which mathematically involves a type of interpolation. The situation gets worse for complex structured materials. On the other hand, a topological approach can help in such analysis, but such an approach has computational costs. At the same time, increasing modern computational capabilities remove this barrier. This study is focused on building a method to analyze material structure in a topological sense. The orientation distribution function was used to describe the structure of the material. The plane case was investigated. Quadratic and biquadratic forms of interpolant were investigated. The persistent homology approach was used for topology analysis. For this purpose, a persistence diagram for quadratic and biquadratic forms was found and analyzed. In this study, it is shown how scaling the origin point cloud influences H1 points in the persistence diagram. It was assumed that the topology of the biquadratic form can be understood as a superposition of quadratic forms. Quantitative estimates are given for ellipticity and H1 points. A dataset of micro photos was processed using the proposed method. Furthermore, the supply criteria for the interpolation choice in quadratic or biquadratic forms was formulated.

Sequence entropy tuples and mean sensitive tuples

AbstractUsing the idea of local entropy theory, we characterize the sequence entropy tuple via mean forms of the sensitive tuple in both topological and measure-theoretical senses. For the measure-theoretical sense, we show that for an ergodic measure-preserving system, the $\mu $ -sequence entropy tuple, the $\mu $ -mean sensitive tuple, and the $\mu $ -sensitive in the mean tuple coincide, and give an example to show that the ergodicity condition is necessary. For the topological sense, we show that for a certain class of minimal systems, the mean sensitive tuple is the sequence entropy tuple.

Generalized Fibonacci shifts in the Lorenz attractor

In this article we deal with symmetric Lorenz attractors having a homoclinic loop that exhibits a well ordered orbit. We show the symmetry implies a very regular behaviour on the dynamic in the topological and metric sense. Let ([−1,1],f) be the one-dimensional reduction Lorenz map satisfying a well ordered orbit and ([−1,0],f̃) be the quotient map, given by the equivalence relation x∼−x, the dynamic of f˜ is described explicitly as a subshift of finite type which generalizes the Fibonacci shifts and this fact is used to compute topological entropy of f.Moreover we show that in general ([−1,0],f̃) is related to a factor of the k-bonacci shift. In particular we found that the 1-dimensional Lorenz map replicates an interesting duplicating behaviour of the k-bonacci shift found in Sirvent (1996, 2011).

Propagation thresholds in a diffusive epidemic model with latency and vaccination.

This paper studies the propagation thresholds in a diffusive epidemic model with latency and vaccination. When the initial condition satisfies proper exponential decaying behavior, we present the spatial expansion feature of the infected. Different leftward and rightward spreading speeds are obtained with respect to different decaying initial values. Moreover, the convergence in the sense of compact open topology is also studied when the spreading speeds are finite. Finally, we show that the minimal spreading speed is the minimal wave speed of traveling wave solutions, which also presents the precisely asymptotic behavior of traveling wave solutions for the infected branch at the disease-free side. Here, the asymptotic behavior plays an important role that distinguishes the minimal spreading speed from all possible spreading speeds. From the definition of possible spreading speeds, we may find some factors affecting the spatial expansion ability, which includes that the vaccination could decrease the spatial expansion ability of the disease.

Arithmetical complexity of the language of generic limit sets of cellular automata

The generic limit set of a dynamical system is the smallest set that attracts most of the space in a topological sense: it is the smallest closed set with a comeager basin of attraction. Introduced by Milnor, it has been studied in the context of one-dimensional cellular automata by Djenaoui and Guillon, Delacourt, and Törmä. In this article we present complexity bounds on realizations of generic limit sets of cellular automata with prescribed properties. We show that generic limit sets have a Π20 language if they are inclusion-minimal, a Σ10 language if the cellular automaton has equicontinuous points, and that these bounds are tight. We also prove that many chain mixing Π20 subshifts and all chain mixing Δ20 subshifts are realizable as generic limit sets. As a corollary, we characterize the minimal subshifts that occur as generic limit sets.

Evolution of tornado-like vortices in three-dimensional compressible rectangular cavity flows

The spatial structure and time evolution of tornado-like vortices in a three-dimensional cavity are studied by topological analysis and numerical simulation. The topology theory of the unsteady vortex in the rectangular coordinate system (Zhang, Zhang & Shu, J. Fluid Mech., vol. 639, 2009, pp. 343–372) is generalized to the curvilinear coordinate system. Two functions $\lambda (q_1,t)$ and $q(q_1,t)$ are obtained to determine the topology structure of the sectional streamline pattern in the cross-section perpendicular to the vortex axis and the meridional plane, respectively. The spiral direction of the sectional streamlines in the cross-section perpendicular to the vortex axis depends on the sign of $\lambda (q_1,t)$ . The types of critical points in the meridional plane depend on the sign of $q(q_1,t)$ . The relation between the critical points of the streamline pattern in the meridional plane and that in the cross-section perpendicular to the vortex axis is set up. The flow in a three-dimensional rectangular cavity is numerically simulated by solving the three-dimensional Navier–Stokes equations using high-order numerical methods. The spatial structures and the time evolutions of the tornado-like vortices in the cavity are analysed with our topology theory. Both the bubble type and spiral type of vortex breakdown are observed. They have a close relationship with the vortex structure in the cross-section perpendicular to the vortex axis. The bubble-type breakdown has a conical core and the core is non-axisymmetric in the sense of topology. A criterion for the bubble type and the spiral type based on the spatial structure characteristic of the two breakdown types is provided.

Filtrations on the representation ring of an affine algebraic group

Abstract Let G be an affine algebraic group over a field. The representation ring $\mathrm{R}(G)$ has three standard filtrations, defining the same topology on $\mathrm{R}(G)$: augmentation, Chern and Chow, each of which contained in the next one. For split reductive G, motivated by potential applications related to spin groups, we introduce and study one more filtration, containing the previous ones, which we call induced because it is induced by any of the filtrations on the representation ring of a maximal split torus of G. In the case of semisimple simply connected G, this fourth filtration turns out to be equivalent (in the above topological sense) to the previous three. However, for the spin group $G=\operatorname{\mathrm{Spin}}(d)$ over the complex numbers with $d=7,8$, the new filtration is shown to be strictly larger than the others. It is also shown that for $G=\operatorname{\mathrm{Spin}}(d)$ over an arbitrary field and with any $d\geq7$, the Chern and Chow filtrations on $\mathrm{R}(G)$ are not the same, giving new counter-examples to an extension of Atiyah’s conjecture.

Rigorous numerical study of the Colpitts oscillator with an exponential nonlinearity.

The dynamics of the Colpitts oscillator with an exponential nonlinearity is investigated using rigorous interval arithmetic based tools. The existence of various types of periodic attractors is proved using the interval Newton method. The main results involve the chaotic case for which a trapping region for the associated return map is constructed and a rigorous lower bound for the value of the topological entropy is computed, thus proving that the system is chaotic in the topological sense. A systematic search for unstable periodic orbits embedded in the chaotic attractor is carried out and the results are used to obtain an accurate approximation of the topological entropy of the system.

A new analytical method to simulate the mutual impact of space-time memory indices embedded in (1 + 2)-physical models

Abstract In the present article, we geometrically and analytically examine the mutual impact of space-time Caputo derivatives embedded in (1 + 2)-physical models. This has been accomplished by integrating the residual power series method (RPSM) with a new trivariate fractional power series representation that encompasses spatial and temporal Caputo derivative parameters. Theoretically, some results regarding the convergence and the error for the proposed adaptation have been established by virtue of the Riemann–Liouville fractional integral. Practically, the embedding of Schrödinger, telegraph, and Burgers’ equations into higher fractional space has been considered, and their solutions furnished by means of a rapidly convergent series that has ultimately a closed-form fractional function. The graphical analysis of the obtained solutions has shown that the solutions possess a homotopy mapping characteristic, in a topological sense, to reach the integer case solution where the Caputo derivative parameters behave similarly to the homotopy parameters. Altogether, the proposed technique exhibits a high accuracy and high rate of convergence.

The Complexity of Drawing a Graph in a Polygonal Region

We prove that the following problem is complete for the existential theory of the reals: Given a planar graph and a polygonal region, with some vertices of the graph assigned to points on the boundary of the region, place the remaining vertices to create a planar straight-line drawing of the graph inside the region. This establishes a wider context for the NP-hardness result by Patrignani on extending partial planar graph drawings. Our result is one of the first showing that a problem of drawing planar graphs with straight-line edges is hard for the existential theory of the reals. The complexity of the problem is open in the case of a simply connected region. We also show that, even for integer input coordinates, it is possible that drawing a graph in a polygonal region requires some vertices to be placed at irrational coordinates. By contrast, the coordinates are known to have bounded bit complexity for the special case of a convex region, or for drawing a path in any polygonal region. In addition, we prove a Mnëv-type universality result - loosely speaking, that the solution spaces of instances of our graph drawing problem are equivalent, in a topological and algebraic sense, to bounded algebraic varieties.

Singularity in the relativistic celestial dynamics

This article studies the singularity in the relativistic celestial dynamics. For the relativistic N-body system, we prove the nonexistence of noncollision-singularities. We demonstrate approaching the light speed within finite time is equivalent to the occurrence of a collision. We also introduce the spiral vector to describe the behavior of 2-collisions. We prove the improbability of the 2-collision singularities with zero spiral vectors. By constructing singularities, we prove the Lebesgue measure of singularities is infinity and singularities are of the second Baire category. The singularities are not negligible in the measure-theoretic or the topological sense, and we thus solve Littlewood's thirteenth problem in the relativistic situation.

The profinite topology of free groups and weakly generic tuples of automorphisms

AbstractLet be a countable first order structure and endow the universe of with the discrete topology. Then the automorphism group of becomes a topological group. A tuple of automorphisms is defined to be weakly generic iff its diagonal conjugacy class (in the algebraic sense) is dense (in the topological sense) and the ‐orbit of each is finite. Existence of tuples of weakly generic automorphisms are interesting from the point of view of model theory as well as from the point of view of finite combinatorics.The main results of the present work are as follows. In Theorem 2.6 we characterize the existence of tuples of weakly generic automorphisms with the aid of the profinite topology of free groups. In Corollary 2.12 we will show that if has finite topological rank r (and satisfies a further, mild technical condition) then the existence of a weakly generic tuple in implies the existence of weakly generic tuples in for all natural number . Finally, in Theorem 3.2 we show that if is a countable model of an ℵ0‐categorical, simple theory in which all types over the empty set are stationary and has a pair of weakly generic automorphisms then it has tuples of weakly generic automorphisms of arbitrary finite length.At the technical level we will combine elementary investigations about the profinite topology of free groups with the results of [11] about topological ranks of the automorphism groups of some structures.

Climate modelling and structural stability

Climate modelling plays a crucial role for understanding and addressing the climate challenge, in terms of both mitigation and adaptation. It is therefore of central importance to understand to what extent climate models are adequate for relevant purposes, such as providing certain kinds of climate change projections in view of decision-making. In this perspective, the issue of the stability of climate models under small relevant perturbations in their structure (or small relevant ‘structural model errors’ with respect to the target system) seems particularly important. Within this framework, a debate has emerged in the philosophy of science literature about the relevance for climate modelling of the mathematical notion of structural stability. This paper adresses several important foundational and epistemological questions that arise in this context, in particular about the the role of abstract mathematical considerations of a qualitative nature (in some precise, topological sense) for concrete modelling projects with mainly quantitative purposes.

Topological bifurcation structure of one-parameter families of C 1 unimodal maps

We consider the bifurcation structure of one-parameter families of unimodal maps whose differentiability is only C 1. The structure of its bifurcation diagram can be a very wild one in such case. However we prove that in a certain topological sense, the structure is the same as that of the standard family of quadratic polynomials. In the case of families of polynomials, irreducible component of the bifurcation diagram can be defined naturally by dividing by the polynomials corresponding to lower periods. We show that such an irreducible component can be defined even if the maps and the family satisfy only a very mild differentiability condition. By removing components of lower periods, the structure of the bifurcation diagram becomes a considerably simplified one. We prove that the symbolic condition for the irreducible components is exactly the same as that of the standard family of quadratic polynomials. When we consider families of maps without strict conditions, the bifurcation diagrams may have infinitely many wild components. We show that such a situation does not affect the irreducible component essentially by proving a separation theorem for compact set in the plane which asserts that a given connected component can be cut out from the compact set by a curve.

Locating Heidegger’s Kotoba between Actuality and Hollowness: the Way towards a Thinking Conversation with Japanese Philosophy

What is the philosophical significance of Heidegger’s interpretation of the Japanese notion of kotoba (言葉) for Japanese philosophy? Was his conversation with Tezuka Tomio a real dialogue or not? To answer to these correlated questions, I elucidate Heidegger’s 1954 essay “A Dialogue on Language” by following a topological mode of thinking, and I inquire into the way-making of a “thinking conversation”. First, I problematize whether Heidegger engaged in a genuine dialogue with Tezuka. To that end, I distinguish the hermeneutic horizon of the actual encounter between Tezuka and Heidegger from Heidegger’s essay which places Tezuka (the Japanese) and Heidegger (the Inquirer) in a fictional philosophical conversation. Second, I argue that Heidegger’s topological method of interpretating kotoba can be read as a poetic means of thematizing East-West dialogue. Third and finally, exploring the topological sense of kotoba, I engage with third generation Kyoto School thinker Ueda Shizuteru’s idea of “hollow words” of language, situated in a twofold view of the world. I conclude that the true character of Heidegger’s conversation with Tezuka can be identified neither in Heidegger’s “actual” encounter with Tezuka, nor merely in Heidegger’s “hollow” essay. Departing from Ueda’s account of kotoba, it appears that a genuine conversation with language can be located in the dialogue of actuality and hollowness, which finds it expression in poetic language.