arXiv subjects

Victoria Rayskin

Publications and source records attributed to Victoria Rayskin.

12 recordsLinked to original sources

arXiv2024

Homoclinic tangencies in $\mathbb{R}^n$

Let $f: M \to M$ denote a diffeomorphism of a smooth manifold $M$. Let $p$ in $M$ be its hyperbolic fixed point with stable and unstable manifolds $W_S$ and $W_U$, respectively. Assume that $W_S$ is a curve. Suppose that $W_U$ and $W_S$ have a degenerate homoclinic crossing at a point $B\ne p$, i.e., they cross at $B$ tangentially with a finite order of contact. It is shown that, subject to $C^1$-linearizability and certain conditions on the invariant manifolds, a transverse homoclinic crossing will arise arbitrarily close to $B$. This proves the existence of a horseshoe structure arbitrarily close to $B$, and extends a similar planar result of Homburg and Weiss.

math.DS↗

arXiv2020

Multivariate time series approximation by multiple trajectories of a dynamical system. Applications to Internet traffic and COVID-19 data

Utilization of multiple trajectories of a dynamical system model provides us with several benefits in approximation of time series. For short term predictions a high accuracy can be achieved via switches to new trajectory at any time. Different long term trends (tendency to different stationary points) of the phase portrait characterize various scenarios of the process realization influenced by externalities. The dynamical system's phase portrait analysis helps to see if the equations properly describe the reality. We also extend the dynamical systems approach (discussed in \cite{R5}) to the dynamical systems with external control. We illustrate these ideas with the help of new examples of the rental properties HOMES.mil platform data. We also compare the qualitative properties of HOMES.mil and Wikipedia.org platforms' phase portraits and the corresponding differences of the two platforms' users. In our last example with COVID-19 data we discuss the high accuracy of the short term prediction of confirmed infection cases, recovery cases and death cases in various countries.

physics.soc-ph↗

arXiv2020

Nonchaotic Models and Predictability of the Users' Volume Dynamics on Internet Platforms

Internet platforms' traffic defines important characteristics of platforms, such as price of services, advertisements, speed of operations. The traffic is usually estimated with the help of the traditional time series models (ARIMA, Holt-Winters, etc.), which are successful in short term extrapolations of sufficiently denoised signals. We propose a dynamical system approach for the modeling of the underlying process. The method allows to discuss the global qualitative properties of the dynamics' phase portrait and long term tendencies. The proposed models are nonchaotic, the long term prediction is reliable, and it explains the fundamental properties and trend of various types of digital platforms. Because of these properties, we call the flow of these models the {\it trending flow}. Utilizing the new approach, we construct the two-sided platform models for the volume of users, that can be applied to Amazon.com, Homes.mil or Wikipedia.org. We consider a generalization of the two-sided platforms' models to multi-sided platforms. If the equations' are cooperative, the flow is trending, and it helps to understand the properties of the platforms and reliably predicts the long term behavior. We show how to reconstruct the governing differential equations from time series data. The external effects are modeled as system's parameters (initial conditions).

physics.soc-ph↗

arXiv2020

Dynamical systems' models for the prediction of multi-variable time series. Wikipedia's traffic example

The models VAR, ARIMA, Holt-Winters, are frequently used for short-term forecasts of multivariate time series. In this paper we consider models constructed with the help of dynamical systems that have relatively simple limiting behavior. Switching between different trajectories of the phase portrait, we obtain a high precision prediction. Moreover, the dynamical system approach provides the global qualitative picture of the model's phase portrait, and allows us to discuss multidimensional patterns and long-term properties of the process. The simple limiting behavior allows us to associate different trends with different process's realization scenarios that can be influenced by externalities. We demonstrate these ideas using the examples of the Wikipedia's traffic of Readers, Contributors and Edits. First, we consider the two-dimensional model, predicting the traffic of Readers and Edits. The prediction precision is higher than the two-dimensional VAR prediction. Different trends (corresponding to different fixed points) can be associated with different platform's incentives. Then, adding the Contributors data, we discuss the three-dimensional model (more precise than the three-dimensional VAR). It provides a more accurate short-term prediction of Edits than the two-dimensional dynamic model. The global picture shows that the number of new Edits tends to decline in the future, while the number of new Contributors and Readers will grow in the long run.

math.DS↗

arXiv2019

Whitney's and Seeley's type of extensions for maps defined on some Banach spaces

Let $X=C[0,1]$, and $Y$ be an arbitrary Banach space. Consider a collection of open segments $\{V_i \}\subset X$. Suppose the map $f: \cup_i V_i \to Y$ has $q$ bounded Fréchet derivatives ($q=0,1,...,\infty$), and $f$ and all its derivatives have continuous bounded limits at the boundary. Then, subject to some non-intercept condition for the segments $V_i$, the map $f$ can be extended to $F: X\to Y$, so that $F_{|\,\cup_i V_i}=f$ and $F$ has $q$ bounded derivatives. We prove similar Whitney's Extension theorem generalizations for some other Banach spaces. We also prove Seeley Extension theorem for $X=C[0,1].$ These results are related to the problems of function approximation, and manifold learning, which are of central importance to many applied fields.

math.FA↗

arXiv2019

New Method of Smooth Extension of Local Maps on Linear Topological Spaces. Applications and Examples

The question of extension of locally defined maps to the entire space arises in many problems of analysis (e.g., local linearization of functional equations). A known classical method of extension of smooth local maps on Banach spaces uses smooth bump functions. However, such functions are absent in the majority of infinite-dimensional spaces. We suggest a new approach to localization of Banach spaces with the help of locally identical maps, which we call blid maps. In addition to smooth spaces, blid maps also allow to extend local maps on non-smooth spaces (e.g., $C^q [0, 1]$, $q=0, 1, 2,...$). For the spaces possessing blid maps, we show how to reconstruct a map from its derivatives at a point (see the Borel Lemma). We also demonstrate how blid maps assist in finding global solutions of cohomological equations having linear transformation of the argument. We present application of blid maps to local differentiable linearization of maps on Banach spaces. We discuss differentiable localization for metric spaces (e.g., $C^{\infty}(\R)$), prove an extension result for locally defined maps and present examples of such extensions for the specific metric spaces. In conclusion, we formulate open problems.

math.DS↗

arXiv2019

Users' traffic on two-sided Internet platforms. Qualitative dynamics

Internet platforms' traffic defines important characteristics of platforms, such as pricing of services, advertisements, speed of operations. One can estimate the traffic with the traditional time series models like ARIMA, Holt-Winters, functional and kernel regressions. When using these methods, we usually smooth-out noise and various external effects in the data and obtain short-term predictions of processes. However, these models do not necessarily help us to understand the underlying mechanism and the tendencies of the processes. In this article, we discuss the dynamical system approach to the modeling, which is designed to discover the underlying mechanism and the qualitative properties of the system's phase portrait. We show how to reconstruct the governing differential equations from data. The external effects are modeled as system's parameters (initial conditions). Utilizing this new approach, we construct the models for the volume of users, interacting through Internet platforms, such as "Amazon.com", "Homes.mil" or "Wikipedia.org". Then, we perform qualitative analysis of the system's phase portrait and discuss the main characteristics of the platforms.

physics.soc-ph↗

arXiv2018

Extension of differentiable local mappings on linear topological spaces

Usually, for extension of local maps, one uses multiplication by so called bump functions. However, majority of infinite-dimensional linear topological spaces do not have smooth bump functions. Therefore, in \cite{BR} we suggested a new approach for Banach spaces, based on the composition with locally identical maps. In the present work we discuss a possibility of generalization of this method for arbitrary spaces and applications of this theory.

math.FA↗

arXiv2018

On a conjecture of the paper Differentiability of the Conjugacy in the Hartman-Grobman Theorem

In this note we show that for the construction of differentiable conjugation, the assumption of the existence of smooth bump function is not necessary, and consequently the corresponding conjecture stated in the paper of W. Zhang, K. Lu and W. Zhang "Differentiability of the Conjugacy in the Hartman-Grobman Theorem" (\cite{ZLZ}) is incorrect. We show that instead of bump functions we can use smooth blid maps. We also propose a construction of the blid map for the space $X=C^0[0,1]$, which does not possess a smooth bump function.

math.DS↗

arXiv2018

A connection between tests for absolute convergence of infinite series, or how to be fair

The Ratio Test and the Root Test for absolute convergence/divergence of series of numbers $\sum_{n=0}^{\infty}a_n$ are frequently discussed and proved independently in Calculus courses. The Root Test is stronger (verifies convergence for more series) than the Ratio Test. This relation inspires introduction of some intermediate strength tests (stronger than the Ratio Test and weaker than the Root Test) that we call Power Mean Tests (they, in particular, include the Arithmetic Mean Test). We show the connection between the Root, the Power Mean and the Ratio Tests. We also note that all these tests are related to the test that we formulate and call Generalized $f$-mean Test (or Kolmogorov-Nagumo-de Finetti mean Test). We provide an example of an infinite series, where the Arithmetic Mean test is the test that should be used for convergence verification, because the Root and the Ratio Tests are not easy to apply. We conclude this work with the statement emphasizing why it is ``fair'' to include the summarizing Corollary 3.1 in our Calculus course.

math.HO↗

arXiv2017

A New Method of Extension of Local Maps of Banach Spaces. Applications and Examples

A known classical method of extension of smooth local maps of Banach spaces uses smooth bump functions. However, such functions are absent in the majority of infinite-dimensional Banach spaces. This is an obstacle in the development of local analysis, in particular in the questions of extending local maps onto the whole space. We suggest an approach that substitutes bump functions with special maps, which we call blid maps. It allows us to extend smooth local maps from non-smooth spaces, such as $C^q[0,1], q=0,1,...$. As an example of applications, we show how to reconstruct a map from its derivatives at a point, for spaces possessing blid maps. We also show how blid maps can assist in finding global solutions to cohomological equations having linear transformation of argument.

math.FA↗

arXiv2017

Extension of local smooth maps of Banach spaces

It is known that smooth bump functions are absent in the majority of infinite-dimensional Banach spaces. This is an obstacle in the development of local analysis, in particular in the questions of extending local maps onto the whole space. We suggest an approach that substitutes bump functions with special maps, which we call K-maps. It allows us to extend smooth local maps from non-smooth spaces, such as $C^q[0,1], q=0,1,...$. We also prove the Borel lemma for spaces possessing K-maps.

math.FA↗