Piecewise linear iterated function systems on the line of overlapping construction

In this paper we consider iterated function systems (IFS) on the real line consisting of continuous piecewise linear functions. We assume some bounds on the contraction ratios of the functions, but we do not assume any separation condition. Moreover, we do not require that the functions of the IFS are injective, but we assume that their derivatives are separated from zero. We prove that if we fix all the slopes but perturb all other parameters, then for all parameters outside of an exceptional set of less than full packing dimension, the Hausdorff dimension of the attractor is equal to the exponent which comes from the most natural system of covers of the attractor.

Identification of system with distributed-order derivatives

The identification problem for system with distributed-order derivative was considered. The order-weight distribution was approximated by piecewise linear functions. Then the discretized order-weight distribution was solved in frequency domain by using the least square technique based on the Moore-Penrose inverse matrix. Finally, five representative numerical examples were used to illustrate the validity of the method. The identification results are satisfactory, especially for the continuous order-weight distributions. In addition, the overlapped Bode magnitude frequency responses from the identified and exact transfer functions imply the effectiveness of the method.

Approximation factor of the piecewise linear functions in Mamdani fuzzy system and its realization process1

Piecewise linear function (PLF) is not only a generalization of univariate segmented linear function in multivariate case, but also an important bridge to study the approximation of continuous function by Mamdani and Takagi-Sugeno fuzzy systems. In this paper, the definitions of the PLF and subdivision are introduced in the hyperplane, the analytic expression of PLF is given by using matrix determinant, and the concept of approximation factor is first proposed by using m-mesh subdivision. Secondly, the vertex coordinates and their changing rules of the n-dimensional small polyhedron are found by dividing a three-dimensional cube, and the algebraic cofactor and matrix norm of corresponding determinants of piecewise linear functions are given. Finally, according to the method of solving algebraic cofactors and matrix norms, it is proved that the approximation factor has nothing to do with the number of subdivisions, but the approximation accuracy has something to do with the number of subdivisions. Furthermore, the process of a specific binary piecewise linear function approaching a continuous function according to infinite norm in two dimensions space is realized by a practical example, and the validity of PLFs to approximate a continuous function is verified by t-hypothesis test in Statistics.

Stability and Bifurcations of Equilibria in Networks with Piecewise Linear Interactions

In this paper, we study equilibria of differential equation models for networks. When interactions between nodes are taken to be piecewise constant, an efficient combinatorial analysis can be used to characterize the equilibria. When the piecewise constant functions are replaced with piecewise linear functions, the equilibria are preserved as long as the piecewise linear functions are sufficiently steep. Therefore the combinatorial analysis can be leveraged to understand a broader class of interactions. To better understand how broad this class is, we explicitly characterize how steep the piecewise linear functions must be for the correspondence between equilibria to hold. To do so, we analyze the steady state and Hopf bifurcations which cause a change in the number or stability of equilibria as slopes are decreased. Additionally, we show how to choose a subset of parameters so that the correspondence between equilibria holds for the smallest possible slopes when the remaining parameters are fixed.

A posteriori error estimates for semilinear optimal control problems

In two and three dimensional Lipschitz, but not necessarily convex, polytopal domains, we devise and analyze a reliable and efficient a posteriori error estimator for a semilinear optimal control problem; control constraints are also considered. We consider a fully discrete scheme that discretizes the state and adjoint equations with piecewise linear functions and the control variable with piecewise constant functions. The devised error estimator can be decomposed as the sum of three contributions which are associated to the discretization of the state and adjoint equations and the control variable. We extend our results to a scheme that approximates the control variable with piecewise linear functions and also to a scheme that approximates the solution to a nondifferentiable optimal control problem. We illustrate the theory with two and three-dimensional numerical examples.

Fast Linear Interpolation

We present fast implementations of linear interpolation operators for piecewise linear functions and multi-dimensional look-up tables. These operators are common for efficient transformations in image processing and are the core operations needed for lattice models like deep lattice networks, a popular machine learning function class for interpretable, shape-constrained machine learning. We present new strategies for an efficient compiler-based solution using MLIR to accelerate linear interpolation. For real-world machine-learned multi-layer lattice models that use multidimensional linear interpolation, we show these strategies run 5-10× faster on a standard CPU compared to an optimized C++ interpreter implementation.

The Reeb Graph Edit Distance is Universal

We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it provides an upper bound to any other stable distance. In contrast, via a specific construction, we show that the interleaving distance and the functional distortion distance on Reeb graphs are not universal.

DEMOGRAPHIC PHASE SPACE OF THE TVER REGION FROM 1989 TO 2020 AND TVER REGION POPULATION

Авторами предложен метод исследования особенностей демографической динамики на основе демографического фазового пространства. Целью работы является анализ динамики народонаселения Тверской области и оценка возможности ее стабилизации в будущем с использованием демографического фазового пространства. Научная новизна работы состоит в применении нового метода исследования особенностей демографической динамики на основе демографического фазового пространства на интересующем промежутке времени. В статье построены кусочно-линейные функции, которые непрерывно аппроксимируют временные ряды численности народонаселения и скорости изменения численности народонаселения Тверской области с 1989 по 2020 гг. Сконструировано демографическое фазовое пространство Тверского региона с 1989 по 2020 гг. Получены аналитические выражения, описывающие динамику тренда численности народонаселения Тверской области. Найдена асимптотическая стабилизация численности народонаселения Тверского региона на уровне 1.0998 млн человек к 2060 году. The article proposes a method for studying the features of demographic dynamics based on the demographic phase space. The aim of the work is to analyze the dynamics of the population of the Tver region and to assess the possibility of its stabilization in the future using the demographic phase space. The scientific novelty of the work consists in the application of a new method for studying the characteristics of demographic dynamics based on the demographic phase space at the specific time interval. The article constructs piecewise linear functions that continuously approximate the time series of the population and the rate of change in the population of the Tver region from 1989 to 2020. The authors present a demographic phase space of the Tver region from 1989 to 2020t. An analytical expression describes the dynamics of the trend in the population of the Tver region. The investigation highlights an asymptotic stabilization of the population of the Tver region at the level of 1.0998 million people by 2060.