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Differential Forms
Published in James K. Peterson, Basic Analysis II, 2020
For the given x0and a given f, let A = f−1(d) where d = f(x0). If y∈ A, then because y is an interior point, there is a neighborhood Bs(y) contained in U. Hence, Bs(y) ∩ A is an open set with zinA for all z in Bs(y) ∩ A. This shows y is an interior point of A and so A is open. Now let B = U \ A. Pick any y in B. Then f(y)≠d and since f is locally constant at y, we know there is a neighborhood about y with corresponding function values not in A. Hence, they are in B which shows y is an interior point of B. We conclude B is an open set and we have written U = A ∪ B with A and B disjoint and open. This shows U is not connected. This is a contradiction and so we must conclude B is empty and U = f−1(d). Thus any such f must be constant on all of U.
Optical Regeneration
Published in Mário F. S. Ferreira, Optical Signal Processing in Highly Nonlinear Fibers, 2020
where f is the coupler power-splitting fraction. This shows that the NOLM can be used to suppress the zero-level noise, which becomes more effective for a coupling ratio near 0.5 [35,36]. On the other hand, the NOLM can also suppress the amplitude fluctuations of the input signal if its power is set at one of the values at which the output power is locally constant. In a 2004 experiment, 42.66 Gb/s signal transmission over 10,000 km with a Q factor of 11 dB in a five-channel wavelength division multiplexing system was demonstrated with NOLM-based 2R regenerators inserted every 240 km [37]. Noise and pedestal suppression have been also demonstrated using the nonlinear amplifying loop mirror (NALM) mechanism [36,38,39].
Partial difference Equations on Graphs for Local and Nonlocal Image Processing
Published in Olivier Lézoray, Leo Grady, Image Processing and Analysis with Graphs, 2012
Abderrahim Elmoataz, Olivier Lézoray, Vinh-Thong Ta, Sébastien Bougleux
The divergence operator, defined by −dw∗, measures the net outflow of a function of ℋ(ℰ) at each vertex of the graph. Each function H∈ℋ(ℰ) has a null divergence over the entire set of vertices: ∑υi∈V(dw∗H)(υi)=0. Another general definition of the difference operator has been proposed by Zhou [21] as (dwf)(υi,υj)=wij(f(υj)deg(υj)−f(υi)deg(υi)). However, the latter operator is not null when the function f is locally constant and its adjoint is not null over the entire set of vertices [22, 23, 24].
Continuity of isomorphisms applied to rigidity problems of entropy spectra
Published in Dynamical Systems, 2023
Let and let . Let λ be an eigenvalue of and a left eigenvector of λ. We define the 1-locally constant function by . Then, , and hence, λ is an eigenvalue of .The Perron root of is equal to the eigenvalue of .
A theoretical study of optical transport properties’ considerations in three variants of one-dimensional Fibonacci-based plasma photonic crystals
Published in Philosophical Magazine, 2022
M. Solaimani, M. Ghalandari, M. Salmani
The Fibonacci sequence-based function can be created using piecewise constant functions. A function is piecewise constant if it is locally constant in connected regions separated by many intervals. The Fibonacci numbers F (i) for generation i obey the relation: F (i) = F (i-1) + F (i-2). The initial conditions are F (0) = 1 and F (1) = 1. In this work, three types of Fibonacci structures are employed. In type (1) structure, the well and barrier thicknesses, independently change based on the Fibonacci sequence. In the type (2) system, the thicknesses of the barriers vary based on the Fibonacci sequence. In type (3) structure, the well and barrier widths change according to the Fibonacci sequence dependently. These types of structures have been illustrated in Figure 1. The layers are alternatively created from plasma and dielectric media. The types of structures only determine the widths of the layers. In our study, the well and barrier layers are created from dielectric and plasma media, respectively. The wells and barrier are the lower and higher refractive index media, respectively. The widths of the wells and barriers can just be obtained using a computer programme. We fix the total system length, create the layers’ width ratios according to the structure type, and calculate each layer’s width.
The K-property for subadditive equilibrium states
Published in Dynamical Systems, 2021
We now introduce two classes of cocycles appearing in Theorems B, C, and D. First is the class of locally constant cocycles. A locally constant cocycle is a cocycle whose generator is locally constant. If is locally constant, then from the compactness of , there exists such that depends only on the word for every . For any locally constant -valued function on , there exists a recoding of to another subshift of finite type such that is carried to a -valued function on depending only on the 0th entry of .