Bootstrap Empirical Mode Decomposition with Application to CIELAB Color Images

This paper proposes an alternative optimization-based EMD based on the notions of: 1. <i>local mean points </i>that impose mode symmetry via a Tikhonov regularized least-square (RLS) problem, and 2. efficient <i>bootstrap sifting</i> that guarantees asymptotic convergence of the mean envelope to the local mean points, regardless of regularization. Mathematical proof of convergence and a straightforward extension to the 2D-multivariate setting and CIELAB color image sare presented. Performance is demonstrated with a univariate signal and two images. Spectral analysis confirms coordinated feature extraction among image components, and separation of spatial spectra among the intrinsic mode functions.

Bootstrap Empirical Mode Decomposition with Application to CIELAB Color Images

This paper proposes an alternative optimization-based EMD based on the notions of: 1. <i>local mean points </i>that impose mode symmetry via a Tikhonov regularized least-square (RLS) problem, and 2. efficient <i>bootstrap sifting</i> that guarantees asymptotic convergence of the mean envelope to the local mean points, regardless of regularization. Mathematical proof of convergence and a straightforward extension to the 2D-multivariate setting and CIELAB color image sare presented. Performance is demonstrated with a univariate signal and two images. Spectral analysis confirms coordinated feature extraction among image components, and separation of spatial spectra among the intrinsic mode functions.

An Algorithm for Linearizing the Collatz Convergence

The Collatz dynamic is known to generate a complex quiver of sequences over natural numbers for which the inflation propensity remains so unpredictable it could be used to generate reliable proof-of-work algorithms for the cryptocurrency industry; it has so far resisted every attempt at linearizing its behavior. Here, we establish an ad hoc equivalent of modular arithmetics for Collatz sequences based on five arithmetic rules that we prove apply to the entire Collatz dynamical system and for which the iterations exactly define the full basin of attractions leading to any odd number. We further simulate these rules to gain insight into their quiver geometry and computational properties and observe that they linearize the proof of convergence of the full rows of the binary tree over odd numbers in their natural order, a result which, along with the full description of the basin of any odd number, has never been achieved before. We then provide two theoretical programs to explain why the five rules linearize Collatz convergence, one specifically dependent upon the Axiom of Choice and one on Peano arithmetic.

Random additions in urns of integers

Consider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, one at a time and with replacement, and noting the labels. Add a new ball labeled with the sum of the two drawn labels. This model was introduced by Siegmund and Yakir (2005) Ann. Prob.33, 2036 for labels taking values in a finite group, in which case the distribution defined by the urn converges to the uniform distribution on the group. For the urn of integers, the main result of this paper is an exponential limit law. The mean of the exponential is a random variable with distribution depending on the starting configuration. This is a novel urn model which combines multi-drawing and an infinite type of balls. The proof of convergence uses the contraction method for recursive distributional equations.

Solution of a Class of Nonlocal Elliptic BVPs Arising in Fluid Flow: An Iterative Approach

The ultimate goal of this study is to implement a fixed point iterative scheme for the numerical solution of an extended class of nonlinear, nonlocal, elliptic boundary value problems. The method is based on applying well-known fixed point procedures such as Mann’s and Picard’s to a tailored linear integral that is expressed in terms of a Green’s function. A proof of convergence that utilizes the contraction principle is included. A number of test examples are presented to confirm the applicability, high accuracy and efficacy of the iterative process.

Sharp Interface Limit of a Stokes/Cahn–Hilliard System, Part II: Approximate Solutions

We construct rigorously suitable approximate solutions to the Stokes/Cahn–Hilliard system by using the method of matched asymptotics expansions. This is a main step in the proof of convergence given in the first part of this contribution, [3], where the rigorous sharp interface limit of a coupled Stokes/Cahn–Hilliard system in a two dimensional, bounded and smooth domain is shown. As a novelty compared to earlier works, we introduce fractional order terms, which are of significant importance, but share the problematic feature that they may not be uniformly estimated in $$\epsilon $$ ϵ in arbitrarily strong norms. As a consequence, gaining necessary estimates for the error, which occurs when considering the approximations in the Stokes/Cahn–Hilliard system, is rather involved.

A Noise Reduction Method Based on Modified LMS Algorithm of Real time Speech Signals

In real time speech de-noising, adaptive filtering technique with variable length filters are used which is used to track the noise characteristics and through those characteristics the filter equations are selected The main features that attracted the use of the LMS algorithm are low computational complexity, proof of convergence in stationary environment. In this paper, modified LMS algorithm is proposed which is used to denoise real time speech signal. The proposed algorithm is made by combining general LMS algorithm with Diffusion least mean-square algorithm which increase the capabilities of adaptive filtering. The performance parameter calculation shows that the proposed algorithm is effective to de-noise speech signal. A full programming routine written in MATLAB software is provided for replications and further research applications.

Deferred Acceptance with Compensation Chains

In a foundational paper, Gale and Shapley (1962) introduced the deferred acceptance algorithm that achieves a stable outcome in a two-sided matching market by letting one side of the market make proposals to the other side. What happens when both sides of the market can propose? In “Deferred Acceptance with Compensation Chains,” Dworczak answers this question by constructing an equitable version of the Gale–Shapley algorithm in which the sequence of proposers can be arbitrary. The main result of the paper shows that the extended algorithm, equipped with so-called compensation chains, is not only guaranteed to converge in polynomial time to a stable outcome, but—in contrast to the original Gale–Shapley algorithm—achieves all stable matchings (as the sequence of proposers vary). The proof of convergence uses a novel potential function. The algorithm may find applications in settings where both stability and fairness are desirable features of the matching process.