Setting Conditions Research Articles

Representations of hermite processes using local time of intersecting stationary stable regenerative sets

AbstractHermite processes are a class of self-similar processes with stationary increments. They often arise in limit theorems under long-range dependence. We derive new representations of Hermite processes with multiple Wiener–Itô integrals, whose integrands involve the local time of intersecting stationary stable regenerative sets. The proof relies on an approximation of regenerative sets and local times based on a scheme of random interval covering.

The Surface Diffusion Flow with Elasticity in Three Dimensions

We establish the short-time existence of a smooth solution to the surface diffusion equation with an elastic term and without an additional curvature regularization in three space dimensions. We also prove the asymptotic stability of strictly stable stationary sets.

Mutual stationarity and singular Jonsson cardinals

We prove that if the sequence $$\langle k_n:1 \le n < \omega\rangle$$ contains a so-called gap then the sequence $$\langle S^{\aleph_n}_{\aleph_{k_n}}:1 \le n < \omega\rangle$$ of stationary sets is not mutually stationary, provided that $$k_n<n$$ for every $$n \in \omega$$ . We also prove a sufficient condition for being singular Jonsson cardinals.

Time series prediction using deep echo state networks

Artificial neural networks have been used for time series modeling and forecasting in many domains. However, they are often limited in their handling of nonlinear and chaotic data. More recently, reservoir-based recurrent neural net systems, most notably echo state networks (ESN), have made substantial improvements for time series modeling. Their shallow nature lends themselves to an efficient training method, but has limitations on nonstationary, nonlinear chaotic time series, particularly large, multidimensional time series. In this paper, we propose a novel approach for forecasting time series data based on an additive decomposition (AD) applied to the time series as a preprocessor to a deep echo state network. We compare the performance of our method, AD-DeepESN, on popular neural net architectures used for time series prediction. Stationary and nonstationary data sets are used to evaluate the performance of the methods. Our results are compelling, demonstrating that AD-DeepESN has superior performance, particularly on the most challenging time series that exhibit non-stationarity and chaotic behavior compared to existing methods.

A functional non-central limit theorem for multiple-stable processes with long-range dependence

A functional limit theorem is established for the partial-sum process of a class of stationary sequences which exhibit both heavy tails and long-range dependence. The stationary sequence is constructed using multiple stochastic integrals with heavy-tailed marginal distribution. Furthermore, the multiple stochastic integrals are built upon a large family of dynamical systems that are ergodic and conservative, leading to the long-range dependence phenomenon of the model. The limits constitute a new class of self-similar processes with stationary increments. They are represented by multiple stable integrals, where the integrands involve the local times of intersections of independent stationary stable regenerative sets.

On the growth of nonlocal catenoids

As well known, classical catenoids in \mathbb R^3 possess logarithmic growth at infinity. In this note we prove that the case of nonlocal minimal surfaces is significantly di¤erent, and indeed all nonlocal catenoids must grow at least linearly. More generally, we prove that stationary sets for the nonlocal perimeter functional which grow sublinearly at infinity are necessarily half-spaces.

Wavelet and multiple linear regression analysis for identifying factors affecting particulate matter PM2.5 in Mumbai City, India

Purpose The purpose of this paper is to identify significant factors of environmental variables and pollutants that have an effect on PM2.5 through wavelet and regression analysis. Design/methodology/approach In order to provide stable data set for regression analysis, multiresolution analysis using wavelets is conducted. For the sampled data, multicollinearity among the independent variables is removed by using principal component analysis and multiple linear regression analysis is conducted using PM2.5 as a dependent variable. Findings It is found that few pollutants such as NO2, NOx, SO2, benzene and environmental factors such as ambient temperature, solar radiation and wind direction affect PM2.5. The regression model developed has high R2 value of 91.9 percent, and the residues are stationary and not correlated indicating a sound model. Research limitations/implications The research provides a framework for extracting stationary data and other important features such as change points in mean and variance, using the sample data for regression analysis. The work needs to be extended across all areas in India and for various other stationary data sets there can be different factors affecting PM2.5. Practical implications Control measures such as control charts can be implemented for significant factors. Social implications Rules and regulations can be made more stringent on the factors. Originality/value The originality of this paper lies in the integration of wavelets with regression analysis for air pollution data.

Square below a non-weakly compact cardinal

In his seminal paper introducing the fine structure of L, Jensen (Ann Math Log 4:229–308, 1972) proved that under $$V=L$$ any regular cardinal that reflects stationary sets is weakly compact. In this paper we give a new proof of Jensen’s result that is straight-forward and accessible to those without a knowledge of Jensen’s fine structure theory. The proof here instead uses hyperfine structure, a very natural and simpler alternative to fine structure theory introduced by Friedman and Koepke (Bull Symb Log 3:453–468, 1997).

The consistency strength of hyperstationarity

We introduce the large-cardinal notions of [Formula: see text]-greatly-Mahlo and [Formula: see text]-reflection cardinals and prove (1) in the constructible universe, [Formula: see text], the first [Formula: see text]-reflection cardinal, for [Formula: see text] a successor ordinal, is strictly between the first [Formula: see text]-greatly-Mahlo and the first [Formula: see text]-indescribable cardinals, (2) assuming the existence of a [Formula: see text]-reflection cardinal [Formula: see text] in [Formula: see text], [Formula: see text] a successor ordinal, there exists a forcing notion in [Formula: see text] that preserves cardinals and forces that [Formula: see text] is [Formula: see text]-stationary, which implies that the consistency strength of the existence of a [Formula: see text]-stationary cardinal is strictly below a [Formula: see text]-indescribable cardinal. These results generalize to all successor ordinals [Formula: see text] the original same result of Mekler–Shelah [A. Mekler and S. Shelah, The consistency strength of every stationary set reflects, Israel J. Math. 67(3) (1989) 353–365] about a [Formula: see text]-stationary cardinal, i.e. a cardinal that reflects all its stationary sets.

On singular stationarity I (mutual stationarity and ideal-based methods)

We study several ideal-based constructions in the context of singular stationarity. By combining methods of strong ideals, supercompact embeddings, and Prikry-type posets, we obtain three consistency results concerning mutually stationary sets, and answer a well-known question of Foreman and Magidor ([7]) concerning stationary sequences on the first uncountable cardinals, ℵn, 1≤n<ω.

ON SINGULAR STATIONARITY II (TIGHT STATIONARITY AND EXTENDERS-BASED METHODS)

AbstractWe study the notion of tightly stationary sets which was introduced by Foreman and Magidor in [8]. We obtain two consistency results showing that certain sequences of regular cardinals ${\langle {\kappa _n}\rangle _{n &lt; \omega }}$ can have the property that in some generic extension, every ground-model sequence of fixed-cofinality stationary sets ${S_n} \subseteq {\kappa _n}$ is tightly stationary. The results are obtained using variations of the short-extenders forcing method.

On Disjoint Stationary Sets in Topological Spaces

On Disjoint Stationary Sets in Topological Spaces

On the existence of skinny stationary subsets

Matsubara–Usuba [13] introduced the notion of skinniness and its variants for subsets of Pκλ and showed that the existence of skinny stationary subsets of Pκλ is related to large cardinal properties of ideals over Pκλ and to Jensen's diamond principle on λ. In this paper, we study the existence of skinny stationary sets in more detail.

On the Numbers of Connected Components in the Solution Sets of Polynomial Vector Variational Inequalities

In this paper, we establish explicit upper bounds for the number of connected components in the proper Pareto solution sets and the weak Pareto solution sets of polynomial vector variational inequalities. Consequently, upper bounds for the numbers of connected components in the stationary point sets, the proper stationary point sets, and the weak Pareto solution sets of polynomial vector optimization problems are obtained.

Tracking Sparse Linear Classifiers.

In this paper, we investigate the problem of sparse online linear classification in changing environments. We first analyze the tracking performance of standard online linear classifiers, which use gradient descent for minimizing the regularized hinge loss. The derived shifting bounds highlight the importance of choosing appropriate step sizes in the presence of concept drifts. Notably, we show that a better adaptability to concept drifts can be achieved using constant step sizes rather than the state-of-the-art decreasing step sizes. Based on these observations, we then propose a novel sparse approximated linear classifier, called sparse approximated linear classification (SALC), which uses a constant step size. In essence, SALC simply rounds small weights to zero for achieving sparsity and controls the truncation error in a principled way for achieving a low tracking regret. The degree of sparsity obtained by SALC is continuous and can be controlled by a parameter which captures the tradeoff between the sparsity of the model and the regret performance of the algorithm. Experiments on nine stationary data sets show that SALC is superior to the state-of-the-art sparse online learning algorithms, especially when the solution is required to be sparse; on seven groups of nonstationary data sets with various total shifting amounts, SALC also presents a good ability to track drifts. When wrapped with a drift detector, SALC achieves a remarkable tracking performance regardless of the total shifting amount.

The approachability ideal without a maximal set

We develop a forcing poset with finite conditions which adds a partial square sequence on a given stationary set, with adequate sets of models as side conditions. We then develop a kind of side condition product forcing for simultaneously adding partial square sequences on multiple stationary sets. We show that certain quotients of such forcings have the ω1-approximation property. We apply these ideas to prove, assuming the consistency of a greatly Mahlo cardinal, that it is consistent that the approachability ideal I[ω2] does not have a maximal set modulo clubs.

Real time high cycle fatigue estimation algorithm and load history monitoring for vehicles by the use of frequency domain methods

Abstract Real time high cycle fatigue estimation problem for vehicles is examined by the use of frequency domain methods. The purpose was twofold: monitoring of fatigue damage and tracking of load history in real time. Firstly, power spectral density functions (PSDs) of acceleration measurement at a selected location are calculated in a piecewise manner by dividing the acceleration-time history into pieces. Following, Frequency Response Functions (FRF’s), whose outputs are the absolute maximum principal stress values at selected components, are calculated by finite element methods to account for multi-axial stress state in fatigue life estimations. Then, fatigue damage intensity at selected output locations is estimated using the FRF results. To this end, the following frequency domain fatigue estimation methods (FDFEMs) proposed for Gaussian and stationary data sets are applied to the selected components of a heavy duty truck: narrow-band approximation, Tovo and Benasciutti, Zhao and Baker, Dirlik and Tovo’s α 0.75 methods. Gaussianity and stationarity of acceleration-time data set used to estimate fatigue life of a component is checked to ensure the validity of FDFEMs. Numerical results are compared with experimental fatigue lives and damage calculations in time domain made by the combination of rainflow counting and Miner-Palmgren rules. There are two difficulties in implementing this approach using on-board equipment in real time such as overcoming the limited memory to store data sets and completing the computations sufficiently fast. To overcome them, the proposed approach is implemented in a piecewise manner and associated normalized PSDs are updated accordingly. Then, spectral moments and damage intensities are calculated in frequency domain. Implementation of the proposed approach is described in detail and numerical results are presented. It is shown that the proposed approach is able to predict the fatigue damage accurately and can keep track of loading conditions in real time.

Towards justifying unsupervised stationary decisions for geostatistical modeling: Ensemble spatial and multivariate clustering with geomodeling specific clustering metrics

The subdivision of samples into stationary sets is one of the first decisions in a resource modeling workflow where geologically and statistically related samples are grouped for further geostatistical modeling. Unsupervised learning algorithms (clustering) with modifications to consider spatial correlation have several benefits in the decision of stationarity, they are: automatic and repeatable; assess uncertainty; and provide a framework for checking existing groupings. However, subjective parameterization remains a critical limitation for the application of spatial clustering in geostatistical workflows. In this work, two main extensions to the current state of research are proposed: 1) a combined spatial-multivariate metric that describes cluster quality in both multivariate and Cartesian space by measuring multivariate compactness and spatial contiguity; and 2) a novel random-path spatial-multivariate ensemble clustering algorithm to reduce the reliance on subjective clustering parameterization. The metrics developed in this work quantify clustering quality based multivariate and spatial properties. Moreover, the proposed clustering algorithm allows the user to control the spatial contiguity and multivariate compactness of the final clusters by modifying a single parameter. The clustering algorithm is introduced and demonstrated on a synthetic test dataset, then further application to more complex datasets are explored to demonstrate the clustering properties and simplicity of tuning the algorithm to prefer spatially contiguous clusters or multivariate compactness. The proposed algorithm outperforms other clustering algorithms for the datasets tested based on the metrics developed.

The Surface Diffusion Flow with Elasticity in the Plane

In this paper we prove short-time existence of a smooth solution in the plane to the surface diffusion equation with an elastic term and without an additional curvature regularization. We also prove the asymptotic stability of strictly stable stationary sets.

Bubbles and droplets in a singular limit of the FitzHugh–Nagumo system

The FitzHugh–Nagumo system gives rise to a geometric variational problem, when its parameters take values in a particular range. A stationary set of the variational problem satisfies an Euler–Lagrange equation that involves the curvature of the boundary of the set and a nonlocal term that inhibits unlimited growth and spreading. The nonlocal term is the solution of a nonhomogeneous modified Helmholtz equation with the characteristic function of the stationary set as the inhomogeneous term. Two types of stationary sets are studied: disc shaped stationary sets in the plane, termed bubbles, and unions of perturbed small discs, termed droplet assemblies, in bounded domains. A complete description of the existence and the stability of bubbles is established. Depending on the parameters, there may be zero, one, two, or even three bubbles. Droplet assemblies are constructed by a reduction argument. Each droplet in a stationary assembly is close in size and shape to the corresponding bubble in the plane. The locations of the droplets in the assembly are determined by the Green’s function of the modified Helmholtz equation.