Wave packet transform and fractional wave packet transform of rapidly decreasing functions

The continuity of wave packet transform and inverse wave packet transform is proved in the suitable Schwartz space and extended to its corresponding dual space of tempered distribution. The consistency, linearity and continuity of the transform with respect to [Formula: see text] topology are proved in this distribution space. Further, the continuity of the fractional wave packet transform and its inverse in the above space is proved. The examples of generalized fractional wave packet transform of certain distributions are given.

Investigating climatic changes of the wind regime over Western Iran

Objective The aim of the present study was to reveal changes in the wind regime by investigating wind-speed data from meteorological stations in western Iran and comparing them in the last three decades (1986–2015). Results Two main groups of daily cycles were identified; one group with a single peak and one group with two or more peaks. Using spectral decomposition technique, it was revealed that the heterogeneity observed in the area in terms of altitude and topography results in differences in the density of the spectra with similar frequencies. Two main daily cycles were also identified for each station. Although there were low frequencies, the intensity of the waves at the examined stations was the consequence of the interaction between the frequency, period, and distribution space. By evaluating harmonics in the area, it was revealed that the variance of the first harmonic is maximized in the south and southwest, while the variance of the second harmonic is maximized in the north and northwest. The positive value ​​of the trend in the first harmonic indicated that the trend of the variance for the first harmonic has increased in the central and eastern parts and has decreased in the northern and western parts.

A Probabilistic Bag-to-Class Approach to Multiple-Instance Learning

Multi-instance (MI) learning is a branch of machine learning, where each object (bag) consists of multiple feature vectors (instances)—for example, an image consisting of multiple patches and their corresponding feature vectors. In MI classification, each bag in the training set has a class label, but the instances are unlabeled. The instances are most commonly regarded as a set of points in a multi-dimensional space. Alternatively, instances are viewed as realizations of random vectors with corresponding probability distribution, where the bag is the distribution, not the realizations. By introducing the probability distribution space to bag-level classification problems, dissimilarities between probability distributions (divergences) can be applied. The bag-to-bag Kullback–Leibler information is asymptotically the best classifier, but the typical sparseness of MI training sets is an obstacle. We introduce bag-to-class divergence to MI learning, emphasizing the hierarchical nature of the random vectors that makes bags from the same class different. We propose two properties for bag-to-class divergences, and an additional property for sparse training sets, and propose a dissimilarity measure that fulfils them. Its performance is demonstrated on synthetic and real data. The probability distribution space is valid for MI learning, both for the theoretical analysis and applications.

Multiplication of Distributions and Algebras of Mnemofunctions

In this paper, we discuss methods and approaches for definition of multiplication of distributions, which is not defined in general in the classical theory. We show that this problem is related to the fact that the operator of multiplication by a smooth function is nonclosable in the space of distributions. We give the general method of construction of new objects called new distributions, or mnemofunctions, that preserve essential properties of usual distributions and produce algebras as well. We describe various methods of embedding of distribution spaces into algebras of mnemofunctions. All ideas and considerations are illustrated by the simplest example of the distribution space on a circle. Some effects arising in study of equations with distributions as coefficients are demonstrated by example of a linear first-order differential equation.

Stochastic elastic equation driven by multiplicative multi-parameter fractional noise

In this paper, we consider the stochastic elastic equation driven by multiplicative multiparameter fractional noise. By using the Wiener chaos expansion and undetermined coefficient methods, we obtain the existence and uniqueness of the solution in a distribution space. The asymptotic behavior and the Hölder index of the solution are also estimated.