Boundedness of the Maximal Function of the Ornstein-Uhlenbeck semigroup on variable Lebesgue spaces with respect to the Gaussian measure and consequences

The main result of this paper is the proof of the boundedness of the Maximal Function T* of the Ornstein-Uhlenbeck semigroup {Tt}t≥ 0 in Rd, on Gaussian variable Lebesgue spaces Lp(.) (γd); under a condition of regularity on p(.) following [5] and [8]. As an immediate consequence of that result, the Lp(.) (γd)-boundedness of the Ornstein-Uhlenbeck semigroup {Tt}t≥ 0 in Rd is obtained. Another consequence of that result is the Lp(.) (γd)-boundedness of the Poisson-Hermite semigroup and the Lp(.) (γd)- boundedness of the Gaussian Bessel potentials of order β > 0.

Yoga Posture Recognition and Quantitative Evaluation with Wearable Sensors Based on Two-Stage Classifier and Prior Bayesian Network

Currently, with the satisfaction of people’s material life, sports, like yoga and tai chi, have become essential activities in people’s daily life. For most yoga amateurs, they could only learn yoga by self-study, like mechanically imitating from yoga video. They could not know whether they performed standardly without feedback and guidance. In this paper, we proposed a full-body posture modeling and quantitative evaluation method to recognize and evaluate yoga postures to provide guidance to the learner. Back propagation artificial neural network (BP-ANN) was adopted as the first classifier to divide yoga postures into different categories, and fuzzy C-means (FCM) was utilized as the second classifier to classify the postures in a category. The posture data on each body part was regarded as a multidimensional Gaussian variable to build a Bayesian network. The conditional probability of the Gaussian variable corresponding to each body part relative to the Gaussian variable corresponding to the connected body part was used as criterion to quantitatively evaluate the standard degree of body parts. The angular differences between nonstandard parts and the standard model could be calculated to provide guidance with an easily-accepted language, such as “lift up your left arm”, “straighten your right forearm”. To evaluate our method, a wearable device with 11 inertial measurement units (IMUs) fixed onto the body was designed to measure yoga posture data with quaternion format, and the posture database with a total of 211,643 data frames and 1831 posture instances was collected from 11 subjects. Both the posture recognition test and evaluation test were conducted. In the recognition test, 30% data was randomly picked from the database to train BP-ANN and FCM classifiers, and the recognition accuracy of the remaining 70% data was 95.39%, which is highly competitive with previous posture recognition approaches. In the evaluation test, 30% data were picked randomly from subject three, subject four, and subject six, to train the Bayesian network. The probabilities of nonstandard parts were almost all smaller than 0.3, while the probabilities of standard parts were almost all greater than 0.5, and thus the nonstandard parts of body posture could be effectively separated and picked for guidance. We also tested separately the trainers’ yoga posture performance in the condition of without and with guidance provided by our proposed method. The results showed that with guidance, the joint angle errors significantly decreased.

Gaussian variable neighborhood search for the file transfer scheduling problem

This paper presents new modifications of Variable Neighborhood Search approach for solving the file transfer scheduling problem. To obtain better solutions in a small neighborhood of a current solution, we implement two new local search procedures. As Gaussian Variable Neighborhood Search showed promising results when solving continuous optimization problems, its implementation in solving the discrete file transfer scheduling problem is also presented. In order to apply this continuous optimization method to solve the discrete problem, mapping of uncountable set of feasible solutions into a finite set is performed. Both local search modifications gave better results for the large size instances, as well as better average performance for medium and large size instances. One local search modification achieved significant acceleration of the algorithm. The numerical experiments showed that the results obtained by Gaussian modifications are comparable with the results obtained by standard VNS based algorithms, developed for combinatorial optimization. In some cases Gaussian modifications gave even better results.

DISTANCE-k GRAPHS OF HYPERCUBE AND q-HERMITE POLYNOMIALS

We will prove that some weighted graphs on the distance-k graph of hypercubes approximate the q-Hermite polynomial of a q-gaussian variable by providing an appropriate matrix model.

Stability and heavy traffic results for the general bulk queue

In this paper we prove that the limiting distribution of the general bulk queue exists and is independent of the initial conditions if and only if the traffic intensity is less than one. We further generalize the following heavy traffic results of the GI/G/1 model to the general bulk queue model. When ρ > 1 or ρ = 1 the waiting time is distributed approximately as a Gaussian variable and the absolute value of a Gaussian variable, respectively. The exponential approximation is derived from the Wiener–Hopf matrix equations established in a previous paper while the unstable case ρ ≧ 1 is treated by means of functional central limit theorems for mixing processes.