An Optimal Estimate for the Anisotropic Logarithmic Potential

This paper introduces the new annulus body to establish the optimal lower bound for the anisotropic logarithmic potential as the complement to the theory of its upper bound estimate which has already been investigated. The connections with convex geometry analysis and some metric properties are also established. For the application, a polynomial dual log-mixed volume difference law is deduced from the optimal estimate.

Fixed time synchronization of a class of chaotic systems based via the saturation control

In this paper, we discuss the fixed time synchronization of a class of chaotic systems based on the backstepping control with disturbances. A new and important fixed time stability theorem is presented. The upper bound estimate formulas of the settling time are also given which are different from the existing results in the literature. Based on the new fixed time stability theorem, a novel saturation controller for the fixed time synchronization a class of chaotic systems is proposed via the backstepping method. Finally, the new chaotic system is taken as an example to illustrate the applicability of the obtained theory.

Fixed time synchronization of a class of chaotic systems based via the saturation control

In this paper, we discuss the fixed time synchronization of a class of chaotic systems based on the backstepping control with disturbances. A new and important fixed time stability theorem is presented. The upper bound estimate formulas of the settling time are also given which are different from the existing results in the literature. Based on the new fixed time stability theorem, a novel saturation controller for the fixed time synchronization a class of chaotic systems is proposed via the backstepping method. Finally, the new chaotic system is taken as an example to illustrate the applicability of the obtained theory.

Fujita modified exponent for scale invariant damped semilinear wave equations

The aim of this paper is to prove a blow-up result of the solution for a semilinear scale invariant damped wave equation under a suitable decay condition on radial initial data. The admissible range for the power of the nonlinear term depends both on the damping coefficient and on the pointwise decay order of the initial data. In addition, we give an upper bound estimate for the lifespan of the solution. It depends not only on the exponent of the nonlinear term and not only on the damping coefficient but also on the size of the decay rate of the initial data.

Second Hankel Determinant for a Certain Subclass of Bi-Close to Convex Functions Defined by Kaplan

In this paper, we consider the class of strongly bi-close-to-convex functions of order α and bi-close-to-convex functions of order β. We obtain an upper bound estimate for the second Hankel determinant for functions belonging to these classes. The results in this article improve some earlier result obtained for the class of bi-convex functions.

Fixed-time synchronization of coupled memristive neural networks with multi-links and application in secure communication

This paper is devoted to investigating the issues of fixed-time synchronization of coupled memristive neural networks with multi-links (MCMNN). Based on the fixed-time stability criterion and the upper bound estimate formula for the settling time, we propose a secure communication scheme. The network with multi-links performance and coupled form increase the complexity of network topology and the unstable of systems, which improve security of communication in the aspect of encrypt the plaintext signal. We design a proper controller and build the Lyapunov function, several effective conditions are obtained to achieve the fixed-time synchronization of MCMNN. Moreover, the settling times can be estimated for fixed-time synchronization without depending on any initial values. Meanwhile, the plaintext signals can be recovered according to the fixed-time stability theorem. Finally, numerical simulations are given to verify the effectiveness of the theoretical results in fixed-time synchronization of MCMNN, and an example of a secure communication scheme is given to show the usability and superiority based on fixed-time stability theorem.

Performance of Regression Models as a Function of Experiment Noise

Background: A challenge in developing machine learning regression models is that it is difficult to know whether maximal performance has been reached on the test dataset, or whether further model improvement is possible. In biology, this problem is particularly pronounced as sample labels (response variables) are typically obtained through experiments and therefore have experiment noise associated with them. Such label noise puts a fundamental limit to the metrics of performance attainable by regression models on the test dataset. Results: We address this challenge by deriving an expected upper bound for the coefficient of determination ( R2) for regression models when tested on the holdout dataset. This upper bound depends only on the noise associated with the response variable in a dataset as well as its variance. The upper bound estimate was validated via Monte Carlo simulations and then used as a tool to bootstrap performance of regression models trained on biological datasets, including protein sequence data, transcriptomic data, and genomic data. Conclusions: The new method for estimating upper bounds for model performance on test data should aid researchers in developing ML regression models that reach their maximum potential. Although we study biological datasets in this work, the new upper bound estimates will hold true for regression models from any research field or application area where response variables have associated noise.

Estimating the Critical Parameter in Almost Stochastic Dominance from Insurance Deductibles

Knowing how small a violation of stochastic dominance rules would be accepted by most individuals is a prerequisite to applying almost stochastic dominance criteria. Unlike previous laboratory-experimental studies, this paper estimates an acceptable violation of stochastic dominance rules with 939,690 real world data observations on a choice of deductibles in automobile theft insurance. We find that, for all policyholders in the sample who optimally chose a low deductible, the upper bound estimate of the acceptable violation ratio is 0.0014, which is close to zero. On the other hand, considering that most decision makers, such as 99% (95%) of the policyholders in the sample, optimally chose the low deductible, the upper bound estimate of the acceptable violation ratio is 0.0405 (0.0732). Our results provide reference values for the acceptable violation ratio for applying almost stochastic dominance rules. This paper was accepted by Manel Baucells, decision analysis.

Blow-up of solutions for a Kirchhoff type equation with variable-exponent nonlinearities

This paper deals with a Kirchhoff type equation with variable exponent nonlinearities, subject to a nonlinear boundary condition. Under appropriate conditions and regarding arbitrary positive initial energy, it is proved that solutions blow up in a finite time. Moreover, we obtain the upper bound estimate of the blow-up time.