Halanay inequality involving Caputo-Hadamard fractional derivative and application

A Halanay inequality with distributed delay of non-convolution type is considered. We establish a decay of solutions as a Mittag-Leffler function composed with a logarithmic function. A general sufficient condition is found and a large class of admissible retardation kernels is provided. This needs the preparation of several lemmas on properties of the Hadamard derivative and some basic fractional differential problems with this kind of derivative. The obtained result is then applied to a Hopfield neural network system to discuss its stability.

Boundary-Aware Hashing for Hamming Space Retrieval

Hamming space retrieval is a hot area of research in deep hashing because it is effective for large-scale image retrieval. Existing hashing algorithms have not fully used the absolute boundary to discriminate the data inside and outside the Hamming ball, and the performance is not satisfying. In this paper, a boundary-aware contrastive loss is designed. It involves an exponential function with absolute boundary (i.e., Hamming radius) information for dissimilar pairs and a logarithmic function to encourage small distance for similar pairs. It achieves a push that is bigger than the pull inside the Hamming ball, and the pull is bigger than the push outside the ball. Furthermore, a novel Boundary-Aware Hashing (BAH) architecture is proposed. It discriminatively penalizes the dissimilar data inside and outside the Hamming ball. BAH enables the influence of extremely imbalanced data to be reduced without up-weight to similar pairs or other optimization strategies because its exponential function rapidly converges outside the absolute boundary, making a huge contrast difference between the gradients of the logarithmic and exponential functions. Extensive experiments conducted on four benchmark datasets show that the proposed BAH obtains higher performance for different code lengths, and it has the advantage of handling extremely imbalanced data.

Predicting Logarithm of Real Numbers to any Real Number Base

This paper aims to propose an algorithm that can predict the values of logarithmic function for any domain and for any bases with minimum possible error using basic mathematical operations. It talks about a univariate quadratic function that overlaps with the graph of the common logarithmic function to some extent, and using this resemblance to fullest advantage. It proposes a formula and certain algorithms based on the same formula.

Coseismic and postseismic deformation from the 2007 Bengkulu earthquake based on GPS Data

On September 12, 2007, a M8.5 megathrust earthquake occurred along the Sunda trench near Bengkulu, West Sumatra. GPS data in Sumatra have indicated the coseismic and postseismic deformations resulting from this earthquake. Our estimate of coseismic displacements suggests that the earthquake displaced up to ~1.8m at GPS stations located north of the epicenter. Moreover, our principal strain estimation in the region suggests that the maximum coseismic extensional strain is ~40 ppm. Our analysis of GPS data in the region suggests that the postseismic decay of the 2007 Bengkulu earthquake was 46 days, estimated using a logarithmic function.

Dependence of death and injury of people in fires on the duration of fires

Introduction. The effectiveness of the actions of the fire protection units depends on their readiness for action, that is, on the state of the forces and means of the station and the ability to perform the main task within the existing tactical capabilities. Problem Statement. When developing mathematical models of the operational activities of the fire department, its effectiveness and the assessment of the readiness of fire protection units to extinguish fires, it is important to study the dependence of the death of people in fires on the duration of the fire. This study is devoted to this task. Theoretical Part. The distribution of fires in the Russian Federation for 2016-2020 by their duration and the dependence of deaths and injuries of people in fires, as well as their ratio to the duration of fires in the Russian Federation for 2016-2020, is studied. It is shown that the average duration of the fire during the studied period was 26 minutes, the median value of the distribution was 18 minutes. The dependences of deaths and injuries of people in fires on the duration of fires are approximated by a logarithmic function. When the duration of the fire is up to 26 minutes. the ratio of the number of people injured in fires to those killed decreases exponentially, but remains stable in the rest of the time range. Conclusions. It is concluded that it is necessary to increase the readiness of the fire department for actions to extinguish fires and rescue people, which includes the development of an information and analytical model of the operational activities of the fire department.

Relation of Some Known Functions in terms of Generalized Meijer G -Functions

The aim of this paper is to prove some identities in the form of generalized Meijer G -function. We prove the relation of some known functions such as exponential functions, sine and cosine functions, product of exponential and trigonometric functions, product of exponential and hyperbolic functions, binomial expansion, logarithmic function, and sine integral, with the generalized Meijer G -function. We also prove the product of modified Bessel function of first and second kind in the form of generalized Meijer G -function and solve an integral involving the product of modified Bessel functions.

An Estimation of Dates Production Function in Karbala Governorate For The Agricultural Season 2020 (Al-jadual Al-gharby District A case Study)

This study aimed to detect the most important factor that affects dates production. About 108 questionary forms collecttted palm orchard farmers in Karbala to estimate the dates production function in Karbala governorate for the agricultural season 2021 (the district of Al-jadual Al-Gharby). The study distributed those formmmsss for about 10% of the total palm orchards in Al-jadual Al-Gharby district of the holy governorate of Karbala. The study used the method of ordinary Least squares (OLS) to estimate the mathematical model of the function. The results showed that the double Logarithmic function in terms of its estimation of the estimated Coefficients by one unit leads to a corresponding change in the produced quality of dates and the same direction by 0.188 0.808) % respectively, and that the capital variable is more in fluently in production than the work variable. As for the total production elasticity (the sum of the partial elasticities of the resources used), which represents returns to scale, it amounted to about (0.996), which indicates a decrease in the return to scale.

A new branch of inflationary speed limits

We present a new mechanism for inflation which exhibits a speed limit on scalar motion, generating accelerated expansion even on a steep potential. This arises from explicitly integrating out the short modes of additional fields coupled to the inflaton ϕ via a dimension six operator, yielding an expression for the effective action which includes a nontrivial (logarithmic) function of (∂ϕ)2. The speed limit appears at the branch cut of this logarithm arising in a large flavor expansion, similarly to the square root branch cut in DBI inflation arising in a large color expansion. Finally, we describe observational constraints on the parameters of this model.

Double Integral of Logarithmic and Quotient Rational Functions Expressed in Terms of the Lerch Function

In this manuscript, the authors derive a double integral whose kernel involves the logarithmic function a polynomial raised to a power and a quotient function expressed it in terms of the Lerch function. All the results in this work are new.