On a Subclass of Analytic Functions That Are Starlike with Respect to a Boundary Point Involving Exponential Function

In the present exploration, the authors define and inspect a new class of functions that are regular in the unit disc D ≔ ς ∈ ℂ : ς < 1 , by using an adapted version of the interesting analytic formula offered by Robertson (unexploited) for starlike functions with respect to a boundary point by subordinating to an exponential function. Examples of some new subclasses are presented. Initial coefficient estimates are specified, and the familiar Fekete-Szegö inequality is obtained. Differential subordinations concerning these newly demarcated subclasses are also established.

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.

A new non-conformable derivative based on Tsallis’s q- exponential function

Neste artigo, uma nova derivada do tipo local é proposta e algumas propriedades básicas são estudadas. Esta nova derivada satisfaz algumas propriedades do cálculo de ordem inteira, por exemplo linearidade, regra do produto, regra do quociente e a regra da cadeia. Devido à função exponencial generalizada de Tsallis, podemos estender alguns dos resultados clássicos, a saber: teorema de Rolle, teorema do valor médio. Apresentamos a correspondente Q-integral a partir da qual surgem novos resultados. Especificamente, generalizamos a propriedade de inversão do teorema fundamental do cálculo e provamos um teorema associado à integração clássica por partes. Finalmente, apresentamos uma aplicação envolvendo equações diferenciais lineares por meio da Q-derivada.

EXCEL-orientated procedure for calculating the values of special functions with interval argument assigned on the hyperbolic form

Aim of the work is to propose the main terms of the EXCEL-orientated procedures for calculating the values of elementary and special functions with interval argument that is assigned on the hyperbolic form. The results of the work. The methods of presenting the interval values in the hyperbolic form and the rules of addition, subtraction, multiplication, and division of this values were considered. The procedures of calculating the function values, whose arguments can be degenerate or interval values were described. Namely, the direct and the reverse functions of the linear trigonometry, the direct and the reverse functions of the hyperbolic trigonometry, exponential function, arbitrary exponential function and power function, Gamma-function, incomplete Gamma-function, digamma-function, trigamma-function, tetragamma-function, pentagamma-function, Beta-function and its partial derivatives, integral exponential function, integral logarithm, dilogarithm, Frenel integrals, sine integral, cosine integral, hyperbolic sine integral, hyperbolic cosine integral. The basic terms of the EXCEL-orientated procedures for calculating the values of elementary and special functions with interval argument that is assigned on the hyperbolic form were proposed. The numerical examples were provided, that illustrate the application of the proposed methods.

ON MODIFIED MOMENT-TYPE OPERATORS

We propose a modification for moment-type operators in order to preserve the exponential function $e^{2cx}$ with $c>0$ on real axis. First, we present moment identities. Then, we prove two weighted convergence theorems. Finally, we present a Voronovskaya-type theorem for the new operators.

Series expansion of the Gamma function and its reciprocal

In this paper we give representations for the coefficients of the Maclaurin series for \Gamma(z+1) and its reciprocal (where \Gamma is Euler’s Gamma function) with the help of a differential operator \mathfrak{D}, the exponential function and a linear functional ^{*} (in Theorem 3.1). As a result we obtain the following representations for \Gamma (in Theorem 3.2): \begin{align*} \Gamma(z+1) & = \big(e^{-u(x)}e^{-z\mathfrak{D}}[e^{u(x)}]\big)^{*}, \\ \big(\Gamma(z+1)\big)^{-1} & = \big(e^{u(x)}e^{-z\mathfrak{D}}[e^{-u(x)}]\big)^{*}. \end{align*} Theorem 3.1 and Theorem 3.2 are our main results. With the help of the first theorem we give our approach for finding the coefficients of Maclaurin series for \Gamma(z+1) and its reciprocal in an explicit form.

ASSESSMENT OF THE TRANSPORT INTENSITY OF THE EUROPEAN ECONOMY

The excessive increase in transport intensity is one of the negative impacts on the economy. The costs borne due to transport activities are indirectly expressed by the volume of carriages (in tons) and by the scope of transport activity (in ton-kilometers). The result of social and economic activities are global product values and national incomes. This article shows the research on transport activity expressed through transport activity (in ton-kilometers) for all means of transport in total, the results of social and economic activities expressed using the gross domestic product, as well as shaping transport intensity of national economies in selected European countries. The analysis of the course of the exponential function curves, as well as polynomial curves has been carried out, and conclusions have been formulated on their bases.

Laboratory studies of suspended particle volume concentration in atmosphere using a smartphone

The article presents the results of laboratory studies of the effect of volume concentration of suspended particles in contrast to the luminous slits image obtained by smartphone cameras of SAMSUNG Galaxy A3 and Honor 8 Lite. Experimentally it was found that a pattern of change in contrast to the luminous slits image from the volumetric concentration of suspended particles appears under ambient light. The pattern of contrast change can be expressed by an exponential function. The correlation coefficient is 0.97. Cigarette smoke was used as suspended particles.