New-type Hoeffding’s inequalities and application in tail bounds

OMS-Vol. 5 (2021), Issue 1, pp. 248 – 261 Open Access Full-Text PDF
Pingyi Fan
Abstract: It is well known that Hoeffding’s inequality has a lot of applications in the signal and information processing fields. How to improve Hoeffding’s inequality and find the refinements of its applications have always attracted much attentions. An improvement of Hoeffding inequality was recently given by Hertz [1]. Eventhough such an improvement is not so big, it still can be used to update many known results with original Hoeffding’s inequality, especially for Hoeffding-Azuma inequality for martingales. However, the results in original Hoeffding’s inequality and its refined version by Hertz only considered the first order moment of random variables. In this paper, we present a new type of Hoeffding’s inequalities, where the high order moments of random variables are taken into account. It can get some considerable improvements in the tail bounds evaluation compared with the known results. It is expected that the developed new type Hoeffding’s inequalities could get more interesting applications in some related fields that use Hoeffding’s results.
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Qualitative analysis of solutions for a parabolic type Kirchhoff equation with logarithmic nonlinearity

ODAM-Vol. 4 (2021), Issue 2, pp. 1 – 10 Open Access Full-Text PDF
Erhan Pişkin, Tuğrul Cömert
Abstract: In this work, we investigate the initial boundary-value problem for a parabolic type Kirchhoff equation with logarithmic nonlinearity. We get the existence of global weak solution, by the potential wells method and energy method. Also, we get results of the decay and finite time blow up of the weak solutions.
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A hybrid method for solution of linear Volterra integro-differential equations (LVIDES) via finite difference and Simpson’s numerical methods (FDSM)

OMA-Vol. 5 (2021), Issue 1, pp. 69 – 75 Open Access Full-Text PDF
Bashir Danladi Garba, Sirajo Lawan Bichi
Abstract: In this paper, a hybrid of Finite difference-Simpson’s approach was applied to solve linear Volterra integro-differential equations. The method works efficiently great by reducing the problem into a system of linear algebraic equations. The numerical results shows the simplicity and effectiveness of the method, error estimation of the method is provided which shows that the method is of second order convergence.
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Trinomial equation: the Hypergeometric way

OMS-Vol. 5 (2021), Issue 1, pp. 236 – 247 Open Access Full-Text PDF
Daniele Ritelli, Giulia Spaletta
Abstract: This paper is devoted to the analytical treatment of trinomial equations of the form \(y^n+y=x,\) where \(y\) is the unknown and \(x\in\mathbb{C}\) is a free parameter. It is well-known that, for degree \(n\geq 5,\) algebraic equations cannot be solved by radicals; nevertheless, roots are described in terms of univariate hypergeometric or elliptic functions. This classical piece of research was founded by Hermite, Kronecker, Birkeland, Mellin and Brioschi, and continued by many other Authors. The approach mostly adopted in recent and less recent papers on this subject (see [1,2] for example) requires the use of power series, following the seminal work of Lagrange [3]. Our intent is to revisit the trinomial equation solvers proposed by the Italian mathematician Davide Besso in the late nineteenth century, in consideration of the fact that, by exploiting computer algebra, these methods take on an applicative and not purely theoretical relevance.
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Study of a unit power-logarithmic distribution

OMS-Vol. 5 (2021), Issue 1, pp. 218 – 235 Open Access Full-Text PDF
Christophe Chesneau
Abstract: This article proposes a new unit distribution based on the power-logarithmic scheme. The corresponding cumulative distribution function is defined by a special ratio of power and logarithmic functions that is dependent on one parameter. We show that this function benefits from great flexibility characterized by a large selection of convex and concave shapes. The other key functions are determined and studied. In particular, we show that the probability density function may take on different decreasing or U shapes, and the hazard rate function has a wide panel of U shapes. These functional capabilities are rare for a one-parameter unit distribution. In addition, we prove certain stochastic order results, provide the expression of the quantile function via the Lambert function, some interesting distributional results, and give simple expressions for the ordinary moments, mean, variance, skewness, kurtosis, moment generating function and incomplete moments. Subsequently, a basic statistical approach is described, to show how the new distribution can be applied in a data analysis scenario. Finally, complementary mathematical findings are presented, yielding new integrals linked to the Euler constant via some well-known moments properties.
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On some iterative methods with frozen derivatives for solving equations

OMS-Vol. 5 (2021), Issue 1, pp. 209 – 217 Open Access Full-Text PDF
Samundra Regmi, Christopher Argyros, Ioannis K. Argyros, Santhosh George
Abstract: We determine a radius of convergence for an efficient iterative method with frozen derivatives to solve Banach space defined equations. Our convergence analysis use \(\omega-\) continuity conditions only on the first derivative. Earlier studies have used hypotheses up to the seventh derivative, limiting the applicability of the method. Numerical examples complete the article.
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Dye ability of henna dye towards cotton fabrics in comparison with reactive dye by following reactive dyeing procedure

EASL-Vol. 4 (2021), Issue 2, pp. 29 – 35 Open Access Full-Text PDF
Pranay Dutta, Md. Razaya Rabbi, Mohammad Abu Sufian, Shahnaz Mahjebin
Abstract: Although synthetic dyes are commonly used, natural dyes are still being utilized and used to improve their intrinsic aesthetic properties as the main material for the body’s beauty. For example, research results have shown that henna plant leaves comprise dye together with other additives. This provides a hint that if color from henna is properly studied, it can be used not only as body decoration but may also have fiber-substrates affinity. This paper explores the dyeing possibility-the ability to dye and the fastness qualities of henna dye extracted from henna leaves on cotton fabric compared to reactive dyeing using the same dyeing technique as reactive dyeing. Also, color fastness tests have been performed according to the ISO test methods. The implications of henna dye have been shown to have poor to moderate dyeing capability towards cotton fabrics as opposed to the reactive dyes when henna dyeing is accompanied by reactive dyeing. Similarly, henna dye demonstrated satisfactory properties of fastness as opposed to reactive dye. For henna dye with 50% shade, it gives an outstanding color tone with a good level of coloration. Taken into account the ability to dye and the fastness of color, the dyeing of matured henna leaves is equally advantageous to the dyeing of cotton fabrics.
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