Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions

Recent advances in mathematical inequalities suggest that bounds of polynomial-exponential-type are appropriate for evaluating key trigonometric functions. In this paper, we innovate in this sense by establishing new and sharp bounds of the form (1−αx2)eβx2 for the trigonometric sinc and cosine functions. Our main result for the sinc function is a double inequality holding on the interval (0, π), while our main result for the cosine function is a double inequality holding on the interval (0, π/2). Comparable sharp results for hyperbolic functions are also obtained. The proofs are based on series expansions, inequalities on the Bernoulli numbers, and the monotone form of the l’Hospital rule. Some comparable bounds of the literature are improved. Examples of application via integral techniques are given.

Finite-Time H ∞ State Estimation for Markovian Jump Neural Networks with Time-Varying Delays via an Extended Wirtinger’s Integral Inequality

This study investigates the finite-time boundedness for Markovian jump neural networks (MJNNs) with time-varying delays. An MJNN consists of a limited number of jumping modes wherein it can jump starting with one mode then onto the next by following a Markovian process with known transition probabilities. By constructing new Lyapunov–Krasovskii functional (LKF) candidates, extended Wirtinger’s, and Wirtinger’s double inequality with multiple integral terms and using activation function conditions, several sufficient conditions for Markovian jumping neural networks are derived. Furthermore, delay-dependent adequate conditions on guaranteeing the closed-loop system which are stochastically finite-time bounded (SFTB) with the prescribed H ∞ performance level are proposed. Linear matrix inequalities are utilized to obtain analysis results. The purpose is to obtain less conservative conditions on finite-time H ∞ performance for Markovian jump neural networks with time-varying delay. Eventually, simulation examples are provided to illustrate the validity of the addressed method.

Some inequalities for exponentially convex functions on time scales

In this paper, we introduce the notion of exponentially convex functions on time scales and then we establish Hermite-Hadamard type inequalities for this class of functions. As special case, we derive this double inequality in the context of classical notion of exponentially convex functions and convex functions. Moreover, we prove some new integral inequalities for n-times continuously differentiable functions with exponentially convex first Δ-derivative.

One of the methods of root selection in solving trigonometric equations

При подготовке учащихся 10-11 классов к профильному ЕГЭ по математике возникают трудности при отборе корней тригонометрического уравнения, которые принадлежат заданному промежутку. Существует несколько методов отбора корней, но идеального не существует – у каждого из этих методов есть свои слабые стороны. Мы хотим предложить метод, который, на наш взгляд, позволяет учащимся более успешно производить отбор корней в тригонометрических уравнениях. В школьном курсе математики для отбора корней чаще всего используются тригонометрический круг или отбор корней с помощью двойного неравенства, определяющего заданный промежуток. Ситуация в реальных заданиях усложняется тем, что заданный диапазон для значений корней выходит за рамки одного круга. Это обстоятельство усложняет отбор корней на самой окружности, т.к. требует от учащихся более сложной ориентации на ней. Если значение корня не может быть явно записано в радианной мере, то отбор корней с помощью двойного неравенства становится проблематичным. Экзаменационная работа по математике базового уровня состоит из одной части, включающей 20 заданий с кратким ответом. Все задания направлены на проверку освоения базовых умений и практических навыков применения математических знаний в повседневных ситуациях. Ответом к каждому из заданий 1-20 является целое число, конечная десятичная дробь, или последовательность цифр. When preparing students in grades 10-11 for the profile USE in mathematics, there are difficulties in selecting the roots of the trigonometric equation that belong to a given interval. There are several methods of root selection, but there is no perfect one – each of these methods has its own weaknesses. We want to propose a method that, in our opinion, allows students to more successfully select the roots in trigonometric equations. In a school mathematics course, the most common way to select roots is to use a trigonometric circle or to select roots using a double inequality that defines a given interval. The situation in real tasks is complicated by the fact that the specified range for the values of the roots goes beyond one circle. This fact complicates the selection of roots on the circle itself, since it requires students to have a more complex orientation on it. If the root value cannot be explicitly written in the radian measure, then selecting the roots using the double inequality becomes problematic. The basic level math exam paper consists of one part, including 20 tasks with a short answer. All tasks are aimed at testing the development of basic skills and practical skills of applying mathematical knowledge in everyday situations. The answer to each of the tasks 1-20 is an integer, a finite decimal, or a sequence of digits.

Some Padé approximations and inequalities for the complete elliptic integrals of the first kind

In this paper, we present Padé approximations of some functions involving complete elliptic integrals of the first kind $K(x)$ K ( x ) , and motivated by these approximations we also present the following double inequality: $$ \frac{1-x^{2}}{1-x^{2}+\frac{x^{4}}{62}}< \frac{2 e^{\frac{2}{\pi }K(x)-1}}{ (1+\frac{1}{\sqrt{1-x^{2}}} )}< \frac{1-\frac{96}{100}x^{2}}{1-\frac{96}{100}x^{2}+\frac{x^{4}}{64}},\quad x\in ( 0,1 ). $$ 1 − x 2 1 − x 2 + x 4 62 < 2 e 2 π K ( x ) − 1 ( 1 + 1 1 − x 2 ) < 1 − 96 100 x 2 1 − 96 100 x 2 + x 4 64 , x ∈ ( 0 , 1 ) . Our results have superiority over some new recent results.

New Sharp Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean

Let I v x be he modified Bessel function of the first kind of order v. We prove the double inequality sinh t t cosh 1 / q q t < I 0 t < sinh t t cosh 1 / p p t holds for t > 0 if and only if p ≥ 2 / 3 and q ≤ ln 2 / ln π . The corresponding inequalities for means improve already known results.

Examining Climate Justice with Regional Disaster Resilience

&lt;p&gt;&amp;#160; &amp;#160; &amp;#160;&quot;Climate Justice&quot; explains &quot;climate change as the source of a double inequality with an inverse distribution of risk and responsibility around the regions.&amp;#8221; It is also represents a &amp;#8220;disproportionate disaster risk burden&amp;#8221; between regions, and focus on the limit of the living conditions in climate change. Recently, the issue of &quot;climate justice&quot; has been highly valued internationally. Before the start of the &amp;#8220;United Nations Climate Change Conference&amp;#8221;(COP24) in 2018, there were 130 countries and 403 nonprofit organization signed a statement and required that all governments needed to pay attention to climate justice and should include in &amp;#8220;Paris Agreetment&amp;#8221;.&lt;br&gt;&amp;#160; &amp;#160; &amp;#160;In recent years, there has been a correlation between climate justice research and &amp;#8220;disaster resilience&amp;#8221;, but it can be found that the research of climate justice is not much different from the general disaster resilience research, and the analysis of the research is less included in the inequality of climate justice. In addition, the meanings and theories of &quot;climate justice&quot; have not been systematically generalized in the past literature.&lt;br&gt;Therefore, in addition to thoroughly understanding the theory and contents of &quot;climate justice&quot; this research will identify areas with &quot;climate injustice&quot; characteristics through quantitative research methods (Spatial Autocorrelation e.g.). Besides, climate change is a &quot; long-term impact &quot;, it is not easy to calculate from a single timing, so this research will join the time factors to analyze the &quot;time lag effect.&quot; &amp;#160;&lt;br&gt;&amp;#160; &amp;#160; &amp;#160;This research will choose Taiwan as the research area and focus on flooding data because of the unfairness between water management budget and the flooding condition of the extreme rainfall. Then the above research results will be incorporated into the &amp;#8220;Climate Justice &amp;#8221; theory as a basis for diagnosing regional disaster resilience and give advice on policy and planning in response to climate justice.&lt;/p&gt;