Interesting Formula Research Articles

Exploring Explicit Definite Integral Formulae with Trigonometric and Hyperbolic Functions

Making use of integration by parts and variable replacement methods, we derive some interesting explicit definite integral formulae involving trigonometric or hyperbolic functions, whose results are expressed in terms of Catalan’s constant, Dirichlet’s beta function, and Riemann’s zeta function, as well as π in the denominator.

CM points, class numbers, and the Mahler measures of 𝑥³+𝑦³+1-𝑘𝑥𝑦

We study the Mahler measures of the polynomial family Q k ( x , y ) = x 3 + y 3 + 1 − k x y Q_k(x,y) = x^3+y^3+1-kxy using the method previously developed by the authors. An algorithm is implemented to search for complex multiplication points with class numbers ⩽ 3 \leqslant 3 , we employ these points to derive interesting formulas that link the Mahler measures of Q k ( x , y ) Q_k(x,y) to L L -values of modular forms. As by-products, some conjectural identities of Samart are confirmed, one of them involves the modified Mahler measure n ~ ( k ) \tilde {n}(k) introduced by Samart recently. For k = 729 ± 405 3 3 k=\sqrt [3]{729\pm 405\sqrt {3}} , we also prove an equality that expresses a 2 × 2 2\times 2 determinant with entries the Mahler measures of Q k ( x , y ) Q_k(x,y) as some multiple of the L L -value of two isogenous elliptic curves over Q ( 3 ) \mathbb {Q}(\sqrt {3}) .

Using the Girard-Newton-Waring Formula to Evaluate Trigonometric Sums: A Blast from the Past

Summary This article shows how two classical results about polynomials with trigonometric roots may be used together to create interesting new formulas for trigonometric sums. Readers are encouraged to find further new trigonometric formulas by applying the ideas in this article to other polynomials having trigonometric roots.

Geometry of Enumerable Class of Surfaces Associated with Mylar Balloons

In this paper, the very fundamental geometrical characteristics of the Mylar balloon like the profile curve, height, volume, arclength, surface area, crimping factor, etc. are recognized as geometrical moments In(x) and In and this observation has been used to introduce an infinite family of surfaces Sn specified by the natural numbers n=0,1,2,…. These surfaces are presented via explicit formulas (through the incomplete Euler’s beta function) and can be identified as an interesting family of balloons. Their parameterizations is achieved relying on the well-known relationships among elliptic integrals, beta and gamma functions. The final results are expressed via the fundamental mathematical constants, such as π and the lemniscate constant ϖ. Quite interesting formulas for recursive calculations of various quantities related to associated figures modulo four are derived. The most principal results are summarized in a table, illustrated via a few graphics, and some direct relationships with other fundamental areas in mathematics, physics, and geometry are pointed out.

Strangeness as home. Georg Simmel in Berlin

Georg Simmel is the sociologist of the city. He spent almost his entire life in Berlin where he tried to make an academic career at his home town university. He had from early on remarkable success and published a number of classical books, most well-known among them his “Philosophy of Money.” But the intellectual merits did not translate into professional success. In Berlin he remained a “Privatdozent”—a teaching person without salary—and an “Extraordinarius”—a professor not remunerated. Simmel’s status inconsistency—a globally famous philosopher and sociologist yet without a full professorship—expressed an interesting meritocratic formula: failure by success. Simmel could endure this complicated fate only by putting on a mask which made strangeness his home. This thesis is developed in three steps: First, by looking on young Simmel who combined intellectual success with critical reception, in short: philosophe maudit. Secondly, by referring to his private life and life style as far as this can be reconstructed at all because Simmel turned all matters private into a secret. Thirdly, by discussing his predominate topics (city, prostitution, sociability, secret, and the stranger) we try to show how he made his personal existence possible philosophically, sociologically, and psychologically.

Мониторинг динамики финансовой грамотности студентов в условиях интеграции учебных дисциплин

The article is devoted to the problem of the formation of financial literacy of students of economic universities in the context of the integration of academic disciplines. The aim of the work is to identify the possibilities of integrative links between academic disciplines in improving the level of financial literacy within the framework of the continuity of educational programs for undergraduate and graduate programs. As a means of integration, it is proposed to use the models of simple and compound percent. The importance of formalized models in the performance of financial operations of accumulation, discounting and, accordingly, in assessing the effectiveness of financial decision-making is noted. The paper presents the results of monitoring the dynamics of the level of financial literacy of students of the University of Economics. An online survey of undergraduate students was conducted in the context of courses of study and undergraduates. Respondents were asked questions related to identifying the objective level of financial literacy (knowledge of financial terms, activity in the financial services market, knowledge of simple and compound interest formulas, etc.). A separate block includes questions about the subjective level of financial literacy of respondents and the need for additional knowledge and skills in financial literacy. he results of the analysis of empirical material indicate a positive correlation between the objective and subjective levels of financial literacy of students, as well as the need for additional knowledge on financial literacy. At the same time, higher levels of financial literacy are associated with skills in mastering simple and compound interest models and demonstrate a positive trend in the cumulative effect of integrating academic disciplines. In its turn, the study of student activity in the financial services market determines the widespread use of digital financial instruments. The findings determine the need for students of economic universities to master formalized methods for assessing financial decision-making, which in the future will allow avoiding costly mistakes in practice and maintaining the required level of digital financial literacy.

A study of double general transform for solving fractional partial differential equations

In this research, we introduce the basic properties of the single and double general transforms. New interesting results related to fractional operators in the sense of Caputo derivative are investigated and proved. In this study, the general double transformation is utilized to study the application on some special functions, such as the Mittag–Leffler function, and on the Caputo's fractional derivative of different and higher orders. Moreover, we establish new interesting formulas and prove some theories that can save efforts and time for researchers as they investigate the properties and applications of integral transformations on fractional operators. The results obtained in this research are tested by solving different types of fractional partial differential equations to obtain exact solutions. The outcomes of this article are valuable to researchers and mathematicians who are interested in the solving fractional partial differential equations via integral transforms, which are considered as important techniques to handle these equations.

Enumeration of spanning trees of complete multipartite graphs containing a fixed spanning forest

AbstractMoon's classical result implies that the number of spanning trees of a complete graph with vertices containing a given spanning forest equals , where is the number of components of , and are the numbers of vertices of component of . Dong and Ge extended this result to the complete bipartite graph, and obtain an interesting formula to count spanning trees of a complete bipartite graph containing a given spanning forest . They also posed the problem to count spanning trees of a complete ‐partite graph containing a given spanning forest for . In this paper, we propose a technique to solve this problem. Using this technique, we obtain closed formulae to count spanning trees of complete ‐partite graphs containing a given spanning forest for and 4. Our technique also results in a new and simple proof of Dong and Ge's formula.

On Time Fractional Partial Differential Equations and Their Solution by Certain Formable Transform Decomposition Method

This paper aims to investigate a new efficient method for solving time fractional partial differential equations. In this orientation, a reliable formable transform decomposition method has been designed and developed, which is a novel combination of the formable integral transform and the decomposition method. Basically, certain accurate solutions for time-fractional partial differential equations have been presented. The method under concern demands more simple calculations and fewer efforts compared to the existing methods. Besides, the posed formable transform decomposition method has been utilized to yield a series solution for given fractional partial differential equations. Moreover, several interesting formulas relevant to the formable integral transform are applied to fractional operators which are performed as an excellent application to the existing theory. Furthermore, the formable transform decomposition method has been employed for finding a series solution to a time-fractional Klein-Gordon equation. Over and above, some numerical simulations are also provided to ensure reliability and accuracy of the new approach.

Hausdorff of a cycle in topological graph

The main purpose of the paper is to bring together two areas in which strong relation, graph theory and topological spaces. And derive interesting formula for the set that contains all minimal dominating sets(MDS) and ( -set). Some separation axioms are discussed in topological graph theory, especially in a cycle graph Cn.

Analisis Anuitas Kredit Motor Honda Revo FI Fit

Along with the many needs of the community for their lives, especially in motorized vehicles, many financial institutions provide credit facilities to make it easier for people who want to own a motorbike in the near future but are constrained by costs. One of the institutions that provide a credit system is Honda. Honda has launched many kinds of products, one of which is Revo FI FIT. To attract buyers, Honda provides various installment offers with loan amounts and terms. The data used in this study is secondary data in the form of qualitative data by searching documents from the Honda Cengkareng website which was updated on November 30, 2022. In this study, we will analyze whether the installment calculation is in accordance with the annuity theory or not. Based on the results of data processing by calculating the theoretical formula of compound interest and ordinary annuities, it is found that the calculation of the motorcycle loan annuity is not in accordance with the ordinary annuity theory, where the installments using the ordinary annuity theory are smaller than the installments listed on the web.

The effect of partial loads on the performance of a funnel solar cooker

Several solar cooker designs have been proposed over the last decades. The funnel cooker is a well-known model, representative of the category of panel-type solar cookers. From the empirical experience of real practice, it is known that the use of partial loads has an impact on the performance of the cooker. However, this effect has not yet been investigated rigorously. This work aims to fill this lack. Extensive experimental work was performed to determine the effect of partial loads on a funnel cooker thermal performance. Tests were conducted on two identical funnel cookers, in Malaga, Spain, with low sun elevation. Cookers were tracked only azimuthally. Experimental protocol was based on ASAE 580.1 Standard for better replicability. Results showed that the standardised power drops by about 15% of the original value when the water volumetric load fraction drops by 25%, for both cooker operations, with and without glass enclosure. This important reduction is explained on how the fill level of the cooking vessel affects its function as a thermal radiation receiver. Results from experimental tests were correlated into a simple formula of practical interest. Finally, a new cooking vessel design, that improves performance at partial loads, i.e., a 25.4% increase in cooker power, was proposed and tested.

Novel Formulae of Certain Generalized Jacobi Polynomials

The main goal of this article is to investigate theoretically a kind of orthogonal polynomials, namely, generalized Jacobi polynomials (GJPs). These polynomials can be expressed as certain combinations of Legendre polynomials. Some basic formulas of these polynomials such as the power form representation and inversion formula of these polynomials are first introduced, and after that, some interesting formulas concerned with these polynomials are established. The formula of the derivatives of the moments of these polynomials is developed. As special cases of this formula, the moment and high-order derivative formulas of the GJPs are deduced. New expressions for the high-order derivatives of the GJPs, but in terms of different symmetric and non-symmetric polynomials, are also established. These expressions lead to some interesting connection formulas between the GJPs and some various polynomials.

Gould-Hopper Based Degenerate Truncated Bernoulli Polynomials

In this study, we consider the truncated degenerate Bernoulli polynomials based on the Gould-Hopper polynomials and examine diverse properties and formulas covering addition formulas, correlations and derivation property. Then, we derive some interesting implicit summation formulas and symmetric identities. Moreover, we define Gould-Hopper based truncated degenerate Bernoulli polynomials of order rr and give some of their properties and relations.

Pricing of A European Call Option in Stochastic Volatility Models

Volatility occupies a strategic place in the financial markets. In this context of crisis, and with the great movements of the markets, traders have been forced to turn to volatility trading for the potential gain it provides. The Black-Scholes formula for the value of a European option to purchase the underlying depends on a few parameters which are more or less easy to calculate, except for the realized volatility at maturity which makes a problem, because there is no single value, nor an established way to calculate it. In this article, we exploit the Martingale pricing method to find the expected present value of a given asset relative to a riskneutral probability measure. We consider a bond-stock market that evolves according to the dynamics of the Black-Scholes model, with a risk-free interest rate varying with time. Our methodology has effectively directed us towards interesting formulas that we have derived from the exact calculation, giving the present value of the volatility realized over a period of maturity for a European option in a stochastic volatility model.

Square numbers, square pyramidal numbers, and generalized Fibonacci polynomials

In this paper, we derive two interesting formulas for square and square pyramidal numbers. We focus on the linear recurrence relation with constant coefficients for square and square pyramidal numbers. Then we deal with the relationship between generalized Fibonacci polynomials and these numbers. Also, we give some determinant representations of these numbers.

Another Method for Proving Certain Reduction Formulas for the Humbert Function ψ2 Due to Brychkov et al. with an Application

Recently, Brychkov et al. established several new and interesting reduction formulas for the Humbert functions (the confluent hypergeometric functions of two variables). The primary objective of this study was to provide an alternative and simple approach for proving four reduction formulas for the Humbert function ψ2. We construct intriguing series comprising the product of two confluent hypergeometric functions as an application. Numerous intriguing new and previously known outcomes are also achieved as specific instances of our primary discoveries. It is well-known that the hypergeometric functions in one and two variables and their confluent forms occur naturally in a wide variety of problems in applied mathematics, statistics, operations research, physics (theoretical and mathematical) and engineering mathematics, so the results established in this paper may be potentially useful in the above fields. Symmetry arises spontaneously in the abovementioned functions.

Графический юмор комикса "Хармсиниада" Алексея Никитина

The concept of the study is to look at the comic through the eyes of a modern person and understand why we might be interested in it. We will draw up a formula of interest on the example of a particular work that has this property. The content of the study is built on three parts - immersion in the context, analysis of graphic techniques and analysis of conceptual techniques that are used in the comic. The purpose of this visual study is to analyze, using the example of the book "Harmsiniad" by Alexey Nikitin, how graphic stories about popular personalities and events should be approached in such a way that it is new, interesting and ironic. This is a study on how to make artistic images from the last century relevant now.

On Generating Functions for Parametrically Generalized Polynomials Involving Combinatorial, Bernoulli and Euler Polynomials and Numbers

The aim of this paper is to give generating functions for parametrically generalized polynomials that are related to the combinatorial numbers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the cosine-Bernoulli polynomials, the sine-Bernoulli polynomials, the cosine-Euler polynomials, and the sine-Euler polynomials. We investigate some properties of these generating functions. By applying Euler’s formula to these generating functions, we derive many new and interesting formulas and relations related to these special polynomials and numbers mentioned as above. Some special cases of the results obtained in this article are examined. With this special case, detailed comments and comparisons with previously available results are also provided. Furthermore, we come up with open questions about interpolation functions for these polynomials. The main results of this paper highlight the existing symmetry between numbers and polynomials in a more general framework. These include Bernouilli, Euler, and Catalan polynomials.

On degenerate gamma matrix functions and related functions

Recently, the applications of special functions of matrix arguments have received more attention in many fields, such as theoretical physics, number theory, probability theory, engineering and theory of group representations. Gaining enlightenment from these works, in this paper, we introduce the degenerate gamma matrix function, the degenerate zeta matrix function, the degenerate digamma matrix function, the degenerate polygamma matrix function and the degenerate Gauss hypergeometric matrix function. Basic properties of these functions are discussed. In addition, we derive some interesting formulas related to these functions and special cases.