The Ostrowski inequality for $ s $-convex functions in the third sense

<abstract><p>In this paper, the Ostrowski inequality for $ s $-convex functions in the third sense is studied. By applying Hölder and power mean integral inequalities, the Ostrowski inequality is obtained for the functions, the absolute values of the powers of whose derivatives are $ s $-convex in the third sense. In addition, by means of these inequalities, an error estimate for a quadrature formula via Riemann sums and some relations involving means are given as applications.</p></abstract>

Particle physics phenomenology with Python 3: $e^- + \bar{\nu}_e \to e^- + \bar{\nu}_e$ scattering in the Fermi theory

In this work, we develop an algorithm in Python 3 to compute the theoretical prediction of the electron and electron anti-neutrino scattering cross section using two different numerical methods: {\it i)} Riemann sums and {\it ii)} Monte Carlo integration. We compare the precision among these two methods and the theoretical result. Finally, the theoretical prediction is compared with the result obtained with MadGraph 5 which is commonly used to provide theoretical predictions for the LHC. With this project, we would like to encourage students to use programming languages as a tool for the study of new physics.

Approximate integration through remarkable points using the Intermediate Value Theorem

Using the Intermediate Value Theorem we demonstrate the rules of Trapeze and Simpson's. Demonstrations with this approach and its generalization to new formulas are less laborious than those resulting from methods such as polynomial interpolation or Gaussian quadrature. In addition, we extend the theory of approximate integration by finding new approximate integration formulas. The methodology we used to obtain this generalization was to use the definition of the integral defined by Riemann sums. Each Riemann sum provides an approximation of the result of an integral. With the help of the Intermediate Value Theorem and a detailed analysis of the Middle Point, Trapezoidal and Simpson Rules we note that these rules of numerical integration are Riemann sums. The results we obtain with this analysis allowed us to generalize each of the rules mentioned above and obtain new rules of approximation of integrals. Since each of the rules we obtained uses a point in the interval we have called them according to the point of the interval we take. In conclusion we can say that the method developed here allows us to give new formulas of numerical integration and generalizes those that already exist.

Divide and conquer, just like Riemann sums in calculus

“Divide and conquer, just like Riemann sums in calculus” offers a basic introduction for how to estimate—with any desired non-zero margin of error—the area of an irregular shape. For an initial underestimate, readers are encouraged to draw a large rectangle that fits inside the irregular shape and then use the grade school formula to calculate the rectangle’s area: area equals length times width. To refine this underestimate, readers learn to draw multiple smaller rectangles inside the shape whose areas they also sum. An analogous method is provided for overestimates. The discussion concerning how to obtain underestimates or overestimates with any desired margin of error is illustrated with numerous hand-drawn sketches. Mathematics students and enthusiasts are encouraged to consider dividing and conquering in all challenges they face in mathematics or life. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.