In this paper, we first recall the concept o f the multiplicative conformable fractional integrals (MCFI) and their several properties. Then, we establish the Ostrowski type inequalities in two distinct senses for multiplicative conformable fractional integrals. The reason for considering two distinct types of Ostrowski-type inequalities is to capture a broader class of functions and provide more general results that can be applied in different settings within the framework of multiplicative conformable fractional calculus For this aim we first prove two new equalities for multiplicative differentiable functions. Then, by advantage of these identities, we prove some Ostrowski-type inequalities by using the concept of multiplicative convex functions and the well-known Hölder inequality. Moreover, we establish Ostrowski type inequalities for functions whose multiplicative derivatives are bounded. By special cases, we present the relations between newly obtained inequalities for MCFI and existing results for multiplicative Riemann-Liouville fractional integrals (MRLFI) and multiplicative integrals. Furthermore, we give some new Ostrowski type inequalities for multiplicative integrals and MRLFI. Finally, we give several examples and 3D graphs to illustrate the main results.
This short note introduce an analytic solution for a generalized Bernoulli non-linear first-order ordinary differential equation (ODE), extending the framework proposed by Azevedo and Valentino [Int J Math Educ Sci, 2016, 47(8), 1271--1276]. The family of analytic solutions are obtained by employing a non-linear substitution that transform the extended Bernoulli ODE into a classical Bernoulli equation. We derive a family of solutions and provide illustrative examples, demonstrating the versatility of our approach for arbitrary values of the Bernoulli non-linear term power α. This work connects to applications in population growth models, such as the θ-logistic and Richards models.
This is a survey of a connection between the distribution of certain power residues modulo p, p a prime, and relative class numbers. The focus lies on quadratic residues and sixth power residues. Dirichlet's class number formula yields a number of results about the distribution of quadratic residues, for instance, the well-known fact that the interval [0,p/2] contains more quadratic residues than nonresidues. This class number formula is also responsible for some properties of the digit expansions of numbers m/p, p m. In a certain sense the results based on Dirichlet's formula can be extended to sixth power residues, where geometry plays an important role.
We define filtered ADHM data and connect a notion of filtered quiver representations to Grothendieck--Springer resolutions. We also provide current developments and give a list of research problems to further study filtered ADHM equation.
One of the most significant contributions to the proof of the Baum-Connes conjecture was made by Higson and Kasparov. Their proof of the conjecture for a-T-menable groups is a highly technical achievement. Some details of this result were later exposed in a survey by Guentner and Higson, where the conjecture for ℤn is approached as an intermediate step to the general case. In this work we review the arguments given for ℤn in the aforementioned survey and show that they apply to the case of the infinite dihedral group.
