This paper is interested in establishing some new reverse Hilbert-type inequalities, by using chain rule on time scales, reverse Jensen’s, and reverse Hölder’s with Specht’s ratio and mean inequalities. To get the results, we used the Specht’s ratio function and its applications for reverse inequalities of Hilbert-type. Symmetrical properties play an essential role in determining the correct methods to solve inequalities. The new inequalities in special cases yield some recent relevance, which also provide new estimates on inequalities of these type.
AbstractThe main aim of the present article is to introduce some new ∇-conformable dynamic inequalities of Hardy type on time scales. We present and prove several results using chain rule and Fubini’s theorem on time scales. Our results generalize, complement, and extend existing results in the literature. Many special cases of the proposed results, such as new conformable fractional h-sum inequalities, new conformable fractional q-sum inequalities, and new classical conformable fractional integral inequalities, are obtained and analyzed.
The triple product rule, also known as the cyclic
chain rule, cyclic relation, cyclical rule or Euler's chain rule,
relates the partial derivatives of three interdependent variables,
and often finds application in thermodynamics. It is shown here
that its derivation is wrong, and that this rule is not correct; hence,
the Mayer's relation and the heat capacity ratio, which describe
the difference between isobaric and isochoric heat capacities, are
also untrue. Also, the relationship linking thermal expansion and
isothermal compressibility is wrong. These results are confirmed
by many experiments and by the previous theoretical findings of
the author.
The triple product rule, also known as the cyclic chain rule, cyclic relation, cyclical rule or Euler’s chain rule, relates the partial derivatives of three interdependent variables, and often finds application in thermodynamics. It is shown here that its derivation is wrong, and that this rule is not correct; hence, the Mayer’s relation and the heat capacity ratio, which describe the difference between isobaric and isochoric heat capacities, are also untrue. Also, the relationship linking thermal expansion and isothermal compressibility is wrong. These results are confirmed by many experiments and by the previous theoretical findings of the author.
The chapter “Mathematics Refresher” provides a brief reminder of operations with logarithms, matrices, and calculus, for student reference. It starts off by reviewing the differences between regular logarithms and natural logarithms and provides some examples of common operations with logarithms. It then introduces derivatives and integrals (although it is never necessary to compute an integral in this book, it is still useful to know what an integral is) and explains the sum rule, the product rule, the quotient rule, and the chain rule. Next, it provides a brief overview of matrices and matrix operations, including matrix dimensions, and addition and multiplication of matrices. It concludes with a discussion of the identity matrix.
AbstractWe define notions of differentiability for maps from and to the space of persistence barcodes. Inspired by the theory of diffeological spaces, the proposed framework uses lifts to the space of ordered barcodes, from which derivatives can be computed. The two derived notions of differentiability (respectively, from and to the space of barcodes) combine together naturally to produce a chain rule that enables the use of gradient descent for objective functions factoring through the space of barcodes. We illustrate the versatility of this framework by showing how it can be used to analyze the smoothness of various parametrized families of filtrations arising in topological data analysis.
AbstractIn this paper, we prove some new Opial-type dynamic inequalities on time scales. Our results are obtained in frame of convexity property and by using the chain rule and Jensen and Hölder inequalities. For illustration purpose, we obtain some particular Opial-type inequalities reported in the literature.
On the autonomous Nemytskii operator between Sobolev spaces in the critical and supercritical cases: Well-definedness and higher-order chain rule