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Young's integral inequality is reformulated with upper and lower bounds for the remainder. The new inequalities improve Young's integral inequality on all time scales, such that the case where equality holds becomes particularly transparent…
Finite differences, as a subclass of direct methods in the calculus of variations, consist in discretizing the objective functional using appropriate approximations for derivatives that appear in the problem. This article generalizes the…
Results of research of possibility of transformation of a difference equation into a system of the first-order difference equation are presented. In contrast to the method used previously, an unknown grid function is split into two new…
We prove multidimensional integration by parts formulas for generalized fractional derivatives and integrals. The new results allow us to obtain optimality conditions for multidimensional fractional variational problems with Lagrangians…
If g and h are functions over some field, we can consider their composition f = g(h). The inverse problem is decomposition: given f, determine the ex- istence of such functions g and h. In this thesis we consider functional decom- positions…
Universal geometric calculus simplifies and unifies the structure and notation of mathematics for all of science and engineering, and for technological applications. This paper treats the fundamentals of the multivector differential…
The well-known mathematical instrument for detection common roots for pairs of polynomials and multiple roots of polynomials are resultants and discriminants. For a pair of polynomials $f$ and $g$ their resultant $R(f,g)$ is a function of…
Differential operators usually result in derivatives expressed as a ratio of differentials. For all but the simplest derivatives, these ratios are typically not algebraically manipulable, but must be held together as a unit in order to…
It is proved that the random integral mappings (some type of functionals of L\'evy processes) are always isomorphisms between convolution semigroups of infinitely divisible measures. However, the inverse mappings are no longer of the random…
Decomposing the domain of a function into parts has many uses in mathematics. A domain may naturally be a union of pieces, a function may be defined by cases, or different boundary conditions may hold on different regions. For any…
Differential Calculus is a staple of the college mathematics major's diet. Eventually one becomes tired of the same routine, and wishes for a more diverse meal. The college math major may seek to generalize applications of the derivative…
We first prove some weighted inequalities for compositions of functions on time scales which are in turn applied to establish some new dynamic Opial-type inequalities in several variables. Some generalizations and applications to partial…
We consider the properties of the second order nonlinear differential equations b''= g(a,b,b') with the function g(a,b,b'=c) satisfying the following nonlinear partial differential equation $$ \frac{d^2…
One of the most common types of functions in mathematics, physics, and engineering is a sum of products, sometimes called a partition function. After "normalization," a sum of products has a natural graphical representation, called a normal…
We state a generalization of the Connes-Tretkoff-Moscovici Rearrangement Lemma and give a surprisingly simple (almost trivial) proof of it. Secondly, we put on a firm ground the multivariable functional calculus used implicitly in the…
Eliminating the arbitrary coefficients in the equation of a generic plane curve of order $n$ by computing sufficiently many derivatives, one obtains a differential equation. This is a projective invariant. The first one, corresponding to…
A generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders. That is, a fractional exterior derivative is defined. This is found to generate new vector…
We construct a new arithmetic-term representation for the function gcd(a,b). As a byproduct, we also deduce a representation gcd(a,b) by a modular term in integer arithmetic.
Fractional variation is defined as the limit of the difference quotient of the increments of a function and its argument raised to a fractional power. Fractional velocity can be suitable for characterizing singular behavior of derivatives…
A gauge invariant partition function is defined for gauge theories which leads to the standard quantization. It is shown that the descent equations and consequently the consistent anomalies and Schwinger terms can be extracted from this…