Related papers: The Natural Logarithm on Time Scales
We discuss a series of 8 energy scales, some of which just speculated by ourselves, and fit the logarithms of these energies as a straight line versus a quantity related to the dimensionalities of action terms in a way to be defined in the…
Bohmian mechanics can be generalized to a relativistic theory without preferred foliation, with a price of introducing a puzzling concept of spacetime probability conserved in a scalar time. We explain how analogous concept appears…
We deal with direct and inverse problems of the calculus of variations on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we give a general form for a variational functional to…
We define a symmetric derivative on an arbitrary nonempty closed subset of the real numbers and derive some of its properties. It is shown that real-valued functions defined on time scales that are neither delta nor nabla differentiable can…
We determine the Lagrange function in Taylor polynomial approximation by solving an appropriate initial-value problem. Hence, we determine the remainder term which we then approximate by means of a natural cubic spline. This results in a…
For every natural number $T,$ we write $\Ln T$ as a series, generalizing the known series for $\Ln 2.$
We study more general variational problems on time scales. Previous results are generalized by proving necessary optimality conditions for (i) variational problems involving delta derivatives of more than the first order, and (ii) problems…
We prove a version of the Euler-Lagrange equations for certain problems of the calculus of variations on time scales with higher-order delta derivatives.
We prove that any power of the logarithm of Fourier series with random signs is integrable. This result has applications to the distribution of values of random Taylor series, one of which answers a long-standing question by J.-P. Kahane.
We establish some nonlinear integral inequalities for functions defined on a time scale. The results extend some previous Gronwall and Bihari type inequalities on time scales. Some examples of time scales for which our results can be…
An introduction to the physical interpretation of the Coulomb logarithm is given with particular emphasis on the quantum-mechanical corrections that are required at high temperatures. Excerpts from the literature are used to emphasize the…
We give a proposal to generalize the concept of the differential equations on time scales, such that they can be more appropriate for the analysis of real world problems, and give more opportunities to increase the theoretical depth of…
We present an $O(n\sqrt{\log n})$ time and linear space algorithm for sorting real numbers. This breaks the long time illusion that real numbers have to be sorted by comparison sorting and take $\Omega (n\log n)$ time to be sorted.
Gosper developed an algorithm for performing arithmetic on continued fractions (CFs), and introduced continued logarithms (CLs) as a variant of continued fractions better suited to representing extremely large (or small) numbers. CLs are…
In this paper, we got some refinements of the norm inequalities related to the Heinz mean and logarithmic mean.
Lanford has shown that Feigenbaum's functional equation has an analytic solution. We show that this solution is a polynomial time computable function. This implies in particular that the so-called first Feigenbaum constant is a polynomial…
The operations of linear algebra, calculus, and statistics are routinely applied to measurement scales but certain mathematical conditions must be satisfied in order for these operations to be applicable. We call attention to the conditions…
We study the process of integration on time scales in the sense of Riemann-Stieltjes. Analogues of the classical properties are proved for a generic time scale, and examples are given.
The Wigner functions on the one dimensional lattice are studied. Contrary to the previous claim in literature, Wigner functions exist on the lattice with any number of sites, whether it is even or odd. There are infinitely many solutions…
This work belongs to the framework of inverse problems with linear model. The resolution of this type of problem consists in minimizing (possibly under constraints) a function of discrepancy between the measurements and a physical model of…