Search results for "Trigonometric functions"
showing 10 items of 16 documents
Turán type inequalities for generalized inverse trigonometric functions
2013
In this paper we study the inverse of the eigenfunction $\sin_p$ of the one-dimensional $p$-Laplace operator and its dependence on the parameter $p$, and we present a Tur\'an type inequality for this function. Similar inequalities are given also for other generalized inverse trigonometric and hyperbolic functions. In particular, we deduce a Tur\'an type inequality for a series considered by Ramanujan, involving the digamma function.
On an Inequality for Trigonometric Polynomials In Several Variables
1990
Publisher Summary This chapter presents trigonometric polynomials in n variables. Using the methods of approximation theory, an inequality can be extended to almost periodic functions and to still more general classes of functions as in the case for Bohr's inequality. However, no analogous result exists in the case of two variables. For the solution of problems containing small divisors, the estimate has to be completed by theorems concerning the best approximation of holomorphic functions by trigonometric polynomials in polystrips. The chapter also presents equations to provide an estimate for a differential operator.
Comments on “Mean velocity and turbulent characteristics of flow over half-cycle cosine sharp-crested weirs” by Salehi S., Esmaili K., Azimi A.H.
2019
Abstract In this paper the stage-discharge equation of a half-cycle cosine weir is theoretically deduced applying the Π-Theorem of dimensional analysis and the self-similarity theory. The coefficients of the new stage-discharge relationships are estimated using the results of the experimental runs by Salehi et al..
On Carlson"s and Shafer"s inequalities
2014
In this paper the authors re ne the Carlson"s inequalities for inverse cosine function, and the Shafer"s inequalities for inverse tangent function.
Millimeter-Scale and Billion-Atom Reactive Force Field Simulation on Sunway Taihulight
2020
Large-scale molecular dynamics (MD) simulations on supercomputers play an increasingly important role in many research areas. With the capability of simulating charge equilibration (QEq), bonds and so on, Reactive force field (ReaxFF) enables the precise simulation of chemical reactions. Compared to the first principle molecular dynamics (FPMD), ReaxFF has far lower requirements on computational resources so that it can achieve higher efficiencies for large-scale simulations. In this article, we present our efforts on scaling ReaxFF on the Sunway TaihuLight Supercomputer (TaihuLight). We have carefully redesigned the force analysis and neighbor list building steps. By applying fine-grained …
Eine Neue Rechnung zur Röntgenkleinwinkelstreuung an Fadenmolekulen. Die Ermittlung der Segmentgestalt
1967
The scattering function of chain molecules in the x-ray small-angle range depends on the shape of the segments. For instance, one obtains quite different scattering curves from solutions of isotactic and syndiotactic poly(methyl methacrylate) (PMMA) in the same solvent. From several models of statistically coiled polymer chains, the scattering functions were calculated with the aid of the Monte Carlo method. Again, a considerable difference between the functions obtained is observed. If the curvature of a thread varies statistically from one point to another, the mean curvature can be determined from the scattering function. A suitable measure for the mean curvature is the persistence lengt…
Ultrarelativistic (Cauchy) spectral problem in the infinite well
2016
We analyze spectral properties of the ultrarelativistic (Cauchy) operator $|\Delta |^{1/2}$, provided its action is constrained exclusively to the interior of the interval $[-1,1] \subset R$. To this end both analytic and numerical methods are employed. New high-accuracy spectral data are obtained. A direct analytic proof is given that trigonometric functions $\cos(n\pi x/2)$ and $\sin(n\pi x)$, for integer $n$ are {\it not} the eigenfunctions of $|\Delta |_D^{1/2}$, $D=(-1,1)$. This clearly demonstrates that the traditional Fourier multiplier representation of $|\Delta |^{1/2}$ becomes defective, while passing from $R$ to a bounded spatial domain $D\subset R$.
Geometric efficiency for a parallel-surface source and detector system with at least one axisymmetric surface
2007
Abstract An exact and numerically friendly method is given to calculate the geometric efficiency G of a planar radiation source and cosine detector system. Either the source or the detector, but not necessarily both, must have axial symmetry. For two non-coaxial disks the results are in exact agreement with a recent generalization of Ruby's formula for G. Detailed formulas and sample numerical results are given for a disk combined with rectangles and triangles. A disk and a general polygon can be solved by dividing the polygon into triangles. The method can also be applied to electrical inductance calculations and a solution recently given for the inductance of circular and elliptic loops c…
A geometrical constructive approach to infinitesimal analysis: epistemological potential and boundaries of tractional motion
2014
Recent foundational approaches to Infinitesimal Analysis are essentially algebraic or computational, whereas the first approaches to such problems were geometrical. From this perspective, we may recall the seventeenth-century investigations of the “inverse tangent problem.” Suggested solutions to this problem involved certain machines, intended as both theoretical and actual instruments, which could construct transcendental curves through so-called tractional motion. The main idea of this work is to further develop tractional motion to investigate if and how, at a very first analysis, these ideal machines (like the ancient straightedge and compass) can constitute the basis of a purely geome…
Shifting of wrapped phase maps in the frequency domain using a rational number
2016
The number of phase wraps in an image can be either reduced, or completely eliminated, by transforming the image into the frequency domain using a Fourier transform, and then shifting the spectrum towards the origin. After this, the spectrum is transformed back to the spatial domain using the inverse Fourier transform and finally the phase is extracted using the arctangent function. However, it is a common concern that the spectrum can be shifted only by an integer number, meaning that the phase wrap reduction is often not optimal. In this paper we propose an algorithm than enables the spectrum to be frequency shifted by a rational number. The principle of the proposed method is confirmed b…