Results 1 - 10 of 1974
Results 1 - 10 of 1974. Search took: 0.032 seconds
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[en] Basic equation of elementary quantum mechanics are investigated in a pedagogical way. A very simple solution for a simple model system such as the particle-in a box is obtained by using Laplace transform technique. (Author)
[en] Integral Transform methods play an extremely important role in many branches of science and engineering. The ease with which many problems may be solved using these techniques is well known. In Electrical Engineering especially, Laplace and Fourier Transforms have been used for a long time as a way to change the solution of differential equations into trivial algebraic manipulations or to provide alternate representations of signals and data. These techniques, while seemingly overshadowed by today's emphasis on digital analysis, still form an invaluable basis in the understanding of systems and circuits. A firm grasp of the practical aspects of these subjects provides valuable conceptual tools. This tutorial paper is a review of Laplace and Fourier Transforms from an applied perspective with an emphasis on engineering applications. The interrelationship of the time and frequency domains will be stressed, in an attempt to comfort those who, after living so much of their lives in the time domain, find thinking in the frequency domain disquieting
[en] The detrended fluctuation analysis (DFA) and its extensions (MF-DFA) have been used extensively to determine possible long-range correlations in time series. However, recent studies have reported the susceptibility of DFA to trends which give rise to spurious crossovers and prevent reliable estimation of the scaling exponents. In this report, a smoothing algorithm based on the discrete laplace transform (DFT) is proposed to minimize the effect of exponential trends and distortion in the log-log plots obtained by MF-DFA techniques. The effectiveness of the technique is demonstrated on monofractal and multifractal data corrupted with exponential trends.
[en] In this paper, we propose a novel probability distribution that asymptotically represents a power-law, ψ(t) ∼ t −α−1, with 0 < α < 2. The main feature of the distribution is that it has a simple expression in the Laplace transform representation, making it suitable for performing calculations in stochastic processes, particularly non-Poissonian processes. (paper)
[en] We found Fuchs-Garnier pairs in 3 x 3 matrices for the first and second Painleve equations which are linear in the spectral parameter. As an application of our pairs for the second Painleve equation we use the generalized Laplace transform to derive an invertible integral transformation relating two of its Fuchs-Garnier pairs in 2 x 2 matrices with different singularity structures, namely, the pair due to Jimbo and Miwa and that found by Harnad, Tracy and Widom. Together with the certain other transformations it allows us to relate all known 2 x 2 matrix Fuchs-Garnier pairs for the second Painleve equation to the original Garnier pair
[en] We obtain necessary and sufficient conditions on a function in order that it be the Laplace transform of an absolutely monotonic function. Several closely related results are also given.
[en] The dual to a weighted space G of infinitely smooth functions on the real axis is described by means of the Fourier-Laplace transformation. This result is used in the study of the surjectivity in G of an infinite-order linear differential operator with constant coefficients
[en] A clarification is given of the difference between the equation adjoint to the Laplace-transformed time-dependent transport equation and the Laplace-transformed time-dependent adjoint transport equation. Proper procedures are derived to obtain the Laplace transform of the instantaneous detector response. (author)
[en] The purpose of this paper is to present some propositions about the Laplace transform related to the first hitting time to piecewise linear functions of a Brownian motion. We introduce also a terminal date and examine the lower between the hitting time and the terminal moment. Regardless of whether the hitting time is before the terminal date or not, we shall only know either the stopping time value or the value of the Brownian motion. This requires the separate examination of both cases. The derived results can be used for pricing financial derivatives related to reaching some boundaries of the underlying asset - for example, barrier or American options.
[en] The first digit law, also known as Benford’s law or the significant digit law, is an empirical phenomenon that the leading digit of numbers from real world sources favors small ones in a form log(1 + 1/d), where d = 1, 2,…, 9. Such a law has been elusive for over 100 years because it has been obscure whether this law is due to the logical consequence of the number system or some mysterious mechanism of nature. We provide a simple and elegant proof of this law from the application of the Laplace transform, which is an important tool of mathematical methods in physics. It is revealed that the first digit law originates from the basic property of the number system, thus it should be attributed as a basic mathematical knowledge for wide applications. (paper)