A RESIDUE TYPE PROCESS FOR SMOOTH FUNCTIONS INVOLVING THE DERIVATIVES OF THE NEWTONIAN POTENTIAL IN R2

In this paper, we present a systematical study of braneworlds of string theory on S1/Z2. In particular, starting with the toroidal compactification of the Neveu–Schwarz/Neveu–Schwarz sector in D + d dimensions, we first obtain an effective D-dimensional action, and then compactify one of the D - 1 spatial dimensions by introducing two orbifold branes as its boundaries. We divide the whole set of the gravitational and matter field equations into two groups, one holds outside the two branes, and the other holds on them. By combining the Gauss–Codacci and Lanczos equations, we write down explicitly the general gravitational field equations on each of the two branes, while using distribution theory we express the matter field equations on the branes in terms of the discontinuities of the first derivatives of the matter fields. Afterwards, we address three important issues: (i) the hierarchy problem; (ii) the radion mass; and (iii) the localization of gravity, the four-dimensional Newtonian effective potential and the Yukawa corrections due to the gravitational high-order Kaluza–Klein (KK) modes. The mechanism of solving the hierarchy problem is essentially the combination of the large extra dimension and warped factor mechanisms together with the tension coupling scenario. With very conservative arguments, we find that the radion mass is of the order of 10-2 GeV. The gravity is localized on the visible brane, and the spectrum of the gravitational KK modes is discrete and can be of the order of TeV. The corrections to the four-dimensional Newtonian potential from the higher order of gravitational KK modes are exponentially suppressed and can be safely neglected in current experiments. In an appendix, we also present a systematical and pedagogical study of the Gauss–Codacci equations and Israel's junction conditions across a (D - 1)-dimensional hypersurface, which can be either spacelike or timelike.

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DEXP: A fast method to determine the depth and the structural index of potential fields sources

Geophysics ◽

10.1190/1.2399452 ◽

2007 ◽

Vol 72 (1) ◽

pp. I1-I11 ◽

Cited By ~ 115

Author(s):

Maurizio Fedi

Keyword(s):

Potential Field ◽

Extreme Points ◽

Power Laws ◽

Potential Fields ◽

Newtonian Potential ◽

Specific Power ◽

Scaling Exponent ◽

Fast Method ◽

Structural Index ◽

Derivatives Of

We show that potential fields enjoy valuable properties when they are scaled by specific power laws of the altitude. We describe the theory for the gravity field, the magnetic field, and their derivatives of any order and propose a method, called here Depth from Extreme Points (DEXP), to interpret any potential field. The DEXP method allows estimates of source depths, density, and structural index from the extreme points of a 3D field scaled according to specific power laws of the altitude. Depths to sources are obtained from the position of the extreme points of the scaled field, and the excess mass (or dipole moment) is obtained from the scaled field values. Although the scaling laws are theoretically derived for sources such as poles, dipoles, lines of poles, and lines of dipoles, we give also criteria to estimate the correct scaling law directly from the data. The scaling exponent of such laws is shown to be related to the structural index involved in Euler Deconvolution theory. The method is fast and stable because it takes advantage of the regular behavior of potential field data versus the altitude [Formula: see text]. As a result of stability, the DEXP method may be applied to anomalies with rather low SNRs. Also stable are DEXP applications to vertical and horizontal derivatives of a Newtonian potential of various orders in which we use theoretically determined scaling functions for each order of a derivative. This helps to reduce mutual interference effects and to obtain meaningful representations of the distribution of sources versus depth, with no prefiltering. The DEXP method does not require that magnetic anomalies to be reduced to the pole, and meaningful results are obtained by processing its analytical signal. Application to different cases of either synthetic or real data shows its applicability to any type of potential field investigation, including geological, petroleum, mining, archeological, and environmental studies.

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Automatic choice of parameters at approximation of functions by rational splines

Researches in Mathematics ◽

10.15421/248709 ◽

1987 ◽

pp. 52

Author(s):

A.D. Malysheva

Keyword(s):

Sufficient Conditions ◽

Order Of Approximation ◽

Approximation Of Functions ◽

Smooth Functions ◽

Higher Derivatives ◽

Necessary And Sufficient ◽

Automatic Choice ◽

Derivatives Of ◽

Best Parameters ◽

Approximation Of A Function

We obtain necessary and sufficient conditions put on the parameters of rational splines that provide given order of approximation of smooth functions. We point out the formulas of asymptotically the best parameters of rational splines that, while providing the best order of approximation of a function by rational splines, do not contain information about the values of higher derivatives of a function.

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The Newtonian Potential and the Demagnetizing Factors of the General Ellipsoid

10.31219/osf.io/fzh93 ◽

2017 ◽

Author(s):

Giovanni Di Fratta

Keyword(s):

Integral Representation ◽

Explicit Expression ◽

Representation Formula ◽

Newtonian Potential ◽

Bounded Domains ◽

Dimensional Problem ◽

Smooth Functions ◽

Integral Representation Formula ◽

Transport Theorem ◽

Stokes Integral

The objective of this paper is to present a modern and concise new derivation for the explicit expression of the interior and exterior Newtonian potential generated by homogeneous ellipsoidal domains in $\mathbb{R}^N$ (with $N \geqslant 3$). The very short argument is essentially based on the application of Reynolds transport theorem in connection with Green-Stokes integral representation formula for smooth functions on bounded domains of$\mathbb{R}^N$, which permits to reduce the N-dimensional problem to a 1-dimensional one. Due to its physical relevance, a separate section is devoted to the derivation of the demagnetizing factors of the general ellipsoid which are one of the most fundamental quantities in ferromagnetism.

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A verified method for solving piecewise smooth initial value problems

International Journal of Applied Mathematics and Computer Science ◽

10.2478/amcs-2013-0055 ◽

2013 ◽

Vol 23 (4) ◽

pp. 731-747 ◽

Cited By ~ 3

Author(s):

Ekaterina Auer ◽

Stefan Kiel ◽

Andreas Rauh

Keyword(s):

Initial Value Problems ◽

System Model ◽

Future Research ◽

Initial Value ◽

Smooth Functions ◽

Solution Processes ◽

Right Hand ◽

Future Research Directions ◽

The Right ◽

Derivatives Of

Abstract In many applications, there is a need to choose mathematical models that depend on non-smooth functions. The task of simulation becomes especially difficult if such functions appear on the right-hand side of an initial value problem. Moreover, solution processes from usual numerics are sensitive to roundoff errors so that verified analysis might be more useful if a guarantee of correctness is required or if the system model is influenced by uncertainty. In this paper, we provide a short overview of possibilities to formulate non-smooth problems and point out connections between the traditional non-smooth theory and interval analysis. Moreover, we summarize already existing verified methods for solving initial value problems with non-smooth (in fact, even not absolutely continuous) right-hand sides and propose a way of handling a certain practically relevant subclass of such systems. We implement the approach for the solver VALENCIA-IVP by introducing into it a specialized template for enclosing the first-order derivatives of non-smooth functions. We demonstrate the applicability of our technique using a mechanical system model with friction and hysteresis. We conclude the paper by giving a perspective on future research directions in this area.

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The Newtonian potential and the demagnetizing factors of the general ellipsoid

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2016.0197 ◽

2016 ◽

Vol 472 (2190) ◽

pp. 20160197 ◽

Cited By ~ 5

Author(s):

Giovanni Di Fratta

Keyword(s):

Integral Representation ◽

Representation Formula ◽

Newtonian Potential ◽

Bounded Domains ◽

Dimensional Problem ◽

Smooth Functions ◽

One Dimensional ◽

Integral Representation Formula ◽

Transport Theorem ◽

Stokes Integral

The objective of this paper is to present a modern and concise new derivation for the explicit expression of the interior and exterior Newtonian potential generated by homogeneous ellipsoidal domains in R N (with N ≥3). The very short argument is essentially based on the application of Reynold's transport theorem in connection with the Green–Stokes integral representation formula for smooth functions on bounded domains of R N , which permits to reduce the N -dimensional problem to a one-dimensional one. Owing to its physical relevance, a separate section is devoted to the the derivation of the demagnetizing factors of the general ellipsoid which are one of the most fundamental quantities in ferromagnetism.

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