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A double iteration process already used to find the nth root of a positive real number is analysed and showed to be equivalent to the Newton's method. These methods are of order two and three. Higher-order methods for finding the nth root are also mentioned.

An iterative algorithm for computing the principal nth root of a positive-definite matrix is presented. The algorithm is based on the Gauss-Legendre approximation of a definite integral. We present a parallelization in which we use as many processors as the order of the approximation. An analysis of the error introduced at each step of the iteration indicates that the algorithm converges more rapidly as the order of the approximation (and thus the number of processors) increases. We describe the results of our implementation of an eight-processor Meiko CS-2, comparing the parallel algorithm to the fastest sequential algorithm, which is the Hoskins-Walton method.

Dynamical reflection electron microscopy (REM) can provide a wealth of time-resolved data pertinent to the initial and intermediate stages of cluster growth. REM allows one to follow and quantify the size and shape evolution of individual clusters. Average cluster size and cluster density data can be obtained as a function of time due to the large field of view produced by REM image foreshortening. We describe here a methodology for extracting these data from dynamical REM experiments, based on a geometrical model for the interpretation of REM images from two- and three-dimensional clusters. This methodology has been applied to studies of In cluster growth on InP (110) surfaces at 650°C. The average In cluster height and length initially increased as a fourth root of time, with a constant contact angle with the surface, consistent with surface-diffusion-limited growth of 3D clusters. The same behavior was found in the later stage of cluster growth, but the intermediate stage showed anomalous power laws for the cluster height and base length, accompanied by a decrease in the contact angle between the In clusters and the InP (110) surface. This anomalous regime can be explained by growth of In clusters into the InP substrate, when the true contact angle is no longer defined with respect to the InP(110) surface.

We consider how to accelerate fermionic molecular dynamics algorithms by introducing n pseudofermion fields coupled with the nth root of the fermionic kernel. This reduces the maximum pseudofermionic force, and thus allows a larger molecular dynamics integration step size without hitting an instability in the integrator., 3 pages, 1 figure

The geometric mean of a series of positive numbers a1,...,an is defined as the nth root of its product. Generalizing this concept to positive definite matrices is not straightforward due to the noncommutativity. Based on a list of desired properties –the ALM-list– initially some recursive algorithms were proposed. Unfortunately, these algorithms were typically quite time consuming. A novel optimization based approach leads to the concept of the Karcher mean and is nowadays favored as the generalization towards matrices of the geometric mean. It is not so well-known that one can design for two scalars an algorithm quadratically convergent to the geometric mean, solely relying on arithmetic and harmonic means. The procedure is straightforward, compute the arithmetic and harmonic mean, and iterate this procedure with these newly computed means. Apparently, this procedure also works for two matrices. In these notes, we will propose a whole series of fast possible generalizations, and conduct numerical experiments with these algorithms. Unfortunately most of them do not satisfy all imposed ALM-constraints, but on the other hand, they converge rapidly and approximate the Karcher mean. As such they can be used to initiate manifold optimization procedures for computing the Karcher mean. ispartof: pages:785-793 ispartof: Lecture Notes in Computer Science vol:8085 pages:785-793 ispartof: Geometric Science of Information location:Paris, France date:28 Aug - 30 Aug 2013 status: published

We construct a C‐algebra C associated with an R‐matrix of the 2‐state chiral Potts model. This algebra will be shown to coincide with the Sklyanin algebra at an 8‐torsion point and, in the self‐dual case, with Uq(sl2) at the fourth root of unity.

The dual mode sorption solubility isotherms assume, and in instances Zimm–Lundberg analysis of the solubilities show, that glassy polymers are heterogeneous and that the distribution of the solute in the polymer is also inhomogeneous. Under some conditions, the heterogeneities cannot be represented as holes. A mathematical model describing diffusion in inhomogeneous polymer membranes is presented using Cahn and Hilliard’s gradient theory. The fractional mass uptake is found to be proportional to the fourth root of time rather than the square root, predicted by Fickian diffusion. This type of diffusion is classified as pseudo‐Fickian. The model is compared with one experimental result available. A negative value of the persistence factor is obtained and the results are interpreted.

We introduce a novel technique for creating maps of bandwidth supported locally by non-uniform sampling schemes of geophysical data. The input data required are solely the coordinates of the actual sampling locations. The term “bandwidth supported locally by non-uniform sampling” refers here to the maximum bandwidth of data with a flat amplitude spectrum that could be reconstructed by an optimum “sinc-function” like linear operator at arbitrary sampling locations within the survey area without unacceptable high reconstruction error. Our technique creates individual maximum wavenumber estimators for selected locations, such that two-dimensional data can be reconstructed at non-sampling locations as long as the square root of the product of the two maximum spatial wavenumbers of the data is below the computed wavenumber estimator. For higher n-dimensional sampling, this extends to a wavenumber support estimator as the nth root of the product of the n maximum wavenumbers of the data. We outline how these maps can be computed from the local average of the size of cells designated to each sampling point, and give an example of a two-dimensional bandwidth support map for the midpoint locations of a part of a land seismic cross spread.

Radial excitations of the quark-antiquark string sweeping the Wilson-loop area are considered in the framework of the effective-action formalism. Identifying these excitations with the daughter Regge trajectories, we find corrections which they produce to the constituent quark mass. The energy of the quark-antiquark pair turns out to be mostly saturated by the constituent quark masses, rather than by the elongation of the quark-antiquark string. Specifically, while the constituent quark mass turns out to increase as the square root of the radial-excitation quantum number, the energy of the string increases only as the fourth root of that number.

Using methods from the theory of commutative graded Banach algebras, we obtain a generalization of the two dimensional Borsuk-Ulam theorem as follows: Let $\phi:S^{2} \rightarrow S^{2}$ be a homeomorphism of order n and $\lambda\neq 1$ be an nth root of the unity, then for every complex valued continuous function $f$ on $S^{2}$ the function $\sum_{i=0}^{n-1} \lambda^{i}f(\phi^{i}(x))$ must be vanished at some point of $S^{2}$. We give a generalization in term of action of compact groups. We also discuss about some noncommutative versions of the Borsuk- Ulam theorem

We present a new orbital localization function based on the sum of the fourth central moments of the orbitals. To improve the locality, we impose a power on the fourth central moment to act as a penalty on the least local orbitals. With power two, the occupied and virtual Hartree-Fock orbitals exhibit a more rapid tail decay than orbitals from other localization schemes, making them suitable for use in local correlation methods. We propose that the standard orbital spread (the square root of the second central moment) and fourth moment orbital spread (the fourth root of the fourth central moment) are used as complementary measures to characterize the locality of an orbital, irrespective of localization scheme.

The mean is the arithmetic average of the data. Outliers pull the mean toward the outliers. A trimmed mean is calculated by discarding data at the extremes. The median is the middle value when the data are arranged in numerical order. One measure of dispersion in the data is the average absolute deviation from the mean. A more popular measure is the variance, the average squared deviation from the mean (or its square root, the standard deviation). A boxplot shows the median, the middle 50 percent of the data, and the maximum and minimum values. The geometric mean is calculated by multiplying the numbers and taking the nth root, and is often used for calculating annual rates of change or annual returns. The covariance and correlation both gauge whether when one variable is above its mean, the other variable tends to be above or below its mean. The correlation coefficient does not depend on the units in which the variables are measured and ranges between −1 and +1.

(1994). Approaches to the Formula for the nth Fibonacci Number. The College Mathematics Journal: Vol. 25, No. 2, pp. 139-142.

A re-examination of the traveling wave interaction (in the helix traveling wave tube and the free-electron laser) is performed using three-dimensional linear stability analyses for an idealized annular electron beam model in order to investigate the validity of the well-known scaling laws in which the gain varies as the cube (fourth) root of the current when space-charge effects are negligible (dominant). The results indicate that these scaling laws are simplistic generalizations which break down for broad bandwidth interactions, and that the actual variation of the pain, with the current can be more complex. A three-dimensional nonlinear analysis of a free-electron laser using a more realistic electron beam model is also discussed in which these scaling laws are also shown to break down,. >

The coherent signal of Small Angle Scattering superimposes to a flat background, which is given by incoherent scattering, by electronic noise and, sometimes, by radiation background around the instrument. In order to analyze the experimental data, the background level has to be taken into account and quite often theoretical estimations do not give values comparable to the experimental evidences. In this paper it is shown a new representation which allows in a simple way the background level determination ; in particular, in a convenient plot, the experimental data are fitted by a straight line passing through the origin, the slope of which is equal to the fourth root of the background.

Representations for quantum groups at q an Nth root of 1 with a structure resembling the algebraic structure in the theory of the chiral Potts models are constructed in the Uq(sp(2n,C)), Uq(so(2n+1,C)), and Uq(so(2n,C)) cases. The dimensions of the given representations and their number of free parameters are N(1/2)(dim g‐rank g), dim g, respectively.

Let K be a subfield of C. We give a criterion for a nonsingular matrix A in MmK to have an nth root in MmK. The number of similarity classes of nth roots of A is given for K = R. Further we indicate which matrices A in MmK have infinitely many nth roots in MmK.

For A(C nxn , let W⊥n(A)-{per(U▴AU)∣UeUn } In [1] Iam conjectured that if n≥3, then W⊥n(A) lies on the positive real line If and only If ξA is positive definite for some nth root of unity ξin this note we prove that Tam's conjecture is true, and determine those A:eCnxn such that lies on the nonegative real line or the real line.

The \nl \cls for the N=1 supersymmetric KdV equation are shown to be related in a simple way to powers of the fourth root of its Lax operator. This provides a direct link between the supersymmetry invariance and the existence of \nl conservation laws. It is also shown that nonlocal conservation laws exist for the two integrable N=2 supersymmetric KdV equations whose recursion operator is known.

The general symmetric n × n determinant with diagonal coefficients unity is expressed as a sum of products of linear forms. The resulting expansion (which generalizes a familiar identity for 3 × 3 determinants) is used to show that if these linear forms are nonnegative, then the nth root of the determinant lies between their arithmetic mean and their geometric mean. In effect, the result gives an expression for the determinant of an n-ary quadratic form with diagonal coefficients unity in terms of the values taken by the form at the vertices of the cube [−1, 1]n, which under certain conditions leads to bounds for the determinant of the form in terms of these values.

Based on idealized interconnect scaling rules, we derive the optimal distribution of linewidths as a function of length for wire-limited layouts utilizing RC-limited interconnections. We show that the width of the wires should be chosen proportional to the cube root of their length for two-dimensional layouts and proportional to the fourth root of their length for full three-dimensional layouts so as to minimize average signal delay.

The quantum group for the ZN model is studied from the braid group representation. The fundamental representation is constructed from the Weyl relation ZX= omega XZ with omega being an Nth root of unity. In the case of N=2 (the eight vertex model), the quantum group is shown to be a homomorphic image of the GLq(2) with q2=1.

Summary We show that, within the family of power transformations of a Chisquare variable, the square and fourth roots minimize Pearson's index of kurtosis. Two new transtormations of the fourth root, a symmetrized-truncated version and its linear combination with the square root are also studied. The first transformation shows a considerable improvement over the fourth root while the second one turns out to be even more accurate than Hilferty-Wilson's cube root transformation.

The most common use for fast Fourier transform is in high speed signal processing, encryption algorithms and other fields, which use the properties of primitive complex nth root of unity in complex number field. In order to implement the FFT-based large integers multiplication arithmetic whose time complexity is O(nlogn), this paper discusses how to select the primitive nth root of unity for FFT-based large integers multiplication under modulo-p arithmetic, where p is a prime or a composite number, further more, some related proofs which the primitive nth root of unit satisfies the condition of discrete Fourier transform(DFT) and its inverse are given too.

Recently some methods have been proposed to find the distance and weight distribution of cyclic codes using Grobner bases (Sala in Appl. Algebra Engrg. Comm. Comput. 13(2):137–162, 2002; Mora and Sala in J. Symbolic Comput. 35(2):177–194, 2003). We identify a class of codes for which these methods can be generalized. We show that this class contains all interesting linear codes (i.e., with d≥2) and we provide variants and improvements.

Let μ be a positive measure with compact support on R. We consider the nth root asymptotic behavior of orthonormal polynomials associated with the measure μ. The main result consists of two theorems: (i) a characterization and (ii) a localization theorem. In the first theorem regular nth root asymptotic behavior on a subset of the support of the measure μ is compared with the asymptotic behavior of other polynomial sequences, and equivalences between the different types of behavior are proved. In the second theorem the asymptotic behavior of the original orthonormal polynomials is characterized by the asymptotic behavior of polynomials orthonormal with respect to restrictions of the measure μ.

Let W(x) = e-Q(z) where Q(x) is even, continuous on ℝ and of slower than polynomial growth at infinity. We establish, under certain conditions on Q, nth root asymptotics for the extremal error associated with W, E np(W), 0 < p ≤ ∞ where

Conventional stacking techniques are not generally suitable for airborne TEM surveys conducted at speeds exceeding 150 km/hr. For late channels (2?50 ms) linear stacking can be tolerated for up to 8 cycles representing lateral translations of 30 to 40 m accompanied by variations in coupling. The Nth root stack developed to reduce the effect of spikes in geophysical data can result in considerable distortion complicating the effects of sferics activity. Consequently other procedures are required including bandpass filters to compensate for bird motion, decay curve analysis, and spike rejection based on observations at a base station or predictive filtering.

[1] In this paper, we discuss the effects of magnetospheric activity (as measured by Kp) on an ion distribution function signature of nonlinear particle dynamics in current sheet-like magnetic fields. The signature manifests itself as a series of peaks in the ion distribution function whose separation depends on the fourth root of the energy and parameters that describe the current sheet structure. We have found that the signature is evident in Geotail CPI data for Kp ≤ 3+ and that the larger the value of Kp, the closer the peaks are together. We have also used the energy resonance signature in conjunction with measurements of the magnetic field to determine a lower bound on the current sheet thickness and have found that this lower bound decreases for increasing levels of magnetospheric activity. For values of Kp > 4−, there are frequently peaks in the ion distribution function; however, their separations do not follow a simple rule.

This paper introduces a novel quantitative framework for measuring the risk and the reward provided by adaptive transmission schemes. In particular, the reward is measured as the expected value of the link spectral efficiency in excess of some predefined threshold. The risk, on the other hand, is the nth root of the nth order lower partial moment of the link spectral efficiency distribution. We apply mathematical tools of finance theory to analyze the risk-reward performance of various state-of-the-art adaptive transmission schemes in generic multi- antenna channels. We identify the maximum-return, minimum- risk, efficient, and optimal risk-reward schemes. The numerical results suggest that in a general case the optimal risk-reward scheme is neither the scheme that maximizes the expected link spectral efficiency nor the scheme that minimizes the risk by minimizing, e.g., the outage probability. The financial risk-reward theory brings a new intuition to the understanding of adaptive transmission in nonergodic channels. (15 refs.)

Many results from lattice QCD of broad importance to particle and nuclear physics are obtained with 2+1 flavors of staggered sea quarks. In the continuum limit, staggered fermions yield four species, called tastes. To reduce the number of tastes to one (per flavor), the simulation employs the fourth root of the four-taste staggered fermion determinant. This talk surveys evidence in favor of this procedure, refutes recent criticisms, and reviews recent algorithmic and technical improvements. Physics results are covered in other plenary talks., Comment: 25 pages; Lattice 2007 (plenary); v2 corrects typos, adds references, moves endnote to footnote, and conforms with version published in PoS

Rainfall occurs as synoptic events, and in many climates annual totals are made up of a moderate to large number of events of varying amounts. Assumption of an underlying Poisson process of event occurrence leads to a simple statistical density, in terms of physically meaningful parameters, for the fourth root of annual totals. Analysis of data for four locations gives estimates of event-recurrence rates in agreement with the number of rain-days in the local climates.

For q a primitive Nth root of unity we give an Nn-1 -dimensional representation of Uq( s l)(n, C )). Restrictin g to the case n = 3, N = 3 we then construct a trigonometric R matrix which intertwines tensor products of two such representations.

The process of step‐mediated growth in molecular beam epitaxy, where deposited atoms move along surface terraces until they eventually bind at a surface step, is currently being explored as a mechanism for forming two‐dimensional periodic structures. To succeed, such schemes require a periodic step distribution, i.e., uniform terrace lengths. A model, based on step‐mediated growth, is presented, which yields an analytical derivation of the approach to uniform terrace lengths on a stepped surface, given a terrace length distribution of finite width at the outset. The results show that the approach to uniform terrace lengths with increasing deposition is quite slow. The width of the terrace length distribution varies approximately as the inverse fourth root of the deposited coverage. This will only occur if the atoms attach themselves predominately at the up‐step of each terrace. Otherwise, the width of the terrace length distribution will grow without bounds.

In general there is no unique relationship between conductor curvature and surface charge density. However by restricting attention to situations for which the potential is a function of a single variable, the authors demonstrate that the magnitude of the surface charge density at any point of the conductor surface is proportional to the fourth root of the magnitude of the Gaussian curvature at this location.

I present a simple, constructive and elementary proof of the arithmetic‐geometric mean inequality. The proof does not require calculus. Applications to the design of a multistage rocket and to the computation of the nth root of a positive real number are presented.

Because the staggered fermion determinant is complex at nonzero mu, taking its fourth root leads to phase ambiguities. These unphysical effects cause the measure to become discontinuous; the problem becomes acute when Re mu exceeds approximately half the pion mass (when T>0 this rough bound probably moves towards larger mu). We show how to overcome the problem, but only very close to the continuum limit. This regime may be beyond reach with current resources., Lattice 2006 (High Temperature and Density), PoS format, 7 pages

[1] In this paper we present a survey of the quiet (Kp < 1+) current sheet thickness for X GSE between -20 R E and -80 R E as determined from an ion distribution function signature of nonlinear particle dynamics in current sheet-like magnetic fields. The signature manifests itself as a series of peaks in the ion distribution function whose separation depend on the fourth root of the energy and parameters that describe the current sheet structure. We have found clear evidence of the distribution function signature throughout the entire region of interest. Analysis of the data shows that the current sheet thickness is remarkably uniform in the region under study with an average thickness of 0.76 ± 0.56 R E . This result is consistent with measurements of the quiet time current sheet thickness made using other techniques.

International audience; In diesem Artikel führen wir einen neuen Regulator, der in Beziehung zu Gross Regulator für CM-Erweiterungen steht, für beliebige Zahlkörper ein. Dieser Regulator steht so in Beziehung mit der Arithmetik der logarithmischen Klassen, dass er nicht trivial ist, genau dann wenn die verallgemeinerte Gross Vermutung gilt. Als eine Anwendung geben wir eine Formel vom Hasse Typ für die logarithmische Klassenzahl biquadratischer Körper, die die vierten Einheitswurzeln enthalten.; In this article we introduce a new regulator for arbitrary number fields, related to Gross’s regulator defined for CM-extensions. More precisely, this regulator is linked to the arithmetic of logarithmic classes, so that its non-triviality is equivalent to Gross’s generalized conjecture. As an application, we give a formula of Hasse type for the logarithmic class number of biquadratic fields containing the fourth root of unity.; Le régulateur introduit dans cet article est, quoique voisin du classique régulateur de Gross qui est spécifique aux extensions de type CM, attaché à un corps de nombres quelconque. Plus précisément, il est défini en liaison avec l’arithmétique des classes logarithmiques, de sorte que sa non-trivialité soit équivalente à la conjecture de Gross généralisée. Comme application, une formule de type Hasse pour les classes logarithmiques des corps biquadratiques contenant les racines quatrièmes de ’unité est donnée.

We discuss the optimal detection strategy for a stochastic background of gravitational waves in the case n detectors are available. In literature so far, only two cases have been considered: 2- and n-point correlators. We generalize these analysises to m-point correlators (with m, Comment: 12 pages, version accepted by Class. & Quant. Grav

An example of interpolation by means of local field theories between the case of normal Kogut-Susskind fermions and the case of keeping just the fourth root of the Kogut-Susskind determinant is given. For the fourth root trick to be a valid approximation certain limits need to be smooth. The question about the validity of the fourth root trick is not resolved, only cast into a local field theoretical framework., 2p

If there is proposed a series,1 of quantities in arithmetic proportion (or as the natural sequence of numbers)2 continually increasing, beginning from a point or 0 (that is, nought, or nothing),3 thus as 0, 1, 2, 3, 4, etc., let it be proposed to inquire what is the ratio of the sum of all of them, to the sum of the same number of terms equal to the greatest.

In this chapter, we consider analogues of Pell’s equation of higher degree, particularly the fourth. Throughout, n is a positive integer and c is an integer which has a real nth root 6 of the same sign. Define $${g_{c}}({x_{1}},{x_{2}},....,{x_{n}}) \equiv N({x_{1}} + {x_{2}}\theta + {x_{3}}{\theta ^{2}} + \cdots + {x_{n}}{\theta ^{{n - 1}}}) \equiv \prod\limits_{{i = 0}}^{n} {({x_{1}} + {x_{2}}({\xi ^{i}}\theta ) + \cdots + {x_{n}}{{({\xi ^{i}}\theta )}^{{n - 1}}})} $$ where \(\varsigma = \cos \frac{{2\pi }}{n} + i\sin \frac{{2\pi }}{n} \) is a primitive nth root of unity. The analogue of Pell’s equation that we want to examine is $${g_{c}}({x_{1}},{x_{2}}...,{x_{n}}) = 1. $$

We propose a new lattice reduction method. Our algorithm approximates shortest lattice vectors up to a factor ≤ (k/6) n/2k and makes use of Grover’s quantum search algorithm. The proposed method has the expected running time O(n 3(k/6) k/8 A + n 4 A). That is about the square root of the running time O(n 3(k/6) k/4 A + n 4 A) of Schnorr’s recent random sampling reduction which in turn improved the running time to the fourth root of previously known algorithms. Our result demonstrates that the availability of quantum computers will affect not only the security of cryptosystems based on integer factorization or discrete logarithms, but also of lattice based cryptosystems. Rough estimates based on our asymptotic improvements and experiments reported in [1] suggest that the NTRU security parameter needed to be increased from 503 to 1277 if sufficiently large quantum computer were available nowadays.

As in Chapter 3, the nth cyclotomic polynomial Φ n is the minimal polynomial of a primitive nth root of unity. Recall that Φ n is given by $$ {\Phi_n}(z) = \prod\limits_{{\begin{array}{*{20}{c}} {1 \leqslant j \leqslant n} \\ {\gcd \left( {j,n} \right) = 1} \\ \end{array} }} {\left( {z - \exp \left( {{{{j2\pi i}} \left/ {n} \right.}} \right)} \right)} $$ .

In embedded computing, it is common to find applications such as signal processing, image processing, computer graphics or data compression that might benefit from hardware implementation for the computation of integer roots of order N≥2. However, the scientific literature lacks architectural designs that implement such operations for different values of N, using a low amount of resources. This article presents a parameterisable field programmable gate array (FPGA) architecture for an efficient Nth root calculator that uses only adders/subtractors and N2 location memory elements. The architecture was tested for different values of N={2,6,10,14,17}, using 64-bit number representation. The results show a consumption up to 10% of the logical resources of a Xilinx XC6SLX45-CSG324C device, depending on the value of N. The hardware implementation improved the performance of its corresponding software implementations in one order of magnitude. The architecture performance varies from several thousands to seven m...