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This paper develops the analysis of discrete-time periodically time-varying linear systems over finite fields. It is shown that the conditions for the existence of Floquet Transform for periodic linear systems over reals (or complex) do not carry forward for this case over finite fields. The existence of Floquet Transform is shown to be equivalent to the existence of an Nth root of the monodromy matrix for the class of non-singular periodic linear systems. As the verification of existence and computation of the Nth root is a computationally hard problem, an independent analysis of the solutions of such systems is carried without the use of Floquet Transform. It is proved that all initial conditions of such systems lie either in a periodic orbit or a chain leading to a periodic orbit. A subspace of the state space is also identified, containing all initial conditions lying on a periodic orbit. Further, whenever the Floquet Transform exists, more concrete results on the orbit lengths are established depending on the coprimeness of the orbit length with the system period.

We propose a new method and hardware architecture to compute the functions expressed as $X^{Y}$ ( $X$ and $Y$ are arbitrary floating-point numbers), which can support arbitrary Nth root, exponential and power operations. Because of the complexity of direct computation, we usually convert it to logarithm, multiplication, and antilogarithm operations. Traditional approaches suffer from long latency, large area and high power consumption. To solve this problem, we propose a symmetric-mapping lookup table (SM-LUT) to be capable of computing $\log _{2}x$ ( $x\in [{1, 2}]$ ) and $2^{x}$ ( $x\in [{0, 1}]$ ) simultaneously. It lays the foundation for computing $X^{Y}$ . To further improve hardware performance of our architecture, we propose a multi-region address searcher to speed up the calculation of SM-LUT. In addition, we use an optimized Vedic multiplier to shorten the critical path and improve the efficiency of multiplication, which is included in computing $X^{Y}$ . Under the TSMC 40nm CMOS technology, we design and synthesize a reference circuit to compute $X^{Y}$ with a maximum relative error of 10−3. The report shows that the reference circuit achieves the area of $14338.50~\mu \text {m}^{2}$ and the power consumption of 4.59 mW at the frequency of 1 GHz. In comparison with the state-of-the-art work under the same input range and similar precision, it saves 78.57% area and 80.42% power consumption for $\sqrt [N]{R}$ computation and 82.89% area and 81.89% power consumption for $R^{N}$ computation averagely. On top of that, our architecture reduces the computation latency by 62.77% averagely and has one more order of magnitude of energy efficiency than others.

In recent studies, it has been shown that the theory of fourth-root metrics plays a very important role in physics, theory of space-time structures, gravitation, and general relativity. The class of weakly Berwald metrics contains the class of Berwald metrics as a special case. We establish necessary and sufficient conditions under which the fourth-root Finsler space with an (α, β)-metric is a weakly Berwald space.

Decision-making problems are more often tainted with uncertainty. Fuzzy numbers play utmost important role to band uncertainty, more especially intuitionistic fuzzy number (IFN) which is the extension of fuzzy number (FN). Different types of IFNs such as normal and generalized trapezoidal, triangular and symmetric hexagonal IFNs are explored. However, based on nature of the data, semi-elliptic type of IFN may exists in real world decision-making problems. In this paper, a maiden attempt has been made to study normal semi elliptic intuitionistic fuzzy number (NSEIFN). Our emphasis has been on arithmetic operations of NSEIFNs and comparing with the other existing IFNs. Also rank of NSEIFNs has been proposed based on mean and value. Apart from that inverse, exponential, logarithm, square root and nth root of NSEIFNs are derived. Finally, the proposed ranking method is applied to the decision making problem where criteria and rating of alternatives are represented in terms of NSEIFN. It is observed that the proposed model produces better results and overcome the drawbacks of existing approaches.

We recently showed how nonhomogeneous second-order difference equations that appear in describing the ABJM quantum spectral curve can be solved using a Mellin space technique. In particular, we provided explicit results for anomalous dimensions of twist-1 operators in the sl(2) sector at arbitrary spin values up to the four-loop order. We showed that the obtained results can be expressed in terms of harmonic sums with additional factors in the form of a fourth root of unity, and the maximum transcendentality principle therefore holds. Here, we show that the same result can also be obtained by directly solving the mentioned difference equations in the space of the spectral parameter u. The solution involves new highly nontrivial identities between hypergeometric functions, which can have various applications. We expect that this method can be generalized both to higher loop orders and to other theories, such as the N=4 supersymmetric Yang–Mills theory.

The theory of 4-th root metrics plays an important role in Biology, General Relativity and Seismic Ray Theory. Using the Shen's sy-separating trick, we prove that every conformally flat weakly Einstein 4-th root ( α , β ) -metric on a manifold M of dimension n ≥ 3 is either a Riemannian metric or a locally Minkowski metric. Then, we show that every conformally flat 4-th root ( α , β ) -metric of almost vanishing Ξ-curvature on a manifold M of dimension n ≥ 3 reduces to a Riemannian metric or a locally Minkowski metric.

There are two types of elliptic curves, ordinary elliptic curves and supersingular elliptic curves. In 2012, Sutherland proposed an efficient and almost deterministic algorithm for determining whether a given curve is ordinary or supersingular. Sutherland’s algorithm is based on sequences of isogenies started from the input curve, and computation of each isogeny requires square root computations, which is the dominant cost of the algorithm. In this paper, we reduce this dominant cost of Sutherland’s algorithm to approximately a half of the original. In contrast to Sutherland’s algorithm using j-invariants and modular polynomials, our proposed algorithm is based on Legendre form of elliptic curves, which simplifies the expression of each isogeny. Moreover, by carefully selecting the type of isogenies to be computed, we succeeded in gathering square root computations at two consecutive steps of Sutherland’s algorithm into just a single fourth root computation (with experimentally almost the same cost as a single square root computation). The results of our experiments using Magma are supporting our argument; for cases of characteristic p of 768-bit to 1024-bit lengths, our algorithm runs 43.6% to 55.7% faster than Sutherland’s algorithm.

An important problem of numerical flow simulation in the frame of an oil field development is the need to improve the quality and quantity of the input data, in particular, the data on cross-well formation properties. Well test is the most informative means of a reservoir borehole-to-borehole survey. When studying the cross-well formation properties based on the interpretation of continuous curves of pressure variations measured by telemetry sensors on down-hole pumps, the impact of adjacent wells and high noise contamination of the obtained data shall be taken into account. In an attempt to solve this issue this article studies multi-well deconvolution as a means to study all components of the pressure variations curve. The multi-well deconvolution can identify the relevant response to a change in the operating mode of a certain well and can process this response in traditional ways. The multi-well deconvolution method makes it possible to assess and account for noise impacts on the pressure curve. This approach also simplifies the curve processing, as the diagnosis of an interpreted formation model is done easier. This work proposes a new approach to building self-influence and influence functions: we propose to set them down as a sum of elementary time functions representing individual filtration modes of the formation. The well bore impact is represented exponentially; the bilinear flow – as a fourth root; the linear flow – as a second root; the radial flow – as a logarithm; the boundary effect – as a linear function. This way the self-influence and influence functions coefficients are set down linearly and Newton method can be used to define them. This approach was tested on two synthetic curves of the bottom-hole pressure obtained by modelling. The simulation and deconvolution curves of the bottom-hole pressure converged closely and this proved that the formation parameters selected for the modelling and retrieved during the self-influence and influence curves processing are close and this, consequently, proves high efficiency of the proposed approach. #COMESYSO1120.

In this chapter, we will introduce exponentiation with integer exponents of real numbers. We will then derive general properties of exponentiation including the negative operator. We will also present root of order n of real numbers. We will then derive general properties of the nth root.

In this chapter, we will present exponentiation in integers. We will then derive general properties of exponentiation including the negative operator. We will also present root of order n in integers and discuss the extension of the nth root to negative numbers.

The limit superior of the nth root of |x(n)|, where x is any nonzero solution of a scalar homogeneous linear difference equation of finite order, is identified in an elementary manner with ...

The set of functions {xq∣q∈R} is linearly independent over R (with respect to any open subinterval of (0, ∞)). The titular result is a corollary for any integer n ≥ 2 (and the domain [0, ∞)). Some more accessible proofs of that result are also given. Let F be a finite field of characteristic p and cardinality pk. Then the pth-root function F → F is a polynomial function of degree at most pk − 2 if pk ≠ 2 (resp., the identity function if pk = 2). Also, for any integer n ≥ 2, every element of F has an nth root in F if and only if, for each prime number q dividing n, q is not a factor of pk − 1. Various parts of this note could find classroom use in courses at various levels, on precalculus, calculus or abstract algebra. A final section addresses educational benefits of such coverage and offers some recommendations to practitioners.

In this work, we present four results for the Laplace inverse transform of functions that involve the nth root of a product of linear factors. In order to find the Laplace inverse transform, we considered a branch cut for the nth root and a region of suitable integration, to avoid the branching points. Due to that, the solution is in terms of integrals, we easily approach this solution for some specific parameters.

Cannabis is the most frequently used illicit drug in Belgium, where it is mainly cultivated indoor. To improve the fight against this drug, cannabis-profiling methods are required. Cannabis is a natural product and its chemical composition depends on many factors, which cause a high heterogeneity and variability in the secondary metabolites, and make this study challenging. The aim of this study is to combine cannabis profiling with statistical methodology to evaluate the intra (within)- and inter (between)–plantation variabilities with the goal to define a suitable approach linking seized marijuana to given plantations. The data set used contains 46 samples from 9 locations. The chemical profiles, consisting of data from eight cannabinoids, are obtained by gas chromatography - mass spectrometry. The raw data (peak areas) is pretreated with different preprocessing methods. The Pearson correlation coefficients between intra-location profiles were calculated after each pre-treatment, and the 95 and 99 % confidence limits determined. All preprocessed data were then compared with the internal standard normalization reference method with the aim to minimize the overlap between intra- and inter-location results, i.e. to reduce the number of false positives, and to obtain the best discrimination. Furthermore, cross-validation was used to evaluate the model originating from the most efficient data pre-treatment technique. The best results were obtained, when the peak areas were normalized to the internal standard with subsequent calculation of the fourth root. It results in a reduction of false positives for both confidence limits to 11 % and 14 % compared to 21 % and 27 % for the reference method. Cross-validation reveals similar false positive results as for the calibration set. In conclusion, when preprocessing the data, an improved model is obtained resulting in a significant decrease in the number of false positives. After studying the predictive performance of the model, it appears to be representative for the entire plantation information.

Ground-penetrating radar (GPR) is an effective tool for subsurface detection. Due to the influence of the environment and equipment, the echoes of GPR contain significant noise. In order to suppress noise for GPR data, a method based on singular value decomposition (SVD) of a window-length-optimized Hankel matrix is proposed in this paper. First, SVD is applied to decompose the Hankel matrix of the original data, and the fourth root of the fourth central moment of singular values is used to optimize the window length of the Hankel matrix. Then, the difference spectrum of singular values is used to construct a threshold, which is used to distinguish between components of effective signals and components of noise. Finally, the Hankel matrix is reconstructed with singular values corresponding to effective signals to suppress noise, and the denoised data are recovered from the reconstructed Hankel matrix. The effectiveness of the proposed method is verified with both synthetic and field measurements. The experimental results show that the proposed method can effectively improve noise removal performance under different detection scenarios.

As part of our ongoing work on the enumeration of symmetry classes of lozenge tilings of hexagons with certain four-lobed structures removed from their center, we consider the case of the tilings which are both vertically and horizontally symmetric. In order to handle this, we need an extension of Kuo's graphical condensation method, which works in the presence of free boundary. Our results allow us to compute exactly the correlation in a sea of dimers of a macroscopic dent in a 90 degree wedge with mixed boundary conditions. We use previous results to compute the correlation of the corresponding symmetrized system with no boundary, and show that its fourth root has the same log-asymptotics as the correlation of the dent in the 90 degree wedge. This is the first result of this kind involving a macroscopic defect. It suggests that the connections between dimer systems with gaps and 2D electrostatics may be deeper that previously thought., 17 pages

Parabolic equation techniques are based on rational approximations of the square root and other functions. The split-step Pade solution is based on an exponential of the square root [1], which takes into account the range numerics and allows large range steps. An initial condition may be obtained using the self-starter [2], which is based on a far-field approximation of a Hankel function of the square root. Accurate solutions to range-dependent problems may be obtained with the energy-conserving parabolic equation [3], which is based on the fourth root. Three-dimensional parabolic equations are derived by factoring the wave equation without making the uncoupled azimuth approximation [4–6]. When horizontal variations in the environment are sufficiently gradual so that energy coupling between modes may be neglected, three-dimensional calculations can be avoided by solving horizontal wave equations for the mode coefficients [7, 8].

In this paper we prove an nth root test for series as well as a Cauchy–Hadamard type formula and Abel's' theorem for power series on universally complete Archimedean complex vector lattices. These results are aimed at developing an alternative approach to the classical theory of complex series and power series using the notion of order convergence.

For a fixed integer $n$, we study the question whether at least one of the numbers $\Re X\omega^k$, $1\leq k\leq n$, is $\varepsilon$-close to an integer, for any possible value of $X\in\mathbb{C}$, where $\omega$ is a primitive $n$th root of unity. It turns out that there is always a $X$ for which the above numbers are concentrated around $1/2\bmod1$. The distance from $1/2$ depends only on the local properties of $n$, rather than its magnitude. This is directly related the so-called "pyjama" problem which was solved recently., Comment: 7 pages

This paper presents a new algorithm to approximate the nth root of a given real number z. The proposed algorithm is a hybrid algorithm between the Bisection method and the combination of the inverse of sine series and Newton’s method. The proposed algorithm will be tested to find the numerical results on Matlab programming. The results showed that if z > 1, the proposed method converges with the initial interval [1, z] and if 0 < z < 1, it converges with initial interval [0,4]. The comparison of the obtained result with Newton’s method and AS-Newton’s method shown that the efficiency of the proposed algorithm may better than both of them concerning the initial value.

Let $q$ be an $n^{th}$ root of unity for $n > 2$ and let $T_n(q)$ be the Taft (Hopf) algebra of dimension $n^2$. In 2001, Susan Montgomery and Hans-J��rgen Schneider classified all non-trivial $T_n(q)$-module algebra structures on an $n$-dimensional associative algebra $A$. They further showed that each such module structure extends uniquely to make $A$ a module algebra over the Drinfel'd double of $T_n(q)$. We explore what it is about the Taft algebras that leads to this uniqueness, by examining actions of (the Drinfel'd double of) Hopf algebras $H$ "close" to the Taft algebras on finite-dimensional algebras analogous to $A$ above. Such Hopf algebras $H$ include the Sweedler (Hopf) algebra of dimension 4, bosonizations of quantum linear spaces, and the Frobenius-Lusztig kernel $u_q(\mathfrak{sl}_2)$., 28 pages, 1 table

Given a knot $K$ we may construct a group $G_n(K)$ from the fundamental group of $K$ by adjoining an $n$th root of the meridian that commutes with the corresponding longitude. For $n\geq2$ these "generalised knot groups" determine $K$ up to reflection (Nelson and Neumann, 2008; arXiv:0804.0807). The second author has shown that for $n\geq2$, the generalised knot groups of the square knot $SK$ and the granny knot $GK$ can be distinguished by counting homomorphisms into a suitably chosen finite group (arXiv:0706.1807). We extend this result to certain generalised knot groups of square and granny knot analogues $SK_{a,b}=T_{a,b}\# T_{-a,b}$, $GK_{a,b}=T_{a,b}\# T_{a,b}$, constructed as connect sums of $(a,b)$-torus knots of opposite or identical chiralities. More precisely, for coprime $a,b\geq2$ and $n$ satisfying a certain coprimality condition with $a$ and $b$, we construct an explicit finite group $G$ (depending on $a$, $b$ and $n$) such that $G_n(SK_{a,b})$ and $G_n(GK_{a},b)$ can be distinguished by counting homomorphisms into $G$. The coprimality condition includes all $n\geq2$ coprime to $ab$. The result shows that the difference between these two groups can be detected using a finite group., Comment: 18 pages, 3 figures

When considering rings acting on groups, one may specialize to subrings of \(\mathbb {Q}.\) A group action by a subring of \(\mathbb {Q}\) may be interpreted as root extraction in the said group, which is the underlying subject of Chap. 5. In Sect. 5.1, we discuss the theory of isolators and isolated subgroups. This topic is related to the study of root extraction in nilpotent groups, which is the theme of Sect. 5.2. We discuss some of the theory of root extraction in groups, emphasizing the main results for nilpotent groups. Residual properties of nilpotent groups are also discussed in Sect. 5.2, as well as embeddings of nilpotent groups into nilpotent groups with roots. Section 5.3 contains an introduction to the theory of localization of nilpotent groups. A group is said to be P-local, for a set of primes P, if every group element has a unique nth root whenever n is relatively prime to the elements of P. The main result in this section is: Given a nilpotent group G, there exists a nilpotent group G P of class at most the class of G which is the best approximation to G among all P-local nilpotent groups.

In this note we consider the sequence whose n th term is the number of M -sequences of length n . We show that the n th term of this sequence is bounded above by the n th Fibonacci number and bounded below by the number of integer partitions of n into distinct parts.

In this paper we propose two original iterated maps to numerically approximate the nth root of a real number. Comparisons between the new maps and the famous Newton-Raphson method are carried out, including fixed point determination, stability analysis and measure of the mean convergence time, which is confirmed by our analytical convergence time model. Stability of solutions is confirmed by measuring the Lyapunov exponent over the parameter space of each map. A generalization of the second map is proposed, giving rise to a family of new maps to address the same problem. This work is developed within the language of discrete dynamical systems.

In this paper new method is introduced for complex numbers; this method does not include imaginary number “i” but produces the same results that occur in Addition, Subtraction, Multiplication & Division of complex numbers, also proof of Eluer Formula eiθ and De Moivre theorem without using imaginary number “i”. Furthermore placing the light on the square root of a negative number, the square root of a negative number is equal to the square root of the same positive real number but with an angle of 90 degree to real line. The intention is that there’s nothing mystical about imaginary number “i”. The square root of minus one is just as real as any other number. Complex numbers exists without imaginary number “i”. This paper is limited to the results which are already established.

We construct a sequence of rational functions, which will be called disc wavelets, parametrized by points in the unit disc λk,n(γ)=(1−γ2n)(ωn)k,n∈N, 0≤k

Based on the Taylor's expansion of an inverse function, we extend the Schroder's process to find and improve high-order fixed-point iteration functions (IFs) for solving a nonlinear equation. We illustrate the extended processes by using them to find better iterative methods to compute the nth root and the logarithm of a strictly positive real number. IFs for inverse trigonometric function evaluations are also considered.

We analyze the asymptotic extremal growth rate of the Betti numbers of clique complexes of graphs on n vertices not containing a fixed forbidden induced subgraph H. In particular, we prove a theorem of the alternative: for any H the growth rate achieves exactly one of five possible exponentials, that is, independent of the field of coefficients, the nth root of the maximal total Betti number over n-vertex graphs with no induced copy of H has a limit, as n tends to infinity, and, ranging over all H, exactly five different limits are attained. For the interesting case where H is the 4-cycle, the above limit is 1, and we prove a slightly superpolynomial upper bound., Comment: 14 pages (32 with appendix). Version 2 newly contains a proof of an optimal bound for I_4-free graphs (in the appendix). Version 3 was rearranged for improved readability and also we now claim only a slightly weaker bound for the case of 4-cycle

In this paper, we study locally projectively flat fourth root Finsler metrics and their generalized metrics. We prove that if they are irreducible, then they must be locally Minkowskian.

For a system of two measures supported on a starlike set in the complex plane, we study the asymptotic properties of the associated multiple orthogonal polynomials Qn and their recurrence coefficients. These measures are assumed to form a Nikishin-type system, and the polynomials Qn satisfy a three-term recurrence relation of order three with positive coefficients. Under certain assumptions on the orthogonality measures, we prove that the sequence of ratios {Qn+1/Qn} has four different periodic limits, and we describe these limits in terms of a conformal representation of a compact Riemann surface. Several relations are found involving these limiting functions and the limiting values of the recurrence coefficients. We also study the nth root asymptotic behavior and zero asymptotic distribution of Qn.

Fourth root metrics are a special and important class of Finsler metrics, which have been applied to physics. In this paper, we study invariant fourth root Finsler metrics on the Grassmannian manifolds S O ( p + q ) / S O ( p ) × S O ( q ) . By using the results from the theory of invariant polynomials of Lie groups, we obtain a complete classification of such metrics. Further, some invariant 2 m -th root Finsler metrics are also given.

This paper continues the study of colourings of the sets of cyclotomic integers ℳ n = ℤ[ξ n ] (ξ n = e 2πi/n , a primitive nth root of unity) with class number one. We present results for the colour symmetry group and colour preserving group for a given ideal colouring of ℳ n , with φ(n) = 8 and 10, thus completing the characterisation of the colour preserving group for the cases φ(n) ≤ 10, where φ is Euler's totient function.

In 2001, Borwein, Choi, and Yazdani looked at an extremal property of a class of polynomial with ±1 coefficients. Their key result was: Theorem (See Borwein, Choi, Yazdani, 2001.) Let f ( z ) = ± z ± z 2 ± ⋯ ± z N − 1 , and ζ a primitive Nth root of unity. If N is an odd positive integer then max i | f ( ζ i ) | ⩾ N with equality if and only if N is an odd prime. Moreover, if equality holds, they gave an explicit construction for f ( z ) . In this paper, we look at the case when N is even. In particular, we investigate the following Conjecture Let f ( z ) and ζ be as above. If N > 2 is an even positive integer then max i | f ( ζ i ) | ⩾ N + 1 with equality if and only if N + 1 is a power of an odd prime. This conjecture was made after extensive computations. Partial results towards proving this conjecture are given.

Problem statement: The need to find an efficient and reliable algorit hm for computing the exact real roots of the steady-state polynomial enc ountered in the investigation of temperature profil es in biological tissues during Microwave heating and other similar cases as found in the literature gave rise to this study. Approach: The algorithm (simply called ERA-Exact Root Algorithm) adopted polynomial deflation technique and uses Newton-Raphson iterative procedure though with a modified termination rule. A general formula was specified f or finding the initial approximation so as to overcome the limitation of local convergence which is inherent in Newton's method. Results: A new algorithm for finding the real roots of an nth degr ee polynomial at a practically low computational co st was obtained. Conclusion/Recommendations: ERA is simple, flexible, easy to use and has clear benefits and preferences to a number of existing me thods.

For a field F and a family of central simple F-algebras we prove that there exists a regular field extension E/F preserving indices of F-algebras such that all the algebras from the family are cyclic after scalar extension by E. Let \( \mathcal{A} \) be a central simple algebra over a field F of degree n with a primitive nth root of unity ρn. We construct a quasi-affine F-variety Symb(\( \mathcal{A} \)) such that for a field extension L/F Symb(\( \mathcal{A} \)) has an L-rational point if and only if \( \mathcal{A}{ \otimes_F}L \) is a symbol algebra. Let \( \mathcal{A} \) be a central simple algebra over a field F of degree n and K/F be a cyclic field extension of degree n. We construct a quasi-affine F-variety C(\( \mathcal{A} \),K) such that, for a field extension L/F with the property [KL : L] = [K : F], the variety C(\( \mathcal{A} \),K) has an L-rational point if and only if KL is a subfield of \( \mathcal{A}{ \otimes_F}L \).

We consider a class of nonlinear matrix equations X n - f ( X ) = 0 where f is a self-map on the convex cone P ( k ) of k × k positive definite real matrices. It is shown that for n ⩾ 2 , the matrix equation has a unique positive definite solution depending continuously on the function f if f belongs to the semigroup of nonexpansive mappings with respect to the GL ( k , R ) -invariant Riemannian metric distance on P ( k ) , which contains congruence transformations, translations, the matrix inversion and in particular symplectic Hamiltonians appearing in Kalman filtering. We show that the sequence of positive definite solutions varying over n ⩾ 2 converges always to the identity matrix.

Two modifications of Newton’s method to accelerate the convergence of the nth root computation of a strictly positive real number are revisited. Both modifications lead to methods with prefixed order of convergence p∈N,p≥2. We consider affine combinations of the two modified pth-order methods which lead to a family of methods of order p with arbitrarily small asymptotic constants. Moreover the methods are of order p+1 for some specific values of a parameter. Then we consider affine combinations of the three methods of order p+1 to get methods of order p+1 again with arbitrarily small asymptotic constants. The methods can be of order p+2 with arbitrarily small asymptotic constants, and also of order p+3 for some specific values of the parameters of the affine combination. It is shown that infinitely many pth-order methods exist for the nth root computation of a strictly positive real number for any p≥3.

Heninger, Rains and Sloane proved that the nth root of the theta series of any extremal even unimodular lattice in Rn has integer coefficients if n is of the form 2i3j5k(i ≥ 3). Motivated by their discovery, we find the congruences between extremal modular forms and theta series of special types modulo powers of 2 and 3. This assertion enables us to prove that the 2nth root and the (3n/2)th root of the extremal modular form of weight n/2 have at least one non-integer coefficient.

The authors give a construction of multiblock space-time block codes from cyclic division algebras with a given nth root of unity as the nonnorm element. The construction uses local class field theory. Multiblock codes with M blocks can achieve an M-fold increase in diversity.

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A vanishing sum , where ζ n is a primitive nth root of unity and the a i s are non-negative integers is called minimal if the coefficient vector (a 0 , … , a n−1 ) does not properly dominate the coefficient vector of any other such non-zero sum. We show that for every c∈ℕ there is a minimal vanishing sum of nth roots of unity with its greatest coefficient equal to c, where n is of the form 3pq for odd primes p, q. This solves an open problem posed by Lenstra Jr.

This study outlines an improved method for calculating dose per monitor unit values for irregularly shaped electron fields using the nth root percent depth dose method. This method calculates the percent depth dose and output factors for an irregularly shaped electron field directly from the measured electron beam percent depth dose curves and output factors for circular fields. The percent depth dose curves and output factors for circular fields are normalized and measured at a fixed depth of maximum dose for a reference field, respectively. When compared with the sector integral lateral buildup ratio method, the percent depth dose data calculated using the nth root method accounts more accurately for the change in lateral scatter with decreasing field size. Therefore, it provides more accurate values of dose per monitor unit at different depths for all type of field shapes and beam energies. For beam energies in the range of 6-21 MeV, the differences between measured and calculated dose per monitor unit values, at different depths, were found to be within {+-}1.0% when the nth root percent depth dose method was used for calculation and 12.6% when the sector integral lateral buildup ratio method was used. The nth root percent depthmore » dose method was tested and compared with the sector integral lateral buildup ratio method for ten clinically used irregularly shaped inserts (cutouts). For small irregularly shaped fields, a maximum difference of 2% was found between calculated dose per monitor unit values and measurements when the nth root percent depth dose method was used; this difference changed to 7% when comparisons were made between measurements and calculations based on the sector integral lateral buildup ration method. For large irregular fields this difference was found to be within 1.5% and 3.5%, respectively.« less

In 1858 Chebyshev, and some years later his students Korkin and Zolotarev, determined the polynomial which deviates least from zero among all polynomials of degree n with leading coefficient one with respect to the maximum- and the L1-norm, respectively; these are now called the Chebyshev polynomial of first and second kind.The next natural step which is to find, at least asymptotically, the minimal polynomial with respect to a given weight function has not been settled until today. Indeed, Bernstein gave asymptotics for the minimum deviation of weighted minimal polynomials, Fekete and Walsh found nth root asymptotics and, recently, Lubinsky and Saff provided asymptotics outside [−1, 1]. But the main point of interest: the asymptotic representation of the weighted minimal polynomials on the interval of approximation [−1, 1] remained open. Here we settle this problem with respect to the maximum norm for weight functions whose second derivative is Lipα, α ∈ (0, 1), and with respect to the L1-norm under somewhat stronger differentiability conditions.

Summary Small differences in slowness and azimuth for overlapping phases especially where the branches of the travel-time curve are triplicated must be resolved for a meaningful inversion of array data. A computer program package has been written in FORTRAN IV which enables a user to determine, automatically, the apparent azimuth and slowness of any portion of the seismic wavetrain recorded at various arrays if he has the raw data on digital tape and has access to any modern computer. These programs make use of two methods, (i) adaptive processing, and (ii) Nth root beam forming which have been compared to determine the apparent azimuth and slowness of the seismic wavelets. The former method is performed by cross correlating the signal on each channel with a velocity and azimuth filtered trace in an iterative manner until the convergence takes place. In the latter method the operation is done by delaying the various channels to align a group of arrivals with a particular velocity and azimuth; taking the Nth root of the signal; summing and then raising the result to the Nth power. The value of apparent velocity and azimuth which produces a maximum filtered signal is determined. Experiments with clean and noisy synthetic data have shown that the adaptive processing method is more successful for resolving small differences in apparent velocity and azimuth of overlapping wavelets. It also has an advantage that a set of residuals may easily be obtained from the analysis. The Nth root method is extremely powerful in enhancing the signal to noise ratio at the expense of signal distortion. The computation time for both methods is about the same.

The study was undertaken to establish the relationship between the roots of the perfect numbers and the nconsecutive odd numbers. Odd numbers were arranged in such a way that their sums were either equal to the perfectsquare number or equal to a cube. The findings on the patterns and relationships of the numbers showed that therewas a relationship between the roots of numbers and n consecutive odd numbers. From these relationships a generalformula was derived which can be used to determine the number of consecutive odd numbers that can add up to getthe given number and this relationship was used to define the other meaning of nth root of a given perfect numberfrom consecutive n odd numbers point of view. If the square root of 4 is 2, then it means that there are 2 consecutiveodd numbers that can add up to 4, if the cube root of 27 is 3, then there are 3 consecutive odd number that add up to27, if the fifth root of 32 is 5, then it means that there are 5 consecutive odd numbers that add up to 32.

We equip the odd nilHecke algebra and its associated thick calculus category with digrammatically local differentials. The resulting differential graded Grothendieck groups are isomorphic to two different forms of the positive part of quantum sl(2) at a fourth root of unity., Comment: 53 pages