Complex Roots Of Unity Research Articles

Characterization of Nonuniform Perfect-Reconstruction Filterbanks Using Unit-Step Signal

An algebraic characterization of nonuniform perfect reconstruction (PR) filterbanks with integer decimation factors is presented. The PR property is formulated in the z domain based on the response of the linear multirate systems to the delayed unit-step signals. This leads to a unique class of characterizing formulas that are necessary and sufficient conditions for the PR property and free from the complex roots of unity. Two related characterizations of nonuniform PR systems, in the form of necessary conditions, are also developed based on these formulas. As a concrete example, the results are then used to derive necessary and sufficient conditions for PR nonuniform delay chain filterbanks. The conditions show that nonuniform delay chain filterbanks are the signal processing realizations of the mathematical notion of the exact covering systems of congruence relations. Important results from the mathematics literature on the exact covering systems are introduced. The results elucidate the admissible factors of decimation for the nonuniform PR delay chain systems in settings with maximally distinct decimation factors. A simple test of the PR property for delay chain systems is also presented. The test is based on the divisibility of certain polynomials by the cyclotomic polynomials. Finally, multirate systems based on the Beatty sequences, which are the irrational generalization of the exact covering systems, are briefly discussed.

Generalised Sarwate bounds on the aperiodic correlation of sequences over complex roots of unity

General aperiodic correlation bounds named generalised Sarwate bounds are derived, for sequence sets over complex roots of unity with zero or low correlation zone (ZCZ/LCZ), with respect to family size, sequence length, maximum autocorrelation sidelobe, maximum cross-correlation value and the zero or low correlation zone. It is shown that the existing aperiodic binary sequence bounds, such as Sarwate bounds, Welch bounds, Levenshtein bounds, Tang–Fan bounds and Peng–Fan bounds, are only special cases of the presented generalised Sarwate bounds. In addition, the new bounds are also, in general, stronger than the existing aperiodic binary sequence bounds, as well as Boztas aperiodic correlation bounds for normal complex roots-of-unity sequences.

A comparison of algorithms for exact analysis of unordered 2 × K contingency tables

We present a comparison of two efficient algorithms for exact analysis of an unordered 2 × K table. First, by considering conditional generating functions, we show that both the network algorithm of Mehta and Patel ( J. Amer. Statist. Assoc. 78 (1983)) and the fast Fourier transform (FFT) algorithm of Baglivo et al. ( J. Amer. Statist. Assoc. 82 (1992)) rest on the same foundation. This foundation is a recursive polynomial relation. We further show that the network algorithm is equivalent to a stage-wise implementation of this recursion while the FFT algorithm is based on performing the same recursion at complex roots of unity. Our empirical results for the Pearson X 2, likelihood ratio, and Freeman-Halton statistics show that the network algorithm, or equivalently, the recursive polynomial multiplication algorithm is superior to the FFT algorithm with respect to computing speed and accuracy.

Equations Admitting Scaling and Discrete Cyclic Symmetries

Some coordinate transformations on two variables are obtained that give a realization of the cyclic group. The differential invariants of scaling transformations are then combined to give quantities that are multiplied by a complex root of unity when the cyclic transformations are applied. The second order ordinary differential equations that are invariant under both scaling and the cyclic transformations can then be written down.

Über Gruppen Von Iterativen Wurzeln Der Formalen Potenzreihe F(x) = x

Let ℂ¦x¦ be the ring of formal power series in one indeterminate x over ℂ, denote by Γ the group of invertible series in ℂ¦x¦, and by EΓ the set of all iterative roots of x in Γ. Then we will show that EΓ is neither a subgroup of Γ nor a family of commuting series. We describe all subgroups of Γ lying in EΓ, they are abelian and isomorphic to subgroups of the group E of complex roots of unity. Furthermore we determine the maximal subgroups of Γ in E{Γ} and use them to investigate how the subgroups in EI are related.

Bemerkung zu einer arbeit von achou über die anzahl zulässiger lösungen eines knapsack-problems

The formula for calculating the number of feasible solutions to a knapsack problem given by O. Achou is converted so that only rational numbers appear and all complex roots of unity are removed.

Mean and variance of the number of renewals of a censored Poisson process

Expressions are derived for computing the mean and variance of the number of renewals in observation times of any length and for any values of the scaling parameter of the censored Poisson process. These results depend on certain identities involving the complex roots of unity. Computations show the oscillatory behavior to be expected when the average interval between renewals of the censored Poisson process is of the order of the observation time. Both ordinary and equilibrium renewal processes are considered.

A note on B- and Br- incomplete topological Abelian groups

From results of Baker (2) it appeared to be unlikely that, in Hausdorff topological Abelian groups, completeness would be implied by Br- or B-completeness (for definitions, see below). We show here (Corollary 1) that the group of all complex roots of unity, though not complete, is B-complete. Another example, for the suggestion of which we are indebted to Dr J. W. Baker, is used to show (Corollary 2) that B-completeness is not a consequence of Br-completeness. It is proved, however, that B- and Br-complete Hausdorff topological Abelian groups are embedded in their completions in special ways in relation to the closed subgroups of the completions. Also, the completions of B- and Br-complete Hausdorff topological Abelian groups are shown to be, respectively, B- and Br-complete.

The Schur Multipliers of the Mathieu Groups

The Mathieu groups are the finite simple groups M11, M12, M22, M23, M24 given originally as permutation groups on respectively 11, 12, 22, 23, 24 symbols. Their definition can best be found in the work of Witt [1]. Using a concept from Lie group theory we can describe the Schur multiplier of a group as the center of a “simply-connected” covering of that group. A precise definition will be given later. We also mention that the Schur multiplier of a group is the second cohomology group of that group acting trivially on the complex roots of unity. The purpose of this paper is to determine the Schur multipliers of the five Mathieu groups.

Generators and Relations for Cyclotomic Units

We prove here an unpublished conjecture of Milnor which gives a complete set of multiplicative relations between the numbers e′(ζ) = 1−ζ,where ranges over complex roots of unity. Information of this type is useful in certain areas of topology as well as in number theory.

Equal Sums of Like Powers

In this note all small latin letters denote rational integers. We write k ≧ 1, s ≧ 1 and consider the simultaneous equationsA solution of these equations is said to be non-trivial if no set {xiu} is a permutation of another set {xiv}. In 1851 Prouhet constructed a non-trivial solution of these equations with j = sk and Lehmer has recently found a parametric solution for the same j. Here I give two alternative elementary proofs of Lehmer's result. Lehmer's own proof depends on the ideas of generating functions, exponentials, differentiation, matrices, and complex roots of unity, though all at a fairly simple level. One of my proofs requires only the factor theorem for a polynomial and the other only the multinomial theorem for a positive integral index.