Power integral bases in prime-power cyclotomic fields

We consider the problem of determining all power integral bases for the maximal real subfieldQ(ζ + ζ−1) of thep-th cyclotomic fieldQ(ζ), wherep≥ 5 is prime and ζ is a primitivep-th root of unity. The ring of integers isZ[ζ+ζ−1] so a power integral basis always exists, and there are further non-obvious generators for the ring. Specifically, we prove thatifor one of the Galois conjugates of these five algebraic integers. Up to integer translation and multiplication by −1, there are no additional generators forp≤ 11, and it is plausible that there are no additional generators forp> 13 as well. Forp= 13 there is an additional generator, but we show that it does not generalise to an additional generator for 13 <p< 1000.

We propose a new method in the spectral analysis of noisy time-series data for damped oscillators. From the Jacobi three terms recursive relation for the denominators of the Padé approximations built on the well-known Z-transform of an infinite time series, we build a Hilbert space operator, a J-operator, where each bound state (inside the unit circle in the complex plane) is simply associated with one damped oscillator while the essential spectrum of the J-operator, which lies on the unit circle itself, is shown to represent the noise. Signal and noise are thus clearly separated in the complex plane. For a finite time series of length 2N, the J-operator is replaced by a finite order J-matrix JN, having N eigenvalues which are time reversal covariant. Different classes of input noise, such as blank (white and uniform), Gaussian and pink, are discussed in detail, the J-matrix formalism allowing us to efficiently calculate hundreds of poles of the Z-transform. Evidence of a universal behavior in the final statistical distribution of the associated poles and zeros of the Z-transform is shown. In particular, the poles and zeros tend, when the length of the time series goes to infinity, to a uniform angular distribution on the unit circle. Therefore at finite order, the roots of unity in the complex plane appear to be noise attractors. We show that the Z-transform presents the exceptional feature of allowing lossless undersampling and how to make use of this property. A few basic examples are given to suggest the power of the proposed method.

I recently came across a very appealing result about the product of the lengths of the chords of a regular polygon with n sides inscribed in a unit circle. If one vertex is selected and the product of the chord lengths from that vertex to each of the others is evaluated, the result is just n . There is a simple proof. Consider such a polygon and denote its vertices by the points A 0 , A 1 , … , A n . Suppose that O is the centre of the circle and the origin of the complex plane. The roots of the equation are just where ω is a primitive nth root of unity. Each of these roots represents one of the vertices. Suppose for convenience that the real number 1 is represented by A 0 .

The article investigates the classification with precision up to equivalence of involutions in the algebra of upper triangular matrices over the ring of integers of algebraic numbers of quadratic fields.The description of involutions in algebras represents one of the classical problems of ring theory. Standard examples of involutions are transposition in matrix algebra and conjugation in the field of complex numbers and the algebra of quaternions.In the case where the field 𝑃 has a characteristic different from two, a complete description of involutions with precision up to their equivalence in the algebra 𝑇𝑛(𝑃) for any natural number 𝑛 was obtained in [15]. In this work [3] involutions in the algebra of upper triangular matrices over commutative rings are studied. If the ring is a field of characteristic 2 or a Boolean ring, then necessary and sufficient conditions for the finiteness of the number of equivalence classesof involutions were found.This article is a continuation of the work of [3]. In the article [3], in particular, the number of equivalence classes of involutions in the algebras of upper triangular matrices over the ring of integers was found. In this regard, the natural result is the problem of describing involutions with precision up to their equivalence in algebras of upper triangular matrices over the ring of algebraic integers of quadratic fields, to which this work is devoted. In the work, the number of equivalence classes of involutions in such algebras is found and the method of finding representatives in each equivalence class is illustrated with examples. Upon receipt the main results in this work, the apparatus of the theory of Pell’s equations is significantly used.

Let $M\subset K$ be number fields. We consider the relation of relative power integral bases of $K$ over $M$ with absolute power integral bases of $K$ over $Q$. We show how generators of absolute power integral bases can be calculated from generators of relative ones. We apply our ideas in infinite families of octic fields with quadratic subfields.

The unstable burning of a bipropellant rocket combustion chamber is investigated and a study made of the requirements for an automatic closed loop control circuit to stabilize the motor. The bipropellant combustion chamber equations developed by Dr. L. Crocco(1) are utilised as the analytical description of the rocket motor burning phenomena. Equations similar to those developed by Dr. H. S. Taiga(2) are used for the oxidizer and fuel supply systems and the two closed loop stabilizing circuits. The stability or instability of the system is demonstrated by the use of a special plotting diagram in the complex plane suggested by M. Satche as a means of handling systems with time lag, and developed for this use by H. S. Tsien. This involves separating the transfer function into two parts. In the complex plane the first portion of the transfer function, the exponential variable containing the time lag, plots as a unit circle as the complex variable p is made to take a contour enclosing the positive half of the p—plane. If the remaining portion of the transfer function intersects this unit circle, the rocket motor can be unstable for large reduced time lag; if it does not intersect the unit circle, the system is generally stable, although the roots of the exponential coefficient in the positive half of the complex plane must be investigated. This latter requirement can be conveniently accomplished by the aid of a Nyquist Diagram. The equations for the feedback circuit are developed and the oxidizer and fuel transfer function requirements are determined. Two cases of stable combustion and two cases of unstable combustion are analyzed. One unstable case is stabilized by the addition of a feedback circuit.

In this section, we obtain theorems on the approximation of functions of the complex variable dened on closed contours bounding a simply connected domain in the complex plane; the approximations are constructed in the space of continuous functions and in Lebesgue spaces with the use of Lozinskii interpolation polynomials. These results are used to obtain a theoretical justication of the subdomain method for the approximate solution of singular integral equations dened on closed contours of various smoothness. The solvability of the computational scheme of this method is proved by the technique in [1]. Note that issues related to the approximate solution of singular integral equations have been considered in numerous papers (see [2{10] and the bibliography therein). However, the subdomain method for singular integral equations on closed contours other than the unit circle has not been studied. Let be a closed smooth contour bounding a connected domain F + containing the origin, and let F = CnfF + [ g ,w hereC is the entire complex plane. We say that belongs to the class (m) ,w herem is a positive integer, if the Riemann function t = (w), w2 0 =f : jj =1 g, (1 )= 1, 0 (1 )= a1 (a1 > 0), of this contour has continuous :

This paper shows that the inverse chirp z-transform (ICZT), which generalizes the inverse fast Fourier transform (IFFT) off the unit circle in the complex plane, can also be used with chirp contours that perform partial or multiple revolutions on the unit circle. This is done as a special case of the ICZT, which in algorithmic form has the same computational complexity as the IFFT, i.e., O(n log n). Here we evaluate the ICZT algorithm for chirp contours on the unit circle and show that it is numerically accurate for large areas of the parameter space. The numerical error in this case depends on the polar angle between two adjacent contour points. More specifically, the error profile for a transform of size n is determined by the elements of the Farey sequence of order n − 1. Furthermore, this generalization allows the use of non-orthogonal frequency components, thus lifting one of the main restrictions of the IFFT.

Let p be an odd prime number and K a number field having a primitive p th root of unity \zeta_p . We prove that Nikshych's non group-theoretical Hopf algebra H_p , which is defined over \mathbb Q(\zeta_p) , admits a Hopf order over the ring of integers \mathcal O_K if and only if there is an ideal I of \mathcal O_K such that I^{2(p-1)} = (p) . This condition does not hold in a cyclotomic field. Hence this gives an example of a semisimple Hopf algebra over a number field not admitting a Hopf order over any cyclotomic ring of integers. Moreover, we show that, when a Hopf order over \mathcal O_K exists, it is unique and we describe it explicitly.

Linearized rational interpolation problems at roots of unity play a crucial role in the fast and superfast Toeplitz solvers that we have developed. Our interpolation algorithm is a sequential algorithm in which a matrix polynomial that satisfies already some of the interpolation conditions is updated to satisfy two additional interpolation conditions. In the algorithm that we have used so far, the updating matrix, which is a matrix polynomial of degree one, is constructed in a two-step process that resembles Gaussian elimination. We briefly recall this approach and then consider two other approaches. The first one is a completely new approach based on an updating matrix that is unitary with respect to a discrete inner product that is based on roots of unity. The second one is an application of an algorithm for solving discrete least squares problems on the unit circle, a problem that has linearized rational interpolation at roots of unity as its limiting case. We conduct a number of numerical experiments to compare the three strategies.

Class group of the group ring of a cyclic group of order p2 over the ring of integers in a quadratic number field

Let / be an integrable valued function on the unit circle in the complex plane, and let g be the rearrangement of/ satisfying g(e") > g(eiv) if 0 < 0 < <p < lit.Define G(0) = (' tie") dq,. J-BIt is shown that some rearrangement of / is in Re H ', that is, the distribution off is the distribution of a function in Re Hl, if and only if J" |G(0)/0| d$ < cc, and that, if any rearrangement of / is in Re H ', then g is.The existence and form of rearrangements minimizing the H ' norm are investigated.It is proved that / is in Re H ' if and only if some rotation of / is in the space dyadic H ' of martingales.These results are extended to other Hp spaces.1. Introduction.Let T be the unit circle in the complex plane C, let the measure m on T be given by dm = d9/2ir, and if A is a function on T let ||A|| = (fT\h\p dmy/p.Functions on T which are equal a.e.(m) are identified.Let /be an integrable real valued function on T, denote the conjugate function of/by/, and if p > 0 define \\f\\H, = ||/ + z/|| .The first part of this paper is concerned primarily with the space Re H ', which consists of those/ such that \\f\\H> < oo.For a discussion of conjugate functions and Hardy spaces see [12], and for a survey of recent developments in this area and the spaces Re Hp, see [6].A real valued function g on T is said to have the same distribution as /, or to be a rearrangement of f, if m{f < X} = m{ g < X) for each real number X, and in this case we write/ g.Let/ be the rearrangement of/ satisfying fd(e'9) > fd(e"p) if 0 < 0 < <p < 2ir, and let M/0) = M(0) = /%/>;-"O dq>.The following theorem characterizes the distributions of functions in Re H1.Theorem 1.1.There is a rearrangement of fin Re Hi if and only if f\M(9)/9\ dB < oo, and in this case fd G Re H '. There are absolute positive constants C and c such that CII/IU.> f\M(9)/e\ d9 > c\\fd\\", -y/11,.(1.1)The quantity f^\M(9)/9\ d9 depends only on the distribution of/ and is 0 if -fivf.A theorem of Zygmund [12, Vol. 1, p. 254] states that if / > 0, then

Clean group rings over localizations of rings of integers

Let G be a tensor category over a field k of characteristic 0. In his paper [-i], w 7, Delinge had established rather general conditions for G to be the category of representations of an algebraic group. He has shown that if any object neC satisfies the condition A " n = 0 for a sufficiently large n, then 0 is equivalent to the category Rep(G) of representations of a proalgebraic group scheme G over k. For tensor categories over fields of positive characteristic this result is no longer true. In the first part of this paper we present a construction of a class of tensor categories over fields of positive characteristic for which an analogue of Deligne's theorem does not hold. To describe these categories, denote by Gz an algebraic Chevalley group over Z and by G the corresponding group over Fp (for a sufficiently large p). Our category C will be a certain "subquotient" of the category R(G) of finite-dimensional representations of G. More precisely, let S be the full subcategory of R(G) whose objects (called small representations) are direct sums of the irreducible representations of G with highest weights inside the fundamental alcove. Let also SL 2 c G be the principal SLz-subgroup in G and (SLz)Im c SL2 be the infinitesimal neighbourhood of the Cartan subgroup H in SL2. Denote by O the full subcategory of R(G) consisting of representations M of G (called quasiprojective representations) such that the restriction of M to (SL2)m) is projective. The category 0 is a quotient of the subcategory S+PcR(G). In the last section of the paper we show that the same approach can be applied to representations of quantum groups. Let U be the quantum group at a root of prime order p of 1 (which is a Hopf algebra over the ring of integers in a cyclotomic field, see [2, 3]). Starting from the category of representations of U, we construct a semisimple braided category ~(G, p) with the finite number of simple objects. On the other hand, in Wess-Zumino-Witten conformal field theory one encounters the category O(G, m) of integrable representations of the correspond-

Using an idea of Doug Lind, we give a lower bound for the Perron–Frobenius degree of a Perron number that is not totally real, in terms of the layout of its Galois conjugates in the complex plane. As an application, we prove that there are cubic Perron numbers whose Perron–Frobenius degrees are arbitrary large, a result known to Lind, McMullen and Thurston. A similar result is proved for bi-Perron numbers.