Showing papers on "Coprime integers published in 1998"

The valence of harmonic polynomials

Abstract: The paper gives an upper bound for the valence of harmonic polynomials An example is given to show that this bound is sharp Interest in harmonic mappings in the complex plane has increased due to the publication of [1] in 1984 Most of this interest has centered on univalent harmonic mappings It is, however, the purpose of this paper to present the fundamental valence results of complex valued polynomials which are harmonic in the plane Specifically a harmonic polynomial is a function P(z) = S(z) + T(z) where S(z) and T(z) are analytic polynomials The degree of P is defined as the larger of the degrees of S and T Lyzzaik studied the local topology of harmonic mappings in [4]; in particular he put a bound on the local valence of P but did not obtain a global valence bound In this paper it will be shown that, apart from certain degenerate examples, P(z) is at most n2-valent, where n is the degree of P An example is exhibited to show this to be the sharp bound To prove the n2 valence bound we need the following classical result from algebraic geometry; a proof may be found in [2] Theorem 1 (Bezout's Theorem) Let A and B be relatively prime polynomials in the real variables x and y with real coefficients, and let deg A = n and deg B = m Then the two algebraic curves A(x, y) = 0 and B(x, y) = 0 have at most mn points in common To apply Bezout's theorem to the valence problem we restate it in an equivalent form Theorem 2 (Alternate form of Bezout's Theorem) Let A and B be polynomials in the real variables x and y with real coefficients If deg A = n and deg B m, then either A and B have at most mn common zeros or have infinitely many common zeros Proof If A and B are relatively prime, then, by Bezout's theorem, the polynomial equations A(x, y) = 0 and B(x, y) = 0 have at most mn common zeros Otherwise A and B are not relatively prime and they have a greatest common factor, say C Let deg C = q If C(x, y) = 0 has no solutions, then A and B have at most (n q)(m q) common zeros But if there exists a point on C(x, y) = 0 with either Oc 74 0 or 4c 04 o, then, by the implicit function theorem, C(x, y) = 0 Received by the editors June 26, 1995 and, in revised form, September 20, 1996 and January 2, 1997 1991 Mathematics Subject Classification Primary 30C55 (?)1998 American Mathematical Society

Coprime Factorizations and Well-Posed Linear Systems

Abstract: We study the basic notions related to the stabilization of an infinite-dimensional well-posed liner system in the sense of Salamon and Weiss. We first introduce an appropriate stabilizability and detectability notion and show that if a system is jointly stabilizable and detectable then its transfer function has a doubly coprime factorization in $H^\infty$. The converse is also true: every function with a doubly coprime factorization in $H^\infty$ is the transfer function of a jointly stabilizable and detectable well-posed linear system. We show further that a stabilizable and detectable system is stable if and only if its input/output map is stable. Finally, we construct a dynamic, possibly non-well-posed, stabilizing compensator. The notion of stability that we use is the natural one for the quadratic cost minimization problem, and it does not imply exponential stability.

Consistency of open-loop experimental frequency-response data with coprime factor plant models

Abstract: The model/data consistency problem for coprime factorization considered is: Given some possibly noisy frequency-response data obtained by running open-loop experiments on a system, show that these data are consistent with a given family of perturbed coprime factor models and a time-domain /spl Lscr//sub /spl infin// noise model. In the noise-free open-loop case, the model/data consistency problem boils down to the existence of an interpolating function in /spl Rscr//spl Hscr//sub /spl infin// that evaluates to a finite number of complex matrices at a finite number of points on the imaginary axis. A theorem on boundary interpolation in /spl Rscr//spl Hscr//sub /spl infin// is a building block that allows one to devise computationally simple necessary and sufficient tests to check if the perturbed coprime factorization is consistent with the data. For standard coprime factorizations, the test involves the computation of minimum-norm solutions to underdetermined complex matrix equations. The Schmidt-Mirsky theorem is used in the case of special factorizations of flexible systems. For /spl Lscr//sub /spl infin// noise corrupting the frequency-response measurements, a complete solution to the open-loop noisy SISO problem using the structured singular value /spl mu/ is given.

A transformation for 2-D linear systems and a generalization of a theorem of Rosenbrock

Abstract: The transformation of zero coprime system equivalence (z.c.s.e.) with its various characterizations is shown to have at least one important role for two dimensional linear systems theory. This paper shows that it is z.c.s.e. which forms the basis of the generalization of Rosenbrock's characterization of all least order polynomial realizations of a transfer function matrix for the case of 2-D systems. The definition of what consistutes the least order is shown to be crucial.

A burge tree of virasoro-type polynomial identities

Abstract: Using a summation formula due to Burge, and a combinatorial identity between partition pairs, we obtain an infinite tree of q-polynomial identities for the Virasoro characters , dependent on two finite size parameters M and N, in the cases where: (1) p and p′ are coprime integers that satisfy 0 < p < p′. (2) If the pair (p′:p) has a continued fraction (c1, c2, …, ct-1, ct+2), where t≥1, then the pair (s:r) has a continued fraction (c1, c2, …, cu-1, d), where 1 ≤ u ≤ t, and 1 ≤ d ≤ cu. The limit M → ∞, for fixed N, and the limit N → ∞, for fixed M, lead to two independent boson-fermion-type q-polynomial identities: in one case, the bosonic side has a conventional dependence on the parameters that characterize the corresponding character. In the other, that dependence is not conventional. In each case, the fermionic side can also be cast in either of two different forms. Taking the remaining finite size parameter to infinity in either of the above identities, so that M → ∞ and N → ∞, leads to the same q-series identity for the corresponding character.

Curves of genus 2 with real multiplication by a square root of 5

Abstract: Our aim in this work is to produce equations for curves of genus 2 whose Jacobians have real multiplication (RM) by $\mathbb{Q}(\sqrt{5})$, and to examine the conjecture that any abelian surface with RM by $\mathbb{Q}(\sqrt{5})$ is isogenous to a simple factor of the Jacobian of a modular curve $X_0(N)$ for some $N$. To this end, we review previous work in this area, and are able to use a criterion due to Humbert in the last century to produce a family of curves of genus 2 with RM by $\mathbb{Q}(\sqrt{5})$ which parametrizes such curves which have a rational Weierstrass point. We proceed to give a calculation of the $\mbox{\ell}$-adic representations arising from abelian surfaces with RM, and use a special case of this to determine a criterion for the field of definition of RM by $\mathbb{Q}(\sqrt{5})$. We examine when a given polarized abelian surface $A$ defined over a number field $k$ with an action of an order $R$ in a real field $F$, also defined over $k$, can be made principally polarized after $k$-isogeny, and prove, in particular, that this is possible when the conductor of $R$ is odd and coprime to the degree of the given polarization. We then give an explicit description of the moduli space of curves of genus 2 with real multiplication by $\mathbb{Q}(\sqrt{5})$. From this description, we are able to generate a fund of equations for these curves, employing a method due to Mestre.

Decompositions and extremal type II codes over Z/sub 4/

Abstract: In previous work by Huffman and by Yorgov (1983), a decomposition theory of self-dual linear codes C over a finite field F/sub q/ was given when C has a permutation automorphism of prime order r relatively prime to q. We extend these results to linear codes over the Galois ring Z/sub 4/ and apply the theory to Z/sub 4/-codes of length 24. In particular we obtain 42 inequivalent [24,12] Z/sub 4/-codes of minimum Euclidean weight 16 which lead to 42 constructions of the Leech lattice.

Coprime factorizations and well-posed linear systems

Abstract: We study the basic notions related to the stabilization of an infinite-dimensional well-posed linear system in the sense of Salamon and Weiss. We first introduce an appropriate stabilizability and detectability notion, and show that if a system is jointly stabilizable and detectable then its transfer function has a doubly coprime factorization in H/sup /spl infin//. The converse is also true: every function with a doubly coprime factorization in H/sup /spl infin// is the transfer function of a jointly stabilizable and detectable well-posed linear system. We show further that a stabilizable and detectable system is stable if and only if its input/output map is stable. Finally, we construct a dynamic, possibly nonwell-posed, stabilizing compensator. The notion of stability that we use is the natural one for the quadratic cost minimization problem, and it does not imply exponential stability.

Entropy and uniform distribution of orbits in $$\mathbb{T}$$

Abstract: We present a class of integer sequences {c n } with the property that for everyp-invariant and ergodic positive-entropy measure μ on L 2 $$\mathbb{T}$$ , {c n x (mod 1)} is uniformly distributed for μ-almost everyx. This extends a result of B. Host, who proved this for the sequence {q n }, forq relatively prime top. Our class of sequences includes, for instance, the sequencec n =Мf i (n)q i n , where the numbersq i are distinct and are relatively prime top andf i are any polynomials. More generally, recursion sequences for which the free coefficient of the recursion polynomial is relatively prime top are in this class as well, provided they satisfy a simple irreducibility condition. In the multi-dimensional case we derive sufficient conditions for a pair of endomorphisms $$\mathbb{T}^d $$ (withA diagonal) and anA-invariant and ergodic measure μ, such thatB-orbits of the form {B n ω} are uniformly distributed for μ-almost every $$\mathbb{T}^d $$ .

On the conjecture of Jesmanowicz concerning Pythagorean triples

Abstract: Let a, b, c be relatively prime positive integers such that a2 + b2 = c2. Jeśmanowicz conjectured in 1956 that for any given positive integer n the only solution of (an)x + (bn)y = (en)z in positive integers is x = y = z = 2. Building on the work of earlier writers for the case when n = 1 and c = b + 1, we prove the conjecture when n > 1, c = b + 1 and certain further divisibility conditions are satisfied. This leads to the proof of the full conjecture for the five triples (a, b, c) = (3, 4, 5), (5, 12, 13), (7, 24, 25), (9, 40, 41) and (11, 60, 61).

Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links

Abstract: We identify 998 closed hyperbolic 3-manifolds whose volumes are rationally related to Dedekind zeta values, with coprime integers $a$ and $b$ giving $$\frac{a}{b}\,{\rm vol}({\cal M})=\frac{(-D)^{3/2}}{(2\pi)^{2n-4}}\,\frac {\zeta_K(2)}{2\zeta(2)}$$ for a manifold ${\cal M}$ whose invariant trace field $K$ has a single complex place, discriminant $D$, degree $n$, and Dedekind zeta value $\zeta_K(2)$. The largest numerator of the 998 invariants of Hodgson--Weeks manifolds is, astoundingly, $a=2^4\times23\times37\times691=9,408,656$; the largest denominator is merely $b=9$. We also study the rational invariant $a/b$ for single-complex-place cusped manifolds, complementary to knots and links, both within and beyond the Hildebrand--Weeks census. Within the censi, we identify 152 distinct Dedekind zetas rationally related to volumes. Moreover, 91 census manifolds have volumes reducible to pairs of these zeta values. Motivated by studies of Feynman diagrams, we find a 10-component 24-crossing link in the case $n=2$ and $D=-20$. It is one of 5 alternating platonic links, the other 4 being quartic. For 8 of 10 quadratic fields distinguished by rational relations between Dedekind zeta values and volumes of Feynman orthoschemes, we find corresponding links. Feynman links with $D=-39$ and $D=-84$ are missing; we expect them to be as beautiful as the 8 drawn here. Dedekind-zeta invariants are obtained for knots from Feynman diagrams with up to 11 loops. We identify a sextic 18-crossing positive Feynman knot whose rational invariant, $a/b=26$, is 390 times that of the cubic 16-crossing non-alternating knot with maximal $D_9$ symmetry. Our results are secure, numerically, yet appear very hard to prove by analysis.

Finding finite B 2 -sequences faster

Abstract: A B 2 -sequence is a sequence a 1 < a 2 <... < a r of positive integers such that the sums a i + aj, 1 < i < j ≤ r, are different. When q is a power of a prime and θ is a primitive element in GF(q 2 ) then there are B 2 -sequences A(q,θ) of size q with a q < q 2 , which were discovered by R. C. Bose and S. Chowla. In Theorem 2.1 I will give a faster alternative to the definition. In Theorem 2.2 I will prove that multiplying a sequence A(q, θ) by integers relatively prime to the modulus is equivalent to varying θ. Theorem 3.1 is my main result. It contains a fast method to find primitive quadratic polynomials over GF(p) when p is an odd prime. For fields of characteristic 2 there is a similar, but different, criterion, which I will consider in Primitive quadratics reflected in B 2 -sequences , to appear in Portugaliae Mathematica (1999).

Counting divisors of Lucas numbers

Abstract: Let {Sn} be a second order linear recurrence consisting of integers only. M. Ward [22] proved that, except for some degenerate cases, there are always an infinite number of distinct primes dividing the terms of {Sn}. A deeper question is whether in the non-degenerate case the set of prime divisors has a prime density. (If S is any set of natural numbers, then S(x) denotes the number of elements n in S with 1 < n ≤ x. In case S is a set of primes we define the prime density of S to be limx→∞ S(x)/π(x), if it exists, where π(x) denotes the number of primes not exceeding x.) It is conjectured that the answer is yes and that the density is in fact positive. In case of what are called torsion sequences, this was recently established by P. Stevenhagen [21], generalizing on results in the earlier papers [9, 11, 13]. Stevenhagen showed, moreover, that the density of a torsion sequence is a rational number. For a large class of non-torsion sequences, the existence and positivity of of the prime density was established by P.J. Stephens [20], under the assumption of the Generalized Riemann Hypothesis. The sequence {Ln} is torsion. Lagarias established that it has prime density 2/3. His method goes back to H. Hasse [6], who expressed the prime density of sequences {a+b}k=1 in terms of degrees of Kummer extensions. This method will be used in Section 3. The analytic aspects of prime divisors of sequences {ak + b}k=1 were explored by K. Wiertelak in several papers [24, 25, 26, 27, 28]. For a survey of results on prime divisors of, not necessarily second order, linear recurrences, see Ballot [1]. The problem of general divisors of second order linear recurrence sequences, in contrast, has not received much attention. Let a and b be fixed coprime integers such that |a| 6= |b|. In [12] the set of divisors, Ga,b, of the sequence {ak + bk} was considered. Some of the results obtained there have

A Shadow identity and an application to isoduality

Abstract: The identity derived here from the theta transformation law replaces the “Atkin-Lehner identity” when the level decomposes into two factors which are not coprime. An application is given to the study of modular lattices of level 4, connected with modular forms for the classical theta group. CONWAY and Sloane have determined the maximal Hermite number of a self-dual lattice in ℝn for alln ≤ 33, and their result generalizes to the isodual case considered here in most of these dimensions.

Regular Orbits and the k ( GV )-Problem

Abstract: We describe some recent results concerning regular orbits of quasisimple groups in coprime representations, and discuss an application to the k(GV)-problem in modular representation theory.

Derived lengths and character degrees

Abstract: Let G be a finite solvable group. Assume that the degree graph of G has exactly two connected components that do not contain 1. Suppose that one of these connected components contains the subset {a1, . . ., an}, where ai and aj are coprime when i =A j. Then the derived length of G is less than or equal to I cd(G)-n + 1.

Brieskorn Lattices and Torelli Type Theorems for Cubics in ℙ3 and for Brieskorn-Pham Singularities with Coprime Exponents

Abstract: The first part is a survey about polarized mixed Hodge structures and Brieskorn lattices for hypersurface singularities. It describes wellknown properties of these objects and contains new results about classification spaces and moduli spaces for these objects. In the second part the Brieskorn lattices of cubics in ℂ4 and of Brieskorn-Pham singularities with coprime exponents are studied. A nice application is a global Torelli theorem for cubics in ℙ3 by some pure Hodge structure.

Coprime factors, strictly positive real functions, LMI and low-order controller design for SISO systems

Abstract: This paper develops a low-order controller design method requiring only the solution of a convex optimization problem. It addresses only linear continuous time-invariant single-input, single-output systems. The technique integrates several well-known results in control theory. An important step is the use of coprime factors so that based on strictly positive real functions, feedback stabilization using low-order controllers becomes a zero-placement problem which is convex. From this result, we develop algorithms to solve several optimal control problems, such as simultaneous pole-placement for several plants, H/sub /spl infin// optimization, and H/sub 2/ optimization. Four example systems are used to illustrate the design algorithms.

Modulation and Sampling Techniques for Multichannel Deconvolution

Abstract: : The problem of recovering information from single component linear translation invariant systems is inherently ill-posed. However, this situation may be circumvented in a multichannel system if the components of the system satisfy the strongly coprime condition. The signal may be completely recovered from a strongly coprime system by filtering the output of each channel with a deconvolver, and adding. This approach has been labeled "multichannel deconvolution." The author explores the state of the art for multichannel theory, and then uses sampling theory to develop an alternative method for creating these systems. Modulation techniques are then used to create a strongly coprime system, for which the corresponding deconvolvers are developed. The report concludes with discussions of several applications of the theory.