Showing papers on "Coprime integers published in 2000"

Model Reduction for Control System Design

Abstract: 1 Methods for Model Reduction.- 2 Multiplicative Approximation.- 3 Low Order Controller Design.- 4 Model and Controller Reduction Based On Coprime Factorizations.

Symmetric powers of modular representations, hilbert series and degree bounds

Abstract: Let G = Z p be a cyclic group of prime order p with a representation G → GL(V) over a field K of characteristic p. In 1976, Almkvist and Fossum gave formulas for the decomposition of the symmetric powers of V in the case that V is indecomposable. From these they derived formulas for the Hilbert series of the invariant ring K[V]G. Following Almkvist and Fossum in broad outline, we start by giving a shorter, self-contained proof of their results. We extend their work to modules which are not necessarily indecomposable. We also obtain formulas which give generating functions encoding the decompositions of all symmetric powers of V into indecomposables. Our results generalize to groups of the type Z p ×H with |H| coprime to p. Moreover, we prove that for any finite group G whose order is divisible by p but not by p 2 the invariant ring A,K[V]G is generated by homogeneous invariants of degrees at most dim (V).(|G| – 1).

Feedback Stabilization over Commutative Rings: Further Study of the Coordinate-Free Approach

Abstract: This paper is concerned with the coordinate-free approach to control systems. The coordinate-free approach is a factorization approach but does not require the coprime factorizations of the plant. We present two criteria for feedback stabilizability for multi-input multi-output (MIMO) systems in which transfer functions belong to the total rings of fractions of commutative rings. Both of them are generalizations of Sule's results in [SIAM J. Control Optim., 32 (1994), pp. 1675--1695]. The first criterion is expressed in terms of modules generated from a causal plant and does not require the plant to be strictly causal. It shows that if the plant is stabilizable, the modules are projective. The other criterion is expressed in terms of ideals called generalized elementary factors. This gives the stabilizability of a causal plant in terms of the coprimeness of the generalized elementary factors. As an example, a discrete finite-time delay system is considered.

Low-order controller design for SISO systems using coprime factors and LMI

Abstract: This paper develops a low-order controller design method for linear continuous time-invariant single-input, single-output systems requiring only the solution of a convex optimization problem. 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 two optimal control problems.

On an irreducibility theorem of A. Schinzel associated with coverings of the integers

Abstract: Let $f(x)$ and $g(x)$ be two relatively prime polynomials having integer coefficients with $g(0) eq 0$. The authors show that there is an $N=N(f,g)$ such that if $n \geq N$, then the non-reciprocal part of the polynomial $f(x)x^{n}+g(x)$ is either irreducible or identically 1 or $-1$ with certain clear exceptions that arise from a theorem of Capelli. A version of this result is originally due to Andrzej Schinzel. The present paper gives a new approach that allows for an improved estimate on the value of $N$.

A complex-number fourier technique for lower bounds on the mod-m degree

Abstract: We say an integer polynomial p, on Boolean inputs, weakly m-represents a Boolean function f if p is nonconstant and is zero (mod m), whenever f is zero. In this paper we prove that if a polynomial weakly m-represents the Modq function on n inputs, where q and m are relatively prime and m is otherwise arbitrary, then the degree of the polynomial is \( \Omega(n) \). This generalizes previous results of Barrington, Beigel and Rudich, and Tsai, which held only for constant or slowly growing m. In addition, the proof technique given here is quite different. We use a method (adapted from Barrington and Straubing) in which the inputs are represented as complex q-th roots of unity. In this representation it is possible to compute the Fourier transform using some elementary properties of the algebraic integers. As a corollary of the main theorem and the proof of Toda's theorem, if q, p are distinct primes, any depth-three circuit that computes the Modq function, and consists of an exact threshold gate at the output, Modp gates at the next level, and AND gates of polylog fan-in at the inputs, must be of exponential size. We also consider the question of how well circuits consisting of one exact gate over ACC(p)-type circuits (where p is an odd prime) can approximate parity. It is shown that such circuits must have exponential size in order to agree with parity for more than 1/2 + o(1) of the inputs.

The hodge numbers of the moduli spaces of vector bundles over a Riemann surface

Abstract: Inductive formulas for the Betti numbers of the moduli spaces of stable holomorphic vector bundles of coprime rank and degree over a fixed Riemann surface of genus at least two were obtained by Harder, Narasimhan, Desale and Ramanan using number theoretic methods and the Weil conjectures and were rederived by Atiyah and Bott using gauge theory In this note we observe that there are similar inductive formulas for determining the Hodge numbers of these moduli spaces

The generalized Borwein conjecture. I. The Burge transform

Abstract: Given an arbitrary ordered pair of coprime integers (a,b) we obtain a pair of identities of the Rogers--Ramanujan type. These identities have the same product side as the (first) Andrews--Gordon identity for modulus 2ab\pm 1, but an altogether different sum side, based on the representation of (a/b-1)^{\pm 1} as a continued fraction. Our proof, which relies on the Burge transform, first establishes a binary tree of polynomial identities. Each identity in this Burge tree settles a special case of Bressoud's generalized Borwein conjecture.

The Hodge Numbers of the Moduli Spaces of Vector Bundles over a Riemann surface

Abstract: Inductive formulas for the Betti numbers of the moduli spaces of stable holomorphic vector bundles of coprime rank and degree over a fixed Riemann surface of genus at least two were obtained by Harder, Narasimhan, Desale and Ramanan using number theoretic methods and the Weil conjectures and were rederived by Atiyah and Bott using gauge theory. In this note we observe that there are similar inductive formulas for determining the Hodge numbers of these moduli spaces.

The number of ideals in a quadratic field, II

Abstract: The asymptotic formula for the number of integral ideals in a quadratic field with norm up to N contains the Gauss circle problem as a special case, when the conductor r is 4. We obtain estimates for the remainder term with similar growth in r and N. The modifications to the Iwaniec-Mozzochi method for the circle problem include choosing the denominators of gradients of sides of the approximating polygon to be relatively prime to r, and allowing shifted characters mod r in the Poisson summation formula.

Estimation of exponential sums of polynomials of higher degrees II

Abstract: In this article we will prove certain identities between the above exponential sum and hyper-Kloosterman sums, generalize the above estimation for the exponential sum to other cases of m when a ≥ 2, and establish new bounds for hyper-Kloosterman sums. Write p ‖n if p |n but ph+1 -n. Theorem 1. Let p be a prime, q = p, a ≥ 2, and k an integer with a ≤ k < φ(q) and p k. We set h by p ‖ (k − 1). Then for any b and c relatively prime to p we have

Partitions with parts in a finite set

Abstract: Let A be a nonempty finite set of relatively prime positive integers, and let PA (n) denote the number of partitions of n with parts in A. An elementary arithmetic argument is used to prove the asymptotic formula PA((n) = (,A a) (k1)! + 0 (nk2)Let A be a nonempty set of positive integers. A partition of a positive integer n with parts in A is a representation of n as a sum of not necessarily distinct elements of A. Two partitions are considered the same if they differ only in the order of their summands. The partition function of the set A, denoted PA (n), counts the number of partitions of n with parts in A. If A is a finite set of positive integers with no common factor greater than 1, then every sufficiently large integer can be written as a sum of elements of A (see Nathanson [3] and Han, Kirfel, and Nathanson [2]), and so PA (n) > 1 for all n > no. In the special case that A is the set of the first k integers, it is known that nk-i PA(n) k!(k 1)! Erdos and Lehner [1] proved that this asymptotic formula holds uniformly for k = o(n1/3). If A is an arbitrary finite set of relatively prime positive integers, then PA (1 (k1)!' (1) PA (2J \HalEAa) (kThe usual proof of this result (Netto [4], Polya-Szeg6 [5, Problem 27]) is based on the partial fraction decomposition of the generating function for PA(n). The purpose of this note is to give a simple, purely arithmetic proof of (1). We define PA(O) = 1. Theorem 1. Let A = {al,... , ak} be a set of k relatively prime positive integers, that is, gcd(A) = (a,,... , ak)=1. Received by the editors June 5, 1998. 2000 Mathematics Subject Classification. Primary 11P81; Secondary 05A17, 11B34.

Diophantine properties of elements of SO(3)

Abstract: A number alpha in R is diophantine if it is not well approximable by rationals, i.e. for some C, nu>0 and any relatively prime p, q in Z we have |alpha q -p|>C q^{-1-\vu}. It is well-known and easy to prove that almost every alpha in R is diophantine. In this paper we address a noncommutative version of the question about diophantine properties. Consider a pair A,B in SO(3) and for each n in Z_+ take all possible words in A, A^{-1}, B, and B^{-1} of length n, i.e. for a multiindex I=(i_1,j_1,..., i_s,j_s) put |I|=\sum_{k=1}^s (|i_k|+|j_k|) and W_{A,B}(n)={W_{I}(A,B)= A^{i_1}B^{j_1}... A^{i_s}B^{j_s}}_{|I|=n}. Gamburd-Jakobson-Sarnak raised the problem: prove that for Haar almost every pair A, B in SO(3) the closest distance of the word of length n to the identity, i.e. s_{A,B}(n)=min_{|I|=n} |W_{I}(A,B)-Id|, is bounded from below by an exponential function in n. This is an analog of diophantine property for elements of SO(3). In this paper we make first step toward understanding diophantine properties of SO(3). We prove that s_{A,B}(n) is bounded from below by an exponential function in n^2. We also exhibit obstructions to prove a simple exponential in n estimate.

Generation of a mathematically constrained key using a one-way function

Abstract: A cryptographic key (K2) is generated using a one-way function (325) and testing for a mathematical constraint. Pre-seed data is obtained by subdividing (320) a random bit string into several segments (PRE-P1, ..., PRE-P8, PRE-Q1, ..., PRE-Q8), then independently processing each segment with a one-way function (325) to obtain respective values (P, Q) which are tested for a mathematical constraint such as primeness (345). If the values do not pass the test, the steps are repeated. Otherwise a modulus N (350), and Euler's Totient 011001 = (P-1) (Q-1) (355) are formed. Segments Pre-K1-1, ..., Pre-K1-16 are also processed through a one way function to form segments K1-1, ..., K1-16, which are assembled to form a value K1. Euclid's Basic Algorithm (360) is used to determine if K1 is relatively prime to &phgr:. If not, a new K1 is formed. If so, a key (K2) is formed from Euclid's Extended Algorithm (365) for encrypting data.

Existence of Right and Left Representations of the Graph for Linear Periodically Time-Varying Systems

Abstract: The graph representation of a system (the set of all input-output pairs) has gained considerable attention in the control literature in view of its usefulness for the analysis of feedback systems. In this paper it is shown that the graph of any stabilizable, linear, periodically time-varying (LPTV), continuous-time system can be expressed as the range and kernel of bounded, causal, LPTV systems that are, respectively, left and right invertible by bounded, causal, LPTV systems. These so-called strong-right and strong-left representations are closely related to the perhaps more common notion of coprime factor representations. As an example of their usefulness, a neat characterization of closed-loop stability is obtained in terms of strong-right and strong-left representations of the plant and controller graphs. This in turn leads to a Youla-style parametrization of stabilizing controllers. All of the results obtained accommodate possibly infinite-dimensional input and output spaces and apply, as a special case, to sampled-data control-systems. Furthermore, they are particularly useful for robustness analysis in terms of the gap metric.

The order of inverses mod q

Abstract: Let q be a prime number and let a = (a 1 ,..., as) be an s-tuple of distinct integers modulo q. For any x coprime with q, let 1 ≤ x < q be such that xx = 1 (mod q). For fixed s and q → ∞ an asymptotic formula is given for the number of residue classes x modulo q for which x + a 1 < x + a 2 < ... < x + a s . The more general case, when q is not necessarily prime and x is restricted to lie in a given subinterval of [1, q], is also treated.

Right coprime factorizations using system upper Hessenberg forms-the multi-input system case

Abstract: Based on a method for right coprime factorizations of linear systems using matrix elementary transformations, it is shown that a very simple iteration formula exists for right coprime factorizations of multi-input linear systems in system upper Hessenberg forms. This formula gives directly the coefficient matrices of the pair of solutions to the right coprime factorization of the system Hessenberg form, and involves only manipulations of inverses of a few triangular matrices and some matrix productions and summations. Based on this formula, a simple, efficient procedure for determining a right coprime factorization of a multi-input linear system is proposed, which first converts a given linear system into its system Hessenberg form using some orthogonal similarity transformations and then applies the iteration formula to the converted system Hessenberg form. An example demonstrates the usage of the approach.

Bloch electron in a magnetic field and the ising model

Abstract: The spectral determinant $\mathrm{det}(H\ensuremath{-}\ensuremath{\varepsilon}I)$ of the Azbel-Hofstadter Hamiltonian $H$ is related to Onsager's partition function of the 2D Ising model for any value of magnetic flux $\ensuremath{\Phi}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2\ensuremath{\pi}P/Q$ through an elementary cell, where $P$ and $Q$ are coprime integers. The band edges of $H$ correspond to the critical temperature of the Ising model; the spectral determinant at these (and other points defined in a certain similar way) is independent of $P$. A connection of the mean of Lyapunov exponents to the asymptotic (large $Q$) bandwidth is indicated.

Coprime Factorizations of Multivariate Rational Matrices

Abstract: Coprime factorization is a well-known issue in one-dimensional systems theory, having many applications in realization theory, balancing, controller synthesis, etc. Generalization to systems in more than one independent variable is a delicate matter: First, several nonequivalent coprimeness notions for multivariate polynomial matrices have been discussed in the literature: zero, minor, and factor coprimeness. Here we adopt a generalized version of factor primeness that appears to be most suitable for multidimensional systems: a matrix is prime iff it is a minimal annihilator. After reformulating the sheer concept of a factorization, it is shown that every rational matrix possesses left and right coprime factorizations that can be found by means of computer algebraic methods. Several properties of coprime factorizations are given in terms of certain determinantal ideals.

Cyclic Structure of Dynamical Systems Associated with $3x+d$ Extensions of Collatz Problem

Abstract: We study here, from both theoretical and experimental points of view, the cyclic structures, both general and primitive, of dynamical systems ${cal D}_d$ generated by iterations of the functions $T_d$ acting, for all $dgeq 1$ relatively prime to 6, on positive integers : $$T_d : {f N} longrightarrow {f N}; qquad T_d(n) = cases{hskip 0.6em elax {n over 2} &, if $n$ is even; {3n+d over 2} &, if $n$ is odd. cr}$$ In the case $d = 1$, the properties of the system ${cal D} = {cal D}_1$ are the subject of the well-known $3x+1$ conjecture. For every one of 6667 systems ${cal D}_d, 1le d le 19999$, we calculate its (complete, as we argue) list of primitive cycles. We unite in a single conceptual framework of primitive memberships, and we experimentally confirm three primitive cycles conjectures of Jeff Lagarias. An in-deep analysis of the diophantine formulae for primitive cycles, together with new rich experimental data, suggest several new conjectures, theoretically studied and experimentally confirmed in the present paper. As a part of this program, we prove a new upper bound to the number of primitive cycles of a given oddlength.

Reciprocity laws for generalized higher dimensional Dedekind sums

Abstract: This is a slight generalization of the usual Dedekind sum which is defined by s(h, k) = s(1, h; k). The Dedekind sum has various formal properties which we shall not list; one is s(a, b; c) = s(ad, bd; c) for d coprime to c. If a is coprime to c this shows that s(a, b; c) equals the classical Dedekind sum s(a′b, c) where aa′ ≡ 1 (mod c). Rademacher [4] proved a three-term reciprocity law for these sums. Theorem 1. Let a, b and c be pairwise coprime positive integers. Then s(a, b; c) + s(b, c; a) + s(c, a; b) = − 4 + 1 12 ( a bc + b ca + c ab ) .

On groups with d-generator subgroups of coprime index

Abstract: Kovacs and Sim ([8], Theorem 2) proved that if a finite soluble group G has a family of d-generator subgroups whose indices have no common divisor, then G can be generated by d + 1 elements. This was generalised by the author [lo] to the larger class of finite groups with zero presentation rank. The hypothesis in the previous results may be seen as a weakening of the hypothesis of a theorem of Kovacs [7], which asserts that if each Sylow subgroup of a finite soluble group can be generated by d elements then the group itself can be generat,ed by d + 1 elements. The solubility hypothesis in the theorem of Kovbcs has been removed by the author [9] and Guralnick [6 ] , while it remains an open problem whether the theorem of Kovacs and Sim holds for arbitrary finite groups. In this paper we prove:

Doubly Coprime Factorizations for General Proper Systems by Using Reduced-Order Observer-Based Controllers

Abstract: By using reduced-order observer-based controllers, explicit state-space realizations of doubly coprime factorizations for general proper systems are derived. The known results for strictly proper systems can be obtained from our factorizations as particular forms. Also, the physical significance of the resulting controller parameterization is analyzed. Finally, the interesting connections between these factorizations in the present paper using an auxiliary stable matrix and those given before without introducing an auxiliary stable matrix are revealed, which show two versions of doubly coprime factorizations can be obtained each other by simple computations.

On Erdos's elementary method in the asymptotic theory of partitions

Abstract: Let m be a positive integer, and let A be the set of all positive integers that belong to a union of r distinct congruence classes modulo m. We assume that the elements of A are relatively prime, that is, gcd(A) = 1. Let p_A(n) denote the number of partitions of n into parts belonging to A. We obtain the asymptotic formula log p_A(n) ~ \pi \sqrt(2rn/3m). The proof is based on Erdos's elementary method to obtain the asymptotic formula for the usual partition function p(n).

The fixed points of coprime action

Abstract: Let P be an odd \(\pi \)-group that acts as a group of automorphisms on the soluble \(\pi '\)-group G. We obtain generators for the fixed points of P on [G, P].