Showing papers on "Coprime integers published in 1994"

Feedback Stabilization Over Commutative Rings: The Matrix Case

Abstract: This paper provides a solution of the feedback stabilization problem over commutative rings for matrix transfer functions. Stabilizability of a transfer matrix is realised as local stabilizability over the entire spectrum of the ring. For stabilizable plants, certain modules generated by its fractions and that of the stabilizing controller are shown to be projective compliments of each other. In the case of rings with irreducible spectrum, this geometric relationship shows that the plant is stabilizable if and only if the above modules of the plant are projective of ranks equal to the number of inputs and the outputs. If the maxspectrum of the ring is Noetherian and of zero (Krull) dimension, then this result shows that the stabilizable plants have doubly coprime fractions. Over unique factorization domains the above stabilizability condition is interpreted in terms of matrices formed by the fractions of the plant. Certain invariants of these matrices known as elementary factors, are defined and it is shown that the plant is stabilizable if and only if these elementary factors generate the whole ring. This condition thus provides a generalization of the coprime factorizability as a condition for stabilizabilty. A formula for the class of all stabilizing controllers is then developed that generalizes the previous well-known formula in factorization theory. For multidimensional transfer functions these results provide concrete conditions for stabilizabilty. Finally, it is shown that the class of polynomial rings over principal ideal domains is an additional class of rings over which stabilizable plants always have doubly coprime fractions.

Normalized coprime factorizations and balancing for unstable nonlinear systems

Abstract: In this paper we first study the normalized right and left coprime factorizations of a nonlinear system from a state-space point of view. In order to do so, we introduce the notion of inner and co-inner nonlinear systems. Secondly we deal with balancing for unstable nonlinear systems. For linear systems there are several ways to approach this problem, and we generalize the ideas of balancing the normalized coprime factorization and of LQG balancing to the nonlinear case. LQG balancing can be seen as measuring the influence of a state component on a certain future and past energy function. We extend this interpretation to the nonlinear case. Furthermore we use the normalized coprime factorization to give another way of balancing the unstable nonlinear system. We give the relation between these two methods of balancing, which is similar to the relation in the linear case.

Systems of convolution equations, deconvolution, Shannon Sampling, and the wavelet and Gabor transforms

Abstract: Linear translation invariant systems (e.g., sensors, linear filters) are modeled by the convolution equation $s = f *\mu $, where f is the input signal, $\mu$ is the system impulse response function (or, more generally, impulse response distribution), and s is the output signal. In many applications, the output s is an inadequate approximation of f, which motivates solving the convolution equation for f, i.e., deconvolving f from $\mu$. If the function $\mu$ is time-limited (compactly supported) and nonsingular, it is proven that this deconvolution problem is ill-posed.A theory of solving such equations has been developed by Berenstein et al. It circumvents ill-posedness by using a multichannel system. If the signal f is overdetermined by using a system of convolution equations, $s_i = f * \mu _i ,\,i = 1, \ldots ,n$, the problem of solving for f is well-posed if the set of convolvers $\{ \mu _i\}$ satisfies the condition of being what is called strongly coprime. In this case, there exist compactly suppor...

An Improved Projection for Cylindrical Algebraic Decomposition

Abstract: It is proved that the projection needed by the CAD algorithm needs only to compute the resultants, the discriminants, the leading coefficients and the constant coefficients of the input polynomials, provided they have be made square—free and relatively prime by GCD computations. This improves [7] first improvement by removing any dimension condition and dropping out from the projection set all coefficients of the input polynomials, other than the leading and the constant ones.

Some remarks on the abc -conjecture

Abstract: Let r(x) be the product of all distinct primes dividing a nonzero integer x . The abc-conjecture says that if a, b, c are nonzero relatively prime integers such that a + b + c = 0, then the biggest limit point of the numbers logmax(lal, ibl, cil) log r(abc) equals 1. We show that in a natural anologue of this conjecture for n > 3 integers, the largest limit point should be replaced by at least 2n 5. We present an algorithm leading to numerous examples of triples a, b, c for which the above quotients strongly deviate from the conjectural value 1.

A key-exchange protocol using real quadratic fields

Abstract: In 1976 Diffie and Hellman first introduced their well-known key-exchange protocol which is based on exponentiation in the multiplicative group GF(p)* of integers relatively prime to a large primep (see [8]). Since then, this scheme has been extended to numerous other finite groups. Recently, Buchmann and Williams [2] introduced a version of the Diffie-Hellman protocol which uses the infrastructure of a real quadratic field. Theirs is the first such system not to require an underlying group structure, but rather a structure which is "almost" like that of a group. We give here a more detailed description of this scheme as well as state the required algorithms and considerations for their implementation.

Stable kernel representations as nonlinear left coprime factorizations

Abstract: A representation of nonlinear systems based on the idea of representing the input-output pairs of the system as elements of the kernel of a stable operator has been previously introduced by the authors (1993, 1994). This has been denoted the kernel representation of the system. In this paper it is demonstrated that the kernel representation is a generalization of the left coprime factorization of a general nonlinear system in the sense that it is a dual operator to the right coprime factorization of a nonlinear system. The results obtainable in the linear case linking left and right coprime factorizations are shown to be reproduced within the kernel representation framework. >

On coprime factorization and minimal realization of transfer function matrices using the pseudo-observability concept

Abstract: A simple computational algorithm is presented for calculating the coprime matrix fraction description and minimal state-space representation of a multivariable linear system specified by a transfer function matrix. The concept of admissibility of pseudo-observability indices is utilized in the algorithm to permit extraction of the most convenient representations, in the sense that the representation elements have the lowest absolute values. A computational example is given to show the feasibility of the suggested method

Representations of graphs modulo n

Abstract: A graph is representable modulo n if its vertices can be labeled with distinct integers between 0 and n, the difference of the labels of two vertices being relatively prime to n if and only if the vertices are adjacent. Erdős and Evans recently proved that every graph is representable modulo some positive integer. We derive a combinatorial formulation of representability modulo n and use it to characterize those graphs representable modulo certain types of integers, in particular integers with only two prime divisors. Other facets of representability are also explored. We obtain information about the values of n modulo which paths and cycles are representable.

On characters of coprime operator groups and the Glaubermann character correspondence.

Abstract: An air injection sub for use with a dual conduit drill pipe string is provided having inner and outer concentric tubular members which are connected to the corresponding inner and outer conduits of a dual conduit string to provide isolated annular and central passageways and is particularly characterized by improved injection means in the inner tubular member which allows fluid to pass from the annular passageway into the central passageway through a plurality of apertures which open into the central passageway at a plurality of levels and angular positions, is field adjustable and provides improved erosion resistance.

Performance-preserving frequency weighted controller approximation: a coprime factorisation approach

Abstract: Given a stabilising controller which satisfies a prespecified H/sub /spl infin// performance bound on the closed loop, we derive sufficient conditions for reduced order controllers to be stabilising and satisfy the same performance bound. The conditions are expressed as norm bounds on the frequency weighted error between coprime factor representations of the given controller and the reduced order controller. Reduced order controllers are then designed by interpreting the sufficient conditions as frequency weighted model reduction problems. >

Tiling a rectangle with the fewest squares

Abstract: We show that a square-tiling of a $p\times q$ rectangle, where $p$ and $q$ are relatively prime integers, has at least $\log_2p$ squares. If $q>p$ we construct a square-tiling with less than $q/p+C\log p$ squares of integer size, for some universal constant $C$.

Residue multipliers using factored decomposition

Abstract: A technique for residue multiplication is described where the modulus is a non-prime integer. If the modulus m can be decomposed into two or more relatively prime factors, then multiplication can be done as a set of concurrent multiplication operations using the relatively prime factors as moduli. If a factor is prime, then multiplication is performed with an index calculus technique, otherwise a direct table look-up is used. An all ROM table look-up implementation of this technique is considered and a specific example is given for a modulo-28 multiplier. Hardware requirements for non-prime moduli up to 1024 are calculated and analyzed and compared with those for prime moduli of similar magnitude. Finally, an extension of the decomposition technique is discussed which allows further levels of modulus decomposition. >

Generalized Schur Methods to Compute Coprime Factorizations of Rational Matrices

Abstract: Numerically reliable state space algorithms are proposed for computing the following stable coprime factorizations of rational matrices: 1) factorizations with least order denominators; 2) factorizations with inner denominators; and 3) factorizations with proper stable factors. The new algorithms are based on a recursive generalized Schur algorithm for pole assignment. They are generally applicable regardless the original descriptor state space representation is minimal or not, or is stabilizable/detectable or not. The proposed algorithms are useful in solving various computational problems for both standard and descriptor system representations.

The number of lattice points within a contour and visible from the origin

Abstract: or with the RH 0 A similar estimate, with the same sort of error, is obtained for the number of relatively prime pairs (α, b) of positive integers so that ab < r. The error term for a general contour depends on the maximal value of the radius of curvature of the bounding contour.

Coprime factorization approach to robust stabilization of control structures interaction evolutionary model

Abstract: Stabilization in the presence of uncertainties is a fundamental requirement in the design of feedback compensators for flexible structures. The search for the largest neighborhood around a given design plant for which a single feedback controller produces closed-loop stability can be formulated as an H^ control problem. It has been shown that the use of normalized coprime factor plant descriptions, where the plant perturbations are defined as additive modifications to the coprime factors, leads to a closed-form expression for the maximal-neighborhood boundary allowing optimal and suboptimal HOC compensators to be computed directly without the usual 7iteration. This paper describes an application of normalized coprime factor stabilization theory to the computation of robustly stable compensators for the NASA Control Structures Interaction Evolutionary Model. Results indicate that the suboptimal version of the theory has the potential of providing low authority compensators that are robustly stable for significant regions of variations in design model parameters and additive unmodeled dynamics.

A new class of infinite products, and Euler's totient

Abstract: We introduce some new infinite products, the simplest being(1−y)∏k=2∞∏j∈ϕk(1−ykqj)1/k=(1−y1−qy)1/(1−q),where ϕk is the set of positive integers less than and relatively prime to k, valid for |y|∧|qy| both less than unity, with q≠1 The idea of a q-analogue for the Euler totient function is suggested

Counting problems relating to a theorem of Dirichlet

Abstract: We study weighted versions of Dirichlet's theorem on the probability that two integers, taken at random, are relatively prime. This leads to a uniform approach in the study of several counting problems in discrete and computational geometry relating to incidences between points and lines.

The bounded real characteristic function and Nehari extensions

Abstract: For the class of bounded real functions in the open right half plane we define a map, we call it the B-characteristic, which associates with each bounded real unction a stable function. The definition is based on normalized coprime factorizations of bounded real functions, the normalization being with respect to an indefinite metric. The construction bears great similarity to the characteristic functions studied in Fuhrmann and Ober [1993] for other classes of functions. We study the properties of the B-characteristic, the inverse map and the connections to balancing. We apply the construction to shed some new light on the problem of suboptimal Nehari complementation.

Pole assignment at infinity and polynomial matrix equations

Abstract: A simple approach to synthesizing a proportional and derivative state feedback gain for assigning poles to infinity is developed. We also apply the synthesized gain to determine a doubly coprime matrix fraction description (MFD) of a state-space system and solve related problems. It is interesting to note that the MFD and the solutions to the related problems could be obtained in an explicit formula by using the synthesized gain.