Showing papers on "Coprime integers published in 1997"

Boundary Solutions of the Classical Yang--Baxter Equation

Abstract: We define a new class of unitary solutions to the classical Yang--Baxter equation (CYBE). These ‘boundary solutions’ are those which lie in the closure of the space of unitary solutions of the modified classical Yang--Baxter equation (MCYBE). Using the Belavin--Drinfel'd classification of the solutions to the MCYBE, we are able to exhibit new families of solutions to the CYBE. In particular, using the Cremmer--Gervais solution to the MCYBE, we explicitly construct for all n ≥ 3 a boundary solution based on the maximal parabolic subalgebra of $${\mathfrak{s}}{\mathfrak{l}}\left( n \right)$$ obtained by deleting the first negative root. We give some evidence for a generalization of this result pertaining to other maximal parabolic subalgebras whose omitted root is relatively prime to n. We also give examples of nonboundary solutions for the classical simple Lie algebras.

Computation of coprime factorizations of rational matrices

Abstract: We propose a numerically reliable state space algorithm for computing coprime factorizations of rational matrices with factors having poles in a given stability domain. The new algorithm is based on a recursive generalized Schur technique for poles dislocation by means of proportional-derivative state feedback. The proposed algorithm is generally applicable regardless the underlying descriptor state space representation is minimal or not, or is stabilizable/detectable or not.

Representation Theory of the Affine Lie Superalgebra at Fractional Level

Abstract: N= 2 noncritical strings are closely related to the \(\) Wess-Zumino-Novikov-Witten model, and there is much hope to further probe the former by using the algebraic apparatus provided by the latter. An important ingredient is the precise knowledge of the \(\) representation theory at fractional level. In this paper, the embedding diagrams of singular vectors appearing in \(\)Verma modules for fractional values of the level (\(\), p and q coprime) are derived analytically. The nilpotency of the fermionic generators in $\hslc$ requires the introduction of a nontrivial generalisation of the MFF construction to relate singular vectors among themselves. The diagrams reveal a striking similarity with the degenerate representations of the N= 2 superconformal algebra.

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 \chi^{p, p'}_{r, s}, dependent on two finite size parameters M and N, in the cases where: (i) p and p' are coprime integers that satisfy 0 < p < p'. (ii) If the pair (p', p) has a continued fraction (c_1, c_2, ... , c_{t-1}, c_t+2), where t >= 1, then the pair (s, r) has a continued fraction (c_1, c_2, ... , c_{u-1}, d), where 1 =< u =< t, and 1 =< d =< c_{u}. The limit M -> infinity, for fixed N, and the limit N -> infinity, 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 characterise 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 -> infinity and N -> infinity, leads to the same q-series identity for the corresponding character.

On two classes of permutations with number-theoretic conditions on the lengths of the cycles

Abstract: Let Λ be an arbitrary set of positive integers andSn(Λ) the set of all permutations of degreen for which the lengths of all cycles belong to the set Λ. In the paper the asymptotics of the ratio |Sn(Λ)|/n! asn→∞ is studied in the following cases: 1) Λ is the union of finitely many arithmetic progressions, 2) Λ consists of all positive integers that are not divisible by any number from a given finite set of pairwise coprime positive integers. Here |Sn(Λ)| stands for the number of elements in the finite setSn(Λ).

On the binary quadratic residue system with noncoprime moduli

Abstract: The residue number system (RNS) appropriate for implementing fast digital signal processors since it can support parallel, carry-free, high-speed arithmetic. A development in residue arithmetic is the quadratic residue number system (QRNS), which can perform complex multiplications with only two integer multiplications instead of four. An RNS/QRNS is defined by a set of relatively prime integers, called the moduli set, where the choice of this set is one of the most important design considerations for RNS/QRNS systems. In order to maintain simple QRNS arithmetic, moduli sets with numbers of forms 2/sup n/+1 (n is even) have been considered. An efficient such set is the three-moduli set (2/sup 2k-2/+1.2/sup 2k/+1.2/sup 2k+2/+1) for odd k. However, if large dynamic ranges are desirable, QRNS systems with more than three relatively prime moduli must be considered. It is shown that if a QRNS set consists of more than four relatively prime moduli of forms 2/sup n/+1, the moduli selection process becomes inflexible and the arithmetic gets very unbalanced. The above problem can be solved if nonrelatively prime moduli are used. New multimoduli QRNS systems are presented that are based on nonrelatively prime moduli of forms 2/sup n/+1 (n even). The new systems allow flexible moduli selection process, very balanced arithmetic, and are appropriate for large dynamic ranges. For a given dynamic range, these new systems exhibit better speed performance than that of the three-moduli QRNS system.

Nonlinear coprime factorizations and parametrization of a class of stabilizing controllers

Abstract: New definitions for right, left and doubly coprime factorizations for nonlinear, input-affine state-space systems are introduced. These definitions are based on the state-to-output stability introduced by Baramov and Kimura (1996) and the chain-scattering formalism. Sufficient conditions for the existence of these factorizations as well as local state-space formulas for factors are given. Finally, these results are applied to obtain a parametrized set of stabilizing controllers to a fairly broad class of plants, for transforming the original feedback control configuration into the open loop model matching configuration and for thus extending the classical Youla-Kucera parametrization to nonlinear (local) cases.

Numbers and Symmetry: An Introduction to Algebra

Abstract: Chapter 1. New Numbers A Planeful of Integers, Z[i] Circular Numbers, Zn More Integers on the Number Line, Z [v2] Notes Chapter 2. The Division Algorithm Rational Integers Norms Gaussian Numbers Q (v2) Polynomials Notes Chapter 3. The Euclidean Algorithm Bezout's Equation Relatively Prime Numbers Gaussian Integers Notes Chapter 4. Units Elementary Properties Bezout's Equations Wilson's Theorem Orders of Elements: Fermat and Euler Quadratic Residues Z [v2] Notes Chapter 5. Primes Prime Numbers Gaussian Primes Z [v2] Unique Factorization into Primes Zn Notes Chapter 6. Symmetries Symmetries of Figures in the Plane Groups The Cycle Structure of a Permutation Cyclic Groups The Alternating Groups Notes Chapter 7. Matrices Symmetries and Coordinates Two-by-Two Matrices The Ring of Matrices M2(R) Units Complex Numbers and Quaternions Notes Chapter 8. Groups Abstract Groups Subgroups and Cosets Isomorphism The Group of Units of a Finite Field Products of Groups The Euclidean Groups E (1), E (2), and E (3) Notes Chapter 9. Wallpaper Patterns One-Dimensional Patterns Plane Lattices Frieze Patterns Space Groups The 17 Plane Groups Notes Chapter 10. Fields Polynomials Over a Field Kronecker's Construction of Simple Field Extensions Finite Fields Notes Chapter 11. Linear Algebra Vector Spaces Matrices Row Space and Echelon Form Inverses and Elementary Matrices Determinants Notes Chapter 12. Error-Correcting Codes Coding for Redundancy Linear Codes Parity-Check Matrices Cyclic Codes BCH Codes CDs Notes Chapter 13. Appendix: Induction Formulating the n-th Statement The Domino Theory: Iteration Formulating the Induction Statement Squares Templates Recursion Notes Chapter 14. Appendix: The Usual Rules Rings Notes Index

Designing commutative cascades of multidimensional upsamplers and downsamplers

Abstract: In multiple dimensions, the cascade of an upsampler by L and a downsampler by M commutes if and only if the integer matrices L and M are right coprime and LM=ML. This letter presents algorithms to design L and M that yield commutative upsampler/dowsampler cascades. We prove that commutativity is possible if the Jordan canonical form of the rational (resampling) matrix R=LM/sup -1/ is equivalent to the Smith-McMillan form of R. A necessary condition for this equivalence is that R has an eigendecomposition and the eigenvalues are rational.

Characters, coprime actions, and operator groups

Abstract: In this paper, we answer the following question posed by G. Navarro: suppose that S acts on G coprimely, and that A acts on SG so that S and G are both A-invariant. If $\chi $ is an S-invariant irreducible character of G, then must $\chi $ extend to AG if and only if $\chi ^*$ extends to A C G (S), where $\chi ^*$ is the Glauberman-Isaacs Correspondent for $\chi $ under the action of S? We prove that the answer to this question is yes.

Normalized coprime factorizations for sampled-data systems

Abstract: The notion of normalized coprime factorization (NCF) for sampled-data systems is studied. The problem is approached by the study of linear systems with discrete jumps at periodic time instants. These systems arise in the study of linear systems with sampled-data control and filtering problems. It is shown that the NCF can be obtained through the solution to Riccati equations with jumps. The properties of the Riccati equations and the corresponding properties of the NCF are studied.

A generalization of rummer’s congruences

Abstract: Suppose thatm, n are positive even integers andp is a prime number such thatp-1 is not a divisor ofm. For any non-negative integerN, the classical Kummer’s congruences on Bernoulli numbersB n(n = 1,2,3,...) assert that (1-p m-1)B m/m isp-integral and (1) $$(1 - p^{m - 1} )\frac{{B_m }}{m} \equiv (1 - p^{n - 1} )\frac{{B_n }}{n}(\bmod p^{N + 1} )$$ ifm ≡ n (mod (p-1)p n). In this paper, we shall prove that for any positive integerk relatively prime top and non-negative integers α, β such that α +jk =pβ for some integerj with 0 ≤j ≤p-l.Then for any non-negative integerN, (2) $$\frac{1}{m}\{ B_m (\frac{\alpha }{k}) - p^{m - 1} B_m (\frac{\beta }{k})\} \equiv \frac{1}{n}\{ B_n (\frac{\alpha }{k}) - p^{n - 1} B_n (\frac{\beta }{k})\} (\bmod p^{N + 1} )$$ ifp-1 is not a divisor ofm andm ≡ n (mod (p-1)p n). HereB n(x) (n = 0,1,2,...) are Bernoulli polynomials. This of course contains the Kummer’s congruences. Furthermore, it contains new congruences for Bernoulli polynomials of odd indices.

Good matrices: Matrices that preserve ideals

Abstract: The case r = 1 is fairly well-known: if A = [al, . . ., an] and B = [bl, . . ., b"]T are such that AB [1], then al, . . ., an are relatively prime. In this case, there is an n X n integral matrix that has an integral inverse and whose first row is al, . . ., an; see [10, Thm. II.1, p. 13]. Hajja's Problem suggests a natural generalization that has not received much attention and will be a main topic of this article: what are the properties that two or more given rows must have so they can serve as the first rows of an invertible matrix? The property central to our paper is the following. A matrix A with entries in a communtative ring R with unity is left good if, for every vector x, the ideal (xA) generated by the entries in the vector xA is the same as the ideal (x) generated by the entries in the vector x. In the context of matrices with integral entries, this is equivalent to requiring that the greatest common divisor of the entries in xA is the same as the greatest common divisor of the entries in x. Since, for any matrix A and any vector x, it is obvious that (xA) c (x), the content of left goodness is in the reverse containment. Our goal is to prove the following.

Two results on arithmetical functions

Abstract: Generalizing two results of Rieger [8] and Selberg [10] we give asymptotic formulas for sums of type $${\matrix {\sum \limits_{n\leq x}\cr n\equiv l({\rm mod}k)\cr f_{\kappa}(n)\equiv s_{\kappa}({\rm mod}p_{\kappa})\cr (\kappa=1,\dots,r)\cr}}\qquad \chi(n)\qquad {\rm and} {\matrix {\sum \limits_{n\leq x}\cr n\equiv l({\rm mod}k)\cr f_{\kappa}(n)\equiv s_{\kappa}({\rm mod}p_{\kappa})\cr (\kappa=1,\dots,r)\cr}}\qquad \chi(n),$$ where χ is a suitable multiplicative function, f1,…, fr are “small” additive, prime-independent arithmetical functions and k, l are coprime. The proofs are based on an analytic method which consists of considering the Dirichlet series generated by \( \chi(n)z_{1}^{f_{1}(n)}\cdot... \cdot z_{r}^{f_{r}(n)},z_{1}\dots z_{r} \) complex.

Model reduction for non-minimal state-space systems

Abstract: This paper is concerned with discrete-time model reduction via order-reduction of a discrete-time left coprime factor representation of a given model or controller. The proposed model reduction scheme is suitable for both, stable and unstable models, and for non-minimal state-space systems. Balanced singular perturbation approximation is considered rather than balanced truncation. Numerical issues are also considered.

Nonprime memory systems and error correction in address translation

Abstract: Using a prime number p of memory banks on a vector processor allows a conflict-free access for any slice of p consecutive elements of a vector stored with a stride not multiple of p. To reject the use of a prime number of memory banks, it is generally advanced that address computation for such a memory system would require systematic Euclidean division by the number p. The Chinese Remainder Theorem allows a simple mapping of data onto the memory banks for which address computation does not require any Euclidean division. However, this requires that the number of words in each memory module m and p be relatively prime. We propose a method based on the Chinese Remainder Theorem for moduli with common factors that does not have such a restriction. The proposed method does not require Euclidean division and also results in an efficient error detection/correction mechanism for address translation.

On computing normalized coprime factorizations of rational matrices

Abstract: We propose a new computational approach based on descriptor state space algorithms to compute normalized coprime factorizations of rational matrices

On Computing Normalized-Coprime Factorizations of

Abstract: This note presents a numerically reliable approach for com- puting the normalized-coprime factorization of an arbitrary (possibly im- proper) rational matrix. We consider both the cases in which the factoriza- tion is coprime and normalized with respect to either the left-half plane or the unit disk, corresponding to systems with continuous- or discrete-time evolutions. The proposed algorithm remedies the numerical drawbacks of alternative methods available thus far in the literature. Index Terms—Descriptor systems, normalized-coprime factorization, numerical algorithms, rational matrices. I. INTRODUCTION We address the problem of computing normalized-coprime fac- torizations of an arbitrary (possibly improper) rational matrix . Normalized-coprime factorizations appear throughout in robust control systems (as for example in model/controller order reduction and robust controller design), identification and signal processing, and circuit theory (as for example in topological descriptions). Normalized-coprime factorizations of proper rational matrices have been extensively studied in the literature (17), (18), (3). In the proper case, the underlying computation involves the solution to a standard continuous or discrete-time algebraic Riccati equation, the so-called normalized Riccati equation associated with a state-space realization of (18), (7). In this paper, we show that even in the general improper case, the normalized-coprime factorization involves the solution of a standard algebraic Riccati equation whose coefficients are obtained by performing solely orthogonal transformations on a generalized state- space realization of . Our method does not require minimality (or irreducibility) of the realization, and works on a realization which is stabilizable and observable on the boundary of the stability domain (imaginary axis or unit circle). The ideas used here are related to the general factorization tech- niques introduced by the author in (9) and (10) where the most general inner-outer and spectral factorization problems have been solved. In this paper, we make further use of the peculiarities of the problem at hand and treat in a unified way the continuous- and discrete-time cases, both theoretically and numerically, without resorting to bilinear or other nonorthogonal transformations. Mostly important, the resulting algorithm remedies all the numerical drawbacks of the alternative methods proposed in (1) and (14).

Analyzing convolutional encoders using realization theory

Abstract: Using realization theory, we develop an efficient way of determining if a k/spl times/n polynomial matrix G(s) has coprime full size minors. This has the immediate application of determining whether a convolutional encoder is non-catastrophic. The approach is to easily find a realization for the behavior of G(s) and then compute the rank of the controllability matrix of the realized system.

Operations on locally free classgroups

Abstract: Let G be a finite group and K a number field. We show that the "k-th Adams operation" defined by Cassou-Nogues and Taylor on the locally free class group Cl(Z_K G) is a symmetric power operation, if k is coprime to the order of G. Using the equivariant Adams-Riemann-Roch theorem, we furthermore give a geometric interpretation of a formula established by Burns and Chinburg for these operations.

Primes at a (somewhat lengthy) glance

Abstract: One can tell that 17 = 23 + 32 19 = 24 + 3 37 = 52 + 22.3 47 = 2 52 3 53 = 32 * 7 _ 2 * 5 97 = 3 5 7 23 are all prime, at a glance, since we have written each n = A + B where each prime < 4 divides exactly one of A and B (and thus n is coprime with every prime < 4). This strange procedure is thoroughly investigated in [1]; in general, it is quite a challenge to so write a given prime n since the product of the primes < 4 is around e{l +° (l )} 5ln . A similar but more complicated method to establish the primality of n goes as follows: let Pl = 2 < P2 = 3 < *** < Pk be the sequence of primes < 4. Write n in the form

Control: A -Dissipative Approach

Abstract: This paper applies the chain-scattering approach for linear control to input affine nonlinear plants with general output structure. The nonlinear -control problem is reformu- lated in terms of the chain representation of a given plant. The state-space characterization of -dissipative ( -lossless) systems together with two special types of coprime factorizations are used to provide a class of controllers which solve locally the -control problem. The problems considered correspond to the general two- and four-block problems in the linear case. The results are given in terms of two uncoupled nonlinear partial differential inequalities with a coupling condition on their solutions.

Using coprime factorizations in partial decoupling of linear multivariable systems

Abstract: Examines the problem of partial (upper or lower triangular) decoupling in a one-degree-of-freedom feedback configuration. The approach taken is based on the use of coprime factorizations over the ring of proper and stable rational functions, which provides a direct understanding of issues such as internal stability. Within this framework, both necessary and sufficient conditions for partial decoupling are studied and, as a consequence, a design method for partial decoupling control is presented. A parametrization of the set of all partial decoupling controllers is also given. Finally, diagonal decoupling is considered as a particular case of partial decoupling, indicating how the corresponding links with related work on diagonal decoupling and coprime factorizations can be obtained.

Separation properties of theta functions

Abstract: In a 1993 article, G. Faltings gave a new construction of the moduli space $U$ of semistable vector bundles on a smooth curve $X$, avoiding geometric invariant theory. Roughly speaking, Faltings showed that the normalisation $B$ of the ring $A$ of theta functions (associated with vector bundles on $X$) suffices to realize $U$ as a projective variety. Describing Faltings' work, C.S. Seshadri asked how close $A$ is to $B$. In this article, we address this question from a geometric point of view. We consider the rational map, $\pi : U @>>> Proj(A)$, and show that, not only is $\pi$ defined everywhere, but also $\pi$ is bijective, and is an isomorphism over the stable locus of $U$, if the characteristic of the ground field is 0. Moreover, we give a direct local construction of $U$ as a fine moduli space, when the rank and degree are coprime, in any characteristic. The methods in the article apply to singular curves as well.

Factorizations of Rational Matrices

Abstract: We propose a numerically reliable state space algorithm for computing coprime factorizations of rational matrices with factors having poles in a given stability domain. The new algorithm is based on a recursive generalized Schur technique for poles dislocation by means of proportional-derivative state feedback. The proposed algorithm is generally applicable regardless the underlying descriptor state space representation is minimal or not, or is stabilizable/detectable or not.

Bezout identities with pseudopolynomial entries

Abstract: Let $ A_p({\Bbb C}^{n-1}) $ be the algebra of entire functions in n - 1 variables satisfying some growth conditions (p is a plurisubharmonic weight). We consider p-pseudopolynomials in (z 1;z ') = (z 1; z 2,..., z n) in the weighted algebra $ A_{\tilde p}({\Bbb C}^n){\tilde p}(z) = (\ln 2+\vert z_1\vert )+ p(z') $ of the form¶¶ $ f(z)= a_0(z')z_1^m+\sum\limits ^n_{k=1} a_k(z')z_1^{m-k} $ ¶¶ where $ a_0,a_1,\ldots, a_m \in A_p({\Bbb C}^{n-1}) $ .¶We establish several results concerning solutions of the Bezout identity $ 1 = q_1 f_1 + \cdots + q_m f_m $ when $ f_1 \ldots f_m \in A _{\tilde p}({\Bbb C}^n) $ are “strongly coprime”.