Noncommutative Galois Extension and Graded q-Differential Algebra

Abstract Identifiability guarantees that the mathematical model of a dynamic system is well defined in the sense that it maps unambiguously its parameters to the output trajectories. This paper casts identifiability in a set-membership (SM) framework and relates recently introduced properties, namely, SM-identifiability, μ-SM-identifiability, and ε-SM-identifiability, to the properties of parameter estimation problems. Soundness and ε-consistency are proposed to characterize these problems and the solution returned by the algorithm used to solve them. This paper also contributes by carefully motivating and comparing SM-identifiability, μ-SM-identifiability and ε-SM-identifiability with related properties found in the literature, and by providing a method based on differential algebra to check these properties.

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Linearization in the parameters via differential algebra techniques

IFAC Proceedings Volumes ◽

10.1016/s1474-6670(17)34924-8 ◽

2003 ◽

Vol 36 (16) ◽

pp. 1209-1214

Author(s):

Maria Pia Saccomani

Keyword(s):

Differential Algebra

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Distribution of points on Abelian covers over finite fields

International Journal of Number Theory ◽

10.1142/s1793042118500860 ◽

2018 ◽

Vol 14 (05) ◽

pp. 1375-1401 ◽

Cited By ~ 1

Author(s):

Patrick Meisner

Keyword(s):

Finite Fields ◽

Galois Extension ◽

Random Variables ◽

Families Of Curves ◽

Abelian Covers ◽

Distribution Of Points ◽

Genus Formula

We determine in this paper the distribution of the number of points on the covers of [Formula: see text] such that [Formula: see text] is a Galois extension and [Formula: see text] is abelian when [Formula: see text] is fixed and the genus, [Formula: see text], tends to infinity. This generalizes the work of Kurlberg and Rudnick and Bucur, David, Feigon and Lalin who considered different families of curves over [Formula: see text]. In all cases, the distribution is given by a sum of [Formula: see text] random variables.

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ON THE RANK OF ABELIAN VARIETIES OVER AMPLE FIELDS

International Journal of Number Theory ◽

10.1142/s1793042110003071 ◽

2010 ◽

Vol 06 (03) ◽

pp. 579-586 ◽

Cited By ~ 3

Author(s):

ARNO FEHM ◽

SEBASTIAN PETERSEN

Keyword(s):

Finite Field ◽

Abelian Variety ◽

Galois Extension ◽

Characteristic Zero ◽

Rational Point ◽

Abelian Varieties ◽

Finitely Generated ◽

Infinite Rank ◽

Finite Set

A field K is called ample if every smooth K-curve that has a K-rational point has infinitely many of them. We prove two theorems to support the following conjecture, which is inspired by classical infinite rank results: Every non-zero Abelian variety A over an ample field K which is not algebraic over a finite field has infinite rank. First, the ℤ(p)-module A(K) ⊗ ℤ(p) is not finitely generated, where p is the characteristic of K. In particular, the conjecture holds for fields of characteristic zero. Second, if K is an infinite finitely generated field and S is a finite set of local primes of K, then every Abelian variety over K acquires infinite rank over certain subfields of the maximal totally S-adic Galois extension of K. This strengthens a recent infinite rank result of Geyer and Jarden.

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A Hopf algebra having a separable Galois extension is finite dimensional

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-08-09557-9 ◽

2008 ◽

Vol 136 (10) ◽

pp. 3405-3408 ◽

Cited By ~ 2

Author(s):

Juan Cuadra

Keyword(s):

Hopf Algebra ◽

Galois Extension ◽

Finite Dimensional

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An embedding of Schwartz distributions in the algebra of asymptotic functions

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171298000593 ◽

1998 ◽

Vol 21 (3) ◽

pp. 417-428 ◽

Cited By ~ 20

Author(s):

Michael Oberguggenberger ◽

Todor Todorov

Keyword(s):

Differential Algebra ◽

Generalized Functions ◽

Schwartz Distributions ◽

Asymptotic Functions ◽

Space Of Distributions ◽

Asymptotic Function

We present a solution of the problem of multiplication of Schwartz distributions by embedding the space of distributions into a differential algebra of generalized functions, called in the paper asymptotic function, similar to but different from J. F Colombeau's algebras of new generalized functions.

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