Search results for "Galois theory"
showing 8 items of 8 documents
Binary Hamming codes and Boolean designs
2021
AbstractIn this paper we consider a finite-dimensional vector space $${\mathcal {P}}$$ P over the Galois field $${\text {GF}}(2),$$ GF ( 2 ) , and the family $${\mathcal {B}}_k$$ B k (respectively, $${\mathcal {B}}_k^*$$ B k ∗ ) of all the k-sets of elements of $$\mathcal {P}$$ P (respectively, of $${\mathcal {P}}^*= {\mathcal {P}} \setminus \{0\}$$ P ∗ = P \ { 0 } ) summing up to zero. We compute the parameters of the 3-design $$({\mathcal {P}},{\mathcal {B}}_k)$$ ( P , B k ) for any (necessarily even) k, and of the 2-design $$({\mathcal {P}}^{*},{\mathcal {B}}_k^{*})$$ ( P ∗ , B k ∗ ) for any k. Also, we find a new proof for the weight distribution of the binary Hamming code. Moreover, we…
A-Codes from Rational Functions over Galois Rings
2006
In this paper, we describe authentication codes via (generalized) Gray images of suitable codes over Galois rings. Exponential sums over these rings help determine--or bound--the parameters of such codes.
Hurwitz spaces of Galois coverings of P1, whose Galois groups are Weyl groups
2006
Abstract We prove the irreducibility of the Hurwitz spaces which parametrize equivalence classes of Galois coverings of P 1 , whose Galois group is an arbitrary Weyl group, and the local monodromies are reflections. This generalizes a classical theorem due to Luroth, Clebsch and Hurwitz.
Bi-homogeneity and integrability of rational potentials
2020
Abstract In this paper we consider natural Hamiltonian systems with two degrees of freedom for which Hamiltonian function has the form H = 1 2 ( p 1 2 + p 2 2 ) + V ( q 1 , q 2 ) and potential V ( q 1 , q 2 ) is a rational function. Necessary conditions for the integrability of such systems are deduced from integrability of dominate term of the potential which usually is appropriately chosen homogeneous term of V. We show that introducing weights compatible with the canonical structure one can find new dominant terms which can give new necessary conditions for integrability. To deduce them we investigate integrability of a family of bi-homogeneous potentials which depend on two integer para…
Group theory aspects of chaotic strings
2014
Chaotic strings are a special type of non-hyperbolic coupled map lattices, exhibiting a rich structure of complex dynamical phenomena with a surprising correspondence to physical contents. Chaotic strings are generated by the Chebyshev maps T2() and T3(). In this paper we connect the Chebyshev maps via the Galois theory to the cyclic groups Z2 and Z3 and give some ideas how this fundamental connection might lead to the emergence of the familiar Lie group structure of particle physics and, finally, even to the emergence of space-time. The Z3-graded cubic and ternary algebras presented here have been introduced by R. Kerner in 1991 and then developed and elaborated in collaboration with many …
Algebraic groups as difference Galois groups of linear differential equations
2019
We study the inverse problem in the difference Galois theory of linear differential equations over the difference-differential field $\mathbb{C}(x)$ with derivation $\frac{d}{dx}$ and endomorphism $f(x)\mapsto f(x+1)$. Our main result is that every linear algebraic group, considered as a difference algebraic group, occurs as the difference Galois group of some linear differential equation over $\mathbb{C}(x)$.
Integrability and Non Integrability of Some n Body Problems
2016
International audience; We prove the non integrability of the colinear 3 and 4 body problem, for any positive masses. To deal with resistant cases, we present strong integrability criterions for 3 dimensional homogeneous potentials of degree −1, and prove that such cases cannot appear in the 4 body problem. Following the same strategy, we present a simple proof of non integrability for the planar n body problem. Eventually, we present some integrable cases of the n body problem restricted to some invariant vector spaces.
A Symplectic Kovacic's Algorithm in Dimension 4
2018
Let $L$ be a $4$th order differential operator with coefficients in $\mathbb{K}(z)$, with $\mathbb{K}$ a computable algebraically closed field. The operator $L$ is called symplectic when up to rational gauge transformation, the fundamental matrix of solutions $X$ satisfies $X^t J X=J$ where $J$ is the standard symplectic matrix. It is called projectively symplectic when it is projectively equivalent to a symplectic operator. We design an algorithm to test if $L$ is projectively symplectic. Furthermore, based on Kovacic's algorithm, we design an algorithm that computes Liouvillian solutions of projectively symplectic operators of order $4$. Moreover, using Klein's Theorem, algebraic solution…