On the essential dimension of a finite group

Let f(x) = aixi be a monic polynomial of degree n whosecoefficients are algebraically independent variables over a base field k of characteristic 0. We say that a polynomial g(x) isgenerating (for the symmetric group) if it can be obtained from f(x) by a nondegenerate Tschirnhaus transformation. We show that the minimal number dk(n) of algebraically independent coefficients of such a polynomial is at least [n/2]. This generalizes a classical theorem of Felix Klein on quintic polynomials and is related to an algebraic form of Hilberts 13th problem.Our approach to this question (and generalizations) is basedon the idea of the essential dimension of a finite group G:the smallest possible dimension of an algebraic G-variety over k to which one can compress a faithful linear representation of G. We show that dk(n) is just the essential dimension of the symmetricgroup Sn. We give results on the essential dimension ofother groups. In the last section we relate the notion of essential dimension to versal polynomials and discuss their relationship to the generic polynomials of Kuyk, Saltman and DeMeyer.     

On the crystalline cohomology of Deligne–Lusztig varieties

Let $X\to Y^0$ be an abelian prime-to-p Galois covering of smooth schemes over a perfect field k of characteristic $p>0$. Let Y be a smooth compactification of $Y^0$ such that $Y-Y^0$ is a normal crossings divisor on Y. We describe a logarithmic F-crystal on Y whose rational crystalline cohomology is the rigid cohomology of X, in particular provides a natural $W[F]$-lattice inside the latter; here W is the Witt vector ring of k. If a finite group G acts compatibly on X, $Y^0$ and Y then our construction is G-equivariant. As an example we apply it to Deligne–Lusztig varieties. For a finite field k, if $\mathbb{G}$ is a connected reductive algebraic group defined over k and $\mathbb{L}$ a k-rational torus satisfying a certain standard condition, we obtain a meaningful equivariant $W[F]$-lattice in the cohomology (ℓ-adic or rigid) of the corresponding Deligne–Lusztig variety and an expression of its reduction modulo p in terms of equivariant Hodge cohomology groups.     

Rank gain of Jacobians over number field extensions with prescribed Galois groups

We investigate the rank gain of elliptic curves, and more generally, Jacobian varieties, over non-Galois extensions whose Galois closure has a Galois group permutation-isomorphic to a prescribed group G (in short, “G-extensions”). In particular, for alternating groups and (an infinite family of) projective linear groups G, we show that most elliptic curves over (for example) $\mathbb{Q}$ gain rank over infinitely many G-extensions, conditional only on the parity conjecture. More generally, we provide a theoretical criterion, which allows to deduce that “many” elliptic curves gain rank over infinitely many G-extensions, conditional on the parity conjecture and on the existence of geometric Galois realizations with group G and certain local properties.     

The Groups H3(F(X)/FM) and CH2(X) for Generic Splitting Varieties of Quadratic Forms

Let F be a field of characteristic different from 2 and be a quadratic form over F. Let X be an arbitrary projective homogeneous generic splitting variety of . For example, we can take X to be equal to the variety X_,m of totally isotropic m-dimensional subspaces of V, where V is the quadratic space corresponding to and < dim V. In this paper, we study the groups CH²(X) and H³(F(X)/F) = ker(H ³(F) H ³(F(X))). One of the main results of this paper claims that the group Tors CH²(X) is always zero or isomorphic to . In many cases we prove that Tors CH²(X) = 0 and compute the group H ³(F(X)/F) completely. As an application of the main results, we give a criterion of motivic equivalence of eight-dimensional forms except for the case where the Schur indices of their Clifford algebras equal 4.     

On K3 of Witt Vectors of Length Two Over Finite Fields

We prove that for W₂ the Witt vectors of length two over the finite field , we have in characteristic at least 5 and for (3,f) = 1. The result is proved by using the identity and calculating the right term with a group homology spectral sequence. Some information on the spectral sequence is achieved by using the action of the outer automorphism of SL on the homology groups and recent results on K-groups of local rings and the ring of dual numbers over finite fields.     

Cohomologie galoisienne des groupes quasi-déployés sur des corps de dimension cohomologique ≤2; Galois cohomology of quasi-split groups over fields of cohomological dimension ≤ 2

Let k be a perfect field with cohomological dimension 2. Serre's conjecture II claims that the Galois cohomology set H ¹(k,G) is trivial for any simply connected semi-simple algebraic G/k and this conjecture is known for groups of type ¹ A _n after Merkurjev–Suslin and for classical groups and groups of type F ₄ and G ₂ after Bayer–Parimala. For any maximal torus T of G/k, we study the map H ¹(k, T) H ¹(k, G) using an induction process on the type of the groups, and it yields conjecture II for all quasi-split simply connected absolutely almost k-simple groups with type distinct from E ₈. We also have partial results for E ₈ and for some twisted forms of simply connected quasi-split groups. In particular, this method gives a new proof of Hasse principle for quasi-split groups over number fields including the E ₈-case, which is based on the Galois cohomology of maximal tori of such groups.     

Invariants Cohomologiques de Rost En Caractéristique PositiveRost''s Cohomological Invariants in Positive Characteristic

Let G/F be a semisimple algebraic group defined over a field F with characteristic . Let us denote by the Galois cohomology group introduced by Kato. If , we show that the p-primary part of Rost's invariant lifts in characteristic 0. This result allows to deduce properties of the Rost invariant in positive characteristic from known properties in characteristic 0. The case of Merkurjev–Suslin's invariant is specially interesting, i.e. if G/F=SL(D) for a central simple algebra D/F with degree p and class , one has and an element is a reduced norm if and only if the cup-product is trivial in ; one characterizes also in positive characteristic fields with p-dimension by the surjectivity of reduced norms.In a second part, we study Rost invariants when the base field is complete for a discrete valuation. As planned by Serre, invariants are then linked with Bruhat–Tits' theory, this yields a new proof of their nontriviality.     

On a Calderón–Zygmund commutator-type estimate

In this paper we present a Calderón–Zygmund commutator-type estimate. This estimate enables us to prove an embedding result concerning weighted function spaces.     

Nilpotent Lie Algebras of Maximal Rank and of Kac–Moody Type: E6

Let be the Kac–Moody algebra associated to the affine Cartan matrix E₆⁽¹⁾. Each nilpotent Lie algebra of type E₆⁽¹⁾ is isomorphic to a quotient of the positive part of . We determine the isomorphism classes of nilpotent Lie algebras of type E₆⁽¹⁾.     

On the period function of reversible quadratic centers with their orbits inside quartics

This paper is concerned with the monotonicity of the period function for a class of reversible quadratic centers with their orbits inside quartics. It is proved that such a system has a period function with at most one critical point.     

Inclusion properties for certain classes of meromorphic functions associated with the Choi–Saigo–Srivastava operator

The purpose of the present paper is to introduce several new classes of meromorphic functions defined by using a meromorphic analogue of the Choi–Saigo–Srivastava operator for analytic functions and investigate various inclusion properties of these classes. Some interesting applications involving these and other classes of integral operators are also considered.     

The Vlasov–Maxwell–Fokker–Planck system in two space dimensions

We consider the Cauchy problem for the Vlasov–Maxwell–Fokker–Planck system in the plane. It is shown that for smooth initial data, as long as the electromagnetic fields remain bounded, then their derivatives do also. Glassey and Strauss have shown this to hold for the relativistic Vlasov–Maxwell system in three dimensions, but the method here is totally different. In the work of Glassey and Strauss, the relativistic nature of the particle transport played an essential role. In this work, the transport is nonrelativistic, and smoothing from the Fokker–Planck operator is exploited. Copyright © 2015 John Wiley & Sons, Ltd.     

On Interpolation of Vector – Valued Banach Lattices and Calderón – Lozanovskii Construction

We establish the connection between the variants of Schechter's complex interpolation methods, Peetre – Gustavsson's interpolation methods, and the Calderón – Lozanovskii construction on vector – valued Banach lattices. As applications, we show that the uniform convexity and the UMD property are stable by interpolation.     

Bundles over Quantum Sphere and Noncommutative Index Theorem

The Noncommutative Index Theorem is used to prove that the Chern numbers of quantum Hopf line bundles over the standard Podle quantum sphere equal the winding numbers of the representations defining these bundles. This result gives an estimate of the positive cone of the algebraic K₀ of the standard quantum sphere.     

A Navier–Stokes–Voight model with memory

In this article, we consider a three‐dimensional Navier–Stokes–Voight model with memory where relaxation effects are described through a distributed delay. We prove the existence of uniform global attractors , where ? ∈ (0,1) is the scaling parameter in the memory kernel. Furthermore, we prove that the model converges to the classical three‐dimensional Navier–Stokes–Voight system in an appropriate sense as ? → 0. In particular, we construct a family of exponential attractors Ξ_? that is robust as ? → 0. Copyright © 2013 John Wiley & Sons, Ltd.     

Linearized quantum and relativistic Fokker–Planck–Landau equations

After a recent work on spectral properties and dispersion relations of the linearized classical Fokker–Planck–Landau operator [8], we establish in this paper analogous results for two more realistic collision operators: The first one is the Fokker–Planck–Landau collision operator obtained by relativistic calculations of binary interactions, and the second is a collision operator (of Fokker–Planck–Landau type) derived from the Boltzmann operator in which quantum effects have been taken into account. We apply Sobolev–Poincaré inequalities to establish the spectral gap of the linearized operators. Furthermore, the present study permits the precise knowledge of the behaviour of these linear Fokker–Planck–Landau operators including the transport part. Relations between the eigenvalues of these operators and the Fourier‐space variable in a neighbourhood of 0 are then investigated. This study is a first natural step when one looks for solutions near equilibrium and their hydrodynamic limit for the full non‐linear problem in all space in the spirit of several works [3, 6, 20, 2] on the non‐linear Boltzmann equation. Copyright © 2000 John Wiley & Sons, Ltd.     

The Characterization of the Derivatives for Linear Combinations of Post–Widder Operators inLp

The aim of the paper is to characterize the global rate of approximation of derivativesf^(l)through corresponding derivatives of linear combinations of Post–Widder operators in an appropriate weightedL_p-metric using a weighted Ditzian and Totik modulus of smoothness, and also to characterize derivatives of these operators in Besov spaces of Ditzian–Totik type.     

On M–multisplittings of singular M–matrices with application to Markov chains

Given a singular M–matrix of a linear system, convergent conditions under which iterative schemes based on M–multisplittings are studied. Two of those conditions, the index of the iteration matrix and its spectral radius are investigated and related to those of the M-matrix. Furthermore, a parallel multisplitting iteration scheme for solving singular linear systems is suggested which can be applied to practical problems such as Poisson and elasticity problems under certain boundary conditions, the Neumann problem, and in Markov chains. A discussion of that multisplitting scheme, based on Gauss–Seidel type splittings is given for computing the stationary distribution vector of Markov chains. In this case a computational viable algorithm can be constructed, since only the nonsingularity of one weighting matrix of the multisplitting is needed. © 1998 John Wiley & Sons, Ltd.