Explore the vast range of titles available.

We prove the McKay conjecture on characters of odd degree. A major step in the proof is the verification of the inductive McKay condition for groups of Lie type and primes ℓ such that a Sylow ℓ-subgroup or its maximal normal abelian subgroup is contained in a maximally split torus by means of a new equivariant version of Harish-Chandra induction. Specifics of characters of odd degree, namely, that most of them lie in the principal Harish-Chandra series, then allow us to deduce from it the McKay conjecture for the prime 2, hence for characters of odd degree.

A theorem of Leibman asserts that a polynomial orbit (g(n)Γ) n∈ℤ on a nilmanifold G/Γ is always equidistributed in a union of closed subnilmanifolds of G/Γ. In this paper we give a quantitative version of Leibman's result, describing the uniform distribution properties of a finite polynomial orbit (g(n)Γ) n∈[N] in a nilmanifold. More specifically we show that there is a factorisation g = εg′γ, where ε(n) is \"smooth,\" (γ(n)Γ) n∈ℤ is periodic and \"rational,\" and (g′(n)Γ) n∈P is uniformly distributed (up to a specified error δ) inside some subnilmanifold G′/Γ′ of G/Γ for all sufficiently dense arithmetic progressions P ⊆ [N]. Our bounds are uniform in N and are polynomial in the error tolerance δ. In a companion paper we shall use this theorem to establish the Möbius and Nilsequences conjecture from an earlier paper of ours.

We study algebraic dynamical systems (and, more generally, σ-varieties) $\\Phi :\\mathbb{A}^n_{\\mathbb{C}}\\rightarrow \\mathbb{A}^n_{\\mathbb{C}}$ given by coordinatewise univariate polynomials by refining an old theorem of Ritt on compositional identities amongst polynomials. More precisely, we find a nearly canonical way to write a polynomial as a composition of \"clusters\" from which one may easily read off possible compositional identities. Our main result is an explicit description of the (weakly) skew-invariant varieties, that is, for a fixed field automorphism σ : ℂ → ℂ those algebraic varieties $X\\subseteq \\mathbb{A}^n_\\mathbb{C}$ for which Π(X) ⊆ Xσ. As a special case, we show that if f(x) ∈ ℂ[x] is a polynomial of degree at least two that is not conjugate to a monomial, Chebyshev polynomial or a negative Chebyshev polynomial, and X\\subseteq \\mathbb{A}^2_\\mathbb{C} is an irreducible curve that is invariant under the action of (x,y) ↦ (f(x), f(y)) and projects dominantly in both directions, then X must be the graph of a polynomial that commutes with f under composition. As consequences, we deduce a variant of a conjecture of Zhang on the existence of rational points with Zariski dense forward orbits and a strong form of the dynamical Manin-Mumford conjecture for liftings of the Frobenius. We also show that in models of ACFA0, a disintegrated set defined by σ(x) = f(x) for a polynomial f has Morley rank one and is usually strongly minimal, that model theoretic algebraic closure is a locally finite closure operator on the nonalgebraic points of this set unless the skew-conjugacy class of f is defined over a fixed field of a power σ, and that nonorthogonality between two such sets is definable in families if the skew-conjugacy class of f is defined over a fixed field of a power of σ.

We discuss the application of virtual elements to linear elasticity problems, for both the compressible and the nearly incompressible case. Virtual elements are very close to mimetic finite differences (see, for linear elasticity, [L. Beirão da Veiga, M2AN Math. Model. Numer. Anal., 44 (2010), pp. 231-250]) and in particular to higher order mimetic finite differences. As such, they share the good features of being able to represent in an exact way certain physical properties (conservation, incompressibility, etc.) and of being applicable in very general geometries. The advantage of virtual elements is the ductility that easily allows high order accuracy and high order continuity.

We present a family of conforming finite elements for the Stokes problem on general triangular meshes in two dimensions. The lowest order case consists of enriched piecewise linear polynomials for the velocity and piecewise constant polynomials for the pressure. We show that the elements satisfy the inf-sup condition and converges with order kk for both the velocity and pressure. Moreover, the pressure space is exactly the divergence of the corresponding space for the velocity. Therefore the discretely divergence-free functions are divergence-free pointwise. We also show how the proposed elements are related to a class of C1C^1 elements through the use of a discrete de Rham complex.

In this paper, we consider the numerical solution of the time fractional diffusion equation. Essentially, the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term. In the former case, the first-order time derivative is replaced by a fractional derivative, making the problem global in time. We propose a spectral method in both temporal and spatial discretizations for this equation. The convergence of the method is proven by providing a priori error estimate. Numerical tests are carried out to confirm the theoretical results. Thanks to the spectral accuracy in both space and time of the proposed method, the storage requirement due to the \"global time dependence\" can be considerably relaxed, and therefore calculation of the long-time solution becomes possible.

In an earlier study (Zhang & Shu 2010b J. Comput. Phys. 229, 3091-3120 (doi:10.1016/j.jcp.2009.12.030)), genuinely high-order accurate finite volume and discontinuous Galerkin schemes satisfying a strict maximum principle for scalar conservation laws were developed. The main advantages of such schemes are their provable high-order accuracy and their easiness for generalization to multi-dimensions for arbitrarily high-order schemes on structured and unstructured meshes. The same idea can be used to construct high-order schemes preserving the positivity of certain physical quantities, such as density and pressure for compressible Euler equations, water height for shallow water equations and density for Vlasov-Boltzmann transport equations. These schemes have been applied in computational fluid dynamics, computational astronomy and astrophysics, plasma simulation, population models and traffic flow models. In this paper, we first review the main ideas of these maximum-principle-satisfying and positivity-preserving high-order schemes, then present a simpler implementation which will result in a significant reduction of computational cost especially for weighted essentially non-oscillatory finite-volume schemes.

A Kakeya set is a subset of Fn\\mathbb {F}^n, where F\\mathbb {F} is a finite field of qq elements, that contains a line in every direction. In this paper we show that the size of every Kakeya set is at least Cn⋅qnC_{n} \\cdot q^{n}, where CnC_{n} depends only on nn. This answers a question of Wolff.

We analyze the simplest and most standard adaptive finite element method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As is customary in practice, the AFEM marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property. We prove that the AFEM is a contraction, for the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops. This geometric decay is instrumental to derive the optimal cardinality of the AFEM. We show that the AFEM yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.

We show that multiple polynomial ergodic averages arising from nilpotent groups of measure preserving transformations of a probability space always converge in the L 2 norm.