Let p be a prime and q = p s and ζk a fixed primitive kth root of unity in some extension of Q. Let χ be a multiplicative character over Fq of order k and J (χ, χ) the associated Jacobi sum. We give examples of χ which satisfy J (χ, χ) ∈p [s/2]Z[ζk]. Moreover, for s = 3, we prove that there is only a finite number of k such that J(χ, χ) is an element of pZ[ζk] except for the case where k is divisible by nine and p ≡1 ± k/3 (mod k).
Let (R, m) be a commutative Noetherian local ring with the maximal ideal m and A an Artinian R-module with N-dimA = d. Let n = (n1,n2,...,nd) be a d-tuple of positive integers. For each system of parameters x = (x1,...,xd) of A, we consider the length ℓ R(0 :A (x 1 n1 ,...,x d nd )R) as a function d-variables on n 1 ,...,n d . Then a necessary and sufficient condition for this length to be polynomial (in n1,...,nd) is given. Moreover, we shall introduce a system of parameters for which the length ℓ R(0 : A (x 1 n 1,...,x d nd )R) can be computed by a nice formula.
This article is devoted to dealing with existence of a solution to a quasilinear elliptic problem with the Hardy potential term up−1 / |x|p and critical growth in the gradient |∇u|p as an absorption term. By considering the interaction between the two terms in the equation, we prove that there exists a positive solution to the problem. Moreover, the existence result for a wider class of weight functions can also be obtained.
In this paper we are concerned with the optimal convergence rates of the global strong solution to the stationary solutions for the compressible Navier-Stokes equations with a potential external force ∇Φ in the whole space Rn for n ≥3. It is proved that the perturbation and its first-order derivatives decay in L 2 norm in O(t −n/4 ) and O(t −n/4−1/2), respectively, which are of the same order as those of the n-dimensional heat kernel, if the initial perturbation is small in H s0(R n ) ∩L 1 (R n) with s0 =[n/2]+ 1 and the potential force Φ is small in some Sobolev space. The results also hold for n ≥ 2 when Φ = 0. When Φ = 0, we also obtain the decay rates of higher-order derivatives of perturbations.
We introduce a method which can be used to establish sharp maximal estimates for functions of bounded lower oscillation (BLO). The technique allows us to deduce such estimates from the existence of certain special functions, and can be regarded as a version of Bellman function method, which has gained considerable interest in the recent literature. As an application, we establish a sharp exponential bound, which can be regarded as a version of integral John-Nirenberg inequality for BLO functions.
Let G be the multiplicative group generated by the gamma functions Γ(ax + 1) (a = 1, 2,...), and H be the subgroup of all elements of G that converge to non-zero constants as x →∞. The quotient group G/H is the group of equivalence classes of G, where ƒand g are equivalent ⇔ ƒ∼Cg (x →∞) for some C ≠ 0. We show that G/H ~ Q+. Similar consideration is possible for the case that the gamma functions Γ(ax + 1) with a ∈R+ are concerned, and we show that G/H ~ Z × R × R. Also, several concrete examples of the elements of H are constructed, e.g. it holds that / → (n →∞), where denotes a multinomial coefficient.
In this paper we define and completely classify three types of constant angle surfaces in a three-dimensional two-parameter solvable Lie group G(μ1, μ2) corresponding to each of the three left-invariant vector fields.
In this paper we present the notion of de Rham cohomology with compact support for diffeological spaces. Moreover, we shall discuss the existence of three long exact sequences. As a concrete example, we show that long exact sequences exist for the de Rham cohomology of diffeological subcartesian spaces.
We recently established a Toponogov-type triangle comparison theorem for a certain class of Finsler manifolds whose radial flag curvatures are bounded below by that of a von Mangoldt surface of revolution. In this article, as its applications, we prove the finiteness of topological type and a diffeomorphism theorem to Euclidean spaces.
Let GS be the Galois group of the maximal pro-p-extension QS of Q unramified outside a finite set S of places of Q not containing the prime p > 2. In this work, we develop a method to produce some examples of mild (and thus FAB) pro-p-groups GS for which some relations are of degree three (according to the Zassenhaus filtration). The key computation are done in some Heisenberg extensions of Q of degree p3. With the help of GP-Pari we produce some examples for p = 3.
It is known that a finite-dimensional reductive Lie algebra has a non-degenerate symmetric invariant bilinear form. In this paper, for a given reductive Lie algebra and its finite-dimensional completely reducible representation, we will construct a graded Lie algebra by using a non-degenerate symmetric invariant bilinear form on the reductive Lie algebra. This graded Lie algebra also has a non-degenerate symmetric invariant bilinear form and, moreover, the reductive Lie algebra, its representation and the bilinear form which are used to construct the graded Lie algebra can be embedded into it.
In this paper, for any multivariable function with a periodicity and a certain distribution relation, we define a multiple Dedekind-type sum. Then by a combinatorialgeometric method, we study generalizations of Knopp's formula for the classical Dedekind sums. The main theorem contains many of the preceding results concerning generalizations of Knopp's formula.
Following the analogies between three-dimensional topology and number theory, we study an idèlic form of class field theory for 3-manifolds. For a certain set K of knots in a 3-manifold M, we first present a local theory for each knot in K, which is analogous to local class field theory, and then, getting together over all knots in K, we give an analogue of idèlic global class field theory for an integral homology sphere M.