We study a second-order ordinary differential equation coming from the Kepler problem on ${{\mathbb{S}}^{2}}$. The forcing term under consideration is a piecewise constant with singular nonlinearity that changes sign. We establish necessary and sufficient conditions to the existence and multiplicity of $T$-periodic solutions.

Consider the n-th order (n ≥ 1 ) neutral differential equation where σ1 < σ 2 < ∞ and μ and η are increasing real-valued functions on [Ƭ1, Ƭ2] and [σ1, σ2] respectively. The function μ is assumed to be not constant on [Ƭ1, Ƭ2] and [Ƭ1, Ƭ2] for every Ƭ ∈ (Ƭ1, Ƭ2) similarly, for each σ ∈ (σ1, σ2), it is supposed that r\ is not constant on [σ1 , σ] and [σ, σ2]. Under some mild restrictions on Ƭ1,- and σ1, (ι = 1,2), it is proved that all solutions of (E) are oscillatory if and only if the characteristic equation of (E) has no real roots.

Suppose that

(1.1)

and define the bounded self-adjoint operators H and J on the Hilbert space L2(0, 1) by

(1.2)

the integral being a Cauchy principal value

It is seen that

(1.3)

or, equivalently,

(1.4)

Since (Cƒ, ƒ) = π–1|(ƒ, ϕ)|2 ≧ 0, A is semi-normal. (For a discussion of such operators, see [4].)

Let R be a free associative ring with identity freely generated by r1, r2,. .. , rk. In analogy to group theory the lower central series for R is defined inductively By

γo = R and γn = [γn-1, R],

where γn is the ideal generated by the indicated ring commutators. Using P. Hall's collection process [2; 1, Chapter 11] γn/γn+1 will be shown to be free as a Z-module and as an R/R''-module for each non-negative integer n. In each case a basis will be exhibited.

Definition 1. Commutators of order zero are the free generators of R. A commutator, c, of order n (denoted by o(c) = n) is of the form [x, y], where x and y are commutators and o(x) + o(y) = n — 1.

In this paper we consider matrix representations of compact groups over the field of the complex numbers. We shall deal mainly with finite groups.

The Kronecker product of two irreducible representations σ1 and σ2 of a group is in general a reducible representation of . The explicit reduction of such a product to irreducible representations σ3 can be performed by means of a unitary matrix, the elements of which are called Wigner coefficients or Clebsch-Gordan coefficients [1; 25; 27].

In this paper we consider Fourier multipliers on local Hardy spaces ${{\mathbf{h}}^{\mathbf{p}}}(0<p\le 1)$ for Chébli-Trimèche hypergroups. The molecular characterization is investigated which allows us to prove a version of Hörmander’s multiplier theorem.

In [3] Remeslennikov proves that a finitely generated metabelian group G has a faithful representation of finite degree over some field F of characteristic zero (respectively, p > 0) if its derived group G’ is torsion-free (respectively, of exponent p). By the Lie-Kolchin-Mal'cev theorem any metabelian subgroup of GL(n, F) has a subgroup of finite index whose derived group is torsion-free if char F = 0 and is a p-group of finite exponent if char F = p > 0. Moreover every finite extension of a group with a faithful representation (of finite degree) has a faithful representation over the same field. Thus Remeslennikov's results have a gap which we propose here to fill.

Let ƒ(x1, … , xn) = Σaijxixj be a positive definite quadratic form of determinant D = |aij|, and let M be the minimum of f for integral x1, … , xn not all zero. The form ƒ is said to be extreme if the ratio Mn/D does not increase when the coefficients aij of f suffer any sufficiently small variation.

We study the asymptotic behavior of the Laplacian on functions when the underlying Riemannian metric is collapsed to a Carnot-Carathéodory metric. We obtain a uniform short time asymptotics for the trace of the heat kernel in the case when the limit Carnot-Carathéodory metric is almost Heisenberg, the limit of which is the result of Beal-Greiner-Stanton, and Stanton-Tartakoff.

All functions will be complex, periodic, integrable (on [0, 2π]) functions of a real variable x. Moreover, we shall require that every function have mean zero on [0, 2π], so that in particular non-zero constants are excluded.

1. Plessner's characterization of absolutely continuous functions. An old theorem of Plessner (4), generalized to arbitrary compact groups by Bochner (1), can be taken as our starting point. Consider the functions f of bounded variation on [0, 2π]. These f form a Banach space F when each f is normed by its total variation on [0, 2π]. And translations define a natural one-parameter group of isometries on F.

Consider the following result.

PROPOSITION. Let D be a finite-dimensional central division algebra over a field F, and let Dn be the algebra (over F) of all n × n matrices with entries in D. Let A and B be in Dn, and suppose that BX = XB for every X in Dn such that XA = AX. Then B is a polynomial in A with coefficients in F.

The case D = F is a well-known classical result. Recently, the particular case where D is the algebra of real quaternions was established by Cullen and Carlson (2). In this note, the general proposition is proved by reduction to the classical case by way of tensor products.

Let Fn be the collection of n-by-n matrices over a field F. For Y in Fn let ΔY be the mapping on Fn given by XΔY = XY — YX. In this paper we study the following

Proposition. Let A and B be in Fn and let m be a positive integer. If BΔxm = 0 whenever XΔAm = 0, then B is a polynomial in A with coefficients in F.

Denote by {t(x)} some linear transform of the sequence

of the form

where x attains continuous or only integer values. The problem of estimating |t(x) — sm| as x and m tend to ∞ with some connection between them was considered first by H.

In this note the Hilbert spaces under consideration are complex, and the operators referred to are bounded, linear operators. If is a Hilbert space, then the algebra of all operators on is denoted by .

It is known (1) that if is any Hilbert space, then the class of commutators on , i.e., the class of all operators that can be written in the form PQ — QP for some , can be exactly described. A similar problem is that of characterizing all operators on that can be written in the form for some .

A purely algebraic solution for the problem of the extreme definite ternary quadratic form has been given by Mordell (2). The equivalent geometric problem of the critical lattice of a sphere was solved by Mrs. Ollerenshaw (3) under a preliminary assumption that was said to be justified by “simple considerations” (p. 297).

For f ∊ L−1(0, T), we define the distribution function

where T is a fixed positive number and |·| denotes Lebesgue measure. Let Φ:[0, T] → [0, m] be a nonincreasing, right continuous function. In an earlier paper [3], we discussed the equation

(0.1)

when the coefficient q was allowed to vary in the class

We were in particular interested in finding the supremum and infimum of y(T) when q was in or in the convex hull Ω(Φ) of (see below).

Let U be a finite dimensional vector space over an infinite field F. Let U(r) denote the r–th symmetric product space over U. Let T: U(r) → U(s) be a linear transformation which sends nonzero decomposable elements to nonzero decomposable elements. Let dim U ≥ s + 1. Then we obtain the structure of T for the following cases: (I) F is algebraically closed, (II) F is the real field, and (III) T is injective.

An f-ring R with zero right annihilator is called a qf-ring if its Utumi maximal left quotient ring Q = Q(R) can be made into and f-ring extension of R. F. W. Anderson [2, Theorem 3.1] has characterized unital qf-rings with the following conditions: For each q ∈ Q and for each pair d1, d2 ∈ R+ such that diq ∈ R

(i) (d1q)+ Λ (d2q)- = 0, and

(ii) d1 Λ d2 = 0 implies (d1q)+ Λ d2 = 0.

We remark that this characterization holds even when R does not have an identity element.

Let $K$ be a complex quadratic extension of $\mathbb{Q}$ and let ${{\mathbb{A}}_{K}}$ denote the adeles of $K$. We find special values at all of the critical points of twisted tensor $L$-functions attached to cohomological cuspforms on $G{{L}_{2}}\left( {{\mathbb{A}}_{K}} \right)$ and establish Galois equivariance of the values. To investigate the values, we determine the archimedean factors of a class of integral representations of these $L$-functions, thus proving a conjecture due to Ghate. We also investigate analytic properties of these $L$-functions, such as their functional equations.