We investigate the function R(T,σ), which denotes the error term in the asymptotic formula for \int_0^T|\log\zeta(σ + it)|^2dt. It is shown thatR(T,σ) is uniformly bounded for σ \ge 1 and almost periodic in the sense of Bohr for fixed σ \ge 1; hence R(T,σ) = Ω(1) when T \to \infty. In case {1 \over 2}<σ<1 is fixed we can obtain the bound R(T,σ) \ll_ϵ T\,^{(9-2σ)/8+ϵ}.

We show that if F is a free Lie algebra of rank at least 2 and if G is a non-trivial finite group of automorphisms of F then the fixed point subalgebra F^G is not finitely generated. Some similar results are proved for relatively free Lie algebras.

Considering odd-dimensional complex projective space as a complex contact manifold, one may ask which of the Calabi (Veronese) imbeddings can be positioned by a holomorphic congruence as integral submanifolds of the complex contact structure. It is first shown that when the first normal space is the whole normal space, this is impossible. It is also shown to be impossibile for a Calabi surface (complex dimension 2) in complex projective space of dimension 9 where one has both a first and second normal space. However when the complex dimension of the submanifold is odd and the whole normal space consists of the first and second normal spaces, then there is a holomorphic congruence positioning the Calabi imbedding as an integral submanifold of the complex contact structure.

We investigate the boundedness of solutions of a second order nonlinear differential system.

Let R be an associative ring with 1, and let I be a nilpotent two-sided ideal of R. Assume further that there exists z \in Z(R) such that z, z^2-1 \in R^*. Let m \in ℕ with m \geq 3. In this paper we describe all liftings of the elementary group{\tf="times-b"E}_m(R/I\,) to the general linear group{\tf="times-b"GL}_m(R) , i.e. all splittings of the natural projection{\tf="times-b"E}_m(R) + {\tf="times-b"M}_m(I\,) \rightarrow {\tf="times-b"E}_m(R/I\,) .

The problem considered is the determination of “lower bounds” of matrix operators on the spaces\ell_p(w) or d(w,p). Under fairly general conditions, the solution is the same for both spaces and is given by the infimum of a certain sequence. Specific cases are considered, with the weighting sequence defined by w_n = 1/n^\alpha . The exact solution is found for the Hilbert operator. For the averaging operator, two different upper bounds are given, and for certain values of p and \alpha it is shown that the smaller of these two bounds is the exact solution.

It is shewn that, if N(P) be the number of solutions of theindeterminate equation

ax^3+by^3+cz^3+dw^3 =0 \qquad (a,b,c,d \neq0)

for which \vert x \vert,\vert y\vert, \vert z\vert, \vert w\vert \leq P, then

N(P) = KP^2 + o(P^2),

where, to within a term O(P),KP^2 is the contribution to N(P) corresponding to therational lines in the projective surface defined by the equation. This proves a conjecture made byHeath-Brown, who has studied N(P) under the assumption of the Riemann Hypothesisfor certain Hasse-Weil L-functions. The remainder term o(P^2)in the formula represents O(P^{ {4 \over 3}+ϵ}),O(P^ {{5 \over 3}+ϵ}),or O(P^{2}/^3 \sqrt{ \log P}) according as the surface contains three, one, or norational lines.

Letu'=f\,(t,u) , u(0)=u_0 be an initial value problem with quasimonotone increasing right-hand side. We prove that if u,v are solutions such that u(t_0)\ll v(t_0) then there is a solution w withu(t_0)<w(t_0)<v(t_0) .

We prove the Löwner-Heinz inequality, via the Cordes inequality, for elements a,b>0 of a unital hermitian Banach *-algebra A. Letting p be a real number in the interval (0,1], the former asserts that a^p \le b^p if a \le b, a^p < b^p ifa<b , provided that the involution of A is continuous, and the latter that s(a^pb^p) \le s(ab)^p, where s(x)=r(x^*x)^{1/2} andr(x) is the spectral radius of an element x.

Elements\alpha\in A\otimes E of the tensor product of a Banach algebra A and a Banach space E induce systems \{\psi(\alpha):\psi\in E^*\} of elements of A indexed by the dual space E^*, whose joint spectrum belongs to the second dual E^{**}. In this note we investigate when the spectrum actually lies in E\subseteq E^{**}, and extend the spectral mapping theorem P\sigma_A(\alpha)=\sigma_AP(\alpha) to polynomial mappings P:E\to F between Banach spaces. When the algebra A is commutative and the Banach space E=B is another algebra we also reach a sort of vector-valued Gelfand theory.

In this paper we give a new bound for the solutionsx of the title equation, provided that k \ge 8. This bound is polynomial in d. Moreover, under the same condition, a similar bound for the number of solutions in (x, k, y, l) is given.

A module M is called a CS-module or an extending module if every submodule is essential in a direct summand of M. A ring R is called a right CS-ring or a right extending ring if R_R is a CS-module. For several types of right CS-rings it is known that either all right ideals or some large class of right ideals inherit the CS property. For example, by a result of Dung-Smith or Vanaja-Purav, a ringR is (right and left) Artinian, serial, and J(R)^2 = 0 if and only if every R-module is CS. In particular, if R is a QF-ring and J(R)^2 = 0 (hence R is serial), then everyR -module is CS. However we exhibit a finite, serial, strongly bounded QF group algebra R with J(R)^3 = 0 for which there is a principal right ideal which is a right essential extension of a CS-module and essential inR_R but not CS itself.

Using classification theorems ofsimple groups, we give a proof of a conjecture on factorized finite groups which is an extension of awell known theorem due to P. Hall.

We show that if S is the maximal ideal space of certain ultrapowers of C_0(L) spaces then: C_0(S, [Copf ]) allows polar decompositions while C_0(S, ℝ) does not, which answers a question of Greim and Rajalopagan [4]. Also, S is almost homogeneous but not transitive, which answers a question of Wood [9].

For a C1 function f:ℝ^n →ℝ\;(n \ge 2), we consider the least numberk of distinct critical points that f must possess when restricted to the sphere S=\{x\in ℝ^n: \Vert x\Vert =1\}. Clearly k \ge 2 (for f attains its absolute minimum and maximum onS ), and a result of Lusternik and Schnirelmann establishes thatk=n if f is even. Here we prove that k=n if, for a given orthonormal system (e_i), \max\limits_{S \cap V_i}\,f<\min\limits_{S \cap V_i^\bot}\,f, for all i=1, …n-1, where V_i is the subspace spanned by e_1, …, e_i andV_i^\bot its orthogonal complement. It is shown that this criterion is satisfied by suitably restricted perturbations of quadratic forms having n distinct eigenvalues.

In its original formulation Lang's theorem referred to a semilinear map on an n-dimensional vector space over the algebraic closure of GF(p): it fixes the vectors of a copy ofV(n, p^h) . In other words, every semilinear map defined over a finite field is equivalent by change of coordinates to a map induced by a field automorphism. We provide an elementary proof of the theorem independent of the theory of algebraic groups and, as a by-product of our investigation, obtain a convenient normal form for semilinear maps. We apply our theorem to classical groups and to projective geometry. In the latter application we uncover three simple yet surprising results.

We prove that if n \geq 2 then A_nd cannot be a maximal left ideal of then -th Weyl algebra A_n over ℂ for almost every operator d of degree 2 of A_n.

An identity property defined for a pair of 2-complexes (Y,X) first arose in 1993 within a strategy for constructing a counterexample of infinite type to Whitehead's Asphericity Conjecture. In this note we make use of the theory of pictures to characterize a more general right N-identity property, where N < \pi _1Y. We also define combinatorial asphericity (CA) for the pair (Y,X) and determine a test for (CA) in the case that Y is obtained from X by the addition of a single 2-cell. This test can be used to determine an explicit generating set for \pi _2Y.