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

We show that if S is the maximal idealspace of certain ultrapowers of C_0(L) spaces then: C_0(S, [Copf]) allows polar decompositions whileC_0(S, ℝ) does not, which answers a question of Greim and Rajalopagan [4]. Also, S isalmost homogeneous but not transitive, which answers a question of Wood[9].

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 thefixed point subalgebra F^G is not finitely generated. Some similarresults are proved for relatively free Lie algebras.

Let R be anassociative ring with 1, and let I be a nilpotent two-sidedideal of R. Assume further that there exists z \in Z(R) suchthat z, z^2-1 \in R^*. Let m \in ℕ with m \geq3. 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\,).

Weinvestigate the function R(T,σ), which denotes the error term in theasymptotic formula for \int_0^T|\log\zeta(σ + it)|^2dt. It is shown thatR(T,σ) is uniformly bounded for σ \ge 1 and almostperiodic in the sense of Bohr for fixed σ \ge 1; hence R(T,σ)= Ω(1) when T \to \infty. In case {1 \over2}<σ<1 is fixed we can obtain the bound R(T,σ) \ll_ϵT\,^{(9-2σ)/8+ϵ}.

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

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

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.

In this paper, we study a statistical property of classes of real-valued functions that we call approximation from interpolated examples. We derive a characterization of function classes that have this property, in terms of their ‘fat-shattering function’, a notion that has proved useful in computational learning theory. The property is central to a problem of learning real-valued functions from random examples in which we require satisfactory performance from every algorithm that returns a function which approximately interpolates the training examples.

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.

A module M iscalled a CS-module or an extending module if every submodule is essentialin a direct summand of M. A ring R is called a rightCS-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 ofright 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 andonly if every R-module is CS. In particular, if R is a QF-ringand J(R)^2 = 0 (hence R is serial), then everyR-module is CS. However we exhibit a finite, serial, strongly bounded QF groupalgebra R with J(R)^3 = 0 for which there is aprincipal right ideal which is a right essential extension of a CS-module and essential inR_R but not CS itself.

Answering a question of Wilf, we show that, if n is sufficiently large, then one cannot cover an n × p(n) rectangle using each of the p(n) distinct Ferrers shapes of size n exactly once. Moreover, the maximum number of pairwise distinct, non-overlapping Ferrers shapes that can be packed in such a rectangle is only Θ(p(n)/log n):

A cladogram is a tree with labelled leaves and unlabelled degree-3 branchpoints. A certain Markov chain on the set of n-leaf cladograms consists of removing a random leaf (and its incident edge) and re-attaching it to a random edge. We show that the mixing time (time to approach the uniform stationary distribution) for this chain is at least O(n2) and at most O(n3).

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

For any fixed l < k we present families of asymptotically good packings and coverings of the l-subsets of an n-element set by k-subsets, and an algorithm that, given a natural number i, finds the ith k-subset of the family in time and space polynomial in log n.

We prove the Löwner-Heinz inequality, via the Cordesinequality, 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 \leb^p if a \le b, a^p < b^p ifa<b, provided that the involution of A is continuous, and the latterthat s(a^pb^p) \le s(ab)^p, where s(x)=r(x^*x)^{1/2} andr(x) is the spectral radius of an elementx.

The twisted odd graphs are obtained from the well-known odd graphs through an involutive automorphism. As expected, the twisted odd graphs share some of the interesting properties of the odd graphs but, in general, they seem to have a more involved structure. Here we study some of their basic properties, such as their automorphism group, diameter, and spectrum. They turn out to be examples of the so-called boundary graphs, which are graphs satisfying an extremal property that arises from a bound for the diameter of a graph in terms of its distinct eigenvalues.