We consider the following nonlinear elliptic equations

\begin{gather*} \begin{cases} \Delta u+u_{+}^{N/(N-2)}=0\amp\quad\text{in }\sOm, \\ u=\mu\amp\quad\text{on }\partial\sOm\quad(\mu\text{ is an unknown constant}), \\ \dsty\int_{\partial\sOm}\biggl(-\dsty\frac{\partial u}{\partial n}\biggr)=M, \end{cases} \end{gather*}

where $u_{+}=\max(u,0)$, $M$ is a prescribed constant, and $\sOm$ is a bounded and smooth domain in $R^N$, $N\geq3$. It is known that for $M=M_{*}^{(N)}$, $\sOm=B_R(0)$, the above problem has a continuum of solutions. The case when $M>M_{*}^{(N)}$ is referred to as supercritical in the literature. We show that for $M$ near $KM_{*}^{(N)}$, $K>1$, there exist solutions with multiple condensations in $\sOm$. These concentration points are non-degenerate critical points of a function related to the Green's function.

AMS 2000 Mathematics subject classification: Primary 35B40; 35B45. Secondary 35J40

The best uniform approximation of a function f on [-1,1] by real algebraic polynomials satisfies

if and only if ƒ is the restriction to [- 1,1] of an entire function (Bernstein [2], p. 113, see also [12], pp. 83–85). For such functions ƒ the rate of best approximation has been characterized by Varga [24], Reddy [24], Shah [21], and Kapoor and Nautiyal [10] in terms of order and type of ƒ , lower order and type, and in terms of more general concepts of order. On the other hand, order and type of ƒ are connected with the Taylor coefficients, i.e. with the rate of growth of the sequence (see [23], p. 41 or [3], pp. 11/12; cf. also [19], [20], [6], [7], [8]) and this has been extended to iterated orders by Schonhage [17], Sato [16], Reddy [14], Juneja, Kapoor, and Bajpai [9] (also [22], [13]), and to generalized orders by Seremeta [18], Bajpai, Gautam, and Bajpai [1] as well as Kapoor and Nautiyal [10]. Combining the two kinds of characterizations (as done, e.g., by Reddy [15], p. 105) approximation theorems in terms of the sequence are obtained. But in such results the rate ofbest approximation is always described by a limit relation, e.g. of the form , and this causes a considerable loss of precision, as will be discussed in more detail in Section 3 (in this respect cf. also the remark by Bernstein [2], pp. 114/115).

If f(z) is analytic in a suitable domain, it is shown how the integral of f(α1x1+…+αnxn) over the interior of a simplex may be reduced to the evaluation of a contour integral, in fact to an exercise in partial fractions.

The contour integral is expressed in two ways, according as the simplex is given in terms of its vertices or faces.

Several structure theorems are proved for groups G having the following property. There is a prime p and a collection of subgroups of G such that the elements of G which lie in the complement of every subgroup of the collection all have order p.

Let $q$ be a positive integer, let $\mathcal{I}=\mathcal{I}(q)$ and $\mathcal{J}=\mathcal{J}(q)$ be subintervals of integers in $[1,q]$ and let $\mathcal{M}$ be the set of elements of $\mathcal{I}$ that are invertible modulo $q$ and whose inverses lie in $\mathcal{J}$. We show that when $q$ approaches infinity through a sequence of values such that $\varphi(q)/q\rightarrow0$, the $r$-spacing distribution between consecutive elements of $\mathcal{M}$ becomes exponential.

AMS 2000 Mathematics subject classification: Primary 11K06; 11B05; 11N69

Algebraic limit cycles in quadratic polynomial differential systems started to be studied in 1958, and a few years later the following conjecture appeared: quadratic polynomial differential systems have at most one algebraic limit cycle. We prove that a quadratic polynomial differential system having an invariant algebraic curve with at most one pair of diametrically opposite singular points at infinity has at most one algebraic limit cycle. Our result provides a partial positive answer to this conjecture.

This paper addresses linear hyperbolic partial differential equations and pseudodifferential equations with strongly singular coefficients and data, modelled as members of algebras of generalized functions. We employ the recently developed theory of generalized Fourier integral operators to construct parametrices for the solutions and to describe propagation of singularities in this setting. As required tools, the construction of generalized solutions to eikonal and transport equations is given and results on the microlocal regularity of the kernels of generalized Fourier integral operators are obtained.

We show that the global dimension, dgA, of every commutative Banach algebra A whose radical is a weighted convolution algebra is strictly greater than one. As an application, we see that in this case H2(A, X) ≠ 0 for some Banach A-bimodule X and thus there exists an unsplittable singular extension of the algebra A.

This paper studies topological and metric rigidity theorems for hypersurfaces in a Euclidean sphere. We first show that an $n({\geq}\,2)$-dimensional complete connected oriented closed hypersurface with non-vanishing Gauss–Kronecker curvature immersed in a Euclidean open hemisphere is diffeomorphic to a Euclidean $n$-sphere. We also show that an $n({\geq}\,2)$-dimensional complete connected orientable hypersurface immersed in a unit sphere $S^{n+1}$ whose Gauss image is contained in a closed geodesic ball of radius less than $\pi/2$ in $S^{n+1}$ is diffeomorphic to a sphere. Finally, we prove that an $n({\geq}\,2)$-dimensional connected closed orientable hypersurface in $S^{n+1}$ with constant scalar curvature greater than $n(n-1)$ and Gauss image contained in an open hemisphere is totally umbilic.

Let A be a unital C*-algebra with the canonical (H) C*-bundle over the maximal ideal space of the centre of A, and let E(A) be the set of all elementary operators on A. We consider derivations on A which lie in the completely bounded norm closure of E(A), and show that such derivations are necessarily inner in the case when each fibre of is a prime C*-algebra. We also consider separable C*-algebras A for which is an (F) bundle. For these C*-algebras we show that the following conditions are equivalent: E(A) is closed in the operator norm; A as a Banach module over its centre is topologically finitely generated; fibres of have uniformly finite dimensions, and each restriction bundle of over a set where its fibres are of constant dimension is of finite type as a vector bundle.

In a paper (1) by Harding there is a tacit invitation to seek the connection between the following two problems:

(i) Find the number, ηk(N), of regions into which a k-dimensional space is partitioned by a set of N (k- l)-dimensional hyperplanes.

(ii) Find the number, vk(N), of distinct partitions of a given set of N points in a k-dimensional space E that can be induced by (k- 1)-dimensional hyperplanes.

We classify all the effective anticanonical divisors on weak del Pezzo surfaces. Through this classification we obtain the smallest number among the log canonical thresholds of effective anticanonical divisors on a given Gorenstein canonical del Pezzo surface.

Let G be any finite group, and p any prime number. (All groups to be considered here are finite, and we assume this without further comment.) We denote by Kp(G) the unique smallest normal subgroup of G for which the quotient G/Kp(G) is a p-group. G/Kp(G) is called the p-residual of G. W. Gaschütz (2, Satz 7) has proved the following

Theorem. Set K = Kp(G). If the Sylow p-subgroups of K are abelian, then G splits over K.

We show that the dynamic asymptotic dimension of an action of an infinite virtually cyclic group on a compact Hausdorff space is always one if the action has the marker property. This in particular covers a well-known result of Guentner, Willett, and Yu for minimal free actions of infinite cyclic groups. As a direct consequence, we substantially extend a famous result by Toms and Winter on the nuclear dimension of $C^{*}$-algebras arising from minimal free $\mathbb {Z}$-actions. Moreover, we also prove the marker property for all free actions of countable groups on finite-dimensional compact Hausdorff spaces, generalizing a result of Szabó in the metrisable setting.

Let K be a field of characteristic p>0, G a finite p-solvable group, P a p-Sylow subgroup of G of order pa, KG the group algebra of G over K, and J(KG) the Jacobson radical of KG. In the present paper we study the nilpotency index t(G) of J(KG), which is the least positive integer t with J(KG)t= 0. Since J(EG) = E⊗KJ(KG) for any extension field E of K (cf. [7, Proposition 12.11]), we may assume that K is algebraically closed.

A ring R is said to satisfy the right Ore condition with respect to a subset C of R if, given a ∈ R and e ∈ C, there exist b ∈ R and D ∈ C such that ad = cb. It is well known that R has a classical right quotient ring if and only if R satisfies the right Ore condition with respect to C when C is the set of regular elements of R (a regular elemept of R being an element of R which is not a zero-divisor). It is also well known that not every ring has a classical right quotient ring. If we make the non-trivial assumption that R has a classical right quotient ring, it is natural to ask whether this property also holds in certain rings related to R such as the ring Mn(R) of all n by n matrices over R. Some answers to this question are known when extra assumptions are made. For example, it was shown by L. W. Small in (5) that if R has a classical right quotient ring Q which is right Artinian then Mn(Q) is the right quotient ring of Mn(R) and eQe is the right quotient ring of eRe where e is an idempotent element of R. Also it was shown by C. R. Hajarnavis in (3) that if R is a Noetherian ring all of whose ideals satisfy the Artin-Rees property then R has a quotient ring Q and Mn(Q) is the quotient ring of Mn(R).

Let a meromorphic differential equation

be given, where r is an integer, and the series converges for |z| sufficiently large. Then it is well known that (0.1) is formally satisfied by an expression

where F( z) is a formal power series in z–1 times an integer power of z, and F( z) has an inverse of the same kind, L is a constant matrix, and

is a diagonal matrix of polynomials qj( z) in a root of z, 1≦ j≦ n. If, for example, all the polynomials in Q( z) are equal, then F( z) can be seen to be a convergent series (see Section 1), whereas if not, then generally the coefficients in F( z) grow so rapidly that F( z) diverges for every (finite) z.

A one parameter family of algorithms is studied, which contains both the arithmetic-geometric mean of Gauss and its generalization by Borchardt, recently studied by J. and P. Borwein. We prove that the presence of an asymptotic formula for such an algorithm is, in view of the Poisson summation formula, equivalent to the vanishing of certain integrals. In the case of Gauss and Borchardt the latter involve theta functions. Finally, we investigate the question of convergence of the algorithm for complex values, thereby generalizing the corresponding result of Gauss.

For an inverse semigroup S we will consider the lattice of inverse subsemigroups of S, denoted L(S). A major problem in algebra has been that of finding to what extent an algebra is determined by its lattice of subalgebras. (See, for example, the survey article [9]). By a lattice isomorphism (L-isomorphism, structural isomorphism, or projectivity) of an inverse semigroup S onto another T we shall mean an isomorphism Φ of L(S) onto L(T). A mapping φ from S to T is said to induce Φ if AΦ = Aφ for all A in L(S). We say that S is strongly determined by L(S) if every lattice isomorphism of S onto T is induced by an isomorphism of S onto T.

An irreducible curve in S4, projective 4-space, may arise as the complete intersection of three given irreducible threefolds. At a simple point P on such a curve there is an osculating solid, and we would like to have its equation. This solid, necessarily containing the tangent line to the curve at P, belongs to the net spanned by the tangent solids at P to the threefolds. We seek the appropriate linear combination of the known equations for these tangent solids.