We compute the global log canonical thresholds of quasi-smooth well-formed complete intersection log del Pezzo surfaces of amplitude 1 in weighted projective spaces. As a corollary we show the existence of orbifold Kähler—Einstein metrics on many of them.

Let A be a quasi-accretive operator defined in a uniformly smooth Banach space. We present a necessary and sufficient condition for the strong convergence of the semigroups generated by – A and of the steepest descent methods to a zero of A.

Generalized Chow forms were introduced by Cayley for the case of 3-space; their zero set on the Grassmannian G(1, 3) is either the set Z of lines touching a given space curve (the case of an ‘honest’ Cayley form), or the set of lines tangent to a surface. Cayley gave some equations for F to be a generalized Cayley form, which should hold modulo the ideal generated by F and by the quadratic equation Q for G(1, 3). Our main result is that F is a Cayley form if and only if Z = G(1, 3) ∩ {F = 0} is equal to its dual variety. We also show that the variety of generalized Cayley forms is defined by quadratic equations, since there is a unique representative F0 + QF1 of F, with F0, F1 harmonic, such that the harmonic projection of the Cayley equation is identically 0. We also give new equations for honest Cayley forms, but show, with some calculations, that the variety of honest Cayley forms does not seem to be defined by quadratic and cubic equations.

We prove sufficient and necessary conditions for compactness of the Sobolev embeddings of Besov and Triebel–Lizorkin spaces defined on bounded and unbounded uniformly E-porous domains. The asymptotic behaviour of the corresponding entropy numbers is calculated. Some applications to the spectral properties of elliptic operators are described.

Let p ≥ 3 be a prime. A generalized multi-edge spinal group

$$G = \langle \{ a\} \cup \{ b_i^{(j)} {\rm \mid }1 \le j \le p,\, 1 \le i \le r_j\} \rangle \le {\rm Aut}(T)$$

$b^{(j)}_{1}, \ldots, b^{(j)}_{r_{j}}$

In the first part of the paper, we use states on $C^{*}$-algebras in order to establish some equivalent statements to equality in the triangle inequality, as well as to the parallelogram identity for elements of a pre-Hilbert $C^{*}$-module. We also characterize the equality case in the triangle inequality for adjointable operators on a Hilbert $C^{*}$-module. Then we give certain necessary and sufficient conditions to the Pythagoras identity for two vectors in a pre-Hilbert $C^{*}$-module under the assumption that their inner product has a negative real part. We introduce the concept of Pythagoras orthogonality and discuss its properties. We describe this notion for Hilbert space operators in terms of the parallelogram law and some limit conditions. We present several examples in order to illustrate the relationship between the Birkhoff–James, Roberts, and Pythagoras orthogonalities, and the usual orthogonality in the framework of Hilbert $C^{*}$-modules.

In 1973, Williams [D. Williams, On Rényi's ‘record’ problem and Engel's series, Bull. London Math. Soc.5 (1973), 235–237] introduced two interesting discrete Markov processes, namely C-processes and A-processes, which are related to record times in statistics and Engel's series in number theory respectively. Moreover, he showed that these two processes share the same classical limit theorems, such as the law of large numbers, central limit theorem and law of the iterated logarithm. In this paper, we consider the large deviations for these two Markov processes, which indicate that there is a difference between C-processes and A-processes in the context of large deviations.

Some years ago Lambe and Ward (1) and Erdélyi (2) obtained integral equations for Heun polynomials and Heun functions. The integral equations discussed by these authors were of the form

Further, as is well known, the Heun equation includes, among its special cases, Lamé's equation and Mathieu's equation and so (1.1) may be considered a generalisation of the integral equations satisfied by Lamé polynomials and Mathieu functions. However, integral equations of the type (1.1) are not the only ones satisfied by Lamé polynomials; Arscott (3) discussed a class of non- linear integral equations associated with these functions. This paper then is concerned with discussing the existence of non-linear integral equations satisfied by solutions of Heun's equation.

The theorem is—

The distance between the circumscribed and the inscribed centres of a triangle it a mean proportional between the circumscribed radius and its excess above the inscribed diameter.

We find conditions which imply that a saturated fusion system over a product of 2-groups splits as a product of fusion systems over the factors.

In this article, we prove a local implication of boundedness of Fano varieties. More precisely, we prove that $d$-dimensional $a$-log canonical singularities with standard coefficients, which admit an $\epsilon$-plt blow-up, have minimal log discrepancies belonging to a finite set which only depends on $d,\,a$ and $\epsilon$. This result gives a natural geometric stratification of the possible mld's in a fixed dimension by finite sets. As an application, we prove the ascending chain condition for minimal log discrepancies of exceptional singularities. We also introduce an invariant for klt singularities related to the total discrepancy of Kollár components.

In a previous paper the author has employed the expansion

where

and

to establish the theorem that, if −π/2<amp γ<π/2, the function Ps/Ts+1 remains finite as γ→∞. This theorem is valid provided that z is not real and ≧ 1.

Let $\varOmega$ be a convex, open subset of $\mathbb{R}^n$ and let $\mathcal{D}'(\varOmega)$ be the space of distributions on $\varOmega$. It is shown that there exist linear embeddings of $\mathcal{D}'(\varOmega)$ into a differential algebra that commute with partial derivatives and that embed $\mathcal{C}^{\infty}(\varOmega)$ as a subalgebra. This embedding appears to be the first one after Colombeau’s to possess these properties. We show that many nonlinear operations on distributions can be defined that are not definable in the Colombeau setting.

AMS 2000 Mathematics subject classification: Primary 46F30. Secondary 13C11

The uniform stability of the zero solution and the asymptotic behaviour of all solutions of the neutral delay differential equation

$$ [x(t)-P(t)x(t-\tau)]'+Q(t)x(t-\sigma)=0,\quad t\ge t_0, $$

are investigated, where $\tau,\sigma\in(0,\infty)$, $P\in C([t_0,\infty),\mathbb{R})$, and $Q\in C([t_0,\infty), [0,\infty))$. The obtained sufficient conditions improve the existing results in the literature.

AMS 2000 Mathematics subject classification: Primary 34K20; 34K15; 34K40

For any sequence (aj) of complex numbers and for any ρ > ½, we construct an entire function F with the following properties. F has order ρ, mean type, each aj is a deficient value of F, and F is given by F(z)=f(g(z)), where f and g are transcendental entire functions. This complements a result of Goldstein. We also construct, for any ρ>½, an entire function G of order p, mean type, such that liminf,→ ∞ T(r, G)/T(r, G′)>1.

Let R be a (commutative Noetherian) local ring (with identity) having maximal ideal and dimension d≧l. It is shown in [5,3.6rsqb; that the local cohomology module may be described as a module of generalized fractions: if x1…,xd is a system of parameters for R, then , where U(x)d+1 is the triangular subset [4,2.1] of Rd+1 given by

By the results of Rickman [7] and Ralston [6], a finite group G admitting a fixed point free automorphism α of order pq, where p and q are primes, is soluble. If p = q, then |G| is necessarily coprime to |α|, and it follows from Berger [1] that G has Fitting height at most 2, the composition length of <α>. The purpose of this paper is to prove a corresponding result in the case when p≠q.

A monoid S is said to be right coherent if every finitely generated subact of every finitely presented right S-act is finitely presented. Left coherency is defined dually and S is coherent if it is both right and left coherent. These notions are analogous to those for a ring R (where, of course, S-acts are replaced by R-modules). Choo et al. have shown that free rings are coherent. In this paper we prove that, correspondingly, any free monoid is coherent, thus answering a question posed by Gould in 1992.

We establish a generalization of the Widder–Arendt theorem from Laplace transform theory. Given a Banach space $E$, a non-negative Borel measure $\mea$ on the set $\Rplus$ of all non-negative numbers, and an element $\bnd$ of $\R\cup\{-\infty\}$ such that $\natres{-\coefl}$ is $\mea$-integrable for all $\coefl>\bnd$, where $\natres{-\coefl}$ is defined by $\natres{-\coefl}(t)=\exp(-\coefl t)$ for all $t\in\Rplus$, our generalization gives an intrinsic description of functions $\f\colon\Set\to E$ that can be represented as $\f(\coefl)=T(\natres{-\coefl})$ for some bounded linear operator $T\colon\Ma\to E$ and all $\coefl> \bnd$; here $\Ma$ denotes the Lebesgue space based on $\mea$. We use this result to characterize pseudo-resolvents with values in a Banach algebra, satisfying a growth condition of Hille–Yosida type.

AMS 2000 Mathematics subject classification: Primary 44A10; 47A10. Secondary 43A20; 46B22; 46G10; 46J25; 47D06

We consider the eigenvalue equation for the largest eigenvalue of certain kinds of non-compact linear operators given as the sum of a multiplication and a kernel operator. It is shown that, under moderate conditions, such operators can be approximated arbitrarily well by operators of finite rank, which constitutes a discretization procedure. For this purpose, two standard methods of approximation theory, the Nyström and the Galerkin method, are generalized. The operators considered describe models for mutation and selection of an infinitely large population of individuals that are labelled by real numbers, commonly called continuum-of-alleles models.

AMS 2000 Mathematics subject classification: Primary 47A58; 45C05. Secondary 47B34