What we cail here the “construction principle” is a principle on the ground of which some functionals can be defined; the domain and the range of such a functional consist of some “computable” functionals of various finite types. The principle above is considered here as the basis of the functional interpretation of transfinite induction up to ε0. It is concretely repesented as the “term-forms”, where every term-form is shown to be “computable” in some sense.

We use a theorem of Loxton and van der Poorten to prove the transcendence of certain real numbers defined by digit patterns. Among the results we prove are the following. If k is an integer at least 2, P is any nonzero pattern of digits base k, and counts the number of occurrences (mod r) of p in the base k representation of n, then is transcendental except when k = 3, P = 1 and r = 2. When (r, k − 1) = 1 the linear span of the numbers has infinite dimension over Q, where P ranges over all patterns base k without leading zeros.

The following question is discussed and evidence for and against it is advanced: is it true that if F is an arbitrary finite subgroup of an arbitrary non-linear simple locally finite group G, then CG(F) is infinite? The following points to an affirmative answer.

Theorem A. Let F be an arbitrary finite subgroup of a non-linear simple locally finite group G. Then there exist subgroups D ◃ C ≤ G such that F centralizes C/D, F∩C ≤ D, and C/D is a direct product of finite alternating groups of unbounded orders. In particular, F centralizes an infinite section of G.

Theorem A is deduced from a “local” version, namely

Theorem B. There exists an integer valued function f(n, r) with the following properties. Let H be a finite group of order at most n, and suppose that H ≤ S, where S is either an alternating group of degree at least f = f(n, r) or a finite simple classical group whose natural projective representation has degree at least f. Then there exist subgroups D ◃ C ≤ S such that (i) [H, C] ≤ D, (ii) H ∩ C ≤ D, (iii) C/D ≅ Alt(r), (iv) D = 1 if S is alternating, and D is a p-group of class at most 2 and exponent dividing p2 if S is a classical group over a field of characteristic p.

The natural “local version” of our main question is however definitely false.

Proposition C. Let p be a given prime. Then there exists a finite group H that can be embedded in infinitely many groups PSL(n, p) as a subgroup with trivial centralizer.

Let p be a prime, let R be a finite p-ring with identity and suppose that the radical of R has pm elements. If R is indecomposable as a ring then there are at most m+1 minimal ideals in R/Rad R.

1980 Mathematics subject classification (Amer. Math. Soc): 16 A 44.

The stability properties of the family ℳ of all intersections of closed balls are investigated in spaces C(K), where K is an arbitrary Hausdorff compact space. We prove that ℳ is stable under Minkowski addition if and only if K is extremally disconnected. In contrast to this, we show that ℳ is always ball stable in these spaces. Finally, we present a Banach space (indeed a subspace of C[0, 1]) which fails to be ball stable, answering an open question. Our results rest on the study of semicontinuous functions in Hausdorff compact spaces.

A double triangle subspace lattice in a Hilbert space H is a 5-element set of subspaces of H, containing (0) and H, with each pair of non-trivial elements intersecting in (0) and spanning H. It is shown that if any pair of non-trivial elements has a closed vector sum the double triangle is both non-reflexive and non-transitive. A double triangle in H⊕H is an operator double triangle if each non-trivial elements is the graph of an operator acting on H. A sufficient condition is given for any operator double triangle to be non-reflexive.

We consider real-valued functions ƒ(x) which are defined for all sufficiently large real numbers x. In discussing the behaviour of such functions as x → + ∞, it is useful to compare ƒ with the functions of some “comparison scale”. The early work in this field was due to Du Bois-Reymond (see, for example, (2), (3)). This was elaborated by Hardy, (6), who was mainly concerned with what he calls the “logarithmico-exponential” scale of functions. This “scale of Hardy” may be defined as the smallest class ℋ of functions ƒ with the following properties:

(i) if ƒ ∈ ℋ, ƒ, is defined and continuous for all sufficiently large values of x;

(ii) (a) the function ƒ(x) ≡ α, where α is any real constant, is in ℋ;

(b) the function ƒ(x) ≡ x is in ℋ;

(iii) if ƒ and g are in ℋ and g is non-zero for all sufficiently large values of x, then

are also in ℋ.

Brizolis asked for which primes p greater than 3 there exists a pair (g,h) such that h is a fixed point of the discrete exponential map with base g, or equivalently h is a fixed point of the discrete logarithm with base g. Various authors have contributed to the understanding of this problem. In this paper, we use p-adic methods, primarily Hensel’s lemma and p-adic interpolation, to count fixed points, two-cycles, collisions, and solutions to related equations modulo powers of a prime p.

It is an open problem to establish whether or not the partial sums operator SNN2f(x, y) of the Fourier series of f ∈ Lp, 1 < p < 2, converges to the function almost everywhere as N → ∞. The purpose of this paper is to identify the operator that, in this problem of a.e. convergence of Fourier series, plays the central role that the maximal Hilbert transform plays in the one-dimensional case. This operator appears to be a singular integral with variable coefficients which is a variant of the maximal double Hilbert transform.

Let x = (x1, x2,…, xn) be a normal random vector with zero expectation vector and with a variance-covariance matrix which has 1 for its diagonal elements and ρ for its off-diagonal elements. Consider the quantity where.

Let {Pn} be any sequence of real or complex numbers subject to the sole restriction And let If tn → s, whenever sn → s we say that the sequence {sn} is summable Nörlund or summable (N, p) to s.

Suppose that ${\Gamma }^{+ } $ is the positive cone of a totally ordered abelian group $\Gamma $, and $(A, {\Gamma }^{+ } , \alpha )$ is a system consisting of a ${C}^{\ast } $-algebra $A$, an action $\alpha $ of ${\Gamma }^{+ } $ by extendible endomorphisms of $A$. We prove that the partial-isometric crossed product $A\hspace{0.167em} { \mathop{\times }\nolimits}_{\alpha }^{\mathrm{piso} } \hspace{0.167em} {\Gamma }^{+ } $ is a full corner in the subalgebra of $\L ({\ell }^{2} ({\Gamma }^{+ } , A))$, and that if $\alpha $ is an action by automorphisms of $A$, then it is the isometric crossed product $({B}_{{\Gamma }^{+ } } \otimes A)\hspace{0.167em} {\mathop{\times }\nolimits }^{\mathrm{iso} } \hspace{0.167em} {\Gamma }^{+ } $, which is therefore a full corner in the usual crossed product of system by a group of automorphisms. We use these realizations to identify the ideal of $A\hspace{0.167em} { \mathop{\times }\nolimits}_{\alpha }^{\mathrm{piso} } \hspace{0.167em} {\Gamma }^{+ } $ such that the quotient is the isometric crossed product $A\hspace{0.167em} { \mathop{\times }\nolimits}_{\alpha }^{\mathrm{iso} } \hspace{0.167em} {\Gamma }^{+ } $.

We consider the following problem arising in agricultural statistics. Suppose that a large number of plants are set out on a regular grid, which may be triangular, square or hexagonal, and that among these plants, half are to be given one and half the other of two possible treatments. For the sake of statistical balance, we require also that, if one plant in every k plants has i of its immediate neighbours receiving the same treatment as itself, then k is constant over all possible values of i. For square and triangular grids, there exist balanced arrays of finite period in each direction, but for the hexagonal grid, we show that no such balanced array can exist. Several related questions are discussed.

Many generalizations of continued fractions, where the reciprocal function has been replaced by a more general function, have been studied, and it is often asked whether such generalized expansions can have nice properties. For instance, we might ask that algebraic numbers of a given degree have periodic expansions, just as quadratic irrationals have periodic continued fractions; or we might ask that familiar transcendental constants such as e or π have periodic or terminating expansions. In this paper, we show that there exist such generalized continued function expansions with essentially any desired behaviour.

Consider a positive regular Markov chain X0, X1, X2,… with s(s finite) number of states E1, E2,… E8, and a transition probability matrix P = (pij) where = , and an initial probability distribution given by the vector p0. Let {Zr} be a sequence of random variables such that and consider the sum SN = Z1+Z2+ … ZN. It can easily be shown that (cf. Bartlett [1] p. 37), where λ1(t), λ2(t)…λ1(t) are the latent roots of P(t) ≡ (pijethij) and si(t) and t′i(t) are the column and row vectors corresponding to λi(t), and so constructed as to give t′i(t)Si(t) = 1 and t′i(t), si(o) = si where t′i(t) and si are the corresponding column and row vectors, considering the matrix .

A Bhaskar Rao design is obtained from the incidence matrix of a partially balanced incomplete block design with m associate classes by negating some elements of the matrix in such a way that the inner product of rows α and β is ci if α and β are ith associates. In this paper we use nested designs constructed from unions of cyclotomic classes to give Bhaskar Rao designs.

We study recurrence and transience for Lévy processes induced by topological transformation groups acting on complete Riemannian manifolds. In particular the transience–recurrence dichotomy in terms of potential measures is established and transience is shown to be equivalent to the potential measure having finite mass on compact sets when the group acts transitively. It is known that all bi-invariant Lévy processes acting in irreducible Riemannian symmetric pairs of noncompact type are transient. We show that we also have ‘harmonic transience’, that is, local integrability of the inverse of the real part of the characteristic exponent which is associated to the process by means of Gangolli’s Lévy–Khinchine formula.

Banach's contraction principle guarantees the existence of a unique fixed point for any contractive selfmapping of a complete metric space. This paper considers generalizations of the completeness of the space and of the contractiveness of the mapping and shows that some recent extensions of Banach's theorem carry over to spaces whose topologies are generated by families of quasi-pseudometrics.

The first and last papers of Harald Bohr deal with ordinary Dirichlet series and their order (or Lindelöf) function μ(σ) (= inf{α;f(σ + it) + 0(|t|α)}). The Lindelöf hypothesis is μ(σ) = inf(0, ½ − t) when an = (−1)n. Are there ordinary Dirichlet series with −l < μ′ (σ) < 0 for some σ? A negative answer would imply Lindelöf's hypothesis. This is the last problem of Harald Bohr. This paper gives (1) a review on Bohr's results on ordinary Dinchlet series; (2) a review on results of the author and of Queffelec on “almost sure” and “quasi sure” properties of series with the solution of a previous problem of Bohr; (3) the following answer to the last problem: μ′(σ) can approach − ½, and necessarily μ(σ + μ(σ) + ½) = 0. The characterization of the order functions of ordinary Dirichlet series remains an open question.

In 1991, Andrews and Hickerson established a new Bailey pair and combined it with the constant term method to prove some results related to sixth-order mock theta functions. In this paper, we study how this pair gives rise to new mock theta functions in terms of Appell–Lerch sums. Furthermore, we establish some relations between these new mock theta functions and some second-order mock theta functions. Meanwhile, we obtain an identity between a second-order and a sixth-order mock theta functions. In addition, we provide the mock theta conjectures for these new mock theta functions. Finally, we discuss the dual nature between the new mock theta functions and partial theta functions.