In continuation of the two previous papers (10; 11), this paper was originally written at the Indian Institute of Technology, Kharagpur and revised at the University of Sydney under the advice of Prof. T. G. Room. Although the altitudes of a general simplex S(A) in n-space (n > 2) do not concur as they do for a triangle (n = 2), yet we observe that its Monge point, M (1; 5), is an appropriate analogue of the orthocentre of a triangle such that M coincides with its orthocentre when it is orthogonal (or orthocentric). In consistency with the previous papers (10; 11; 13; 15) we shall call M as the S-point of S(A) and denote it as S as explained in § 1.2. The altitudes of S(A) are all met by the (n − 2)-spaces normal to its plane faces at their orthocentres, each parallel to of them, thus indicating the associated character of the altitudes as discussed separately in 2 other papers (12; 16). Before we introduce an orthogonal simplex and develop its properties in regard to its γ-altitudes and associated hyperspheres, we come across a number of intermediate ones of special interest. Two special types are treated here and the other two are developed in 2 other papers (13; 15).

The results described herein were obtained in connection with the development of some lattice theoretic machinery needed by Randall and Foulis in their treatment of the so-called “logic of empirical science” (See [3] for an introduction to this subject). The final product, however, turned out to be of sufficient interest to warrant its independent presentation.

The Redheffer matrix An = (aij)n×n defined by aij = 1 when i|j or j = 1 and aij = 0 otherwise has many interesting number theoretic properties. In this paper we give fairly precise estimates for its eigenvalues in punctured discs of small radius centred at 1.

Let Bn(x) denote the number of 1’s occurring in the binary expansion of an irrational number x>0. A difficult problem is to provide nontrivial lower bounds for Bn(x) for interesting numbers such as , e or π: their conjectural simple normality in base 2 is equivalent to Bn(x)∼n/2. In this article, amongst other things, we prove inequalities relating Bn(x+y), Bn(xy) and Bn(1/x) to Bn(x) and Bn(y) for any irrational numbers x,y>0, which we prove to be sharp up to a multiplicative constant. As a by-product, we provide an answer to a question raised by Bailey et al. (D. H. Bailey, J. M. Borwein, R. E. Crandall and C. Pomerance, ‘On the binary expansions of algebraic numbers’, J. Théor. Nombres Bordeaux16(3) (2004), 487–518) concerning the binary digits of the square of a series related to the Fibonacci sequence. We also obtain a slight refinement of the main theorem of the same article, which provides a nontrivial lower bound for Bn(α) for any real irrational algebraic number. We conclude the article with effective or conjectural lower bounds for Bn(x) when x is a transcendental number.

In his paper [1], Thron introduced a concept of lattice-equivalence of topological spaces. Let C(X) denote the lattice of all closed sets of a topological space X. Two topological spaces X and Y are said to be lattice-equivalent if there exists a lattice-isomorphism between C(X) and C(Y). It is clear that for any continuous function f: X → Y, the induced map ψf: C(Y) → C(X), defined by ψ(F)=f−1(F), is a lattice-homomorphism. Furthermore, if h: X→ Y is a homeomorphism then ψh: C(Y) → C(X) is a lattice-isomorphism. Thron proved among others that for TD-spaces X and Y, any lattice-isomorphism: C(Y) → C(X) can be induced by a homeomorphism f: X → Y in the above way.

We give a detailed exposition of the theory of decompositions of linearised polynomials, using a well-known connection with skew-polynomial rings with zero derivative. It is known that there is a one-to-one correspondence between decompositions of linearised polynomials and sub-linearised polynomials. This correspondence leads to a formula for the number of indecomposable sub-linearised polynomials of given degree over a finite field. We also show how to extend existing factorisation algorithms over skew-polynomial rings to decompose sub-linearised polynomials without asymptotic cost.

Burkill (1957) introduced extended Riemann-Stieltjes integrals of the form ∫baƒ(x)dkg(x)/dxk-1 to provide an alternative approach to the theory of distributions. We will denote Burkill's integral by (B)∫baƒ(x)dkg(x)/dxk-1 Burkill's paper (1957) partially motivated the study of the Riemann-Stieltjes integral in Russell (1970). He showed that the two integrals are not equivalent.

We obtain analogues, in the setting of semigroups with zero, of McAlister's convering theoren and the structure theorems of McAlister, O'Carroll, and Margolis and Pin. The covers come from a class C of semigroups defined by modifying one of the many characterisations of E-unitary inverse semigroups, namely, that an inverse semigroups is E-unitary if and only if it is an inverse image of an idempotent-pure homomorphism onto a group. The class C is properly contained in the class of all E*-unitary inverse semigroups introduced by Szendrei but properly contains the class of strongly categorical E*-unitary semigroups recently considered by Gomes and Howie.

The following paper is a sequel to the author's earlier paper [2]. In that paper some general results were obtained which described the motion of a fluid with a free surface subsequent to a given initial state and prescribed boundary conditions of a certain type. The analysis was based on a linearized theory but gravity effects were included. Viscosity, compressibility and surface tension effects were neglected. Among the problems treated was that of the normal symmetric entry of a thin wedge into water at rest. This Water entry problem has attracted a considerable amount of attention since the pioneer paper by Wagner [5]. Both linear and non-linear approximations have been used but all papers apart from [2] neglect gravity on the assumption that in the early stages of the penetration this is unimportant. One of the objects of [2] was to determine the solution with the gravity terms retained. A formal solution was obtained but no attempt was made to analyse this quantitatively. In the present paper we examine the extent of this effect in some detail. It will be of help to the reader to have some familiarity with the first three or four sections of [2] but in order to make the present paper self-contained we shall first reintroduce the notation used there and quote the necessary results from that paper without proof.

An inhomogeneous version of a general form of the Jarník-Besicovitch Theorem is proved.

Let Ω be the weakly pseudoconvex domain

and let ∂Ω be its boundary. If ϕ ∈ L∞ (∂Ω), we denote by Tϕ, the Toephtz operator with symbol ϕ acting on the Hardy space H2(∂Ω), and by J(∂Ω) the C*-subalgebra of B(H2(∂Ω)) generated by the Toeplitz operators with continuous symbol. Our main theorem asserts that J(∂Ω) contains the ideal K of all compact operators on H2(∂Ω), and that the symbol map ϕ→Tϕ induces an isomorphism of C(∂Ω) onto the quotient C*-algebra ℑ(∂Ω)/K. Similar results have been established before for other domains, and in particular when Ω is strongly pseudoconvex. The main interest of our results lies in their proofs: ours are elementary, whereas those used in the strongly pseudoconvex case depend heavily on the theory of the tangential Cauchy-Riemann operator.

Maillet proved that the set of Liouville numbers is preserved under rational functions with rational coefficients. Based on this result, a problem posed by Mahler is to investigate whether there exist entire transcendental functions with this property or not. For large parametrized classes of Liouville numbers, we construct such functions and moreover we show that they can be constructed such that all their derivatives share this property. We use a completely different approach than in a recent paper, where functions with a different invariant subclass of Liouville numbers were constructed (though with no information on derivatives). More generally, we study the image of Liouville numbers under analytic functions, with particular attention to $f(z)=z^{q}$, where $q$ is a rational number.

An inverse semigroup S is said to be modular if its lattice 𝓛𝓕 (S) of inverse subsemigroups is modular. We show that it is sufficient to study simple inverse semigroups which are not groups. Our main theorem states that such a semigroup S is modular if and only if (I) S is combinatorial, (II) its semilattice E of idempotents is “Archimedean” in S, (III) its maximum group homomorphic image G is locally cyclic and (IV) the poset of idempotents of each 𝓓-class of S is either a chain or contains exactly one pair of incomparable elements, each of which is maximal. Thus in view of earlier results of the second author a simple modular inverse semigroup is “almost” distributive. The bisimple modular inverse semigroups are explicitly constructed. It is remarkable that exactly one of these is nondistributive.

If L is a pseudocomplemented distributive lattice which is generated by a finite set X, then we will show that there exists a subset G of L which is associated with X in a natural way that ¦G¦ ≦ ¦X¦ + 2¦x¦ and whose structure as a partially ordered set characterizes the structure of L to a great extent. We first prove in Section 2 as a basic fact that each element of L can be obtained by forming sums (joins) and products (meets) of elements of G only. Thus, L considered as a distributive lattice with 0,1 (the operation of pseudocomplementation deleted), is generated by G. We apply this to characterize for example, the maximal homomorphic images of L in each of the equational subclasses of the class Bω of pseudocomplemented distributive lattices, and also to find the conditions which have to be satisfied by G in order that X freely generates L.

The order cycle system of stock control can be formulated as follows: demands for stock occur according to some pattern which we call the demand process and specify in detail later in this section. At fixed intervals of time orders are placed to replenish stock. Let orders for stock be placed at the instants jN, j = 1,2, …; the interval [(j — 1)(N, jN) which we suppose closed at its lower end-point and open at its upper end-point, is called the j-th order cycle and N the length of this interval is called the order cycle period. We suppose that an order placed at time jN is delivered into stock at time jN + li where {li} is a sequence of non-negative random variables independent of the demand process and the order cycle period, we suppose also that the li are mutually independent and identically distributed with common distribution function L(x) with L(0 +) = 0 and finite expectation . The quantity li is called the lead time of the j-th order, that is of the order placed at jN and is supposed independent of the amount ordered. That portion of a demand, if any, which cannot be satisfied immediately is satisfied from future deliveries, thus every demand is satisfied eventually and a negative inventory or back orders can be held. We shall suppose that all order cycle periods under consideration are multiples of some fixed time interval of length τ which we shall take as our unit of time. For example, the interval of length τ could be one day and we would consider order cycle periods which were integral multiples of days. For convenience we take τ = 1 and suppose that the order cycle peiod N is an integer. In this paper we shall consider the following two demand processes.

By using pairwise relatively complete families of functions, which we define here, we obtain two characterizations of quasi-pseudometrizable bitopological spaces from which some known theorems can be derived as easy corollaries.

Sharp results for the cofficient estimates, distortion theorems, radius of convexity, arc-length and area of the image curve are obtained for the class R(A, B) of regular functions whose derivative is subordinate to (1+AZ)/(1+Bz), -1 ≤ B ≤ A ≤ 1, in the unit disc E = {z:|z| < 1}. We also establish a convolution theorem for this class.

We supply proofs that are simple, and possibly partly new, for two theorems that appear in Rankin's book [6].

In this paper, we show some results involving classical geometric concepts. For example, we characterize rotundity and Efimov-Stechkin property by mean of faces of the unit ball. Also, we prove that every almost locally uniformly rotund Banach space is locally uniformly rotund if its norm is Fréchet differentiable. Finally, we also provide some theorems in which we characterize the (strongly) exposed points of the unit ball using renormings.

Let F(X, Y) be an absolutely irreducible polynomial with coefficients in an algebraic number field K. Denote by C the algebraic curve defined by the equation F(X, Y) = 0 and by K[C] the ring of regular functions on Cover K. Assume that there is a unit ϕ in K[C] − K such that 1 − ϕ is also a unit. Then we establish an explicit upper bound for the size of integral solutions of the equation F(X, Y) = 0, defined over K. Using this result we establish improved explicit upper bounds on the size of integral solutions to the equations defining non-singular affine curves of genus zero, with at least three points at ‘infinity’, the elliptic equations and a class of equations containing the Thue curves.