The interesting results arising from the study of ‘Four intersecting spheres’ [9] in a solid made the author think of an analogous picture in higher spaces too and thus the present paper arose.

The main purpose of this paper is to prove the proposition: “A set of r mutually perspective (m.p.) (s—1)-simplexes have the same [s−2] (say x) of perspectivity, if and only if their centres of perspectivity (c.p.) lie in an [r−2] (say y); there then arises another such set of s m.p. (r—1)-simplexes, having the same rs vertices, which have y as their common [r−2] of perspectivity such that their c.p. lie in x.” The proposition is true in any [k] for k = s−1, s,…, r+s−2 (r ≦ s). The configuration of the proposition in [r+s−2] arises from the incidences of any r+s arbitrary primes therein and is therefore invariant under the symmetric group of permutations of r+s objects, and that in [r+s−3] is self-dual and therefore self- polar for a quadric therein. Some special cases of some interest for r = s are deduced. The treatment is an illustration of the elegance of the Möbius Barycentric Calculus ([15], pp. 136–143; [1], p. 71).

Let S = A0 hellip An be an n-simplex and Aih the foot of its altitude from its vertex Ai to its opposite prime face Si; O, G the circumcentre and centroid of S and Oi, Gi of Si. Representing the position vector of a point P, referred to O, by p, Coxeter [2] defines the Monge pointM of S Collinear with O and G by the relation so that the Monge point Mi of Si is given by If the n+1 vectors a are related by oi be given by Aih is given by Since Aih lies in Si, If Ti be a point on MiAih such that i.e. That is, MTi is parallel to ooi or normal to Si at Ti:. Or, the normals to the prime faces Si of S at their points Ti concur at M. In fact, this property of M has been used to prove by induction [3] that an S-point S of S lies at M. Thus M = 5, M = S or .

The well-known Taylor expansion of a function around a point a can be formally written as The last expression is just a symbolic form and is valid, as we know, under certain restrictive conditons. The last expression is just a symbolic form and is valid, as we know, under certain restrictive conditons. We shall study the situation when the differential operator d/da is replaced by the finite difference operator Δh/h, where the operator Δh, is defined by In general, Then we have the following theorem.

In this paper we shall discuss infinite capacity storage processes in which periods of input and output alternate. The length of a period of input and the length of the period of output immediately following may be statistically dependent and the change in storage level during an input or output period may depend on the length of the time interval in a rather general manner. However, we do not exploit either of these facts in the present paper.

Let G be a group. With each element a in G we associate the mappings ρ(a) and λ(a) of G into itself defined as follows, The product of mappings is defined as usual. Let P(G) and ∧(G) denote respectively the semigroups generated by the set of all ρ's and λ's. These semigroups will be called the commutation semigrowps of G.

We denote lattice join and meet by ∨ and ∧ respectively and the associated partial order by ≧. A lattice L with 0 and I is said to be orthocomplemented if it admits a dual automorphism x → x′, that is a one-one mapping of L onto itself such that which is involutive, so that for each x in L and, further, is such that for each x in L.

The behaviour of waves in elastic solids with linear stress strain curves is expressed, for plane strain, by a pair of simultaneous partial differential equations of hyperbolic type. Detailed behaviour of the waves is examined by solving these equations numerically.

Let G denote a Hausdorff locally compact Abelian group which is nondiscrete and second countable. The main results (Theorems (2.2) and (2.3)) assert that, for any closed subset E of G there exists a pseudomeasure s on G whose singular support is E; and that if no portion of E is a Helson set, then such an s may be chosen having its support equal to E. There follow (Corollaries (2.2.4) and (2.3.2)) sufficient conditions for the relations to hold for some pseudomeasure s, E and F being given closed subsets of G. These results are analogues and refinements of a theorem of Pollard [4] for the case G = R, which asserts the existence of a function in L∞(R) whose spectrum coincides with any preassigned closed subset of R.

Let k be a field of characteristic 2 and let G be a finite group. Let A(G) be the modular representation algebra1 over the complex numbers C, formed from kG-modules2. If the Sylow 2-subgroup of G is isomorphic to Z2×Z2, we show that A(G) is semisimple. We make use of the theorems proved by Green [4] and the results of the author concerning A(4) [2], where 4 is the alternating group on 4 symbols.

In this paper we give some algorithms for determining αw(T) and βw(T), the generalized internal and external stability numbers respectively, of a finite directed tree graph T whose nodes are weighted by a function w. We define αw(T) and βw in section 2. When w gives every node of T the weight 1 then αw(T) = α(T) and βw(T) = β(T) where α(T) and β(T) are the usual stability numbers.

It has been shown that the assumption of a specified cosmological substratum leads to the Lorentz-equivalencc of observers whose uniform motion in the substratum lies along a common straight line. The result is generalised for any pair of unaccelerated observers thus confirming that only the relative velocity of such observers is relevant in relating their observations of an event.

Let A be an associative algebra over the field F. We denote by ℒ(A) the lattice of all subalgebras of A. By an ℒ-isomorphism (lattice isomorphism) of the algebra A onto an algebra B over the same field, we mean an isomorphism of ℒ(A) onto ℒ(B). We investigate the extent to which the algebra B is determined by the assumption that it is ℒ-isomorphic to a given algebra A. In this paper, we are mainly concerned with the case in which A is a finite- dimensional semi-simple algebra.

In a recent study of generalized transfinite dimeters [4, 5] some geometric extremal problems were encountered. These form the subject matter of this note.