Let Cr, be the intersection of n — r quadrics with a common self-polar simplex S in projective n-space [n]. Let Γr be a Cr that can be taken in coordinate form as Every C1 is a Γ1, and its points of hyperosculation have special properties: they are the points of intersection of C1 with the faces of S each counting (n—1)(n — 2)/2 times, and the osculating [s], for s≦n–1, has 2s-point contact. Here we show that if r≧2 and n>2r then every point of Γ, has exceptional higher tangent spaces: the s-tangent space at a point P of an r-dimensional variety V is the intersection of all primes that cut V in a variety having an (s + l)-fold point (at least) at P, and normally has dimension if this is less than n. The s-tangent space to Γ, at a point not in a face of S is an [rs] (provided rs <n). Usually it is the existence of lines on V through P that cause a lower than expected s-tangent dimension. Not so on Γr, since its lines form a subvariety. If n≧5 not every C2 is a Γ2. Take n≧5. We show that C2 is Γ2 if and only if C2 contains a line. Also C2 is a Γ2 if and only if at some one point of C2 off the faces of S the second-tangent space is a [4]. Thus, unexpectedly, we have: if one point of C2 off the faces of S has a [4] for second-tangent space, then so do all such points of C2. We obtain results for points of Γ, in the faces of S.

The proposed mechanism explained here is based on the geometrical properties of Fig. 1.

0ABO′ is a straight line having 0A = BO′ = 1, and OB = AO′ = x and AH, BK are lines perpendicular to AB.

The science of the solution of Differential Equations has been in great measure systematized by the aid of ideas borrowed from the Theory of Functions, the equations being classified according to the singularities possessed by their solutions. In the case of linear Differential Equations of the second order

the solutions can have no singularities except at the singularities of the functions q(x) and r(x) (and possibly also at x = ∞ ): these equations may therefore be classified simply according to the number and nature of these singularities.

In this paper we investigate certain aspects of the multiparameter spectral theory of systems of singular ordinary differential operators. Such systems arise in various contexts. For instance, separation of variables for a partial differential equation on an unbounded domain leads to a multiparameter system of ordinary differential equations, some of which are defined on unbounded intervals. The spectral theory of systems of regular differential operators has been studied in many recent papers, e.g. [1, 3, 6, 9, 19, 21], but the singular case has not received so much attention. Some references for the singular case are [7, 8, 10, 13, 14, 18, 20], in addition general multiparameter spectral theory for self adjoint operators is discussed in [3, 9, 19].

In this paper we study near-rings of functions on Ω-groups which are compatible with all congruence relations. Polynomial functions, for instance, are of this type. We employ the structure theory for near-rings to get results for the theory of compatible and polynomial functions (affine completeness, etc.). For notations and results concerning near-rings see e.g. (10). However, we review briefly some terminology from there. (N, +,.) is a near-ring if (N, +) is a group and . is associative and right distributive over +. For instance, M(A): = (AA, +, °) is a near-ring for any group (A, +) (° is composition). If N is a near-ring then N0: = {n ∈ N/n0 = 0}. A group (Γ, +) is an N-group (we write NΓ) if a “product” ny is defined with (n + n‛)γ = nγ + n‛γ and (nn‛)γ = n(n‛γ). Ideals of near-rings and N-groups are kernels of (N-) homomorphisms. If Γ is a vector-space, Maff (Γ) is the near-ring of all affine transformations on Γ. N is 2-primitive on NΓ if NΓ is non-trivial, faithful and without proper N-subgroups. The (2-) radical and (2-) semisimplicity are defined similarly to the ring case.

This paper owes its origin to the following question posed by A. M. Sinclair, “If a linear algebra with identity has two equivalent unital algebra norms, |.|1 and |.|2, whose corresponding numerical radii, v1 and v2, are equal on the whole algebra, are |.|1 and |.|2 related? Are they, for example, necessarily equal?” We do not give a complete answer to this question but are able to give sufficient conditions on algebras of operators for v1 = v2 to imply |.|1 = |.|2 That this implication does not hold for an arbitrary algebra with identity is demonstrated by means of a counter-example. The result for operator algebras is used to deduce some essentially non numerical range results for equivalent operator norms.

The purpose of this paper is to extend to locally convex spaces and to uncountable systems several well-known results concerning infinite series, biorthogonal sequences, and Schauder bases. Section 2 gives three extensions of the theorem of Orlicz (10) and Pettis (11) and some lemmas that will be needed later. The third section introduces the notions of a summability basis and a summability basis of subspaces, and two main theorems are proved, including a simplification of Retherford and McArthur's proof (12) of a theorem of Nikol'skiĭ (9). Section 4 investigates the positive cone of an uncountable biorthogonal system, particularly conditions equivalent to the regularity of this cone.

In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an “octad generator”; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.

In this work we prove the existence and uniqueness of positive solutions of the nonlinear singular boundary value problem

where 0<σ<1.

Extensions of the above results to the case of Δ2u−f(x, u) = 0 with appropriate singularity built into f are also given.

Normal right submodules and right ideals need not coincide in an arbitrary near-ring. Berman and Silverman (1) have shown that in a near-ring (N, +, ·) with a two-sided zero (i.e. x · 0 = 0 · x = 0, for all x ∈ N) a right ideal is also a right submodule. If (N, +, ·) is in fact a distributively generated near-ring, then all normal right submodules are also right ideals. (See (5).)

In recent years considerable attention has been given to problems of thermal stress in isotropic materials. Much of this work has been devoted to statical problems although there has been some work on problems with time dependence, for example the quasi-static solutions obtained by Sternberg (1) and Eason and Sneddon (2). A good deal of interest has also been shown in statical thermal stress problems when the material is anisotropic. For the type of material considered here statical problems have been investigated by Grechushnikov and Brodovskii (3) and Sirotin (4) among others. Little attention has been given, however, to time dependent thermal stress problems when the material is anisotropic.

In this paper we give a classification up to isomorphism of Jordan nilalgebras whose lattices of subalgebras are modular when the ground field is algebraically closed.