In 1946 Baer (Polarities infinite projective planes, Bull. Am. Math. Soc. 52, 77–93) showed that the absolute points of a polarity in a finite projective plane of odd non-square order always form an oval, that is, in a plane of order n there are exactly n+ 1 absolute points and no three are collinear. It is well known that the absolute points of polarities in planes of odd square order form ovals in some cases.

If the oval is a subset of the set of absolute points, then the oval itself determines the polarity, and this makes it appear unlikely that the oval could be a proper subset. Among other results in the paper it is to be proved that in the regular Hughes plane there is a polarity which is determined by an oval which is a relatively small subset of the set of absolute points. Explicitly, if Ω is the Hughes plane of order q2 and A is the central subplane of order q, then every conic in Δ can be extended to an oval in Ω, and this oval determines a polarity in which there are ½(q3–q) additional absolute points.

In 1904 Poincaré [11] conjectured that every compact, simply-connected closed 3-dimensional manifold is homeomorphic to a 3-sphere. The corresponding result for dimension 2 is classical; for dimension ≧ 5 it was proved by Smale [12] and Stallings [13], but for dimensions 3 and 4 the question remains open. It has been discovered in recent years that the 3-dimensional Poincaré conjecture could be reformulated in purely algebraic terms [6, 10, 14, 15] however the algebraic problems which are posed in the references cited above have not, to date, proved tractable.

A celebrated result, due to Jacobi, says that the number of representations of the positive integer n as a sum of four squares is equal to eight times the sum of the divisors of n which are not divisible by 4. We give a new and simple proof of this result which depends only on Jacobi's triple product identity.

Using the terminology in 2 (where the expression m-type is also explained) we will prove the following theorems:Theorem 1. If there exist(i) a skew-Hadamard matrix H = U+I of order h,(ii)m-type matrices M = W+I and N = NT of order m,(iii) three matrices X, Y, Z of order x = 3 (mod 4) satisfying(a) XYT, YZT and ZXT all symmetric, and(b) XXT = aIx+bJxthen is an Hadamard matrix of order mxh.

The variety of quasigroups is universal for varieties of algebras of the most general kind in the sense that each such variety can be interpreted in a natural way in a suitably chosen subvariety of quasigroups. More precisely, for any algebra 〈A, f0, f1, f2, …〉 where f0, f1, f2, … is an arbitrary finite or infinite sequence of operations of finite rank, there exists a quasigroup 〈B,.〉 and polynormial operations F0, F1, F2, … over 〈B,·〉 such that 〈A, f0, f1, …〉 is a subalgebra of 〈B, F0, F1, …〉 satisfying exactly the same identities. Moreover, if there are only finitely many f0, f1, …, then 〈B1〉 can be taken so that its identities are recursive in those of 〈A, f0, f1, …〉, If 〈A, f0, f1, …〉 is a free algebra with an infinite number of free generators, then B can also be taken to coincide with A. This universal property of quasigroups has a number of consequences for their equational metatheory.

Let be a non-cyclic free presentation of a group G, the lower central series of R. Then Rn/Rn+1, n ≧ 1, can be regarded as a right G-module by defining the action of G via conjugation in F. We wish to investigate the annihilators of these modules which we call higher relation modules.

Let Lp(Ω, K) denote the Banach space of weakly measurable functions F defined on a finite measure space and taking values in a separable Hilbert space K for which ∥ F ∥p = ( ∫ | F(ω) |p)1/p < + ∞. The bounded Hermitian operators on Lp(Ω, K) (in the sense of Lumer) are shown to be of the form , where B(ω) is a uniformly bounded Hermitian operator valued function on K. This extends the result known for classical Lp spaces. Further, this characterization is utilized to obtain a new proof of Cambern's theorem describing the surjective isometries of Lp(Ω, K). In addition, it is shown that every adjoint abelian operator on Lp(Ω, K) is scalar.

The ellipsoid is characterised among all convex bodies in n-dimensional Euclidean space, Rn, by many different properties. In this paper we give a characterisation which generalizes a number of previous results mentioned in [2], p. 142. The major result will be used, in a paper yet to be published, to prove some results concerning generalizations of the Minkowski theory of reduction of positive definite quadratic forms.

We provide a comprehensive study of the function $h=h(q)$ defined by

$$\begin{eqnarray}h=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{12j-1})(1-q^{12j-11})}{(1-q^{12j-5})(1-q^{12j-7})}\end{eqnarray}$$

$k=k(q)$

$$\begin{eqnarray}k=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{10j-1})(1-q^{10j-2})(1-q^{10j-8})(1-q^{10j-9})}{(1-q^{10j-3})(1-q^{10j-4})(1-q^{10j-6})(1-q^{10j-7})}.\end{eqnarray}$$

We provide a wedge decomposition of the homotopy type of the p-subgroup complex in the case of a finite solvable group G. In particular, this includes a new proof of the result of Quillen which says that this complex is contractible if and only if there is a non-trivial normal p-subgroup in G. We also provide reduction formulas for the G-module structure of the homology groups. Our results are obtained with diagram-methods by gluing the p-subgroup complex of G along the p-subgroup complex of = G/N for a normal p′-subgroup of G.

The object of this paper is to consider the existence of the double-six over GF(2n) and particularly over GF(4).

The problem of finding the most economical coverings of n-dimensional Euclidean space by equal spheres whose centres form a lattice, which is equivalent to a problem concerning the inhomogeneous minima of positive definite quadratic forms, has been discussed recently by Barnes and Dickson [1]. The reader is referred to [1] for a complete background on the problem. Terms and notations used will be as in that paper.

The concept of a permutation polynomial function over a commutative ring with 1 can be generalized to multiplace functions in two different ways, yielding the notion of a k-ary permutation polynomial function (k > 1, k ∈ N) and the notion of a strict k-ary permutation polynomial function respectively. It is shown that in the case of a residue class ring Zm of the integers these two notions coincide if and only if m is squarefree.

We address the problem of describing all graphs Γ such that Aut Γ is a symmetric group, subject to certain restrictions on the sizes of the orbits of Aut Γ on vertices. As a corollary of our general results, we obtain a classification of all graphs Γ on v vertices with Aut Γ ≅ Sn, where ν < min{5n, ½n(n – 1)}.

In a topological vector space, duality invariant is a very important property, some famous theorems, such as the Mackey-Arens theorem, the Mackey theorem, the Mazur theorem and the Orlicz-Pettis theorem, all show some duality invariants.

In this paper we would like to show an important improvement of the invariant results, which are related to sequential evaluation convergence of function series. Especially, a very general invariant result is established for an abstract mapping pair (Φ, B(Φ, X)) consisting of a nonempty set Φ and B(Φ, X) = {f ∈ XΦ: f (Φ) is bounded}, where X is a locally convex space.

An elementary and self-contained account of analytic Jordan decomposition of matrix-valued analytic functions is given. An integral representation for their eigenvalues is obtained. This leads to estimates of the differences in eigenvalues and the number of points of degeneracy.

Mrowka and Engleking [1] have recently introduced a notion more general than that of compactness. Perhaps the most convenient direction at departure is the following: for spaces X and E, X is said to be E-compact if X is topologically embeddable as a closed subset of a product Em for some cardinal m, in which case we write X ⊂cl Em. More generally, X is said to be E-completely regular if X is topologically embeddable in a product Em for some m. For example, if we take E to be the unit interval I, we obtain the class of compact spaces and completely regular spaces, respectively, as is well-known. The question then arises, of course, given a space E, what spaces are compact with respect to it? A related question, to which we address ourselves in this note, is the following. Denote by K[E] all those topological spaces which are E-compact. Then we ask: are there very many distinct E-compact classes? It will develop that there are indeed quite a large number of such classes.

The Hamiltonian system Y′ = BY + CZ, Z′ = – AY – B*Z is considered where the coefficients are continuous on I = [a, ∞, C = C* ≧ 0, and A = A* ≦ 0. A solution (Y, Z) satisfying Y*Z = Z*Y is defined to be principal (coprincipal) provided that (i) Y−1 exists on I (Z−1 exists on I) and (ii) as t→∞ ( as t → ∞). Three conditions are given which are separtely equivalent to the condition that a solution is principal iff it is coprincipal. For a self-adjoint scalar operator L of order 2n, this problem is related to the deficiency index problem and to a problem of Anderson and Lazer (1970) which concerns the number of lnearly independent solutions of L (y) =0 satisfying y(k) ∈ (a, ∞) (k = 0, …, n).

Spitzbart [1] has considered a generalization of Hermite's interpolation formula in one variable and has obtained a polynomial p(x) of degree n + Σnj=0 = rj in x which interpolates to the values of a function and its derivatives up to order rj at xj, j = 0, 1,···n. Ahlin [2] has considered a bivariate generalization of Hermite's interpolation formula. He has developed a bivariate osculatory interpolation polynomial which agrees with f(x, y) and its partial and mixed partial derivatives up to a specified order at each of the nodes of a Cartesian grid. However, the above interpolation problem considered by Ahlin assumes that the values of partial and mixed partial derivatives of the same fixed order k – 1 are available at every point of the rectangular grid. It may also be observed that Ahlin's formula is essentially a Cartesian product of a special case of Spitzbart's formula in one variable.

We prove that the p-adic zeta-function constructed by Kubota and Leopoldt has the Dirichlet series expansion Where the convergence of the first summation is for the p-adic topology. The proof of this formula relates the values of p(–s, ω1+σ) for s ∈ Zp, with a branch of the ‘sth-fractional derivative’, of a suitable generating function.