Let L be an odd degree extension field of the field K, char K ≠ 2. Let U* denote the natural extension map from W(K) to W(L) where W(K), resp. W(L) denotes the Witt group of quadratic forms over K, resp. L. It is well-known that U* is injective [4, p. 198]. In fact Springer [10] proved a stronger theorem, namely that if φ is anisotropic over K then it remains anisotropic on extension to L. Rosenberg and Ware [8] proved that if L is a Galois extension then the image of U* is precisely the subgroup of W(L) fixed by the Galois group of L over K, this Galois group having a natural action on W(L). See [4, p. 214] and [3] for a quick proof. See also Dress [1] who extended these results to equivariant forms. In this article we investigate the corresponding map U* when we replace the field L by a central simple K-algebra of odd degree and indeed more generally by any finite dimensional K-algebra which becomes odd-dimensional on factoring out by the radical. Our algebras are equipped with an involution of the second kind, i.e. one which is non-trivial on the centre, and we replace quadratic forms by hermitian forms with respect to the involution. We show that U* is injective for all the algebras mentioned above and that a weaker version of Springer's theorem holds for central simple algebras of odd degree provided we make a suitable restriction on the nature of the involution. We show that the analogue of the Rosenberg-Ware result is valid for hermitian forms over odd-dimensional Galois field extensions but that for central simple algebras of odd degree a result as nice as the Rosenberg–Ware one cannot hold. Indeed the group of all K-automorphisms of such an algebra which commute with the involution fixes all of the Witt group. However the map U* is not surjective in general even for division algebras of odd degree.

A period of a function f(z) is defined to be a number w(≠0) such that

is identically zero; and it is not a difficult matter to show that an integral function may either have no periods or else a single sequence kλ, k = ±1, ±2, ….

Parallel to sides.

(a) Let O and 0' be two reciprocal points,

DE(α), FG(β), HI(γ) the intercepts made on the sides by lines drawn parallel to the sides through O;

D'E'(α'), F'G' (β'), H'I' (γ') similar intercepts for O'.

The aim of this paper is to prove the Theorem: Let M be a complete non compact surface without boundary in the euclidean space 3. We suppose that all geodesies of M are congruent. Then M is an affine plane in 3.

Rainville (6) has discussed the following Euler transformation of the hypergeometric function

We refer to (1) for the definitions of Un and Hn. Our object is to find asymptotic expansions for Un and Hn for large n. This enables us to improve the approximations to Un and Hn for large n found in the last two pages of (1).

We relax the growth condition in time for uniqueness of solutions of the Cauchy problem for the heat equation as follows: Let u(x, t) be a continuous function on ℝn × [0, T] satisfying the heat equation in ℝn × (0, t) and the following:

(i) There exist constants a > 0, 0 < α < 1, and C > 0 such that

(ii) u(x, 0) = 0 for x ∈ ℝn.

Then u(x, t)≡ 0 on ℝn × [0, T]

We also prove that the condition 0 < α < 1 is optimal.

The reciprocity, , has been shown to hold when g(x) is in B* [Proc. Edin. Math. Soc., 12 (1960), 85]. The purpose of the present note is to show that it holds more generally and in particular when g(x) is in a subclass B″, of functions of bounded variation B, such that B* ⊂ B″ ⊂ B′ ⊂ B; it is assumed that f(x) is any function in D1 and that f(x) and g(x) are both continuous on the left and possibly with simultaneous points of discontinuity.

In the first section of Kneser's book on Integral Equations and their Applications to Mathematical Physics, he applies that theory to the solution of some of the problems which arise in the Linear Flow of Heat. The object of this paper is to illustrate Kneser's use of Integral Equations in the Mathematical Theory of the Conduction of Heat by the discussion of one of the classical problems of Linear Flow which he leaves untouched.

Let Μ be a Bade complete (or σ-complete) Boolean algebra of projections in a Banach space X. This paper is concerned with the following questions: When is Μ equal to the resolution of the identity (or the strong operator closure of the resolution of the identity) of some scalar-type spectral operator T (with σ(T) ⊆ ℝ) in X? It is shown that if X is separable, then Μ always coincides with such a resolution of the identity. For certain restrictions on Μ some positive results are established in non-separable spaces X. An example is given for which Μ is neither a resolution of the identity nor the strong operator closure of a resolution of the identity.

Let P be the point at which the potential has to be found (Fig. 7). Let the uniform surface density be σ, and take two radii PQ, PQ′ differing in length by a small quantity Q′K. Let QR be drawn perpendicular to BD, the diameter through P, and let CL be drawn perpendicular to PQ.

In the Proceedings of the National Academy of Sciences of the U.S.A., Vol. II. (1916), page 171, Professor F. Morley has established a theorem which both extends and simplifies the theorem of Feuerbach, viz., All curves of class three which (i) touch the six lines OP, OQ, OR, QR, RP, PQ joining four orthocentric points, O, P, Q, R, and (ii) pass through the circular points, also touch the common nine-points-circle of the triangles PQR, OQR, ORP, OPQ. Sixteen of these curves of class three break up into one of the four points aud a circle touching the sides of the triangle formed by the other three. Thus the sixteen instances of Feuerbach's theorem derivable from the four triangles are included as special cases, in Morley's theorem. A purely geometrical proof of the theorem may be worth consideration.

The object of the paper was to draw attention to a few important and well known cases of the harmonic section of a straight line, and to show their application to one or two problems of interest, more especially the method of drawing tangents to a conic by the ruler only. The effort throughout was to secure clearness, brevity, and freshness of proof, coupled with purely geometrical treatment.

Let S be a semigroup. An element a of S is called right (resp. left) regular if a=a2x (resp. a=xa2) for some x∈S. If a is regular and right (resp. left) regular, a is called strongly right (resp. left) regular. As is well known, if a is strongly regular (i.e., right and left regular) then it is regular, more precisely, there exists uniquely an element x such that a= a2x,x= x2a and ax=xa, and a is contained in a subgroup of S (and conversely).

This paper deals with a system of tetrahedra in a sphere corresponding to the co-symmedian system of triangles in a circle. Such a system of tetrahedra, so far as the writer knows, has not been hitherto discussed. The condition that a tetrahedron may have a symmedian point is given in Wolstenholme's “Problems” (1878).