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).

Let S and T be compact Hausdorff spaces and G and H finite-dimensional subspaces of C(S) and C(T) respectively. Suppose μ and ν are regular Borel measures on S and T respectively such that μ(S)= ν(T)= 1. The product measure μ × ν will be denoted by σ. Set U = G⊗C(T), V =C(S)⊗H and W = U + V. If G and H possess continuous proximity maps, then U and V are proximinal subspaces of C(S × T) when this linear space is equipped with the L1-norm, [4, Lemma 2]. That is, every z∈C(S × T) possesses at least one best approximation from U and from V. A metric selection Au:C(S × T →U is a mapping which associates each z ∈ C(S × T) with one of its best approximations in U.

This paper deals with the construction of exact and analytical-numerical solutions with a priori error bounds for systems of the type ut = Auxx, A1u(0, t) + B1ux (0, t) = 0, A2u (1, t) + B2ux (1, t) = 0, 0 < x < 1, t > 0, u(x, 0) = f(x), where A1, A2, B1 and B2 are matrices for which no simultaneous diagonalizable hypothesis is assumed, and A is a positive stable matrix. Given an admissible error ε and a bounded subdomain D, an approximate solution whose error with respect to an exact series solution is less than ε uniformly in D is constructed.

1. If I is an inflexion on a non-singular plane cubic curve, a variable line IPP′ establishes a (1, 1) correspondence between points P, P′ on the curve. This correspondence defines a perspective transformation of the whole plane, with I for pole, and the harmonic polar of I for axis, of perspective; for, when I is projected to infinity on the y-axis, and its harmonic polar taken for x-axis, the resulting equation

indicates a curve symmetrical with respect to the latter.

The object of the present note is to obtain expansions of the square of Whittaker's M-Functions in series of M-Functions and also other expansions involving M-Functions.

In a given circle let the arc AP subtend an angle 3a at the centre O, it is required to trisect the angle AOP, or the arc AP.

The three trisectors will be OQ1, OQ2, OQ3, where AOQ1 = a,

∴ Q1 Q2 Q3 form an equilateral triangle. (See Figs. 10 and 11.)

We proceed to solve the problem by drawing a conic through Q1, Q2, Q3, a nd we wish to find in what cases such a conic can be drawn, a conic cutting the circle in four points, three of which form an equilateral triangle.

Let

where the {lν} are numbers near to an arithmetic progression of common difference unity. Let

b being a constant. Write

so that k(z) is an integral function, and

is meromorphic with simple poles at {lν}.