The object of this note is to deduce from first principles the probability differential of Pearson's x2, and to account for “degrees of freedom.”

For a class of elliptic equations in the entire space and with nonlinear terms having a possibly uncountable (but of Lebesgue measure zero) set of discontinuities, the existence of strong solutions is established. Two simple applications are then developed. The approach taken is strictly based on set-valued analysis and fixed-points arguments.

Some results are presented relating to questions raised in a recent paper by Anderson, Hayman and Pommerenke regarding the size of the set of boundary points of the unit disc at which a univalent function has a prescribed radial growth.

AMS 2000 Mathematics subject classification: Primary 30C55

A formally self-adjoint differential operator L is said to be of limit circle type at infinity if its highest order coefficient is zero-free and all solutions x of L(x) = 0 are square-integrable on [a, ∞). (We will drop reference to “at infinity” in what follows.)

For the second-order case

Dunford and Schwartz (3) p. 1409 prove that given

then L is of limit circle type if and only if

It is assumed that the sines of the sides are proportional to the sines of the opposite angles.

ACB (Fig. 9) is a spherical Δ, with C a right angle.

Produce AC, AB to D and E so that

Draw the great ⊙ DEF and produce CB to meet it in F.

Then F is the pole of AD and A the pole of DF.

Recently J. M. Osborn has investigated the structure of a simple commutative non-associative algebra with unity element satisfying a polynomial identity, (4), (5) and (6). From his work it seems likely that if such an algebra is of degree three or more it is necessarily power-associative. In (4) he establishes a hierarchy of identities with the property that each identity is satisfied by an algebra satisfying no preceding identity. Following (5), (6),the next identity to consider is

where a, c and h are elements of the ground field.

If, in any triangle ABC, the angle A > the angle B, and lines AD and BE be drawn so that the angles BAD and DAC are, respectively, greater than the angles ABE and EBC, then is BE > AD. (Fig. 6.)

Proof: Draw AD′ and AD″ making angles BAD′ and D″AC, respectively, equal to the angles ABE and EBC.

A ℤG-module A is said to have an f-decomposition if in which A∫ is a ℤG-submodule of A such that each irreducible ℤG-factor of A∫ as an abelian group is finite and the ℤG-submodule A∫ has no finite irreducible ℤG-factors. In this paper, we prove that: if G is a hyperfinite group then any artinian ℤG-module A has an f-decomposition, which gives a positive answer to the question raised by D.I. Zaitzev in 1986.

The following problem is a slight generalisation of one posed partly solved by H. Nagler.1 We shall use A* for the conjugate transpose of a matrix A. A projection is an idempotent matrix, its latent roots consist of units and zeros.

Let k be a field of characteristic zero and let $\Omega_{A/k}$ be the universally finite differential module of a k-algebra A, which is the local ring of a closed point of an algebraic or algebroid curve over k. A notorious open problem, known as Berger’s Conjecture, predicts that A must be regular if $\Omega_{A/k}$ is torsion-free. In this paper, assuming the hypotheses of the conjecture and observing that the module ${\rm Hom}_A(\Omega_{A/k}, \Omega_{A/k})$ is then isomorphic to an ideal of A, say $\mathfrak{h}$, we show that A is regular whenever the ring $A/a\mathfrak{h}$ is Gorenstein for some parameter a (and conversely). In addition, we provide various characterizations for the regularity of A in the context of the conjecture.

In this paper, we show how to give a geometric interpretation of the modular correspondence T3 on the modular curve X(11) of level 11 using projective geometry. We use Klein's theorem that X(11) is isomorphic to the nodal curve of the Hessian of the cubic threefold Λ defined by V2W + W2X + X2Y + Y2Z + Z2V = 0 in P4(C) and geometry which we learned from a paper of W. L. Edge. We show that the correspondence T3 is essentially the correspondence which associates to a point p of the curve X(11) the four points where the singular locus of the polar quadric of p with respect to Λ meets X(11). Our control of the geometry is good enough to enable us to compute the eigenvalues of T3 acting on the cohomology of X(11). This is the first example of an explicit geometric description of a modular correspondence without valence. The results of this article will be used in subsequent articles to associate two new abelian varieties to a cubic threefold, to desingularize the Hessian of a cubic threefold and to study self-conjugate polygons formed by the quadrisecants of the nodal curve of the Hessian.

Kilmister (1) has discussed the existence of linear integrals of a dynamical system specified by generalized coordinates qα(α = 1, 2, …, n) and a Lagrangian

repeated indices being summed from 1 to n. He derived covariant conditions for the existence of such an integral, conditions which do not imply the existence of an ignorable coordinate. Boyer (2) discussed the conditions and found the most general Lagrangian satisfying the conditions for the case of two degrees of freedom (n = 2).

A semigroup S is said to be normal if aS = Sa for each a in S. Thus the class of normal semigroups includes the class of groups and the class of Abelian semigroups. Given a compact semigroup S we write P(S) for the convolution semigroup of probability regular Borel measures on S. In (3), Theorem 7, Lin asserts that a compact semigroup S is normal if and only if P(S) is normal. We show in this paper that Lin's result is false. In fact, if S is the union of subsemigroups each of which has an identity element, we show that P(S) is normal if and only if S is Abelian. Thus any compact non-Abelian group contradicts Lin's result. What Lin's argument does establish is that if P(S) is normal then S is normal, and if S is normal then μP(S) = P(S)μ for each point mass measure μ.

In a perfect sequence space α, on which a norm is defined, we can consider three types of convergence, namely projective convergence, strong projective convergence and distance convergence. In the space σ∞, when distance is defined in the usual way, the last two types of convergence coincide and are distinct from projective convergence ((2), p. 316). In the space σ1 all three types of convergence coincide ((2), p. 316). It will be shown in this paper that, if distance convergence and projective convergence coincide, then all three types of convergence coincide. It will not be assumed that the limit under one convergence is also the limit under the other convergence.

In a recent paper, motivated by the study of central extensions of associative algebras, George Janelidze introduces the notion of weakly action representable category. In this paper, we show that the category of Leibniz algebras is weakly action representable and we characterize the class of acting morphisms. Moreover, we study the representability of actions of the category of Poisson algebras and we prove that the subvariety of commutative Poisson algebras is not weakly action representable.

In a previous paper [1] we constructed a free resolution for a class of groups which include Fuchsian groups with compact orbit spaces [2, 3], infinite polyhedral groups, plane crystallographic groups p2, p3, p4 and p6 and Dyck's groups [4], and used this resolution for computation of the integral homology and cohomology of these groups. Lyndon [5] determined the cohomology of groups with a single defining relation. The plane crystallographic groups p1 and pg and Artin's braid group B3 are among these groups. In this paper we have constructed free resolutions for certain classes of groups–resolutions which are particularly suitable for direct computation of the homology and the cohomology of these groups for any coefficient module. These classes of groups include the plane crystallographic groups pm, cm and pgg. We have computed the integral homology and cohomology from each of the free resolutions obtained.

In a recent paper Einstein has proposed a new type of geometry in which it is possible to give an arbitrary vector in the field a displacement which conserves parallelism at a distance. It may be briefly stated as follows:

Let the metric ofthe space, supposed to be of n dimensions, be given by

At each point of the space we consider a set of n unit vectors, denoted by the n2 quantities phq, which form an orthogonal ennuple such that

A C*-algebra A is said to be approximately finite dimensional (AF) if it is the inductive limit of a sequence of finite dimensional C*-algebras(see [2], [5]). It is said to be nuclear if, for each C*-algebra B, there is a unique C*-norm on the *-algebraic tensor product A ⊗B [11]. Since finite dimensional C*-algebras are nuclear, and inductive limits of nuclear C*-algebras are nuclear [16];,every AF C*-algebra is nuclear. The family of nuclear C*-algebras is a large and well-behaved class (see [12]). The AF C*-algebras for a particularly tractable sub-class which has been completely classified in terms of the invariant K0 [7], [5].

Let G be a group having presentation

a braid relator and a commutator respectively, and define the graph of such a group as having vertices labelled x1,…, xn + 1 and such that xi and xj are joined if and only if rij = (xi, xj).

One of the well known properties of the number 7 is that when is reduced to a decimal, the periods of two digits are obtained to infinity by successive doubling. It is interesting to find for what other numbers this property is true.

Let n be any number, and r the base of notation, then