Nonnegative Degree Research Articles

Algorithm AS 231: The Distribution of a Noncentral Χ 2 Variable with Nonnegative Degrees of Freedom

Algorithm AS 231: The Distribution of a Noncentral Χ 2 Variable with Nonnegative Degrees of Freedom

Minimal R operation in the 1/N expansion of ? models

The renormalization structure of the 1/N expansion in sigma models is investigated. It is shown that the theory is renormalized by an R operation in which subtractions are made only for some of the subgraphs with non-negative degree (of divergence). In contrast to the standard renormalization, such a renormalization manifestly preserves the nonlinear symmetry of the sigma model.

Homogeneous Generalized Temperatures

For the generalized heat equation $u_{xx} + ({{2v} / x})u_x = u_t ,\, v > 0$, there are two basic kinds of homogeneous solutions of nonnegative integral degree n and two of negative degree $ - n - 1 - 2v,0 < \nu < \frac{1}{2}$. Criteria are established for the expansion of generalized homogeneous temperatures in series of these basic solutions. The results parallel, in part, those of D. V. Widder for the classical heat equation.

On the Künneth theorem

Introduction. The Kiinneth theorem asserts that the homology H(X0 Y) of the product of two free abelian positively graded chain complexes is isomorphic to Tor(HX, HY) = HX 0HY+Tor1(HX, HY). This isomorphism however gives no insight into the effect of chain maps; that is to say, H(f 0g) cannot be computed in terms of Hf and Hg. Alternatively the difficulty may be expressed by saying that the isomorphism in question is not canonical. It appears that in order to produce a functorial Kiinneth theorem some additional invariant of a chain complex is needed, and indeed Bockstein [I I and Palermo [4] have produced such theorems using as the starting point the homology or cohomology spectra. In this paper we adopt the alternate course of introducing as an invariant the affine space of orientations (?2) of a chain complex. Having defined the notion of skew-isomorph (?3) of a group with respect to such an affine space we then see that the homology of the product, as well as cohomology groups, groups of homotopy classes and so forth, are just skew-isomorphs of those produced by the usual functors of homological algebra. In ?6 we use the same device to sharpen the notion of derived functor, and with these sharpened functors we are able to compute the space of orientations of a product complex (?7). There is an obvious application, which however we omit, to the triple (etc.) product studied by MacLane [3]. The external product in cohomology, and also the cup-product, have straightforward expressions in the language here adduced. These are given in ?8. 1. Unfolding chain complexes. By chain complexes we shall mean free abelian graded groups with derivations of degree -1. We shall further require that these groups be graded by nonnegative degrees, i.e., that their homogeneous components of negative degree be 0. Some of our results remain valid without the latter restriction; the reader may easily supply the generalizations for himself. We shall also have occasion to consider bicomplexes, by which we understand free abelian bigraded groups A = E,,oo A,q with derivations of degree (0, 1). We shall call p the principal, q the resolvent degree of Apq. The notions of chain map and chain homotopy of chain complexes are the usual ones; we extend them by analogy to bicomplexes. Observe that in the latter case a chain homotopy of degree (p, q) connects maps of degree (p, q1).

A simple proof of the Künneth theorem

The following jfroof of a much-proved theorem is offered in the belief that it is simpler than the many existing ones (cf. for example [1; 2, ?12]). It has also the virtue of a form suitable to generalization to abstract categories, which will be undertaken in another place [4]. By a chain complex we shall mean a graded abelian group with a derivation of degree 1. A chain complex X is positive if Xi=0 for i<0. If X and Y are positive chain complexes then so is X0 Y. If at least one of X, Y is a free abelian group then the Kiinneth theorem relates the homology of X 0 Y to the homology of X and Y. Recall that for abelian groups A, B graded by non-negative degrees (A OB)n= Zi+j=n Aj0Bj. We write also