Let be a finite group. Consider a pair of linear characters of subgroups of with and agreeing on . Naturally associated with is a finite monoid . Semigroup representation theory then yields a representation of . If is irreducible, we say that is a weight for . When the underlying field is the field of complex numbers, we obtain a formula for the character of in terms of and . We go on to construct weights for some familiar group representations.
Let be a principal bundle over a manifold of dimension , and let be its -dimensional Pontrjagin class. In this paper, we aim at answering the following question: Which representatives of the class can be realised as the Pontrjagin form of some connection on ?
Let be a vector lattice of real functions on a set with , and let be a linear positive functional on . Conditions are given which imply the representation , , for some bounded charge . As an application, for any bounded charge on a field , the dual of is shown to be isometrically isomorphic to a suitable space of bounded charges on . In addition, it is proved that, under one more assumption on , is the integral with respect to a -additive bounded charge.
Let be the polynomial algebra over a field of characteristic . We call a polynomial coordinate (or a generator) if for some polynomials . In this note, we give a simple proof of the following interesting fact: for any polynomial of the form where is a polynomial without constant and linear terms, and for any integer , there is a coordinate polynomial such that the polynomial has no monomials of degree . A similar result is valid for coordinate -tuples of polynomials, for any . This contrasts sharply with the situation in other algebraic systems.
On the other hand, we establish (in the two-variable case) a result related to a different kind of density. Namely, we show that given a non-coordinate two-variable polynomial, any sufficiently small perturbation of its non-zero coefficients gives another non-coordinate polynomial.
Let be a holomorphic function in the unit ball. Then is a Nevanlinna function if and only if there exist Smirnov functions , such that and has no zeros in the ball.
A corollary to our result will be that for any weight and any finitely homotopy dominated CW-complex , there exists a Hausdorff compactum with weight and which is universal for the property and weight . The condition means that for every closed subset of and every map , there exists a map which is an extension of . The universality means that for every compact Hausdorff space whose weight is and for which is true, there is an embedding of into .
We shall show, on the other hand, that there exists a CW-complex which is not finitely homotopy dominated but which has the property that for each weight , there exists a Hausdorff compactum which is universal for the property and weight .
Let be a commutative ring, let be an indeterminate, and let . There has been much recent work concerned with determining the Dedekind-Mertens number =min , especially on determining when = . In this note we introduce a universal Dedekind-Mertens number , which takes into account the fact that deg() + for any ring containing as a subring, and show that behaves more predictably than .
We study the behavior of Mañé's action potential associated to a convex superlinear Lagrangian, for bigger than the critical value . We obtain growth estimates for the action potential as a function of . We also prove that the action potential can be written as where is a smooth function and is the distance function associated to a Finsler metric.
The problem of existence and uniqueness of polynomial solutions of the Lamé differential equation where and are polynomials of degree and , is under discussion. We concentrate on the case when has only real zeros and, in contrast to a classical result of Heine and Stieltjes which concerns the case of positive coefficients in the partial fraction decomposition , we allow the presence of both positive and negative coefficients . The corresponding electrostatic interpretation of the zeros of the solution as points of equilibrium in an electrostatic field generated by charges at is given. As an application we prove that the zeros of the Gegenbauer-Laurent polynomials are the points of unique equilibrium in a field generated by two positive and two negative charges.
The following has been proven by Brauer and Nesbitt. Let be a finite group, and let be a prime. Assume is an irreducible complex character of such that the order of a -Sylow subgroup of divides the degree of . Then vanishes on all those elements of whose order is divisible by . The two only known proofs of this theorem use profound methods of representation theory, namely the theory of modular representations or Brauer's characterization of generalized characters. The purpose of this paper is to present a more elementary proof.
Under suitable assumptions on , we prove that generates a positive -semigroup on and, hence, many previous (linear or nonlinear) results are extended substantially.
For a knot in the -sphere, by using the linking form on the first homology group of the double branched cover of the -sphere, we investigate some numerical invariants, -genus , nonorientable -genus and -dimensional clasp number , defined from the four-dimensional viewpoint. T. Shibuya gave an inequality , and asked whether the equality holds or not. From our result in this paper, we find that the equality does not hold in general.
We give an example of a positive operator in a Krein space with the following properties: the nonzero spectrum of consists of isolated simple eigenvalues, the norms of the orthogonal spectral projections in the Krein space onto the eigenspaces of are uniformly bounded and the point is a singular critical point of
Let be the field obtained by adjoining to all -power roots of unity where is a prime number. We prove that the theory of is undecidable.