To the Sobolev embedding theorem for the limiting exponent

We establish embeddings of the Sobolev space W _p ^s and the space B _pq ^s (with the limiting exponent) in certain spaces of locally integrable functions of zero smoothness. This refines the embedding of the Sobolev space in the Lorentz and Lorentz-Zygmund spaces. Similar problems are considered for the case of irregular domains and for the potential space.     

An embedding theorem for rings

Kaplansky [2] proved that if P is a projective module, then every f.g. submodule of P is contained in a finitely generated direct summand iff P is the direct sum of f.g. projectives. We show that in order that all injectives have the dual property to the above statement,, for ach pair of simples (S₁, S₂), Hom(?₁,?₂) must be an Artinian and Noet. i.ian End(?₁) module where _i. is the injective hull of ?_i. This leads to a study of universally cotorsionless modules.     

An embedding theorem

We show in this paper that given any reduced, cancellative, torsion-free, atomic monoid, it is possible to construct a possibly non-atomic domain with atomic factorization structure isomorphic to the given monoid. This is significant, since atomic monoids are known to have more freedom in the factorization properties they may possess than atomic domains. This construction is motivated by the paper written by Coykendall and Zafrullah (2004) [5], in which a non-atomic domain was constructed with factorization structure isomorphic to a singly-generated monoid.     

An embedding theorem for Eichler orders

Let B be a quaternion algebra over number field K. Assume that B satisfies the Eichler condition (i.e., there is at least one archimedean place which is unramified in B). Let Ω be an order in a quadratic extension L of K. The Eichler orders of B which admit an embedding of Ω are determined. This is a generalization of Chinburg and Friedman's embedding theorem for maximal orders.     

Another note on the embedding of the Sobolev space for the limiting exponent

The embedding of the Sobolev spaces W _p ^s (? ⁿ ) in a Lizorkin-type space of locally summable functions of zero smoothness is established. This result is extended to the case of the embedding of Sobolev spaces on nonregular domains of n-dimensional Euclidean space. The formulation of the theorem depends on the geometric parameters of the domain of the functions.     

An embedding theorem for integral equations

In this paper, an embedding theorem is established for a system of nonlinear integral equations of the Volterra type. The main result is basic in the development of a maximum principle for an optimal control problem in which the state variables are determined as solutions to integral equations.     

An embedding theorem for Lie algebras

In this paper we give a sufficient condition for a restricted enveloping algebra to be quasi-elementary. We also prove that every finite dimensional -nilpotent Lie algebra can be embedded in a finite dimensional -nilpotent quasi-elementary Lie algebra.

     

An embedding theorem for differential equations

The present paper establishes a general embedding theorem for differential equations, which is useful in establishing the maximum principle for optimal control problems.The preparation of this paper was sponsored by the Office of Naval Research and the US Army Research Office.     

An embedding theorem for polycyclic groups

It is shown that every polycyclic group embeds in a two-generated polycyclic group.Translated from Matematicheskie Zametki, Vol. 14, No. 5, pp. 741–744, November, 1973.In conclusion the author thanks V. N. Remeslennikova for posing the problem and for helpful discussions.     

An embedding theorem for Lasnev spaces

A Lasnev space is a space which is the image of a metric space under a closed, continuous function. We show that every Lasnev space can be densely embedded in a Lasnev space which has a dense, metrizable, complete, G_δ subspace A. With this, we extend a theorem of K.R. Van Doren to say that for every Lasnev space X there is a Lasnev space Y each open set of which has subspace homeomorphic to X.By considering additional conditions on A, we find a necessary and sufficient conditional that an arbitrary space be a Lasnev space.     

On the Sobolev embedding theorem for variable exponent spaces in the critical range

In this paper we study the Sobolev embedding theorem for variable exponent spaces with critical exponents. We find conditions on the best constant in order to guaranty the existence of extremals. The proof is based on a suitable refinement of the estimates in the Concentration–Compactness Theorem for variable exponents and an adaptation of a convexity argument due to P.L. Lions, F. Pacella and M. Tricarico.     

An embedding theorem for affine resolvable designs

In this note, we consider a question raised by S.S. Shrikhande [6] concerning the embedding of an ARD with deficiency d into ARBIBD. We show that such an embedding is always possible if we assume that and in addition that there is one good class.     

An embedding theorem for finite planar spaces

LetS be a finite planar space such that any two distinct planes intersect in a line. We show thatk≤n ²+1 for anyk-cap ofS, wheren is the order ofS. Moreover, if a (n ²+1)-cap exists inS, a necessary and sufficient condition is provided forS to be embeddable in a 3-dimensional projective space. Work supported by the National Research Project “Strutture geometriche, Combinatoria e loro applicazioni” of the italian M.U.R.S.T.