The shape of an electoral district may suggest whether it was drawn with political motivations, or gerrymandered. For this reason, quantifying the shape of districts, in particular their compactness, is a key task in politics and civil rights. A growing body of literature suggests and analyzes compactness measures mathematically, but little consideration has been given to how these scores shoul...

In the Theory of the Seiberg-Witten Equations, the configuration space is Cα = Aα × Γ (S + α ), where Aα is a space of u1-connections and Γ (S + α ) is the space of sections of the complex spinor bundle over X. Since the SWequation fits in a variational approach, invariant by the action of the Gauge Group Gα = Map(X,U1), which satisfies the Palais-Smale Condition, our aim is to obtain existence...

We study nonlinear eigenvalue problems of the type −div(a(x)∇u) = g(λ,x,u) in RN , where a(x) is a degenerate nonnegative weight. We establish the existence of solutions and we obtain information on qualitative properties as multiplicity and location of solutions. Our approach is based on the critical point theory in Sobolev weighted spaces combined with a Caffarelli-Kohn-Nirenberg-type inequal...

The Schrödinger operator −∆+V , of a compact Riemannian manifold M , has pure point spectrum. Suppose that V0 is a smooth reference potential. Various criteria are given which guarantee the compactness of all V satisfying spec(−∆+V ) = spec(−∆+V0). In particular, compactness is proved assuming an a priori bound on the Ws,2(M) norm of V , where s > n/2 − 2 and n = dimM . This improves earlier wo...

In this paper, we present a study of various concepts pertaining to compactness properties of a lattice. Special cases include such notions as countably compactness, almost countably compactness, and countably paracompactness. Some investigations in these matters have been done in [2, 7, 14]. We go beyond these results and introduce new subsets of I(L) and their lattices to investigate. This wa...

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We point out a connection between reflection principles and generic large cardinals. One principle of pure reflection is introduced that is as strong as generic supercompactness of ω2 by σ-closed forcing. This new concept implies CH and extends the reflection principles for stationary sets in a canonical way.

An improvement of the author’s result, proved in 1961, concerning necessary and sufficient conditions for the compactness of embedding operators is given. A discussion of the necessity of the compatibility of the norms of the Banach spaces X2 and X3, where X2 ⊂ X3, is given. The injectivity of the embedding operator J : X2 → X3 implies this compatibility.

Let (S3, c) be the standard 3-sphere, i.e., the 3-sphere equipped with the standard metric. Let K be a C2 positive function on S3. The Kazdan-Warner problem [l] is the problem of finding suitable conditions on K such that K is the scalar curvature for a metric g on S3 conformally equivalent to c. The metric g then reads g=u4c and u is a positive function on S3 satisfying the partial differentia...

We prove that for a uniformly convex Lagrangian system L on a compact manifold M , almost all energy levels contain a periodic orbit. We also prove that below Mañé’s critical value of the lift of the Lagrangian to the universal cover, cu(L), almost all energy levels have conjugate points. We prove that if the energy level [E = k] is of contact type and M 6= T then the free time action functiona...