In this paper, we investigate the existence of infinitely many solutions for a bi-nonlocal equation with sign-changing weight functions. We use some natural constraints and the Ljusternik-Schnirelman critical point theory on C1-manifolds, to prove our main results.

In this paper, completeness problem for a class of fractional rational basis functions with incommensurable differentiation orders is studied. Using the Müntz-Szász theory, it is established that the completeness problem for this class of the basis functions is equivalent to another completeness problem for a particular set of uncountably many basis functions. This equivalence allows one to dra...

We study the C∗-algebra generated by Teoplitz operators with symbols of vanishing (mean) oscillation on the Bergman space of the unit ball. We show that the index calculation for Fredholm operators in this C∗-algebra can be easily and completely reduced to the classic case of Toeplitz operators with symbols that are continuous on the closed unit ball. Moreover, in addition to a number of other ...

It is a well-known fact that every Riemann surface with negative Euler characteristic admits a hyperbolic metric. But this metric is by no means unique – indeed, there are uncountably many such metrics. In this paper, we study the space of all such hyperbolic structures on a Riemann surface, called the Teichmüller space of the surface. We will show that it is a complete metric space, and that i...

This project will study the concept of chaos from a mathematical viewpoint. Qualities which characterise chaos (e.g., sensitive dependence on initial conditions, presence of periodic orbits of all periods, uncountably many aperiodic orbits, and dense orbits) will be studied. Methods for analytically proving that a system is chaotic (presence of a “horseshoe,” intersection of stable and unstable...

During the Notre Dame workshop on Vaught’s Conjecture, Hjorth and Kechris asked which Borel equivalence relations can arise as the isomorphism relation for models of a first order theory. In particular, they asked if the isomorphism relation can be essentially countable but not tame. We show this is not possible if the theory has uncountably many types. I am grateful to the logic group at Notre...

Croke and Kleiner [5] gave a construction for a family Xa : 0 < ac p 2 of CAT(0) spaces that each admit a geometric action by the same group G. They showed that qXa T qXp=2 for all a < p 2 . We show that in fact qXa T qXb for all a0 b, so that G is a CAT(0) group with uncountably many non-homeomorphic boundaries.

Let K be a finite simplicial complex and f:K—>K be a "skeletal" map. A digraph D is defined whose vertices correspond to the Simplexes of K and whose arcs give information about the behavior of / on the Simplexes. For every walk in D there exists a point of K whose iterates under / mimic the walk. Periodic walks are mimicked by a periodic point. Digraphs with uncountably many infinite walks are...