We survey the techniques used in our recent resolution of the Kadison– Singer problem and proof of existence of Ramanujan Graphs of every degree: mixed characteristic polynomials and the method of interlacing families of polynomials. To demonstrate the method of interlacing families of polynomials, we give a simple proof of Bourgain and Tzafriri’s restricted invertibility principle in the isotr...

The amount of information produced in the last decades has grown notably. Much of this information only exists in the form of electronic documents and it has often to be stored for long periods. Therefore, digital archives are increasingly needed. However, for the documents to remain trustworthy while they are archived, they need to be protected by the archivists. Important protection goals tha...

In this paper the problem of optimal derivative design, profit maximization and risk minimization under adverse selection when multiple agencies compete for the business of a continuum of heterogenous agents is studied. In contrast with the principal–agent models that are extended within, here the presence of ties in the agents’ best–response correspondences yields discontinuous payoff function...

We consider two-dimensional water waves of infinite depth, periodic in the horizontal direction. It has been proven by Wu (in the slightly different nonperiodic setting) that solutions to this initial value problem exist in the absence of surface tension. Recently Ambrose has proven that solutions exist when surface tension is taken into account. In this paper, we provide a shorter, more elemen...

The semantics of name-passing calculi is often defined employing coalgebraic models over presheaf categories. This elegant theory lacks finiteness properties, hence it is not apt to implementation. Coalgebras over named sets, called history-dependent automata, are better suited for the purpose due to locality of names. A theory of behavioural functors for named sets is still lacking: the semant...

We study the stability/instability of the subsonic travelling waves of the Nonlinear Schrödinger Equation in dimension one. Our aim is to propose several methods for showing instability (use of the GrillakisShatah-Strauss theory, proof of existence of an unstable eigenvalue via an Evans function) or stability. For the later, we show how to construct in a systematic way a Liapounov functional fo...

Let G be a string graph (an intersection graph of continuous arcs in the plane) with m edges. Fox and Pach proved that G has a separator consisting of O(m √ logm) vertices, and they conjectured that the bound of O( √ m) actually holds. We obtain separators with O( √ m logm) vertices. Let G = (V,E) be a graph with n vertices. A separator in G is a set S ⊆ V of vertices such that there is a parti...

We consider the Clay Institute Prize Problem asking for a mathematical analytical proof of existence, smoothness and uniqueness (or a converse) of solutions to the incompressible Navier-Stokes equations. We argue that the present formulation of the Prize Problem asking for a strong solution is not reasonable in the case of turbulent flow always occuring for higher Reynolds numbers, and we propo...

In his book on convex polytopes [2] A.D. Aleksandrov raised a general question of finding variational formulations and solutions to geometric problems of existence of convex polytopes in Rn+1, n ≥ 2, with prescribed geometric data. Examples of such problems for closed convex polytopes for which variational solutions are known are the celebrated Minkowski problem [2] and the Gauss curvature [18]...

data envelopment analysis (dea) is a nonparametric technique that includes models to evaluate the relative efficiency of decision making units (dmus). it has the ability to separate efficient units and inefficient units. one of the applications of this mathematical technique is evaluating performance of supply chain. according to such as the inefficiency factors of one dmu, is existence congest...