Generalized solutions of nonlocal elliptic problems

An elliptic equation of order $2m$ with general nonlocal boundary-value conditions, in a plane bounded domain $G$ with piecewise smooth boundary, is considered. Generalized solutions belonging to the Sobolev space $W_2^m(G)$ are studied. The Fredholm property of the unbounded operator corresponding to the elliptic equation, acting on $L_2(G)$, and defined for functions from the space $W_2^m(G)$ that satisfy homogeneous nonlocal conditions is proved.Comment: 18 pages, 2 figure

Distance Oracles for Time-Dependent Networks

We present the first approximate distance oracle for sparse directed networks with time-dependent arc-travel-times determined by continuous, piecewise linear, positive functions possessing the FIFO property. Our approach precomputes $(1+\epsilon)-$approximate distance summaries from selected landmark vertices to all other vertices in the network. Our oracle uses subquadratic space and time preprocessing, and provides two sublinear-time query algorithms that deliver constant and $(1+\sigma)-$approximate shortest-travel-times, respectively, for arbitrary origin-destination pairs in the network, for any constant $\sigma > \epsilon$. Our oracle is based only on the sparsity of the network, along with two quite natural assumptions about travel-time functions which allow the smooth transition towards asymmetric and time-dependent distance metrics.Comment: A preliminary version appeared as Technical Report ECOMPASS-TR-025 of EU funded research project eCOMPASS (http://www.ecompass-project.eu/). An extended abstract also appeared in the 41st International Colloquium on Automata, Languages, and Programming (ICALP 2014, track-A

On systems with finite ergodic degree

In this paper we study the ergodic theory of a class of symbolic dynamical systems (\O, T, \mu) where T:{\O}\to \O the left shift transformation on \O=\prod_0^\infty\{0,1\} and $\mu$ is a \s-finite $T$-invariant measure having the property that one can find a real number $d$ so that $\mu(\tau^d)=\infty$ but $\mu(\tau^{d-\epsilon})0$, where $\tau$ is the first passage time function in the reference state 1. In particular we shall consider invariant measures $\mu$ arising from a potential $V$ which is uniformly continuous but not of summable variation. If $d>0$ then $\mu$ can be normalized to give the unique non-atomic equilibrium probability measure of $V$ for which we compute the (asymptotically) exact mixing rate, of order $n^{-d}$. We also establish the weak-Bernoulli property and a polynomial cluster property (decay of correlations) for observables of polynomial variation. If instead $d\leq 0$ then $\mu$ is an infinite measure with scaling rate of order $n^d$. Moreover, the analytic properties of the weighted dynamical zeta function and those of the Fourier transform of correlation functions are shown to be related to one another via the spectral properties of an operator-valued power series which naturally arises from a standard inducing procedure. A detailed control of the singular behaviour of these functions in the vicinity of their non-polar singularity at $z=1$ is achieved through an approximation scheme which uses generating functions of a suitable renewal process. In the perspective of differentiable dynamics, these are statements about the unique absolutely continuous invariant measure of a class of piecewise smooth interval maps with an indifferent fixed point.Comment: 42 page

Standard finite elements for the numerical resolution of the elliptic Monge-Ampere equation: Aleksandrov solutions

We prove a convergence result for a natural discretization of the Dirichlet problem of the elliptic Monge-Ampere equation using finite dimensional spaces of piecewise polynomial C0 or C1 functions. Standard discretizations of the type considered in this paper have been previous analyzed in the case the equation has a smooth solution and numerous numerical evidence of convergence were given in the case of non smooth solutions. Our convergence result is valid for non smooth solutions, is given in the setting of Aleksandrov solutions, and consists in discretizing the equation in a subdomain with the boundary data used as an approximation of the solution in the remaining part of the domain. Our result gives a theoretical validation for the use of a non monotone finite element method for the Monge-Amp\`ere equation

Multilevel refinable triangular PSP-splines (Tri-PSPS)

A multi-level spline technique known as partial shape preserving splines (PSPS) (Li and Tian, 2011) has recently been developed for the design of piecewise polynomial freeform geometric surfaces, where the basis functions of the PSPS can be directly built from an arbitrary set of polygons that partitions a giving parametric domain. This paper addresses a special type of PSPS, the triangular PSPS (Tri-PSPS), where all spline basis functions are constructed from a set of triangles. Compared with other triangular spline techniques, Tri-PSPS have several distinctive features. Firstly, for each given triangle, the corresponding spline basis function for any required degree of smoothness can be expressed in closed-form and directly written out in full explicitly as piecewise bivariate polynomials. Secondly, Tri-PSPS are an additive triangular spline technique, where the spline function built from a given triangle can be replaced with a set of refined spline functions built on a set of smaller triangles that partition the initial given triangle. In addition, Tri-PSPS are a multilevel spline technique, Tri-PSPS surfaces can be designed to have a continuously varying levels of detail, achieved simply by specifying a proper value for the smoothing parameter introduced in the spline functions. In terms of practical implementation, Tri-PSPS are a parallel computing friendly spline scheme, which can be easily implemented on modern programmable GPUs or on high performance computer clusters, since each of the basis functions of Tri-PSPS can be directly computed independent of each other in parallel