Related papers: An introduction to higher walks
Recently an extension to higher-order logic -- called DHOL -- was introduced, enriching the language with dependent types, and creating a powerful extensional type theory. In this paper we propose two ways how choice can be added to DHOL.…
The monadic second-order theory of trees allows quantification over elements and over arbitrary subsets. We classify the class of trees with respect to the question: does a tree T have a definable choice function (by a monadic formula with…
We develop a theory of higher order structures in compact abelian groups. In the frame of this theory we prove general inverse theorems and regularity lemmas for Gowers's uniformity norms. We put forward an algebraic interpretation of the…
In this paper we construct uniformly expanding random walks on smooth manifolds. In higher dimensions, our definition of uniform expansion measures the growth of subspaces rather than single vectors. Potrie showed that given any open set…
This survey is concerned with random walks on mapping class groups. We illustrate how the actions of mapping class groups on Teichm\"uller spaces or curve complexes reveal the nature of random walks, and vice versa. Our emphasis is on the…
We present a detailed circuit implementation of Szegedy's quantization of the Metropolis-Hastings walk. This quantum walk is usually defined with respect to an oracle. We find that a direct implementation of this oracle requires costly…
We introduce a variation on Barthe et al.'s higher-order logic in which formulas are interpreted as predicates over open rather than closed objects. This way, concepts which have an intrinsically functional nature, like continuity,…
A method for introducing the higher order terms in the potential expansion to study the continuous limits of the Toda hierarchy is proposed in this paper. The method ensures that the higher order terms are differential polynomials of the…
In recent years, computer simulations are playing a fundamental role in unveiling some of the most intriguing features of prime numbers. In this work, we define an algorithm for a deterministic walk through a two-dimensional grid that we…
We consider walks on the edges of the square lattice $\mathbb Z^2$ which obey \emph{two-step rules,} which allow (or forbid) steps in a given direction to be followed by steps in another direction. We classify these rules according to a…
A canonical formalism for higher-derivative theories is presented on the basis of Dirac's method for constrained systems. It is shown that this formalism shares a path integral expression with Ostrogradski's canonical formalism.
Exponential integrators based on contour integral representations lead to powerful numerical solvers for a variety of ODEs, PDEs, and other time-evolution equations. They are embarrassingly parallelizable and lead to global-in-time…
Random walks conditioned to stay positive are a prominent topic in fluctuation theory. One way to construct them is as a random walk conditioned to stay positive up to time $n$, and let $n$ tend to infinity. A second method is conditioning…
Recent research in molecular discovery has primarily been devoted to small, drug-like molecules, leaving many similarly important applications in material design without adequate technology. These applications often rely on more complex…
In the eighties, A. Connes and E. J. Woods made a connection between hyperfinite von Neumann algebras and Poisson boundaries of time dependent random walks. The present paper explains this connection and gives a detailed proof of two…
We study extensions of expressive decidable fragments of first-order logic with circumscription, in particular the two-variable fragment FO$^2$, its extension C$^2$ with counting quantifiers, and the guarded fragment GF. We prove that if…
Opetopes are algebraic descriptions of shapes corresponding to compositions in higher dimensions. As such, they offer an approach to higher-dimensional algebraic structures, and in particular, to the definition of weak $\omega$-categories,…
This work presents new asymptotic formulas for family of walks in Weyl chambers. The models studied here are defined by step sets which exhibit many symmetries and are restricted to the first orthant. The resulting formulas are very…
We study arithmetic properties of short uniform random walks in arbitrary dimensions, with a focus on explicit (hypergeometric) evaluations of the moment functions and probability densities in the case of up to five steps. Somewhat to our…
In Chapter 3 of his Notes on constructive mathematics, Martin-L{\"o}f describes recursively constructed ordinals. He gives a constructively acceptable version of Kleene's computable ordinals. In fact, the Turing definition of computable…