Harmonic and analytic functions have natural discrete analogues. Harmonic functions can be defined on every graph, while analytic functions (or, more precisely, holomorphic forms) can be defined on graphs embedded in orientable surfaces. Many important properties of the “true” harmonic and analytic functions can be carried over to the discrete setting.

ExPosition is a new comprehensive R package providing crisp graphics and implementing multivariate analysis methods based on the singular value decomposition (svd). The core techniques implemented in ExPosition are: principal components analysis, (metric) multidimensional scaling, correspondence analysis, and several of their recent extensions such as barycentric discriminant analyses (e.g., di...

We present in expository style the main ideas in SGGS, which stands for Semantically-Guided GoalSensitive theorem proving. SGGS uses sequences of constrained clauses to represent models, instance generation to go from a candidate model to the next, and resolution as well as other inferences to repair the model. SGGS is refutationally complete for first-order logic, DPLL-style model based, seman...

The aim of this short paper is to give a practical introduction to functional interpretation of proofs in arithmetic for computer scientists interested in synthesis. Towards this, we will define our own notion of functional interpretation which differs (sometimes only inessentially) from those used in the literature, but has the advantage (in the opinion of the author) of being very natural. No...

We provide an exposition of three lemmas that relate general properties of distributions over bit strings to the exclusive-or (xor) of values of certain bit locations. The first XOR-Lemma, commonly attributed to Umesh Vazirani (1986), relates the statistical distance of a distribution from the uniform distribution over bit strings to the maximum bias of the xor of certain bit positions. The sec...