Related papers: Fun with "Analysis I": basic theorems in calculus …
Encodings or the proof of their absence are the main way to compare process calculi. To analyse the quality of encodings and to rule out trivial or meaningless encodings, they are augmented with quality criteria. There exists a bunch of…
In intuitionistic mathematics, the Brouwer Continuity Theorem states that all total real functions are (uniformly) continuous on the unit interval. We study this theorem and related principles from the point of view of Reverse Mathematics…
Context: A growing amount of code is written to explore and analyze data, often by data analysts who do not have a traditional background in programming, for example by journalists. Inquiry: The way such data anlysts write code is different…
In this workshop, we present a compact but rigorous introduction to the basic language of nonlinear programming, variational inequalities, and complementarity systems. The goal is twofold. First, we explain the mathematical logic of…
The evaluation of a matrix exponential function is a classic problem of computational linear algebra. Many different methods have been employed for its numerical evaluation [Moler C and van Loan C 1978 SIAM Review 20 4], none of which…
We establish essentially optimal bounds on the complexity of initial-value problems in the randomized and quantum settings. For this purpose we define a sequence of new algorithms whose error/cost properties improve from step to step. These…
A new viewpoint of the G\"odel's incompleteness theorem be given in this article which reveals the deep relationship between the logic and computation. Upon the results of these studies, an algorithm be given which shows how to search a…
We introduce a set of eight universal Rules of Inference by which computer programs with known properties (axioms) are transformed into new programs with known properties (theorems). Axioms are presented to formalize a segment of Number…
Recurrence equations have played a central role in static cost analysis, where they can be viewed as abstractions of programs and used to infer resource usage information without actually running the programs with concrete data. Such…
One of the elegant achievements in the history of proof theory is the characterization of the provably total recursive functions of an arithmetical theory by its proof-theoretic ordinal as a way to measure the time complexity of the…
The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. The application of this undervalued formalism has been hampered by the absence of well-behaved proof systems on the one hand, and…
Retrograde analysis reads programs from the end to the beginning: treat statements as constraints on prior states, propagate sets of states backward, and compare the reachable inputs with the intended specification. This tutorial condenses…
We demonstrate that techniques of Weihrauch complexity can be used to get easy and elegant proofs of known and new results on initial value problems. Our main result is that solving continuous initial value problems is Weihrauch equivalent…
Integration at a point is a new kind of integration derived from integration over an interval in infinitesimal and infinity domains which are spaces larger than the reals. Consider a continuous monotonic divergent function that is…
We give an algorithm for testing the extremality of minimal valid functions for Gomory and Johnson's infinite group problem that are piecewise linear (possibly discontinuous) with rational breakpoints. This is the first set of necessary and…
Interval calculus is a relatively new branch of mathematics. Initially understood as a set of tools to assess the quality of numerical calculations (rigorous control of rounding errors), it became a discipline in its own rights today.…
Quantitative logic reasons about the degree to which formulas are satisfied. This paper studies the fundamental reasoning principles of higher-order quantitative logic and their application to reasoning about probabilistic programs and…
Usually the first course in mathematics is calculus. Its a core course in the curriculum of the Business, Engineering and the Sciences. However many students face difficulties to learn calculus. These difficulties are often caused by the…
We present an iterative inverse reinforcement learning algorithm to infer optimal cost functions in continuous spaces. Based on a popular maximum entropy criteria, our approach iteratively finds a weight improvement step and proposes a…
This is the first of a series of papers in which we study deep computations (ultracomputations) and deep iterates, formalizing the ideas of "asymptotic limit" of computations and compositional iterates, respectively. In this first paper of…