Related papers: Mathematical Proof
Our introductory classes in statistics and data science use too much mathematics. The key causal effect which our students want our classes to have is to improve their future performance and opportunities. The more professional their…
Message passing is a key ingredient of concurrent programming. The purpose of this paper is to describe the equivalence between the proof theory, the categorical semantics, and term calculus of message passing. In order to achieve this we…
Artificial intelligence assisted mathematical proof has become a highly focused area nowadays. One key problem in this field is to generate formal mathematical proofs from natural language proofs. Due to historical reasons, the formal proof…
Moore introduced a method for graduate mathematics instruction that consisted primarily of individual student work on challenging proofs (Jones, 1977). Cohen (1982) described an adaptation with less explicit competition suitable for…
Semi-algebraic proof systems such as sum-of-squares (SoS) have attracted a lot of attention recently due to their relation to approximation algorithms: constant degree semi-algebraic proofs lead to conjecturally optimal polynomial-time…
This is a set of lecture notes used in a graduate topic class in applied mathematics called ``Quantum Algorithms for Scientific Computation'' at the Department of Mathematics, UC Berkeley during the fall semester of 2021. These lecture…
Fourier Transforms is a first in a series of monographs we present on harmonic analysis. Harmonic analysis is one of the most fascinating areas of research in mathematics. Its centrality in the development of many areas of mathematics such…
These lecture notes, suitable for a two-semester introductory course or self-study, offer an elementary and self-contained exposition of the basic tools and concepts that are encountered in practical computations in perturbative thermal…
In order to help students learn how to write mathematical proofs, we adapt the Coq proof assistant into an educational tool we call Waterproof. Like with other interactive theorem provers, students write out their proofs inside the software…
Here I share a few notes I used in various course lectures, talks, etc. Some may be just calculations that in the textbooks are more complicated, scattered, or less specific; others may be simple observations I found useful or curious.
We present a study of the implementation of the Electronic Preparatory Test for beginning undergraduates reading mathematics. The Test comprises two elements: diagnostic and self-learning. The diagnostic element identifies gaps in the…
These notes are a self-contained introduction to Galois theory, designed for the student who has done a first course in abstract algebra.
This paper presents mathematics as a general science of computation in a way different from the tradition. It is based on the radical philosophical standpoint according to which the content, meaning and justification of experience lies in…
This paper constructs a cirquent calculus system and proves its soundness and completeness with respect to the semantics of computability logic (see http://www.cis.upenn.edu/~giorgi/cl.html). The logical vocabulary of the system consists of…
We describe the development of a junior-senior level course for Physics majors designed to teach Mathematica skills in support of their undergraduate coursework, but also to introduce students to modern research level results. Standard…
We extend the theoretical framework of proof mining by establishing general logical metatheorems that allow for the extraction of the computational content of theorems with prima facie "non-computational" proofs from probability theory,…
We introduce real induction, a proof technique analogous to mathematical induction but applicable to statements indexed by an interval on the real line. More generally we give an inductive principle applicable in any Dedekind complete…
I formalize important theorems about classical propositional logic in the proof assistant Coq. The main theorems I prove are (1) the soundness and completeness of natural deduction calculus, (2) the equivalence between natural deduction…
We discuss a version of the fundamental theorem of calculus in several variables and some applications, of potential interest as a teaching material in undergraduate courses.
Proof Blocks is a software tool which enables students to write proofs by dragging and dropping prewritten proof lines into the correct order. These proofs can be graded completely automatically, enabling students to receive rapid feedback…