相关论文: On proof and progress in mathematics

Estimation algebras have been extensively studied in Euclidean space, where finite-dimensional estimation algebras form the foundation of the Kalman and Benes filters, and have contributed to the discovery of many other finite-dimensional…

While proof is a central component of postsecondary mathematical study, proof construction has historically posed significant difficulties for students who intend to earn mathematics degrees at the undergraduate level. This work is…

Throughout this book, we discuss some open problems in various branches of science, including mathematics, theoretical physics, astro-physics, geophysics etc. It is of our hope that some of the problems discussed in this book will find…

We explore the use of expert iteration in the context of language modeling applied to formal mathematics. We show that at same compute budget, expert iteration, by which we mean proof search interleaved with learning, dramatically…

Deeper insight leads to better practice. We show how the study of the foundations of quantum mechanics has led to new pictures of open systems and to a method of computation which is practical and can be used where others cannot. We…

In the paper it is shown that, even without a knowledge of the concrete form of the equations of mathematical physics and field theories, with the help of skew-symmetric differential forms one can see specific features of the equations of…

This is a draft of a chapter on mathematical logic and foundations for an upcoming handbook of computational proof assistants.

We study open book foliations on surfaces in 3-manifolds, and give applications to contact geometry of dimension 3. We prove a braid-theoretic formula of the self-linking number of transverse links, which reveals an unexpected link to the…

The formalisation of mathematics is starting to become routine, but the value of this technology to the work of mathematicians remains to be shown. There are few examples of using proof assistants to verify brand-new work. This paper…

We illustrate the concept of mathematical proof.

Some notes about quantum physics, an interpretation if one wishes, are put forward, insisting on `closely following the mathematics/formalism, the `nuts and bolts of what quantum physics says'. These, basically well-known, issues seem to…

In this paper, we study the Birkhoff sections in a 3-manifold foliated by invariant tori. We establish the necessary and sufficient conditions for various types of periodic orbits to serve as boundary orbits of a Birkhoff section. The…

Deformation theory of complex manifolds is a classical subject with recent new advances in the noncompact case using both algebraic and analytic methods. In this note, we recall some concepts of the existing theory and introduce new notions…

The work presents two examples of simple mathematical formulas which are natural nonlinear modifications (one being a generalization) of Gielis' formula. These formulas involve a comparable number of parameters and provide non-Platonic…

The capture calculus is an extension of System F<: that tracks free variables of terms in their type, allowing one to represent capabilities while limiting their scope. While previous calculi had mechanized soundness proofs -- notably…

A new elementary proof of the prime number theorem presented recently in the framework of a scale invariant extension of the ordinary analysis is re-examined and clarified further. Both the formalism and proof are presented in a much more…

The main aim of the paper is to develop the "Floer theory" associated to Calabi-Yau 3-folds, exending the analogy of Thomas' "holomorphic Casson invariant". The treatment in the body of the paper is largely formal, assuming appropriate…

We discuss some important issues arising from computational efforts in dynamical systems and fluid dynamics. Various individuals have misunderstood these issues since the onset of these problem areas; indeed, they have been routinely…

A classification theorem is given of projective threefolds that are covered by a two-dimensional family of lines, but not by a higher dimensional family.

We present a geometrical way of understanding the dynamics of wavefunctions in a free space, using the phase-space formulation of quantum mechanics. By visualizing the Wigner function, the spreading, shearing, the so-called "negative…