相关论文: Lectures on differentials, generalized differentia…

Let $R$ be a commutative noetherian ring. The $n$-semidualizing modules of $R$ are generalizations of its semidualizing modules. We will prove some basic properties of $n$-semidualizing modules. Our main result and example shows that the…

This monograph discusses dualities in physics: what dualities are, their main examples--from quantum mechanics and electrodynamics to statistical mechanics, quantum field theory and string theory--and the philosophical questions they raise.…

The study of Complex Systems is considered by many to be a new scientific field, and is distinguished by being a discipline that has applications within many separate areas of scientific study. The study of Neural Networks, Traffic…

We survey some important properties of fields of generalized series and of exponential-logarithmic series, with particular emphasis on their possible differential structure, based on a joint work of the author with S. Kuhlmann [KM12b,KM11].

Lecture notes from the course given by Professor Sara A. Solla at the Les Houches summer school on "Statistical physics of Machine Learning". The notes discuss neural information processing through the lens of Statistical Physics. Contents…

An algebraic technique adapted to the problems of the fundamental theoretical physics is presented. The exposition is an elaboration and an extension of the methods proposed in previous works by the aut

The purpose of this manuscript is to review my recent activity on three main research topics. The first concerns the nature of low temperature amorphous solids and their relation with the spin glass transition in a magnetic field. This is…

Differentiable physics provides a new approach for modeling and understanding the physical systems by pairing the new technology of differentiable programming with classical numerical methods for physical simulation. We survey the rapidly…

We develop the theory of mixed finite elements in terms of special inverse systems of complexes of differential forms, defined over cellular complexes. Inclusion of cells corresponds to pullback of forms. The theory covers for instance…

We present a unified formulation for higher gauge theory using generalized forms, encompassing higher connections, curvatures, and gauge transformations. We begin by developing the calculus of generalized forms valued in higher algebras and…

In this note we describe how some objects from generalized geometry appear in the qualitative analysis and numerical simulation of mechanical systems. In particular we discuss double vector bundles and Dirac structures. It turns out that…

Simple physical models of a measuring rod and of a clock are used to demonstrate the contraction of objects and clock retardation in special relativity. It is argued that the models could help in promoting student understanding of special…

We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but…

These notes are the outgrowth of a series of lectures given at MSRI in January 1995 at the beginning of the special semester in complex dynamics and hyperbolic geometry. In these notes, the primary aim is to motivate the study of complex…

We discuss and provide nontrivial evidence for a large class of dualities in three-dimensional field theories with different gauge groups. We match the full partition functions of the dual phases for any value of the couplings to underpin…

These lecture notes for the IAS/Park City Graduate Summer School in Geometric Combinatorics (July 2004) provide an overview of root systems, generalized associahedra, and the combinatorics of clusters. Lectures 1-2 cover classical material:…

Formulations of some Grassmann-valued systems of ordinary differential equations invariant under (infinitesimal) supersymmetry transformations, including $N$-superspace extended types, are reviewed and discussed, with use of superfields.…

We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…

$N$-complexes have been argued recently to be algebraic structures relevant to the description of higher spin gauge fields. $N$-complexes involve a linear operator $d$ that fulfills $d^N = 0$ and that defines a generalized cohomology. Some…

These are notes for a very rapid introduction to the basics of exterior differential systems and their connection with what is now known as Lie theory, together with some typical and not-so-typical applications to illustrate their use.