Related papers: Induction, Coinduction, and Fixed Points: Intuitio…
The Lecture-Tutorials for Introductory Astronomy have been designed to help introductory astronomy instructors actively engage their students in developing their conceptual understandings and reasoning abilities across a wide range of…
Optimization problems, generalized equations, and the multitude of other variational problems invariably lead to the analysis of sets and set-valued mappings as well as their approximations. We review the central concept of set-convergence…
Metaphysical interpretations of set theory are either inconsistent or incoherent. The uses of sets in mathematics actually involve three distinct kinds of collections (surveyable, definite, and heuristic), which are governed by three…
Theory revision integrates inductive learning and background knowledge by combining training examples with a coarse domain theory to produce a more accurate theory. There are two challenges that theory revision and other theory-guided…
In these lectures we review the motivation, principles of and (circumstantial) evidence for the program of unification of the fundamental forces. In an appendix, we review the group theory pertinent to the program.
The main purpose of this paper is to find the fixed point in such cases where existing literature remain silent. In this paper we introduce partial completeness, a new type of contraction and many other definitions. Using this approach the…
Students in introductory calculus-based physics not only have difficulty understanding the fundamental physical concepts, they often have difficulty relating those concepts to the mathematics they have learned in math courses. This produces…
Over the last century, the principle of "induction on the continuum" has been studied by different authors in different formats. All of these different readings are equivalent to one of the three versions that we isolate in this paper. We…
The concept of a visible point of a convex set relative to a given point is introduced. A number of basic properties of such visible point sets is developed. In particular, it is shown that this concept is useful in the study of best…
We provide a simple proof for the union-closed sets conjecture, a long-standing open problem in set theory with immediate applications to graph theory, number theory, and order-theory.
We introduce INDUCTION, a benchmark for finite structure concept synthesis in first order logic. Given small finite relational worlds with extensionally labeled target predicates, models must output a single first order logical formula that…
The paper is an introduction to intuitionistic mathematics.
In these 4 lectures, I give a brief introduction to the principles of effective field theory and discuss their application via 3 examples: (i) the Standard Model as an effective theory; (ii) non-linear sigma models and the composite Higgs;…
In recent work we have shown how it is possible to define very precise type systems for object-oriented languages by abstractly compiling a program into a Horn formula f. Then type inference amounts to resolving a certain goal w.r.t. the…
We explain the notion of the {\em entropy} of a discrete random variable, and derive some of its basic properties. We then show through examples how entropy can be useful as a combinatorial enumeration tool. We end with a few open…
In these self-contained low prerequisite introductory notes we first present (in part 1) basic concepts of set theory and algebra without explicit category theory. We then present (in part 2) basic category theory involving a somewhat…
Building models of the world from observation, i.e., induction, is one of the major challenges in machine learning. In order to be useful, models need to maintain accuracy when used in novel situations, i.e., generalize. In addition, they…
In this paper we show that an intuitionistic theory for fixed points is conservative over the Heyting arithmetic with respect to a certain class of formulas. This extends partly the result of mine. The proof is inspired by the quick…
This chapter presents some of the links between automata theory and symbolic dynamics. The emphasis is on two particular points. The first one is the interplay between some particular classes of automata, such as local automata and results…
The process of cognition is analysed to adjust the set theory to physical description. Postulates and basic definitions are revised. The specific sets of predicates, called presets, corresponding to the physical objects identified by an…