Related papers: Textbook examples of recursion
This chapter opens with a review of classic tools for regression, a subset of machine learning that seeks to find relationships between variables. With the advent of scientific machine learning this field has moved from a purely data-driven…
In this thesis, we investigate the computational content and the logical strength of Ramsey's theorem and its consequences. For this, we use the frameworks of reverse mathematics and of computable reducibility. We proceed to a systematic…
We use a generalized Lambert series identity due to the first author to present q-series proofs of recent results of Imamoglu, Raum and Richter concerning recursive formulas for the coefficients of two 3rd order mock theta functions.…
We consider an extension of first-order logic with a recursion operator that corresponds to allowing formulas to refer to themselves. We investigate the obtained language under two different systems of semantics, thereby obtaining two…
The review of modern study of algebraic, geometric and differential properties of quaternionic (Q) numbers with their applications. Traditional and "tensor" formulation of Q-units with their possible representations are discussed and groups…
We show that probabilistic computable functions, i.e., those functions outputting distributions and computed by probabilistic Turing machines, can be characterized by a natural generalization of Church and Kleene's partial recursive…
This book can be seen either as a text on theorem proving that uses techniques from general algebra, or else as a text on general algebra illustrated and made concrete by practical exercises in theorem proving. The book considers several…
We use elementary methods to establish three key recurrence relations: one for derangement numbers, a second for harmonic numbers, and a third for degenerate harmonic numbers. Our results not only contribute to the understanding of the…
Many semantical aspects of programming languages, such as their operational semantics and their type assignment calculi, are specified by describing appropriate proof systems. Recent research has identified two proof-theoretic features that…
We study the structure of families of theories in the language of arithmetic extended to allow these families to refer to one another and to themselves. If a theory contains schemata expressing its own truth and expressing a specific Turing…
The following article summarizes research where theorems and their respective demonstrations are postulated based on quadratic equations with special properties given by the Pythagorean triplets and the Fibonacci sequence given the second…
We prove a recursive identity involving formal iterated logarithms and formal iterated exponentials. These iterated logarithms and exponentials appear in a natural extension of the logarithmic formal calculus used in the study of…
Circular (or cyclic) proofs have received increasing attention in recent years, and have been proposed as an alternative setting for studying (co)inductive reasoning. In particular, now several type systems based on circular reasoning have…
In the paper, we give partition-theoretic results for the coefficients of some mock theta functions and prove their congruence properties. Some recurrence relations connecting the coefficients of the mock theta functions with certain…
Two classical results characterizing regularity of a convergence space in terms of continuous extensions of maps on one hand, and in terms of continuity of limits for the continuous convergence on the other, are extended to…
Methods for specifying Moore type state machines (transducers) abstractly via primitive recursive functions and for defining parallel composition via simultaneous primitive recursion are discussed. The method is mostly of interest as a…
Recursive self-improving (RSI) systems have been dreamed of since the early days of computer science and artificial intelligence. However, many existing studies on RSI systems remain philosophical, and lacks clear formulation and results.…
This graduate textbook on machine learning tells a story of how patterns in data support predictions and consequential actions. Starting with the foundations of decision making, we cover representation, optimization, and generalization as…
For which sets A does there exist a mapping, computed by a total or partial recursive function, such that the mapping, when its domain is restricted to A, is a 1-to-1, onto mapping to $\Sigma^*$? And for which sets A does there exist such a…
Over 300 sequences and many unsolved problems and conjectures related to them are presented herein together with theorems corollaries, formulae, examples, mathematical criteria, etc. (about integer sequences, numbers, quotients, residues,…