Related papers: Recursive definitions on surreal numbers

This paper proposes bimorphic recursion, which is restricted polymorphic recursion such that every recursive call in the body of a function definition has the same type. Bimorphic recursion allows us to assign two different types to a…

Non-linear recurrences which generate integers in a surprising way have been studied by many people. Typically people study recurrences that are linear in the highest order term. In this paper I consider what happens when the recurrence is…

In this paper we discuss the notion of universality for classes of candidate common Lyapunov functions of linear switched systems. On the one hand, we prove that a family of absolutely homogeneous functions is universal as soon as it…

A theory of recursive definitions has been mechanized in Isabelle's Zermelo-Fraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other…

In his monograph On Numbers and Games, J. H. Conway introduced a real-closed field No of surreal numbers containing the reals and the ordinals, as well as a vast array of less familiar numbers. A longstanding aim has been to develop…

For each odd prime power q, we construct an infinite sequence of rational functions f(X) in F_q(X), each of which is exceptional, which means that for infinitely many n the map c-->f(c) induces a bijection of P^1(F_{q^n}). Moreover, each of…

In this article we show the following result: if $C$ is an $n$-dimensional convex and compact subset, $f:C\rightarrow[0,\infty)$ is concave, and $\phi:[0,\infty)\rightarrow[0,\infty)$ is a convex function with $\phi(0)=0$, we then…

An order theoretic and algebraic framework for the extended real numbers is established which includes extensions of the usual difference to expressions involving $-\infty$ and/or $+\infty$, so-called residuations. Based on this,…

Given a continuous real-valued function on [0, 1], and a closed subset E \subset [0, 1] we denote by f E the restriction of f to E, that is, the function defined only on E that takes the same values as f at every point of E >. The…

Simple methods permit to generalize the concepts of iteration and of recursive processes. We shall see briefly on several examples what these methods generate. In additive sequences, we shall encounter not only the golden or the silver…

We investigate the existence of a class of ZFC-provably total recursive unary functions, given certain constraints, and apply some of those results to show that, for $\Sigma_1$-sound set theory, ZFC$\not\vdash P<NP$.

Bernoulli numbers are usually expressed in terms of their lower index numbers (recursive). This paper gives explicit formulas for Bernoulli numbers of even index. The formulas contain a remarkable sequence of determinants. The value of…

Several classes of DNR functions are characterized in terms of Kolmogorov complexity. In particular, a set of natural numbers A can wtt-compute a DNR function iff there is a nontrivial recursive lower bound on the Kolmogorov complexity of…

We present a concept of uniform encodability of theories and develop tools related to this concept. As an application we obtain general undecidability results which are uniform for large families of structures. In the way, we define…

This work proposes a unified theory of regularity in one hypercomplex variable: the theory of $T$-regular functions. In the special case of quaternion-valued functions of one quaternionic variable, this unified theory comprises…

A class of spherical functions is studied which can be viewed as the matrix generalization of Bessel functions. We derive a recursive structure for these functions. We show that they are only special cases of more general radial functions…

This paper investigates functions from $\mathbb{R}^d$ to $\mathbb{R} \cup \{\pm \infty\}$ that satisfy axioms of linearity wherever allowed by extended-value arithmetic. They have a nontrivial structure defined inductively on $d$, and…

To a function with values in the power set of a pre-ordered, separated locally convex space a family of scalarizations is given which completely characterizes the original function. A concept of a Legendre-Fenchel conjugate for set-valued…

In this paper we provide an identity between determinant and generalized matrix function. Also, a criterion of positive semi-definite matrices affirming the permanent dominant conjecture is given. As a consequence, infinitely many infinite…

We present a theorem on taking the repeated indefinite summation of a holomorphic function $\phi(z)$ in a vertical strip of $\mathbb{C}$ satisfying exponential bounds as the imaginary part grows. We arrive at this result using transforms…