Related papers: A note on ordinal exponentiation and derivatives o…

Let $K$ be a compact metric space. A real-valued function on $K$ is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. In this paper, we study two well known ordinal indices of…

Let $\alpha:[0,1]\to [0,1]$ be a measurable function. It was proved by P. Marchal \cite{Mar15} that the function $$ \phi^{(\alpha)}(\lambda):=\exp\left[ \int_0^1\frac{\lambda-1}{1+(\lambda-1)x}\,\alpha(x)\,d x \right],\quad \lambda>0 $$ is…

We prove a functional extension of an exponential inequality originally proposed by Bin Zhao and proved by Xiaosheng Mou. The main result asserts that if $\alpha_1\leq \cdots\leq \alpha_n$ and $\sum_{k=1}^n \alpha_k=0$, then \[ \sum_{k=1}^n…

We investigate the log-concavity on the half-line of the Wright function $\phi(-\alpha,\beta,-x),$ in the probabilistic setting $\alpha\in (0,1)$ and $\beta \ge 0.$ Applications are given to the construction of generalized entropies…

We examine the computable part of the differentiability hierarchy defined by Kechris and Woodin. In that hierarchy, the rank of a differentiable function is an ordinal less than omega_1 which measures how complex it is to verify…

Let $A$ be an excellent two-dimensional normal local ring containing an algebraically closed field and let $X\to \mathrm{Spec} (A)$ be a resolution of singularity. We prove a theorem giving a condition under which the dimension of the…

This is a follow up to a paper by the author where the disjointness relation for (the graphs of) definable functions from ${^\omega \omega}$ to ${^\omega \omega}$ is analyzed. In that paper, for each $a \in {^\omega \omega}$ we defined a…

We show that Caratheodory's conjecture, on umbilical points of closed convex surfaces, may be reformulated in terms of the existence of at least one umbilic in the graphs of functions f: R^2-->R whose gradient decays uniformly faster than…

A real number is called simply normal to base $b$ if its base-$b$ expansion has each digit appearing with average frequency tending to $1/b$. In this article, we discover a relation between the frequency that the digit $1$ appears in the…

We exhibit a way of "forcing a functional to be an effective operation" for arbitrary partial combinatory algebras (pcas). This gives a method of defining new pcas from old ones for some fixed functional, where the new partial functions can…

Simpson and the second author asked whether there exists a characterization of the natural numbers by a second-order sentence which is provably categorical in the theory RCA$^*_0$. We answer in the negative, showing that for any…

Let $k \geq 1$ be a natural number and $f \in \mathbb{F}_q[t]$ be a monic polynomial. Let $\omega_k(f)$ denote the number of distinct monic irreducible factors of $f$ with multiplicity $k$. We obtain asymptotic estimates for the first and…

We consider the untyped lambda calculus with constructors and recursively defined constants. We construct a domain-theoretic model such that any term not denoting bottom is strongly normalising provided all its `stratified approximations'…

A derivative expansion of the effective average action beyond first order yields renormalization group functional flow equations which are used for the computation of critical exponents of the Ising universality class. The critical exponent…

We consider a piecewise analytic real expanding map $f: [0,1]\to [0,1]$ of degree $d$ which preserves orientation, and a real analytic positive potential $g: [0,1] \to \mathbb{R}$. We assume the map and the potential have a complex analytic…

Boyer and Moore have discussed a recursive function that puts conditional expressions into normal form [1]. It is difficult to prove that this function terminates on all inputs. Three termination proofs are compared: (1) using a measure…

A function $U:\left[ \omega_{1}\right] ^{2}\longrightarrow\omega$ is called $\left( 1,\omega_{1}\right) $\emph{-weakly universal }if for every function $F:\left[ \omega_{1}\right] ^{2}\longrightarrow\omega$ there is an injective function…

In this paper we introduce hyperations and cohyperations, which are forms of transfinite iteration of ordinal functions. Hyperations are iterations of normal functions. Unlike iteration by pointwise convergence, hyperation preserves…

The main purpose of this paper is to give characterization theorems on derivations as well as on linear functions. Among others the following problem will be investigated: Let $n\in\mathbb{Z}$, $f, g\colon\mathbb{R}\to\mathbb{R}$ be…

For g < f in omega^omega we define c(f,g) be the least number of uniform trees with g-splitting needed to cover a uniform tree with f-splitting. We show that we can simultaneously force aleph_1 many different values for different functions…