Related papers: Luzin's (N) and randomness reflection
We give an example of a reflected diffferential equation which may have infinitely many solutions if the driving signal is rough enough (e.g. of infinite $p$-variation, for some $p>2$). For this equation, we identify a sharp condition on…
Frequentist inference typically is described in terms of hypothetical repeated sampling but there are advantages to an interpretation that uses a single random sample. Contemporary examples are given that indicate probabilities for random…
We investigate the continuous function $f$ defined by $$x\mapsto \sum_{\sigma\le_L x }2^{-K(\sigma)}$$ as a variant of Chaitin's Omega from the perspective of analysis, computability, and algorithmic randomness. Among other results, we…
Intrinsic random functions (IRF) provide a versatile approach when the assumption of second-order stationarity is not met. Here, we develop the IRF theory on the circle with its universal kriging application. Unlike IRF in Euclidean spaces,…
Let $\mathbb{K}$ be an uncountable field of characteristic zero and let $f$ be a function from $\mathbb{K}^n$ to $\mathbb{K}$. We show that if the restriction of $f$ to every affine plane $L\subset\mathbb{K}^n$ is regular, then $f$ is a…
We define a class of functions termed "Computable in the Limit", based on the Machine Learning paradigm of "Identification in the Limit". A function is Computable in the Limit if it defines a property P_p of a recursively enumerable class A…
In this paper, some sufficient conditions for the differentiability of the $n$-variable real-valued function are obtained, which are given based on the differentiability of the $n-1$-variable real-valued function and are weaker than…
We say that a function is rare-case hard against a given class of algorithms (the adversary) if all algorithms in the class can compute the function only on an $o(1)$-fraction of instances of size $n$ for large enough $n$. Starting from any…
We consider whether given a simple, finite description of a group in the form of an algorithm, it is possible to algorithmically determine if the corresponding group has some specified property or not. When there is such an algorithm, we…
Different questions lead to the same class of functions from natural integers to integers: those which have integral difference ratios, i.e. verifying $f(a)-f(b)\equiv0 \pmod {(a-b)}$ for all $a>b$. We characterize this class of functions…
We consider the problem of the uniform (in $L_\infty$) recovery of ridge functions $f(x)=\varphi(\langle a,x\rangle)$, $x\in B_2^n$, using noisy evaluations $y_1\approx f(x^1),\ldots,y_N\approx f(x^N)$. It is known that for classes of…
By the sometimes so-called 'Main Theorem' of Recursive Analysis, every computable real function is necessarily continuous. We wonder whether and which kinds of HYPERcomputation allow for the effective evaluation of also discontinuous…
A real $x$ is $\Delta^1_1$-Kurtz random ($\Pi^1_1$-Kurtz random) if it is in no closed null $\Delta^1_1$ set ($\Pi^1_1$ set). We show that there is a cone of $\Pi^1_1$-Kurtz random hyperdegrees. We characterize lowness for…
We show that in the setting of fair-coin measure on the power set of the natural numbers, each sufficiently random set has an infinite subset that computes no random set. That is, there is an almost sure event $\mathcal A$ such that if…
We study the computational power of randomized computations on infinite objects, such as real numbers. In particular, we introduce the concept of a Las Vegas computable multi-valued function, which is a function that can be computed on a…
Compactness is one of the core notions of analysis: it connects local properties to global ones and makes limits well-behaved. We study the computational properties of the compactness of Cantor space $2^{\mathbb{N}}$ for uncountable covers.…
In this paper, we study Bernoulli random sequences, i.e., sequences that are Martin-L\"of random with respect to a Bernoulli measure $\mu_p$ for some $p\in[0,1]$, where we allow for the possibility that $p$ is noncomputable. We focus in…
According to a celebrated result by L\"owner, a real-valued function $f$ is operator monotone if and only if its L\"owner matrix, which is the matrix of divided differences $L_f=(\frac{f(x_i)-f(x_j)}{x_i-x_j})_{i,j=1}^N$, is positive…
A type-2 computable real function is necessarily continuous; and this remains true for relative, i.e. oracle-based computations. Conversely, by the Weierstrass Approximation Theorem, every continuous f:[0,1]->R is computable relative to…
We prove a functional limit theorem for vector-valued functionals of the fractional Ornstein-Uhlenbeck process, providing the foundation for the fluctuation theory of slow/fast systems driven by such a noise. Our main contribution is on the…