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We study properties of Euclidean affine functions (EAFs), namely those of the form $f(r) = (\alpha\cdot r + \beta)/\delta$, and their closely related expression $\mathring{f}(r) = (\alpha\cdot r + \beta)\%\delta$, where $r$, $\alpha$,…
In this paper we study number fields which are Euclidean with respect to a function different from the absolute value of the norm. We also show that the Euclidean minimum with respect to weighted norms may be irrational and not isolated.
The problem is considered of arranging symbols around a cycle, in such a way that distances between different instances of a same symbol be as uniformly distributed as possible. A sequence of moments is defined for cycles, similarly to the…
Let $p(n)$ denote the smallest prime divisor of the integer $n$. Define the function $g(k)$ to be the smallest integer $>k+1$ such that $p(\binom{g(k)}{k})>k$. So we have $g(2)=6$ and $g(3)=g(4)=7$. In this paper we present the following…
When people mention the mathematical achievements of Euclid, his geometrical achievements always spring to mind. But, his Number-Theoretical achievements (See Books 7, 8 and 9 in his magnum opus \emph{Elements} [1]) are rarely spoken. The…
We propose constructions of codes over quotient rings of Eisenstein integers equipped with the Euclidean, square Euclidean, and hexagonal distances as a generalization of codes over Eisenstein integer fields. By set partitioning, we…
The Euclidean algorithm is one of the oldest algorithms known to mankind. Given two integral numbers $a_1$ and $a_2$, it computes the greatest common divisor (gcd) of $a_1$ and $a_2$ in a very elegant way. From a lattice perspective, it…
We consider Gaussian distributions on certain Riemannian symmetric spaces. In contrast to the Euclidean case, it is challenging to compute the normalization factors of such distributions, which we refer to as partition functions. In some…
For the each of the five Euclidean rings of complex quadratic integers, we consider a complex continued fraction algorithm with digits in the ring. We show for each algorithm that the maximal digit obeys a Fr\'echet distribution. We use…
We consider the fundamental task of optimising a real-valued function defined in a potentially high-dimensional Euclidean space, such as the loss function in many machine-learning tasks or the logarithm of the probability distribution in…
For each $d \in {1,2,3,7,11}$, let $T_d$ be the nearest-integer complex continued fraction map associated with the Euclidean ring $\mathcal{O}*d$, and let $(a_n)$ be its digit sequence. We prove two metric results for this five-system…
In this paper, we define Euclidean minima for function fields and give some bound for this invariant. We furthermore show that the results are analogous to those obtained in the number field case.
We introduce a generalization of the Euclidean algorithm for rings equipped with an involution, and completely enumerate all isomorphism classes of orders over definite, rational quaternion algebras equipped with an orthogonal involution…
A new algorithm for one-dimensional minimization is described in detail and the results of some tests on practical cases are reported and illustrated. The method requires only punctual computation of the function, and is suitable to be…
We study the complexity of estimating the partition function $\mathsf{Z}(\beta)=\sum_{x\in\chi} e^{-\beta H(x)}$ for a Gibbs distribution characterized by the Hamiltonian $H(x)$. We provide a simple and natural lower bound for quantum…
A central problem in computational statistics is to convert a procedure for sampling combinatorial from an objects into a procedure for counting those objects, and vice versa. Weconsider sampling problems coming from *Gibbs distributions*,…
We present a new tool to compute the number $\phi_\A (\b)$ of integer solutions to the linear system $$ \x \geq 0 \qquad \A \x = \b $$ where the coefficients of $\A$ and $\b$ are integral. $\phi_\A (\b)$ is often described as a \emph{vector…
An index $e$ in a numbering of partial-recursive functions is called minimal if every lesser index computes a different function from $e$. Since the 1960's it has been known that, in any reasonable programming language, no effective…
Consider the partition function S(\epsilon) associated in theory of Renyi dimension to a finite Borel measure \mu on Euclidean d-space. This partion function S(\epsilon) is the sum of the q-th powers of the measure applied to a partition of…
We study surfaces in Euclidean space ${\mathbb R}^3$ that are minimal for a log-linear density $\phi(x,y,z)=\alpha x+\beta y+\gamma y$, where $\alpha,\beta,\gamma$ are real numbers not all zero. We prove that if a surface is $\phi$-minimal…