Search results for "Arithmetic"
showing 10 items of 271 documents
A tool for a first analysis of architectural façades
1997
Abstract This work presents a tool for analysing the figurative structure of architectural facades. The procedure begins by singling out the elementary shapes which make up the facade image; it detects and identifies them as “area objects”, even if present in combination in virtual or mental form and groups them into classes of equal objects. A second step is the analysis of the inner structure of the classes: equidistant, arithmetical and geometrical sequences, or alternate distances are distinguished. The procedure ends by singling out the symmetries which structure the facade image and displaying them, pointing out their implied hierarchy through a thickness differentiation.
On new means with interesting practical applications: Generalized power means
2021
Means of positive numbers appear in many applications and have been a traditional matter of study. In this work, we focus on defining a new mean of two positive values with some properties which are essential in applications, ranging from subdivision and multiresolution schemes to the numerical solution of conservation laws. In particular, three main properties are crucial—in essence, the ideas of these properties are roughly the following: to stay close to the minimum of the two values when the two arguments are far away from each other, to be quite similar to the arithmetic mean of the two values when they are similar and to satisfy a Lipchitz condition. We present new means with these pr…
On the arithmetic of a family of degree-two K3 surfaces
2018
Let $\mathbb{P}$ denote the weighted projective space with weights $(1,1,1,3)$ over the rationals, with coordinates $x,y,z,$ and $w$; let $\mathcal{X}$ be the generic element of the family of surfaces in $\mathbb{P}$ given by \begin{equation*} X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2. \end{equation*} The surface $\mathcal{X}$ is a K3 surface over the function field $\mathbb{Q}(t)$. In this paper, we explicitly compute the geometric Picard lattice of $\mathcal{X}$, together with its Galois module structure, as well as derive more results on the arithmetic of $\mathcal{X}$ and other elements of the family $X$.
Boosting Textual Compression in Optimal Linear Time
2005
We provide a general boosting technique for Textual Data Compression. Qualitatively, it takes a good compression algorithm and turns it into an algorithm with a better compression performance guarantee. It displays the following remarkable properties: (a) it can turn any memoryless compressor into a compression algorithm that uses the “best possible” contexts; (b) it is very simple and optimal in terms of time; and (c) it admits a decompression algorithm again optimal in time. To the best of our knowledge, this is the first boosting technique displaying these properties.Technically, our boosting technique builds upon three main ingredients: the Burrows--Wheeler Transform, the Suffix Tree d…
Some models of inductive syntactical synthesis from sample computations
2005
The paper is a survey of several models of inductive program synthesis from sample computations. Synthesis tools are basically syntactical: the synthesis is based on the detection of "regular" fragments related with "shuffled" arithmetical progressions. Input sample computations are supposed to be "representative": they have to "reflect" all loops occurring in the target program. Programs are synthesized in nontraditional form of "generalized" regular expressions having Cleene stars and unions for loops and CASE-like operators. However, if input samples are somehow "annotated" (we consider two different approaches), then loops can be synthesized in more traditional WHILE-form, where loop co…
Approximate supervised learning of quantum gates via ancillary qubits
2018
We present strategies for the training of a qubit network aimed at the ancilla-assisted synthesis of multi-qubit gates based on a set of restricted resources. By assuming the availability of only time-independent single and two-qubit interactions, we introduce and describe a supervised learning strategy implemented through momentum-stochastic gradient descent with automatic differentiation methods. We demonstrate the effectiveness of the scheme by discussing the implementation of non-trivial three qubit operations, including a Quantum Fourier Transform (QFT) and a half-adder gate.
The Burrows-Wheeler Transform between Data Compression and Combinatorics on Words
2013
The Burrows-Wheeler Transform (BWT) is a tool of fundamental importance in Data Compression and, recently, has found many applications well beyond its original purpose. The main goal of this paper is to highlight the mathematical and combinatorial properties on which the outstanding versatility of the $BWT$ is based, i.e. its reversibility and the clustering effect on the output. Such properties have aroused curiosity and fervent interest in the scientific world both for theoretical aspects and for practical effects. In particular, in this paper we are interested both to survey the theoretical research issues which, by taking their cue from Data Compression, have been developed in the conte…
Robustness and Randomness
2008
The study of robustness problems for computational geometry algorithms is a topic that has been subject to intensive research efforts from both computer science and mathematics communities. Robustness problems are caused by the lack of precision in computations involving floating-point instead of real numbers. This paper reviews methods dealing with robustness and inaccuracy problems. It discusses approaches based on exact arithmetic, interval arithmetic and probabilistic methods. The paper investigates the possibility to use randomness at certain levels of reasoning to make geometric constructions more robust.
Multiple Usage of Random Bits in Finite Automata
2012
Finite automata with random bits written on a separate 2-way readable tape can recognize languages not recognizable by probabilistic finite automata. This shows that repeated reading of random bits by finite automata can have big advantages over one-time reading of random bits.
The development of “junk”. Irregularization strategies of have and say in the Germanic languages
2001
Although it is a wellknown fact that the most frequent verbs are the most irregular ones (if not suppletive), it is rarely asked how they became irregular. This article deals with the irregularization process of two originally regular (weak) verbs, HAVE and SAY in the Germanic languages, e.g. have, but has/’s and had/’d (instead of regular *haves/*haved) or say [sei], but says [sez] and said [sed] in English. Other verbs, such as DO, GO, STAND, BE, COME, and so on, also tend to irregularizations again and again without any apparent reason. In contrast to HAVE and SAY these verbs have always been rather irregular, at least dating from their first written records.