Search results for "proof"
showing 10 items of 187 documents
Towards a characterization of explanations in mathematics textbooks
2013
This work is part of a broader study which aims to examine the relation between explanation and proof in teachers’ practices and the way in which this relation emerges from the texts in the mathematics schoolbooks. Our hypothesis is that explanation and proof share some of their characterizing parameters and this makes possible an overlapping of meaning which favors a switch of roles during teaching/learning activities. First results from textbooks analysis are presented pointing out remarkable features of explanation arisen to date.
Combinatorial proofs of two theorems of Lutz and Stull
2021
Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed to be Borel, or even analytic. One of the theorems states that if $K \subset \mathbb{R}^{n}$ is any set with equal Hausdorff and packing dimensions, then $$ \dim_{\mathrm{H}} π_{e}(K) = \min\{\dim_{\mathrm{H}} K,1\} $$ for almost every $e \in S^{n - 1}$. Here $π_{e}$ stands for orthogonal projection to $\mathrm{span}(e)$. The primary purpose of this paper is to present proofs for Lutz and Stull's projection theorems which do not refer to information theoretic concepts. Instead, they will rely on combinatori…
Inductive types in homotopy type theory
2012
Homotopy type theory is an interpretation of Martin-L\"of's constructive type theory into abstract homotopy theory. There results a link between constructive mathematics and algebraic topology, providing topological semantics for intensional systems of type theory as well as a computational approach to algebraic topology via type theory-based proof assistants such as Coq. The present work investigates inductive types in this setting. Modified rules for inductive types, including types of well-founded trees, or W-types, are presented, and the basic homotopical semantics of such types are determined. Proofs of all results have been formally verified by the Coq proof assistant, and the proof s…
A permutation code preserving a double Eulerian bistatistic
2016
Visontai conjectured in 2013 that the joint distribution of ascent and distinct nonzero value numbers on the set of subexcedant sequences is the same as that of descent and inverse descent numbers on the set of permutations. This conjecture has been proved by Aas in 2014, and the generating function of the corresponding bistatistics is the double Eulerian polynomial. Among the techniques used by Aas are the M\"obius inversion formula and isomorphism of labeled rooted trees. In this paper we define a permutation code (that is, a bijection between permutations and subexcedant sequences) and show the more general result that two $5$-tuples of set-valued statistics on the set of permutations an…
Mahonian STAT on words
2016
In 2000, Babson and Steingrimsson introduced the notion of what is now known as a permutation vincular pattern, and based on it they re-defined known Mahonian statistics and introduced new ones, proving or conjecturing their Mahonity. These conjectures were proved by Foata and Zeilberger in 2001, and by Foata and Randrianarivony in 2006.In 2010, Burstein refined some of these results by giving a bijection between permutations with a fixed value for the major index and those with the same value for STAT , where STAT is one of the statistics defined and proved to be Mahonian in the 2000 Babson and Steingrimsson's paper. Several other statistics are preserved as well by Burstein's bijection.At…
Cryogenic nanoelectromechanical switch enabled by Bi2Se3 nanoribbons
2022
Abstract Nanoelectromechanical (NEM) switches are potential candidates for memory and logic devices for low standby-current and harsh environment applications. Cryogenic operation of these devices would allow to use them, e.g., in space probes and in conjunction with quantum computers. Herein, it is demonstrated that cryogenic application requirements such as good flexibility and conductivity are satisfied by using Bi2Se3 nanoribbons as active elements in NEM switches. Experimental proof of principle NEM switching at temperatures as low as 5 K is achieved in volatile and non-volatile reversible regimes, exhibiting distinct ON and OFF states, backed by theoretical modelling. The results open…
New Acoustic Test Chambers at the University of Palermo
2008
Acoustic test chambers at the University of Palermo have been completed after a design optimization of existent building structures. A unique building houses five reverberation chambers, differently coupled each other in order to achieve a large flexibility on testing procedures, and offices for the staff. Main measurements offered to national and international standards include determination of sound power, sound insulation and sound absorption. Reverberation chambers present peculiar geometries in order to achieve the desidered sound fields. Chambers structure and layout within the whole building, housing all the acoustic facilities, has been designed in order to achieve strong sound insu…
Flood proofing measures cost-efficiency analysis for hydraulic risk mitigation in an urbanized riverine area.
2020
<div> <div> <div> <p><strong>Flood proofing measures cost-efficiency analysis for hydraulic risk mitigation in an urbanized riverine area</strong></p> <p>Ugo Ventimiglia <sup>2</sup>, Angela Candela <sup>1,</sup> Giuseppe Tito Aronica <sup>2</sup></p> <p><sup>1 </sup>Department of Engineering, University of Palermo, Palermo, Italy</p> <p><sup>2 </sup>Department of Engineerin…
Finite Alphabet Control of Logistic Networks with Discrete Uncertainty
2014
We consider logistic networks in which the control and disturbance inputs take values in finite sets. We derive a necessary and sufficient condition for the existence of robustly control invariant (hyperbox) sets. We show that a stronger version of this condition is sufficient to guarantee robust global attractivity, and we construct a counterexample demonstrating that it is not necessary. Being constructive, our proofs of sufficiency allow us to extract the corresponding robust control laws and to establish the invariance of certain sets. Finally, we highlight parallels between our results and existing results in the literature, and we conclude our study with two simple illustrative exampl…
Amount of nonconstructivity in deterministic finite automata
2010
AbstractWhen D. Hilbert used nonconstructive methods in his famous paper on invariants (1888), P. Gordan tried to prevent the publication of this paper considering these methods as non-mathematical. L.E.J. Brouwer in the early twentieth century initiated intuitionist movement in mathematics. His slogan was “nonconstructive arguments have no value for mathematics”. However, P. Erdös got many exciting results in discrete mathematics by nonconstructive methods. It is widely believed that these results either cannot be proved by constructive methods or the proofs would have been prohibitively complicated. The author (Freivalds, 2008) [10] showed that nonconstructive methods in coding theory are…