The Pigeonhole Principle, Two Centuries Before Dirichlet

Rittaud, Benoît; Heeffer, Albrecht

doi:10.1007/s00283-013-9389-1

Cited by 14 publications

(4 citation statements)

References 3 publications

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“…3 I. e., both seats of such a doubleseat are unoccupied. 4 Newcomers, particularly when alone, prefer to sit down on an unoccupied doubleseat to not sitting next to someone other.…”

Section: ) a Solution By The Inverse Dirichlet's Principlementioning

confidence: 99%

“…W HILE the Dirichlet's principle is applied in various fields such as number theory and calculus [1], data compression [2], quantum mechanics [3] and many others, in this work, we focus only on the original definition in a pure discrete fashion, and mainly on a derived, inverse form. The original version of the Dirichlet's principle [4] claims that if there are more items than containers so that the items are put into the containers, then there is at least one container containing two or more items. Thus, due to the idea's relative simplicity, applications of the original Dirichlet's principle are usually limited to rather fancy problems coming from recreational mathematics.…”

Section: Introductionmentioning

confidence: 99%

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Dirichlet's principle revisited: An inverse Dirichlet's principle definition and its bound estimation improvement using stochastic combinatorics

Štěpánek¹,

Habarta²,

Malá³

et al. 2022

Communication Papers of the 17th Conference on Computer Science and Intelligence Systems

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Dirichlet's principle, also known as a pigeonhole principle, claims that if n ∈ N item are put into m ∈ N containers, with n > m, then there is a container that contains more than one item. In this work, we focus rather on an inverse Dirichlet's principle (by switching items and containers), which is as follows: considering n ∈ N items put in m ∈ N containers, when n < m, then there is at least one container with no item inside. Furthermore, we refine Dirichlet's principle using discrete combinatorics within a probabilistic framework. Applying stochastic fashion on the principle, we derive the number of items n may be even greater than or equal to m, still very likely having one container without an item. The inverse definition of the problem rather than the original one may have some practical applications, particularly considering derived effective upper bound estimates for the items number, as demonstrated using some applied mini-studies.

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“…3 I. e., both seats of such a doubleseat are unoccupied. 4 Newcomers, particularly when alone, prefer to sit down on an unoccupied doubleseat to not sitting next to someone other.…”

Section: ) a Solution By The Inverse Dirichlet's Principlementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dirichlet's principle revisited: An inverse Dirichlet's principle definition and its bound estimation improvement using stochastic combinatorics

Štěpánek¹,

Habarta²,

Malá³

et al. 2022

Communication Papers of the 17th Conference on Computer Science and Intelligence Systems

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“…Given N exp total expected neutrino events in RES-NOVA occurring in the smallest time window that a single detector can resolve, we use the Pigeonhole principle 1 [62] to derive the probability P that at least one out of the 500 detectors will count more that one event in such a time window:…”

Section: Res-nova Detection Significancementioning

confidence: 99%

RES-NOVA sensitivity to core-collapse and failed core-collapse supernova neutrinos

Pattavina,

Iachellini,

Pagnanini

et al. 2021

Preprint

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RES-NOVA is a new proposed experiment for the investigation of astrophysical neutrino sources with archaeological Pb-based cryogenic detectors. RES-NOVA will exploit Coherent Elastic neutrino-Nucleus Scattering (CEνNS) as detection channel, thus it will be equally sensitive to all neutrino flavors produced by Supernovae (SNe). RES-NOVA with only a total active volume of (60 cm) 3 and an energy threshold of 1 keV will probe the entire Milky Way Galaxy for (failed) core-collapse SNe with > 3 σ detection significance. The high detector modularity makes RES-NOVA ideal also for reconstructing the main parameters (e.g. average neutrino energy, star binding energy) of SNe occurring in our vicinity, without deterioration of the detector performance caused by the high neutrino interaction rate. For the first time, distances < 3 kpc can be surveyed, similarly to the ones where all known past galactic SNe happened. We discuss the RES-NOVA potential, accounting for a realistic setup, considering the detector geometry, modularity and background level in the region of interest. We report on the RES-NOVA background model and on the sensitivity to SN neutrinos as a function of the distance travelled by neutrinos. Contents 1 Introduction 1 2 Supernovae as neutrino sources 3 3 The RES-NOVA detector 4 4 Detector background model 6 5 Detector response to SN neutrinos 10 6 RES-NOVA detection significance 11 6.1 SN neutrino signals at close distances (d < 3 kpc) 12 6.2 SN neutrino signals in absence of pile-up (d > 3 kpc) 13 7 Conclusions 14

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“…. , q |α| are states of the NFA A, q i is reachable from q i−1 for all integers 0 < i ≤ |α|, and there exist 0 ≤ j < k ≤ |α| such that q j = q k by the pigeon-hole principle [34]. Hence, q |α| is reachable from the cycle q j , .…”

Section: B Detectability Of Symbolic Modelsmentioning

confidence: 99%

A Framework for Output-Feedback Symbolic Control

Khaled¹,

Zhang²

2020

Preprint

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Symbolic control is a an abstraction-based controller synthesis approach that provides, algorithmically, certifiable-by-construction controllers for cyber-physical systems. Current methodologies of symbolic control usually assume that full-state information is available. This is not suitable for many real-world applications with partially-observable states or output information. This article introduces a framework for output-feedback symbolic control. We propose relations between original systems and their symbolic models based on outputs. They enable designing symbolic controllers and refining them to enforce complex requirements on original systems. To demonstrate the effectiveness of the proposed framework, we provide three different methodologies. They are applicable to a wide range of linear and nonlinear systems, and support general logic specifications.

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The Pigeonhole Principle, Two Centuries Before Dirichlet

Cited by 14 publications

References 3 publications

Dirichlet's principle revisited: An inverse Dirichlet's principle definition and its bound estimation improvement using stochastic combinatorics

Dirichlet's principle revisited: An inverse Dirichlet's principle definition and its bound estimation improvement using stochastic combinatorics

RES-NOVA sensitivity to core-collapse and failed core-collapse supernova neutrinos

A Framework for Output-Feedback Symbolic Control

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