scholarly journals Duality Pairs and Homomorphisms to Oriented and Unoriented Cycles

10.37236/9747 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Santiago Guzmán-Pro ◽  
César Hernández-Cruz
Keyword(s):  
Undirected Graph ◽  
Linear Size ◽  
Directed Path ◽  
Duality Pairs ◽  
Finite Set ◽  
Oriented Path ◽  
Transitive Tournament

 In the homomorphism order of digraphs, a duality pair is an ordered pair of digraphs $(G,H)$ such that for any digraph, $D$, $G\to D$ if and only if $D\not \to H$. The directed path on $k+1$ vertices together with the transitive tournament on $k$ vertices is a classic example of a duality pair. In this work, for every undirected cycle $C$ we find an orientation $C_D$ and an oriented path $P_C$, such that $(P_C,C_D)$ is a duality pair. As a consequence we obtain that there is a finite set, $F_C$, such that an undirected graph is homomorphic to $C$, if and only if it admits an $F_C$-free orientation. As a byproduct of the proposed duality pairs, we show that if $T$ is an oriented tree of height at most $3$, one can choose a dual of $T$ of linear size with respect to the size of $T$.

scholarly journals Subdivisions in Digraphs of Large Out-Degree or Large Dichromatic Number

10.37236/6521 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Pierre Aboulker ◽  
Nathann Cohen ◽  
Frédéric Havet ◽  
William Lochet ◽  
Phablo F. S. Moura ◽  
...  
Keyword(s):  
Directed Path ◽  
Directed Paths ◽  
Oriented Path ◽  
Transitive Tournament ◽  
Dichromatic Number

In 1985, Mader conjectured the existence of a function $f$ such that every digraph with minimum out-degree at least $f(k)$ contains a subdivision of the transitive tournament of order $k$. This conjecture is still completely open, as the existence of $f(5)$ remains unknown. In this paper, we show that if $D$ is an oriented path, or an in-arborescence (i.e., a tree with all edges oriented towards the root) or the union of two directed paths from $x$ to $y$ and a directed path from $y$ to $x$, then every digraph with minimum out-degree large enough contains a subdivision of $D$. Additionally, we study Mader's conjecture considering another graph parameter. The dichromatic number of a digraph $D$ is the smallest integer $k$ such that $D$ can be partitioned into $k$ acyclic subdigraphs. We show that any digraph with dichromatic number greater than $4^m (n-1)$ contains every digraph with $n$ vertices and $m$ arcs as a subdivision. We show that any digraph with dichromatic number greater than $4^m (n-1)$ contains every digraph with $n$ vertices and $m$ arcs as a subdivision.

scholarly journals Connections in Randomly Oriented Graphs

2016 ◽  
Vol 27 (4) ◽  
pp. 667-671 ◽  
Author(s):  
BHARGAV NARAYANAN
Keyword(s):  
Undirected Graph ◽  
Directed Path ◽  
Random Orientation ◽  
Oriented Graphs

Given an undirected graph G, let us randomly orient G by tossing independent (possibly biased) coins, one for each edge of G. Writing a → b for the event that there exists a directed path from a vertex a to a vertex b in such a random orientation, we prove that for any three vertices s, a and b of G, we have ℙ(s → a ∩ s → b) ⩾ ℙ(s → a) ℙ(s → b).

Improving the efficiency of hardening of tillage machine working bodies by arc surfacing using wear-resistant rollers

2020 ◽  
pp. 176-185
Author(s):  
Dmitriy B. Slinko ◽  
Vyacheslav A. Denisov ◽  
Dmitriy A. Dobrin ◽  
Andrey V. Afanas’yev ◽  
Pavel M. Kislov
Keyword(s):  
Wear Resistance ◽  
Operating Time ◽  
Field Tests ◽  
Electric Arc ◽  
Cutting Edge ◽  
Linear Size ◽  
Research Purpose ◽  
Cored Wire ◽  
Wear Resistant ◽  
Working Bodies

Reducing operating costs during soil processing and increasing the wear resistance of parts and components is an important condition for reducing the cost of agricultural products, increasing its efficiency and competitiveness. The development of materials with increased wear resistance and new effective technologies for strengthening working bodies in their manufacture is now becoming an urgent task. (Research purpose) The research purpose is increasing the wear resistance of the working bodies of soil-processing machines by electric arc surfacing. (Materials and methods) During the experimental study, it has been performed testing of modes and surfacing of a pilot batch of working bodies from Kverneland for field tests in VIM on an automated installation for electric arc surfacing. Authors used eutectic CastolinEnDotec DO*30 powder wire with a diameter of 1.2 millimeters and a boron content of up to 4 percent for surfacing wear-resistant rollers, which allows to obtain wear-resistant rollers with a hardness of up to 65 HRC without pores and cracks. (Results and discussion) It has been revealed that hardened ploughshares that have passed field tests are subject to lower wear rates compared to non-hardened ones. It was found that when operating time is 24.785 hectares per ploughshare, the wear of the linear size of the cutting edge along the width of hardened ploughshares is on average up to 10-11 millimeters less than that of non-hardened ones. It was found that when processing 228 hectares, the wear of the linear size of the cutting edge along the width of hardened bits is on average up to 9-10 millimeters less than that of non-hardened ones. (Conclusions) The technology of surfacing with intermittent wear-resistant rollers provides an increase in the efficiency of hardening of Kverneland working bodies according to the criterion of wear resistance by an average of 20-30 percent. The adjusted technological parameters of the surfacing process will reduce the wear rate and increase the service life of the blade part of the working bodies, as well as reduce the amount of surfaced material by an average of 60 percent. The continuation of work on strengthening the working bodies should be aimed at changing the surfacing scheme and choosing a cheaper domestic cored wire.

scholarly journals On the star forest polytope for trees and cycles

2019 ◽  
Vol 53 (5) ◽  
pp. 1763-1773
Author(s):  
Meziane Aider ◽  
Lamia Aoudia ◽  
Mourad Baïou ◽  
A. Ridha Mahjoub ◽  
Viet Hung Nguyen
Keyword(s):  
Undirected Graph ◽  
Dominating Set ◽  
Discrete Math ◽  
Complete Characterization ◽  
Facial Structure ◽  
Maximum Weight ◽  
Single Node ◽  
End Node ◽  
Vertex Disjoint

Let G = (V, E) be an undirected graph where the edges in E have non-negative weights. A star in G is either a single node of G or a subgraph of G where all the edges share one common end-node. A star forest is a collection of vertex-disjoint stars in G. The weight of a star forest is the sum of the weights of its edges. This paper deals with the problem of finding a Maximum Weight Spanning Star Forest (MWSFP) in G. This problem is NP-hard but can be solved in polynomial time when G is a cactus [Nguyen, Discrete Math. Algorithms App. 7 (2015) 1550018]. In this paper, we present a polyhedral investigation of the MWSFP. More precisely, we study the facial structure of the star forest polytope, denoted by SFP(G), which is the convex hull of the incidence vectors of the star forests of G. First, we prove several basic properties of SFP(G) and propose an integer programming formulation for MWSFP. Then, we give a class of facet-defining inequalities, called M-tree inequalities, for SFP(G). We show that for the case when G is a tree, the M-tree and the nonnegativity inequalities give a complete characterization of SFP(G). Finally, based on the description of the dominating set polytope on cycles given by Bouchakour et al. [Eur. J. Combin. 29 (2008) 652–661], we give a complete linear description of SFP(G) when G is a cycle.

Non-repetitive sequences

1970 ◽  
Vol 68 (2) ◽  
pp. 267-274 ◽  
Author(s):  
P. A. B. Pleasants
Keyword(s):  
Repetitive Sequences ◽  
Finite Set ◽  
Infinite Sequences

This note is concerned with infinite sequences whose terms are chosen from a finite set of symbols. A segment of such a sequence is a set of one or more consecutive terms, and a repetition is a pair of finite segments that are adjacent and identical. A non-repetitive sequence is one that contains no repetitions.

The inverse problem of recovering the coefficients of a differential equations on a graph

2020 ◽  
Vol 28 (5) ◽  
pp. 727-738
Author(s):  
Victor Sadovnichii ◽  
Yaudat Talgatovich Sultanaev ◽  
Azamat Akhtyamov
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Finite Number ◽  
Characteristic Equation ◽  
Arbitrary Number ◽  
New Class ◽  
Finite Set

AbstractWe consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

scholarly journals A Novel Simplified Implementation of Finite-Set Model Predictive Control for Power Converters

IEEE Access ◽  
2021 ◽  
pp. 1-1
Author(s):  
Jose J. Silva ◽  
Jose R. Espinoza ◽  
Jaime A. Rohten ◽  
Esteban S. Pulido ◽  
Felipe A. Villarroel ◽  
...  
Keyword(s):  
Model Predictive Control ◽  
Predictive Control ◽  
Power Converters ◽  
Finite Set

scholarly journals Robust Universal Inference

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 773
Author(s):  
Amichai Painsky ◽  
Meir Feder
Keyword(s):  
Robust Estimation ◽  
Minimax Estimator ◽  
Worst Case ◽  
Capacity Theory ◽  
Alternative Approach ◽  
True Model ◽  
Finite Set ◽  
Universal Prediction ◽  
Estimation Scheme

Learning and making inference from a finite set of samples are among the fundamental problems in science. In most popular applications, the paradigmatic approach is to seek a model that best explains the data. This approach has many desirable properties when the number of samples is large. However, in many practical setups, data acquisition is costly and only a limited number of samples is available. In this work, we study an alternative approach for this challenging setup. Our framework suggests that the role of the train-set is not to provide a single estimated model, which may be inaccurate due to the limited number of samples. Instead, we define a class of “reasonable” models. Then, the worst-case performance in the class is controlled by a minimax estimator with respect to it. Further, we introduce a robust estimation scheme that provides minimax guarantees, also for the case where the true model is not a member of the model class. Our results draw important connections to universal prediction, the redundancy-capacity theorem, and channel capacity theory. We demonstrate our suggested scheme in different setups, showing a significant improvement in worst-case performance over currently known alternatives.

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