scholarly journals Extremal Topological and Geometric Problems for Polyominoes

10.37236/9086 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Greg Malen ◽  
Erika Berenice Roldan-Roa
Keyword(s):  
Structural Properties ◽  
Isoperimetric Inequality ◽  
Complete Solution ◽  
Dual Graph ◽  
Combinatorial Problem ◽  
Topological Properties ◽  
Geometric Problems ◽  
Minimum Number ◽  
Minimum Perimeter

 We give a complete solution to the extremal topological combinatorial problem of finding the minimum number of tiles needed to construct a polyomino with $h$ holes. We denote this number by $g(h)$ and we analyze structural properties of polyominoes with $h$ holes and $g(h)$ tiles, characterizing their efficiency by a topological isoperimetric inequality that relates minimum perimeter, the area of the holes, and the structure of the dual graph of a polyomino. For $h\leqslant 8$ the values of $g(h)$ were originally computed by Tomas Olivera e Silva in 2015, and for the sequence $h_l=(2^{2l}-1)/3$ by Kahle and Róldan-Roa in 2019, who also showed that asymptotically $g(h) \approx 2h$. Here we also prove that the sequence of polyominoes constructed by Kahle and Róldan-Roa that have $h_l=(2^{2l}-1)/3$ holes and $g(h_l)$ tiles, are in fact unique up to isometry with respect to attaining these extremal topological properties; that is, having the minimal number of tiles for $h_l$ holes.

scholarly journals On the effects of memory and topology on the controllability of complex dynamical networks

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Panagiotis Kyriakis ◽  
Sérgio Pequito ◽  
Paul Bogdan
Keyword(s):  
Complex Networks ◽  
Temporal Dynamics ◽  
Complex Dynamical Networks ◽  
Long Term Memory ◽  
Topological Properties ◽  
Control Effort ◽  
Dynamical Networks ◽  
Term Memory ◽  
Minimum Number

Abstract Recent advances in network science, control theory, and fractional calculus provide us with mathematical tools necessary for modeling and controlling complex dynamical networks (CDNs) that exhibit long-term memory. Selecting the minimum number of driven nodes such that the network is steered to a prescribed state is a key problem to guarantee that complex networks have a desirable behavior. Therefore, in this paper, we study the effects of long-term memory and of the topological properties on the minimum number of driven nodes and the required control energy. To this end, we introduce Gramian-based methods for optimal driven node selection for complex dynamical networks with long-term memory and by leveraging the structure of the cost function, we design a greedy algorithm to obtain near-optimal approximations in a computationally efficiently manner. We investigate how the memory and topological properties influence the control effort by considering Erdős–Rényi, Barabási–Albert and Watts–Strogatz networks whose temporal dynamics follow a fractional order state equation. We provide evidence that scale-free and small-world networks are easier to control in terms of both the number of required actuators and the average control energy. Additionally, we show how our method could be applied to control complex networks originating from the human brain and we discover that certain brain cortex regions have a stronger impact on the controllability of network than others.

COMPUTING THE REVERSAL DISTANCE BETWEEN GENOMES IN THE PRESENCE OF MULTI-GENE FAMILIES VIA BINARY INTEGER PROGRAMMING

2007 ◽  
Vol 05 (01) ◽  
pp. 117-133 ◽  
Author(s):  
JAKKARIN SUKSAWATCHON ◽  
CHIDCHANOK LURSINSAP ◽  
MIKAEL BODÉN
Keyword(s):  
Integer Programming ◽  
Combinatorial Problem ◽  
Gene Families ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
High Accuracy ◽  
Biological Data ◽  
Binary Integer Programming ◽  
A Genome ◽  
Minimum Number

Hannenhalli and Pevzner developed the first polynomial-time algorithm for the combinatorial problem of sorting signed genomic data. Their algorithm determines the minimum number of reversals required for rearranging a genome to another — but only in the absence of gene duplicates. However, duplicates often account for 40% of a genome. In this paper, we show how to extend Hannenhalli and Pevzner's approach to deal with genomes with multi-gene families. We propose a new heuristic algorithm to compute the nearest reversal distance between two genomes with multi-gene families via binary integer programming. The experimental results on both synthetic and real biological data demonstrate that the proposed algorithm is able to find the reversal distance with high accuracy.

A Classification of Spatial Stiffness Based on the Degree of Translational–Rotational Coupling

1999 ◽  
Vol 123 (3) ◽  
pp. 353-358 ◽  
Author(s):  
Shuguang Huang ◽  
Joseph M. Schimmels
Keyword(s):  
Stiffness Matrix ◽  
Full Rank ◽  
Rotational Behavior ◽  
Topological Properties ◽  
Number Of Components ◽  
Stiffness Matrices ◽  
Minimum Number ◽  
Rotational Coupling

Previously, we have shown that, to realize an arbitrary spatial stiffness matrix, spring components that couple the translational and rotational behavior along/about an axis are required. We showed that, three such coupled components and three uncoupled components are sufficient to realize any full-rank spatial stiffness matrix and that, for some spatial stiffness matrices, three coupled components are necessary. In this paper, we show how to identify the minimum number of components that provide the translational-rotational coupling required to realize an arbitrarily specified spatial stiffness matrix. We establish a classification of spatial stiffness matrices based on this number which we refer to as the “degree of translational–rotational coupling” (DTRC). We show that the DTRC of a stiffness matrix is uniquely determined by the spatial stiffness mapping and is obtained by evaluating the eigenstiffnesses of the spatial stiffness matrix. The topological properties of each class are identified. In addition, the relationships between the DTRC and other properties identified in previous investigations of spatial stiffness behavior are discussed.

scholarly journals Implementasi Algoritma Greedy Pada Pewarnaan Wilayah Kecamatan Sukodadi Lamongan

2020 ◽  
Vol 6 (2) ◽  
pp. 29-38
Author(s):  
Umi Maftukhah ◽  
Siti Amiroch ◽  
Mohammad Syaiful Pradana
Keyword(s):  
Graph Theory ◽  
Greedy Algorithm ◽  
Communication Networks ◽  
Graph Coloring ◽  
Operations Research ◽  
Dual Graph ◽  
Transportation Problems ◽  
Minimum Number ◽  
Different Color ◽  
The Greedy Algorithm

Graph theory can be applied in various fields of science such as transportation problems, communication networks, operations research, chemistry, cartography and so on. Graph theory does not only represent structure but in its application, a graph can also be colored. Many problems have graph coloring characteristics such as regional coloring. This regional coloring theory was applied to the map area of ​​Sukodadi District which consists of 20 villages. In this area coloring uses the Greedy algorithm by first making a dual graph consisting of 20 vertices and 43 edges. Based on the results of regional coloring, the minimum number of colors is 4, namely red, blue, green and yellow, with each neighboring village having a different color.

Evolving Combinatorial Problem Instances That Are Difficult to Solve

2006 ◽  
Vol 14 (4) ◽  
pp. 433-462 ◽  
Author(s):  
Jano I. van Hemert
Keyword(s):  
Evolutionary Computation ◽  
Structural Properties ◽  
Constraint Satisfaction ◽  
Travelling Salesman Problem ◽  
Combinatorial Problem ◽  
Boolean Satisfiability ◽  
Combinatorial Optimisation ◽  
Travelling Salesman ◽  
Binary Constraint ◽  
Problem Instances

This paper demonstrates how evolutionary computation can be used to acquire difficult to solve combinatorial problem instances. As a result of this technique, the corresponding algorithms used to solve these instances are stress-tested. The technique is applied in three important domains of combinatorial optimisation, binary constraint satisfaction, Boolean satisfiability, and the travelling salesman problem. The problem instances acquired through this technique are more difficult than the ones found in popular benchmarks. In this paper, these evolved instances are analysed with the aim to explain their difficulty in terms of structural properties, thereby exposing the weaknesses of corresponding algorithms.

scholarly journals Topological Properties of Benzenoid Systems. IX *. On the Sextet Polynomial

1982 ◽  
Vol 37 (1) ◽  
pp. 69-73
Author(s):  
Ivan Gutman
Keyword(s):  
Structural Properties ◽  
Combinatorial Background ◽  
Topological Properties ◽  
Benzenoid System ◽  
Sextet Polynomial ◽  
Mathematical Relations

Abstract A number of mathematical relations for the sextet polynomial are derived. A graph has been introduced (the so called C-graph), representing those properties of a benzenoid system which are essential in the sextet theory of Clar. The main structural properties of the C-graph are deter-mined. The obtained results contribute towards a better understanding of the algebraic and combinatorial background of Clar's theory of the aromatic sextet.

scholarly journals Workspace exploration and protection with multiple robots assisted by sensor networks

2018 ◽  
Vol 15 (4) ◽  
pp. 172988141879217 ◽  
Author(s):  
Jonghoek Kim
Keyword(s):  
Sensor Network ◽  
Voronoi Diagram ◽  
Dual Graph ◽  
Upper Bounds ◽  
Sensor Nodes ◽  
Multiple Robots ◽  
Global Localization ◽  
Topological Map ◽  
Minimum Number ◽  
Exchange Data

This article introduces multi-robot strategies making multiple robots explore an unknown environment in a cooperative manner. Our exploration strategies do not require global localization of a robot or a node. Multiple robots build a Voronoi diagram as a topological map of the environment, while deploying sensor nodes which can sense and communicate. As the sensor network built by one robot meets the network built by another robot, both robots can exchange data with each other. The robots then use the merged sensor network to protect the environment. We introduce an intruder capture algorithm assuming that a robot is able to access any intruder’s location utilizing the sensor network. This algorithm is robust to time delay in information sharing utilizing the sensor network. Utilizing the algorithm, we derive upper bounds on the number of robots needed to capture every intruder in the environment. This article proves that the minimum number of robots needed can be computed by finding proper edge covers of the dual graph of the Voronoi diagram.

scholarly journals Algorithms for Sorting by Reversals or Transpositions, with Application to Genome Rearrangement

2020 ◽  
Author(s):  
Gustavo Rodrigues Galvão ◽  
Zanoni Dias
Keyword(s):  
Comparative Genomics ◽  
Genome Rearrangement ◽  
Heuristic Algorithms ◽  
Combinatorial Problem ◽  
Sorting By Reversals ◽  
Sorting Problem ◽  
Minimum Number ◽  
Phd Thesis

The problem of finding the minimum sequence of rearrangements that transforms one genome into another is a well-studied problem that finds application in comparative genomics. Representing genomes as permutations, in which genes appear as elements, that problem can be reduced to the combinatorial problem of sorting a permutation using a minimum number of rearrangements. Such combinatorial problem varies according to the types of rearrangements considered. The PhD thesis summarized in this paper presents exact, approximation, and heuristic algorithms for solving variants of the permutation sorting problem involving two types of rearrangements: reversals and transpositions.

Electron Microscopy of Nucleic Acids

1974 ◽  
Vol 32 ◽  
pp. 302-303
Author(s):  
Norman Davidson
Keyword(s):  
Electron Microscopy ◽  
Nucleic Acids ◽  
Basic Protein ◽  
Molecular Biologist ◽  
Topological Properties ◽  
Protein Film ◽  
Polynucleotide Chain

The basic protein film technique for mounting nucleic acids for electron microscopy has proven to be a general and powerful tool for the working molecular biologist in characterizing different nucleic acids. It i s possible to measure molecular lengths of duplex and single-stranded DNAs and RNAs. In particular, it is thus possible to as certain whether or not the nucleic acids extracted from a particular source are or are not homogeneous in length. The topological properties of the polynucleotide chain (linear or circular, relaxed or supercoiled circles, interlocked circles, etc. ) can also be as certained.

Information capacity and bandwidth limitations in the SEM

1989 ◽  
Vol 47 ◽  
pp. 84-85
Author(s):  
D. C. Joy ◽  
R. D. Bunn
Keyword(s):  
Signal To Noise Ratio ◽  
Critical Dimension ◽  
Information Capacity ◽  
Acquisition Time ◽  
Signal To Noise ◽  
Sem Image ◽  
Probe Size ◽  
Minimum Number ◽  
Noise Ratio ◽  
Signal Channel

The information available from an SEM image is limited both by the inherent signal to noise ratio that characterizes the image and as a result of the transformations that it may undergo as it is passed through the amplifying circuits of the instrument. In applications such as Critical Dimension Metrology it is necessary to be able to quantify these limitations in order to be able to assess the likely precision of any measurement made with the microscope.The information capacity of an SEM signal, defined as the minimum number of bits needed to encode the output signal, depends on the signal to noise ratio of the image - which in turn depends on the probe size and source brightness and acquisition time per pixel - and on the efficiency of the specimen in producing the signal that is being observed. A detailed analysis of the secondary electron case shows that the information capacity C (bits/pixel) of the SEM signal channel could be written as :

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