scholarly journals On the Richter–Thomassen Conjecture about Pairwise Intersecting Closed Curves

2016 ◽  
Vol 25 (6) ◽  
pp. 941-958
Author(s):  
JÁNOS PACH ◽  
NATAN RUBIN ◽  
GÁBOR TARDOS
Keyword(s):  
Real Function ◽  
Important Ingredient ◽  
Continuous Real Function ◽  
Closed Curves ◽  
Jordan Curves ◽  
Family Of Curves ◽  
The Family ◽  
Intersection Points ◽  
Pass Through

A long-standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane, so that any pair of them intersect and no three curves pass through the same point, is at least (1−o(1))n2.We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class touches every curve from the second class. (Two closed or open curves are said to be touching if they have precisely one point in common and at this point the two curves do not properly cross.)An important ingredient of our proofs is the following statement. Let S be a family of n open curves in ℝ2, so that each curve is the graph of a continuous real function defined on ℝ, and no three of them pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is $\Omega(nt\sqrt{\log t/\log\log t})$.

scholarly journals The Tritangent Circles of a Circular Quartic Curve

1932 ◽  
Vol 3 (1) ◽  
pp. 46-52
Author(s):  
H. W. Richmond
Keyword(s):  
Fixed Points ◽  
Space Curve ◽  
Cubic Surface ◽  
Quartic Curve ◽  
Family Of Curves ◽  
The Family ◽  
The Subject ◽  
Simple Notation ◽  
Pass Through

§1. In a recent paper with this title Prof. W. P. Milne has discussed the properties of the conics which pass through two fixed points of a plane quartic curve and touch the curve at three other points. In dealing with a numerous family of curves such as this it is very desirable to have a scheme of marks or labels to distinguish the different members of the family; Hesse's notation for the double tangents of a C4 illustrates this. By using another line of approach to the subject, by projecting the curve of intersection of a quadric and a cubic surface from a point at which (under exceptional circumstances) the surfaces touch, I find that a fairly simple notation for the 64 conics, in harmony with that for the bitangents, can be obtained. This paper, let it be said, from start to finish is no more than an adaptation of results known for the sextic space-curve referred to; it will be sufficient therefore to state results with short explanations.

A Review of Phytochemical and Pharmacological Studies of Inula Species

2020 ◽  
Vol 16 (5) ◽  
pp. 557-567
Author(s):  
Aparoop Das ◽  
Anshul Shakya ◽  
Surajit Kumar Ghosh ◽  
Udaya P. Singh ◽  
Hans R. Bhat
Keyword(s):  
Bacterial Infections ◽  
Chemical Constituents ◽  
Valuable Insight ◽  
Important Ingredient ◽  
Pharmacological Activities ◽  
Medicinal Uses ◽  
The Family ◽  
Pharmacological Studies ◽  
Perennial Herbs ◽  
Insight Into

Background: Plants of the genus Inula are perennial herbs of the family Asteraceae. This genus includes more than 100 species, widely distributed throughout Europe, Africa and Asia including India. Many of them are indicated in traditional medicine, e.g., in Ayurveda. This review explores chemical constituents, medicinal uses and pharmacological actions of Inula species. Methods: Major databases and research and review articles retrieved through Scopus, Web of Science, and Medline were consulted to obtain information on the pharmacological activities of the genus Inula published from 1994 to 2017. Results: Inula species are used either alone or as an important ingredient of various formulations to cure dysfunctions of the cardiovascular system, respiratory system, urinary system, central nervous system and digestive system, and for the treatment of asthma, diabetes, cancers, skin disorders, hepatic disease, fungal and bacterial infections. A range of phytochemicals including alkaloids, essential and volatile oils, flavonoids, terpenes, and lactones has been isolated from herbs of the genus Inula, which might possibly explain traditional uses of these plants. Conclusion: The present review is focused on chemical constituents, medicinal uses and pharmacological actions of Inula species and provides valuable insight into its medicinal potential.

scholarly journals Duality of Moduli and Quasiconformal Mappings in Metric Spaces

2020 ◽  
Vol 8 (1) ◽  
pp. 166-181
Author(s):  
Rebekah Jones ◽  
Panu Lahti
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Poincaré Inequality ◽  
Quasiconformal Mappings ◽  
Duality Relation ◽  
Doubling Measure ◽  
Poincare Inequality ◽  
Family Of Curves ◽  
The Family

AbstractWe prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincaré inequality. Then we apply this to show that quasiconformal mappings can be characterized by the fact that they quasi-preserve the modulus of certain families of surfaces.

scholarly journals A family of quadrics

1938 ◽  
Vol 5 (4) ◽  
pp. 207-210
Author(s):  
L. M. Brown
Keyword(s):  
General Position ◽  
Common Part ◽  
The Family ◽  
The Common ◽  
Pass Through

In a recent paper I have discussed the families of quadrics in [2n] which are obtained by causing the members to have the greatest possible number of fixed [n – 1]'s or “generators.” It was found possible to fix four [n – 1]'s in general position; the family of quadrics through these possessed a “base” variety, common to all the members, which consisted of a highly degenerate Vn–1. Here I consider the same problem for quadrics in [2n + 1], find how many generators may be assigned arbitrarily and discuss the common part of the quadrics which pass through such generators.

Abstract Voronoi Diagrams from Closed Bisecting Curves

2017 ◽  
Vol 27 (03) ◽  
pp. 221-240
Author(s):  
Cecilia Bohler ◽  
Rolf Klein ◽  
Chih-Hung Liu
Keyword(s):  
Voronoi Diagrams ◽  
Connected Components ◽  
Voronoi Region ◽  
Jordan Curves ◽  
Intersection Points ◽  
Number Of Faces

We present the first algorithm for constructing abstract Voronoi diagrams from bisectors that are unbounded or closed Jordan curves. It runs in expected [Formula: see text] many steps and [Formula: see text] space, where [Formula: see text] is the number of sites, [Formula: see text] denotes the average number of faces (connected components) per Voronoi region in any diagram of a subset of [Formula: see text] sites, and [Formula: see text] is the maximum number of intersection points between any two related bisectors.

On the Family of Curves z=(t-tn)/2

1962 ◽  
Vol 35 (4) ◽  
pp. 211
Author(s):  
M. Stephanie Sloyan
Keyword(s):  
Family Of Curves ◽  
The Family

scholarly journals Apoderus coryli (L.) – a Biologically Little Known Species of the Attelabidae (Coleoptera)

2014 ◽  
Vol 62 (5) ◽  
pp. 1141-1160 ◽  
Author(s):  
Jaroslav Urban
Keyword(s):  
First Generation ◽  
Second Generation ◽  
Individual Development ◽  
Leaf Roller ◽  
The Family ◽  
Males And Females ◽  
Pass Through ◽  
The Individual ◽  
First And Second Generation ◽  
Cylindrical Roll

Hazel-leaf roller weevil (Apoderus coryli /L./) is a noteworthy species from the perspective of biology and forestry; it belongs to the family Attelabidae. Its occurrence, development and harmfulness were studied in surroundings of Brno city in 2011 and 2012. Imagoes of the first generation and those of not very numerous second generation are observed to winter in the area under study. From the beginning of May to the end of July they occur on the host woody plants (mainly on Carpinus betulus and Corylus avellana). The males and females consume on average 21 and 33 cm2 of leaves, respectively. The fertilized females cut into the leaf blade in an original manner, and bite into the main and side leaf veins. They fold the withering part of the blade lengthwise to the adaxial face first, and then forming the folded blade into a short cylindrical roll. In the initial phase of rolling, the females lay up on average 1.0 egg into the leaf rolls on C. betulus (1.2 on C. avellana). In total, they make around 30 rolls. The larvae emerge on average within 10 days. In the course of 3 to 4 weeks, they pass through two instars only and damage on average 4 cm2 of leaves. This work describes the occurrence and development of the beetles of the first and second generation. It provides an assessment of the mortality of the individual development stages of A. coryli within the rolls. It was demonstrated that rolling of the leaves causes on average 9 times more damage to the trees than maturation feeding of the beetles.

scholarly journals Engagement Ceremony in the Sothern Coastal Area of Albania, An Ethno-Folkloric View

2016 ◽  
Vol 1 (1) ◽  
pp. 132
Author(s):  
Blegina Bezo (Hasko)
Keyword(s):  
Coastal Area ◽  
Moral Issue ◽  
Educational Preparation ◽  
Moral Importance ◽  
The Family ◽  
Pass Through ◽  
Seven Generations

One of the most important rituals in this area is also the engagement ceremony, which was considered like a prenuptial agreement of both young ones and it didn't have only a moral importance but also a juridical one between both families. But this tradition had to pass through some rules that each family had to follow starting from the matchmaker who was supposed to be a respected person to the gifts that was made according to the tradition. Also, in this tradition existed some strict rules where the consanguinity of the young ones, who were going to be engaged was forbidden. The engagement was allowed only if they were cousins after seven generations had passed. This rules was to avoid problems with the baby's health and also that the baby would be pure blood. Anyway the engagement could never be broken by the girls or the girl's family as I was considered to be a great offense to the boy and this could bring blood feud. The engagement ceremonial started from the day that the word was given and the setting date of the day. That rings were exchanged, that happened on Monday or Thursday, which was accompanied by giving gifts. But engagement can be broken from the family of the boy only for some reasons, if the girl was sick or deterioration of relations between the families up in enmity to property issues, if separate for moral issue then the family of the boy had to give explanations for claims that they had based on facts, otherwise that was a great insult to the girl's family amounting to enmity. A key step in this ritual is the moral and educational preparation for daughter as well for the men, and also the preparation of the girl for this holy day from the preparation and decoration of the girl from dyeing hair with henna and to the preparation of the dowry.

scholarly journals Venn Diagrams with Few Vertices

10.37236/1382 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Bette Bultena ◽  
Frank Ruskey
Keyword(s):  
Lower Bound ◽  
Planar Graph ◽  
Connected Region ◽  
Venn Diagram ◽  
Closed Curves ◽  
Venn Diagrams ◽  
Intersection Points ◽  
Few Vertices

An $n$-Venn diagram is a collection of $n$ finitely-intersecting simple closed curves in the plane, such that each of the $2^n$ sets $X_1 \cap X_2 \cap \cdots \cap X_n$, where each $X_i$ is the open interior or exterior of the $i$-th curve, is a non-empty connected region. The weight of a region is the number of curves that contain it. A region of weight $k$ is a $k$-region. A monotone Venn diagram with $n$ curves has the property that every $k$-region, where $0 < k < n$, is adjacent to at least one $(k-1)$-region and at least one $(k+1)$-region. Monotone diagrams are precisely those that can be drawn with all curves convex. An $n$-Venn diagram can be interpreted as a planar graph in which the intersection points of the curves are the vertices. For general Venn diagrams, the number of vertices is at least $ \lceil {2^n-2 \over n-1} \rceil$. Examples are given that demonstrate that this bound can be attained for $1 < n \le 7$. We show that each monotone Venn diagram has at least ${n \choose {\lfloor n/2 \rfloor}}$ vertices, and that this lower bound can be attained for all $n > 1$.

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