scholarly journals The Bernardi Formula for Nontransitive Deformations of the Braid Arrangement

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Ankit Bisain ◽  
Eric Hanson
Keyword(s):  
General Formula ◽  
Plane Tree ◽  
Plane Trees ◽  
Counting Formula

Bernardi has given a general formula for the number of regions of a deformation of the braid arrangement as a signed sum over boxed trees. We prove that each set of boxed trees which share an underlying (rooted labeled plane) tree contributes 0 or $\pm 1$ to this sum, and we give an algorithm for computing this value. For Ish-type arrangements, we further construct a sign-reversing involution which reduces Bernardi's signed sum to the enumeration of a set of (rooted labeled plane) trees. We conclude by explicitly enumerating the trees corresponding to the regions of Ish-type arrangements which are nested, recovering their known counting formula.

The Plane Trees of Diomedes: Staging the Islands of the Blessed in the Adriatic Sea

Numen ◽  
2015 ◽  
Vol 62 (5-6) ◽  
pp. 495-518
Author(s):  
Matthias Egeler
Keyword(s):  
Adriatic Sea ◽  
Conclusive Evidence ◽  
Primary Sources ◽  
Plane Tree ◽  
Historical Case ◽  
Direct Relevance ◽  
Starting Point ◽  
Plane Trees ◽  
New Interpretation ◽  
Adriatic Region

Taking its starting point from the current theoretical debate about “religious aesthetics” (Religionsästhetik), this article offers a new interpretation of the landscaping of the Island of Diomedes. On the basis of a survey of the primary sources, it demonstrates that the descriptions of the Island of Diomedes have to be seen in the context of the mythological motif of the Islands of the Blessed. This in turn suggests an explanation for Theophrastus’ statement that in the Adriatic region of his day, the plane tree grew only around the shrine of Diomedes: given the typical associations of the plane tree, the planting of planes on the sanctuary island of Diomedes may have been intended to bring the appearance of the Island of Diomedes in line with contemporary stereotypes about the appearance of the Islands of the Blessed; thus, the plane trees of Diomedes constitute a historical case of direct relevance for the question of “religious aesthetics.” The article goes on to demonstrate that the plane trees of Diomedes constitute conclusive evidence against the identification of the Island of Diomedes with the island of Pelagosa (recently suggested on the basis of archaeological finds). Instead, the Island of Diomedes should be sought among the Isole Tremiti.

scholarly journals Reflexes of a Hurrian Word in Armenian: A Theonym, a Dendronym, an Anthroponym

2021 ◽  
Vol 18 (3) ◽  
pp. 100-109
Author(s):  
Armen Ye. Petrosyan ◽  
Keyword(s):  
Local Version ◽  
Plane Tree ◽  
Plural Form ◽  
Great Goddess ◽  
Plane Trees ◽  
The Gift ◽  
Bright Color ◽  
Armenian Highland ◽  
The City

In Old Armenian, saws means ‘proud, luxurious, great,’ ‘some (bright) color,’ and saws and sawsi mean ‘oriental plane tree’. The word has no etymology. Hurrian has the word šauša [sausa] ‘big, great’ and the theonym Šauša / Šauška for the local version of the great goddess Ištar. The article undertakes to find a single etymon looking for the clue in comparative mythology. It is known that Anušavan, one of the ancient Armenian mythical patriarchs, was referred to as Sawsanuēr which can be interpreted as “The gift of plane trees” (with a reference to the cult of the plane trees of Armawir, the earliest capital of Armenia). According to mythology, Anushavan’s father and grandfather were related to Šamiram (Greek Semiramis), the queen of the city of Nineveh (capital of Assyria) that is seen as a historicized version of the local goddess Šauš(k)a otherwise called “Ishtar of Nineveh.” The Armenian saws ‘great, magnificent’ quite correlates with this name as a loan from the Hurrian šauša ‘great,’ with a regular apocope. The plane trees were probably symbols of the goddess. Thus, it is natural to assume that the dendronym saws / sawsi (the second form with the Indo-European suffix *-iyā, characteristic of Armenian dendronyms, cf. the genitive plural form sawseac‘) is of Hurrian origin. The first meaning of the Hurrian word ‘great, magnificent’ subsequently turned into theonym and then to the Armenian dendronym, the name of the largest and most luxurious tree in the Armenian Highland and adjacent territories.

scholarly journals Equipopularity Classes of 132-Avoiding Permutations

10.37236/3675 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Lynn Chua ◽  
Krishanu Roy Sankar
Keyword(s):  
Generating Functions ◽  
Complete Classification ◽  
Connected Components ◽  
Plane Tree ◽  
Plane Trees ◽  
Set Of Permutations ◽  
Spine Structure

The popularity of a pattern $p$ in a set of permutations is the sum of the number of copies of $p$ in each permutation of the set. We study pattern popularity in the set of 132-avoiding permutations. Two patterns are equipopular if, for all $n$, they have the same popularity in the set of length-$n$ 132-avoiding permutations. There is a well-known bijection between 132-avoiding permutations and binary plane trees. The spines of a binary plane tree are defined as the connected components when all edges connecting left children to their parents are deleted, and the spine structure is the sorted sequence of lengths of the spines. Rudolph shows that patterns of the same length are equipopular if their associated binary plane trees have the same spine structure. We prove the converse of this result using the method of generating functions, which gives a complete classification of 132-avoiding permutations into equipopularity classes.

scholarly journals Protection number in plane trees

2017 ◽  
Vol 11 (2) ◽  
pp. 314-326
Author(s):  
Clemens Heuberger ◽  
Helmut Prodinger
Keyword(s):  
Generating Functions ◽  
Singularity Analysis ◽  
Random Tree ◽  
Minimal Distance ◽  
Plane Tree ◽  
Plane Trees

The protection number of a plane tree is the minimal distance of the root to a leaf; this definition carries over to an arbitrary node in a plane tree by considering the maximal subtree having this node as a root. We study the the protection number of a uniformly chosen random tree of size n and also the protection number of a uniformly chosen node in a uniformly chosen random tree of size n. The method is to apply singularity analysis to appropriate generating functions. Additional results are provided as well.

scholarly journals Hook Length Formulas for Trees by Han's Expansion

10.37236/151 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
William Y.C. Chen ◽  
Oliver X.Q. Gao ◽  
Peter L. Guo
Keyword(s):  
Weight Function ◽  
General Formula ◽  
Generating Functions ◽  
Binary Trees ◽  
Hook Length ◽  
Labeled Trees ◽  
Plane Trees

Recently Han obtained a general formula for the weight function corresponding to the expansion of a series in terms of hook lengths of binary trees. In this paper, we present weight function formulas for $k$-ary trees, plane trees, plane forests, labeled trees and forests. We also find appropriate generating functions which lead to unifications of the hook length formulas due to Du and Liu, Han, Gessel and Seo, and Postnikov.

scholarly journals Neoteny of the European plane tree

2012 ◽  
Vol 64 (2) ◽  
pp. 771-776
Author(s):  
Mirjana Ocokoljic ◽  
Mirjana Sijacic-Nikolic ◽  
Dragica Vilotic ◽  
D. Vujicic
Keyword(s):  
Sexual Maturity ◽  
Reproductive Organs ◽  
Juvenile Stage ◽  
Scientific Study ◽  
Flower Structure ◽  
Plane Tree ◽  
Practical Utilization ◽  
Plane Trees ◽  
Significant Process ◽  
Tree Development

The neoteny of 64-day-old European plane tree autovegetative progeny is described, starting from the fact that sexual maturity in the juvenile stage of tree development is a rare and significant process by which the period of synthesis of new lower taxonomic units is shortened. The observed formation of European plane tree reproductive organs has been classified as induced neoteny. Compared to the inflorescences of mature European plane trees, the number of precocious inflorescences was lower, flower structure was simpler, and the diameters of inflorescences were the same. As neoteny is a genetic and physiological marker for the tree genotype and the species, and as it is classified as an example of parallelinherited variability explained by changed interactions of trophic, hormonal and genetic systems, this analysis of neoteny can be a contribution to its further scientific study and its practical utilization at a larger scale.

X-ray Structure Refinement and Vibrational Spectroscopy of Ca8Gd2 (PO4)6 O2

2013 ◽  
Vol 12 (10) ◽  
pp. 719-726
Author(s):  
R. Ayadi ◽  
Mohamed Boujelbene ◽  
T. Mhiri
Keyword(s):  
Solid State ◽  
General Formula ◽  
Vibrational Spectroscopy ◽  
Powder Diffraction ◽  
Infrared Spectra ◽  
Solid State Reaction ◽  
Structure Refinement ◽  
Unit Cell ◽  
Cell Group ◽  
X Ray

The present paper is interested in the study of compounds from the apatite family with the general formula Ca10 (PO4)6A2. It particularly brings to light the exploitation of the distinctive stereochemistries of two Ca positions in apatite. In fact, Gd-Bearing oxyapatiteCa8 Gd2 (PO4)6O2 has been synthesized by solid state reaction and characterized by X-ray powder diffraction. The site occupancies of substituents is0.3333 in Gd and 0.3333 for Ca in the Ca(1) position and 0. 5 for Gd in the Ca (2) position.  Besides, the observed frequencies in the Raman and infrared spectra were explained and discussed on the basis of unit-cell group analyses.

Toxicity of Para-Chloro Substituted Benzene Derivatives in the Microtox™ Test

1985 ◽  
Vol 20 (2) ◽  
pp. 36-43 ◽  
Author(s):  
Klaus L.E. Kaiser ◽  
Juan M. Ribo ◽  
Brian M. Zaruk
Keyword(s):  
General Formula ◽  
Functional Group ◽  
Structural Parameters ◽  
Systematic Investigation ◽  
Photobacterium Phosphoreum ◽  
Benzene Derivatives ◽  
Quantitative Structure ◽  
Chlorinated Benzenes ◽  
Chloro Derivatives ◽  
Derivatives Of

Abstract This paper gives the results of part of a systematic investigation into contaminant toxicity to Photobacterium phosphoreum in the Microtox™ test. Reported are the toxicity values for 39 para-chloro substituted benzene derivatives of the general formula l-Cl-C6h4-4-X=CH2CH(NH2)COOH, F, SO2NH2, OCH2COOH, CH2COOH, CONHNH2, NHCOCH3, CONH2, CH=CHCOOH, SeOOH, CH2NH2, CH2CH2NH2, NO2, H, CF3, CHO, CH2OH, OH, CH3, CCl3, COCH3, COOH, NH2, SO2C6H5, Cl, CH2COCH3, COCl, CN, OCH3, NCO, NHCH3, I, COC6H5, CH2Cl, SH, CH2SH, NCS, CH2CN and SO2C6H4Cl. Except for the last compound, whose solubility is below the required concentration, the toxicities increase in the presented order with a total range of more than three orders of magnitude. The data are discussed in terms of quantitative structure-toxicity correlations with compound-specific structural parameters. In combination with a previously developed submodel on chlorinated benzenes, phenols, nitrobenzenes and anilines, the observed relationships allow the prediction of the toxicity of some 780 possible chloro derivatives of the general formula C6H5-nClnX, where n=<5 and X is a functional group as listed above.

Synthesis of isomeric 1,1'-phenylene-bis(6-azauracil-5-carbonitriles)

1990 ◽  
Vol 55 (12) ◽  
pp. 2967-2976 ◽  
Author(s):  
Jan Slouka
Keyword(s):  
General Formula ◽  
Mutual Distance ◽  
Planar Molecules ◽  
Amino Derivatives

The described synthesis of all three isomeric 1,1'-phenylene-bis(6-azauracil-5-carbonitriles) IVa-IVc starts from the respective 1-nitrophenyl-6-azauracil-5-carbonitriles Ia-Ic which were reduced to the corresponding amino derivatives IIa-IIc, diazotized, and coupled with ethyl cyanoacetylcarbamate to give the isomeric hydrazones IIIa-IIIc which were finally cyclized to the title compounds containing two 6-azauracil rings. A general formula is presented for calculation of mutual distance of arbitrary atoms in any planar molecules.

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