scholarly journals Prismatic Sets Associated with Planar Rooted Trees

2019 ◽  
Vol 2019 ◽  
pp. 1-6
Author(s):  
S. Kaan Gürbüzer ◽  
Bedia Akyar
Keyword(s):  
Rooted Trees ◽  
Prismatic Structure ◽  
Algebraic Operations ◽  
Planar Rooted Trees

We construct the almost strong prismatic structure on the set of planar rooted trees and the bicomplex of planar rooted trees. Furthermore, we study the prismatic properties of Loday’s algebraic operations on the set of planar rooted trees.

scholarly journals A combinatorial non-commutative Hopf algebra of graphs

2014 ◽  
Vol Vol. 16 no. 1 (Combinatorics) ◽  
Author(s):  
Adrian Tanasa ◽  
Gerard Duchamp ◽  
Loïc Foissy ◽  
Nguyen Hoang-Nghia ◽  
Dominique Manchon
Keyword(s):  
Hopf Algebra ◽  
Rooted Trees ◽  
Quantum Field ◽  
International Audience ◽  
The One ◽  
Planar Rooted Trees

Combinatorics International audience A non-commutative, planar, Hopf algebra of planar rooted trees was defined independently by one of the authors in Foissy (2002) and by R. Holtkamp in Holtkamp (2003). In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we use a quantum field theoretical (QFT) idea, namely the one of introducing discrete scales on each edge of the graph (which, within the QFT framework, corresponds to energy scales of the associated propagators). Finally, we analyze the associated quadri-coalgebra and codendrifrom structures.

scholarly journals Constructing combinatorial operads from monoids

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Samuele Giraudo
Keyword(s):  
Rooted Trees ◽  
Parking Functions ◽  
Combinatorial Objects ◽  
Dyck Paths ◽  
International Audience ◽  
Motzkin Paths ◽  
Planar Rooted Trees

International audience We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operad obtained from the additive monoid. These involve various familiar combinatorial objects: parking functions, packed words, planar rooted trees, generalized Dyck paths, Schröder trees, Motzkin paths, integer compositions, directed animals, etc. We also retrieve some known operads: the magmatic operad, the commutative associative operad, and the diassociative operad.

scholarly journals Hopf algebras on planar trees and permutations

2021 ◽  
Author(s):  
Diego Arcis ◽  
Sebastián Márquez
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Rooted Trees ◽  
Labeled Trees ◽  
Different Types ◽  
Structure Lead ◽  
Planar Rooted Trees

We endow the space of rooted planar trees with the structure of a Hopf algebra. We prove that variations of such a structure lead to Hopf algebras on the spaces of labeled trees, [Formula: see text]-trees, increasing planar trees and sorted trees. These structures are used to construct Hopf algebras on different types of permutations. In particular, we obtain new characterizations of the Hopf algebras of Malvenuto–Reutenauer and Loday–Ronco via planar rooted trees.

scholarly journals On the Lie enveloping algebra of a pre-Lie algebra

2008 ◽  
Vol 2 (1) ◽  
pp. 147-167 ◽  
Author(s):  
J.-M. Oudom ◽  
D. Guin
Keyword(s):  
Hopf Algebra ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Rooted Trees ◽  
Enveloping Algebra ◽  
Associative Product ◽  
Planar Rooted Trees ◽  
Symmetric Brace Algebras

AbstractWe construct an associative product on the symmetric module S(L) of any pre-Lie algebra L. It turns S(L) into a Hopf algebra which is isomorphic to the enveloping algebra of LLie. Then we prove that in the case of rooted trees our construction gives the Grossman-Larson Hopf algebra, which is known to be the dual of the Connes-Kreimer Hopf algebra. We also show that symmetric brace algebras and pre-Lie algebras are the same. Finally, we give a similar interpretation of the Hopf algebra of planar rooted trees.

scholarly journals Typed Angularly Decorated Planar Rooted Trees and Ω-Rota-Baxter Algebras

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 190
Author(s):  
Yi Zhang ◽  
Xiaosong Peng ◽  
Yuanyuan Zhang
Keyword(s):  
Rooted Trees ◽  
Integral Analysis ◽  
Dendriform Algebra ◽  
Dendriform Algebras ◽  
Definition Of ◽  
The Relationship ◽  
Planar Rooted Trees

As a generalization of Rota–Baxter algebras, the concept of an Ω-Rota–Baxter could also be regarded as an algebraic abstraction of the integral analysis. In this paper, we introduce the concept of an Ω-dendriform algebra and show the relationship between Ω-Rota–Baxter algebras and Ω-dendriform algebras. Then, we provide a multiplication recursion definition of typed, angularly decorated rooted trees. Finally, we construct the free Ω-Rota–Baxter algebra by typed, angularly decorated rooted trees.

scholarly journals A Comparative Evaluation of Nanohydroxyapatite-Enriched Hydrogen Peroxide Home Bleaching System on Color, Hardness and Microstructure of Dental Enamel

Materials ◽  
2021 ◽  
Vol 14 (11) ◽  
pp. 3072
Author(s):  
Riccardo Monterubbianesi ◽  
Vincenzo Tosco ◽  
Tiziano Bellezze ◽  
Giampaolo Giuliani ◽  
Mutlu Özcan ◽  
...  
Keyword(s):  
Hydrogen Peroxide ◽  
Particle Diameter ◽  
Dental Enamel ◽  
Sem Analysis ◽  
Control Group ◽  
Average Particle ◽  
Color Analysis ◽  
Enamel Microstructure ◽  
Prismatic Structure ◽  
20 Nm

This study aimed to evaluate two hydrogen peroxide (HP)-based at-home bleaching systems in order to analyze whether nano-hydroxyapatite (nHA) addition may represent a reliable and safe solution for tooth whitening without altering dental microstructure and hardness. Human third molars (N = 15) were treated with two bleaching agents, one containing 6%HP (6HP) and the other 6% HP nHA-enriched (6HP-nHA) with average particle diameter ranging from 5–20 nm. Their effects on enamel were assessed using a spectrophotometer, Vickers microhardness (VMH) test and Scanning Electron Microscopy (SEM), comparing the treated groups with the non-treated control group (CTR). Color analysis revealed improvement in whiteness in both groups compared to CTR. VMH test results showed no differences among the groups. SEM analysis highlighted no evident changes in the enamel microstructure of tested groups compared to CTR. At high magnification, in 6HP group, a slight increase in irregularities of enamel surface morphology was observed, while 6HP-nHA group displayed removal of the aprismatic layer but preservation of the intact prismatic structure. These results suggest that the 6HP-nHA agent may be recommended to provide reliable whitening treatment, without damaging the enamel micromorphology and hardness.

Closed Sets of Boolean Terms in Relational Databases

1991 ◽  
Vol 14 (3) ◽  
pp. 367-385
Author(s):  
Andrzej Jankowski ◽  
Zbigniew Michalewicz
Keyword(s):  
Computational Complexity ◽  
Incomplete Information ◽  
Relational Databases ◽  
The Other ◽  
New Approach ◽  
Closed Sets ◽  
Null Value ◽  
Algebraic Operations

A number of approaches have been taken to represent compound, structured values in relational databases. We review a few such approaches and discuss a new approach, in which every set is represented as a Boolean term. We show that this approach generalizes the other approaches, leading to more flexible representation. Boolean term representation seems to be appropriate in handling incomplete information: this approach generalizes some other approaches (e.g. null value mark, null variables, etc). We consider definitions of algebraic operations on such sets, like join, union, selection, etc. Moreover, we introduce a measure of computational complexity of these operations.

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