scholarly journals Free dendriform algebras. Part I. A parenthesis setting

2006 ◽  
Vol 2006 ◽  
pp. 1-16
Author(s):  
Philippe Leroux
Keyword(s):  
Free Probability ◽  
Dendriform Algebra ◽  
Dendriform Algebras

We propose both a reformulation of some known results on the free dendriform algebra on one generator from a parenthesis setting instead of using permutations and some developments as well. Moreover, by introducing the concept of NCP-operad, we show how to use the free dendriform algebra on one generator to reformulate some results obtained by Speicher in free probability.

scholarly journals Embedding of dendriform algebras into Rota-Baxter algebras

2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Vsevolod Gubarev ◽  
Pavel Kolesnikov
Keyword(s):  
Recent Work ◽  
Dendriform Algebra ◽  
Dendriform Algebras

AbstractFollowing a recent work [Bai C., Bellier O., Guo L., Ni X., Splitting of operations, Manin products, and Rota-Baxter operators, Int. Math. Res. Not. IMRN (in press), DOI: 10.1093/imrn/rnr266] we define what is a dendriform dior trialgebra corresponding to an arbitrary variety Var of binary algebras (associative, commutative, Poisson, etc.). We call such algebras di- or tri-Var-dendriform algebras, respectively. We prove in general that the operad governing the variety of di- or tri-Var-dendriform algebras is Koszul dual to the operad governing di- or trialgebras corresponding to Var!. We also prove that every di-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of weight zero in the variety Var, and every tri-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of nonzero weight in Var.

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 Plate-like structure damage location identification based on Lamb wave baseline-free probability imaging method

2017 ◽  
Vol 9 (1) ◽  
pp. 168781401668570 ◽  
Author(s):  
Dongsheng Li ◽  
Zihao Jing ◽  
Mengdao Jin
Keyword(s):  
Lamb Wave ◽  
Time Window ◽  
Free Probability ◽  
Structural Response ◽  
Guided Wave ◽  
Time Of Arrival ◽  
Imaging Method ◽  
Detection Techniques ◽  
Damage Location ◽  
Ultrasonic Guided Wave

Damage-scattering signal extraction using conventional ultrasonic guided wave–based damage detection techniques requires the measurement of baseline data under pristine condition. This study proposes a baseline-free ultrasonic guided wave damage localization and imaging method based on Lamb wave baseline-free probability imaging method. Although traditional Lamb wave probability imaging can monitor damage location in plate-like structures, the absolute time of arrival and magnitude of the signal are affected by several factors and are therefore difficult to obtain. This study also proposes a probability-based hyperbola diagnostic imaging method that is based on different times of arrival and has no magnitude information. A distributed active sensor network conforming to a pulse-echo configuration and time window functions is developed to separate damage-scattering signals from structural response signals. Continuous wavelet transform is used to calculate the time of flight of damage signal waves. The numerical simulation and experiments validate the effectiveness of the proposed method in identifying damage.

scholarly journals Dual Creation Operators and a Dendriform Algebra Structure on the Quasisymmetric Functions

2017 ◽  
Vol 69 (1) ◽  
pp. 21-53 ◽  
Author(s):  
Darij Grinberg
Keyword(s):  
Hopf Algebras ◽  
Young Tableau ◽  
Vertex Operators ◽  
Schur Function ◽  
Algebra Structure ◽  
Quasisymmetric Functions ◽  
Combinatorial Hopf Algebras ◽  
Dendriform Algebra ◽  
Natural Analogues ◽  
Creation Operators

AbstractThe dual immaculate functions are a basis of the ring QSym of quasisymmetric functions and form one of the most natural analogues of the Schur functions. The dual immaculate function corresponding to a composition is a weighted generating function for immaculate tableaux in the same way as a Schur function is for semistandard Young tableaux; an immaculate tableau is defined similarly to a semistandard Young tableau, but the shape is a composition rather than a partition, and only the first column is required to strictly increase (whereas the other columns can be arbitrary, but each row has to weakly increase). Dual immaculate functions were introduced by Berg, Bergeron, Saliola, Serrano, and Zabrocki in arXiv:1208.5191, and have since been found to possess numerous nontrivial properties.In this note, we prove a conjecture of M. Zabrocki that provides an alternative construction for the dual immaculate functions in terms of certain “vertex operators”. The proof uses a dendriform structure on the ring QSym; we discuss the relation of this structure to known dendriformstructures on the combinatorial Hopf algebras FQSym andWQSym.

scholarly journals A rigidity result for crossed products of actions of Baumslag–Solitar groups

2015 ◽  
Vol 26 (14) ◽  
pp. 1550117
Author(s):  
Niels Meesschaert
Keyword(s):  
Probability Measure ◽  
Free Probability ◽  
Crossed Product ◽  
Crossed Products ◽  
Orbit Equivalence ◽  
Rigidity Result ◽  
Profinite Completions ◽  
Abelian Subgroups ◽  
Measure Equivalence ◽  
Measure Preserving Actions

Let [Formula: see text] and [Formula: see text] be two ergodic essentially free probability measure preserving actions of nonamenable Baumslag–Solitar groups whose canonical almost normal abelian subgroups act aperiodically. We prove that an isomorphism between the corresponding crossed product II1 factors forces [Formula: see text] when [Formula: see text] and [Formula: see text] when [Formula: see text]. This improves an orbit equivalence rigidity result obtained by Houdayer and Raum in [Baumslag–Solitar groups, relative profinite completions and measure equivalence rigidity, J. Topol. 8 (2015) 295–313].

scholarly journals Dual Lukacs regressions of negative orders for noncommutative variables

2014 ◽  
Vol 17 (03) ◽  
pp. 1450021 ◽  
Author(s):  
Kamil Szpojankowski
Keyword(s):  
Free Probability ◽  
Random Variable ◽  
Probability Measures ◽  
Classical Probability ◽  
Binomial Distributions ◽  
Beta Distributions

In the paper we study characterizations of probability measures in free probability. By constancy of regressions for random variable 𝕍1/2(𝕀 - 𝕌)𝕍1/2 given by 𝕍1/2𝕌𝕍1/2, where 𝕌 and 𝕍 are free, we characterize free Poisson and free binomial distributions. Our paper is a free probability analogue of results known in classical probability,3 where gamma and beta distributions are characterized by constancy of 𝔼((V(1 - U))i|UV), for i ∈ {-2, -1, 1, 2}. This paper together with previous results18 exhaust all cases of characterizations from Ref. 3.

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