Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota-Baxter Operators

We will introduce an associative (or quantum) version of Poisson structure tensors. This object is defined as an operator satisfying a "generalized" Rota-Baxter identity of weight zero. Such operators are called generalized Rota-Baxter operators. We will show that generalized Rota-Baxter operators are characterized by a cocycle condition so that Poisson structures are so. By analogy with twisted Poisson structures, we propose a new operator "twisted Rota-Baxter operators" which is a natural generalization of generalized Rota-Baxter operators. It is known that classical Rota-Baxter operators are closely related with dendriform algebras. We will show that twisted Rota-Baxter operators induce NS-algebras which is a twisted version of dendriform algebra. The twisted Poisson condition is considered as a Maurer-Cartan equation up to homotopy. We will show the twisted Rota-Baxter condition also is so. And we will study a Poisson-geometric reason, how the twisted Rota-Baxter condition arises.Comment: 18 pages. Final versio

Rota-Baxter algebras and new combinatorial identities

The word problem for an arbitrary associative Rota-Baxter algebra is solved. This leads to a noncommutative generalization of the classical Spitzer identities. Links to other combinatorial aspects, particularly of interest in physics, are indicated.Comment: 8 pages, improved versio

Time-ordering and a generalized Magnus expansion

Both the classical time-ordering and the Magnus expansion are well-known in the context of linear initial value problems. Motivated by the noncommutativity between time-ordering and time derivation, and related problems raised recently in statistical physics, we introduce a generalization of the Magnus expansion. Whereas the classical expansion computes the logarithm of the evolution operator of a linear differential equation, our generalization addresses the same problem, including however directly a non-trivial initial condition. As a by-product we recover a variant of the time ordering operation, known as T*-ordering. Eventually, placing our results in the general context of Rota-Baxter algebras permits us to present them in a more natural algebraic setting. It encompasses, for example, the case where one considers linear difference equations instead of linear differential equations

Generalized shuffles related to Nijenhuis and TD-algebras

Shuffle and quasi-shuffle products are well-known in the mathematics literature. They are intimately related to Loday's dendriform algebras, and were extensively used to give explicit constructions of free commutative Rota-Baxter algebras. In the literature there exist at least two other Rota-Baxter type algebras, namely, the Nijenhuis algebra and the so-called TD-algebra. The explicit construction of the free unital commutative Nijenhuis algebra uses a modified quasi-shuffle product, called the right-shift shuffle. We show that another modification of the quasi-shuffle product, the so-called left-shift shuffle, can be used to give an explicit construction of the free unital commutative TD-algebra. We explore some basic properties of TD-operators and show that the free unital commutative Nijenhuis algebra is a TD-algebra. We relate our construction to Loday's unital commutative dendriform trialgebras, including the involutive case. The concept of Rota-Baxter, Nijenhuis and TD-bialgebras is introduced at the end and we show that any commutative bialgebra provides such objects.Comment: 20 pages, typos corrected, accepted for publication in Communications in Algebr

Evaluating the effects of language on international trade in MENA countries: A gravity-model approach

Prior studies have investigated the role of economic and noneconomic variables on international trade. A major factor, which has been studied less, is the language used in transactions and negotiations. We explore the effects of language connectedness and the Arabic language on international trade in thirteen countries in the Middle East and North Africa (MENA) region. We used a panel of bilateral data and gravity model for the countries of the region over the 2000 to 2018 period. Our analytic technique was the Poisson pseudo-maximum-likelihood (PPML) estimation method. The empirical outcomes indicate that speaking Arabic leads to an increase in export, that is, Arab nations prefer to export to the countries whose people speak their language. In addition, the language connectedness index, which depends on the extent to which the country's languages are spoken outside the country, is positively associated with the levels of exports and imports. Results further show that the GDP, population of the destination country, and political co-stability have significant positive impacts on the bilateral exports. Additionally, GDP, the population of the source country, political co-stability, and a common border have had significant positive influences on bilateral imports. The major contribution of this research is that the Arabic language has a significant and positive impact on trade among MENA countries

Renormalization: a quasi-shuffle approach

In recent years, the usual BPHZ algorithm for renormalization in perturbative quantum field theory has been interpreted, after dimensional regularization, as a Birkhoff decomposition of characters on the Hopf algebra of Feynman graphs, with values in a Rota-Baxter algebra of amplitudes. We associate in this paper to any such algebra a universal semi-group (different in nature from the Connes-Marcolli "cosmical Galois group"). Its action on the physical amplitudes associated to Feynman graphs produces the expected operations: Bogoliubov's preparation map, extraction of divergences, renormalization. In this process a key role is played by commutative and noncommutative quasi-shuffle bialgebras whose universal properties are instrumental in encoding the renormalization process

Mixable Shuffles, Quasi-shuffles and Hopf Algebras

The quasi-shuffle product and mixable shuffle product are both generalizations of the shuffle product and have both been studied quite extensively recently. We relate these two generalizations and realize quasi-shuffle product algebras as subalgebras of mixable shuffle product algebras. As an application, we obtain Hopf algebra structures in free Rota-Baxter algebras.Comment: 14 pages, no figure, references update

Preparation of Gold Nanoparticles for Biomedical Applications Using Chemometric Technique

Purpose: To study the effect of process conditions on the size of gold nanoparticles (AuNPs) prepared by chemometric technique.Methods: AuNPs were prepared by adding 5 ml of 5 mM of gold (III) chloride hydrate HAuCl4 (2 mg/mL) to 85 ml of filtered deionized water, then refluxed in a 250 mL flask over a hot plate and heated to boiling point. Five milliliters of sodium citrate solution of varying concentrations were quickly added to the boiling solution and stirred for 30 min until the color turned to wine red. Chemometric approach, based on multivariant analysis, was applied to the optimization of iron oxide nanoparticle size in respect of three parameters, viz, concentration of sodium citrate solution, stirrer speed and ionic strength of the medium. The experiments were performed according to Box-Behnken experimental design.Results: The regression model obtained was characterized by both descriptive and predictive ability. The method was optimized with respect to average diameter as a response. The average diameter of the nanoparticles produced under different conditions were between 17.7 up to 168.8 nm. The criteria for the evaluation of the descriptive capability of a polynomial were R2 = 0.974, standard error = 13.994 and F-ratio = 18.4.Conclusion: It can be concluded that the Box-Behnken experimental design provides a suitable approach for optimizing and testing the robustness of gold nanoparticle preparation method.Keywords: Box-Behnken design, Optimization, Nanoparticles, Gold, Biomedical, Chemometric