Baxter Type Research Articles

Quasi-triangular, factorizable Leibniz bialgebras and relative Rota–Baxter operators

Abstract We introduce the notion of quasi-triangular Leibniz bialgebras, which can be constructed from solutions of the classical Leibniz Yang–Baxter equation (CLYBE) whose skew-symmetric parts are invariant. In addition to triangular Leibniz bialgebras, quasi-triangular Leibniz bialgebras contain factorizable Leibniz bialgebras as another subclass, which lead to a factorization of the underlying Leibniz algebras. Relative Rota–Baxter operators with weights on Leibniz algebras are used to characterize solutions of the CLYBE whose skew-symmetric parts are invariant. On skew-symmetric quadratic Leibniz algebras, such operators correspond to Rota–Baxter type operators. Consequently, we introduce the notion of skew-symmetric quadratic Rota–Baxter Leibniz algebras, such that they give rise to triangular Leibniz bialgebras in the case of weight 0, while they are in one-to-one correspondence with factorizable Leibniz bialgebras in the case of nonzero weights.

On Baxter Type Theorems for Generalized Random Gaussian Processes with Independent Values

We construct suitable families of basic functions and prove theorems of Baxter type for generalized Gaussian random processes with independent values. These theorems are used to divide families of such processes into classes. The singularity of probability measures corresponding to representatives of different classes is proved.

Unification Theories: New Results and Examples

This paper is a continuation of a previous article that appeared in AXIOMS in 2018. A Euler’s formula for hyperbolic functions is considered a consequence of a unifying point of view. Then, the unification of Jordan, Lie, and associative algebras is revisited. We also explain that derivations and co-derivations can be unified. Finally, we consider a “modified” Yang–Baxter type equation, which unifies several problems in mathematics.

Anti-corrosion coating for metal surfaces based on superhydrophobic electrosprayed carbon layers

The electrospray deposition of a carbon-polymer ink has been used for the preparation of corrosion protective coatings for metal substrates. The electrosprayed carbon layers are adherent and have highly porous structure that renders Cassie–Baxter type superhydrophobicity. The analysis of their morphology with SEM shows changes in the shape of aggregates with thickness that slightly affects their interaction with water and superhydrophobicity, as a consequence of changing electrospray deposition conditions. Stainless steel immersed in acidic aqueous solutions is efficiently protected from chemical and electrochemical corrosion when coated with electrosprayed carbon layers. The carbon layers with loads above 0.4mgcm−2 are able to avoid water contact with the metallic surface, permanently and during electrochemical polarization, thanks to their internal superhydrophobicity. Electrosprayed carbon coatings convey cheap and durable protection against aqueous corrosive environments. This coating procedure is very appropriate for the protection of internal components in fuel cells (plates, contacts), electrolyzers, and batteries working with aqueous solutions, due to its superhydrophobicity and good electronic conductivity.

Gauge transformations for quasitriangular quasi-Turaev group coalgebras

The purpose of this paper is to introduce an equivalence relation on quasitriangular quasi-Turaev group coalgebras such that the categories of representations of two quasitriangular quasi-Turaev group coalgebras are tensor [Formula: see text]-equivalent. As an application, we construct a new Turaev braided group category by a gauge transformation and give a new solution of Yang–Baxter type equation.

Thermal form-factor approach to dynamical correlation functions of integrable lattice models

We propose a method for calculating dynamical correlation functions at finite temperature in integrable lattice models of Yang–Baxter type. The method is based on an expansion of the correlation functions as a series over matrix elements of a time-dependent quantum transfer matrix rather than the Hamiltonian. In the infinite Trotter-number limit the matrix elements become time independent and turn into the thermal form factors studied previously in the context of static correlation functions. We make this explicit with the example of the XXZ model. We show how the form factors can be summed utilizing certain auxiliary functions solving finite sets of nonlinear integral equations. The case of the XX model is worked out in more detail leading to a novel form-factor series representation of the dynamical transverse two-point function.

Twisted algebras and Rota–Baxter type operators

We define the concept of weak pseudotwistor for an algebra [Formula: see text] in a monoidal category [Formula: see text], as a morphism [Formula: see text] in [Formula: see text], satisfying some axioms ensuring that [Formula: see text] is also an algebra in [Formula: see text]. This concept generalizes the previous proposal called pseudotwistor and covers a number of examples of twisted algebras that cannot be covered by pseudotwistors, mainly examples provided by Rota–Baxter operators and some of their relatives (such as Leroux’s TD-operators and Reynolds operators). By using weak pseudotwistors, we introduce an equivalence relation (called “twist equivalence”) for algebras in a given monoidal category.

On q-deformed symmetries as Poisson–Lie symmetries and application to Yang–Baxter type models

Yang–Baxter type models are integrable deformations of integrable field theories, such as the principal chiral model on a Lie group G or σ-models on (semi-)symmetric spaces G/F. The deformation has the effect of breaking the global G-symmetry of the original model, replacing the associated set of conserved charges by ones whose Poisson brackets are those of the q-deformed Poisson–Hopf algebra . Working at the Hamiltonian level, we show how this q-deformed Poisson algebra originates from a Poisson–Lie G-symmetry. The theory of Poisson–Lie groups and their actions on Poisson manifolds, in particular the formalism of the non-abelian moment map, is reviewed. For a coboundary Poisson–Lie group G, this non-abelian moment map must obey the Semenov-Tian-Shansky bracket on the dual group , up to terms involving central quantities. When the latter vanish, we develop a general procedure linking this Poisson bracket to the defining relations of the Poisson–Hopf algebra , including the q-Poisson–Serre relations. We consider reality conditions leading to q being either real or a phase. We determine the non-abelian moment map for Yang–Baxter type models. This enables to compute the corresponding action of G on the fields parametrising the phase space of these models.

Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from separation of variables

We solve the longstanding problem of defining a functional characterization of the spectrum of the transfer matrix associated with the most general spin-1/2 representations of the six-vertex reflection algebra for general inhomogeneous chains. The corresponding homogeneous limit reproduces the spectrum of the Hamiltonian of the spin-1/2 open XXZ and XXX quantum chains with the most general integrable boundaries. The spectrum is characterized by a second order finite difference functional equation of Baxter type with an inhomogeneous term which vanishes only for some special but yet interesting non-diagonal boundary conditions. This functional equation is shown to be equivalent to the known separation of variables (SOV) representation, hence proving that it defines a complete characterization of the transfer matrix spectrum. The polynomial form of the Q-function allows us to show that a finite system of generalized Bethe equations can also be used to describe the complete transfer matrix spectrum.

The effects of surface energy and roughness on the hydrophobicity of woven fabrics

The wetting behavior of a hydrophobic rough surface is investigated on a surface fabricated by applying low surface tension materials such as silicone or fluoropolymer to polyester woven fabric consisting of multifilament yarns. The roughness factor of various woven fabrics can be calculated by Wenzel’s and Cassie–Baxter’s equations. For the fabrics treated with silicone or fluoropolymer, the Cassie–Baxter model was applied, showing a level of agreement for the fabric specimens non-textured filament fibers between the predicted contact angles and the measured values. More precisely, the fabrics treated with silicone or fluoropolymer represent the transitional state between the Wenzel type and the Cassie–Baxter type; that is, the fractional contact area between the water and air f2 is greater than zero, and the sum of the fractional contact areas for solid–water f1 and air–water f2 is greater than 1. A surface with lower energy and higher roughness gave f1 + f2 close to 1 with smaller f1 and larger f2, which resulted in a high contact angle.

Clustered copper nanorod arrays: a new class of adhesive hydrophobic materials

Clustered copper nanorod arrays constitute single-component, two-level hierarchical structures that mimic rose petals by showing high contact angles as well as considerable contact angle hysteresis (CAH). The cluster morphology of the nanorod arrays – electrochemically grown within porous alumina templates – is determined by the template geometry and the drying conditions. The CAH (a measure of the adhesion) is a function of the nanorod length and the cluster size to inter-cluster separation ratio. The measured pinning force was 176 μN, among the highest reported. The first level of the hierarchical nano-microstructure consists of the nanorod clusters and the air trapped between them – leading to a Cassie–Baxter type hydrophobicity. The second level comprises the tips of the clustered copper nanorods which pin the water droplets due to the inherently hydrophilic nature of metals. This conjecture is supported by fluorescence imaging and surface polymerization experiments. Thus, a clustered metal nanorod array provides a simple model system for studying the coexistence and complex interplay of hydrophobicity and adhesion, while also providing some degree of control over these mutually contradictory properties.

Capillary origami: superhydrophobic ribbon surfaces and liquid marbles

In the wetting of a solid by a liquid it is often assumed that the substrate is rigid. However, for an elastic substrate the rigidity depends on the cube of its thickness and so reduces rapidly as the substrate becomes thinner as it approaches becoming a thin sheet. In such circumstances, it has been shown that the capillary forces caused by a contacting droplet of a liquid can shape the solid rather than the solid shaping the liquid. A substrate can be bent and folded as a (pinned) droplet evaporates or even instantaneously and spontaneously wrapped on contact with a droplet. When this effect is used to create three dimensional shapes from initially flat sheets, the effect is called capillary origami or droplet wrapping.In this work, we consider how the conditions for the spontaneous, capillary induced, folding of a thin ribbon substrate might be altered by a rigid surface structure that, for a rigid substrate, would be expected to create Cassie–Baxter and Wenzel effects. For smooth thin substrates, droplet wrapping can occur for all liquids, including those for which the Young’s law contact angle (defined by the interfacial tensions) is greater than 90° and which would therefore normally be considered relatively hydrophobic. However, consideration of the balance between bending and interfacial energies suggests that the tendency for droplet wrapping can be suppressed for some liquids by providing the flexible solid surface with a rigid topographic structure. In general, it is known that when a liquid interacts with such a structure it can either fully penetrate the structure (the Wenzel case) or it can bridge between the asperities of the structure (the Cassie–Baxter case).In this report, we show theoretically that droplet wrapping should occur with both types of solid–liquid contact. We also derive a condition for the transition between the Cassie–Baxter and Wenzel type droplet wrapping and relate it to the same transition condition known to apply to superhydrophobic surfaces. The results are given for both droplets being wrapped by thin ribbons and for solid grains encapsulating droplets to form liquid marbles.

LevyBaxter theorems for one class of non-Gaussian stochastic processes

The theorems of Baxter type were established for a class of random processes with K-increments. The obtained results can be used in the statistics of random processes, in particular for obtaining sufficient conditions of singularity of measures generated by random processes. This study was focused on the LevyBaxter limit theorems for random processes. A new class of random processes was created. This method of investigation can be used to establish the Baxter type theorems for random fields.

Enumeration and Generating Functions of Rota–Baxter Words

In this paper, we prove results on enumerations of sets of Rota–Baxter words ( $${{{\tt RBWs}}}$$ ) in a single generator and one unary operator. Examples of operators are integral operators, their generalization to Rota–Baxter operators, and Rota–Baxter type operators. $${{{\tt RBWs}}}$$ are certain words formed by concatenating generators and images of words under the operators. Under suitable conditions, they form canonical bases of free Rota–Baxter type algebras which are studied recently in relation to renormalization in quantum field theory, combinatorics, number theory, and operads. Enumeration of a basis is often a first step to choosing a data representation in implementation. We settle the case of one generator and one operator, starting with the idempotent case (more precisely, the exponent 1 case). Some integer sequences related to these sets of $${{{\tt RBWs}}}$$ are known and connected to other sequences from combinatorics, such as the Catalan numbers, and others are new. The recurrences satisfied by the generating series of these sequences prompt us to discover an efficient algorithm to enumerate the canonical basis of certain free Rota–Baxter algebras.

Twisting on associative algebras and Rota–Baxter type operators

We introduce an operation called “twisting” on the Hochschild complex by analogy with Drinfeld’s twisting operations. By using the twisting and derived bracket constructions, we study differential graded Lie algebra structures associated to the bi-graded Hochschild complex. We show that Rota–Baxter type operators are solutions of Maurer–Cartan equations. As an application of twisting, we give a construction of associative Nijenhuis operators.

Superhydrophobic polymer films via aerosol assisted deposition — Taking a leaf out of nature's book

Aerosol assisted deposition of three sets of polymer films based on commercially available resins was achieved on various substrates. The films were characterised using a range of methods, including water contact and slip angle to determine water repellent properties. The aerosol assisted deposition inside the chemical vapour deposition reactor was unique in generating a highly rough superhydrophobic surface with water contact angles up to 170°. During the deposition process, two of the polymers were cured resulting in the development of high surface morphology. It was observed that the polymer that did not cure did not develop such a rough surface resulting in a lower water contact angle (∼ 99°). The superhydrophobic films had a Cassie–Baxter type wetting with water failing to penetrate the surface porosity, water spraying on the surface would bounce off. These films had exceptionally low slide angles of ca 1–2° from the horizontal.

Generalized Shuffles Related to Nijenhuis and TD-Algebras

Shuffle type products are well known in mathematics and physics. They are intimately related to Loday's dendriform algebras and were extensively used to give explicit constructions of free 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, 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. Our construction is related to Loday's unital commutative tridendriform algebra, 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.

Solving colored Yang-Baxter equation by Wu's method

In this article, we discuss nonsymmetric solutions of the colored Yang-Baxter equation dependent on spectral as well as colored parameters and give all seven-vertex solutions by Wu's method. It is also proved that the solutions are composed of six groups of basic solutions up to five solution transformations. Moreover, all solutions can be classified into two categories called Baxter type and free-fermion type.

Non-Noether symmetries and their influence on phase space geometry

In the present paper some geometric aspects of the concept of non-Noether symmetry are discussed. It is shown that in regular Hamiltonian systems such a symmetry canonically leads to a Lax pair on the algebra of linear operators on cotangent bundle over the phase space. Correspondence between the non-Noether symmetries and other wide spread geometric methods of generating conservation laws such as bi-Hamiltonian formalism, bidifferential calculi and Frölicher–Nijenhuis geometry is considered. It is proved that the integrals of motion associated with the continuous non-Noether symmetry are in involution whenever the generator of the symmetry satisfies a certain Yang–Baxter type equation.

Convex Rearrangements of Gaussian Processes

In this paper, we consider the asymptotic behavior of convex rearrangements for regularizations of paths of Gaussian processes with stationary increments, and we use the concentration principle to prove the almost sure convergence of these rearrangements to a nonrandom convex line, the so-called Lorentz curve, corresponding to the standard Gaussian law. Moreover, we obtain the same type of result for the Gaussian bridges of such processes. We also discuss relations with the recent results of Azais and Wschebor about the almost sure weak convergence of oscillations of Gaussian processes. As an application of our basic theorem we prove a theorem of Baxter type for p-variations of the paths and introduce a new family of consistent estimators of the fractal index.