Synthesis of New 2‐Phenylpyrano[3,2‐b]phenothiazin‐4(6H)‐one Derivatives

A new synthesis of 2‐phenylpyrano[3,2‐b]phenothiazin‐4(6H)‐one derivatives was reported. First 2,10‐diacetyl‐3‐hydroxyphenothiazine ( 2 ) was converted into their benzoyloxy esters ( 3a – 3j ) using different aromatic carboxylic acids in the presence of phosphorous oxychloride in pyridine. Benzoyloxy esters were converted into their 1,3‐diones ( 4a – 4j ) by using dry KOH in pyridine via Baker‐Venkataraman transformation reaction. The 1,3‐diones thus obtained were cyclised to pyranophenothiazines ( 5a – 5j ) by refluxing in an acetic acid/HCl mixture.     

Integrable symplectic maps associated with discrete Korteweg‐de Vries‐type equations

In this paper, we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg‐de Vries‐type (KdV‐type). As a consequence it is demonstrated that several distinct Hamiltonian systems lead to one and the same difference equation by means of the Liouville integrability framework. Thus, these integrable symplectic maps may provide an efficient tool for characterizing, and determining the integrability of, partial difference equations.     

Relatively Free Nilpotent Torsion-Free Groups and Their Lie Algebras

Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ? _K (G), grad^(?)(? _K (G)), grad^(g)(exp ? _K (G)), and L _K (G). Let 𝔗 _c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F _n (𝔗 _c ) be the relatively free group of rank n in 𝔗 _c . We prove that ? _K (F _n (𝔗 _c )) is relatively free in some variety of nilpotent Lie algebras, and ? _K (F _n (𝔗 _c )) ? L _K (F _n (𝔗 _c )) ? grad^(?)(? _K (F _n (𝔗 _c ))) ? grad^(g)(exp ? _K (F _n (𝔗 _c ))) as Lie algebras in a natural way. Furthermore, F _n (𝔗 _c ) is a Magnus nilpotent group. Let G ₁ and G ₂ be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G ₁ and G ₂ are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ? of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ? ?_?(H) ? L _?(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ? _K and L _K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ?_?(G) is not isomorphic to L _?(G) as Lie algebras.     

Hybrid Control of Digital Baker Map with Application to Pseudo-Random Number Generator

Dynamical degradation occurs when chaotic systems are implemented on digital devices, which seriously threatens the security of chaos-based cryptosystems. The existing solutions mainly focus on the compensation of dynamical properties rather than on the elimination of the inherent biases of chaotic systems. In this paper, a unidirectional hybrid control method is proposed to improve the dynamical properties and to eliminate the biases of digital chaotic maps. A continuous chaotic system is introduced to provide external feedback control of the given digital chaotic map. Three different control modes are investigated, and the influence of control parameter on the properties of the controlled system is discussed. The experimental results show that the proposed method can not only improve the dynamical degradation of the digital chaotic map but also make the controlled digital system produce outputs with desirable performances. Finally, a pseudorandom number generator (PRNG) is proposed. Statistical analysis shows that the PRNG has good randomness and almost ideal entropy values.     

Fatou sets of entire solutions of linear differential equations

This paper is devoted to studying the dynamical properties of solutions of f^″+A(z)f=0$f^{″}+A\left(z\right)f=0$, where A(z)$A\left(z\right)$ is an entire function of finite order. With some added conditions on A(z)$A\left(z\right)$, we prove that all the Fatou components of E(z)$E\left(z\right)$ are simply-connected provided that E=_f1f2$E=f_1{f}_2$ and _f1,_f2$f_1,f_2$ are two linearly independent solutions of such equations.     

Pseudodifferential Operators of Several Variables and Baker Functions

The KP hierarchy consists of an infinite system of nonlinear partial differential equations and is determined by Lax equations, which can be constructed using pseudodifferential operators. The KP hierarchy and the associated Lax equations can be generalized by using pseudodifferential operators of several variables. We construct Baker functions associated to those generalized Lax equations of several variables and prove some of the properties satisfied by such functions.     

The setereoselective preparation of β-keto esters using baker＇s yeast in the presence of rice wine koji

By the addition of rice wine koji, enhancement of the reactivity was observed for the baker＇s yeast reduction of β-keto esters into （S）-β-hydroxy esters with high enantiomeric purity （73-98%）.     

2D LOCUS CONFIGURATIONS AND THE TRIGONOMETRIC CALOGERO–MOSER SYSTEM

A central hyperplane arrangement in ?² with multiplicity is called a "locus configuration" if it satisfies a series of "locus equations" on each hyperplane. Following [4], we demonstrate that the first locus equation for each hyperplane corresponds to a force-balancing equation on a related interacting particle system on ?*: the charged trigonometric Calogero-Moser system. When the particles lie on S¹ ? ?*, there is a unique equilibrium for this system. For certain classes of particle weight, this is enough to show that all the locus equations are satisfied, producing explicit examples of real locus configurations. This in turn produces new examples of Schrödinger operators with Baker–Akhiezer functions.