Descent distribution on Catalan words avoiding a pattern of length at most three

Catalan words are particular growth-restricted words over the set of non-negative integers, and they represent still another combinatorial class counted by the Catalan numbers. We study the distribution of descents on the sets of Catalan words avoiding a pattern of length at most three: for each such a pattern $p$ we provide a bivariate generating function where the coefficient of $x^n{y}^k$ in its series expansion is the number of length $n$$p$-avoiding Catalan words with $k$ descents. As a byproduct, we enumerate the set of Catalan words avoiding $p$, and we provide the popularity of descents on this set.     

Stochastic nonlinear Perron–Frobenius theorem

We establish a stochastic nonlinear analogue of the Perron–Frobenius theorem on eigenvalues and eigenvectors of positive matrices. The result is formulated in terms of an automorphism T of a probability space and a random transformation D of the non-negative cone of an n-dimensional Euclidean space. Under assumptions of monotonicity and homogeneity of D, we prove the existence of scalar and vector measurable functions α > 0 and x > 0 satisfying the equation αTx = D(x) almost surely. We apply the result obtained to the analysis of a class of random dynamical systems arising in mathematical economics and finance (von Neumann–Gale dynamical systems).     

The Hyperplanes of the U 4(3) Near Hexagon

Combining theoretical arguments with calculations in the computer algebra package GAP, we are able to show that there are 27 isomorphism classes of hyperplanes in the near hexagon for the group U ₄(3). We give an explicit construction of a representative of each class and we list several combinatorial properties of such a representative.     

Comments on: Polling: past,present and perspective

In their exposé on the past, present and perspective of the analysis of polling systems, professors Borst and Boxma give a first-hand account on the advancement of this particular research field. As any good historical account, the paper highlights some of the key turning points in the advancement of our current understanding of the dynamics of polling systems. Their paper, however, goes far beyond a classical literature review, as it also emphasises recent developments and points to many open problems. In the first section of this present note, we give some introductory comments on polling systems and queueing systems in general, followed by a discussion of Borst and Boxma’s survey of the past and the present in Sect. 2. Finally, we discuss some of their suggestions for further research in Sect. 3.     

Regular and Chaotic Dynamics of a Chaplygin Sleigh due to Periodic Switch of the Nonholonomic Constraint

The main goal of the article is to suggest a two-dimensional map that could play the role of a generalized model similar to the standard Chirikov–Taylor mapping, but appropriate for energy-conserving nonholonomic dynamics. In this connection, we consider a Chaplygin sleigh on a plane, supposing that the nonholonomic constraint switches periodically in such a way that it is located alternately at each of three legs supporting the sleigh. We assume that at the initiation of the constraint the respective element (“knife edge”) is directed along the local velocity vector and becomes instantly fixed relative to the sleigh till the next switch. Differential equations of the mathematical model are formulated and an analytical derivation of mapping for the state evolution on the switching period is provided. The dynamics take place with conservation of the mechanical energy, which plays the role of one of the parameters responsible for the type of the dynamic behavior. At the same time, the Liouville theorem does not hold, and the phase volume can undergo compression or expansion in certain state space domains. Numerical simulations reveal phenomena characteristic of nonholonomic systems with complex dynamics (like the rattleback or the Chaplygin top). In particular, on the energy surface attractors associated with regular sustained motions can occur, settling in domains of prevalent phase volume compression together with repellers in domains of the phase volume expansion. In addition, chaotic and quasi-periodic regimes take place similar to those observed in conservative nonlinear dynamics.     

Synthesis, structure, and photoisomerization of derivatives of 2-(2-quinolyl)-1,3-tropolones prepared by the condensation of 2-methylquinolines with 3,4,5,6-tetrachloro-1,2-benzoquinone

A series of novel derivatives of the 1,3-tropolone (β-tropolone) system—2-(2-quinolyl)-5,6,7-trichloro-1,3-tropolones and 2-(2-quinolyl)-4,5,6,7-tetrachloro-1,3-tropolones have been prepared by the acid-catalyzed reaction of 2-methylquinolines with 3,4,5,6-tetrachloro-1,2-benzoquinone. The molecular structures of two compounds, 2-(4-chloro-6,8-dimethyl-5-nitro-2-quinolyl)-5,6,7-trichloro-1,3-tropolone 8 and 2-(4-chloro-7,8-dimethyl-5-nitro-2-quinolyl)-4,5,6,7-tetrachloro-1,3-tropolone 9, have been determined using X-ray crystallography. According to the performed DFT B3LYP/6-311++G^∗∗ calculations the tautomeric (OH) and (NH) forms of β-tropolones 8 and 9 are nearly energy equivalent, the latter being more stabilized in polar media. Photolysis of 2-(2-quinolyl)-1,3-tropolones in heptane solution leads to the disrotatory electrocyclic rearrangement resulting in the formation of a mixture of E- and Z-isomers of 3-[2(1H)-quinolinylyden]-bicyclo[3.2.0]hept-6-en-2,4-dione derivatives.     

First synthesis of 2-aminosubstituted-2-perfluoroalkyl-3,6-dihydro-2H-thiopyrans by hetero-Diels-Alder reactions of fluorinated thioamides under microwave heating

This Letter presents the first examples of hetero-Diels-Alder reactions of polyfluoroalkanethiocarboxylic acid amides and 2,3-dimethylbutadiene under microwave heating. Cycloaddition reactions proved to be dependent on the nature of perfluoroalkyl chain and on the substituents attached to the nitrogen atom. Formation of ammonium salts was also performed by simple treatment of the corresponding cycloadducts with trifluoromethanesulfonic acid. In the case of octafluorobutyl-substituted derivative, one spontaneous desamination reaction took place leading to new 2H-thiopyran.     

A novel synthesis of 1,5-disubstituted fluorinated tetrazoles from 1,1-difluoroazides

A new one-step reaction between 1,1-difluoroazides and primary amines is reported as an efficient synthetic approach for tetrazole formation. Examples include the syntheses of fluorine-containing 1,5-substituted tetrazoles from three fluorinated azide precursors and various amines. Greater substituent diversity in comparison with conventional methods is achieved.     

Immunomodulatory action of monosulfated triterpene glycosides from the sea cucumber Cucumaria okhotensis: stimulation of activity of mouse peritoneal macrophages

Six monosulfated triterpene glycosides, frondoside A1 (1), okhotoside B1 (2), okhotoside A1-1 (3), frondoside A (4), okhotoside A2-1 (5) and cucumarioside A2-5 (6), isolated from Cucumaria okhotensis Levin et Stepanov, stimulate spreading and lysosomal activity of mouse macrophages and ROS-formation in the macrophages. The highest macrophage spreading and stimulation of their lysosomal activity was induced by glycosides 1, 4 and 6. All glycosides similarly stimulate ROS formation in macrophages, but glycoside 2 caused minimal stimulation.     

Diversity-oriented synthesis of N-aryl-N-thiazolyl compounds

An efficient approach toward the parallel solid-phase synthesis of highly diversified N-aryl-N-thiazolyl compounds is presented. The treatment of resin-bound aniline derivatives with Fmoc-isothiocyanate generated aryl thioureas which, following Hantzsch’s reaction with a variety of α-haloketones and cleavage of the solid support, led to the desired N-aryl-N-thiazolyl compounds in good yield and high purity.