Factorization of Lipschitz functions and absolute convergence of Vilenkin-Fourier series

LetG be a Vilenkin group (i.e., an infinite, compact, metrizable, zero-dimensional Abelian group). Our main result is a factorization theorem for functions in the Lipschitz spaces \(\mathfrak{L}\mathfrak{i}\mathfrak{p}\) (α,p; G). As colloraries of this theorem, we obtain (i) an extension of a factorization theorem ofY. Uno; (ii) a convolution formula which says that \(\mathfrak{L}\mathfrak{i}\mathfrak{p} (\alpha , r; G) = \mathfrak{L}\mathfrak{i}\mathfrak{p} (\beta , l; G)*\mathfrak{L}\mathfrak{i}\mathfrak{p} (\alpha - \beta , r; G)\) for 0<β<α<∞ and 1≤r≤∞; and (iii) an analogue, valid for allG, of a classical theorem ofHardy andLittlewood. We also present several results on absolute convergence of Fourier series defined onG, extending a theorem ofC. W. Onneweer and four results ofN. Ja. Vilenkin andA. I. Rubinshtein. The fourVilenkin-Rubinshtein results are analogues of classical theorems due, respectively, toO. Szász, S. B. Bernshtein, A. Zygmund, andG. G. Lorentz.     

Heterocyclic Substitutions Greatly Improve Affinity and Stability of Folic Acid towards FRα. an In Silico Insight

Folate receptor alpha (FRα) is known as a biological marker for many cancers due to its overexpression in cancerous epithelial tissue. The folic acid (FA) binding affinity to the FRα active site provides a basis for designing more specific targets for FRα. Heterocyclic rings have been shown to interact with many receptors and are important to the metabolism and biological processes within the body. Nineteen FA analogs with substitution with various heterocyclic rings were designed to have higher affinity toward FRα. Molecular docking was used to study the binding affinity of designed analogs compared to FA, methotrexate (MTX), and pemetrexed (PTX). Out of 19 FA analogs, analogs with a tetrazole ring (FOL03) and benzothiophene ring (FOL08) showed the most negative binding energy and were able to interact with ASP81 and SER174 through hydrogen bonds and hydrophobic interactions with amino acids of the active site. Hence, 100 ns molecular dynamics (MD) simulations were carried out for FOL03, FOL08 compared to FA, MTX, and PTX. The root mean square deviation (RMSD) and root mean square fluctuation (RMSF) of FOL03 and FOL08 showed an apparent convergence similar to that of FA, and both of them entered the binding pocket (active site) from the pteridine part, while the glutamic part was stuck at the FRα pocket entrance during the MD simulations. Molecular mechanics Poisson-Boltzmann surface accessible (MM-PBSA) and H-bond analysis revealed that FOL03 and FOL08 created more negative free binding and electrostatic energy compared to FA and PTX, and both formed stronger H-bond interactions with ASP81 than FA with excellent H-bond profiles that led them to become bound tightly in the pocket. In addition, pocket volume calculations showed that the volumes of active site for FOL03 and FOL08 inside the FRα pocket were smaller than the FA–FRα system, indicating strong interactions between the protein active site residues with these new FA analogs compared to FA during the MD simulations.     

Synthesis of novel acetylenic cyclophanes with helical chirality: potential new structures for liquid crystals

The synthesis of a series of novel acetylenic cyclophanes is described. X-ray crystallographic analysis of the core structure revealed a twisted conformation with helical chirality. Preliminary results suggest that these cyclophanes, with appropriate functionality, have the potential to act as unique liquid crystalline materials.     

A Paramagnetic Diniobium Complex with a Very Short Nb-Nb Distance: Evidence for a Pseudo Nb-Nb Triple Bond?

In spite of the short Nb-Nb distance (2.268 ?) and the presumable existence of an Nb identical withNb bond, the paddle-wheel-shaped diniobium(II) complex 1 is paramagnetic. Theoretical calculations indicate that the presence of LiCl moieties on the intermetallic axis lowers the Nb-Nb bond order and is responsible for the observed paramagnetism.     

Manganese(I) poly(mercaptoimidazolyl)borate complexes: spectroscopic and structural characterization of Mn...H-B interactions in solution and in the solid state

The manganese(I) tricarbonyl complexes (Bm(R))Mn(CO)3(R = Me, Bz, But, p-Tol) and (PhBmMe)Mn(CO)3, the first bis(mercaptoimidazolyl)borate derivatives for this metal, have been readily prepared and fully characterized. In particular, the presence of three-center-two-electron Mn...H-B interactions in these species, both in solution and in the solid state, has been investigated using a combination of IR and NMR spectroscopies and, in the case of the methyl-, tert-butyl- and para-tolyl-substituted derivatives, by X-ray crystallography. To complement these synthetic and structural studies, the tris(mercaptoimidazolyl)borate complexes (TmMe)Mn(CO)3(R = Me, Bz, But, p-Tol) and (PhTm(Me))Mn(CO)3, as well as the related pyrazolylbis(mercaptoimidazolyl)borate (pzBmMe)Mn(CO)3, have also been synthesized and characterized by a combination of analytical and spectroscopic techniques.     

Pseudo Approximation Algorithms with Applications to Optimal Motion Planning

We introduce a technique for computing approximate solutions to optimization problems. If $X$ is the set of feasible solutions, the standard goal of approximation algorithms is to compute $x\in X$ that is an $\varepsilon$-approximate solution in the following sense: $$d(x) \leq (1+\varepsilon)\, d(x^*),$$ where $x^* \in X$ is an optimal solution, $d\colon\ X\rightarrow {\Bbb R}_{\geq 0}$ is the optimization function to be minimized, and $\varepsilon>0$ is an input parameter. Our approach is first to devise algorithms that compute pseudo $\varepsilon$-approximate solutions satisfying the bound $$d(x) \leq d(x_R^*) + \varepsilon R,$$ where $R>0$ is a new input parameter. Here $x^*_R$ denotes an optimal solution in the space $X_R$ of $R$-constrained feasible solutions. The parameter $R$ provides a stratification of $X$ in the sense that (1) $X_R \subseteq X_{R}$ for $R < R$ and (2) $X_R = X$ for $R$ sufficiently large. We first describe a highly efficient scheme for converting a pseudo $\varepsilon$-approximation algorithm into a true $\varepsilon$-approximation algorithm. This scheme is useful because pseudo approximation algorithms seem to be easier to construct than $\varepsilon$-approximation algorithms. Another benefit is that our algorithm is automatically precision-sensitive. We apply our technique to two problems in robotics: (A) Euclidean Shortest Path (3ESP), namely the shortest path for a point robot amidst polyhedral obstacles in three dimensions, and (B) $d_1$-optimal motion for a rod moving amidst planar obstacles (1ORM). Previously, no polynomial time $\varepsilon$-approximation algorithm for (B) was known. For (A), our new solution is simpler than previous solutions and has an exponentially smaller complexity in terms of the input precision.     

Catalytic construction and reconstruction of guanidines: Ti-mediated guanylation of amines and transamination of guanidines

Bis(guanidinate) titanium imido complexes [{(Me2N)C(NiPr)2}2TiNAr'] (Ar' = 2,6-Me2C6H3 (1a); C6F5 (1b)) are competent catalysts for the guanylation of a variety of arylamines with carbodiimide. The reversible [2 + 2] addition of iPrN=C=NiPr to 1b is demonstrated and is proposed to be part of the catalytic cycle. Compounds 1a and 1b are also effective precatalysts for the transamination of trialkylguanidines with arylamines to yield aryldialkylguanidines.