Bisected theta series,least r-gaps in partitions,and polygonal numbers

The Ramanujan Journal - The least r-gap, $$g_r(\lambda )$$, of a partition $$\lambda $$ is the smallest positive integer that does not appear at least r times as a part of $$\lambda $$. In this...     

Optimal control of a two-dimensional contact problem

We consider a frictionless contact problem with unilateral constraints for a 2D bar. We describe the problem, then we derive its weak formulation, which is in the form of an elliptic variational inequality of the first kind. Next, we establish the existence of a unique weak solution to the problem and prove its continuous dependence with respect to the applied tractions and constraints. We proceed with the study of an associated control problem for which we prove the existence of an optimal pair. Finally, we consider a perturbed optimal control problem for which we prove a convergence result.     

On the restricted partition function

For a vector \(\mathbf a = (a_1,\ldots ,a_r)\) of positive integers, we prove formulas for the restricted partition function \(p_{\mathbf a}(n): = \) the number of integer solutions \((x_1,\dots ,x_r)\) to \(\sum _{j=1}^r a_jx_j=n\) with \(x_1\ge 0, \ldots , x_r\ge 0\) and its polynomial part.     

α-(3,7-Dioxa-r-1-azabicyclo[3.3.0]oct-c-5-ylmethoxy)-diazines. Part 2: Functionalisation via directed ortho-metallation and cross-coupling reactions

The functionalisation of the title compounds via regioselective Directed ortho-Metallation (DoM) and cross-coupling reactions is studied. The compatibility of the 3,7-DiOxa-r-1-AzaBicyclo[3.3.0]Oct-c-5-ylmethoxy system (DOABO-CH₂O) to typical reaction conditions is established. Its role as Directed ortho-Metallation Group (DoMG) is examined, including competition with classical DoMGs, chlorine and methoxy. The chelating ability of some functionalised terms such as DOABO-CH₂O substituting chiral diarylmethanols and polyaza analogues of 2,6-terpyridine is discussed as intramolecular steric relationships determining configuration and aptitude to bind selectively transition metals, respectively.     

Electron impact mass spectra of 3,5,7-triaryl-and -trialkyl-4H-1,2-diazepines

Electron impact mass spectra of 3,5,7-trisubstituted 4H-1,2-diazepines indicate that aryl substituents lead to N₂ expulsion while alkyl substitutents do not. A common fragmentation pattern is observed and discussed for all alkyldiazepines, most of which are newly reported compounds. Assignments are based on electron impact mass spectra of deuteriated substrates and high resolution mass spectra. A previous interpretation of N₂ expulsion is corrected.     

The Chirality of Icosahedral Fullerenes: a Comparison of the Tripling (leapfrog), Quadrupling (chamfering), and Septupling (capra) Transformations

Fullerene polyhedra of icosahedral symmetry have the midpoints of their 12 pentagonal faces at the vertices of a macroicosahedron and can be characterized by the patterns of their hexagonal faces on the (triangular) macrofaces of this macroicosahedron. The numbers of the vertices in fullerene polyhedra of icosahedral symmetry satisfy the Goldberg equation v=20(h ²+hk+k ²), where h and k are two integers and 0 <h≥ k≥ 0 and define a two-dimensional Goldberg vector G = (h, k). The known tripling (leapfrog), quadrupling (chamfering), and septupling (capra) transformations correspond to the Goldberg vectors (1, 1), (2, 0), and (2, 1), respectively. The tripling and quadrupling transformations applied to the regular dodecahedron generate achiral fullerene polyhedra with the full I _h point group. However, the septupling transformation destroys the reflection operations of the underlying icosahedron to generate chiral fullerene polyhedra having only the I icosahedral rotational point group. Generalization of the quadrupling transformation leads to the fundamental homologous series of achiral fullerene polyhedra having 20 n ² vertices and Goldberg vectors (n, 0). A related homologous series of likewise achiral fullerene polyhedra having 60 n ² vertices and Goldberg vectors (n, n) is obtained by applying the tripling transformation to regular dodecahedral C₂₀ to give truncated icosahedral C₆₀ followed by the generalized operations (as in the case of quadrupling) for obtaining homologous series of fullerenes. Generalization of the septupling (capra) transformation leads to a homologous series of chiral C_20m fullerenes with the I point group and Goldberg vectors G=(h, 1) where m=h ²+h+1.     

A quasistatic contact problem with slip‐dependent coefficient of friction

We consider a mathematical model which describes the bilateral quasistatic contact of a viscoelastic body with a rigid obstacle. The contact is modelled with a modified version of Coulomb's law of dry friction and, moreover, the coefficient of friction is assumed to depend either on the total slip or on the current slip. In the first case, the problem depends upon contact history. We present the classical formulations of the problems, the variational formulations and establish the existence and uniqueness of a weak solution to each of them, when the coefficient of friction is sufficiently small. The proofs are based on classical results for elliptic variational inequalities and fixed point arguments. We also study the dependence of the solutions on the perturbations of the friction coefficient and obtain a uniform convergence result. Copyright © 1999 John Wiley & Sons, Ltd.     

Graft copolymers of polyethylene by atom transfer radical polymerization

Poly(ethylene‐g‐styrene) and poly(ethylene‐g‐methyl methacrylate) graft copolymers were prepared by atom transfer radical polymerization (ATRP). Commercially available poly(ethylene‐co‐glycidyl methacrylate) was converted into ATRP macroinitiators by reaction with chloroacetic acid and 2‐bromoisobutyric acid, respectively, and the pendant‐functionalized polyolefins were used to initiate the ATRP of styrene and methyl methacrylate. In both cases, incorporation of the vinyl monomer into the graft copolymer increased with extent of the reaction. The controlled growth of the side chains was proved in the case of poly(ethylene‐g‐styrene) by the linear increase of molecular weight with conversion and low polydispersity (M_w /M_n < 1.4) of the cleaved polystyrene grafts. Both macroinitiators and graft copolymers were characterized by ¹H NMR and differential scanning calorimetry. © 2000 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 38: 2440–2448, 2000