Mathematical foundations of a cultural project or Ramchandra's treatise “through the unsentimentalised light of mathematics”

The nineteenth century witnessed a number of projects of cultural rapprochement between the knowledge traditions of the East and West. This paper discusses the attempt to render elementary calculus amenable to an Indian audience in the indigenous mathematical idiom, undertaken by an Indian polymath, Ramchandra. The exercise is specifically located in his book A Treatise on the Problems of Maxima and Minima. The paper goes on to discuss the “vocation of failure” of the book within the context of encounter and the pedagogy of mathematics.     

On independent sets in hypergraphs

The independence number of a hypergraph H is the size of a largest set of vertices containing no edge of H. In this paper, we prove that if H_n is an n‐vertex ‐uniform hypergraph in which every r‐element set is contained in at most d edges, where , then where satisfies as . The value of c_r improves and generalizes several earlier results that all use a theorem of Ajtai, Komlós, Pintz, Spencer and Szemerédi (J Comb Theory Ser A 32 (1982), 321–335). Our relatively short proof extends a method due to Shearer (Random Struct Algorithms 7 (1995), 269–271) and Alon (Random Struct Algorithms 9 (1996), 271–278). The above statement is close to best possible, in the sense that for each and all values of , there are infinitely many H_n such that where depends only on r. In addition, for many values of d we show as , so the result is almost sharp for large r. We give an application to hypergraph Ramsey numbers involving independent neighborhoods.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 224‐239, 2014     

Me2AlCH2PMe2: a new, bifunctional cocatalyst for the Ni(II)-catalyzed oligomerization of PhSiH3

The role of methylaluminoxane (MAO) in the Ni-catalyzed dehydrogenative homologation of PhSiH3 has been investigated with a view to designing new cocatalysts possessing well-defined chemical compositions and structures. These studies show that species such as the bifunctional reagent (Me2PCH2AlMe2)2, 3, should act as co-catalyst for the Si-Si bond formation reactions. Thus, it was found that the combination of (1-Me-indenyl)Ni(PPh3)Me, 2a, and 3 (Ni/Al ratio of 1:1) converts PhSiH3 to cyclic oligomers (PhSiH)n with a turnover frequency (TOF) of >500 h(-1), 50 times faster than with 2a alone. Detailed NMR studies have indicated that this acceleration is due to the formation of the intermediate (1-Me-indenyl)Ni(Me)(Me2PCH2AlMe2), 4. Coordination of the PMe2 moiety in this complex to the Ni center allows the tethered AlMe2 moiety to interact with the Ni-Me moiety in such a way that promotes fairly slow Al-Me/Ni-CD3 exchange (t(1/2) ca. 12 h) but accelerates the Si-H bond activation and Si-Si bond formation reactions. The catalysis promoted by 2a/3 proceeds even faster in the presence of NEt3 or THF (TOF > 1600 h(-1)), because these Lewis bases favor the monomeric form of 3, which in turn favors the formation of 4. On the other hand, the much more nucleophilic base quinuclidine suppresses the catalysis (TOF < 300 h(-1)) by hindering the Ni.R.Al interactions. These observations point to an emerging strategy for using bifunctional reagents such as 3 to place geometrically constrained Lewis acid moieties adjacent to metal centers, thereby activating certain metal-ligand bonds.     

Hamilton cycles in quasirandom hypergraphs

We show that, for a natural notion of quasirandomness in k‐uniform hypergraphs, any quasirandom k‐uniform hypergraph on n vertices with constant edge density and minimum vertex degree Ω(n^k‐1) contains a loose Hamilton cycle. We also give a construction to show that a k‐uniform hypergraph satisfying these conditions need not contain a Hamilton ?‐cycle if k– ? divides k. The remaining values of ? form an interesting open question. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 363–378, 2016     

The edge‐bandwidth of theta graphs

An edge‐labeling f of a graph G is an injection from E(G) to the set of integers. The edge‐bandwidth of G is B′(G) = min_f {B′(f)} where B′(f) is the maximum difference between labels of incident edges of G. The theta graph Θ(l₁,…,l_m) is the graph consisting of m pairwise internally disjoint paths with common endpoints and lengths l₁ ≤ ··· ≤ l_m. We determine the edge‐bandwidth of all theta graphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 89–98, 2000     

Synthesis of a New Cu(II)-Ion Imprinted Polymer for Solid Phase Extraction and Preconcentration of Cu(II)

A new Cu(II)-ion imprinted polymer (IIP) has been synthesized by copolymerizing salicylic acid and formaldehyde as a monomer and crosslinker, respectively in the presence of Cu(II)-4-(2-pyridylazo) resorcinol complex. The imprinted Cu(II) ions were completely removed by leaching the IIP with 0.05 M EDTA. The maximum adsorption capacity for Cu(II) ions was 310 μg g^?1 at pH 6. The IIP was repeatedly used in adsorption–desorption experiments for seven times with recoveries ~95%. The relative selectivity factor (α _r) values of Cu(II)/Zn(II), Cu(II)/Cd(II), Cu(II)/Ni(II) and Cu(II)/Co(II) are 3.17, 2.90, 2.47 and 3.37, respectively. The detection limit corresponding to three times the standard deviation of the blank was found to be 3.0 μg L^?1. The developed IIP has also been tested for preconcentration and recovery of Cu(II) ions from water samples.     

New Asymptotically Sharp Korn and Korn-Like Inequalities in Thin Domains

It is well known that Korn inequality plays a central role in the theory of linear elasticity. In the present work we prove new asymptotically sharp Korn and Korn-like inequalities in thin curved domains with a non-constant thickness. This new results will be useful when studying the buckling of compressed shells, in particular when calculating the critical buckling load.     

The Chromatic Spectrum of Mixed Hypergraphs

 A mixed hypergraph is a triple ℋ=(X, ?, ?), where X is the vertex set, and each of ?, ? is a list of subsets of X. A strict k-coloring of ℋ is a surjection c:X→{1,…,k} such that each member of ? has two vertices assigned a common value and each member of ? has two vertices assigned distinct values. The feasible set of H is {k: H has a strict k-coloring}. Among other results, we prove that a finite set of positive integers is the feasible set of some mixed hypergraph if and only if it omits the number 1 or is an interval starting with 1. For the set {s,t} with 2≤s≤t−2, the smallest realization has 2t−s vertices. When every member of ?∪? is a single interval in an underlying linear order on the vertices, the feasible set is also a single interval of integers. Received: May 24, 1999 Final version received: August 31, 2000