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Smith EG Webber GB Sakai K Biggs S Armes SP Wanless EJ 《The journal of physical chemistry. B》2007,111(20):5536-5541
The first in situ proof of the presence of true nanoscale micelle structure within alternating layers of a self-assembled cationic and anionic diblock copolymer micelle-micelle multilayer film is presented using atomic force microscopy. Apparently similar layer morphologies are distinguished by the interaction force curves normal to each layer. The three-dimensional order in these low Tg diblock copolymer films, together with their mechanical resilience and strongly hydrated nature, suggests many new technological applications. 相似文献
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We prove quadratic upper bounds on the order of any autotopism of a quasigroup or Latin square, and hence also on the order of any automorphism of a Steiner triple system or 1‐factorization of a complete graph. A corollary is that a permutation σ chosen uniformly at random from the symmetric group will almost surely not be an automorphism of a Steiner triple system of order n, a quasigroup of order n or a 1‐factorization of the complete graph . Nor will σ be one component of an autotopism for any Latin square of order n. For groups of order n it is known that automorphisms must have order less than n, but we show that quasigroups of order n can have automorphisms of order greater than n. The smallest such quasigroup has order 7034. We also show that quasigroups of prime order can possess autotopisms that consist of three permutations with different cycle structures. Our results answer three questions originally posed by D. Stones. 相似文献
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The original article to which this erratum refers was correctly published online on 1 December 2011. Due to an error at the publisher, it was then published in Journal of Combinatorial Designs 20: 124–141, 2012 without the required shading in several examples. To correct this, the article is here reprinted in full. The publisher regrets this error. We prove that for all odd there exists a latin square of order 3m that contains an latin subrectangle consisting of entries not in any transversal. We prove that for all even there exists a latin square of order n in which there is at least one transversal, but all transversals coincide on a single entry. A corollary is a new proof of the existence of a latin square without an orthogonal mate, for all odd orders . Finally, we report on an extensive computational study of transversal‐free entries and sets of disjoint transversals in the latin squares of order . In particular, we count the number of species of each order that possess an orthogonal mate. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 344–361, 2012 相似文献
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The HallPaige conjecture deals with conditions underwhich a finite group G will possess a complete mapping, or equivalentlya Latin square based on the Cayley table of G will possess atransversal. Two necessary conditions are known to be: (i) thatthe Sylow 2-subgroups of G are trivial or non-cyclic, and (ii)that there is some ordering of the elements of G which yieldsa trivial product. These two conditions are known to be equivalent,but the first direct, elementary proof that (i) implies (ii)is given here. It is also shown that the HallPaige conjecture impliesthe existence of a duplex in every group table, thereby provinga special case of Rodney's conjecture that every Latin squarecontains a duplex. A duplex is a double transversal,that is, a set of 2n entries in a Latin square of order n suchthat each row, column and symbol is represented exactly twice.2000 Mathematics Subject Classification 05B15, 20D60. 相似文献
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Nicholas J. Cavenagh Catherine Greenhill Ian M. Wanless 《Random Structures and Algorithms》2008,33(3):286-309
Let L be chosen uniformly at random from among the latin squares of order n ≥ 4 and let r,s be arbitrary distinct rows of L. We study the distribution of σr,s, the permutation of the symbols of L which maps r to s. We show that for any constant c > 0, the following events hold with probability 1 ‐ o(1) as n → ∞: (i) σr,s has more than (log n)1?c cycles, (ii) σr,s has fewer than 9 cycles, (iii) L has fewer than n5/2 intercalates (latin subsquares of order 2). We also show that the probability that σr,s is an even permutation lies in an interval and the probability that it has a single cycle lies in [2n‐2,2n‐2/3]. Indeed, we show that almost all derangements have similar probability (within a factor of n3/2) of occurring as σr,s as they do if chosen uniformly at random from among all derangements of {1,2,…,n}. We conjecture that σr,s shares the asymptotic distribution of a random derangement. Finally, we give computational data on the cycle structure of latin squares of orders n ≤ 11. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008 相似文献
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Blom A Drummond C Wanless EJ Richetti P Warr GG 《Langmuir : the ACS journal of surfaces and colloids》2005,21(7):2779-2788
The effect of the uptake of a low-molecular-weight amphiphilic diblock copolymer on the morphology of didodecyldimethylammonium bromide (DDAB) adsorbed layers on mica, the interactions between two coated surfaces, and the frictional properties of the boundary film have been studied using an atomic force microscope and a dynamic surface forces apparatus nanotribometer. When DDAB-coated surfaces in aqueous solution were compressed, hemifusion or removal of the adsorbed surfactant bilayers could not be induced, and no frictional force could be measured between the surfaces, which display superior lateral cohesion and lubricant properties. Coadsorbing octadecyl end modified poly(ethylene oxide) chains at low density facilitates hemifusion, generating significant shear stress and leading to stick-slip instabilities. The mixed films regain their lateral cohesion at higher adsorbed copolymer densities, but an extra short-range attraction brings the adsorbed layers into adhesive contact without causing bilayer hemifusion. Here, noticeable frictional forces are also measured. 相似文献