Large Distinct Part Sizes in a Random Integer Partition

A partition of a positive integer n is a way of writing it as the sum of positive integers without regard to order; the summands are called parts. The number of partitions of n, usually denoted by p(n), is determined asymptotically by the famous partition formula of Hardy and Ramanujan [5]. We shall introduce the uniform probability measure P on the set of all partitions of n assuming that the probability 1/p(n) is assigned to each n-partition. The symbols E and V ar will be further used to denote the expectation and variance with respect to the measure P . Thus, each conceivable numerical characteristic of the parts in a partition can be regarded as a random variable.

Capacity limits of systems employing multiple optical phase conjugators

We extend the theory of parametric noise amplification to the case of transmission systems employing multiple optical phase conjugators, demonstrating that the excess noise due to this process may be reduced in direct proportion to the number of phase conjugation devices employed. We further identify that the optimum noise suppression is achieved for an odd number of phase conjugators, and that the noise may be further suppressed by up to 3dB by partial digital back propagation (or fractional spans at the ends of the links).

Constructing 7-Clusters

ACM Computing Classification System (1998): G.2, G.4.

Generalized Backscattering and the Lax-Phillips Transform

2000 Mathematics Subject Classification: 35P25, 35R30, 58J50.

New Coefficient Conditions for Functions Starlike with Respect to Symmetric Points

2000 Mathematics Subject Classification: Primary 30C45, secondary 30C80.

Oscillation Criteria of Second-Order Quasi-Linear Neutral Delay Difference Equations

2000 Mathematics Subject Classification: 39A10.

Local Energy Decay in Even Dimensions for the Wave Equation with a Time-Periodic Non-Trapping Metric and Applications to Strichartz Estimates

2000 Mathematics Subject Classification: 35B40, 35L15.

On strongly regular graphs with m2 = qm3 and m3 = qm2

2010 Mathematics Subject Classification: 05C50.

Editorial

The year so far has been a slow start for many businesses, but at least we have not seen the collapse of as many businesses that we were seeing around two years ago. We are, however, still well and truly in the midst of a global recession. Interest rates are still at an all time low, UK house prices seem to be showing little signs of increase (except in London where everyone still seems to want to live!) and for the ardent shopper there are bargains to be had everywhere. It seems strange that prices on the high street do not seem to have increased in over ten years. Mobile phones, DVD players even furniture seems to be cheaper than they used to be. Whist much of this is down to cheaper manufacturing and the rest could probably be explained by competition within the market place. Does this mean that quality suffered too? Now that we live in a world when if a television is not working it is thrown away and replaced. There was a time when you would take it to some odd looking man that your father would know who could fix it for you. (I remember our local television fix-it man, with his thick rimmed bifocal spectacles and a poor comb-over; he had cardboard boxes full of resistors and electrical wires on the floor of his front room that smelt of soldering irons!) Is this consumerism at an extreme or has this move to disposability made us a better society? Before you think these are just ramblings there is a point to this. According to latest global figures of contact lens sales the vast majority of contact lenses fitted around the world are daily, fortnightly or monthly disposable hydrogel lenses. Certainly in the UK over 90% of lenses are disposable (with daily disposables being the most popular, having a market share of over 50%). This begs the question – is this a good thing? Maybe more importantly, do our patients benefit? I think it is worth reminding ourselves why we went down the disposability route with contact lenses in the first place, and unlike electrical goods it was not just so we did not have to take them for repair! There are the obvious advantages of overcoming problems of breakage and tearing of lenses and the lens deterioration with age. The lenses are less likely to be contaminated and the disinfection is either easier or not required at all (in the case of daily disposable lenses). Probably the landmark paper in the field was the work more commonly known as the ‘Gothenburg Study’. The paper, entitled ‘Strategies for minimizing the Ocular Effects of Extended Contact Lens Wear’ published in the American Journal of Optometry in 1987 (volume 64, pages 781-789) by Holden, B.A., Swarbrick, H.A., Sweeney, D.F., Ho, A., Efron, N., Vannas, A., Nilsson, K.T. They suggested that contact lens induced ocular effects were minimised by: •More frequently removed contact lenses •More regularly replaced contact lenses •A lens that was more mobile on the eye (to allow better removal of debris) •Better flow of oxygen through the lens All of these issues seem to be solved with disposability, except the oxygen issue which has been solved with the advent of silicone hydrogel materials. Newer issues have arisen and most can be solved in practice by the eye care practitioner. The emphasis now seems to be on making lenses more comfortable. The problems of contact lens related dry eyes symptoms seem to be ever present and maybe this would explain why in the UK we have a pretty constant contact lens wearing population of just over three million but every year we have over a million dropouts! That means we must be attracting a million new wearers every year (well done to the marketing departments!) but we are also losing a million wearers every year. We certainly are not losing them all to the refractive surgery clinics. We know that almost anyone can now wear a contact lens and we know that some lenses will solve problems of sharper vision, some will aid comfort, and some will be useful for patients with dry eyes. So if we still have so many dropouts then we must be doing something wrong! I think the take home message has to be ‘must try harder’! I must end with an apology for two errors in my editorial of issue 1 earlier this year. Firstly there was a typo in the first sentence; I meant to state that it was 40 years not 30 years since the first commercial soft lens was available in the UK. The second error was one that I was unaware of until colleagues Geoff Wilson (Birmingham, UK) and Tim Bowden (London, UK) wrote to me to explain that soft lenses were actually available in the UK before 1971 (please see their ‘Letters to the Editor’ in this issue). I am grateful to both of them for correcting the mistake.

Early spatial vision:a view through two eyes

Simple features such as edges are the building blocks of spatial vision, and so I ask: how arevisual features and their properties (location, blur and contrast) derived from the responses ofspatial filters in early vision; how are these elementary visual signals combined across the twoeyes; and when are they not combined? Our psychophysical evidence from blur-matchingexperiments strongly supports a model in which edges are found at the spatial peaks ofresponse of odd-symmetric receptive fields (gradient operators), and their blur B is givenby the spatial scale of the most active operator. This model can explain some surprisingaspects of blur perception: edges look sharper when they are low contrast, and when theirlength is made shorter. Our experiments on binocular fusion of blurred edges show that singlevision is maintained for disparities up to about 2.5*B, followed by diplopia or suppression ofone edge at larger disparities. Edges of opposite polarity never fuse. Fusion may be served bybinocular combination of monocular gradient operators, but that combination - involvingbinocular summation and interocular suppression - is not completely understood.In particular, linear summation (supported by psychophysical and physiological evidence)predicts that fused edges should look more blurred with increasing disparity (up to 2.5*B),but results surprisingly show that edge blur appears constant across all disparities, whetherfused or diplopic. Finally, when edges of very different blur are shown to the left and righteyes fusion may not occur, but perceived blur is not simply given by the sharper edge, nor bythe higher contrast. Instead, it is the ratio of contrast to blur that matters: the edge with theAbstracts 1237steeper gradient dominates perception. The early stages of binocular spatial vision speak thelanguage of luminance gradients.

Eigen-Adaptation and Distributed Representation of 2-D Phase, Energy, Scale and Orientation in Spatial Vision

Distributed representations (DR) of cortical channels are pervasive in models of spatio-temporal vision. A central idea that underpins current innovations of DR stems from the extension of 1-D phase into 2-D images. Neurophysiological evidence, however, provides tenuous support for a quadrature representation in the visual cortex, since even phase visual units are associated with broader orientation tuning than odd phase visual units (J.Neurophys.,88,455–463, 2002). We demonstrate that the application of the steering theorems to a 2-D definition of phase afforded by the Riesz Transform (IEEE Trans. Sig. Proc., 49, 3136–3144), to include a Scale Transform, allows one to smoothly interpolate across 2-D phase and pass from circularly symmetric to orientation tuned visual units, and from more narrowly tuned odd symmetric units to even ones. Steering across 2-D phase and scale can be orthogonalized via a linearizing transformation. Using the tiltafter effect as an example, we argue that effects of visual adaptation can be better explained by via an orthogonal rather than channel specific representation of visual units. This is because of the ability to explicitly account for isotropic and cross-orientation adaptation effect from the orthogonal representation from which both direct and indirect tilt after-effects can be explained.

Firefly curios and sundry lights

Firefly Curios and Sundry Lights contains 33 poems and 55 pages, mostly free verse lyric narratives issuing from various geographic, emotional, and temporal landscapes. The book is divided into four sections which might roughly be titled: "before," examining themes of childhood and death: "on-the-road," relaying the compulsion to travel, "odd-and- ends-limbo," including pieces which have no context within the time line; and "in-one- place-for-now," reflecting modes of communication, ordering, and longing. Other concerns include speculations about existence, observations of nature, and the importance of science as a means of apprehending the world. The work reveals a belief in the interconnectedness of mind and matter, combines seriousness and humor, and displays a sonic sensibility. These poems of solitude and observation are themselves vehicles, their motion a means of dislocation in order to find the self. Firefly Curios and Sundry Lights is smaller than a bread box and you can dance to it.

(Table) Carbon, nitrogen and n-alkane content and isotopic composition of bottom sediments obtained in the East Siberian Sea

The chemical composition of organic matter (Corg, Norg, d13C, d1SN, and n-alkanes) was studied in the top layer of bottom sediments of the East Siberian Sea. Possible ways were proposed to estimate the amount of the terrigenous component in their organic matter (OM). The fraction of terrigenous OM estimated by the combined use of genetic indicators varied from 15% in the eastern part of the sea, near the Long Strait, to 95% in the estuaries of the Indigirka and Kolyma rivers, averaging 62% over the sea area.

Data on the biogeochemical cycle of methane in the ocean

Geological, mineralogical and microbiological aspects of the methane cycle in water and sediments of different areas in the oceans are under consideration in the monograph. Original and published estimations of formation- and oxidation rates of methane with use of radioisotope and isotopic methods are given. The role of aerobic and anaerobic microbial oxidation of methane in production of organic matter and in formation of authigenic carbonates is considered. Particular attention is paid to processes of methane transformation in areas of its intensive input to the water column from deep-sea hydrothermal sources, mud volcanoes, and cold methane seeps.

The role of parametric assumptions in adaptive Bayesian estimation.

Variants of adaptive Bayesian procedures for estimating the 5% point on a psychometric function were studied by simulation. Bias and standard error were the criteria to evaluate performance. The results indicated a superiority of (a) uniform priors, (b) model likelihood functions that are odd symmetric about threshold and that have parameter values larger than their counterparts in the psychometric function, (c) stimulus placement at the prior mean, and (d) estimates defined as the posterior mean. Unbiasedness arises in only 10 trials, and 20 trials ensure constant standard errors. The standard error of the estimates equals 0.617 times the inverse of the square root of the number of trials. Other variants yielded bias and larger standard errors.