Central Charge and the Andrews-Bailey Construction

From the equivalence of the bosonic and fermionic representations of finitized characters in conformal field theory, one can extract mathematical objects known as Bailey pairs. Recently Berkovich, McCoy and Schilling have constructed a `generalized' character formula depending on two parameters \ra and $\r2$, using the Bailey pairs of the unitary model $M(p-1,p)$. By taking appropriate limits of these parameters, they were able to obtain the characters of model $M(p,p+1)$, $N=1$ model $SM(p,p+2)$, and the unitary $N=2$ model with central charge $c=3(1-{\frac{2}{p}})$. In this letter we computed the effective central charge associated with this `generalized' character formula using a saddle point method. The result is a simple expression in dilogarithms which interpolates between the central charges of these unitary models.Comment: Latex2e, requires cite.sty package, 13 pages. Additional footnote, citation and reference

John Locke and the scientific revolution: a study of the essay concerning human understanding in relation to seventeenth century science

This work aims to set Locke's Essay Concerning Human Understanding in the context of the scientific revolution of the seventeenth century. It does this by showing that the problem of the extent of human understanding to which Locke tried to find an answer was an issue which itself arose out of developments in the sciences in the seventeenth century, and that Locke's answer was itself indebted in a variety of ways to the work of the scientists. Locke's Essay, therefore, cannot be appreciated fully unless its background is also understood. There is, therefore, a dual aspect to the programme: the scientific background, and the Essay itself. For this reason the thesis is divided into two parts. Part One traces the course of a series of scientific and epistemological issues in the scientific revolution, and shows how scientists answered the problems, sometimes rightly, sometimes wrongly, but, nearly always within the confines of natural philosophy. Part Two is concerned with the intimate link between Locke's answers and the problems faced and the method practised by the scientists. It emerges that not only is much of the content of Locke's answer indebted to the scientific revolution, but also that much of the form of his argument borrows from the new science. The two parts of the work are preceded by an Introduction. Part One has seven chapters. Each chapter is concerned with a particular period or a key figure in the scientific revolution. A common theme is the outlook of the various scientists to the possibility of science achieving knowledge, as opposed to either belief, or its function being only to 'save the appearances', to be, that is, merely a useful calculating device. Other aspects of the thought of the scientists are also considered, but the strong central theme is whether or not they believed that man could obtain knowledge. Chapter One is about a group I call the Neoplatonists. These include Copernicus, Kepler, and the Englishman John Dee. With Copernicus and Kepler I have concentrated particularly on their attitude towards the truth of the heliocentric system. I have used this example to draw out their attitude towards the possibility of achieving truth in the science of astronomy, and I have contrasted their attitude with that of Osiander. Dee's general attitude to science and knowledge, as revealed in his Preface to the first English edition of Euclid's Elements, is shown to have important likenesses with the outlook of Copernicus and Kepler. In Chapter Two I consider the general scientific outlook of Galileo, beginning with his acceptance of the heliocentric theory, and relating his views on astronomy to his larger conception of the nature of science and the possibility of achieving truth in the sciences. Galileo, like the earlier Neoplatonists, emerges as a man who saw the natural sciences as achieving a degree of certainty which is in fact unjustified; it is an attitude which stands in marked contrast to most pre-Renaissance thinkers. Chapter III looks at the nature and limits of science as conceived by Descartes. The paradigm of mathematical certainty which Descartes propounded is assessed, and the limitations of the programme which Descartes mapped out are identified. The chapter also includes a section on Rohault, a Cartesian who did not follow Descartes in all his recommendations, and who was influential in science in the latter part of the century. In Chapter IV we return to the work of Francis Bacon whose influence over the development of the English conception of the right method to be pursued by scientists was central. Bacon laid down the general empirical approach to nature which was to become the dominant procedure of the scientists of the Royal Society; but Bacon never set the method in a general epistemological context. The general approach of Englishmen in the period from Bacon to Newton towards the proper extent and nature of the scientific enterprise is examined in Chapter V. Generally, there were two rather different approaches. Some men, like Sir Kenelm Digby and Robert Hooke, expected certainty from science in the true Baconian fashion. Others, like Joseph Glanvill, had no such high expectations. The position, therefore, remained uncertain as to the possible extent of human knowledge. In Chapter VI we look at the approach to this problem of one of the most important thinkers of seventeenth century England, Robert Boyle. Boyle, we discover, was not so optimistic in outlook as the true Baconians, for he did not expect science to give absolute certainty. Indeed he saw good reason why it could never do so. But he did believe that science could and did achieve Borne form of knowledge about the physical world. Similar attitudes towards the scope of science were exhibited by Isaac Newton whose views on the possibility of knowledge are considered in Chapter VII. Newton did not expect science to be able to go much beyond experience, but he did hold that there was a correct method to be followed in scientific.procedure which could and did lead to knowledge. This knowledge might have to be revised in the light of experience, but it should not be rejected out of hand simply because that possibility existed. In the second part of this work I have attempted to show how John Locke's Essay was a very real attempt to answer the question which had arisen largely as a result of the new science, namely: What was the extent and the limits of human knowledge? We see also how right Locke was to characterize himself as an underlabourer of the new science, for not only was he concerned to clarify issues which scientists had raised about the nature and limits of knowledge, but he also took Over many of their presuppositions. Further, Locke's method of approach to his problem is one which itself borrows much from the method of contemporary science. In Chapter VIII we look at Lockets own scientific background to discover the extent of Lockets acquaintance with the new science. Locke emerges as a man who was well established in at least two important branches of contemporary science, medicine and chemistry, and familiar with most others. He was, therefore, eminently well suited to try to relate science to epistemological issues. In Chapter IX we turn to the first Book of Locke's Essay in which we find that Locke's argument is one that takes a great deal from the method of contemporary science. The theory of innate ideas is treated by Locke as an hypothesis which is to be rejected as not being substantiated by the empirical data. Locke's positive thesis, that the mind is furnished with ideas which are all drawn from experience, is considered in Chapter X. Locke's arguments to show that certain of our ideas are the result of experiences are found to be lacking. Thus we find that Locke fails to give an adequate account of such central notions to the scientific revolution as those either of causation or of material object. But the effect of Lockets programme is to emphasise the importance of the contingent and the empirical for man's knowledge. Chapter XI examines Locke's treatment of the notion of a material object. I consider the reasons why he is led to postulate the existence of material substance, his arguments to show that we cannot know the essences of physical objects, and the implications which these points have for the possibility of achieving knowledge in the sciences. Finally, in Chapter XII, we turn to Locke's positive answers to the question of the limit and extent of human knowledge. We find that although Lockets definition of knowledge is excessively narrow, the general points which he has to make about the possible extent of knowledge are substantial, and substantially correct. Furthermore, they reflect very accurately the views expressed, usually only with regard to science, by Locke's contemporaries such as Boyle and Newton. Fundamentally, Locke's message was that whilst man should not have expectations about the degree of certainty which science could give which went beyond the contingent and the empirical, it was, nevertheless, right to have such expectations: there was indeed a middle course between the two evils of scepticism and dogmatism

Ignition of mixtures of SiH sub 4, CH sub 4, O sub 2, and Ar or N sub 2 behind reflected shock waves

Ignition delay times in mixtures of methane, silane, and oxygen diluted with argon and nitrogen were measured behind reflected shock waves generated in the chemical kinetic shock tube at Langley Research Center. The delay times were inferred from the rapid increase in pressure that occurs at ignition, and the ignition of methane was verified from the emission of infrared radiation from carbon dioxide. Pressures of 1.25 atm and temperatures from 1100 K to 1300 K were generated behind the reflected shocks; these levels are representative of those occurring within a supersonic Ramjet combustor. Expressions for the ignition delay time as a function of temperature were obtained from least squares curve fits to the data for overall equivalence ratios of 0.7 and 1.0. The ignition delay times with argon as the diluent were longer than those with nitrogen as the diluent. The infrared wavelength observations at 4.38 microns for carbon dioxide indicated that silane and methane ignited simultaneously (i.e., within the time resolution of the measurement). A combined chemical kinetic mechanism for mixtures of silane, methane, oxygen, and argon or nitrogen was assembled from one mechanism that accurately predicted the ignition of methane and a second mechanism that accurately predicted silane hydrogen ignition. Comparisons between this combined mechanism and experiment indicated that additional reactions, possibly between silyl and methyl fragments, are needed to develop a good silane methane mechanism

Analysis of a turbine rotor containing a transverse crack at Oak Creek Unit 17

Transient increases in one, two and three per revolution vibration characteristics of a low pressure steam turbine were observed during steam temperature reduction operations. Vibration and fracture mechanics analyses suggested the presence of a transverse shaft crack which was eventually identified by ultrasonic inspection and confirmed by destructive sectioning. Signature analyses of vibration data recorded over a two-year period prior to crack identification are correlated with fatigue crack growth, which occurred intermittently during transient temperature decreases. The apparent increased response of the rotor to vibration is due to asymmetric stiffness changes introduced by the growing transverse crack. The vibration response is predicted to increase with increasing crack depths in excess of 10% of the shaft diameter. Fracture mechanics analyses predict that fatigue crack growth occurred during periods of steam temperature decrease, when high surface tensile stresses are present. These same transient thermal stresses are shown to have retarded and prevented subsequent fatigue crack growth during steady operation

Estimates of the optimal density and kissing number of sphere packings in high dimensions

The problem of finding the asymptotic behavior of the maximal density of sphere packings in high Euclidean dimensions is one of the most fascinating and challenging problems in discrete geometry. One century ago, Minkowski obtained a rigorous lower bound that is controlled asymptotically by $1/2^d$, where $d$ is the Euclidean space dimension. An indication of the difficulty of the problem can be garnered from the fact that exponential improvement of Minkowski's bound has proved to be elusive, even though existing upper bounds suggest that such improvement should be possible. Using a statistical-mechanical procedure to optimize the density associated with a "test" pair correlation function and a conjecture concerning the existence of disordered sphere packings [S. Torquato and F. H. Stillinger, Experimental Math. {\bf 15}, 307 (2006)], the putative exponential improvement was found with an asymptotic behavior controlled by $1/2^{(0.77865...)d}$. Using the same methods, we investigate whether this exponential improvement can be further improved by exploring other test pair correlation functions correponding to disordered packings. We demonstrate that there are simpler test functions that lead to the same asymptotic result. More importantly, we show that there is a wide class of test functions that lead to precisely the same exponential improvement and therefore the asymptotic form $1/2^{(0.77865...)d}$ is much more general than previously surmised.Comment: 23 pages, 4 figures, submitted to Phys. Rev.

Flight-based chemical characterization of biomass burning aerosols within two prescribed burn smoke plumes

Biomass burning represents a major global source of aerosols impacting direct radiative forcing and cloud properties. Thus, the goal of a number of current studies involves developing a better understanding of how the chemical composition and mixing state of biomass burning aerosols evolve during atmospheric aging processes. During the Ice in Clouds Experiment-Layer Clouds (ICE-L) in the fall of 2007, smoke plumes from two small Wyoming Bureau of Land Management prescribed burns were measured by on-line aerosol instrumentation aboard a C-130 aircraft, providing a detailed chemical characterization of the particles. After ~2–4 min of aging, submicron smoke particles, produced primarily from sagebrush combustion, consisted predominantly of organics by mass, but were comprised primarily of internal mixtures of organic carbon, elemental carbon, potassium chloride, and potassium sulfate. Significantly, the fresh biomass burning particles contained minor mass fractions of nitrate and sulfate, suggesting that hygroscopic material is incorporated very near or at the point of emission. The mass fractions of ammonium, sulfate, and nitrate increased with aging up to ~81–88 min and resulted in acidic particles. Decreasing black carbon mass concentrations occurred due to dilution of the plume. Increases in the fraction of oxygenated organic carbon and the presence of dicarboxylic acids, in particular, were observed with aging. Cloud condensation nuclei measurements suggested all particles >100 nm were active at 0.5% water supersaturation in the smoke plumes, confirming the relatively high hygroscopicity of the freshly emitted particles. For immersion/condensation freezing, ice nuclei measurements at −32 °C suggested activation of ~0.03–0.07% of the particles with diameters greater than 500 nm

Spanning tree generating functions and Mahler measures

We define the notion of a spanning tree generating function (STGF) $\sum a_n z^n$, which gives the spanning tree constant when evaluated at $z=1,$ and gives the lattice Green function (LGF) when differentiated. By making use of known results for logarithmic Mahler measures of certain Laurent polynomials, and proving new results, we express the STGFs as hypergeometric functions for all regular two and three dimensional lattices (and one higher-dimensional lattice). This gives closed form expressions for the spanning tree constants for all such lattices, which were previously largely unknown in all but one three-dimensional case. We show for all lattices that these can also be represented as Dirichlet $L$-series. Making the connection between spanning tree generating functions and lattice Green functions produces integral identities and hypergeometric connections, some of which appear to be new.Comment: 26 pages. Dedicated to F Y Wu on the occasion of his 80th birthday. This version has additional references, additional calculations, and minor correction

Fracture toughness of the cancellous bone of FNF femoral heads in relation to its microarchitecture

This study considers the relationship between microarchitecture and mechanical properties for cancellous bone specimens collected from a cohort of patients who had suffered fractured necks of femur. OP is an acute skeletal condition with huge socioeconomic impact [1] and it is associated with changes in both bone quantity and quality [2], which affect greatly the strength and toughness of the tissue [3].Support was provided by the EPSRC (EP/K020196: Point-ofCare High Accuracy Fracture Risk Prediction), the UK Department of Transport under the BOSCOS (Bone Scanning for Occupant Safety) project, and approved by Gloucester and Cheltenham NHS Trust hospitals under ethical consent (BOSCOS – Mr. Curwen CI REC ref 01/179G)

Driving and damping mechanisms in hybrid pressure-gravity modes pulsators

We study the energetic aspects of hybrid pressure-gravity modes pulsations. The case of hybrid beta Cephei-SPB pulsators is considered with special attention. In addition to the already known sensitivity of the driving mechanism to the heavy elements mixture (mainly the iron abundance), we show that the characteristics of the propagation and evanescent regions play also a major role, determining the extension of the stable gap in the frequency domain between the unstable low order pressure and high order gravity modes. Finally, we consider the case of hybrid delta Sct-gamma Dor pulsators.Comment: 7 pages, 9 figures, in the proceedings of the Helas II Conference: "Helioseismology, Asteroseismology and MHD Connections", Goettingen, August 200