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This paper introduces strong bisimulation for continuous-time Markov decision processes (CTMDPs), a stochastic model which allows for a nondeterministic choice between exponential distributions, and shows that bisimulation preserves the validity of CSL. To that end, we interpret the semantics of CSL - a stochastic variant of CTL for continuous-time Markov chains - on CTMDPs and show its measure theoretic soundness. The main challenge faced in this paper is the proof of logical preservation that is substantially based on measure theory.

The paper explores the uniqueness properties of the subjective probabilities in two axiomatizations of state-dependent preferences. Karni, Schmeidler, and Vind's (KSV 1983) system depends on selecting an arbitrary auxiliary probability, and as such, does not guarantee the uniqueness of the derived subjective probability. However, an axiom system initially designed by Karni and Schmeidler (KS 1981) and further elaborated upon here does guarantee the desired uniqueness as well as a useful property of “stability” of the derived solution. When the preference relation displays state-independence, even the KS probabilities may not agree with those derived from the classic Anscombe-Aumann (AA 1963) theorem. However, we claim that, in this case, the KS rather than the AA probabilities are the appropriate representation of the agent's beliefs.

In this paper, the problem of approximating a function defined on a finite subset of the real line by a family of generalized rational functions whose numerator and denominator spaces satisfy the Haar conditions on some closed interval [a, b] containing the finite set is considered. The pointwise closure of the family restricted to the finite set is explicitly determined. The representation obtained is used to analyze the convergence of best approximations on discrete subsets of [a, b] to best approximations over the whole interval (as the discrete subsets become dense) in the case that the function approximated is continuous on [a, b] and the rational family consists of quotients of algebraic polynomials. It is found that the convergence is uniform over [a, b] if the function approximated is a so-called normal point. Only L p {L_p} norms with 1 ⩽ p > ∞ 1 \leqslant p > \infty are employed.

In this paper, several characterizations for s-closed spaces are obtained using regular semiclosed sets and some sets having two properties of near openness and near closedness in the same time. Images of s-closed spaces under some noncontinuous mappings are investigated. The relations between s-closedness and near compactness, co-compactness, almost co-compactness, light compactness, mild compactness are obtained. S-closed subsets relative to a semi-T2-spaces are also discussed. في هذا البحث بعض الخواص الجديدة لفراغات س - المغلقة التي قدمها طومسون عام 1967 ، وذلك باستخدام بعض المجموعات التي لها خاصية قريبة الانفتاح وقريبة الانغلاق في آن واحد ، كما تم مناقشة صور الفراغات س - المغلقة تحت تأثير بعض الرواسم غير المتصلة ، كما تضمن البحث دراسة عن علاقة الانغلاق السيني ببعض أنواع الأصماط ، واحتوى البحث كذلك على مناقشة للمجموعات الجزئية السينية المغلقة بالنسبة لفراغ T2 النصفي .

In this paper, we characterize the class of all spaces X whose products with a fixed non-Archimedan space Y are normal. In particular, we give a characterization of all spaces X whose products with every ω μ -metrizable space are normal.

The main motivation of this paper is to investigate some properties of the generating functions for the numbers Yn(λ) and the polynomials Yn(x; λ), which were recently introduced by Simsek [9] and so we give some identities and relations including the numbers Yn(λ) and the polynomials Yn(x; λ), the Bernoulli numbers and polynomials, the Apostol-Bernoulli numbers and polynomials, the Apostol-Euler numbers and polynomials, the Stirling numbers.

Five years ago in San Diego, USA, a Wheat Industry Utilization Conference took place under the direction of Y. Pomeranz who published the presented papers under the title Wheat is Unique.1 The information gathered in that publication still represents today the present state of knowledge on the structure, composition, processing, end-use properties and products of wheat. This is also true of the topics discussed here, the extraction of starch and gluten from wheat and extrusion cooking, on which the present author contributed two conference papers.2,3 Therefore, as only a few new findings can be reported, this chapter will concentrate on presenting the current state of knowledge on these topics with special reference to the functional characteristics of wheat gluten and starch. In so doing, the process technological possibilities for the extraction of starch and gluten from wheat flours and wheat wholemeal flours as well as the application of wheat starch and wheat flours for the manufacture of extrusion cooking products will be emphasized. Special emphasis is also given to the gluten-forming properties of wheat proteins and the two starch fractions present in the endosperm which differ not only in size but also in physical and chemical properties.

This paper introduces a parametric method for identifying the somatosensory evoked potentials (SEPs). The identification was carried out by using pole-zero modeling of the SEPs in the discrete cosine transform (DCT) domain. It was found that the DCT coefficients of a monophasic signal can be sufficiently approximated by a second-order transfer function with a conjugate pole pair. The averaged SEP signal was modeled by the sum of several second-order transfer functions with appropriate zeros and poles estimated using the least square method in the DCT domain. Results of the estimation demonstrated that the model output was in an excellent agreement with the raw SEPs both qualitatively and quantitatively. Comparing with the common autoregressive model with exogenous input modeling in the time domain, the DCT domain modeling achieves a high goodness of fitting with a very low model order. Applications of the proposed method are possible in clinical practice for feature extraction, noise cancellation and individual component decomposition of the SEPs as well as other evoked potentials.

A path in an edge-colored graph is called conflict-free if it contains a color that is used by exactly one of its edges. An edge-colored graph G is conflict-free connected if for any two distinct vertices of G, there is a conflict-free path connecting them. For a connected graph G, the conflict-free connection number of G, denoted by cfc(G), is defined as the minimum number of colors that are required to make G conflict-free connected. In this paper, we investigate the conflict-free connection numbers of connected claw-free graphs, especially line graphs. We use L(G) to denote the line graph of a graph G. In general, the k-iterated line graph of a graph G, denoted by \(L^k(G)\), is the line graph of the graph \(L^{k-1}(G)\), where \(k\ge 2\) is a positive integer. We first show that for an arbitrary connected graph G, there exists a positive integer k such that \(cfc(L^k(G))\le 2\). Secondly, we get the exact value of the conflict-free connection number of a connected claw-free graph, especially a connected line graph. Thirdly, we prove that for an arbitrary connected graph G and an arbitrary positive integer k, we always have \(cfc(L^{k+1}(G))\le cfc(L^k(G))\), with only the exception that G is isomorphic to a star of order at least 5 and \(k=1\). Finally, we obtain the exact values of \(cfc(L^k(G))\), and use them as an efficient tool to get the smallest nonnegative integer \(k_0\) such that \(cfc(L^{k_0}(G))=2\).

We analyze the explicit contribution of fluid inertia and fluid unsteadiness to the force acting on a solid sphere moving in a vertical solid-body rotation flow, in the limit of small Reynolds and Taylor numbers. This problem can be thought of as a test case where the flow induced by the particle is both unsteady (in the laboratory frame) and convected by the unperturbed flow. Many authors assume that the contributions of these two effects can be approximately superposed, and postulate that the particle motion equation is composed of the classical Boussinesq-Basset-Oseen equation (obtained by neglecting the fluid inertia) plus an additive lift force. In the present paper the simplicity of the unperturbed flow enables one to calculate analytically the explicit contribution of each term appearing in the perturbed flow equation (by using matched asymptotic expansions). Our results show how the convective terms and the unsteady term do contribute to the particle drag and lift coefficients in a very complex and nonadditive manner.

In this paper our aim is to give refinements of Jensen?s type inequalities for the convex function defined on the co-ordinates of the bidimensional interval in the plane.

A compact microwave bandpass filter with λ/4 stepped impedance resonators (SIRs) is presented in this paper. λ/4 SIRs are employed here not only to miniaturize the circuit size, but also to eliminate spurious resonance at 2f 0 . To achieve the maximal number of finite transmission zeros, this filter is synthesized based on the 4th-order fully canonical coupling topology. As a result, four finite transmission zeroes are analytically achieved, and used to sharpen filter rejection skirts. Three capacitive J couplings are systematically designed to meet the required coupling strengths for the filter frequency response. Besides, they also adjust their coupling zeros for 3f 0 spurious suppression. Lastly, a 4th-order fully canonical bandpass filter is designed and fabricated with central frequency of 1.8 GHz and fractional bandwidth of 11.0% The proposed filter possesses preferred merits: compact size, 4 transmission zeros and up to 4.8f 0 20 dB attenuated stopband.

Wide field application of curve number (CN) methodology in design flood estimation and forecasting studies is a well-accepted reality. The parameter CN is often regarded as a constant, but it is in fact a random variable, for it relies on a number of factors governing the process of rainfall-runoff. This paper suggests an alternative procedure to estimate design runoff (Q) using design storm (P) and design curve number (CN) values. To this end, design CN values for different durations and return periods have been derived from 25 years of daily annual maximum P-Q data of 10 Indian catchments. When compared, the computed design Q from design CN values was either very close to or slightly greater than the observed Q. The performance is also evaluated using coefficient of determination (R2), percentage of bias (PBIAS), and normalized Nash-Sutcliffe efficiency (NNSE). The magnitude of design CN is found to decrease for longer durations and smaller return periods, and vice versa, and such a behavior invoked the development of an empirical relation for estimation of design CN from given return period and duration for gauged as well as ungauged watersheds. The statistical analysis revealed the general extreme value (GEV) to fit best the P-CN series, whereas multiple probability distributions, viz., GEV, Gumbel max, log Pearson 3, and Gen Pareto fitted best the Q series. The 50-year, 100-year, and 200-year runoff estimated from the P-CN series exceeded only marginally those derived from the conventional Q series approach.

One of the objects of the paper which last year I had the honour of presenting to the Royal Society, was to shew the in­ sufficiency in mathematical reasoning, of a principle of analogy, by which the properties demonstrated for one figure were to be transferred to another, to which the former was supposed to bear a resemblance; and the argument for the insufficiency of the principle was this, that the analogy between the two figures was neither antecedent to calculation, nor independent of it, and consequently could not regulate it; that analogy was the object of investigation, not the guide; the result of demonstra­tion, not its directing principle. Having shewn that analogy could not establish the truth of certain mathematical conclusions, I next endeavoured to shew why such conclusions had been rightly inferred; not by pro­posing any new excogitated principle, nor by pointing out an hitherto unobserved intellectual process; but I conceived they might be obtained by operations conducted in a manner similar to that by which all reasoning with general terms is conducted, and that the relations between the symbols or general terms were to be established by giving the true meaning to the con­necting signs, which indicate not so much the arithmetical computation of quantities, as certain algebraical operations. It was further observed, that, from certain established formulas, abridged symbols or general terms might be formed, which consequently must have their signification dependent on such formulas; and that, although the parts of certain abridged ex­pressions could not separately be arithmetically computed, yet the expressions themselves might be legitimately employed in all algebraic operations.

My Dear Sir, In a Memoir which I had the honour to present to you two years ago, I related some experiments on trees, from which I inferred, that their sap, having been absorbed by the bark of the root, is carried up by the alburnum or white wood, of the root, the trunk, and the branches; that it passes through what are there called the central vessels, into the succulent part of the annual shoot, the leaf-stalk, and the leaf; and that it returns to the bark, through the returning vessels of the leaf-stalk. The principal object of this Paper is, to point out the causes of the descent of the sap through the ’bark, and the consequent formation of wood. These causes appear to be, gravitation, motion communicated by winds or other agents, capillary attraction, and probably something in the conformation of the vessels themselves, which renders them better calculated to carry fluids in one direction than in another. I shall begin with a few observations on the leaf, from which all the descending fluids in the tree appear to be derived. This organ has much engaged the attention of naturalists, particularly of M. Bonnet: but their experiments have chiefly been made on leaves severed from the tree; and, therefore, whatever conclusions have been drawn, stand on very questionable ground. The efforts which plants always make to turn the upper surfaces of their leaves to the light, have with reason induced naturalists to conclude, that each surface has a totally distinct office; and the following experiments tend strongly to support that conclusion.

My Dear Sir, It is well known that the fluid, generally called the Sap in trees, ascends in the spring and summer from their roots, and that in the autumn and winter it is not, in any considerable quantity, found in them; and I have observed in a former Paper, that this fluid rises wholly through the alburnum, or sap-wood. But Du Hamel and subsequent naturalists have proved, that trees contain another kind of sap, which they have called the true, or peculiar juice, or sap of the plant. Whence this fluid originates does not appear to have been agreed by naturalists; but I have offered some facts to prove that it is generated by the leaf; and that it differs from the common aqueous sap owing to changes it has undergone in its circulation through that organ: and I have contended that from this fluid ( which Du Hamel has called the suc propre , and which I will call the true sap,) the whole substance, which is annually added to the tree, is derived. I shall endeavour in the present Paper to prove that this fluid, in an inspissated state, or some concrete matter deposited by it, exists during the winter in the alburnum, and that from this fluid, or substance, dissolved in the ascending aqueous sap, is derived the matter which enters into the composition of the new leaves in the spring, and thus furnishes those organs, which were not wanted during the winter, but which are essential to the further progress of vegetation. Few persons at all conversant with timber are ignorant, that the alburnum, or sap-wood of trees, which are felled in the autumn or winter, is much superior in quality to that of other trees of the same species, which are suffered to stand till the spring, or summer: it is at once more firm and tenacious in its texture, and more durable. This superiority in winterfelled wood has been generally attributed to the absence of the sap at that season; but the appearance and qualities of the wood seem more justly to warrant the conclusion, that some substance has been added to, instead of taken from it, and many circumstances induced me to suspect that this substance is generated, and deposited within it, in the preceding summer and autumn.

When I had ascertained that carbonaceous substances, whether vegetable, animal, or mineral, were capable of being converted into a product, which, by its effects on earthy and metallic solutions, on dissolved gelatine, and on skin, resembled the natural vegetable principle called Tannin , I was at first inclined to give it the name of artificial or factitious tannin ; but some eminent chemists of this country, for whose opinions I have the highest respect, considered this name as objectionable; for although the artificial substance resembles tannin in the parti­culars above stated, yet in one character there appears to be a very considerable difference, namely, the effect of nitric acid; for by this, the artificial substance is, whilst the varieties of natural tannin are destroyed . Such an objection, sanctioned by such authority, induced me to alter the title of my Paper, and to expunge the word tannin wherever it had been applied to the artificial product. In order to satisfy myself more fully on this point, I have, since the communication of my former Paper, made a few ex­periments on the comparative effects produced by nitric acid on those substances which contain the most notable quantities of tannin , and of these I shall now give a succinct account, and shall also cursorily notice other experiments, in which a tanning substance has been produced, under circumstances different, in some measure, from those which have been already described.

Having some years back, when reading the learned Mr. Landen's fifth Memoir, discovered the manner of applying a method of differences, to the species of series, whose sums are there obtained by the help of impossible quantities, and having since extended that application, I now venture to offer it to the consideration of others. The practice of this method, in most cases, appears to me extremely simple; and on that account, I am almost induced to imagine, that they have already been considered by mathe­maticians; indeed since the greatest part of this Paper was written, I met with Euler's Institutiones Calculi integralis ; two simple series are in that work summed by multiplications similar to those employed in the investigation of the principal theorems contained in this Paper; but whether that learned mathematician has farther pursued the method, in that or in any other work, I have not as yet been able to ascertain.

The observations which accompany this Paper were made at Westbury in Somersetshire, in the years 1800 and 1801, with an Astronomical Circle of two feet and a half diameter, constructed by Mr. Troughton, and considered by him as one of the best divided instruments he had ever made; a drawing of it, with a short description, is annexed to the ob­servations. (Plate XX.) When this instrument came into my possession, I thought I could not employ it in a more advantageous manner, than in endeavouring to determine the declinations of some of the principal fixed stars. The various catalogues differed so much from each other, and such doubt existed as to the accuracy of those which were thought most perfect; that the declinations of few stars could be considered as sufficiently well ascertained for the more accurate purposes of astronomy.

The theorems relative to finite differences, given by M. Lagrange in the Berlin Memoirs, for 1772, have much engaged the attention of mathematicians. M. Laplace has been particularly successful in his investigations respecting them; yet an important difficulty remained, to endeavour to surmount which is the principal object of this Paper. The theorems alluded to may be thus stated. Let u represent any function of x . Let x + h , x +2 h , x +3 h , &c. be successive values of x , and 1 u , 2 u , 3 u &c. corresponding successive values of u . Let Δ n u represent the first term of the n th order of differences of the quantities u , 1 u , 2 u &c. And let also S n u represent the first term of a series of quantities, of which the first term of the n th order of differences is u . Then ( e representing the series 1 + 1 + 1/1.2 + 1/1.2.3 +, &c.)

Mr. William Garrard having shewn me a curious property of the tangents of the three angles of a plane triangle, or in other words, of the tangents of three arches trisecting a semicircle, in a paper which I have communicated to this Society, I was led to consider whether a similar property might not belong to the tangents of three arches trisecting the whole circumference; and, on examination, found it be so. Let the circumference of a circle be divided any how into three arches A, B, C; that is, let A + B + C be equal to the whole circumference. I say, the square of the radius multiplied into the sum of the tangents of the three arches A, B, C, is equal to the product of the tangents multiplied together. I shall demonstrate this by symbolical calculation, now commonly called (especially by foreign mathematicians) analytic calculation.

In the first part of this paper, I have pointed out a variety of methods that will give us coloured concentric rings between two glasses of a proper figure applied to each other, and it has been proved that only two surfaces, namely, those that are in contact with each other, are essential to their formation; it will now be necessary to enlarge the field of prismatic phe­nomena, by showing that their appearance in the shape of rings has been owing to our having only used spherical curves to produce them. 35. Cylindrical Curves produce Streaks . As soon as it occurred to me, that the cause of the figure of any certain prismatic appearance must be looked for in the nature of the curvature of one or both of the surfaces, that are essential to its production, I was prepared to expect that if a spherical curve, when applied to a plain surface of glass, produces coloured rings, a cylindrical one applied to the same would give coloured lines or streaks. To put this to the proof of an experiment, I ground one side of a plate of glass into a cylindrical curve, and after having given it a polish, I laid a slip of plain glass upon it, and soon perceived a beautiful set of coloured streaks. The broadest of them was at the line of contact, and on each side they were gradually narrower and less bright. The colours in the streaks were similar to those in the rings, and they were in the same manner changeable by pressure as in them. Their order was likewise the same, if we reckon from the line of contact, as with rings we do from the center; so that these streaks differed in no respect from rings, except in their linear instead of circular arrange­ment.

In this discourse I propose to investigate the attractions of a very extensive class of spheroids, of which the general description is, that they have their radii expressed by rational and integral functions of three rectangular co-ordinates of a point in the surface of a sphere. Such spheroids may be characterized more precisely in the following manner: conceive a sphere of which the radius is unit, and three planes intersecting one another at right angles in the centre; from any point in the surface of the sphere draw three perpendicular co-ordinates to the fixed planes, and through the same point in the surface likewise draw a right line from the centre, and cut off from that line a part equal to any rational and integral function of the three co-ordinates: then will the extremity of the part so cut off be a point in the surface of a spheroid of the kind alluded to; and all the points in the same surface will be determined by making the like construction for every point in the surface of the sphere. The term of a rational and integral function is not to be strictly confined here to such functions only as consist of a finite number of terms; it may include infinite serieses, provided they are converging ones; and it may even be extended to any algebraic expressions that can be expanded into such serieses. This class of spheroids comprehends the sphere, the ellipsoid, both sorts of elliptical spheroids of revolution, and an infinite number of other figures, as well such as can be described by the revolving of curves about their axes, as others which cannot be so generated. In the second chapter of the third book of the Mécanique Céleste , Laplace has treated of the attractions of spheroids of every kind; and in particular he has given a very ingenious method for computing the attractive forces of that class which in their figures approach nearly to spheres. In studying that work, I discovered that the learned author had fallen into an error in the proof of his fundamental theorem; in consequence of which he has represented his method as applicable to all spheroids whatever, provided they do not differ much from spheres; whereas in truth, when the error of calculation is corrected, and the demonstration made rigorous, his analysis is confined exclusively to that particular kind, described above, which it is proposed to make the subject of this discourse. I have already treated of this matter in a separate paper, in which I have pointed out the source of Laplace’s mistake, and likewise have strictly demonstrated his method for the instances that properly fall within its scope. In farther considering the same subject, it occurred to me that the investigation in the second chapter of the third book of the Mécanique Céleste , however skilfully and ingeniously conceived, is nevertheless indirect, and is besides liable to another objection of still greater weight; it does not exhibit the several terms of the series for the attractive force in separate and independent expressions: it only points out in what manner they may be derived successively, one after another; in so much that the terms of the series near the beginning cannot be found without previously computing all the rest. This remark gave occasion to the following paper, in which it is my design to give a solution of the problem which is not chargeable with the imperfections just mentioned: the analysis is direct, and every term of the series for the attractive force is deduced immediately from the radius of the spheroid. As the ellipsoid, which comprehends both sorts of elliptical spheroids of revolution, falls within the class of figures here treated of, I have derived, as a corollary from my investigation, the formulas for the attractions of that figure which are required in the theory of the earth: this paper therefore will contain all that is useful on the subject of the attractions of spheroids, as far as our knowledge at present extends, deduced by one uniform mode of analysis.

Several years have elapsed, since my very highly esteemed friend, now Rev. Doctor Mooney, Fellow of Trinity College, Dublin, presented to the Royal Irish Academy a paper on the Extermination of Radicals from Equations. He has illustrated, by sundry examples, the extermination of quadratic surds. As he has rightly observed, the method is universal. Any number of quadratic surds, independent, or dependent, on each other, may be removed from an equation; because, 1. Any quantity, or factor of a quantity, necessarily subjected to the radical sign, is but of one dimension. 2. This quantity or factor being brought to one side of the equation, while the quantities unaffected with it remain at the other, may, by squaring both sides, be freed from the radical sign. 3. By a repetition of these reductions for each remaining independent surd quantity, any number of surd quantities may be converted into rational.

The term function has long been introduced into analysis with great advantage, for the purpose of designating the result of every operation that can be performed on quantity. This extent of signification has rendered it of essential use, but the various applications of which it admits, and the questions to which it gives rise, do not appear to have met with sufficient attention. I propose in the following paper to present an outline of a new calculus, which naturally results from it. It comprehends questions of the greatest generality and difficulty, and will probably require the invention of new methods for its improvement.

In the year 1772, Lagrange, in a Memoir, published among those of the Berlin Academy, announced those celebrated theorems expressing the connection between simple exponential indices, and those of differentiation and integration. The demonstration of those theorems, although it escaped their illustrious discoverer, has been since accomplished by many analysts, and in a great variety of ways. Laplace set the first example in two Memoirs presented to the Academy of Sciences,* and may be supposed in the course of these researches, to have caught the first hint of the Calcul des Fonctions Generatrices with which they are so intimately connected ; as, after an interval of two years, another demonstration of them, drawn solely from the principles of that calculus appeared, together with the calculus itself, in the memoirs of the Academy. This demonstration, involving, however, the passage from finite to infinite, is therefore (although preferable perhaps in a systematic arrangement, where all is made to flow from one fundamental principle) less elegant ; not on account of any confusion of ideas, or want of evidence ; but, because the ideas of finite and infinite, as such, are extraneous to symbolic language, and, if we would avoid their use, much circumlocution, as well as very unwieldy formulæ must be introduced. Arbogast also, in his work on derivations, has given two most ingenious demonstrations of them, and added greatly to their generality ; and lastly, Dr. Brinkley has made them the subject of a paper in the Transactions of this Society,* to which I shall have occasion again to refer. Considered as insulated truths, unconnected with any other considerable branch of analysis, the method employed by the latter author seems the most simple and elegant which could have been devised. It has however the great inconvenience of not making us acquainted with the bearings and dependencies of these important theorems, which, in this instance, as in many others, are far more valuable than the mere formulæ.

In a former Paper which the Royal Society honoured with a place in the last Volume of their Transactions, I endeavoured to explain the nature of the calculus of functions, and I proposed means of solving a variety of functional equations containing only one variable quantity. My subsequent enquiries have produced several new methods of solving these, and much more complicated functional equations, and have convinced me of the importance of the calculus, particularly as an instrument of discovery in the more difficult branches of analysis; nor is it only in the recesses of this abstract science, that its advantages will be felt: it is peculiarly adapted to the discovery of those laws of action by which one particle of matter attracts or repels another of the same or of a different species; consequently, it may be applied to every branch of natural philosophy, where the object is to discover by calculation from the results of experiment, the laws which regulate the action of the ultimate particles of bodies. To the accomplishment of these desirable purposes, it must be confessed that it is in its present state unequal; but should the labours of future enquirers give to it that perfection, which other methods of investigation have attained, it is not too much to hope, that its maturer age shall unveil the hidden laws which govern the phenomena of magnetic, electric, or even of chemical action. When functional equations containing two or more variables occur, their solution presents still greater difficulties than those we have already considered; the new relations which arise, necessarily require a new notation to distinguish them. I shall endeavour, as far as I am able, to apply or extend that already in use; but, as it is almost impossible in the infancy of a calculus to foresee the extent to which it may be carried, or the new views which it may be necessary to take of it, the notation I have used should only be considered as of a temporary nature; it may be employed until some more convenient one be devised: perhaps, how ever, it might be more advantageous that it should not be altered until our acquaintance with this subject becomes more intimate, and until the infinitely varied and comprehensive relations displayed in the doctrine of functions, have been more minutely examined.

1. I have endeavoured, in this short Paper, to give a simple and connected theory of the construction of logarithms, which I think has not hitherto been done. Prop. I. To find the Logarithm of 1 + x . It is not difficult to see that we may assume L (1 + x ) = ′A x + ″A x 2 + ‴A x 3 + ‴′A x 4 + &c., whence L (1 + y ) = ′A y + ″A y 2 + ‴A y 3 + ‴′A y 4 + &c., and L {(1+ x ) (1+ y )} = L(1+ x + y + xy ), or putting 1+ x = π , =L{1+( x + πy )=′A( x + πy )+″A( x + πy ) 2 +‴A( x + π y) 3 +&c.

In looking into Mr. Spence’s ingenious “Essay on Loga­rithmic Transcendents,” a work published in 1809, but which I have been so unfortunate as never to have seen till within the last fortnight, I was not a little surprised to find that a demonstration of the binomial theorem, similar to the one I had the honour to present to the Royal Society, had been already given by that writer. The same may be said of the first proposition of the preceding Paper on the construction of Logarithms. Having made this acknowledgment, I shall perhaps be pardoned for observing, that Mr. Spence is not particularly happy in the manner of developing the kind of functions he treats of in his preface. I shall endeavour to give the solu­tion of a class of equations of which he (Pref. p. vii.) has considered a particular case: with this we will begin.

The processes which it is the object of this paper to explain, were discovered several years since; but certain difficulties connected with the subject, which I was at that time unable to explain, and which were equally inexplicable to several of my friends, to whom I had communicated these methods, induced me to defer publishing them, until I could offer some satisfactory solution. These observations refer more particularly to the second method which I have detailed in this paper, and which may not inappropriately be called the method of expanding hori­zontally and summing vertically . Some traces of this method may, perhaps, be found in former writers, and particularly in a paper by Professor Vince, “On the Summation of Se­ries,“ printed in the Philosophical Transactions for 1791; but there exists this peculiarity in that which I have employed, that after a certain number of the vertical columns are summed, all the remainder either vanish, or else have some common factor. This method, which I employed about the year 1812, gave the values of a variety of series whose sums had not hitherto been known, most of which were apparently correct, but some of the consequences which followed were evidently erroneous. About this time, Mr. Herschel, to whom I had communicated these anomalous results, by following a very different course, arrived at several general theorems, which, when applied to the series I had obtained, gave the same results. This coincidence at first increased my confidence in the values so discovered, and I continued to examine the reason why my own formulæ were in some cases defective.

Dr. Ure commences this paper by adverting to the fallacies to which the modes of analysing organic substances hitherto practised are subject; and in detailing the peculiar methods adopted in his own researches, he shows the means of obviating them, and of diminishing the various sources of inaccuracy to which these complicated processes of analytical chemistry are necessarily more or less liable. Where oxide of copper is used, its hygrometric quality has generally been overlooked, or not duly allowed for; and the animal and vegetable substances have not in general been exposed to any process of desiccation sufficiently exact or uniform; the author therefore always used the oxide of copper in some known or ascertained degree of humidity; and he dried the organic bodies in the air.pump vacuum, aided by the absorbent powers of a surface of sulphuric acid in the apparatus, and with precautions which he fully describes. He then details the best means of applying heat for the decomposition of organic substances, and describes a drawing representing the construction of his furnace, and other implements. Lastly, he points out the method of examining the results and products, and gives in detail the analysis of sulphuric ether, as illustrating the mode of computing the relations of the constituents, while the results of the other analyses are, for the sake of brevity, thrown into a tabular form. Dr. Ure concludes his paper with some general remarks on the analytical details. In respect to sugar, he observes, that on comparing pure crystalline sugar with diabetic sugar, the latter exhibits a notable excess of oxygen; and he considers weak sugars (as the refiners call them), in general, to exhibit the same peculiarity. In applying the atomic theory to his experimental results, the author enlarges on the different views which may be taken of the ultimate constitution of a variety of organic products, and enters at considerable length into details relating to the vegetable acids, with a view of determining with exactness their prime equivalents, and the relative proportions of combined water which they contain in their crystalline states.

I have recently received from Captain Kater a printed paper from the Philosophical Transactions for 1821, giving an account of his experiments in examining and comparing the different standard scales. I have read with great attention and satisfaction the whole of his results, and am glad to find that the Commissioners for considering the subject of weights and measures, have adopted Mr. Bird's scale of 1760, as by that means there is now a universal standard of comparison, which applies to the French metre, and to all the measures used on the Continent. From Captain Kater’s results it appears, that my standard scale requires a multiplier of —, 000018 to make it agree with the above scale of Mr. Bird; and that Ramsden's bar, used in the Trigonometrical Survey of Great Britain, requires the multiplier +, 00007. That is to say, with respect to a measurement on the meridian, the degree depending on my brass scale must be multiplied by, 000018 and the product subtracted from the measure given by the scale, to reduce it to what it would have been, had it been measured by what is now the parliamentary standard; and the degree depending on Ramsden's bar, must be multiplied by , 00007, and the product added to the measure given by the bar, to reduce it to the standard measure.

It was originally my intention, in entering on this inquiry, merely to ascertain the composition of the ores, the mineralogical characters of which have been so ably delineated by Mr Haidinger in the preceding paper. I had advanced however but a short way in the investigation, when my progress was arrested by doubts both as to the manner of conducting the analyses, and as to the mode of calculating their results. In this uncertainty I found it necessary to extend my original plan, with the view of supplying by my own researches what appeared to be not sufficiently established by the labours of other chemists. I have accordingly divided the essay into two parts; attempting in the first division to ascertain the atomic weight of manganese, and the composition of the artificial oxides of that metal; and in the second, applying the facts thus established to illustrate the chemical constitution of the native oxides described by Mr Haidinger.

The fruit in this natural order does not appear to have been well understood by most botanists; and I shall therefore attempt to give a general view of what appears to me to be its structure; and most of the parts are visible in the section which is here given (Plate IX. Fig. 1.) of the beautiful but insipid Indian Melon (Cucumis Melo) called Phuti.The outer parietes (Fig. 1. a,) when young, are thick, fleshy, and undivided by sutures, with an uniform rind, not separable from the fleshy part. As the fruit ripens, the rind in some cases becomes so thin as to be unable to contain the pulpy matter, and bursts either gradually, as in the melon, or with elasticity as in the Momordica and Elaterium of Tournefort. At other times, the rind hardens either into a thin substance like leather or strong paper, as in the Luffa, or into a strong ligneous covering, as in the Cucurbita leucanthema or gourd. In these cases, it sometimes opens horizontally, by means of an operculum, which falls off and leaves an aperture for the seeds, as in Fig. 2. representing the summit of the Luffa called Picinna in the Hortus Malabaricus.

Free Disposal Hull (FDH) model was proposed in the end of 20-th century for performance measurement and management of complex units. Production possibility set of FDH model is a nonconvex set. For this reason, nonlinear integer programming methods and enumeration methods have been used for computations of various characteristics of units’ performance. However, as Cesaroni, Kerstens and Van de Woestyne noted, there are no methods in the world scientific literature for frontier visualization in nonconvex FDH models. In this paper, an approach is proposed for frontier visualization with the use of methods of purposeful enumeration. Computational experiments, conducted with the use of real-life data sets, confirm reliability and effectiveness of proposed algorithms.

About three months ago I wrote a paper intitled "Consideration of the objections raised against the geometrical representation of the square roots of negative quantities,” which paper was communicated to the Royal Society by Dr. Young, and read on the 19th of February last. At that time I had only discovered the manner of representing geometrically quantities of the form a + b √— 1, and of geometrically adding and multiplying such quantities, and also of raising them to powers, either whole or fractional, positive or negative; but I was not then able to represent geometrically quantities of the form a + b √ — m + n √ — 1 , that is, quantities raised to powers, whose indices involve the square roots of negative quantities. My attention, however, has since been drawn to these latter quantities in consequence of an observation which I met with in M. Mourey’s work on this subject (the work which I mentioned in my former paper); the observation is as follows: "Les limites dans lesquelles je me suis restreint m’ont forcé à passer sous silence plusieurs espéces de formules, telles sont celles-ci a √ — 1 , a √ — 1 sin (√ — 1) &c., &c., &c. Je les discute amplement dans mon grand ouvrage, et je démontre que toutes expriment des lignes directives situees sur le mêrae plan que 1 et 1.” where a √ — 1 and 1 1 in M. Mourey’s notation signify respectively a ( 1 1 ) √ — 1/4 and ( 1 1 )according to my notation.

In the course of the adjustment and verification of the copies of the Imperial standard yard, destined for the Exchequer, Guildhall, Dublin, and Edinburgh, I discovered a source of error till then, I believe, wholly unsuspected, arising from the thickness of the bar upon the surface of which measures of linear dimension are traced. The difficulties which I experienced, and the remedy which suggested itself upon that occasion, and which was found efficient, are shortly detailed in the account of the construction and adjustment of the new standards of weights and measures of the United Kingdom of Great Britain and Ireland, published in the Philosophical Transactions for 1826. But as the notice there given occupies little more than a single page, and might therefore pass unremarked, I cannot but think that a fact of such importance in inquiries where linear measures are concerned, and which may be sufficient to account for the discrepancies to be found in the experiments of different observers, ought to be placed before the Royal Society in a more pro­minent point of view than that which it at present occupies. I shall, therefore, first extract from the paper alluded to the part to which I refer, and then add an account of such experiments as I have since made on the subject; and describe a scale which I have caused to be constructed so as almost entirely to obviate the source of error of which I am treating.

Reliability engineers often face failure data with suspensions. The rank regression method with an approach introduced by Johnson has been commonly used to handle data with suspensions in engineering practice and commercial software. However, the Johnson method makes partial use of suspension information only—the positions of suspensions, not the exact times to suspensions. A new approach for rank regression with censored data is proposed in this paper, which makes full use of suspension information. Taking advantage of the parametric approach, the refined rank regression obtains the ‘exact’ mean order number for each failure point in the sample. With the ‘exact’ mean order number, the proposed method gives the ‘best’ fit to sample data for an assumed times-to-failure distribution. This refined rank regression is simple to implement and appears to have good statistical and convergence properties. An example is provided to illustrate the proposed method. Copyright © 2004 John Wiley & Sons, Ltd.

The following Table is supplementary to those accompanying the paper “On the Mathematical Theory of Suspension Bridges,” printed in the Philosophical Transactions for 1826. It is deduced from the first Table there given, by the plain operations of common arithmetic; but this admits of a far more ready application than the former, to all cases of practical investigation. The first column contains the deflections or versed sines of the curve, expressed in fractional parts of the double ordinate or Span. It is therefore 2 y divided by x , and their reciprocals are added under each.

This peculiarity was first observed in the axillary arteries and in the iliacs of the Lemur tardigradus, which at their entrance into the upper and lower limbs were found to be suddenly divided into a considerable number of equal-sized cylinders, which occasionally anastomosed with each other, and were regularly distributed on the muscles; whilst the arteries proceeding to the other parts of the body divided in the usual arborescent form. Upon prosecuting this inquiry, it was found that the Bradypus tridactylus, and in some measure also the didactylus, has a similar distribution of these arteries.

This paper relates to an improved solution of a problem by which swiftly converging series are obtained, which are useful in computing the mutual perturbations of the planets; and contains some further remarks and improved formulæ for computing the coefficients, by which the arithmetical work is considerably shortened and facilitated.

The light which is the object of the present inquiry must be distinguished not only from that which we derive from the sun, but also from the brightness exhibited by artificial phosphori, electricity, meteors, and other lucid emanations. The principal bodies which afford the light here treated of, are, 1) Some vegetable and earthy substances, such as rotten wood, and peat earth. 2) Marine animals, some in a living state, viz. the Medusa phosphorea , the Pholas , the Pennatula phosphorea , and the Cancer fulgens ; and most of the marine fishes soon after they are deprived of life. 3) Animal flesh in general, some time after the extinction of life. And, 4) Among insects the creeping glow-worm , the flying glow-worm , the fire-fly , and the great lantern-fly . The numerous experiments described in this paper were chiefly made on marine animals, particularly herrings and mackerel, which were exposed either in the air, or in water impregnated with different salts, and of different temperatures, in a dark vault to which the Doctor assigns the name of his laboratory.

The mode of conducting this important survey having been already noticed in the Journals of the Society on various former occasions, it will only be necessary here to state the progress of the operation, which we find has now been carried on over Essex, the western part of Kent, Suffolk, and Hertfordshire, and portions of the counties contiguous to them.

The object of this paper is to show, that we may be assured of the justness and accuracy of conclusions obtained by means of imaginary quantities, without verifying such conclusions by separate investigations, or without inferring their truth from analogy. In the first part the author premises at some length certain arguments, to show that the operations with impossible quantities must have a logic equally strict and certain with the logic that appertains to real quantities, and that the aid obtained by these quantities would be perfectly useless if such conclusions rested only on the frail basis of analogy. The author proceeds next to show that operations with imaginary quantities are by no means mechanical, but that they are conducted according to the rules of strict and rigorous logic; and that, although strictly speaking no proposition concerning them can be true or false, yet, after the demonstrations of certain formulæ for real quantities, demonstrations with impossible quantities may be legitimately and logically conducted. The series, for instance, for the development of an exponential, when the exponent is an impossible quantity, can never, independently of certain arbitrary assumptions, be duly established; and yet, when the exponent is the sign of a real quantity, the formulæ for the development may be rigorously proved. With regard to demonstration, it is shown, as in the case of real quantities, it actually proceeds by a series of transformation, each proved to be the same as the foregoing, not by any arguments grounded on the properties of real quantities, but by reference to the forms certain abridged symbols are made to represent, and to the nature of the operations directed to be performed with certain signs.