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Because the dynamics of complex systems is the result of both decisive local events and reinforced global effects, the prediction of such systems could not do without a genuine multilevel approach. This paper proposes to found such an approach on information theory. Starting from a complete microscopic description of the system dynamics, we are looking for observables of the current state that allows to efficiently predict future observables. Using the framework of the information bottleneck (IB) method, we relate optimality to two aspects: the complexity and the predictive capacity of the retained measurement. Then, with a focus on agent-based models (ABMs), we analyze the solution space of the resulting optimization problem in a generic fashion. We show that, when dealing with a class of feasible measurements that are consistent with the agent structure, this solution space has interesting algebraic properties that can be exploited to efficiently solve the problem. We then present results of this general framework for the voter model (VM) with several topologies and show that, especially when predicting the state of some sub-part of the system, multilevel measurements turn out to be the optimal predictors.

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.

SUMMARY The advent of standard algorithms for the numerical solution of partial differential equations has given researchers a new tool for fluid flow calculations. In this paper, single-phase flow in curved ducts is numerically simulated by imposing a spatially varying centrifugal force on a fluid flowing in a straight tube. The resulting set of partial differential equations is solved using the HARWELL-FLOW3D computer program. Comparison with other numerical and experimental results shows that this simplified formulation gives accurate results. The model neglects certain geometric terms of the order d/D, the duct-to-coil diameter ratio. The effect of these terms is investigated by considering the flow in a 90" bend for large d/D. It is shown that while there may be significant error in the prediction of the local variables for large d/D, the circumference-averaged quantities are well predicted.

The history of the computer science department at the Institute of Mathematics and Informatics of Bulgarian Academy of Sciences since 1963 is briefly described. Starting with small groups of enthusiastic people initiating the new for Bulgaria research in the field of Informatics, nowadays the department can report some significant achievements. The paper presents some of them in three areas—software quality evaluation and management, data mining and semantic analysis, information systems and communities—which mark some key areas of research during the years.

Group-based cryptography is viewed as a modern cryptographic candidate solution to blocking quantum computer attacks, and key exchange protocols on the Internet are one of the primitives to ensure the security of communication. In 2016 Habeeb et al proposed a “textbook” key exchange protocol based on the semidirect product of two groups, which is insecure for use in real-world applications. In this paper, after discarding the unnecessary disguising notion of semidirect product in the protocol, we establish a simplified yet enhanced authenticated key agreement scheme based on cyclic automorphism subgroups of random orders by making hybrid use of certificates and symmetric-key encryption as challenge-and-responses in the public-key setting. Its passive security is formally analyzed, which is relative to the cryptographic hardness assumption of a computational number-theoretic problem. Cryptanalysis of this scheme shows that it is secure against the intruder-in-the-middle attack even in the worst case of compromising the signatures, and provides explicit key confirmation to both parties.

The paper employs fuzzy automata for emotion control by audio-visual stimulus. In the proposed design, emotions are represented by states, and transition of emotions is controlled by selective audio-visual stimulus. Experiments are undertaken to determine the fuzzy membership for state transitions in the automata. A fuzzy relational algebra has been used to determine the current membership distribution of the emotions from their previous value and fuzzy state transition memberships. Experiments with fifty audio-visual stimulus and one hundred subjects confirm that automatic control of emotion to a desired state from a given state can be attained with a success rate of approximately 86%.

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 show how Guided Local Search (GLS) can be applied to the SAT problem and show how the resulting algorithm can be naturally extended to solve the weighted MAX-SAT problem. GLS is a general, penalty-based meta-heuristic, which sits on top of local search algorithms to help guide them out of local minima. GLS has been shown to be successful in solving a number of practical real-life problems, such as the traveling salesman problem, BT"s workforce scheduling problem, the radio link frequency assignment problem, and the vehicle routing problem. We present empirical results of applying GLS to instances of the SAT problem from the DIMACS archive and also a small set of weighted MAX-SAT problem instances and compare them with the results of other local search algorithms for the SAT problem.

Multiphysics systems are driven by multiple processes acting simultaneously, and their simulation leads to partitioned systems of differential equations. This paper studies the solution of partitioned systems of differential equations using exponential Runge-Kutta methods. We propose specific multiphysics implementations of exponential Runge-Kutta methods satisfying stiff order conditions that were developed in Hochbruck et al. (1998) [22] and Luan and Osterman (2014) [25] . We reformulate stiffly–accurate exponential Runge–Kutta methods in a way that naturally allows of the structure of multiphysics systems, and discuss their application to both component and additively partitioned systems. The resulting partitioned exponential methods only compute matrix functions of the Jacobians of individual components, rather than the Jacobian of the full, coupled system. We derive modified formulations of particular methods of order two, three and four, and apply them to solve a partitioned reaction-diffusion problem. The proposed methods retain full order for several partitionings of the discretized problem, including by components and by physical processes.

In this paper, we show that for a fixed k, there is an NC algorithm that separates the graphs of rank-width at most k from those with rank-width at least, by Bireswar Das, Anirban Dasgupta, Murali Krishna Enduri, and Vinod I. Reddy, 2018-08-10 12:16:50, https://doi.org/10.1016/j.ipl.2018.07.007

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.

Let p be a prime. A famous theorem of Lucas states that (np+t) (m) (t) (modp) if m, n, s, t are nonnegative integers with s, t < p. In this paper we aim to prove a similar result for generalized binomial coefficients defined in terms of second order recurrent sequences with initial values 0 and 1.

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.

This paper investigates the stability of second-order stochastic neutral partial functional differential equations driven by impulsive noises. Some sufficient conditions ensuring $p$th moment exponential stability of the second-order stochastic neutral partial functional differential equations driven by impulsive noises are obtained by establishing a new impulsive-integral inequality. These existing results are generalized and improved by the present study. Finally, an example is given to show the effectiveness of our results.

Kertas kerja ini dikemukakan untuk menjelaskan tentang pembetulan pencongan dan erotan imej jawi. Satu algoritma mudah dan pantas ini menggunakan histogram orientasi cerunan sama seperti yang telah dicadangkan oleh Sun dan Si(1997) terhadap teks Latin telah dicadangkan untuk mengatasi masalah pencongan dan erotan terhadap teks Jawi. Algoritma bermula dengan melaksanakan operasi cerunan imej; diikut oleh penentuan histogram orientasi dan licinkannya menggunakan penapis median. Nilai maksimum histogram dicari untuk mendapatkan sudut pencong; diikuti oleh pembetulan awal dengan menggunakan fungsi polinominal. Sudut dihitung secara analitik; dan akhir sekali imej diputarkan mengikut sudut pencong yang diperoleh. Erotan teks Jawi juga telah menggunakan kaedah yang sama untuk mendapatkan sudut erotan dan kemudian diricih supaya dapat menghasilkan imej yang betul. Hasil daripada algoritma ini menjanjikan keputusan yang baik. Kata kunci: Histogram orientasi cerunan; topeng penjejak sisi sobel; pencong; erotan; penapis median; dan sudut pencong This paper discusses an enhancement algorithm of skew and slant orientation of the Jawi images. A simple and quick algorithm that uses a gradient orientation histogram, similar to the one proposed by Sun and Si(1997) for the Latin text is suggested to overcome this skew and slant problem in Jawi text. This algorithm start by carrying out the image gradient operation, followed by determining the orientation histogram and smoothing algorithm with a median filter. The histogram´s maximum value is generated to obtain the skew angle followed by the correcting procedure of the early value obtained by using polynomial function. The angle is calculated analytically and finally converting the image accordingly. The slanted Jawi text uses the same approach as the above to get the slant angle. This image is sheared in order to get the righ image. The outputs of this study show promising and convincing results. Key words: Gradient orientation histogram; the mask of sobel edge detector; skew; slant; median filter; and skew angle

Let be a 2×2 upper triangular block matrix over a principal ideal domain D with square diagonal blocks A 1 and A 2. We define a cyclic six-term exact sequence e(A) in terms of the kernels and the cokernels of A 1 and A 2 with a connecting map defined by the off-diagonal block X. This cyclic sequence e(A), under a variety of block-preserving matrix equivalences, is an invariant strictly finer than the Smith normal forms of A 1 A 2 and A combined. As one example of how this new K-theoretic invariant is used in classical linear algebra, we prove that 2×2 upper triangular block matrices A and B over a field F are block-preserving similar if and only if , that is, there is a chain F[t]-module isomorphism between the two cyclic sequences. We conclude with an application e(A) to the flow equivalence classification of two-component shifts of finite type in symbolic dynamics. This paper is self-contained and presented in a matrix-theoretic form.

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.

The stabilized Gauge-Uzawa method (SGUM), which is a second order projection type algorithm to solve the time-dependent Navier-Stokes equations, has been newly constructed in 2013 Pyo's paper. The accuracy of SGUM has been proved only for time discrete scheme in the same paper, but it is crucial to study for fully discrete scheme, because the numerical errors depend on discretizations for both space and time, and because discrete spaces between velocity and pressure can not be chosen arbitrary. In this paper, we find out properties of the fully discrete SGUM and estimate its errors and stability to solve the evolution Navier-Stokes equations. The main difficulty in this estimation arises from losing some cancellation laws due to failing divergence free condition of the discrete velocity function. This result will be extended to Boussinesq equations in the continuous research (part II) and is essential in the study of part II.

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 order to obtain a limitation of the number of repetitions in physico-chemical measurements, the author proposes formula 8 of this paper. delta is the difference between the maximum and minimum value of the measurements (form. 3) and A the approximation desired or imposed.

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 this paper we prove that the initial value problem associated to the following higher-order Benjamin–Ono equation∂tv−bH∂x2v+a∂x3v=cv∂xv−d∂x(vH∂xv+H(v∂xv)), where x,t∈R, v is a real-valued function, H is the Hilbert transform, a∈R, b, c and d are positive constants, is locally well-posed for initial datav(0)=v0∈Hs(R),s⩾2orv0∈Hk(R)∩L2(R;x2dx),k∈Z+,k⩾2.

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.

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.

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.