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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.

Hassin et al. [9] consider the Max-Exp-Cover-R problem to study the facility location problem on a graph in the presence of unreliable links when the link failure is according to the Linear Reliability Order (LRO) model. They showed that for unbounded R the problem is polynomial time solvable and for \(R=1\) and planar graphs the problem is NP-Complete. In this paper, we study the Max-Exp-Cover-1 problem under the LRO edge failure model. We obtain a fixed parameter tractable algorithm for Max-Exp-Cover-1 problem for bounded treewidth graphs, parameterized by the treewidth. We extend the Baker’s technique (Baker, J. ACM 1994) to obtain PTAS for Max-Exp-Cover-1 problem under the LRO model on planar graphs. We observe that the coverage function of the Max-Exp-Cover-R problem is submodular and the problem admits a \((1-1/e)\)-approximation for any failure model in which the expected coverage of a set by another set can be computed in polynomial time.

The object of this paper professes to be to ascertain whether the principle of Fermat’s theorem of the polygonal numbers may not be extended to all orders of series whose ultimate differences are con­stant. The polygonal numbers are all of the quadratic form, and they have (according to Fermat’s theorem) this property, that every number is the sum of not exceeding, 3 terms of the triangular num­bers, 4 of the square numbers, 5 of the pentagonal numbers, &c. It is stated in this paper that the series of the odd squares 1,9,25,49, &c. has a similar property, and that every number is the sum of not exceeding 10 odd squares. It is also stated, that a series con­sisting of the 1st and every succeeding 3rd term of the triangular series, viz. 1,10,28,35, &c., has a similar property; and that every number is the sum of not exceeding 11 terms of this last series, and that this may be easily proved [it was proved in a former paper by the same author]. The term “Notation-limit” is applied to the num­ber which denotes the largest number of terms of a series necessary to express any number; and the writer states that 5,7,9,13,21 are respectively the notation-limits of the tetrahedral numbers, the octa­hedral, the cubical, the eicosahedral and the dodecahedral numbers; that 19 is the notation-limit of the series of the 4th powers; that 11 is the notation-limit of the series of the triangular numbers squared, viz. 1,9,36,100, &c., and 31 the notation-limit of the series 1,28,153, &c. (the sum of the odd cubes), whose general expression is 2 n 4 — n 2 .

This paper relates to the second theorem, viz. that which asserts that every number is composed of 4 square numbers (0 [or zero] being considered as an even square). If every odd number be composed of 4 square numbers, then every even number must also be composed of 4 square numbers; for every even number must, on a continued division by 2, ultimately become an odd number. The paper relates chiefly to the Table which accompanies it, from which it appears that a remarkable law obtains as to the division of odd numbers (2 n + 1) into 4 square numbers—when a number of the form 4 n + 1 is divisible into 2 square numbers, which (as 4 n + 1 is an odd number) must be one of them odd, the other even. Before explaining the Table, it is proper to state that if an odd number be divisible into 4 square numbers, three of them must be odd, and one of them even, or one of them must be odd, and 3 of them even, otherwise their sum cannot be an odd number; it follows from this that if the difference between any two of them be an odd number, the difference between the other two must be an even number, and vice versâ ; for let a 2 + b 2 + c 2 + d 2 = 2 n + 1, then if a 2 - b 2 = 2 p , c 2 - d 2 must equal 2 q + 1; if possible let c 2 — d 2 = 2 r , then a 2 - b 2 + c 2 - d 2 = 2 p +2 r ; add 2 b 2 + 2 d 2 (an even number) to each, and a 2 + b 2 + c 2 + d 2 will be an even number, which by the hypothesis it is n o t; if, therefore, a 2 — b 2 be an even number, c 2 — d 2 cannot also be an even number, and therefore must be an odd one. If, therefore, the four roots of the squares into which any odd number may be divided are arranged in any order there will be three differences; the two exterior differences will be one odd, the other even; the middle difference may be either odd or even. The Table is arranged thus:—the lowest row of figures is the series 1, 5, 9, 13, 17, &c. (4 n + 1); the next row above is the series of natural numbers, 0, 1, 2, 3, 4, &c. ( n ), &c.; the next row is 1, 3, 5, 7, 9, &c. (2 n + 1) the odd numbers; each of the odd numbers is the first term in a series increasing upwards by the numbers 2, 4, 6, 8, 10, &c., forming an arithmetic series of the second order (the first and second differences being respectively 2 each); when the number in the lowest row cannot be divided into 2 squares the arithmetic series is not formed, and the squares are marked with an asterisk, but when the number 4 n + 1 is divisible into 2 square numbers, the roots of these squares constitute the two exterior differences of the roots into which the odd number may be divided, and also of the roots into which each term of the series increasing upward may be divided; the middle difference of the roots will be the smaller half of the sum of the 2 roots of the square numbers into which 4 n + 1 may be divided, with a negative sign, and will increase by 1 in each successive term of the upward series.

The object of this paper is to show the result of combining the three series (which have been the subject of previous communications) in a square, in such manner that the division into 4 squares of certain terms in each series, may produce a division into 4 squares of every term of other series, and thus each term in the whole square will at last be divided into 4 squares, and the first term will be so divided into 4 square numbers that two of the roots will be equal to each other; two of them will differ by 1, and the algebraic sum of all the roots will be equal to 1. It is not offered (at present) as a proof that it must be so, but as a method by which that result may always (in fact) be obtained.

In this paper experimental measurements are compared with Navier-Stokes calculations using seven different turbulence models for the internal flow in a two-dimensional U-duct. The configuration is representative of many internal flows of engineering interst that experience strong curvature. In an effort to improve agreement, this paper tests several versions of the two-equation k-epsilon turbulence model including the standard version, an extended version with a production range time scale, and a version that includes curvature time scales. Each is tested in its high and low Reynolds number formulations. Calculations using these new models and the original mixing length model are compared here with measurements of mean and turbulence velocities, static pressure and skin friction in the U-duct at two Reynolds numbers. The comparisons show that only the low Reynolds number version of the extended k-epsilon model does a reasonable job of predicting the important features of this flow at both Reynolds numbers tested.

The purpose of this note is to extend to complex quadratic forms some important investigations of Gauss relating to real quadratic forms. We shall consider in order (I.) the definition of the Genera, (II.) the theory of Composition, (III.) the determination of the number of Ambiguous Classes, (IV .) the representation of forms of the principal genus by ternary quadratic forms of determinant 1. For the comparison of the numbers of classes of different orders, we may refer to a paper by M. Lipschitz (Crelle’s Journal, vol. liv. p. 193) ; and for the principles of the theory of complex numbers and complex quadratic forms, to Lcjeune Dirichlet’s Memoir, “ Recherches sur les formes quadratiques à coefficients et à indéterminées complexes ” (Crelle, vol. xxiv. p. 291). I. The Definition of the Genera . Let f =( a, b, c ) be an uneven primitive form of determinant D, and m = ax 2 + 2 bxy + cy 2 , m' = ax’ 2 + 2 bx'y' + cy' 2 two numbers represented by f . The generic characters of f are deducible from the equation ( ax 2 + 2 bxy + cy 2 ) ( ax’ 2 + 2 bx'y' + cy' 2 ) = ( axx’ + b [ xy' + x'y ] + cyy' ) 2 - D ( xy' - x'y ) 2 , or, as we shall write it, mm' = P 2 - DQ 2 .

The aim of this paper is to introduce a novel phase-based feature representation for robust speech recognition. This method consists of four main parts: autoregressive (AR) model extraction, group delay function (GDF) computation, compression, and scale information augmentation. Coupling GDF with an AR model results in a high-resolution estimate of the power spectrum with low frequency leakage. The compression step includes two stages similar to MFCC without taking a logarithm of the output energies. The fourth part augments the phase-based feature vector with scale information which is based on the Hilbert transform relations and complements the phase spectrum information. In the presence of additive and convolutional noises, the proposed method has led to 15% and 12% reductions in the averaged error rates, respectively (SNR ranging from 0 to 20 dB), compared to the standard MFCCs.

The Fourier method is a rather effective and very simple regularization method for solving some ill-posed problems. The known works on this method are all limited to the a priori choice of the regularization parameter. In this paper, we will systematically consider the a posteriori choice of the regularization parameter, and the corresponding error estimates between the exact solution and its approximation are given. Numerical examples show the effectiveness of the a posteriori method, and the comparisons of the numerical effect between the a posteriori and the a priori Fourier methods are also taken into account.

In the year 1853 Dr. Rutherford, of the Royal Military Academy, Woowich, sent a paper on the Computation of the value of π to the Royal Society, and the paper was published in the 'Proceedings’ of that learned dy. The value of π is there given to 607 decimals, the first 440 being the joint production of Dr. Rutherford and the author of this paper, and the remaining 167 decimals having been calculated by the present writer, the accuracy of which he alone is responsible. Subsequently, the Astronomer Royal, G. B. Airy, Esq., kindly presented the author’s paper the Calculation of the value of e , the base of Napier’s logarithms, to wards of 200 decimals; the aforesaid paper also contained the Napierian logarithms of 2, 3, and 5, as well as the modulus of the common system, to upwards of 200 places of decimals. This paper was not, however, published, but deposited in the Archives of the Royal Society; but an abstract, containing the numerical results, was printed in the ‘Proceedings’. In a paper sent by the author to the Astronomer Royal, and forwarded by m to the Royal Society, will, the author believes, be found the reciprocal the prime number 17389, consisting of a circulating period of no less an 17388 decimals, the largest on record. Some few remarks are also ven touching circulates generally, and the easiest modes of obtaining them. The writer now desires to supplement what he then did, by giving the numerical value of Euler’s constant, which is largely employed in "Infinitesimal Calculus,” to a greater extent than has hitherto been found, a free from error. In Crelle’s Journal for 1860, vol. lx. p. 375, M. Oettinger has contributed an article on Euler’s constant, and especially on “certain discrepancies ” in the value given by former mathematical writers. Adopt the formula there employed, as being well adapted for the purpose, the writer of this paper has both corrected and extended what has been previously done; and as very great care has been bestowed upon the calculations, so as to exclude error, he confidently believes that his results are as far as they go, absolutely correct. He may remark that, since the separate values of n in the formula (which, see below) produce identical results as far as they go, and the higher the value of n the more near we can approximate to the value of the constant, we thus have sufficient proof afforded of the correctness of the value found when n is 10, 20, 50 or 100. If the writer can command sufficient leisure, he may resume the calculation by and by, and, making n 1000, he may thus verify, as well extend, the value of Euler’s constant given in this paper. The number 10, 20, 50, 100, 200, and 1000, especially 10 and its integral powers, are more easily handled than others, particularly in those terms of the formula which contain Bernoulli’s numbers. The harmonic progression is he "summed” much further than was requisite for finding E to 50 or 55 decimals; but this was of some importance in ensuring correctness in the decimal expression of each of the higher terms of S 100 and S 200 . It may be observed that the numbers of decimal places in E, obtained from n being 10, 20, 50, 100, and 200, are nearly proportional to 10 ⅓ , 20 ⅓ , 50 ⅓ 100 ⅓ and 200 ⅓ — a rather curious coincidence.

When n = 2000, we have 1 + ½ + ⅓ = 8.17836 81036 10282 40957 76565 71641 69368 79354 66740 91251 77402 20409 26320 14205 58039 78429 87946 27554 87631 13645 + E = .57721 56649 01532 86060 65120 90082 40243 10421 59335 93995 35988 05772 51046 48794 94723 80546 (last term is —B 12 /24.2000 24 ). Here the 60th decimal place in the value of E is the same when n is 2000 as it is when n is 1000.

DR. INGLEBY is curious to know what Prof. Helmholtz would say on this vexed question. If Dr. Ingleby will turn to page 630 of the “Physiologische Optik,” he will find that Prof. Helmholtz has anticipated his wishes. As others of your readers may be interested in seeing how the matter is treated by one who is facile princeps in this department, I subjoin a translation of the passage. If the curious experiment mentioned by Dr. Ingleby had referred only to the vertical diameter of the disc, it would have seemed to be another illustration of our inveterate tendency to ascribe an exaggerated value to vertical lines or angles, at or near the horizon. It is said that if ten men be required to fix off-hand on a star half way between the zenith and horizon, nine, at least, will choose one very much too low. If an exact square be cut out in paper and pinned against the wall opposite to the eye, the sides will appear longer than the top or bottom. If an equilateral triangle be placed in the same position, the angles at the base will appear larger than the angle at the vertex. If a line be drawn parallel to the bottom of a sheet of paper, and a second line, making with it an angle of 20° or 30°, any one attempting, without moving the paper, to draw a third line through the point of intersection, so as to make an angle with the second line equal to that which the second makes with the first, will make the second angle too large. (This experiment is guaranteed by Helmholtz.) After reading Helmholtz's theory, metaphysicians may be willing to allow that all these illusions are to be derived, after his example, from the clouds. As metaphysicians have, before now, contributed a good deal to the clouds, it is perhaps only fair that the clouds should contribute something to the metaphysicians.

When n = 10000, we have 1 + ½ + ⅓ + . . . 1/1000 = 9·78760 60360 44382 26417 84779 04851 60533 48592 62945 57772 17183 89460 97673 221 + Log ϵ 10000+ 1/2000 = 9·21039 03719 76182 73607 19658 18737 45683 04044 05954 51509 19041 33305 21764 185 + Result of "Bemouilli’s" = + ·00000 00008 33333 33250 00000 03968 25392 65873 02344 87732 37845 49617 88207 355, &c. E = ·57721 56649 01532 86060 65120 90082 40243 10421 59335 93995 35988 05773 64116 391.

The main object of the present communication is to correct some inaccuracies both of reasoning and calculation contained in two papers by Mr. Shanks, viz. “On the Extension of the Value of the Base of Napier’s Logarithms; of the Napierian Logarithms of 2, 3, 5, and 10; and of the Modulus of Briggs on the common System of Logarithms; all to 205 places of Decimals,” in the Proc. Roy. Soc. vol. vi. p. 397; and “On the Calculation of the Numerical Value of Euler’s Constant, in the Proc. Roy. Soc. vol. xv. p. 429 (1867).

Three cases and sources of inaccuracy in finding the value of E in the former paper (Proc. Roy. Soc. vol. xv. p. 429) having been pointed out by Mr. Glaisher, and some other minor errors not noticed by him having since been detected by the author, and these having vitiated the results, but only in a slight degree the inferences drawn from them (for in the former paper the last and leading conclusion as to the value of E, though limited , was certainly correct ), the author has been led, from a deep sense of obligation to the Royal Society, to revise, correct, and extend what he had previously done. And it will be seen, from comparing Mr. Glaisher’s remarks and results with what follows in this paper, that the supplementary matter herein given, including the extension of E &c. to 110 places of decimals, can scarcely be without interest to mathematicians, and especially as regards the summation of the harmonic series in the formula for finding the value of E. Not having seen M. Oettinger’s article in Crelle’s ‘Journal,’ “On Computing the value of E,” the author is unable to state what artifices he used in summing the harmonic series. Mr. Glaisher gives a very simple and obvious one from M. Oettinger, which the author could not but see and employ for calculating the values of the reciprocals of the even numbers.

This paper presents a lower and upper solution approach for singular second order boundary value problems on the half line and establishes the existence of positive, unbounded and monotone solutions.

The substance of part at least of this paper was given in 1876 as an evening lecture to the British Association at its Glasgow meeting.While engaged in writing the article “Mechanics” for the Ency. Brit., I had to consider carefully what basis to adopt, and decided that the time had not yet come in which (at least in a semi-popular article) Newton's laws of motion could be modified. The article was therefore based entirely on these laws, with a mere hint towards the end that in all probability they would soon require essential modification. It is well, however, that the question of modification should now be considered; and this should be done, not in a popular essay but, before a scientific society.

In my former paper on Fourier's double-integral I remarked that Poisson's form of the integral gave the same incorrect; result as Fourier's form in an example by which I tested it, and seemed subject to the same limitations.

The communication on this subject, as originally made to the society, consisted of a series of theorems, giving (1) expressions for the radii of a great many sets of circles, (2) identities connecting several sets of these radii, and (3) miscellaneous identities closely related thereto. As, however, the paper culminated in a general theorem which may be looked upon as fundamental, and the proof of which makes evident the mode of arriving at the said expressions for radii, and as the relations connecting sets of radii are easily found when attention has been directed to their existence, I have thought it best to print little more than the fundamental theorem and a few auxiliary notes.

The third report of the U. S. Entomological Commission, Washington, 1883, contains an excellent paper on the Hessian Fly, by Prof. A. S. Packard, p. 195-248, and in the appendix, p. 3-49, the translation of German papers by Loew, Wagner, Cohn, Koeppen, formerly not known by the Professor; also a reprint of my paper in the Can. Ent., Oct., 1880, “The Hessian Fly not imported from Europe.” The Professor accepts now (p. 238): That the Hessian Fly had appeared in the Eastern U. S. before the Revolutionary War, and that it could not have been introduced by the Hessians. He was not able to find in the files of the Pennsylvania Mercury and Philadelphia Packet some reference to the earlier occurrence of the H. Fly (App. p. 3).

In the year 1786 Erland Samuel Bring, Professor at the University of Lund in Sweden, showed how by an extension of the method of Tschirnhausen it was possible to deprive the general algebraical equation of the 5th degree of three of its terms without solving an equation higher than the 3rd degree. By a well-understood, however singular, academical fiction, this discovery was ascribed by him to one of his own pupils, a certain Sven Gustaf Sommelius, and embodied in a thesis humbly submitted to himself for approval by that pupil, as a preliminary to his obtaining his degree of Doctor of Philosophy in the University. The process for effecting this reduction seems to have been overlooked or forgotten, and was subsequently re-discovered many years later by Mr. Jerrard. In a report contained in the ‘Proceedings of the British Association’ for 1836, Sir William Hamilton showed that Mr. Jerrard was mistaken in supposing that the method was adequate to taking away more than three terms of the equation of the 5th degree, but supplemented this somewhat unnecessary refutation of a result, known à priori to be impossible, by an extremely valuable discussion of a question raised by Mr. Jerrard as to the number of variables required in order that any system of equations of given degrees in those variables shall admit of being satisfied without solving any equation of a degree higher than the highest of the given degrees. In the year 1886 the senior author of this memoir showed in a paper in Kronecker'e (better known as Crelle’s ) ‘Journal that the trinomial equation of the 5th degree, upon which by Bring’s method the general equation of that degree can be made to depend, has necessarily imagmaiy coefficients except in the case where four of the roots of the original equation are imaginary, and also pointed out method of obtaining the absolute minimum degree M of an equation from which an given number of specified terms can be taken away subject to the condition of no having to solve any equation of a degree higher than M. The numbers furnished be Hamilton’s method, it is to be observed, are not minima unless a more stringer condition than this is substituted, viz., that the system of equations which have to be resolved in order to take away the proposed terms shall be the simplest possible i. e ., of the lowest possible weight and not merely of the lowest order; in the memo: in ‘Crelle,’ above referred to, he has explained in what sense the words weight an order are here employed. He has given the name of Hamilton’s Numbers to these relative minima (minima, i. e ., in regard to weight) for the case where the terms to be taken away from the equation occupy consecutive places in it, beginning with the second.

In the year 1786 Erland Samuel Bring, Professor at the University of Lund in Sweden, showed how by an extension of the method of Tschirnhausen it was possible to deprive the general algebraical equation of the 5th degree of three of its terms without solving an equation higher than the 3rd degree. By a well-understood, however singular, academical fiction, this discovery was ascribed by him to one of his own pupils, a certain Sven Gustaf Sommelius, and embodied in a thesis humbly submitted to himself for approval by that pupil, as a preliminary to his obtaining his degree of Doctor of Philosophy in the University. The process for effecting this reduction seems to have been overlooked or forgotten, and was subsequently re-discovered many years later by Mr. Jerrard. In a report contained in the ‘Proceedings of the British Association’ for 1836, Sir William Hamilton showed that Mr. Jerrard was mistaken in supposing that the method was adequate to taking away more than three terms of the equation of the 5th degree, but supplemented this somewhat unnecessary refutation of a result, known a priori to be impossible, by an extremely valuable discussion of a question raised by Mr. Jerrard as to the number of variables required in order that any system of equations of given degrees in those variables shall admit of being satisfied without solving any equation of a degree higher than the highest of the given degrees. In the year 1886 the senior author of this memoir showed in a paper in Kronecker'e (better known as Crelle’s ) ‘Journal that the trinomial equation of the 5th degree, upon which by Bring’s method the general equation of that degree can be made to depend, has necessarily imagmaiy coefficients except in the case where four of the roots of the original equation are imaginary, and also pointed out method of obtaining the absolute minimum degree M of an equation from which an given number of specified terms can be taken away subject to the condition of no having to solve any equation of a degree higher than M. The numbers furnished be Hamilton’s method, it is to be observed, are not minima unless a more stringer condition than this is substituted, viz., that the system of equations which have to be resolved in order to take away the proposed terms shall be the simplest possible i. e ., of the lowest possible weight and not merely of the lowest order; in the memo: in ‘Crelle,’ above referred to, he has explained in what sense the words weight an order are here employed. He has given the name of Hamilton’s Numbers to these relative minima (minima, i. e ., in regard to weight) for the case where the terms to be taken away from the equation occupy consecutive places in it, beginning with the second.

WITH reference to the remarks of “R. B. H.” (NATURE, June 28, p. 197) on my interpretation of the differential equation to all conics, I wish to point out that the objections he seems to take do not appear to be well founded. The difficulty he finds is that the geometrical interpretation given amounts to the fact that “a conic is a conic.” But it is easy to see that there is no peculiarity in this; it arises simply from the well-known fact that all the geometrical properties of any given figure are inter-dependent: one of them being given, the others may be deduced as legitimate consequences from it. “R. B. H.” takes the proposition which constitutes my interpretation, and then, coupling it with the other theorem that the osculating conic of any conic is the given conic, comes to the conclusion that a conic is a conic, and, apparently, he takes it to be very strange; bat, as a matter of fact, given any two properties of a conic (or of any other curve), we can only come to the conclusion that the conic is a conic (or that the given curve is what it professes to be). Take, for example, the geometric interpretation of the differential equation of all right lines, which is q = o; it simply means that the curvature vanishes at every point of every right line, which is equivalent to the fact that a straight line is not curved, or that a straight line is a straight line. There is certainly nothing strange in this: it is the legitimate effect of the process employed. Would “R. B. H.,” on this ground, reject the geometrical interpretation of the differential equation of all straight lines? Surely the process is nothing but a piece of quite unobjectionable verification. Similarly, the differential equation of all circles, (1 + p2)r 3pq2 = o, means that the angle of aberrancy vanishes at every point of every circle. Combining this with the self-evident proposition that the normal and the axis of aberrancy coincide in the case of a circle, we may come to the conclusion that a circle is a circle; but I submit that this is really a verification, and surely no ground for rejecting the interpretation. Indeed, the question whether such processes are to be regarded as verifications or not seems to me to be much the same question whether every syllogism is a petitio principii or not. But as I have elsewhere, in the papers referred to in my last letter (p. 173, ante), fully discussed what a geometrical interpretation properly ought to be, I need not enlarge further on this point.

Of late years several determinations of the electrical resistance of mercury have been made, and the differences between the results arrived at have been greater than would be expected at first sight from the nature of the observations involved. The results of the experiments have been expressed either in terms of the ohm (10 9 absolute C. G. S. units) or of the B. A. unit, which, according to the determinations of Lord Rayleigh and one of the authors of this paper (R. T. G.), is equal to ·98667 ohm. In the case of Lord Rayleigh’s observations, a direct comparison was made between the mercury unit and the original B. A. standards. Other observers have constructed copies of their mercury resistances in German-silver wire, which have been compared with the B. A. standards at the Cavendish Laboratory by one of us, or have compared their tubes directly with copies in platinum-silver wire of the B. A. units which have been sent from Cambridge after careful testing. The result of these various comparisons of recent years is as follows, and may conveniently be put in tabular form, giving the value in B. A. units of the resistance of a column of mercury 1 metre long, 1 square millimetre in cross section, at 0° Centigrade.

In the course of verifying some of the mathematical work in the third instalment of Professor Tait's paper on the Kinetic Theory, the following simplification of his proof of Maxwell's theorem occurred to me.

IN answer to a correspondent who has met with a difficulty in the consideration of Doppler's principle, I may say that I think I fairly solved the difficulty in a paper delivered last year before the University College Chemical and Physical Society. In cases (1) and (2) of your correspondent, viz. approach or recession of observer, source and medium being at rest, the correct formula is and in cases (3) and (4), viz. approach and recession of source, observer and medium at rest, it is, n′ = the new, and n the old frequency of vibration of the note heard. It should be remarked, however, that in all practical cases, the two formulae give very nearly the same result; but if the velocity v is very great, the case is entirely different. Suppose, for instance, in cases (1) and (2) that v = velocity of sound a, then n′ = 2n for approach, and O for recession of the observer. The correctness of these results is obvious without the aid of any formula. Again, in cases (3) and (4) suppose v = a, then n′ = ∞ for approach, and for recession of the source of sound. The effect of an infinite number of waves striking the ear at the same moment would be simply that nothing would be heard. It would be interesting to notice the change in pitch of the whistle of a rifle-bullet passing near an observer. Ganot's formula is correct for cases (1) and (2), and Prof. Everett's for cases (3) and (4); the proofs are very simple, and may be easily thought out. When the observer and the source of sound both move, the two formulae should be applied separately when a, very accurate result is desired. These conclusions have been confirmed by Dr. Fison, of University College. I had not considered the effect of the motion of the medium, but it appears to me, after a little reflection, that this would increase or diminish the velocity of the sound, and the wave-length, in the same proportion, leaving the pitch unaltered; the velocity of the medium should therefore be added to or subtracted from a in the formula.

In this paper we present a new notion of curvature for cell complexes. For each p , we define a p th combinatorial curvature function, which assigns a number to each p -cell of the complex. The curvature of a p -cell depends only on the relationships between the cell and its neighbors. In the case that p=1 , the curvature function appears to play the role for cell complexes that Ricci curvature plays for Riemannian manifolds. We begin by deriving a combinatorial analogue of Bochner's theorems, which demonstrate that there are topological restrictions to a space having a cell decomposition with everywhere positive curvature. Much of the rest of this paper is devoted to comparing the properties of the combinatorial Ricci curvature with those of its Riemannian avatar.

The attention of mathematicians was first called to the subject of the present paper by a memoir of Helmholtz’s in 1868, on “Discontinuous Fluid Motion.” In discussing the steady motion of liquids past salient edges of fixed obstacles, it is found that the assumptio of continuity of the motion leads to negative pressures in the liquid. Helmholtz showed, in the paper above-mentioned, that some cases of this kind could be solved by assuming a surface of discontinuity, on one side of which the liquid is at rest, and he gave a mathematical solution of one case where the motion is in two dimensions.

“ On the whole I am of opinion that if it is desirable at the present time to construct apparatus on the most favourable scale, so as to reach the highest attain­ able accuracy, the modification of Lorenz’s method last described is the one that offers the best prospect of success. Before this is done however, it appears to me important that the value now three times obtained in the Cavendish Laboratory by distinct methods should be approximately verified (or disproved) by other physicists. To distinguish between this value and those obtained for instance by Kohlrausch, by Lorenz, or by the First B. A. Committee, should not require the construction of unusually costly apparatus. Until the larger question is disposed of it appears premature to discuss the details of arrangements from which the highest degree of precision is to be expected.” The above passage, which concludes a paper communicated by Lord Rayleigh to the ‘Philosophical Magazine, a little before the Electrical Congress at Paris, at which the legal ohm was defined to be the resistance of a column of mercury of 1 sq. mm. section and 1060 mm. long, seems not to have met with adequate response in this country. So far as experiments in English Laboratories are concerned the determination of the ohm remains where Lord Rayleigh left it, except for the contribution made by Glazebrook and Fitzpatrick in their re-measurement of the Specific Resistance of Mercury in terms of the B. A. Unit, which is one of the elements in the determination of the Specific Resistance of Mercury in Absolute Measure by Lord Rayleigh’s adaptation of Lorenz’s method.

DR. WATSON has shown in his letter to NATURE of March 31 (p. 512) how the general methods of Maxwell and Boltzmann may be applied to the particular problem discussed by Prof. Burnside. He has also pointed out an error in Burnside's reasoning—namely, the non-introduction of the factor u − U + c?, whereby Burnside's conclusions at variance with the Maxwell-Boltzmann law of partition of energy are vitiated.

IN last month's Philosophical Magazine there is a paper by Lord Rayleigh criticizing a demonstration by Maxwell of the equality of the products dp1…dpn, dq1…dqn, and dP1…dPn, dQ1…dQn, where the p's and P's are the momenta, and the q's and Q's the co-ordinates, of a system at the beginning and end of any interval of time.

Accurate estimation of noise and signal power is of fundamental interest in a wide variety of vision applications as it is critical to thresholding and decision processes. This paper proposes two methods for the estimation of nonstationary noise based upon models of image structure which locally separate signal from noise. The resulting algorithms are noniterative and thereby fast. The accuracy of the proposed and existing methods is compared, first separately and then in application to two common image processing tasks: edge and corner detection. It is demonstrated that the proposed model can be used to improve the stability of both, in the presence of contrast change and nonstationary noise.

I REMEMBER that in the discussion of eighteen years ago it was understood that you could get 1˙4 for the ratio, if the molecules had each five degrees of freedom only—if they were, for instance, perfectly smooth, elastic spheroids. Probably the ultimate source of our knowledge in this respect was Boltzmann's paper, to which Mr. Bryan refers us. The difficulty at the time seemed to be mainly one of faith. One could not believe that the molecules were solid elastic bodies, however useful the discussion of such bodies might be in defining a limiting case. As the white posts along a road are put to show you where you should not go, not where you should go. It was further supposed, perhaps without sufficient reason, that the phenomena, of the spectroscope required us to attribute many degrees of freedom to the molecules.

(1.) If measurements be made of the same part or organ in several hundred or thousand specimens of the same type or family, and a curve be constructed of which the abscissa x represents the size of the organ and the ordinate y the number of specimens falling within a definite small range δx of organ, this curve may be termed a frequency-curve . The centre or origin for measurement of the organ may, if we please, be taken at the mean of all the specimens measured. In this case the frequency-curve may be looked upon as one in which the frequency—per thousand or per ten thousand, as the case may be—of a given small range of deviations from the mean, is plotted up to the mean of that range. Such frequency-curves play a large part in the mathematical theory of evolution, and have been dealt with by Mr. F. Galton, Professor Weldon, and others. In most cases, as in the case of errors of observation, they have a fairly definite symmetrical shape and one that approaches with a close degree of approximation to the well-known error or probability-curve. A frequency-curve, which, for practical purposes, can be represented by the error curve, will for the remainder of this paper be termed a normal curve . When a series of measurements gives rise to a normal curve, we may probably assume something approaching a stable condition; there is production and destruction impartially round the mean. In the case of certain biological, sociological, and economic measurements there is, however, a well-marked deviation from this normal shape, and it becomes important to determine the direction and amount of such deviation. The asymmetry may arise from the fact that the units grouped together in the measured material are not really homogeneous. It may happen that we have a mixture of 2, 3, . . . n homogeneous groups, each of which deviates about its own mean symmetrically and in a manner represented with sufficient accuracy by the normal curve. Thus an abnormal frequency-curve may be really built up of normal curves having parallel but not necessarily coincident axes and different parameters. Even where the material is really homogeneous, but gives an abnormal frequency-curve the amount and direction of the abnormality will be indicated if this frequency-curve can be split up into normal curves. The object of the present paper is to discuss the dissection of abnormal frequency-curves into normal curves. The equations for the dissection of a frequency-curve into n normal curves can be written down in the same manner as for the special case of n = 2 treated in this paper; they require us only to calculate higher moments. But the analytical difficulties, even for the case of n = 2, are so considerable, that it may be questioned whether the general theory could ever be applied in practice to any numerical case. There are reasons, indeed, why the resolution into two is of special importance. A family probably breaks up first into two species, rather than three or more, owing to the pressure at a given time of some particular form of natural selection; in attempting to procure an absolutely homogeneous material, we are less likely to have got a mixture of three or more heterogeneous groups than of two only. Lastly, even where the heterogeneity may be threefold or more, the dissection into two is likely to give us, at any rate, an approximation to the two chief groups. In the case of homogeneous material, with an abnormal frequency-curve, dissection into two normal curves will generally give us the amount and direction of the chief abnormality. So much, then, may be said of the value of the special case dealt with here.

An asymmetrical frequency curve may arise from two quite distinct classes of causes. In the first place the material measured may be heterogeneous and may consist of, a mixture of two or more homogeneous materials. Such frequency curves, for example, arise when we have a mixed population of two different races, a homogeneous population with a sprinkling of diseased or deformed members, a curve for tbe frequency of matrimony covering more than one class of the population, or in economics a frequency of interest curve for securities of different types of stabilit — railways and government stocks mixed with mining and financial companies. The treatment of this class of frequency curves requires us to break up the original curve into component parts, or simple frequency curves. This branch of the subject (for the special case of the compound being the sum of two normal curves) has been treated in a paper presented to the Poyal Society by the author, on October 18,1893. 1 lie second class of frequency curves arises in the case of homogeneous material when the tendency to deviation on one side of the mean is unequal to the tendency to deviation on the other side. Such curves arise in many physical, economic and biological investigations, for example, in frequency curves for the height of the barometer, in those for prices and for rates of interest of securities of the same class, in mortality curves, especially the percentage of deaths to cases in all kinds of fevers, in income tax and house duty returns, and in various types of anthropological measurements. I t is this class of curves, which are dealt with in the present paper. The general type of this class of frequency curve will be found to vary (see Plate 7, fig. 1) through all phases from the form close to the negative exponential curve: y = Ce - px , to a form close to the normal frequency curve y = Ce - px2 where C and p are constants.

The following paper is occupied by an attempt to investigate the distribution of the electric and magnetic forces which are called into play when certain electro­magnetic systems are made to move with uniform velocity through the ether. Maxwell’s theory will be employed throughout, and will be applied to th e exact solution of several problems, and to the establishment of some results of a general nature. Professor J. J. Thomson was, I believe, the first to consider a problem of this sort, his paper giving the solution for the motion of a single electrical point-charge at a speed small compared with that of light. Mr. Oliver Heaviside next considered the question, and in his paper obtained an exact solution for the motion of a point-charge. He got the solution by means of the “vector potential” of the convection current formed by the moving charge. The mathematical analysis is of the symbolical kind, but it is shown that the result obtained by means of it satisfies all the necessary conditions. By integrations Mr. Heaviside obtains solutions for the motion of some simple cases of electrical distribution.

Given a graph G, a total k-coloring of G is a simultaneous coloring of the vertices and edges of G with k colors. Denote χve (G) the total chromatic number of G, and c(Σ) the Euler characteristic of a surfase Σ. In this paper, we prove that for any simple graph G which can be embedded in a surface Σ with Euler characteristic c(Σ), χve (G) = Δ (G) + 1 if c(Σ) > 0 and Δ (G) ≥ 13, or, if c(Σ) = 0 and Δ (G) ≥ 14. This result generalizes results in [3], [4], [5] by Borodin.

This paper presents an analog implementation of the wavelet transform (WT) in an ultra low power environment, using the dynamic translinear (DTL) circuit techniques. The circuit consists of an analog filter whose impulse response is the first derivative of a Gaussian. A convenient method to provide the transfer function of the filter is given by the Pade approximation. In order to fulfil the low-power requirement, the filter's state space description is optimized with respect to dynamic range. Finally, the DTL principle is applied to implement the state space filter. Simulations indicate a good approximation of the wavelet transform.

The general feature of most methods for the integration of partial differential equations in two independent variables is, in some form or other, the construction of a set of subsidiary equations in only a single independent variable; and this applies to all orders. In particular, for the first order in any number of variables (not merely in two), the subsidiary system is a set of ordinary equations in a single independent variable, containing as many equations as dependent variables to be determined by that subsidiary system. For equations of the second order which possess an intermediary integral, the best methods (that is, the most effective as giving tests of existence) are those of Boole, modified and developed by Imschenetsky, and that of Goursat, initially based upon the theory of characteristics, but subsequently brought into the form of Jacobian systems of simultaneous partial equations of the first order. These methods are exceptions to the foregoing general statement. But for equations of the second order or of higher orders, which involve two independent variables and in no case possess an intermediary integral, the most general methods are that of Ampere and that of Darboux, with such modifications and reconstruction as have been introduced by other writers; and though in these developments partial differential equations of the first order are introduced, still initially the subsidiary system is in effect a system with one independent variable expressed and the other, suppressed during the integration, playing a parametric part. In oilier words, the subsidiary system practically has one independent variable fewer than the original equation. In another paper I have given a method for dealing with partial differential equations of the second order in three variables when they possess an intermediary integral; and references will there be found to other writers upon the subject. My aim in the present paper has been to obtain a method for partial differential equations of the second order in three variables when, in general, they possess no intermediary integral. The natural generalisation of the idea in Darboux’s method has been adopted, viz., the construction of subsidiary equations in which the number of expressed independent variables is less by unity than the number in the original equation; consequently the number is two. The subsidiary equations thus are a set of simultaneous partial differential equations in two independent variables and a number of dependent variables.

In the ‘Transactions of the Cambridge Philosophical Society’ (vol. 16, Part IV., p. 262), I brought forward a new instrument of research in Combinatorial Analysis, and applied it to the complete solution of the great problem of the “Latin Square,” which had proved a stumbling block to mathematicians since the time of Euler. The method was equally successful in dealing with a general problem of which the Latin Square was but a particular case, and also with many other questions of a similar character. I propose now to submit the method to a close examination, to attempt to establish it firmly, and to ascertain the nature of the questions to which it may be successfully applied. We shall find that it is not merely an enumerating instrument but a powerful reciprocating instrument, from which a host of theorems of algebraical reciprocity can be obtained with facility. We will suppose that combinations defined by certain laws of combination have to be enumerated; the method consists in designing, on the one hand, an operation and, on the other hand, a function in such manner that when the operation is performed upon the function a number results which enumerates the combinations. If this can be carried out we, in general, obtain far more than a single enumeration; we arrive at the point of actually representing graphically all the combinations under enumeration, and solve by the way many other problems which may be regarded as leading up to the problem under consideration. In the case of the Latin Square it was necessary to design the operation and the function the combination of which was competent to yield the solution of the problem. It is a much easier process, and from my present standpoint more scientific, to start by designing the operation and the function, and then to ascertain the questions which the combination is able to deal with.

It is to be expected that the systematic introduction of irrational covariants will enrich geometry with curves and surfaces, not previously observed or discussed, allied to any given fundamental system by projective relations. Well known irrationalities also must be expected often to reappear. In 1888 Dr. HILBERT t remarked the existence of two nets of conics covariantly related to a general curve of third order in a plane. Two additional systems, nets of curves of the second class, can easily be defined by anl equation closely analogous to Hilbert's. These systems of conics and the four cubics whose polars they are prove to be not entirely unknowvn hitherto, for their dually equivalent loci and envelopes form the basis of the one-to-one correspondences upon the point-cubic and the line-cutbic respectively. That only two of every three such correspondences are found here is due to the domain of rationality that is assumed. By employing the irrationality that occurs in HESSE'S canonical fornm of the cubic I ani able to identify Hilbert's two systems of irrational covariant conics and to exhibit their relation to the other two systems just mentioned. As a consequence it is found possible to give explicitly covariant equations of definition for the two cubics which have the same Hessian and for those which have the same Cayleyan as a given fundamental cubic. These results are here derived by the aid of a canonical form of the cubic containing Hesse's irrationality. The desirable invariantive proofs of these results are given by Professor GORDAN in a paper presented by him to the Americani Mathematical Society for publication in its Transactions.4

1. Since the announcemnent t by KRONECKER, in 1882, of his now well-known theorem regarding linear relations between the minors of an axisymmetric deterininant various papers t have appeared treating of the subject. Dr. MUIR in his paper of 1888 showed that a similar relation exists between the muinors of a centrosymmetric determinant and in his paper of 1900 he gives the following two theorems: THEOREM A: If , and v be any integers, , being the less, taken fromn the series n, n + 1, n + 2, .., 2n and a, 3, ry, *.., w be what the series becomes when , is removed, and a, /3, ry, * , 4 what it becomes when both are removed; then in connection with any even-ordered determinant 112 2n we have

The investigations on the foundation of geometry of RIEMANN, HELMHOLTZ and LIE have as a basal notion the distance between two points of space, and the analytic representation of this distance plays an important r6le. A point in space is determined by means of three independent coordinates; and the distance between two points is a certain function of their six coordinates. Rigid bodies are used and the continuous motions of such, determined by the continuous variation of the co6rdinates of their points. This notion of three independent and continuous co6rdinates of a point is, in reality, not simple, as has been pointed out by LIE and others. In an article entitled Ein Beitrag zur Jllannigfaltigkeitslehre, Borchhardt's Journal, vol. 84 (1877), G. CANTOR describes a space in which the co6rdinates of a point are neither independent nor continuous. It does not seem easy, however, to determine the distance between two points of space if the co6rdinates of a point are neither independent nor continuous. It would not be evident, a priori, that certain loci (as spheres and circles) exist, defined by means of certain equations in the coordinates of their points. Such loci are used in the processes employed by HELMHOLTZ and LIE, and the question arises whether their axioms involving such loci and the motion of points in space, etc., should be modified or replaced by others. One such new axiom, used in the following paper, refers to distance-relations, i. e., it is assumed that certain definite relations connect the mutual distances between any n + 2 different points in space of n dimensions.t One notices that the assumption, used by HELMHOLTZ and LIE and their successors, that the distance between two points in space is a certain function of the

The object of this paper is to investigate certain series and differential equations, generalizations of the series of Bessel and Legendre. Throughout the paper reducing, when p = 1, to n . The series discussed are the following:— a generalized form of P n ( x ), the relation between coefficients of x in successive terms being

IN the January number of the Philosophical Magazine I published a discussion of the general theory underlying the experiment of Messrs. Michelson and Morley on the drift of the aether. As one result, it appeared that the effect to be expected in their special case was just the opposite of that usually supposed, and that consequently the FitzGerald-Lorentz explanation of the observed null effect would not hold. Mr. H. M. Macdonald has pointed out the source of this discrepancy in an algebraic slip in my paper; when this is corrected, the result comes into agreement with the special case treated by Michelson and Morley. The exact effect on the displacement of the interference-bands arising from a vertical component in the rether-drift has not been hitherto directly considered. It is probably null; but this requires verification, which I hope to be able to take up shortly on the basis of my analysis. As the question stands at present, the corrected result shows that the FitzGerald-Lorentz shrinkage would completely annul the shift when the drift is tangential. Although Dr. Larmor has not directly discussed the effect of an oblique drift in his “Aether and Matter,” I understand from him that he has come to the conclusion (cf. loc. cit. § 34) that complete annulment results in all cases on the FitzGerald-Lorentz hypothesis. I think further discussion on the lines of my own method of analysis will verify that this is the case.

Helmholtz's attention was directed to the subjects to be discussed in this paper while very young. Even while attending the gymnasium and reading Cicero and Virgil, for which he had little taste, Helmholtz was pondering on various problems in optics not met with in text-books, and which served as a basis some years later for the construction of the ophthalmoscope. The mathematics of physical problems always came easy to him, not that he seems to have had any special love for pure mathematics for its own sake, but only as a means of solving the questions which interested him. His mathematical ability was of a very high order, and this fact seems all the more remarkable when it is noted that he never had any systematic training in this branch of science, as have most other eminent mathematical physicists. His knowledge in the more advanced departments of mathematics was acquired

By aid of the methods of a paper on “ Ellipsoidal Harmonic Analysis ” (‘ Phil. Trans., A, vol. 197, pp. 461-557), I here resume the subject of a previous paper (‘ Phil. Trans.' A, vol. 198, pp. 301-331). These papers will be referred to hereafter by the abridged titles of " Harmonics ” and " The Pear-shaped Figure.” At the end of the latter of these it was stated that the stability of the figure could not he proved definitely without approximation of a higher order of accuracy. After some correspondence with M. Poincaré during the course of my work, I made an attempt to carry out this further approximation, but found that the expression for a certain portion of the energy entirely foiled me. Meanwhile he had turned his attention to the subject, and he has shown (‘ Phil. Trans.,’A, vol. 198, pp. 333-373) by a method of the greatest ingenuity and skill how the problem may be solved. He has not, however, pursued the arduous task of converting his analytical results into numbers, so that he left the question as to the stability of the pear still unanswered.

By aid of the methods of a paper on “ Ellipsoidal Harmonic Analysis ” (‘ Phil. Trans., A, vol. 197, pp. 461-557), I here resume the subject of a previous paper (‘ Phil. Trans.' A, vol. 198, pp. 301-331). These papers will be referred to hereafter by the abridged titles of " Harmonics ” and " The Pear-shaped Figure.” At the end of the latter of these it was stated that the stability of the figure could not he proved definitely without approximation of a higher order of accuracy. After some correspondence with M. Poincare during the course of my work, I made an attempt to carry out this further approximation, but found that the expression for a certain portion of the energy entirely foiled me. Meanwhile he had turned his attention to the subject, and he has shown (‘ Phil. Trans.,’A, vol. 198, pp. 333-373) by a method of the greatest ingenuity and skill how the problem may be solved. He has not, however, pursued the arduous task of converting his analytical results into numbers, so that he left the question as to the stability of the pear still unanswered.

It seems likely to be of interest at the present time to consider rather at length a fairly full mathematical theory of the pure gamete. We do not venture to call this theory a generalised Mendelian theory of inheritance, partly because it is not even the most general theory of the pure gamete conceivable, partly because Mendel’s original theory of heredity was perfectly clear and perfectly simple, and is not the theory here developed. The pure and simple Mendelian theory seems to have been discarded in the light of recent experimental results by more than one Mendelian, both in this country and abroad. The original Mendelian theory has been replaced by what are termed “ Mendelian Principles.” In this aspect of investigation the fundamental principles propounded by Mendel are given up, and for each individual case a pure gamete formula of one kind or another is suggested as describing the facts, This formula is then emphasised, modified or discarded, according as it fits well, badly, or not at all with the growing mass of experimental data. It is quite clear that it is impossible while this process is going on to term anything whatever Mendelian as far as theory is concerned.

The group (F) which is the object of the present investigation has already been briefly discussed by HuRWIEZ.t His paper deals chiefly with the groups of signs (0, 3; 2, 4, oc) and (0, 3; 2, 6, cc)4 He proves them "commensurable " with the modular-group (0, 3; 2, 3, oc) and hence derives certain algebraic relations between the simplest functions belonging to his groups on the one hand and the invariant J of the modular group on the other. He also obtains the arithmetic character of the substitutions of his groups. The group r is also mentioned briefly in FRICKE-KLEIN'S treatise, ? where it appears as the " reproducing group " of a certain ternary quadratic form. In Part I, we derive r as the monodromy group of the Riemann surface R (p = 2) defined by the equation