Singular Character Research Articles

Nonlinear motion of a symmetric missile acted on by a double valued static moment.

The static moment of a symmetric missile is usually assumed to be single-valued although double-valued moments can occur. For example, lee side separation can occur at an angle of attack that is greater than that for reattachment and the moment for angles between these values can be multivalued. If a mathematically simple double-valued static moment consisting of two linear segments and two jump points is assumed (Fig. 1), an exact solution for planar motion and zero aerodynamic damping can be obtained. The effects of nonplanar motions and both linear and nonlinear damping moments are studied by the use of a quasilinear analysis. First regions in an amplitude plane are determined for which the multivalued moment is brought into play as well as regions for which the motion is only affected by the single-valued linear segments of the static moment curve. Next, quasilinear approximations are derived for frequencies and damping rates of the motion caused by this multivalued static moment coupled with one linear and two cubic damping moments. The accuracy of these approximations is determined by comparison with the exact solution for planar motion with no aerodynamic damping and is found to be better than 1% for most cases. The character of the planar singularity for linear damping is investigated and found to be a stable limit cycle.

Elastostatic problems of curvilinear cracks in bonded dissimilar materials

Boundary problems for a circular elastic insert bonded partially to the interior of an infinite medium of another material are formulated and solved in closed form. The unbonded portions of the interface may be regarded as circular arcs of discontinuities or curved cracks. By application of the complex function theory dealing with sectionally holomorphic functions, the present problem is reduced to the solution of the problem of linear relationship or Hilbert problem. Exact solutions to several examples of theoretical and practical importance are obtained. They include the cases of uni-axial tension applied at infinity and concentrated forces located in the insert and/or the surrounding material. Contact stresses are computed and compared to those for perfect bonding. It is found that the stresses near the tips of a curved crack possess the same trig-log character of singularity as those obtained for a straight crack between dissimilar media. Formulas for finding the crack-tip stress-intensity factors, employed m the Griffith-Irwin theory of fracture, are also derived.

The Canonical Quantization of the Free Electromagnetic Field in the Landau Gauge

The canonical formalism of the electromagnetic field in the Landau gauge is studied. Unlike the case of Nakanishi, the dipole ghost states are not introduced. The singular character of the longitudinal and scalar components are pointed out, and the difficulty originated from the unnormalizable norm of the state vectors are evaded by an appropriate limiting procedure. The interacting case is also discussed in the interaction representation.

Initial value problem for elementary and composite particles

Equivalence between bound states (composite particles) and elementary particles with vanishing wave function renormalization constant Z is re-examined by solving explicitly the initial value problem for a simple model. The highly singular character of the limit Z = 0 is revealed and is investigated in detail. A singular perturbation scheme is developed in the context of quantum field theory.

Asymptotic behaviour of essentially inelastic collisions

The asymptotic value of particle production amplitudes at s → ∞ is investigated for “essentially inelastic” collisions when it is especially large. These collisions are characterized by the production in the c.m. system of two groups of particles at high primary energies, each moving nearly parallel to the momenta p a and p b =− p a of the colliding particles. The momenta of all particles in each group differ considerably by their order of magnitude: p 1 ⪢ p 2 ⪢ … ⪢ p n1 , p n ⪢ p n−1 ⪢ … ⪢ p n1+1 , where the subscripts 1, 2 … n 1 refer to the particles of the first group and n 1+1, n 1+2 … n−1, n to those of the second. The mathematically corresponding limit of a particle production amplitude is determined by the requirement that the squares s ik of the energies of any pair of particles in their c.m. system should increase (or remain large as compared with particle masses squared) as the primary energy increases, s → ∞. It is also necessary that the transverse components k i of all momenta p i of the particles produced and momentum transfers squared t i =( p a− p 1− … − p i ) 2, i=1, 2, …, n should be small. It is shown that in this limit the asymptotic value of the amplitude takes the form of a product of Regge factors Π i=1 n−1 ( s i, i+1 ) jo( ti) γ i (where j 0( t i is a vacuum pole) and can in a sense be compared with the contribution of Feynman-type graphs with n−1 “reggeons”. The quantities γ i=γ i( K′ i, K′ i+1) depending on the projections K′ i and K′ i+1 of the transverse momenta of reggeons before and after the emission of a particle correspond to the vertex parts designating in these graphs the emission of the particle by a reggeon. In a preliminary report on this question 2, 3) the dependence of the quantities γ on the angle between K′ i and K′ i+1 was overlooked. Thus the coefficient of the product of the Regge factors in the asymptotic value of the amplitude depends on the transverse components of the momenta of all the particles produced. As a result the asymptotic value of a particle production amplitude proves to be dependent on the same number of variables as in the general case (i.e. is sensitive to any change of the momenta). The asymptotic behaviour of mixed “essentially inelastic” and “quasi-elastic” collisions is also considered when not particles but n groups of particles are produced and the energy of each particle in the c.m.s. system of a group is low. Collisions of this type are mainly responsible for multiple production at high energies. In conclusion brief consideration is given to the “complicated” types of asymptotic behaviour of “essentially inelastic” processes when the extreme right singularity in the j plane is a branching point or a point of accumulation of singularities etc. It is shown that in this case instead of the factor s i, i+1) j o ( ti) one ought to expect in the asymptotic values of essentially inelastic amplitudes expressions like Σ ν λ ν ( t; ξ i, i+1 )( S i, i+1 ) jv( ti) , where ξ i, i+1 =In( S i, i+1 / m 2) and λ ν ( t, ξ i, i+1 is a function depending on the character of the vth singularity and perfectly analogous to those which determine the corresponding asymptotic values of elastic scattering amplitudes.

Coordination synthesis on organomagnesium centers: octaphenylcubane

Diphenylacetylene is tetramerized uniquely by arylmagnesium to a C 56H 40 hvdrocarbon for which the octaphenylcubane structure has been proposed. Synthesis of the cubane ring system cannot be duplicated when methyl or methyl-phenyl substituents are present on acetylene or various other organic groups on magnesium. The singular character of cubane synthesis on magnesium centers is related in a peculiar manner to an environment of aryl groups.

Strength of Stress Singularities at Crack Tips for Flexural and Torsional Problems

Continuing a series of studies on the fracture analysis of elastic bodies, the present paper provides a complex variable method for evaluating the strength of stress singularities at crack tips encountered in the flexure and torsion of cylindrical bars. As a result of the complex flexure functions being sectionally holomorphic for crack problems, the singular character of the shearing stresses is found to be of the type r−1/2, where r is radial distance from the crack point. Crack-tip stress-intensity-factor solutions are given for problems involving both imbedded and surface cracks in bars. The results suggest the possibility of extending the Griffith-Irwin theory of static fracture to flexural and torsional problems.

On the Singular Character of Thermal Stresses Near a Crack Tip

On the Singular Character of Thermal Stresses Near a Crack Tip

Combined Stresses in an Orthotropic Plate Having a Finite Crack

Using a formulation in integral equations, a solution for the combined extension-classical bending stress and displacement solution is presented for the case of an infinite orthotropic flat plate containing a finite crack. While the solution can be expressed in closed form for the entire field, primary emphasis is placed upon the stresses near the crack point. Qualitatively, no major difference in behavior due to orthotropy was found although certain quantitative features are noted, mainly as a function of the characteristic rigidity ratio (Ex/Ey)1/2. The inverse square-root character of the isotropic stress bending and extension is not changed by orthotropy, although amplitudes and distribution are affected. Account is taken of recent important work by Knowles and Wang dealing with Reissner bending of the plate which shows that the extensional and surface bending stresses are identical in singular character and circumferential distribution. A bending-extension interaction curve for fracture initiation is derived and shown to be linear when based upon the more exact bending theory.

The Relation between the Character of Singularity and the Sun-spot Numbers

In this paper the variation of weather situation on the singularity accompanied with increase or decrease of annual mean sun-spot numbers is discussed. For this investigation, Lettaus concept of “specific singularity” is applied. We have found that the variation of weather situation accompanied with sun-spot numbers before and after the singularities such as December 30th, February 5th and April 6th is almost similar on the respective cases. (Fig. 8) The principal characters obtained are as follows. (I) On the singularity as divided by sun-spot numbers into three cases a), b) and, c), a) When the annual mean value of sun-spot numbers is larger than 60, the daily preciptatatatiion amout over Japan is somewhat plentiful than the normal value (Fig. 4, B), and the positive anomaly of daily mean temperature is observed over the western Japan. (Fig. 5, B). b) When the sun-spot number is smaller than 15, the negative anomaly of daily precipitation amount is especially remarkable along the Japan Sea side. (Fig. 4, E) On the same calendar day, the anomaly of daily mean temperature is negative over the eastern Japan. (Fig. 5, E) Over Japan and her neighbourhood, the number of cyclone and anticyclone is remarkably small. (Fig. 6, E) c) When the sun-spot number is between 15 and 60, the daily mean temperature over Japan is colder than the normal temperature. (Fig. 4, I) It may be considerd that this distribution relates closely to anticyclones locating over Tohoku district. (Fig. 5, I) (II) On the day before or after the temperature singularity showing the maximum range of frequency distribution of daily mean temperature, the anomalies of precipitation and temperature are also examind as related to the magnitude of sun-spot numbers. a) When the sun-spot number is larger than 60, the negative anomaly of daily precipitation amount is observed over the southwestern Japan with its center near Shikoku district. (Figs. 4, (A, C)) The temperature anomaly is positive over almost the whole Japan which its center over the northern Japan. (Figs. 5, (A, C)) It is probable that above characters become distinct when the anticyclones appear over the eastern China or the East China Sea. (Figs. 6, (A, C)) b) When the sun-spot number is smaller than 15, the positive anomaly of daiy mean precipitation is observed along the southern coast of Japan. (Figs. 4, (D, F)) The distribution of temperature anomaly is not so distinct in spite of the positive anomaly over the western Japan. (Figs. 5, (D, F)) Above distributions are primarily explained by anticyclones over Japan. (Figs. 6, (D, F)) c) When the sun-spot number is between 15 and 60, the positive anomaly of precipitation is observed over the western Japan, (Figs. 4, (G. I)) and the temperature anomaly is negative over the area which extends from the west of Tohoku district through Shikoku district. (Figs. 5, (G, I)) It may be seen that these results are caused by numerous cyclones over the East China Sea. (Figs. 6, (G, I)) As already noted, variation of the distriburions of daily precipitator amount and daily mean temperature corresponding with that of the sun-spot numbers are caused indirectly by the passage of cyclones and anticyclones over Japan and her neighbourhood. This result is also verified by comparing the frequency distributions of daily mean temperature at Hiroshima. (Fig. 7)

A Political Sherd

Through the kindness of Dr. Charles Seltman I am enabled to publish an inscribed sherd in his possession, the singular character of which has not hitherto been noticed. Its history is obscure; it was bought, so Dr. Seltman informs me, from a dealer who said that it had been found on the North Slope of the Athenian Acropolis. This provenance may well be correct; the sherd is certainly Attic, and the dealer made no financial gain from his statement; nor, apparently, did he regard the inscriptions as more than a curiosity. The sherd measures 11·6 cm. at its greatest length, and is 4·8 cm. wide at 1. 22 of the inscription on the upper surface. It apparently forms part of the rim and body of a plate of the type known as a fish-plate, of which a considerable number exist, both Attic and South Italian, in a complete or fragmentary state, though they have not received much attention from scholars.

The Influence of Science

The ingenious letter under the above heading, on page 180 of NATURE of August 5, by that industrious astronomer of Stonyhurst College, Father Cortie, S.J., seems to require some brief notice because of the singular character of the statements made in it. We are asked to believe that Copernicus's “heliocentric doctrine was freely taught, even in ecclesiastical colleges, until Galileo interested himself as a champion of the system”; in spite of the admission that after this “truculent and hot-headed controversialist” had endeavoured to get the Church to realise that the doctrine was not really antagonistic to Scripture when reasonably interpreted, and after the offended Pope had brought the matter before the Holy Office, that authority determined that “the Copernican system was false and absurd philosophically.” And we are also asked to believe that the outcome was merely that Galileo had as a penance “to recite certain prayers, and was sent to a beautiful villa at Arcetri”; the implication being that there was really no punishment, and that there was no call for anxiety or distress on the part of either him or his daughter throughout the proceedings.

AN OBJECT LESSON IN THE ADVANTAGES OF ORGANIZATION.

According to the<i>British Medical Journal</i>, a medico-economic dispute of a somewhat singular character has recently taken place in England. The Halifax Board of Guardians, in November of last year, changed the terms of their advertisement calling for a resident medical officer for the Union Hospital, from one without restriction to sex and with a salary of $700 a year to a request for a "lady doctor" at a salary of $500 a year. The position was sought and obtained by a woman physician. The London School of Medicine for Women, however, took the matter up and communicated with the local division of the British Medical Association, which, on investigation, found that the guardians were under the impression that the services of a woman should be obtainable at a less rate than those of a man. A warning notice was published in the<i>British Medical Journal</i>, and the appointee resigned

The Stability of Submarines

THE mathematical investigation which Mr. Jeans puts forward is of great interest, but avowedly rests on the assumption of the complete immersion of a submarine in an infinite ocean. The concluding paragraph of his letter indicates that a great number of factors, which ought to be taken into account, are not represented in the mathematical investigation, the most important being near proximity to the surface and the consequent formation of surface-waves. It will suffice, therefore, for me to say that my insistence on the necessity for direct experiment, rather than mathematical investigation, had relation to the case where the submarine was moving at the surface with a small reserve of buoyancy. The slides which I exhibited at the Royal Society reproduced photographs taken in these circumstances, and showed the singular and irregular character of the surface-waves produced by the headway of submarines under these conditions. These slides furnished conclusive evidence of the impossibility of representing the conditions of practice by purely mathematical investigation, and the absolute necessity for experiments on models and full-sized submarines.

The Stability of Submarines

THE mathematical investigation which Mr. Jeans puts forward is of great interest, but avowedly rests on the assumption of the complete immersion of a submarine in an infinite ocean. The concluding paragraph of his letter indicates that a great number of factors, which ought to be taken into account, are not represented in the mathematical investigation, the most important being near proximity to the surface and the consequent formation of surface-waves. It will suffice, therefore, for me to say that my insistence on the necessity for direct experiment, rather than mathematical investigation, had relation to the case where the submarine was moving at the surface with a small reserve of buoyancy. The slides which I exhibited at the Royal Society reproduced photographs taken in these circumstances, and showed the singular and irregular character of the surface-waves produced by the headway of submarines under these conditions. These slides furnished conclusive evidence of the impossibility of representing the conditions of practice by purely mathematical investigation, and the absolute necessity for experiments on models and full-sized submarines.

IX. The Manavulu-Sandesaya Text and Translation

The Mānāvuḷu - sandesaya, or in Pali Mahānāgakula - sandesa, which to-day first reaches the Western reader, is a little Pali poem of somewhat singular character. It is in the form of a poetical epistle, in the style of high kāvya. It is dated from Mahānāgakula (Mānāvuḷu), a city of Ceylon. After long and occasionally complicated panegyrics upon this city, a local Buddhist monastery, and the mahāthera Nāgasena residing in the latter, it proceeds to describe in similar strains the city of Arimaddanapura (Pugāma, the modern Pagan), the emperor Siri-Dhammarāja who bears rule therein, a monastery built by the latter near his capital, and a distinguished mahāthera named Kassapa - Saṅgha - rakkhita who dwells there. Then follows an address from Nāgasena to Kassapa, in which Nāgasena mentions that he has received a letter from Kassapa through a minister Ñāṇa, apparently containing a request. The poem here practically comes to an end. Five verses follow, containing greetings to a certain Sāriputta and an exhortation to reform the Church in Pagan as it had been reformed in Ceylon by Parākrama-bāhu; but as these verses are in part grossly corrupt, as they are singularly feeble and debased in style, and as they are ignored by the Sinhalese translator of the rest of the poem, we are justified in regarding them with suspicion. Either they are altogether spurious, or they are a rough draft which the poet never worked out. The poem is thus a mere fragment.

THE CAISSON EXPLOSION OF BATTERY F, SECOND REGIMENT OF THE U. S. LIGHT ARTILLERY.

The writer is at present in charge of a number of soldiers wounded by an accident so rare, that hardly any officer in the United States service has seen an example of it. The singular character of the wounds produced, and the rarity of the causes producing them, render the cases most instructive for study. The late riots in Chicago, by obstructing the United States mails, and arresting commerce on interstate railways obliged the general government to send to Chicago about 2,000 regular troops to maintain order. On the 16th of July, Battery F of the Second Regiment of U. S. Light Artillery was moving at a trot on Grand Boulevard, supported by a company of the Seventh Regular Cavalry, when suddenly the three ammunition chests of the first caisson exploded with terrific violence. The caisson was drawn by four horses, which were guided by two mounted drivers. Two cannoneers

Notes on some Singular Cavities in the Magnesian Limestone. (Plate vi.)

The cavities or burrows in the Permian Limestone near Wormald Green Station, to which I wish to draw your attention, are of such obscure origin that at one time one is inclined to refer them to the excavations of some early form of Gastrochaena; at another to an annelid; then to one of the algae. Without attempting to say what they really are, I venture only to describe their forms and their position in the limestone, and what I have been able to gather from the observations of others, hoping that the members here present may be able to throw so much light upon the matter that we may come to some probable conclusion as to their origin. An old quarry, about a quarter of a mile west of Wormald Green, which had been for a long time disused, was opened again a few months ago. In the old and new section about 60 or 70 feet of the middle beds of the Lower Magnesian Limestone have been exposed. Throughout 30 feet of the upper portion of the new part there occur innumerable grooves and cavities of a singular character, agreeing with certain fossil remains which are mentioned by Mr. J. W. Kirkby in a paper read before the Geological Society in 1861,* as occurring in the Lower Permian Limestone of South Yorkshire, at Hampole Stubbs. He says, “the occurrence of another Fossil, of somewhat obscure affinities, though possibly an annelid, tuay here be noticed. It is the cast of ...

On Platysiagum sclerocephalum , Egerton, and Palæospinax priscus , Egerton

In the 13th Decade of the Memoirs of the Geological Survey of the United Kingdom, published in 1872, I described the above-named fossil fishes from the Lias formation of Lyme Regis. Since the issue of that publication I have obtained specimens elucidating parts which were defective in the former examples, and which I consider of sufficient importance to warrant description. On referring to the figure of the former ( Platysiagum ), given on plate 6 of the Decade, it will be seen that the dorsal and anal fins are wanting. This also happens with the several specimens described in the text but not figured. These desiderata are furnished by a fine specimen recently obtained. The two extremities are wanting; the remainder of the fish measures 15 1/2 inches in length. It corresponds very closely in size with the larger of the two specimens described in the Decade. The total length of the fish, if restored, would probably be about 22 inches. The vertical diameter of the trunk is very uniform between the occiput and the dorsal fin, and measures 4 inches. This is a singular character in a Sauroid fish, in which family the body is generally fusiform, tapering gradually from the thoracic arch to the commencement of the caudal fin. The dorsal fin which is here preserved is very singular both in position and structure. It is situated 9 1/2 inches behind the occiput, and occupies a position vertically opposed to the interval between the ventral and anal fins, and extends thence nearly

II. On the theory of probability, applied to random straight lines

This paper relates to the Theory of Local Probability—that is, the application of Probability to geometrical magnitude. This inquiry seems to have been originated by the great naturalist Buffon, in a celebrated problem proposed and solved by him. Though the subject has been more than once touched upon by Laplace, yet the remarkable depth and beauty of this new Calculus seem to have been little suspected till within the last few years, when the attention of several English mathematicians has been directed to it, and results of a most singular character have been obtained. The problems on Local Probability which have been hitherto treated relate almost exclusively to points taken at random. The object of the present paper is to show how the Theory of Probability is to be applied to straight lines whose position is unknown, or, in other words, which are taken at random.