Gravitation and Thermodynamics

In 1952, the late Professor T. Nakayama succeeded in constructing the Galois theory for finite dimensional simple ring extensions [7]. And, we believe, the theory was essentially due to the following proposition: If a simple ring A is Galois and finite over a simple subring B then A is B′-A-completely reducible for any simple intermediate ring B′ of A/B [7, Lemmas 1.1 and 1.2]. Moreover, as was established in [5], Nakayama’s idea was still efficient in considering the infinite dimensional Galois theory of simple rings.

It is generally accepted that the vorticity transport theory of turbulent motion initiated by G. I. Taylor takes some important place in the mechanism of momentum transfer in turbulent chaos. Its superior points have already been affirmed by various papers published by Taylor, but the mechanism is not yet affirmed satisfactorily enough to describe the details of turbulent motion as detected by L. Prandtl.The application of Taylor's theory to the atmospheric field has been examined by the present writer with some success, though the principle of discussion adopted was not rigorous. Anyhow we see here that the vorticity transport theory is found to be reliable fairly well among the various theories hitherto postulated.Prof. S. Fujiwhara has pointed out the existence of transverse resistance accompanying eddy motion long since the publication of his elaborate work on the vortical motion. His intention has partly been accomplished by S. Sakakibara and Y. Isimaru in their equations of transverse eddy viscosity.In the present paper the writer shows that the term of transverse resistance is detectable from the vorticity transport theory under gradient wind condition. Denoting the mean motion by bar and the eddy motion by dash, the equations of motion of perfect fluid adopted by Taylor are, in steady state, where The deformation of vorticity of incompressible fluid is expressed by, in Lagrangian form. and the condition of conservation of vorticity is as follows: Thus Taylor arrived at the following equations: etc. Assuming the relations as realizable in the problem of gradient wind we have as the x-component of frictional resistance and as the y-component. Here If μ1=μ2, ν1=ν2and the coefficients are positive, the result becomes to coincide with Sakakibara-Isimaru's term.Next the effect of compressibility has been considered. Discarding the terms of small order, the frictional term is approximately given by etc., where

We examine the metaphysical theories of motion and matter developed by Democritus, Plato, and Aristotle, as well as other theories that appeared later on and were great influenced by Aristotle. We distinguish two subsequent developments. One is the impetus theory for the projectile motion and the other is the void theory for the rise of liquids by suction. We analyze the Galileo theory of motion, emphasizing the crucial step in its development which was the precise definition of velocity. We also analyze the events related to the experiments of Torricelli, which was understood as the production of void, and his explanation based on the weight of the atmospheric air.

The determination of the tension in cable-stayed and tied-arched bridges has been a concern for authorities for decades in order to verify the integrity of the structures. The increasing age of the bridges and the recent collapse of several of them across Europe is making their monitoring more important than ever. Nowadays, the analyses usually rely on sparse on-site measurements of the strain or of the direct force when hangers are replaced (i.e., a few times in the lifetime of the structure) or on the use of accelerometers and the simple taut string theory, which ignores the bending stiffness of the cable and its boundary conditions, to measure the tension (usually at best once a year). As a result, damages can be detected late and the measured data depends on the environmental conditions during the measurement (traffic, temperature, wind,…) In this paper, we first describe LYRA, a novel autonomous bridge monitoring solution based on wireless accelerometers and on a robust mathematical model of the hangers. The accelerometers measure the response of the hangers to environmental solicitations six times a day and dispatch the data to a central station. Then, the station computes the tension of the cables from the time response of the sensors by means of a robust algorithm, which is an extension of the simple taut string theory for flexibles anchors and/or non-negligible bending stiffness. Finally, the measured tensions are uploaded on a webserver, where they can be monitored from any apparatus with an internet access, and alerts are automatically generated if tensions outside of pre-set boundaries are measured. Then, we demonstrate the capabilities of the system on the Lanaye bridge, located at the border between Belgium and the Netherlands, whose 30 hangers have been monitored since July 2021.

In a recent paper2 we have called a ring 2 primitive if 2 contains a maximal right ideal a such that the quotient (a: W) = (0). The quotient (: 1) is defined to be the largest two-sided ideal of 21 that satisfies the condition 21(a: 21) C a. Thus if 21lias an identity, (a: 21) is the largest two-sided ideal of 21 contained in a. Primitive rings appear to play a role in the general structure theory of rings analogous to that played by simple rings in the Wedderburn-Artin theory. Primitive rings are also fundamental in general representation theory since it is known that a necessary and sufficient condition that a ring 21 be primitive is that 21 be isomorphic to an irreducible ring W of endomorphisms in a commutative group T. If 2 is irreducible in T and Zi is the division ring of endomorphisms commutative with the elements of A, then R is a dense ring of linear transformations in T regarded as a vector space over Z.3 We shall call 2 a left primitive ring if 2 contains a maximal left ideal a' such that the quotient (}': 21)l = (0). Here (s': 2) 1 is the largest two-sided ideal in 2 having the property (a': 21)1 21 C a'. Evidently, a ring is left primitive if and only if it is anti-isomorphic to an irreducible ring of endomorphisms. It is an open question whether or not a primitive ring is necessarily left primitive. This is indeed the case if 2 contains minimal ideals. In many other respects the theory of primitive rings that contain minimal ideals constitutes the most satisfactory part of the theory of primitive rings. An appropriate tool for studying rings of this type is a generalization of the duality theory previously used by Dieudonn6 in studying simple rings that possess minimal ideals.4 In this note we develop this generalization of Dieudonn6's results. We also investigate certain natural topologies that can be defined in any primitive ring.

An outstanding unsolved problem in the theory of rings is the existence or non-existence of a simple nil ring. Such a ring cannot be locally nilpotent as is well known [ 5 ]. Hence, if a simple nil ring were to exist, it would follow that there exists a finitely generated nil ring which is not nilpotent. This seemed an unlikely situation until the appearance of Golod's paper [ 1 ] where a finitely generated, non-nilpotent ring is constructed for any d ≥ 2 generators over any field.

Light scattering measurement and theoretical interpretation of mutual diffusion coefficients in binary liquid mixtures

Formation of a vortex ring in the process of rising a heated light volume of air medium in the Earth’s gravity field is investigated. The results of calculations carried out on various computational grids are compared both with each other and with solutions of the approximate analytical theory of formation and motion of a buoyant vortex ring approved by the available experimental data. A good agreement between all the data is obtained.

In this paper we examine the simple theory of π -electron ring currents in conjugated systems (devised originally by London (1937) and extended by People (1958) and McWeeny (1958)), with particular reference to the topological or graph-theoretical aspects of it (all the necessary graph-theoretical ideas and terminology are explained in the text). There is a close correspondence between the adjacency matrix of the graph representing the σ-bond skeleton of the carbon atoms comprising a given conjugated system, and the secular equations which arise in the theory (a relation now well known to be common to all formalisms based on Hückel ‘topological’ molecular orbitals), but in addition we here emphasize that several other graph-theoretical ideas–notably those concerning circuits and spanning trees–specifically underlie the ring current concept. In this connexion, the question of whether any given molecular graph is semi-Hamiltonian or non-Hamiltonian is of prime importance, and it is pointed out that a unitary transformation originally proposed by McWeeny applies to semi-Hamiltonian molecular graphs, whereas one recently devised by Gayoso & Boucekkine can be applied to any simple, connected graph–as also can an explicit ring current formula (based on the London–McWeeny theory) just published by the present author. These ideas are illustrated by some simple numerical calculations, and an example is given of a conjugated system (decacyclene) whose molecular graph is apparently non-Hamiltonian. It is emphasized that although much graph theory is inherent in the ring current concept, the ring current index itself is not a completely topological quantity–even when a purely topological wavefunction (such as the simple Hückel one) has been used to calculate it.

The evolution of the symmetric circulation of a moving hurricane-scale vortex on a beta-plane is investigated and interpretations of this investigation are given in terms of vorticity fluxes. the study, which is fundamental to understanding vortex motion, is based largely on numerical integrations of the barotropic vorticity equation, using a finite-difference method. In the absence of any large-scale flow, the north-westward drift of an initially symmetric, cyclonic vortex in the northern hemisphere is accompanied by a decrease in the tangential circulation at most radii, and a consequent deceleration of the azimuthally averaged tangential velocity. This behaviour is not simply a consequence of the increase in the Coriolis parameter over the domain of the cyclone as is often supposed, but may be explained as the sum of three effects: the outwards radial flux of relative vorticity associated with the asymmetric component of flow, the corresponding flux of planetary vorticity, and the rate of change in planetary vorticity due to the meridional displacement of the vortex. In particular, the net rate of change in absolute vorticity, the sum of the last two effects, makes the largest contribution to the circulation changes. the flux of absolute vorticity is associated, inter alia, with the generation of Rossby waves by the vortex. the behaviour is different from that in recent calculations by Carr and Williams in which the flux of planetary vorticity was omitted. It is pointed out that in the analytic theory for vortex motion developed by the first author and co-workers, an initially symmetric vortex with zero net relative circulation at large radial distances subsequently develops a finite negative relative circulation whose strength increases linearly with time; this is because the net flux of planetary vorticity at large radial distances is finite. In contrast, in the numerical model the flux is close to zero so that the symmetric circulation decays more rapidly with radius than the 1/(radius) decay rate in the analytic model. At inner radii there is good qualitative agreement between the predictions of the two methods, even though there are quantitative differences between them. It is noted that the occurrence of the finite circulation in the analytic theory violates a theorem of Flierl et al. and that it represents a limitation of the long-term validity of the theory.

The effects of Zn impurities on the electronic state in the vortex core were investigated systematically in almost optimally doped YBa 2 (Cu 1- x Zn x ) 3 O y (0≤ x ≤0.06) using the microwave complex surface impedance ( Z s ) measurement technique. We estimated the viscosity, η, and the pinning constant, κ p , of a vortex as functions of temperature and x on the basis of a mean-field theory of the vortex motion. κ p as a function of Zn concentration, x , suggests that Zn doping is not an effective procedure for pinning the vortex motion. η( x ) depended very weakly on x for x ≤0.003, while it decreased rapidly with increasing x for x >0.003. These finding suggest that the scattering of QPs in the vortex core is much larger than that in the Meissner (zero-field) state for x ≤0.003. On the other hand, for x >0.003, the scattering of QPs in the vortex core is governed by Zn impurity, showing that the presence of the vortex is not important for the scattering in this regime. The large scattering in the low-do...

This paper examines two-dimensional vortex motion in a shear-flow with nonuniform vorticity. Typically, a vortex travels to an extremum in the background vorticity distribution. In general, the rate of this migration increases with the magnitude of the background vorticity gradient; however, a retrograde vortex, which rotates against the local shear, moves orders of magnitude faster than a prograde vortex of equal strength. Retrograde and prograde vortices travel at different speeds because they perturb the background vorticity differently. Linearized equations accurately describe the background vorticity perturbation that is created by a weak retrograde vortex, whereas nonlinear effects dominate for a prograde vortex of any strength. An analytic theory is developed for the velocity of a retrograde vortex, based on the linearized equations. The velocity of a prograde vortex is obtained by a simple “mix-and-move” estimate, which takes into account the nonlinear trapping of fluid around the vortex. Both velocity predictions are verified by vortex-in-cell simulations. If the ratio of background shear to background vorticity gradient exceeds a critical level, there is no vortex motion up or down the background vorticity gradient. Estimates of the critical shear are obtained for both prograde and retrograde vortices. These estimates compare favorably to vortex-in-cell simulations.

A Galois theory of commutative rings

Theory of effect of potential gradient on measurements with the Ebert ion-counter.---In a former paper, the theory of the action of the negative charge induced on the ion-counter by the normal atmospheric potential gradient in causing error in the measurements of the negative ion-density was worked out for the case where the induced charge is insufficient to prevent the entry of ions at any point. The theory led to a linear relation between the reduction in measured negative ionic density and the potential-gradient. The simple theory is now extended, to take account of the action of the induced charge in not only changing the speed of the ions but in reducing the area of the surface through which negative ions can enter the instrument. Assuming a simple ring distribution of the induced charge, an expression for the variation of percentage error or reduction in the measured negative ionic density with potential-gradient, is derived, and shows a non-linear law, in general agreement with the observations of Mackell. In measurements of positive ion-density, the extended theory, like the simple one, indicates that no error is to be expected, for a potential-gradient of normal sign, a conclusion confirmed by recent results of Mackell and of Norinder.

Herstein's Lie theory revisited