Tangential Equations Research Articles

Regularity of the tangential Cauchy-Riemann complex and applications

In a series of papers Webster ([26], [27], [28]) has shown how C-estimates for the tangential Cauchy-Riemann complex can be applied to several non-linear problems in Complex Analysis. For example, he gave a simplification of the proof of Kuranishi’s embedding theorem and an application to the integrability problem for almost CR vector bundles. Connected with Webster’s approach are some regularity assertions, whose parameters follow from the C-estimates. Therefore, an improvement of these estimates would lead to an improvement in the applications. Now the author of this article, together with Lan Ma, has given in [12] such an improvement. In two subsequent papers this has been applied to Webster’s approach. The results are formulated in two theorems at the end of this introduction. Here we want to briefly describe the estimates for the tangential CR equations. Let G be a strictly pseudoconvex domain in C with C-smooth boundary and 0 ∈ bG. By a local homotopy formula for the tangential Cauchy-Riemann operator ∂b we mean the following. There exists a neighborhood base {M} of relatively open sets M , with 0 ∈ M ⊂ bG, and on each M there are given operators Rq (q = 1, 2, . . . , n− 2), which fullfill the equation

Some new surfaces within the general three-system of screws

The general three-system is the two-parameter system of linearly dependent screws that defines most generally first order instantaneous rigid-body motion with three degrees of freedom. Through a general point there pass three screw axes, and in a general plane there lie two screw axes. The theory is here developed further than hitherto by finding three apparently distinct surfaces: (i) The SPS, the sextic point surface that bounds those points through which pass three screw axes that are all real; (ii) the QES, the quartic envelope surface that, by considering its tangent planes, bounds those planes in which lie two screw axes that are both real; and (iii) the quartic point surface S τ at the points of which two real screws intersect one another at right angles. A closer inspection, however, reveals the QES to be none other than the SPS expressed as its tangential equation. The surfaces are described in some detail and convincingly illustrated with the help of computer-graphics techniques.

Oblique impact of similar bodies with circular contact

The general solution for tangential loading histories of similar bodies with circular areas of contact is outlined and applied to the tangential impact. Classical numerical methods use an interpolation function for the stress distribution on a set of points and obtain the stress distribution for given displacements by inversion of a linear equation system. Our solution avoids large systems of simultaneous equations and gives exact solutions for loading histories in form of finite, successive displacement increments. The general equations of motion for the impact are derived and applied to the case, where the normal and the tangential equations of impact are uncoupled. The Nassi-Shneiderman diagrams of the contact algorithm and the impact algorithm are presented. The solution is compared with other results from earlier publications. A tangential coefficient of restitution, which determines the behaviour of the bodies after impact, is plotted as a function of two parameters.

A Hermite interpolation algorithm for hypersingular boundary integrals

AbstractThis paper presents a conforming C1 boundary integral algorithm based on Hermite interpolation. This work is motivated by the requirement that the surface function multiplying a hypersingular kernel be differentiable at the collocation nodes. The unknown surface derivatives utilized by the Hermite approximation are determined, consistent with other boundary values, by writing a tangential hypersingular equation. Hypersingular equations are primarily invoked for solving crack problems, and the focus herein is on developing a suitable approximation for this geometry. Test calculations for the Laplace equation in two dimensions indicate that the algorithm is a promising technique for three‐dimensional problems.

Simulation of geophysical inputs for solar architecture

We present a thermodynamic model for a planet interacting with the solar radiation, our aim being to obtain a numerical simulation for the regular part of the geophysical inputs of an optimally controlled solar dwelling model that we studied elsewhere. Ours is not an approach to the climatology of a realistic planet. If climatology were indeed our main business, we should recognize from the outset the importance of fluid motions. Rather, we show the consequences of a purely linear dynamics of the temperature field and the related heat flows. We start with a system of two tangential field equations and we perform a discretization with a finite number of components. Albedo and greenhouse absorption, are introducedad hoc by fitting the outcome of the numerical calculations to the experimental values of the surface air temperature.

Nonadiabatic and three-dimensional effects in compressible turbulentboundary layers

A defect stream function formulation for nonadiabatic flow with small crossflow is developed. The first-integral property of this formulation provides for two removal of the streamline curvature term in the governing equation so that the form of the reduced equation for small crossflow is the same as that for two dimensional flow. The combined law of the wall and wake is used in place of the no-slip boundary condition. The tangential velocity equation for law-of-the-wall flow is shown to be the same for three-dimensions as for two when the Boussinesq approximation applies, and a closed form solution for the crossflow angle in the inner region is obtained. Analytic solutions for nonadiabatic, compressible, equilibrium flow with a Clauser outer-region eddy-viscosity model are obtained, and excellent agreement with experimental skin friction and velocity profile data for nonadiabatic, compressible flat-plate flow is achieved. An analytic solution for a linear inner-region eddy-viscosity model is also obtained; the wake function part of this solution is found to be inconsistent with the empirically established law of the wake.

繰返し荷重を受けるRC部材の引張硬化係数と増分型構成則

The present paper proposes the general form of the tangential constitutive equation of a reinforced concrete uniaxial member, emphasizing on the cracks' opening and closing behavior. In the model, the tension stiffening factor makes the most important role in describing the varying stiffness due to cracks generation and slippage between concrete and reinforcement. The proposed constitutive model can realistically represent the cyclic behavior in tension and in reversed loading to compressive zone and agrees relatively well with experimental results.

Normal configurations and the nonlinear elastostatic problems of bending, torsion, expansion, and eversion for compressible bodies

First, we decompose a (deformed) equilibrium configuration into a disjoint union of a one-parameter family of surfaces, called lamina surfaces. We require that all of these lamina surfaces be two-dimensional universal solutions for homogeneous isotropic elastic membranes. Then the two tangential equations of equilibrium are satisfied without any body force, universally for all isotropic elastic (membrane) materials

On the coupler-great-circle envelopes of spherical four-bar linkages

With use of the techniques developed for coupler-point curves, the tangential equation is derived for the envelope of a great circle attached to the coupler of a general spherical four-bar linkage in the motion of this mechanism. The class of this envelope is determined to be 8. The dual relationships between the coupler-point curve and the coupler-great-circle envelope are discussed. With this concept, the techniques for designing a spherical four-bar coupler-great-circle envelope generator are introduced.

An Examination of the Fibrogram

The theory of the fibrogram has been derived based on the probability of randomly catching and holding a fiber. The amount axis has been shown to be proportional to the relative mass when related to the original specimen. Tangential equations are used to show that theoretically, for a length l or longer, the proportional mass, the mean length by number, the relative number, and the mean length by mass of fibers in the sample can be obtained from the fibrogram, as well as the number and mass arrays. Finally, the theoretical percentage of fiber by number or by mass extending a length l or longer from a clamped random sliver of the sample is obtainable from the fibrogram.

The heating field in an asymmetric hurricane— Part II: Results of computations

This is the second part of a paper on the distribution of heating fields in a hurricane. The first part dealt with the mathematical framework. The second part, i. e. the present paper deals with numerical calculations for an actual hurricane. The following sequence of calculations has been performed after the analysis and tabulation of an initial field of the tangential velocityU (r, ϑ, p): (1) the radial equation of motion is used to determine the geopotential heights; (2) the hydrostatic equation is used to determine the temperature field; (3) the tangential equation and the mass continuity equation are combined to obtain an omega equation whose solution determines the vertical velocity; (4) the radial velocity is next determined from the mass continuity equation; and (5) the heating function is finally determined from the first law of thermodynamics. The results of this study show an asymmetric banded structure (eye wall and rainband) of the vertical motion field as well as the heating field; these show close resemblence to observations. An analysis of the non-linearities of the asymmetric momentum distribution is shown to be crucial in the analysis of the hurricane heat sources.

Diffusion of non-swelling gases through dry conifer wood

The conditions under which mutual, Knudsen and transition-range diffusion occur are described, and the corresponding gaseous diffusion coefficients are stated. The coefficient of hindered diffusion through pit membrane pores is shown to be about one third of the mutual diffusion coefficient and not about one thirtieth as postulated by earlier workers. A new longitudinal diffusion equation is developed, and results of a new gas permeability technique are used to derive coefficients for Picea sitchensis and Abies grandis. Only a small proportion of the tracheids is assumed to be conducting, owing to pit aspiration. About 98% of the resistance to longitudinal diffusion is in the tracheid lumina, 1% in the pit apertures and 0.2% in the pit membrane pores. From similar considerations a new tangential diffusion equation is developed, and coefficients are calculated. Roughly 90% of the resistance to tangential diffusion occurs in the pit apertures, 10% in the pit membrane pores and only 0.3% in the tracheid lumina. The agreement between the theoretical and published experimental values is satisfactory. For tangential diffusion the agreement is closer by an order of magnitude than that obtained by earlier workers. The experimentally determined temperature dependence of the diffusion coefficients is shown to be consistent with the above results. A hindered diffusion coefficient of one thirtieth of the mutual coefficient would lead to a temperature dependence not observed experimentally.

Drag coefficients of inert and burning particles accelerating in gas streams

The effects of burning and acceleration on the drag coefficients of particles suspended and accelerating in gas streams were studied in a Reynolds number range extending from 250 to 1600. The model chosen for the analytical study was a spherical particle with mass flux through the surface to simulate burning. The governing equation was the integrodifferential representation of the tangential equation of motion of a thin boundary layer on a sphere. The solutions indicated that burning and acceleration tend to reduce the drag coefficient. The fractional reduction was found to be a function of the ratio of mass flux from the surface to that in the free stream f for the burning particle and the acceleration modulus Ac for the accelerating particle. The effects of burning and acceleration were studied experimentally by subjecting burning particles (gunpowder) and nonburning particles to the convective flow behind a shock wave in a shock tube. The variation in particle size and displacement with time were obtained by photographing particle shadows with a high speed framing camera and concentrated light source. The particle density, the shock wave velocity, and the atmospheric conditions together with the photographic data provided sufficient information to calculate the particle's drag coefficient. The analytical expressions predicted and experimental results verified that the particle drag coefficient was insensitive to burning and accelerative effects if Ac ≤10 −2 and f <0.025. When these conditions are met, other phenomena such as free stream turbulence, particle rotation and roughness can create large variations in the drag coefficient than the mechanisms considered in this study.

Intrinsic equations of motion in dynamical meteorology

Tangential and normal equations of horizontal motion along and normal to the characteristic lines (for example: stream lines, isobars, isotherms, etc.) are derived in general form. Then the later section of this paper is devoted to applications to natural coordinates and the coordinates chosen to lie parallel and normal to the isobars.

自然座標による運動方程式

§ 1 General equations of motion parallel and normal to the characteristic lines.§ 2 The tangential and normal equations of motion in the system of natural coordinates.§ 3 Equations of motion parallel and normal to the isobars.

APPLICATION OF BRIANCHON'S THEOREM TO CONSTRUCTION OF GEOLOGIC PROFILES

Brianchon9s theorem states that the three diagonals joining opposite vertices of a hexagon circumscribed about a conic are concurrent. A corollary of this theorem applies to a pentagon so that the points of tangency of an inscribed conic may be located. Any five non-concurrent straight lines in a plane, no three of which are parallel, will ordinarily form some kind of a pentagon; and if considered as tangents to a conic, they will define its shape and position. If these five lines are also normals to the traces of parallel stratigraphic surfaces having a constant strike, the derived conic may be regarded as an evolute, from which a set of involutes can be drawn that will constitute a structural profile. A method is thus afforded for constructing a profile normal to the strike of the rocks, and for measuring stratigraphic thickness, by the utilization of five observations of dip along a suitable linear traverse. Graphical methods are also given for constructing a parabolic evolute from four observations and a circular evolute from three observations. Additional points on the conic evolutes are obtained by the application of Pascal9s theorem. A mathematical analysis is presented of the relationship between a conic and five of its tangents; and the conic evolute, rather than its involutes, is recommended as a satisfactory record of the structure of a parallel fold. To obtain the equation of this evolute, the equations of the five tangents are first derived, using trilinear coordinates. Thereafter the tangential and trilinear equations of the general conic are deduced. Criteria are given for classifying the conic as a hyperbola, ellipse, parabola, or circle. These graphical and analytical methods are adaptations of the general method of evolute and involutes. They are offered, not as substitutes for the general method, but as quicker, though somewhat less accurate, means of obtaining similar results, where structural conditions justify their use.

Analytic systems of central conics in space

The amount of literature dealing with conic sections, individual curves and systems of curves, in one plane, is vast. When however we are dealing with a number of conics, not in the same plane, the situation is quite different. Certain figures, as the focal conics of a set of confocal quadrics, are familiar enough, but very little has been done in the way of a systematic study of more general systems. There are some studies carried out with the aid of purely synthetic methods; the algebraic or analytic treatment lags behind. The first writer to suggest a reasonable set of coordinates for a conic in space was Spottiswoode(l). The totality of straight lines that intersect a conic in three-space generates a very special sort of quadratic complex. The coefficients determining the equation of this complex, when a straight line has the usual Plucker line coordinates, may be taken as the coordinates of the conic, a clumsy enough system. A much better technique, perhaps the best for algebraic purposes, was developed by Johnson(2). Here a conic is looked upon not as a locus, but as the envelop of its tangent planes. Thus its tangential equation aiiuiuuO =0 gives ten homogeneous coordinates, connected by a quartic identity aii = aii, aiil = 0 i, j = 1, 2, 3, 4.

Leçons d'algèbre et de géométrie à l'usage des étudiants des Facultés des Sciences

THIS book has been specially written by Prof. Gamier for mathematical students taking a course in the faculty of science at Paris. It is essentially a theoretical treatise embracing four chapters on linear algebra; two on the classic theorems of both plane and solid protective geometry and one on tangential equations. The treatment is wholly analytical and, as the author points out in the preface, the most general methods have been applied in establishing the fundamental propositions. Centres and diameters of conies, for example, are derived as special cases of poles and polars.

An Elementary Introduction to Line Geometry

The study of line geometry by analytical methods involving the use of tangential coordinates and tangential equations is often found difficult by the junior Honours student. In the writer’s view the reason for this is to be found in the fact that, according to the course followed by a good many textbooks, the student finds himself, so to speak, pitchforked into the middle of the subject without any just appreciation of its bearing, or indeed of the need for such a study at all. Some modern writers on analytical geometry have attempted to remedy this by adopting the view that line geometry should be studied from the outset pari passu with point geometry. Unfortunately, however, experience tends to show that this is a mistake, inasmuch as it creates confusion in minds which have had insufficient practice in, and consequently have obtained an inadequate grasp of, analytical methods, which are never at any time found easy of assimilation by the student of elementary mathematics.

Note on Tangential Equations of Conics

Note on Tangential Equations of Conics