Two-parameter Family Of Curves Research Articles

ON ONE PROBLEM OF INTEGRAL GEOMETRY

The paper is devoted to a problem of integral geometry on a two-dimensional plane. In the band of the two-dimensional space , a two-parameter family of curves is given. Each of these curves consists of two arcs , which are adjacent to each other at the vertex of the curve at an angle , and the other ends rest on the axis. With respect to the branches of the curve , certain smoothness conditions are assumed. The family of curves can be parametrized using the coordinates of the vertices of the curves, that is, each curve from this family can be put in an unambiguous correspondence with the coordinates of its vertex. For each point , there is only one curve belonging to the specified family and having the point as its vertex. It is required to determine the function from the following equation for a given function where – region bounded curves and the axis . The theorem (uniqueness of the solution) is proved. If , then the solution of equation (*) is unique in the class of continuous finite functions with a carrier in . In proving the theorem, the invariance of the family of curves L(ξ,n) and weight functions to shifts along the abscissa axis is significantly used. Further, using the Fourier transform from the function with respect to the variable and the permissible transformations, a family of Volterra integral equations of the second kind is obtained with respect to the Fourier image from the function with respect to the first variable. The paper also provides an estimate of the conditional stability of the considered integral geometry problem

Three-dimensional inverse problem of geometrical optics: a mathematical comparison between Fermat's principle and the eikonal equation.

In the framework of geometrical optics, we consider the following inverse problem: given a two-parameter family of curves (congruence) (i.e., f(x,y,z)=c1,g(x,y,z)=c2), construct the refractive-index distribution function n=n(x,y,z) of a 3D continuous transparent inhomogeneous isotropic medium, allowing for the creation of the given congruence as a family of monochromatic light rays. We solve this problem by following two different procedures: 1. By applying Fermat's principle, we establish a system of two first-order linear nonhomogeneous PDEs in the unique unknown function n=n(x,y,z) relating the assigned congruence of rays with all possible refractive-index profiles compatible with this family. Moreover, we furnish analytical proof that the family of rays must be a normal congruence. 2. By applying the eikonal equation, we establish a second system of two first-order linear homogeneous PDEs whose solutions give the equation S(x,y,z)=const. of the geometric wavefronts and, consequently, all pertinent refractive-index distribution functions n=n(x,y,z). Finally, we make a comparison between the two procedures described above, discussing appropriate examples having exact solutions.

On zero curves of bivariate polynomials

Two conditions on the signs of the coefficients of a bivariate polynomial which ensure that the zero set is a single curve are derived. The first condition demands that all but one of the coefficients has the same sign. The second requires that the signs are ‘split’ by any straight line. Both properties are demonstrated by generalizing the set of isoparametric lines to a certain two-parameter family of curves. Equivalent curves are found for power, tensor-product Bernstein, exponential and triangular Bernstein polynomials. The second property will allow greater freedom when using algebraic curves for geometric modelling.

Frictional Resistance of Flat Plates in Dilute Polymer Solutions

For a flat plate moving in a dilute polymer solution, effects on boundary-layer characteristics, shear stress, drag reduction and maximum drag reduction are considered for polymers which satisfy the Meyer-Elata law. Results are derived from a model in which the velocity profiles satisfy the law-of-the-wall and a velocity-defect law, and the polymer has no effect in the range in which the latter law is valid. It is also assumed that the polymer affects the law of variation of the mixing length, and a family of velocity profiles representing this effect is adopted. This model then yields a curve of maximum drag reduction as well as a two-parameter family of curves of drag reduction, consequences of the nonoverlapping or overlapping of the two velocity-profile laws. The results are compared with those of Granville for drag reduction, and with the predicted curve of Virk-Granville and an experimental result of Levy and Davis for maximum drag reduction.

On streams of charged particles in an electromagnetic field

Problems of two kinds arise in the consideration of streams of charged particles in an electromagnetic field. In the first kind of problem (direct problems) the magnetic and electric fields are given, and it is required to study the trajectories which are possible in these fields. This type of problem reduces to the examination of the solutions of the following system of equations with partial derivatives: (v▽)v = e m E + e mc v × H .(1) The electric and magnetic fields in these equations are taken to be known solutions of the Maxwell equations ▽ × H = 4π c ϱτ , ▽ × E = 0, ▽ H = 0, ▽ E = 4 πϱ. (2) For dense streams, the effect of the space charge is taken into account (the effect of the magnetic proper field is not usually considered) and the problem is much more complicated, since equations (1) must be considered together with equations (2) in which the first equation is replaced by ▽( ϱv) = 0. (3) However, in constructing electronic instruments it is often necessary to create a stream of charged particles with given properties. In problems of this kind the stream of charged particles must be considered to be given and the solutions of Maxwell's equations (2) which correspond to such a stream must be examined. This problem is the converse of the previous problem. It has been studied in its application to electron optics in [1] and [2], where in [1] streams close to an arbitrary curve are considered, and in [2] streams close to an arbitrary surface. The problem has been studied in [3] for streans in an electric field, with the restriction that the lines of flow must be coordinate lines of some orthogonal curvilinear coordinate system. An equation is obtained which the Lamb coefficients of this coordinate system must satisfy, and is then solved. The theory of the formation of streams in an arbitrary electromagnetic field is constructed in [4], with similar restrictions. In this work a method is devised to determine the fields from the properties of the stream, given the condition that the considered stream exists. In this paper we consider the following converse problem: we are given a two-parameter family of curves; it is required to find when this family can be the lines of flow of a stream of charged particles in an electromagnetic field.

Potential Distribution between two Plane Emitting Electrodes II. Thermionic Engines

ABSTRACT This paper gives expressions for the potential distribution between two plane parallel emitting electrodes, extending the work of a previous paper (Lindsay and Parker 1959). It is shown that all potential distributions can be represented by a two-parameter family of curves, the parameters being the ratio of the electrode temperatures θ=T2/T1 and a constant A which depends on 6 and on the potentials and work functions of the two electrodes (A reduces to the parameter of Lindsay and Parker (1959) for θ=1). The results show rather clearly the relative influence of all these quantities on the position and depth of the potential minimum between the electrodes.

The invariant integral and the inverse problem in the calculus of variations

must be taken in order that the value of the integral may be an extremum. The differential equation which defines the desired curve is of the second order and its general solution is a two-parameter family of curves known as the extremals of the problem. By the successive substitution of the coordinates of the end points, P1 and P2, in the equation of the extremals we obtain two equations for the determination of the values of the parameters in terms of these coordinates. 2. The inverse problem is, as its name suggests, that of determining the form of the integrand function which has as its extremals a given twoparameter family of curves. 3. Besides the simplest problem as we have stated it there is the problem of determining the extremals of an integral,

Properties of a Certain Projectively Defined Two-Parameter Family of Curves on a General Surface

Properties of a Certain Projectively Defined Two-Parameter Family of Curves on a General Surface