Familiar Theorem Research Articles

Injection locking of microwave solid-state oscillators

Injection locking of microwave solid-state oscillators is discussed, based on the familiar theorem that the total impedance times the current is equal to the applied voltage. Both quasi-static and dynamic analyses of the locking range, large-signal injection, locking stability, and AM and FM noise are given, and recent experimental work is reviewed briefly. No applications of injection locking are discussed in detail.

Two theorems about projective sets

In this paper we prove two (rather unrelated) theorems about projective sets. The first one asserts that subsets of ℵ_1 which are ∑^1_2 in the codes are constructible; thus it extends the familiar theorem of Shoenfield that ∑^1_2 subsets of ω are constructible. The second is concerned with largest countable ∑^1_(2n) sets and establishes their existence under the hypothesis of Projective Determinacy and the assumption that there exist only countably many ordinal definable reals.

A program for computing identity-kernels from certain directed graphs

A directed graph may be represented in LISP as a list of lists of the form (X X1X2 ...XN) whereX,X1, ...,XN are vertices of the graph and there is an arc fromX to each ofX1, ...,XN. LISP functions are described which formally evaluate the number of spanning trees of certain directed graphs in the above representation. The expressions so derived, the so-called “identity-kernels”, may be convoluted to form binomial or multinomial integral identities which are, in a sense, generalizations of the familiar Binomial and Multinomial Theorems.

Indirect abelian theorems and a linear Volterra equation

where k and xo are real, c _ 0, and a(t) satisfies (H1) a(t) E C(O, oo) n L,(O, 1). a(t) is nonnegative and nonincreasing, limt, , a(t) =0, and O<a(O+)<oo; (H2) a(t) is convex downward; i.e., for 0<8< 1 and 0< tl < t3 <cc, ea(tl) + (1 -e)a(t3) a(t2), where t2 = etl + (1 -e)t3. By a familiar theorem on Volterra equations, (1.1) has a unique solution in C'[O, oo). We define u(t) as the solution of (1.1) with k=O, xo= 1, and we let w(t) = ft u(Q) dJ. It is easily checked that the solution of (1.1) is given by xou(t) + kw(t). Treating (1.1) as a special case of a nonlinear equation, Levin proved in [5] that if a(t) E C [0, oo), a(t) X a(O), and (-1)ka(k)(t) _ O for O < t < oo, k =O, 1, 2, 3, then

A Geometric Application of f(n)=n/(n+1)

Introduction. The elementary geometric theorem which states that the diagonals of a parallelogram bisect each other is well known. It can, however, be placed in a much broader framework providing an interesting example of the limit of the real function f(n)n = n/(n + 1). The purpose of this paper is to develop the essentials utilizing the notion of points of division. Preliminaries. In the figure below, the ratio in which C divides DC is undefined (oo), and divides diagonal DB in the ratio 1/1. This is the familiar theorem on the diagonals of a parallelogram. It can easily be proved that if is the midpoint of DC, i.e., divides DC in the ratio 1/1), then divides diagonal DBT in the ratio 1/2 (i.e., is a point of trisection). Similarly, if E divides DC in the ratio -3/1, then F divides DB7 in the ratio 3/2. And if E' divides D-C in the ratio 1/5, then F' divides D?B (externally) in the ratio 1/4. G~~ B

A generalization of the Mikusinski operational calculus

0. Introduction. In his version of the operational calculus Mikusinski uses as a starting point the familiar theorem from algebra tlhat every integral domain can be embedded isomorphically in a field. He shows that the class of complex-valued continuous functions defined on [0, cO) forms an integral domain when addition and multiplication are taken to be pointwise addition and convolution of functions, respectively. The resulting field is called the field of Mikusinski operators. In the present paper we generalize Mikusinski's basic results by developing an operational calculus for strongly continuous functions defined on [0, cc) and having values in any Banach algebra with unit element. The Mikusinski operational calculus will then be a special case of our theory, obtained by taking the Banach algebra to be the complex numbers. The paper is divided into four parts. The first gives an algebraic construction; the second shows how the algebraic construction may be used to develop the operational calculus mentioned above; the third indicates some basic results which carry over from the (Mikusinski) operational calculus to this more general setting. Finally, in the last section, we generalize within the framework of the operational calculus a recent theorem of T. K. Boehme. We assume the reader is familiar with the basic notations and results in the books by Mikusinski [3] and Erdelyi [2].

Quantization of Fields with Infinite-Dimensional Invariance Groups. III. Generalized Schwinger-Feynman Theory

The formal methods of Schwinger and Feynman are applied to nonlinear field theories having elementary vertex functions of arbitrarily high order. In the first half of the paper, familiar theorems are rederived by noncanonical methods. Emphasis is given to purely formal aspects of the theory which may be expected to survive generalization to situations in which standard asymptotic conditions are inapplicable. Since the context in which the field nonlinearities are assumed to appear is that of a non-Abelian infinite-dimensional invariance group, detailed attention is given to the question of a group invariant measure for the Feynman functional integral. It is shown that the physically important part of the measure is not determined by the group. The second half of the paper is devoted to the theory of the propagators and correlation functions which characterize the system when invariant variables are introduced. The existence of a c-number action functional Γ which contains a complete description of all quantum processes is proved. The second variational derivatives of this functional constitute the wave operator for the one-particle propagators (including all radiative corrections), and its higher derivatives are the renormalized vertex functions. A description of the renormalization process is easily carried out in terms of Γ. Finally, the implications which its existence has for quantum gravidynamics are discussed. Because it leads to nonlocal covariant equations for a complex metric tensor the way is open to transmutations of topology at the quantum level.

Intuition, geometry and physics in relativity

A familiar theorem about chords of a circle has an intersting analogue in Minkowskian geometry, the circle being replaced by a null cone. In curved space-time the theorem is not true, and the extent to which it fails provides (with the help of the world-function) a measure of the curvature. The results are interpreted physically in terms of times measured by clooks carried by free particles.

Theories of Consumer's Behaviour: A Cynical View

Since the popularization in the Eighteen Seventies of the concept of marginal utility, theories of consumer's behaviour have come to occupy one of the central positions in that corpus of economic analysis which, before the war at any rate, was customarily referred to as the theory of value. It is surely a tribute to the influence of a powerful tradition raised on the authority of such names as Jevons, Marshall and Edgeworth, and, more recently, Hicks and Samuelson, to mention only those in the English-speaking assemblage, that although frequently questioned and sometimes challenged, theories of consumer's behaviour continue to form a necessary part of the syllabus of all students specializing in economics. And if with the older techniques, marginal utility and indifference-preference fields, the interest in such theories eventually appeared to languish, the invention of revealed preference has revived our spirits wonderfully by providing us with a rather impressive technique in which to express familiar theorems and in which to carry on the staple controversies of old. I must, therefore, reconcile myself to being regarded as something of a killjoy when I propose that we abandon all this; that we seek to dislodge such theorems from their hold on our imagination and to face the fact that though fascinating in themselves they form an unnecessary adornment on the apparatus of economic analysis. Recognizing the strong intellectual vested interest in the current doctrines I cannot, of course, expect the reader to fall in with these proposals unless cogent reasons are adduced for them. To that extent this paper constitutes an essay in persuasion. My aim is to convince the reader that, after all the display of technical virtuosity associated with such theorems, there is nothing the practising economist can take away with him to help him come to grips with the complexity of the real world. Indeed, he would be no worse off if he remained ignorant of all theories of consumer's behaviour, accepting the obviously indispensable Law of Demand on trust2.

Cartesian Geometry of the Triangle and Hexagon

The present paper is a sequel to two others which have appeared in the Gazette (July, 1941 and December, 1957) and in which the possibility of attacking a number of familiar triangle theorems by cartesian methods was successfully investigated. The same method is here applied to Morley’s and Pascal’s Theorems.

Matrix Elements of Symmetric Operators

In this note we discuss an extension of a familiar theorem concerning the matrix elements of vector and tensor operators between states of definite angular momentum to the case of other groups than the full rotation group and to the case of operators of other symmetries. It is found that there is an extension to this theorem but in some cases there are nontrivial complications. In addition, a method is given for reducing a reducible representation of a group which gives explicitly the matrix elements in the appropriate similarity transformation.

Liouville's Theorem and the Intensity of Beams

The meaning of Liouville's theorem on the flow of points in phase space is discussed. The intensity for particles or for radiation is defined in the most general case. It is shown that the familiar theorem which states that an optical instrument cannot change the brightness of any object which it images is a special case of Liouville's theorem. The generalization of this result to classical particle- and radiation-handling systems of many kinds is discussed. The result holds even for situations in which the fields or refractive index distributions are arbitrarily varying in time, though in such situations the trajectories of “constant brightness” may move around by bending and distorting so that a stationary observer may thereby see some changes. The problem of point sources is considered. Simple examples of phase space motions are exhibited.

Orthogonal Arrays of Index Unity

In this paper we shall proceed to generalize the notion of a set of orthogonal Latin squares, and we term this extension an orthogonal array of index unity. In Section 2 we secure bounds for the number of constraints which are the counterpart of the familiar theorem which states that the number of mutually orthogonal Latin squares of side $s$ is bounded above by $s - 1$. Curiously, our bound depends upon whether $s$ is odd or even. In Section 3 we give a method of constructing these arrays by considering a class of polynomials with coefficients in the finite Galois field $GF(s)$, where $s$ is a prime or a power of a prime. In the concluding section we give a brief discussion of designs based on these arrays.

On Converse Theorems of Summability

This paper, while it does not merit the attention of workers in the field of summability, may be of some interest to readers of the Gazette who are using Bromwich's classical work on infinite series as a textbook. Its main purpose is to show that Bromwich's proof of the Hardy-Landau converse of Cauchy's limit theorem can be made to yield, without any modification of the underlying idea (due to A. E. Jolliffe), all the familiar converse theorems on Rieszian summability of the first order. The paper also shows that after having obtained these theorems, one can pass on, by a natural transition suggested by Szász, to the corresponding results in the theory of summability by Dirichlet's series

A proof of the “Theorem of the Means.”

Numerous proofs have been given of this familiar theorem, which states that if a1, a2, …, an are positive, and not all equal, thenThe following is an elementary proof by induction, which I have not seen used before. It is, of course, not claimed to be novel, and not likely to be so.

The Concurrence of Triads of Common Tangents to Three Circles

“If pairs of common tangents be drawn to three circles, and if one triad of common tangents be concurrent, the other triad will also be concurrent.”This is Casey’s enunciation * of a familiar theorem; Salmon † has the same unqualified assertion, and appends a footnote: “This principle is employed by Steiner in his solution of Malfatti’s problem…. [His] construction is: Inscribe circles in the triangles formed by each side of the triangle and the two adjacent bisectors of angles; these circles having three common tangents meeting in a point will have three other common tangents meeting in a point, and these are common tangents to the circles required.”

On the Roots of a Derivative of a Rational Function

This paper extends some familiar theorems concerning the relations between the roots of a polynomial and those of its first derivative to the more general case of the rational function with a pole at a single point.

On a familiar theorem of the theory of functions

On a familiar theorem of the theory of functions