Parallelogram Law Research Articles

Some theorems on Fourier coefficients

where P is given by (1.1)? If one allows the coefficients Cn to be complex numbers of absolute value 1, an affirmative answer to the question is furnished by the partial sums of the series ZE inlogneinO; this example is due to Hardy and Littlewood [4, pp. 116-118]. A theorem of Salem and Zygmund [2, pp. 270, 278] shows, roughly speaking, that (N log N)12 is the most probable order of magnitude for ||P||X if C-n = ? 1. During the summer of 1958, Salem drew my attention to the problem stated in the first paragraph. It turns out that an affirmative answer can be given by a construction which uses nothing more sophisticated than the parallelogram law

Pythagorean Orthogonality in a Normed Linear Space

This note presents a proof of the following proposition:Theorem. If Pythagorean orthogonality is homogeneous in a normed linear space T then T is an abstract Euclidean space.The theorem was originally stated and proved by R. C. James ([1], Theorem 5. 2) who systematically discusses various characterisations of a Euclidean space in terms of concepts of orthogonality. I came across the result independently and the proof which I constructed is a simplified version of that of James. The hypothesis of the theorem may be stated in the form:Since a normed linear space is known to be Euclidean if the parallelogram law:is valid throughout the space (see [2]), it is evidently sufficient to show that (l) implies (2).

The Concept of Force in Dislocation Theory

The concept of ``force acting on a dislocation'' (or other imperfection) is intrinsically different from the concept of force as used in newtonian mechanics, and must be defined. A definition is given which preserves the basic property (1) that the direction of the resultant force gives the direction in which the imperfection will actually move if the atomic constraints do not prevent motion. For imperfections in an infinite crystal, and assuming linear elasticity, two further properties are established: (2) the parallelogram law of adding forces is valid, and (3) Newton's law of equal and opposite action and reaction is valid. A critical review of previous work relating to forces between crystal imperfections is given.

Blind guidance by ultrasonics: (Frank H. Slaymaker and Willard F. Meeker, Electronics, May, 1948)

Some physical and computational aspects related to the intuitive notion of time in its connection with natural and technological processes are considered. The phenomenon of finite speed of information transmittal relative to measurement and computation is analyzed. It is argued that this phenomenon creates inevitable and irreversible time delays (uncertainties) that affect all measurements and computations, makes the exact synchronization of clocks impossible, shifts our knowledge to the past, and limits the accuracy of experiments. Some past experiments for the measurement of the speed of light are revisited, the possibility of a finite time of mirror reflection is discussed, and a stand for the experimental measurement of time spent in mirror reflection is proposed by a modification of Fizeau experiments. The positive orientation of the flow of time in its relation to the mathematical concept of time derivative is considered. It is demonstrated that right time-derivatives normally used to describe physical processes actually set forth non-causal representations of physical realities and may severely restrict the possibility of control and optimization in real life systems. The use of (causal) left time-derivatives produced by measurements and computations and consideration of variable masses lead to new representations of the second Newton’s law of motion where forces may contain controls depending on accelerations and higher order left time-derivatives of velocity. Such controls are actually required in jet-propelled space vehicles with variable masses, as demonstrated in the space shuttle example. The parallelogram rule does not apply to forces depending on higher order derivatives, so the concept of effective forces is considered within original Newton’s representation in which effective forces can be recovered in the process of integration; thereby the parallelogram law stays intact for effective forces. Consideration of forces with left higher order time derivatives alters classical methods in mechanics that were developed on the basis of absolute time and assuming no higher order derivatives in the forces of Newtonian equations of motion. Inclusion of such forces and consideration of natural time delays and time orientation opens new avenues for investigation and control of processes in physics, economy, medicine and mechanics. It is demonstrated that Lagrange and Hamilton equations stay intact in some generalized forms which forms can be used to derive higher order dynamical equations that exclude geometric constraints, thus having the minimum number of independent generalized coordinates.

The Graphical Equivalent of the Principle of Moments

Statics is founded upon two outstanding principles the Parallelogram of Forces and the Principle of Moments. The Parallelogram Law is universally acknowledged as fundamental and is given its rightful place in both analysis and graphics; in the former it furnishes two equations connecting the forces keeping a body at rest while the Principle of Moments furnishes a third equation. In graphical statics the Funicular Polygon is used as the equivalent of the third equation derived from the Principle of Moments.

Die technische Mechanik: elementares Lehrbuch für mittlere maschienentechnische Fachschulen und Hilfsbuch für studierende höherer technischer Lehranstalten

IN the very early part of this excellent work there is a certain lack of system, inasmuch as, although the author very properly treats first of the equilibrium of a particle, he assumes the nature of the stress exerted in such rigid bodies as the bars of a framework, the crank and connecting rod of an engine, &c. The nature of such forces is never properly appreciated by the student who is truly a beginner in the subject of dynamics—and, indeed, there is no part of statics in which students of even very considerable experience are so apt to go wrong as that relating to the forces exerted by jointed bars. The author treats from the outset the equilibrium of forces acting in space of three dimensions without having previously disposed of the simpler two dimensional case, a course which meets with the approval of many teachers, although it seems to the reviewer to be the less simple method. Herr Stephan enunciates the parallelogram law for the composition of forces (or vectors generally) at the outset, and assumes it as a result of experiment—which, on the whole, is perhaps the wisest plan for a teacher. Near the end of the book, however, he gives the ordinary Newtonian proof of the proposition. Die technische Mechanik: elementares Lehrbuch für mittlere maschienentechnische Fachschulen und Hilfsbuch für studierende höherer technischer Lehranstalten. By P. Stephan, &c. Erster Teil: Mechanik Starrer Körper. Pp. viii + 344. (Leipzig: Teubner, 1904.) Price 7 marks.

Etude sur les Quantités mathématiques Grandeurs dirigées, Quaternions Introduction to Quaternions Bibliography of Quaternions and Allied Systems of Mathematics

IT may perhaps be rather an exaggerated statement, but it is none the less to a great extent true, that mathematicians tend to divide themselves into two classes, quaternionists and non quaternionists, and that these two classes frequently become involved in polemical controversies. But at the present time the notion of vector quantities is of frequent occurrence in physics, and it is important that everyone should have an opportunity of understanding the laws and nature of vector operations. It is not unfrequently stated that forces are vectors, because they are directed quantities, and therefore they are compounded by the parallelogram law. But the moment of inertia of a body about a line is also a directed quantity in the sense that its magnitude depends on the direction of the line, although moments of inertia, as every mathematician knows, are not compounded by the parallelogram law. Clearly dogmatic statements about vectors are dangerous for teaching purposes.

Vectors and Quaternions

PROF. MACFARLANE claims that his “fundamental rules for vectors are based on physical considerations, the principal one of which is that the square of a vector is essentially positive.” His proof is virtually this:—The expression for the kinetic energy (½ mv2) is an essentially positive quantity. It contains one factor ½m evidently positive. Hence the other factor v2 must also be positive. “But v denotes the velocity which is a directed quantity.” Unfortunately for this argument v does not denote the velocity in its complete conception—it simply measures the speed. The physicist may think of velocity as being a vector quantity; but in ordinary analysis the vector is not symbolised. We deal only with tensors and scalars. It would be well, I think, if the strict meaning of vector were clearly borne in mind. A vector is a directed line in space, and may be used to symbolise all physical quantities which can be compounded according to the well-known parallelogram law. Displacement is perhaps the simplest conception that can be so symbolised. Velocities, concurrent forces, couples, &c, are in the same sense vector quantities. Now it can be proved rigorously that quadrantal versors are compounded according to this very addition law. On what grounds, then, are they refused admittance to the order of vectors? If a vector cannot be a versor in product combinations, what is the significance of the equation ij = k? Regarding this Dr. Macfarlane vouchsafes no remark, save that it is possible to get along without its use. As he himself has not done so, such a possibility lies altogether outside our consideration. Again, I fail to see what “physical considerations” have to do with mathematics of the fourth dimension.