Formula Manipulation Research Articles

On the integral ∫ ∞0 χ v−1 (1 + βχ) −λln mχ dχ

A recurrence relation is given for the integral in the title. Formulae which allows easy evaluation by formula manipulation on a computer are given for integer and half-integer values of v and λ. A comparison with formulae given in integral tables is made. Using results obtained, four infinite series are expressed in closed form, generalizing well-known results for infinite series of reciprocal powers. Tables of explicit expressions for the above integral for small values of m, with v = n, v = n + 1 2 , λ = l, λ = l + 1 2 , and certain values of β, are also presented.

Passing-Off, Goodwill and False Advertising: New Wine in Old Bottles

Commentators upon the English common law relating to the protection of trade symbols have frequently remarked the uncertainty concerning the true scope of the action for passing-off: its place in the scheme of English tort law has been said not to have been finally determined, and its capacity for growth not to have been exhausted. It is true that this uncertainty has not prevented greatly increased use of the action in recent years; but this is no reason to be complacent, for the judicial extensions of liability under this head owe more to the manipulation of verbal formulae than to a clarification of its foundations. Inconsequence, as passing-off has come to be employed as the vehicle for novel assertions of right-such as those of groups to a shared symbol, of foreign traders to international reputations that have preceded them to these shores, or of pop stars and other ephemera of our culture to the value of their names and likenxsesses in the marketplace—the risk of important interests being ignored, simply through failing to be recognised, has increased rather than decreased.

Fifth RIMS Conference on FORMULA MANIPULATION AND ITS APPLICATION TO MATHEMATICAL STUDY

The conference of the above title was held on October 24 to 26 in Kyoto. The conference chairman is Prof. Sin Hitotumatu, RIMS, Kyoto University, and the conference organizer is Dr. Tateaki Sasaki, The Institute of Physical and Chemical Research. Presented were twelve papers on algorithm study, application to pure and applied mathematics, etc., and system implementation. The number of attendees is 56, from the fields of pure mathematics, applied mathematics, physics, and computer science. The papers presented and their abstracts are given below.

Numerical Simulation and Experimental Verification of the German Roller Rig for Rail Vehicles*

SUMMARY A numerical simulation model of the roller test stand located at Munich and loaded by a bogie is discussed including its technical structure, the governing physical equations of motion and the structure of the simulation program. Both, the set up of the mathematical and numerical models time and the computation time of simulation runs have been considerably reduced (by a factor of 20) using formula manipulation programs. Simulation results concerning the starting behaviour of a bogie, stationary limit cycle behaviour of bogies with ideal and wear profile and influence of gauge changes and spring/damper modifications on limit cycle behaviour of a bogie are presented, some of which are compared with experimental results gained from test facility measurements. The simulation results are in good agreement with the experimental results and provide an experimental verification of the roller rig simulation model presented.

Formula manipulation in the bond graph modelling and simulation of large mechanical systems

A multibond graph element for a general single moving body is derived. A multibody system can easily be described as an interconnection of these elements. 3-D mechanical systems usually contain dependent inertias having both differential and integral causality. A method is described for the transformation of inertias with differential causality to an integral form, using formula manipulation. The program also helps to find experimentally the optimal choice for the generalized coordinates. The resulting explicit differential equation may be solved using a standard integration routine or simulation program.

Classroom experience with interactive formula manipulation

Some experiences with the use of an interactive formula manipulator in classroom situations are reported. Although a kind of script must be prepared in advance, it is not necessary to program the sessions in detail.

On representation and manipulation of mathematical formulas using general array data structure

APL2, one of the major extensions of APL, has capablities of symbol manipulation and nested data handling as well as numerical computations. This paper discusses an experimental implementation of a formula manipulation system using APL2. A simple, but easy-to-use algebraic system has been realized, primarily due to APL2's ability to define complicated data structures and algorithms. The overall design concept, representation and normalization of mathematical formulas, and various advantages of using APL2 general matrices are described.

Decomposition of systems of nonlinear algebraic equations

AbstractA new method for decomposing irreducible subsets, in the solution of systems of nonlinear algebraic equations, is presented. This method consists of two steps: (1) elimination of the nonlinearity from some of the equations by replacing nonlinear expressions by new variables; and (2) formulation of a problem of smaller dimension by tearing the linear subset of equations. It is shown that these modifications do not change considerably the convergence rate of the Newton‐Raphson and Broyden's methods while reducing the problem's dimension. Computer time reduction up to 80% is demonstrated in the examples solved. An algorithm for elimination of nonlinear expressions, which uses Boolean matrices instead of formula manipulation, is also presented.

Symbol manipulation and formula manipulation

For years, Chinese research workers have noticed that there are systems for symbol manipulation and formula manipulation abroad. But use of computers in China is later. In early years, only assemblers, ALGOL, BASIC and FORTRAN languages were used, which were not suited to help manipulate formulas. Someone has considered extending FORTRAN by manipulation functions. Many people hope to use LISP language and some formula manipulation languages.

A general precompiler for algebraic manipulation

A generalized precompiler for systems performing algebraic manipulation of Poisson series has been written. It accepts a trigonometric superset of FORTRAN IV similar to Jefferys' TRIGRUN language (Jefferys, 1972) and generates a valid FORTRAN IV program which drives an abstract formula manipulation machine. This machine is designed to be generally compatible with any manipulation system, and has been implemented with two such systems. The precompiler is written in standard FORTRAN IV and was designed to allow simple conversion for use on most computers.

On the integral ∫^{∞}₀𝑒^{-𝜇𝑡}𝑡^{𝜈-1}𝑙𝑜𝑔^{𝑚}𝑡𝑑𝑡

A recurrence relation is given for the integral in the title. Formulae which allow easy evaluation by formula manipulation are given for integer and half-integer values of ν \nu . Explicit expressions for these integrals for small values of m and ν = n \nu = n , ν = n + 1 2 \nu = n + \frac {1}{2} are also presented.

On the integral ∫₀^{𝜋/2}𝑙𝑜𝑔ⁿ𝑐𝑜𝑠𝑥𝑙𝑜𝑔^{𝑝}𝑠𝑖𝑛𝑥𝑑𝑥

A formula is derived for the integral in the title which allows easy evaluation by formula manipulation on a computer.

Closed expressions for ∫¹₀𝑡⁻¹𝑙𝑜𝑔ⁿ⁻¹𝑡log^{𝑝}(1-𝑡)𝑑𝑡

Closed expressions for the integral ∫ 0 1 t − 1 log n − 1 t log p ( 1 − t ) d t \smallint _0^1{t^{ - 1}}{\log ^{n - 1}}t{\log ^p}(1 - t)\;dt , whose general form is given elsewhere, are listed for n = 1 ( 1 ) 9 n = 1(1)9 , p = 1 ( 1 ) 9 p = 1(1)9 . A formula is derived which allows an easy evaluation of these expressions by formula manipulation on a computer.

A simple theory for the analysis of pulse trains from neutron counters

The theory of branching processes was applied to the analysis of neutron counting statistics in a previous note. A purely algebraic manipulation of the general formulae leads to a surprisingly simple structured explicit solution for the experimentally measurable quantities (moments of the probability generating function). In this note we show the essential points of this method on a practically important example.

Polynomial type Padé approximants

Some results are established giving conditions on f ( x ) f(x) so that its main diagonal Padé approximation R n ( x ) {R_n}(x) is of the form P n ( x ) / P n ( − x ) {P_n}(x)/{P_n}( - x) , where P n ( x ) {P_n}(x) is a polynomial in x of degree n. A number of applications to special functions are presented. Numerical computations are given for the gamma function using the "bignum" arithmetical facilities of formula manipulation languages REDUCE2, FORMAC.

Formula manipulation in ALGOL 68 and application to Routh's algorithm

This paper presents an implementation of Routh's algorithm and the Schur criterion, in order to determine the location of the zeros of a complex polynomial in one variable, over the ring of multivariate polynomials with integral coefficients. Both algorithms are applied to the characteristic polynomials of multistep methods. Moreover, procedures are given for the arithmetic operations +,−,*,÷ with arbitrarily long integral numbers as operands.

State-space sets, state-space graphs and n-prefix expressions

In this paper we intend to discuss some methods for solving a crucial problem in formula manipulation. This problem arises when a tree is elected -- as the best-fitting data structure - for the representation of a formula.

Higher Order Calculation of Finite Periodic Standing Gravity Waves by Means of the Computer

A computer-aided calculation is made of the standing water waves which can be produced in liquid contained between two parallel vertical plane walls. The periodic solution is obtained analytically up to the eighth order of an amplitude by the use of the formula manipulation language REDUCE-2. This result agrees with that of Penney and Price (Phil. Trans. Roy. Soc. A 244 (1952) 254) up to the fourth order, but not in the fifth order. It is clue to their algebraic error in this order. The present solution gives 0.138λ as the maximum crest height of the greatest stable standing wave, where λ is the wavelength. The wave profile from our solution is in better agreement with experiments than that from Penney and Price.

Symbol calculus by symbolic computation and semi-classical expansions

We present a survey of our work about applications of symbol calculus to quantum mechanics. Symbol calculus embodies the rules of algebra for the linear partial differential operators with variable coefficients, when such operators are expressed as functions of the non-commuting quantities xj and ~/~xj (j=l ..... n). In quantum mechanics the symbol calculus generates S~-C~sica£ expansions, i.e. perturbative series in powers of Planck's constant ~, that are both numerically useful and mathematically interesting. As the complexity of such expansions increases very rapidly with the order, algebraic computer systems have proven most useful in their study [l] . We present here a few of our results, obtained with the extensive use of the formula manipulation language REDUCE (except when specified) : in particular we have found new algorithms to solve the stationary SchrBdinger equation formally in powers of ~. Basic facts about quantum mechanics and semi-classical methods are found in [3] any textbook such as . More precise references, about symbol calculus for instance, are listed in our earlier articles [4j.~ ~

Efficient Synthesis and Implementation of Large Discrete Fourier Transformations

A systematic technique is presented for synthesizing and efficiently performing large discrete Fourier transformations (DFT’s) in the range from 60 to 5000 points. The technique is termed the mutual prime factor cyclic algorithm (MPFCA). The mutual prime factor portion of the algorithm is attributed originally to L. H. Thomas, with generalization supplied by I. J. Good; the cyclic aspect of the algorithm has recently been formalized by S. Winograd. Three methods are described for implementing the MPFCA; computational complexity (multiplications and additions) is estimated for each method and compared with the fast Fourier transform (FFT). For special purpose hardware, the MPFCA is at least twice as efficient as the FFT. A major result of importance is the realization that the three considerably different implementation methods presented lead to rather similar multiplication complexities for large size DFT’s; furthermore, the resulting multiplication complexity is considerably higher than that achieved for small size DFT’s. It is felt that further substantial improvements for large size DFT’s built up using mutually prime factors will require more general theoretical results in addition to long, tedious hours spent with computer based formula manipulation systems.