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Three mathematical procedures were devised which allowed for the interconversion and interpretation of protein—drug binding data in the literature. A minicomputer program enabled the calculation of the fraction of drug bound for a specified total drug and protein concentration, given the protein binding constants and the number of sites. The corollary problem of finding the total drug concentration for a particular fraction drug bound was included. In addition, the concept of partial fraction bound for binding systems with multiple classes of sites was introduced and a formula was derived which made possible the calculation of this value from the usual binding constants. Finally, an equation was derived which allowed the total drug concentration and the fraction bound to be obtained from the binding information for selected r values. These were obtained from the Scatchard plot and the associated binding constants. Examples of the utility of these procedures are supplied based on data for human serum albumin binding of four drugs of clinical interest. Evidence is given to show the prominence of the contribution by the secondary sites to the total binding at therapeutic levels of the drugs.

Letp, q, u, andvbe any four positive integers, and let further δ be a number in the interval 0 < δ ≤ 2. In this note an effective lower bound forqwill be obtained which insures thatIn the special case whenu=v= 1, it was shown by J. Popken,Math. Z. 29 (1929), 525–541, thatHerecandCare two positive absolute constants which, however, were not determined explicity. A similarly non-effective result was given in my paper,J. reine angew. Math. 166 (1932), 118–150.The method of this note depends again on the classical formulae by Hermite which I applied alsoop. cit.

An algorithm is developed to derive the partial fraction expansion of the ratio of two polynomials. A unified approach using tables is presented in order to solve both single and multiple pole cases.

Brugia's work on the noniterative computation of high order derivatives of rational functions with application to multiple-pole fraction expansion is simplified by the use of operators. The formulas for the derivatives are given in a recursive scheme and finally the Laurent expansion is developed and applied to the partial fraction expansion of improper rational functions.

The vector integro-differential equation d d t (X - X*) = p exp (−pτ)KT(−D−1/2 K√) x (X(t-τ)-X*) dτ where T(x) = exp (x2) erfe (x), accounts for transport-transformation interactions in mulitcomponent suspended particle systems and represents the operation of batch processes. Its solution is given for general values of the parameters p, D and K using the method of matrix partial fraction expansions. The approximatic d d t (X0-X*) = K(I-p−1/2D)−1K(X0-X*) is investigated and shown to give useful estimates of the sensitivity of the selectivity of a chemical network to the parameters controlling transport conditions in sluny systems. Predictions for two chemical networks are given. If , we show that the paths arising in the small p (diffusion controlled) and large p (chemically controlled) modes of operation bound a large fraction of the state space and thereby reflect a highly mixing- sensitive selectivity. If 4 ⇌ 1, 2 ⇌s 3 and if components 2 and 3 are dilute, we show that via diffusive interactions the secondary step can either accelerate or inhibit the primary step.

This paper presents a method of network synthesis for a prescribed unit-impulse response f(t) , taken to be zero outside the interval (0, t_{0}) . The imposed response f(t) is approximated by another one h(t) , whose Laplace transform H(s) is always physically realizable as the transfer function of a passive network with lumped parameters. A finite number of equidistant values of the specified function f(t) is used for the approximation. The pulse transfer function that has been used in elaborating the method no longer appears in the final design relations. The approximation criterion is the mean-square error in the interval (0, t_{0}) , and outside this interval h(t) is smaller than some imposed value. The transfer function H(s) results are expanded into partial fractions. The necessary computations are simple. As a particular case, the method permits the determination of the response h(t) in such a way, that it takes certain given values at regular intervals of time, thus making the method also useful for sampled-data systems. Illustrative examples and experimental results are given.

The powerful Laplace transform method for transient analysis by the partial fraction expansion technique has become quite popular in the fields of circuit and servo design. The theory of residues is usually used to find the coefficients of these fractions. The process is quite simple until second and higher order poles are included in the denominator. Previously, this has required a return to the calculus to find the additional coefficients required. This paper discloses a simple technique for finding these additional coefficients by algebraic processes. As a result both manual and machine computation can be performed more easily. The technique is described and its mathematical basis is rigorously proven.

Using Newton-type corrections, a simultaneous improvement of a complete set of approximate factors of arbitrary degree of a polynomial is developed, which can form the basis for an iterative method of factoring the polynomial. The simultaneous corrections for a given step are a generalization of a method introduced by Weierstrass [5] in the case of all linear factors. The present paper includes, therefore, an independent derivation of the Weierstrass formulas. From a different point of view the resulting iterative method is a generalization of a method of Samelson [3], recently analyzed by Stewart [4]. Basic properties of the iteration include the fact that the correction terms are related to a certain partial fraction decomposition. Emphasis is placed on the case of approximate quadratic factors, where formulas for the corrections are derived suitable for use in computation with the practically important case of a polynomial with real coefficients whose zeros are not necessarily all real. The formulas so...

This paper presents some very simplified general treatments which will allow workers to derive equations for any linear mammillary compartment model with any first- or zero-order or impulse input process. This is done through the use of general input and disposition functions, the method of partial fractions for solving Laplace transforms, and the use of a multiple-dosing function. A disposition function defines the model necessary to describe accurately drug body concentrations after the drug has entered the blood circulation. A general equation is presented to describe the disposition function in Laplace operators for the central compartment of a linear mammillary model having n driving force compartments with elimination occurring from any of the compartments. Input functions describing intravenous bolus injection, zero-order infusion, or GI absorption and first-order processes for oral dosing and intramuscular injection are presented. The Laplace transform for the amount of drug in the central compartment is given by the products of the input and disposition functions. These Laplace transforms may usually be solved in one step using the method of partial fractions, without the necessity of referring to extensive tables. By using a multiple-dosing function, general equations may be derived covering the entire dosage regimen of a drug in a patient.

If f(z) is analytic in a suitable domain, it is shown how the integral of f(α1x1+…+αnxn) over the interior of a simplex may be reduced to the evaluation of a contour integral, in fact to an exercise in partial fractions.The contour integral is expressed in two ways, according as the simplex is given in terms of its vertices or faces.

A. Baker's theorems on linear forms in the logarithms of algebraic numbers imply information on the arithmetic nature of definite integrals of rational functions. This paper pro vides a convenient formulation of these implied results. 1. In his monograph on transcendental numbers, C. Siegel [3, p. 97] observed that the theory of transcendental numbers at that time (1949) was as yet unable to determine the arithmetic nature of such integrals as rl dx I / r\ Jo +X = _ 3 log2++/) More recently (1966), A. Baker mentions that his theorems on linear forms in the logarithms of algebraic numbers [1] imply the transcendence of the above integral. Baker's results have led to further "folk theorems" on the arithmetic nature of certain integrals but no general statement appears to have been formulated. In this paper we establish a convenient criterion for determining the arithmetic nature of such definite integrals of rational functions as we are able to deal with by virtue of currently established theorems. THEOREM. Let P(z), Q(z) denote polynomials with algebraic coefficients and with no common polynomnialfactor. Denote by a1, a2, an the distinct zeros of Q(z) and by ri, r2, * * *, rn the residues respectively at the poles of the rational function P(z)/Q(z). Further let P be some contour in the complex plane for which the definite integral (' P(z) dz r Q(z) exists, and suppose that r is either closed, or has endpoints which are algebraic or infinite. Received by the editors February 20, 1970 and, in revised form, October 5, 1970. AMS 1969 subject classifications. Primary 1032.

In the present paper two problems on approximation by rational functions will be treated. The one concerns rational functions whose poles are of any order but lie at two preassigned points. The other problem relates to rational functions which have simple poles only.

In my former article on this subject in THE MATHEMATICS TEACHER of June, 1914, I did not bring out one point, which, on further consideration, seems to me to be the key to this process. Dr. G. F. Metzler’s excellent article in the issue of June, 1915, further emphasizes this need, for while he uses the point of which I speak in various ways, he does not bring the whole question to this focus.

(1916). XV. Note on the partial fraction problem. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science: Vol. 31, No. 182, pp. 108-111.

The transient response of a linear lumped-constant time-invariant network, which is excited by a driving force which has the ratio of two polynomials in the complex frequency p for its Laplace transform, can be tabulated easily using a simple linear difference equation. It is not necessary to know the location or multiplicity of the roots of the denominator, to expand in partial fractions, to use complex variables or contour integrals, or even to find the equation for the response. Any desired accuracy can be obtained easily. The new method uses simple algebra and real numbers only. It can be used to generate any mathematical function which has the ratio of two polynomials in p for its Laplace transform.

The partial-fractions expansion of a function F(s)/(s-a)m, m > 1, involves the computation of m coefficients, namely (1 /i!)(diF(a)/dsi), 0 ≤ i ≤ m-1. Wehrhahn [1] and Karni [3] have provided a method for computing these coefficients algebraically. A new approach is taken here which involves approximating a multiple pole by neighboring simple poles. The theory developed turns out to have a very interesting resemblance to the FFT algorithm. The algorithm is illustrated by several examples. Typical applications are finding the inverse Laplace transform of a function having multiple poles and the evaluation of higher order derivatives of an arbitrary function H(z) at some arbitrary z=z0.

For the two and three layer cases geo-electrical sounding graphs can be rapidly and accurately evaluated by comparing them with an adequate set of standard model graphs. The variety of model graphs required is reasonably limited and the use of a computer is unnecessary for this type of interpretation. For more than three layers a compilation of model graphs is not possible, because the variety of curves required in practice increases immensely. To evaluate a measured graph under these conditions, a model graph is calculated by computer for an approximately calculated resistivity profile which is determined, for example, by means of the auxiliary point methods. This model graph is then compared with the measured curve, and from the deviation between the curves a new resistivity profile is derived, the model graph of which is calculated for another comparison procedure, etc. This type of interpretation, although exact, is very inconvenient and time-consuming, because there is no simple method by which an improved resistivity profile can be derived from the deviations between a model graph and a measured graph. The aim of this paper is, on the one hand, to give a simple interpretation method, suitable for use during field work, for multi-layer geo-electrical sounding graphs, and, on the other hand, to indicate an automatic evaluation procedure based on these principles, suitable for use by digital computer. This interpretation system is based on the resolution of the kernel function of Stefanescu's integral into partial fractions. The system consists of a calculation method for an arbitrary multi-layer case and a highly accurate approximation method for determining those partial fractions which are important for interpretation. The partial fractions are found by fitting three-layer graphs to a measured curve. Using the roots and coefficients of these partial fractions and simple equations derived from the kernel function of Stefanescu's integral, the thicknesses and resistivities of layers may be directly calculated for successively increasing depths. The system also provides a simple method for the approximative construction of model graphs.

This paper presents in terms of a three-dimensional surface a technique for a visual as well as mathematical treatment of broad generality of the RL, RC and LC one-ports. An examination of the partial fraction terms of all the two-element series and parallel components which make up the canonic one-ports indicates that seven different types of terms are all that occur. The real and imaginary parts of all such functions for s = ? + j? may be represented geometrically by a surface in three-space. It is found that all the real and imaginary part functions are geometrically 1) a plane parallel or inclined to Aar the a, o plane; 2) the basic surface, given by 2 + 2' A > O, located in various positions with respect to the axes; or 3) a combination of the plane and basic surface. The driving function of any two-element-kind one-port is the sum of some combination of the real and imaginary part functions and so is represented by a sum of some combination of the canonic surfaces.

In the course of the past 25 years Heaviside's direct operational method of solving differential equations has, to a large extent, been replaced by solutions involving functional transformations. While mathematically sound this trend has resulted in placing a powerful analytical tool far above the mathematical equipment of the average engineer. With the exception of the celebrated expansion theorem or the method of partial fractions no direct method appears to have been discovered by which the average engineer or engineering student could solve even relatively simple differential equations by Heaviside's calculus. Several expansion theorems have been developed1,2,3,4 to cover cases where repeated roots are present but none of these have been set up in a form which could be used by undergraduates. It does not appear that any direct general procedure has ever been suggested to cover the usual type of asymptotic solutions which appear in problems in electrical engineering.

This paper presents a complete solution for the optimum linear system which operates on n stationary and correlated random processes so as to minimize error variance in filtering or prediction. A simple closed-form answer results if the matrix \Phi(s) of spectra of the input signals can be factored such that \Phi(s) = G(-s)G^{T}(s) where G(s) and G^{1}(s) represent matrices of stable transforms in the Laplace variables. A general factoring procedure for rational matrices is presented. G(s) can be viewed as the system which would reproduce signals with the spectrum of \Phi(s) when excited by n uncorrelated unit-density white-noise sources. In the case of a multidimensional filter, when G(s) is separated by partial fractions into two terms, S(s) + N(s) , having 1hp poles from the signal and noise spectra, respectively, the optimum unity-feedback filter is shown to have a forward-loop transference of S(s)N^{-1}(s) .

I believe the following method of decomposition will commend itself to any who will use it enough to become familiar with its application.

The purpose of this paper is to evaluate the displacement of the zeros pk0 of the impedance Z(p) due to the variations ?z in the impedance of any network element. It is shown that the displacement is approximately proportional to the mth root of (-?z) evaluated at p = pk0, where m is the multiplicity of pk0, and the proportionality factor equals the coefficient Ak of (p-pk0)-m in the partial fraction expansion of 1/Z(p).

Die Methode gestattet die Approximation komplexer Funktionen F(z), die durch ein Integral oder einen Quotienten ϕ(z)/Ψ(z) definiert werden. Der Integrand muβ bestimmten Differenzierbarkeitsbedingungen genugen, die Funktionen ϕ and Ψ mussen gewissen Differentialgeichungen zugeordnet sein. Als einfaches Beispiel wird die Entwicklung von Quotienten elementar transzendenter Funktionen behandelt. Als Sonderfalle treten bekannte Potenzreihen auf, jedoch liefert das hier dargestellte Verfahren bei gleichem Rechenaufwand um mehrere Stellen genauere Ergebnisse und bleibt sowohl in der Nahe von Singularitaten als auch auserhalb des Konvergenzkreises der Potenzreihen numerisch brauchbar. Die Methode fuhrt zugleich auf Verallgemeinerungen bekannter Partialbruchzerlegungen meromorpher Funktionen wie z/sin z. The method allows for the approximation of two classes of complex functions F(z), which have to be defined either by an integral or a quotient ϕ(z)/Ψ(z). The class of integrands is restricted by conditions of differentiability whereas the functions ϕ and Ψ have to be related to a class of differential equations. As example the expansion of quotients of elementary transcendental functions is shown in some detail. The formulae contain well-known power series as special cases, but they yield numerical results of superior accuracy remaining useful too near singularities and outside the circle of convergence of the related power series. Besides the method leads to generalizations of classical relations expanding meromorphic functions as z/sin z by means of a sum of partial fractions.

Let F(z) be a rational function of z which is regular at z = 0 and so possesses a convergent power seriesThe problem arises of characterizing those rational functions F(z) that have infinitely many vanishing Taylor coefficientsfh. After earlier and more special results by Siegel(2) and Ward(4) I applied in 1934(1) a p-adic method due to Skolem(3) to the problem and obtained the following partial solution.

For the class of potentials vanishing beyond a certain point we have derived a new dispersion formula in which the scattering matrix S(k) is written as the ratio of two conditionally convergent partial fraction sums. The numerator displays the poles of S(k) at the complex eigenvalues kn and the denominator displays poles at -kn, the zeros of S(k). The expression is manifestly unitary and remains so when an approximation depending only on a finite number of poles is made. Besides the S-wave theory, higher-wave scattering is treated, and among particular illustrations is a generalization to higher waves of the effective range approximation.

The paper shows that the calculation of transients by Tricomi's method using Laguerre functions is a practical alternative to the use of partial fractions. By taking numerical examples of Laplace transforms ranging from quadratic to sixth-power polynomials it shows that the approximation offered by this method would be acceptable for many practical uses. It analyses the composition of the coefficients of the Laguerre series, thus disclosing the conditions for quick convergence; for some transforms these conditions are not attainable.

Cauchy's Integral for the function and region (if the Jordan curves bounding R are rectifiable) in Theorem II splits up the function /(z) into n functions. If Theorem I is applied to each of these in turn (a simple transformation w = l/(z —a,) is first necessary for « —1 of them), we find Theorem II immediately. If the Jordan curves bounding R are not rectifiable, we readily obtain the same result. Consider an auxiliary variable region R' of the same connectivity as R, which lies within R, which is bounded by n rectifiable Jordan curves, and which approaches R as the bounding curves change continuously. For the region R', Cauchy's Integral for/(z) splits that function into « functions which do not change as R' varies. Each of these « functions is, when properly defined, analytic not merely in R but in a region bounded by a single Jordan curve and which contains R. Theorem II then follows as before from Theorem I. On the possibility of approximation of arbitrary functions by polynomials, no deeper lying results than Theorems I and II, except for approximation by certain restricted types of polynomials, appear to have been found until

Es wird ein Algorithmus zur unvollstandigen Zerlegung einer rationalen Funktion in Partialbruche angegeben, mit dem alle auftretenden Sonderfalle in einheitlicher und okonomischer Weise behandelt werden konnen. Der zu einem quadratischen Nennerfaktor einer reellen rationalen Funktion gehorige Partialbruch kann ohne Kenntnis der Nullstellen und ohne Koeffizientenvergleich in reeller Arithmetik berechnet werden.

A simple expansion method, based on partial fractions of a special type, is described for the construction of bi-variate polynomials of interpolation. A distinctive feature is the choice of a system of straight lines, no three of which are concurrent, in order to define the points for interpolation. The expansion formula, which contains considerably fewer terms than others in general use, can be extended to more variables. The applications considered relate to the expansion and solution of determinantal equations of a type frequently encountered in investigations on the flutter and stability of aeroplanes. Reference is made to the possible use of a d.c. network analyser, based on an electrical interpretation of the expansion formulae, as an aid to rapid solution.

It has always seemed to the author of this paper that there should be some extension of the method of solving partial fractions by means of substitution, so that this method would be applicable to all cases without introducing any undue complexity. After several abortive attempts along the line of substitutions, the following method by the equivalent process of taking remainders was found.

The well-known technique of the Heaviside partial fraction expansion is generalized and extended to the state transition matrix determination. The new approach is more convenient than the direct inversion of a square matrix, on one hand, and is much simpler than Sylvester's expansion method, on the other. Two examples (one is a system with distinct roots and one with multiple roots) are included for illustrating the procedures and showing the simplicity of the method.

The paper describes a method for obtaining the derivative of a certain desired order of a proper or improper rational function without working out any of its lower order derivatives.This method is then applied to compute the partial fraction expansion coefficients of a rational function with high order poles.The outlined procedure is suitable both for hand and computer calculation.

amixi + am2X2 + * + amrXr = Um in r>m unknowns, having a row-regular coefficient matrix: (2) Rank A = m = number of equations. We are concerned with the problem of how to investigate the consistency of an arbitrarily given single linear equation (3) am+1,lxl + am+1,2x2 + + am+,,rxr = Um+l with respect to (1). The method mostly used in practice is the Gaussian process of successive eliminations. Thus we use the m equations (1) as pivotal equations in order to eliminate m unknowns in (3) to give a reduced equation

The rotational energy of a link in a mechanism is proportional to the square of its angular velocity. This ‘square’, in contrast to the angular velocity itself, can be expanded into a Fourier series, having finite values for its coefficients. The problem is solved by splitting it into partial fractions, leading to the harmonic analysis of the slotted lever mechanism. Further related series are given at the end.

The known electrical network design methods of partial fraction expansion and synthetic division are used to synthesize translational and rotational mechanical systems having specified, realizable, driving point mechanical impedances or mobilities. The systems are assumed to be formed from lumped, linear, time-invariant masses, springs and dampers. Only two of these component kinds are used in the synthesis of any one system. The synthesis methods are related to those of electrical network theory through the current-force analogy.

In looking for a compact way of writing down the partial fraction formula in general, with repeated factors, I noticed how the expansion of a determinant by its top or bottom row suggested a method. The following gives a formula perfectly easy to write down in any given case where the factors of the denominator of the fraction are known. Incidentally it gives, as a determinant, the integral of a rational fraction f(x)/Q(x) where f(x) and Q(x) are polynomials, Q(x) having higher order.