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G Sager

Publications and source records attributed to G Sager.

At least 145 records · Page 8Linked to original sources
PubMed1981

[Length growth of the North Sea turbot (Scophthalmus maximus L. male) (author's transl)].

The testing of variety of growth functions of traditional or recently established status as started with the Alaskan razor clam (Sager 1980a) is continued for the length/age relation of the North Sea turbot. Whilst well known functions as the Gompertz must be abandoned in this case, the Richards function and the reciprocal function preferred by Krüger (1973) are in a good position but surpassed by a newly developed function of the author resulting from the increase ansatz W = kWm/(t + to)p (Sager 1980c). Thus the already traced line with excellent results from W = kwm/tp for the approximation of the length growth of the razor clam has found its direct extension although the resulting growth functions differ in their structure due to different ways of integration.

Animals↗
PubMed1981

[Allometry and the increase function dW/dt = kWm/(t + to)p (author's transl)].

After investigations of the author into allometry relations of general concern and the interaction of allometry and the growth functions of Bertalanffy, Gompertz and Janoschek in their original and modified forms the ansatz dW/dt - kWm/(t + to)p is taken into consideration. The solutions of the integrals differ for m = 1 and m not equal to 1 with m greater than 0 and p greater than m but are both mathematically rather well suited for applying the principle of allometry. The main characteristics of the growth functions are recalled, supplemented and contrasted with the equivalent properties of the allometric functions. Graphs give a limited idea of the variability of the functions of allometry including 4 or 5 parameters for W (value of growth) and w (allometric expression) respectively.

Animals↗
PubMed1981

[Mathematical considerations concerning the height/weight tables of Maaser (author's transl)].

The height/weight relation of human growth as given in the tables of Maaser (1974) and other authors shows a loosely linear behaviour from the second year up to puberty when plotted in semilogarithmic coordinates (Krüger 1975). Basing on this property mathematical tests are discussed to include the first year of life in the representation of the height/weight relation by adding but one term to the linear equation in the semilogarithmic system. After less successful suggestions a modified form of the exponential function was found to meet requirements relatively well as is shown by the results of nonlinear regressions for male and female following the Paul-method. In a second attempt, the more distinct linear course from about 6 1/2 years on is used as a basis for approximation by adding a function for the difference values against the height/weight tables. Results of the Paul-method are given and compared with the first test.

Body Height↗
PubMed1981

[Mathematical formulation of human height growth (author's transl)].

Gaining a mathematical expression for the height development of girls and boys is all the more desirable since acceleration has been observed for some decades. At the first look the height curve is not unsimilar to a simple parabola when considered up to prepuberty age. In fact the relation can be well approximated when applying a power function: W = W0 + ktr; with W0 representing the birth length, t the time, and k and r being coefficients. When reaching about ten years of age, this formula must be modified by a restricting term proposed as: (formula: see text) with tE as time of adolescence whilst WE would be the final height when omitting the puberal growth spurt which can approximately be taken into account by adding the term (formula: see text), where t is the half-spurt time. Calculations mainly based on the height/age tables of Maaser (1974) and using non linear regression show a close agreement with natural development. Graphs with the growth and increase functions for female and male added for better comprehension.

Adolescent↗
PubMed1980

[Allometry and the original and modified Janoschek growth function. (author's transl)].

After a summary of the author's work on the allometry principle with regard to the original and generalized Bertalanffy and Gompertz functions of organic growth attention is drawn to the analogous topic concerning the Janoschek growth function. At first the Janoschek function is resettled a little as to fit also for starting growth with values unequal to zero. After giving the main characteristics illustrated by graphs the allometric principle is applied to the growth function enforcing a repetition of all derivations due to the changed structure against the growth function proper. Secondly, the increase ansatz is changed with integration now leading to curves of finite growth time. Examples for allometry are added taking up values of recent investigations thus allowing for comparison. Finally an approximation method for the calculation of the inflexion points rounds up the paper.

Biometry↗
PubMed1980

[The function W = (a--b e-ct)n, a generalization of the classical growth functions (author's transl)].

The generalization of Bertalanffy's function of length and mass growth and the extension of the Gompertz function to 4 parameters as exercised by the author (Sager 1979 b, a) leads to the equation W = (a--b e-ct)n with positive values of n, and b in the first case and negative ones for n and b in the second case. It is shown that with the range of values for n the above expression comprises all classical functions of organic growth, namely the logistic or Verhulst function (1838), the Mitscherlich function (1919), and the Pütter-Bertalanffy functions (1920, 1934) of length and mass development. Moreover, the limiting case n leads to +/- infinity leads to the Gompertz-function (1825), as can be verified by the increase ansatz of the universal expression. Beside graphs examples are given showing the evaluation of the parameters a, b, c and n in the case of free choice as well as c, d and n for the transformed notation W = W0 [(1--de-ct)/(1--d)]n with a given initial value W0 leaving three parameters for choice.

Animals↗
PubMed1980

[The testing of growth functions for the length of Siliqua patula (Bivalvia) (author's transl)].

After preceding investigations of the author into increase and growth functions of organic matter the gained formulae are tested together with the classical functions. As an example the razor clam siliqua patula from the Alaskan coast seems rather well fitted because of the number of length measurements and the existence of an inflexion point of the growth curve. Non linear regressions are exercised following the Paul-method. Results are not seldom unexpected, sometimes leaving firmly introduced functions with smaller chances than recently developed ones. The number of parameters of the functions will not necessarily increase the quality of approximation. The best results arise from the increase ansatz W = k Wm/tp proposed by the author Sager 1979c).

Animals↗
PubMed1980

[Allometry and growth functions (author's transl)].

The concept of allometry is traced back to its origin and development during 140 years including opinions of quite different consequences. As a mean topic the allometric relation is regarded in its application to organic growth of parts to that of a body as a whole. For general considerations two simplified growth functions after Janoschek (1957) and the author are submitted to detailed investigation in order to demonstrate the mathematical treatment supported by graphs. Finally the question is raised whether the allometric relation may have a chance when growth procedures do not occur synchronously. For this case mathematical ideas are taken into account which can only be varified or negated by pursuing a broad variety of cases belonging to this category.

Animals↗
PubMed1980

[Allometry and the original and generalized Gompertz function (author's transl)].

After a review of the traditional Gompertz function of organic growth the mathematical consequences in applying the allometry principle are studied. In graphs examples are presented showing the relation between the Gompertz function proper and several cases of allometry having the same functional structure. Moreover the investigations are extended to a generalized form of the Gompertz function resulting from changing the increase function from dW/dt = kWe-rt to dW/dt = kWme-rt thus allowing for one more parameter. This form has recently been developed by the author (SAGER 1979a), giving more flexibility to the growth curve and providing the inflexion point with a broad range of ordinate values thus being superior to the rigidity of the original behaviour. After giving a graphical impression of the effect of the additional parameter allometry relations are considered once more and applied to an example already used when treating interrelations of allometry and the Bertalanffy growth functions (SAGER 1980b).

Biometry↗
PubMed1980

[Increase functions of the type dW/dt=k Wm/(t + to)p and their integrals (author's transl)].

The testing of a dozen functions for the length of the Alaskan razor clam Siliqua Patula (Bivalvia) by the author (Sager 1980) has given the best results when applying the increase ansatz dW/dt=k Wm/tp proposed after a series of investigations into mathematical properties of organic growth (Sager 1979a). The resulting growth function is restricted to vanishing values of the birth length however as is practically the case with the razor clam measuring 0,01 cm at birth and reaching 15 to 16 cm after at least 12 years of age. When the quotient of initial and final length or weight will go up with changing to other species in future investigations, the increase function must be extended to dW/dt=k Wm/(t + to)p, giving quite different solutions for m = 1 and 0 < m < infinity but m not equal to 1 respectively. The properties of the growth functions are discussed, and the evaluation of the parameters shown putting speical attention to the influence of to.

Animals↗
PubMed1979

[Increase functions of the type dw/dt = kwm(te--t)q and their integrals (author's transl)].

In consequent continuation of the investigations into the increase functions dW/dt = kWm(En--Wn), dW/dt = kWm(E--W)n and dW/dt = ktp (E--W)n the type dW/dt = kWm(tE--t)q with q greater than 0 is added. As in the last paper, 2 cases of integration have to be distinguished, namely for m = 1 and 0 less than m less than 1 respectively. In both cases adultness W = E is reached after a finite time tE. A quite different aspect results for the turning point of the growth function with the ordinate covering values between E/e and E for m = 1 and from 0 up to a maximum and down to E/e when 0 less than m less than 1. As in former papers graphs give an illustration of the variability of the growth curves and the increase functions. The already used example is taken up and shows a fargoing neighbourhood of the growth functions resulting from dW/dt = kWm (tE--t)q and dW/dt = ktp (E--W)n, thus hinting that quite different increase equations can lead to similar results.

Animals↗
PubMed1979

[Increase functions of the type dW/dt = ktp-1(tEp--tp)q and their integrals (author's transl)].

After the investigations into 4 increase functions of organic growth a fifth ansatz should be added as dW/dt = ktp(tE--t)q with only the time t involved on the right side of the equation. In the course of researches about approximations the author has dealed with this type of functions, which can only be integrated within the limits t = 0 and t = tE leading to the Beta-function with W = E as a restricted result. Therefore the basic equation was thoroughly modified as to fit the type dW/dt = cf(t) df(t)/dt. Surprisingly the integration results in a growth function formally equal to the solution for the increase function dW/dt = ktp(E--W)n. Although the determination of the coefficients runs along other lines, the traditional example given in the preceding papers shows values so close to each other as to suppose genuine identity, which is finally proved. The fact, that seemingly quite different increase functions can lead to identical results must be given serious account: judging about increase functions should be taken as a rather delicate matter.

Animals↗
PubMed1979

[The growth function w = e (sin (pi/2(t/te)(p))) (2q) and its properties (author's transl)].

Preceding investigations into increase functions of organic growth and their integration to growth functions have repeatedly shown curves looking like axially deformed harmonic functions. Arising from this experience the function W = E (sin (pi/2)t/tE)(p)))(2q) is considered and shown to be settled between growth functions of the increase types dW/dt = ktp(E--W)n and dW/dt = kWm(tE--t)q respectively. After gaining relations of the parameters to the basic values of growth and time graphs are given for the growth and increase functions in general and for 1 = 1 and p = 1 as special cases.

Growth↗
PubMed1979

[Increase functions of the type dW/dt = k Wm/tp and their integrals (author's transl)].

Continuing systematical investigations into increase functions the type dW/dt = k Wm/tp is treated. This case yields 2 types of integrals or growth functions according to m = 1 and m greater than 1 respectively, both reaching adultness after infinite time. Approximation to the final value W = E can be quite different following the amount of m and p especially. Examples are given for comparing the growth function with that of Janoschek (1957) implying 3 parameters instead of four.

Growth↗
PubMed1979

[A generalized form of the Bertalanffy functions of organic growth (author's transl)].

A historical review of the derivation of the Bertalanffy functions of lenght and mass growth (1934) with a critical evaluation of the concept is given. After extending the development to a surface growth function a generalization of the 3 formulas based on critical ideas of Bertalanffy is established giving room for intermediate growth behaviour. Finally the properties of the generalized function are presented, and the limits of application illustrated with the aid of graphs.

Body Height↗