Abstract Generated abstract
This letter comments on G. E. James-Levy’s paper on general anamorphosis, arguing that several of its assertions and constructions are incorrect or insufficiently justified. The authors identify problems with assumed functional representations, criteria for excluding rectilinear scales, and the derivation of differential equations for functions used in nomographic construction, giving a counterexample to show that the stated equations need not hold. They further argue that the proposed differential system can generally yield only local solutions, that global existence and practical determination of solutions are not established, and that the author’s method relies on an unjustified appeal to a general integral. The letter concludes that the posed problem is of interest only when its solution actually leads to a realizable nomogram, which is not guaranteed.
Full Text
Letter to the Editor
In Doklady AN SSSR there was published a paper by G. E. James-Levy, “On the Problem of General Anamorphosis” (¹). In the indicated paper there are a large number of incorrect and unclear assertions. The author’s constructions, after certain restrictions and corrections, can at best be justified only from a “local” point of view.
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The article says: “Equation (1) we shall sometimes write in the form \(x=\theta(y,z)\) or \(y=\Phi(x,z)\).” However, it is obvious that the possibility of such a notation in no way follows from the assumptions previously made by the author.
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Wishing to exclude the case when, in the nomogram under consideration, the scale \(x\) is a straight line, the author asserts that the curvilinearity of the scale \(x\) “is characterized by conditions (5).” These conditions are erroneous. They are not satisfied, for example, at any point of the parabola \(u=x,\ v=x^2\).
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The article derives equations (10) and (12), which the functions \(\varphi_3(z)\) and \(f_3(z)\) must satisfy. However, these equations are derived incorrectly.
In fact, it is not difficult to verify that, for example, the function \(z=\dfrac{x+y-xy}{1-(x+y)}\), for \(x+y\ne1\), makes the determinant
\[ \begin{vmatrix} x & x^2 & 1\\ y & y^2 & 1\\ 1+z & z & 1 \end{vmatrix} \]
identically equal to zero; moreover, the functions \(f_1(x)=x^2\) and \(\varphi_1(x)=x\) satisfy all the conditions required in the article, and nevertheless the functions \(\varphi_3(z)=1+z\) and \(f_3(z)=z\) do not satisfy equation (12).
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Having derived the system of differential equations (10) and (12), the author asserts that “in the general case, from this system are determined the functions \(f_3(z)\) and \(\varphi_3(z)\), depending on 4 constants of integration.” In reality, however, the system of equations (10), (12) determines a solution only locally; a solution defined on the entire interval of variation of the variable \(z\) for this system exists only in exceptional cases. Even when it is known that the system (10), (12) has a solution defined over the whole domain of variation of \(z\), it remains unclear how to find it. The method proposed by the author for selecting the required particular solutions is based on a misunderstanding: the author proposes to use a general integral of the system; but a general integral of the system (10), (12) in the entire domain under consideration will, generally speaking, not exist.
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The reasoning in the derivation of equations (10) and (12) applies only to those values of \(z\) that are assumed by the function \(F(x_0,y)\). The author nowhere indicates this.
Similar remarks also apply to all the cases considered by the author.
In conclusion, it should be noted that the solution of the problem posed by the author—of finding functions \(\varphi_1(x), f_1(x), \varphi_2(y), f_2(y), \varphi_3(z)\), and \(f_3(z)\) satisfying identity (4)—is of real interest only under the assumption that from its solution it is possible to obtain an actual nomogram, which is not always the case.
I. Weinstein, M. Kreines
Cited Literature
¹ G. E. James-Levy, DAN, 113, No. 2, 258 (1957).
From the Editors. The authors of the letter first drew the editors’ attention to the errors indicated in the letter in G. E. James-Levy’s article in June 1958.