Operations Research and Economic Theory: Essays in Honor of by Horst Albach (auth.), Prof. Dr. H. Hauptmann, Prof. Dr. Drs.

By Horst Albach (auth.), Prof. Dr. H. Hauptmann, Prof. Dr. Drs. h. c. W. Krelle, Priv.-Doz. Dr. K. C. Mosler (eds.)

Martin Beckmann goes to rejoice his 60th birthday. this can be nearly unimaginable contemplating the power, ingenuity, and task which he keeps to teach as he regularly did. it's an honor­ capable and delightful responsibility for the complete economics group to teach hirn the distinction, gratitude, and affection which he merits. therefore, these' who've contributed to this festschrift should be regarded as a dele­ gation from a far better group during which we all are joined; the editors particularly suppose deeply hooked up with and enriched via the character and clinical paintings of Martin Beckmann. Martin Beckmann is a kind of infrequent students who're now not slim­ minded experts in a single box; he has been lively in lots of parts of economics and operations study which speedily built seeing that international battle eleven, and he has contributed unique and fruitful principles in just about all of them. the range of themes handled during this quantity goals to re­ flect the awesome width of his medical interests.

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Using the result in Sato [1981, p. 276, eq. 4) af + ät L under extended group: af ao'(t))·k} di. ~ äk + {a~ ät + (~_ ak ak o. 1 We have worked out several interesting cases. f. 5a) f f( k ,k) • After some somewhat complicated calculations, we obtain the results which are summarized in the following: ~ R. SATO / T. NONO / F. 6a) T * 0, f (k, k) f «t , k) = e at+1jJ(k) d1jJ dk = "'(k) 'f .. 5b) f where 8(k) I; (t, + cp(k)k) (a f[k8'(k) - = 1jI(k) ( 2 • 6b) 0, I; _ if = ak a8(k)] jcp(k)dk + c. k) = Ae at /8' ( k) , That is to say, 0 A const.

Rut it turns out--and this is by no means obvious--that the Kepler problem admits not only the rotation group in three-dimensional space; but even in fourdimensional space. This additional symmetry was fully understood first for the quantum theoretical Kepler problem (Moser [1979]). 2see Klein [1918], Lie [1891], Noether [1918], Gelfand and Fomin [1963], Rund [1966], Logan [1977], N~O [1968], N~no and Mimura [1975, 1976, 1977, 1978], Sagan [1969], Lovelock and Rund [1975], Whittaker [1937] and Moser [1979].

Also when "technical change" is explicitly introduced along with a positive discount rate the current value of welfare measure of income is constant if and only if the consumption possibility curve has technical progress multiplicative to k ( S a to [1 9 81, p. 27 9] ) • In Sato [1982] and in Sato, Nono and Mimura [1982], the authors studied the WKL invariance conditions in a more general framework with the use of Noether's Theorem. One of the interesting results answers Samuelson's question [1982] affirmative.

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