Celebratio Mathematica

G. A. Bliss: The geodesic lines on the anchor ring. Ph.D. thesis, University of Chicago, 1900. Advised by O. Bolza. MR 2936782

G. A. Bliss: “The second variation of a definite integral when one end-point is variable,” Trans. Amer. Math. Soc. 3 : 1 (1902), pp. 132–141. MR 1500591 JFM 33.0385.01

G. A. Bliss: “Jacobi’s criterion when both end-points are variable,” Math. Ann. 58 : 1–2 (1903), pp. 70–80. MR 1511229 JFM 34.0402.01

G. A. Bliss: “An existence theorem for a differential equation of the second order, with an application to the Calculus of Variations,” Trans. Amer. Math. Soc. 5 : 2 (1904), pp. 113–125. MR 1500665 JFM 35.0340.01

G. A. Bliss: “The solutions of differential equations of the first order as functions of their initial values,” Ann. of Math. (2) 6 : 2 (January 1905), pp. 49–68. MR 1503546 JFM 36.0389.01

The proof of the existence of solutions of the differential equation \[ \frac{dx}{dt} = f(t,x), \] where the funtion \( f \) is expansible into a power-series in \( t \) and \( x \), is included in most of the text books on differential equations. Methods have also been devised for proving the existence of a integral taking a given value \( \xi \) for \( t=\tau \), when the function \( f \) satsifies much less stringent restrictions [Osgood 1898]. In many applications however the mere existence theorem is of little value unless accompanied by a proof of the continuity and differentiability of the solution considered as a function of the initial constants \( \tau, \xi \). As the literature dealing with these latter questions is somewhat scattered and the proofs complicated, it is proposed in the following pages to collect and simplify as far as possible the results of the different writers. Solutions are defined reaching from boundary to boundary of the region \( R \) over which the properties of the function \( f \) are assumed provided \( R \) is closed, and the method of Peano for proving the existence and continuity of the first partial derivatives with respect to the initial constants has been extended by a simple device to the higher derivatives.

G. A. Bliss: “A generalization of the notion of angle,” Trans. Amer. Math. Soc. 7 : 2 (1906), pp. 184–196. MR 1500741 JFM 37.0490.01

G. A. Bliss and M. Mason: “A problem of the Calculus of Variations in which the integrand is discontinuous,” Trans. Amer. Math. Soc. 7 : 2 (1906), pp. 325–336. MR 1500752 JFM 37.0402.01

G. A. Bliss: “A new form of the simplest problem of the Calculus of Variations,” Trans. Amer. Math. Soc. 8 : 3 (1907), pp. 405–414. MR 1500795 JFM 38.0408.01

M. Mason and G. A. Bliss: “The properties of curves in space which minimize a definite integral,” Trans. Amer. Math. Soc. 9 : 4 (1908), pp. 440–466. MR 1500821 JFM 39.0441.01

G. A. Bliss: “On the inverse problem of the Calculus of Variations,” Ann. of Math. (2) 9 : 3 (April 1908), pp. 127–140. MR 1502361 JFM 39.0445.01

G. A. Bliss and M. Mason: “Fields of extremals in space,” Trans. Amer. Math. Soc. 11 : 3 (1910), pp. 325–340. MR 1500866 JFM 41.0437.01

G. A. Bliss: “A new proof of Weierstrass’s theorem concerning the factorization of a power series,” Bull. Amer. Math. Soc. 16 : 7 (1910), pp. 356–359. MR 1558920 JFM 41.0286.03

G. A. Bliss: “A generalization of Weierstrass’ preparation theorem for a power series in several variables,” Trans. Amer. Math. Soc. 13 : 2 (April 1912), pp. 133–145. MR 1500910 JFM 43.0504.02

G. A. Bliss: “A new proof of the existence theorem for implicit functions,” Bull. Amer. Math. Soc. 18 : 4 (1912), pp. 175–179. MR 1559180 JFM 43.0483.04

G. A. Bliss: “Fundamental existence theorems,” pp. i–ii,1–107 in The Princeton Colloquium (Princeton University, 15–17 September 1909), part 1. American Mathematical Society (New York), 1913. JFM 44.0365.01

G. A. Bliss and A. L. Underhill: “The minimum of a definite integral for unilateral variations in space,” Trans. Amer. Math. Soc. 15 : 3 (1914), pp. 291–310. MR 1500981 JFM 45.0605.03

G. A. Bliss: “Generalizations of geodesic curvature and a theorem of Gauss concerning geodesic triangles,” Amer. J. Math. 37 : 1 (January 1915), pp. 1–18. MR 1507894 JFM 45.0859.02

G. A. Bliss: “Jacobi’s condition for problems of the Calculus of Variations in parametric form,” Trans. Amer. Math. Soc. 17 : 2 (April 1916), pp. 195–206. MR 1501037 JFM 46.0758.03

G. A. Bliss: “Solutions of differential equations as functions of the constants of integration,” Bull. Amer. Math. Soc. 25 : 1 (1918), pp. 15–26. MR 1560139 JFM 47.0940.01

The purpose of this note is to prove the differentiability of the solutions of a system of differential equations with respect to the constants of integration by a method which seems more natural and simpler than those which have hitherto been published. Incidentally a restatement of the so-called “imbedding theorem” for differential equations is given, a theorem which is frequently applied in the Calculus of Variations, and which has been useful, and could be made still more so, in many other connections. It is analogous to the fundamental theorem for implicit functions in its statement that a solution of a system of differential equations given in advance is always a member of a continuous family of such solutions.

G. A. Bliss: “A method of computing differential corrections for a trajectory,” J. U.S. Artillery 51 (1919), pp. 445–449. Corrected copy of earlier printing in vol. 50 (pp. 455–460).

G. A. Bliss: “The use of adjoint systems in the problem of differential corrections for a trajectory,” J. U.S. Artillery 51 (1919), pp. 296–311.

G. A. Bliss: Differential corrections for anti-aircraft guns. Technical report, Aberdeen Proving Ground, 1919.

G. A. Bliss: “Differential equations containing arbitrary functions,” Trans. Amer. Math. Soc. 21 : 2 (1920), pp. 79–92. MR 1501137

G. A. Bliss: “Functions of lines in ballistics,” Trans. Amer. Math. Soc. 21 : 2 (1920), pp. 93–106. MR 1501138 JFM 47.0382.02

G. A. Bliss: “Birational transformations simplifying singularities of algebraic curves,” Trans. Amer. Math. Soc. 24 : 4 (1922), pp. 274–285. MR 1501226 JFM 50.0266.02

G. A. Bliss: “The reduction of singularities of plane curves by birational transformation,” Bull. Amer. Math. Soc. 29 : 4 (1923), pp. 161–183. Presidential address delivered before the American Mathematical Society, December 28, 1922. MR 1560693 JFM 49.0264.01

G. A. Bliss: “Algebraic functions and their divisors,” Ann. of Math. (2) 26 : 1–2 (September–October 1924), pp. 95–124. MR 1502680 JFM 50.0699.03

G. A. Bliss: “A boundary value problem for a system of ordinary linear differential equations of the first order,” Trans. Amer. Math. Soc. 28 : 4 (1926), pp. 561–584. MR 1501366 JFM 52.0453.13

G. A. Bliss: “A boundary value problem in the Calculus of Variations,” Bull. Amer. Math. Soc. 32 : 4 (1926), pp. 317–331. MR 1561219 JFM 52.0509.02

G. Bliss: “The transformation of Clebsch in the Calculus of Variations,” pp. 589–603 in Proceedings of the International Mathematical Congress (Toronto, 11–16 August 1924), vol. 1. Edited by J. C. Fields. University of Toronto Press, 1928. JFM 54.0532.03

G. A. Bliss: “The problem of Lagrange in the Calculus of Variations,” Amer. J. Math. 52 : 4 (October 1930), pp. 673–744. MR 1506783 JFM 56.0435.01

G. A. Bliss and I. J. Schoenberg: “On separation, comparison and oscillation theorems for self-adjoint systems of linear second order differential equations,” Amer. J. Math. 53 : 4 (October 1931), pp. 781–800. MR 1506854 JFM 57.0528.01 Zbl 0003.25702

Topics of the Calculus of Variations. Edited by G. Bliss. University of Chicago, 1932. Principally a report on lectures by E. J. McShane and Richard Courant.

Edward James McShane	Related
Richard Courant	Related

G. A. Bliss: The Calculus of Variations, multiple integrals. University of Chicago, 1933. MR 2936825

G. A. Bliss: Algebraic functions. AMS Colloquium Publications 16. American Mathematical Society (New York), 1933. JFM 59.0384.03 Zbl 0008.21004

G. A. Bliss: “Definitely self-adjoint boundary value problems,” Trans. Amer. Math. Soc. 44 : 3 (1938), pp. 413–428. MR 1501974 JFM 64.0438.02 Zbl 0020.03204

G. A. Bliss: “The Calculus of Variations for multiple integrals,” Amer. Math. Monthly 49 : 2 (February 1942), pp. 77–89. MR 0006022 Zbl 0063.00455

G. A. Bliss: Lectures on the calculus of variations. University of Chicago Press (Chicago, IL), 1946. MR 0017881 Zbl 0063.00459

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Gilbert Ames Bliss

Gilbert Ames Bliss: 1876–1951

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