Part I: Elementare Bemerkungen uber die Losungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus by E. Hopf Commentary by J. B. Serrin A remark on linear elliptic differential equations of …
Part I: Elementare Bemerkungen uber die Losungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus by E. Hopf Commentary by J. B. Serrin A remark on linear elliptic differential equations of second order by E. Hopf Commentary by J. B. Serrin Zum analytischen Charakter der Losungen regularer zweidimensionaler Variationsprobleme by E. Hopf Commentary by H. Weinberger Uber eine Klasse singularer Integralgleichungen by N. Wiener and E. Hopf Commentary by H. Widom Uber den funktionalen, insbesondere den analytischen Charakter der Losungen elliptscher Differentialgleichungen zweiter Ordnung by E. Hopf Commentary by H. Weinberger Abzweigung einer periodischen Losung von einer stationaren Losung eines Differentialsystems by E. Hopf Commentary by M. Golubitsky and P. H. Rabinowitz Repeated branching through loss of stability. An example by E. Hopf A mathematical example displaying features of turbulence by E. Hopf Commentary by R. Temam On S. Bernstein's theorem on surfaces $z(x,y)$ of nonpositive curvature by E. Hopf Commentary by L. Nirenberg The partial differential equation $u_t+uu_x=\mu_{xx}$ by E. Hopf Commentary by P. D. Lax Uber die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen by E. Hopf Commentary by J. B. Serrin Hamilton's theory and generalized solutions of the Hamilton-Jacobi equation by E. D. Conway and E. Hopf Commentary by C. S. Morawetz Part II: Statistik der geodatischen Linien in Mannigfaltigkeiten negativer Krummung by E. Hopf Statistik der Losungen geodatischer Probleme vom unstabilen Typus. II by E. Hopf Commentary by Ya. G. Sinai Closed surfaces without conjugate points by E. Hopf Commentary by Ya. G. Sinai Statistical hydromechanics and functional calculus by E. Hopf Commentary by Ya. G. Sinai On the ergodic theorem for positive linear operators by E. Hopf Commentary by D. Ornstein Acknowledgments.
3. G. H. Hardy, Notes on some points in the integral calculus, Messenger of Mathematics vol. 58 (1929) pp. 50-52. 4. Einar Hille and J. D. Tamarkin, On summability of …
3. G. H. Hardy, Notes on some points in the integral calculus, Messenger of Mathematics vol. 58 (1929) pp. 50-52. 4. Einar Hille and J. D. Tamarkin, On summability of Fourier series. II, Ann. of Math. (2) vol. 34 (1933) pp. 329-348. 5. S. Verblunsky, On some classes of Fourier series, Proc. London Math. Soc. vol. 33 (1932) pp. 287-327. 6. F. H. Young, A matrix transformation of Fourier coefficients, Thesis, University of Oregon, 1950. 7. Antoni Zygmund, Trigonometrical series, Warsaw-Lw6w, 1935, 331 pp.
eberhard hopfLet fl be a bounded open set in ordinary w-space.Denote by p a point of that space.A closed subset of the boundary of A is called accessible from within …
eberhard hopfLet fl be a bounded open set in ordinary w-space.Denote by p a point of that space.A closed subset of the boundary of A is called accessible from within fl if fl contains a semi-open Jordan curve p{t), 0^f< », whose points of accumulation, for t-><», precisely constitute that boundary part.In this note we prove the following theorem.
The following classical theorem is due to S. Bernstein: If z=z(x, y) is of class C" in the whole jc-y-plane and if 2 S(1) ZxzZyy Zxy ^5 0, ZxxZyy Zxy …
The following classical theorem is due to S. Bernstein: If z=z(x, y) is of class C" in the whole jc-y-plane and if 2 S(1) ZxzZyy Zxy ^5 0, ZxxZyy Zxy ^ 0, then z(x, y) cannot be bounded.1The original proof was found to contain a gap of topological nature.It is the purpose of this note to bridge this gap and to prove a somewhat more general theorem.Theorem.If z(x, y) belongs to C" in the whole x-y-plane and satisfies (1) then z(x, y) cannot be o(r) where r is the distance of (x, y) from an arbitrarily chosen fixed point.That this estimate of the order of magnitude at infinity cannot be essentially improved is shown by examples of the form z=f(x) -g(y), /">0, g">0, where/ and g can be chosen such that the order is just 0(r).A still open question is whether z(x, y) can or cannot be o(r) along a special sequence of radii r ->».In proving the theorem we shall, essentially, follow Bernstein's original arguments.For the sake of completeness the arguments will be repeated.Lemma 1 of Bernstein.Let z(x, y) be of class C" in a bounded open set R and let ZxxZyy-zlyg0 in R.Ifzis continuous on the boundary B of R and if z -0 on B, then z=0 in the whole of R.Proof (according to M. Shiftman).Let C be a circle in the plane z = 0 whose interior contains R-\-B.Consider the parts z = 0 of all possible spheres which intersect the plane z = 0 in C.They form a
Introduction.Let ß be the phase space of a dynamical system.We suppose that every motion can be continued along the entire time-axis.Thus we are concerned with a steady flow in ß.The …
Introduction.Let ß be the phase space of a dynamical system.We suppose that every motion can be continued along the entire time-axis.Thus we are concerned with a steady flow in ß.The following concepts are of fundamental significance for the study of dynamical flows.(a) There exists a curve of motion everywhere dense on ß.The existence of such a motion is known under the name of regional transitivity.We now suppose that a measure m in the sense of Lebesgue invariant under the flow exists on ß.Such a measure is usually defined by an invariant phase element dm.The following property is stronger than (a).(b) The curves of motion not everywhere dense on ß form a point set on ß of w-measure zero.Still stronger and more important than (b) is strict ergodicity.We suppose m(ti) to be finite.(c) Let f(P) be an arbitrary w-summable function on ß.The time-average oif(P) along a curve of motion is then,in general, equal to faf(P)dm/m(Q), the exceptional curves forming a point set on ß of m-measure zero.How these concepts are interrelated is seen most clearly if we state them in the following way.(a') Every open point set on ß that is invariant under the flow is everywhere dense on ß. (b') Every open point set on ß that is invariant under the flow has the measure m(ß).(c') Every m-measurable point set on ß that is invariant under the flow has either the m-measure zero orm(û).The latter property of a flow is called metric transitivity.%Its importance rests
Introduction.Let ß be the phase space of a dynamical system.We suppose that every motion can be continued along the entire time-axis.Thus we are concerned with a steady flow in ß.The …
Introduction.Let ß be the phase space of a dynamical system.We suppose that every motion can be continued along the entire time-axis.Thus we are concerned with a steady flow in ß.The following concepts are of fundamental significance for the study of dynamical flows.(a) There exists a curve of motion everywhere dense on ß.The existence of such a motion is known under the name of regional transitivity.We now suppose that a measure m in the sense of Lebesgue invariant under the flow exists on ß.Such a measure is usually defined by an invariant phase element dm.The following property is stronger than (a).(b) The curves of motion not everywhere dense on ß form a point set on ß of w-measure zero.Still stronger and more important than (b) is strict ergodicity.We suppose m(ti) to be finite.(c) Let f(P) be an arbitrary w-summable function on ß.The time-average oif(P) along a curve of motion is then,in general, equal to faf(P)dm/m(Q), the exceptional curves forming a point set on ß of m-measure zero.How these concepts are interrelated is seen most clearly if we state them in the following way.(a') Every open point set on ß that is invariant under the flow is everywhere dense on ß. (b') Every open point set on ß that is invariant under the flow has the measure m(ß).(c') Every m-measurable point set on ß that is invariant under the flow has either the m-measure zero orm(û).The latter property of a flow is called metric transitivity.%Its importance rests
EBERHARD HOPFJ1. Introduction.Let M be an analytic manifold of any number of dimensions, and let the volume measure on M be denoted generally by m.Let T be an analytic one-to-one …
EBERHARD HOPFJ1. Introduction.Let M be an analytic manifold of any number of dimensions, and let the volume measure on M be denoted generally by m.Let T be an analytic one-to-one transformation of M into itself.Such a transformation may have a positive invariant integral, m*(a) = ff(P)dm where/(P) > 0 for almost all points P of M. The invariance property means that m*(ax) = m*(a) holds for any measurable subset a of M, ax being the image of a under T.Such transformations are known to play an important rôle in dynamics.The motions of a dynamical system, considered in the manifold of states of motion, are equivalent to a one-parameter group of one-to-one transformations.In the case of a conservative system these transformations always possess a positive invariant integral; for instance in the case of a Hamiltonian system the phase volume itself is invariant.The integral m*(a) may be regarded as another measure on M ; thus a transformation of that kind is measure-preserving for a suitably chosen measure.The following paper deals with such transformations, for which the invariant measure m*(M) of the whole manifold M is finite, and is devoted to the characterization of these transformations by their intrinsic properties.Necessary conditions for the existence of a finite invariant measure can be easily derived.For instance no point set a of positive measure can be transformed into a "proper" part of itself, i.e. ai c a, m(ax) < m(a), for this would imply m*(ax) <m*(a) in contradiction to the invariance.This intrinsic property of those transformations plays an important rôle in Poincaré's and Birkhoff's work on the motions of dynamical systems.However, this is not the only intrinsic property of those transformations.It is equally
EBERHARD HOPFJ1. Introduction.Let M be an analytic manifold of any number of dimensions, and let the volume measure on M be denoted generally by m.Let T be an analytic one-to-one …
EBERHARD HOPFJ1. Introduction.Let M be an analytic manifold of any number of dimensions, and let the volume measure on M be denoted generally by m.Let T be an analytic one-to-one transformation of M into itself.Such a transformation may have a positive invariant integral, m*(a) = ff(P)dm where/(P) > 0 for almost all points P of M. The invariance property means that m*(ax) = m*(a) holds for any measurable subset a of M, ax being the image of a under T.Such transformations are known to play an important rôle in dynamics.The motions of a dynamical system, considered in the manifold of states of motion, are equivalent to a one-parameter group of one-to-one transformations.In the case of a conservative system these transformations always possess a positive invariant integral; for instance in the case of a Hamiltonian system the phase volume itself is invariant.The integral m*(a) may be regarded as another measure on M ; thus a transformation of that kind is measure-preserving for a suitably chosen measure.The following paper deals with such transformations, for which the invariant measure m*(M) of the whole manifold M is finite, and is devoted to the characterization of these transformations by their intrinsic properties.Necessary conditions for the existence of a finite invariant measure can be easily derived.For instance no point set a of positive measure can be transformed into a "proper" part of itself, i.e. ai c a, m(ax) < m(a), for this would imply m*(ax) <m*(a) in contradiction to the invariance.This intrinsic property of those transformations plays an important rôle in Poincaré's and Birkhoff's work on the motions of dynamical systems.However, this is not the only intrinsic property of those transformations.It is equally
Die Differentialgeometrie „im Grosen“ beschaftigt sich mit Eigenschaften „ganzer“ Flachen, d.h. solcher, die sich nicht durch Hinzufugung neuer Flachenstucke oder Punkte vergrosern lassen. Zu ihnen gehoren alle geschlossenen Flachen. Bei …
Die Differentialgeometrie „im Grosen“ beschaftigt sich mit Eigenschaften „ganzer“ Flachen, d.h. solcher, die sich nicht durch Hinzufugung neuer Flachenstucke oder Punkte vergrosern lassen. Zu ihnen gehoren alle geschlossenen Flachen. Bei einer offenen, unberandeten, mit einer uberall regularen inneren Differentialgeometrie versehenen Flache dagegen erhebt sich stets die Frage, ob sie bereits derart „vollstandig“ ist, das sie ein fur die Betrachtung „im Grosen“ geeigneter Gegenstand ist.
The result presented in this paper is generalization of the celebrated ergodic theorem of G. D. Birkhoff.' Birkhoff's theorem, von Neumann's mean ergodic theorem, and other known results of ergodic …
The result presented in this paper is generalization of the celebrated ergodic theorem of G. D. Birkhoff.' Birkhoff's theorem, von Neumann's mean ergodic theorem, and other known results of ergodic theory deal with measure-preserving flows and describe their statistical properties. The condition of measure invrariance seems to play a very essential role in the entire theory, and the question arises whether it is possible to make any general statements about statistical behaviour of flows without assuming the invariance of the underlying measure. We shall answer this question in the affirmative by introducing an averaging process that can be applied to any flow which is free from dissipative parts. In the special case of a measure-preserving flow, our result coincides with Birkhoff's theorem.