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On Sturm's separation theorem

Paul R. Beesack
1972 Canadian mathematical bulletin  
The classical theorem (cf.  ...  In case y(a)=0, does the separation theorem hold? A complete answer to this question is given in Theorem 1 below, and an application of the extended theorem is given in Theorem 2. 2.  ... 
doi:10.4153/cmb-1972-086-7 fatcat:mxbeyblvpnbjnmz3ttprrg5fsu

A generalization of Sturm's comparison theorem [article]

G. A. Grigorian
2021 arXiv   pre-print
This result is based on a, so called, concept of "null-classes", and is a generalization of Sturm's comparison theorem.  ...  The Riccati equation method is used to establish a new comparison theorem for systems of two linear first order ordinary differential equation.  ...  The Sturm's famous comparison theorem states (see [1] , p. 334) Theorem 1.1 (Sturm).  ... 
arXiv:2005.12538v3 fatcat:rn2rdj7ysnhjddo7uxda547o5m

Sturm's theorem on zeros of linear combinations of eigenfunctions [article]

Pierre Bérard
2018 arXiv   pre-print
Motivated by recent questions about the extension of Courant's nodal domain theorem, we revisit a theorem published by C.  ...  Although not specialists in History of Sciences, we have tried to put these theorems into the context of nineteenth century mathematics.  ...  Sturm's proof of Theorem 1.3.  ... 
arXiv:1706.08247v4 fatcat:4xiq7t3pjbahhes7vz7v233wna

The converse of Sturm's Separation Theorem [article]

L. Gholizadeh, A. B. Mingarelli
2021 arXiv   pre-print
We show that Sturm's classical separation theorem on the interlacing of the zeros of linearly independent solutions of real second order two-term ordinary differential equations necessarily fails in the  ...  This is the actual converse of Sturm's Separation Theorem.  ...  To the best of our knowledge, a converse of Sturm's Separation Theorem has not been addressed.  ... 
arXiv:2109.06953v1 fatcat:b64our27jbgu3glbiebziuwijm

A generalization of Sturm's comparison theorem

C.A Swanson
1966 Journal of Mathematical Analysis and Applications  
INTRODUCTION Comparison theorems of Sturm's type will be obtained for a pair of linear self-adjoint elliptic partial differential equations.  ...  following example illustrates that Theorem 5 is more general than Kreith's theorem.  ... 
doi:10.1016/0022-247x(66)90105-3 fatcat:rzmogrnzy5en3ilcndki4gkyai

Sturm's theorem on the zeros of sums of eigenfunctions: Gelfand's strategy implemented [article]

Pierre Bérard
2018 arXiv   pre-print
In the second section "Courant-Gelfand theorem" of his last published paper (Topological properties of eigenoscillations in mathematical physics, Proc.  ...  Courant 2 partly generalized Assertion 2, in Sturm's theorem, to higher dimensions. Theorem 1.3.  ...  This is a weak form of Sturm's upper bound, Assertion 3a in Theorem 1.1.  ... 
arXiv:1807.03990v3 fatcat:7rquptykvnfzhe455zy6lbta7e

A generalization of Sturm's separation theorem

W. Allegretto
1970 Proceedings of the American Mathematical Society  
A comparison theorem for linear equations and a separation theorem for quasilinear equations are then obtained as applications.  ...  A Sturmian separation theorem is established for elliptic-parabolic equations with minimal assumptions on the coefficients and none on the regularity of the domain.  ...  The contradiction establishes the theorem. The separation theorem now follows for operators for which there exists a function/ such that L(/) <0.  ... 
doi:10.1090/s0002-9939-1970-0255959-3 fatcat:aot35mxyp5gdfl6qjim34u2oc4

Sturm's theorem for equations with delayed argument

A. Domoshnitsky
1994 Georgian Mathematical Journal  
Sturm's type theorems on separation of zeros of solutions are proved for the second order linear differential equations with delayed argument.  ...  It is also known [1] that nonvanishing of the Wronskian is equivalent to the validity of Sturm's theorem about separation of zeros.  ...  This article deals with the extension of the Sturm's theorem to equation (1) with delayed argument. The first result of this type was obtained by N.V.Azbelev [1] .  ... 
doi:10.1007/bf02254675 fatcat:mzrberkhcjfcvgrzdwagbqifde

A New Proof of Sturm's Theorem via Matrix Theory [article]

Kaiwen Hou, Bin Li
2021 arXiv   pre-print
Therefore, this paper offers a new proof of Sturm's theorem using matrix theory.  ...  By the classical Sturm's theorem, the number of distinct real roots of a given real polynomial $f(x)$ within any interval $(a,b]$ can be expressed by the number of variations in the sign of the Sturm chain  ...  Theorem (Sturm's Theorem [1] ).  ... 
arXiv:2110.15364v1 fatcat:sbxokwg4bjdlhcovphocjk6vya

Note on the practical application of Sturm's theorem

J. E. Wright
1906 Bulletin of the American Mathematical Society  
NOTE ON STURM'S THEOREM. [April, As three particular solutions, take S=2U/(u+ v)-Cr', v = 2V l /(u + v)-V' u Ç=2VJ(u + v)-V 2 .  ...  NOTE ON THE PRACTICAL APPLICATION OF STURM'S THEOREM. BY J. E. WRIGHT, M.A. Neue Beitrdge zur Frage des mathematischen und physikalischen Unterrichts an den höheren Schulen. F. KLEIN und E. Teil I.  ... 
doi:10.1090/s0002-9904-1906-01348-9 fatcat:v5ua23e335gpxffiswwxd6akky

358. Note on Sturm's Theorem in the Theory of Equations

A. C. Dixon
1911 Mathematical Gazette  
Note on Sturm's theorem in the theory of equations.  ...  Note on Sturm's theorem in the theory of equations.  ... 
doi:10.2307/3602856 fatcat:w4f3gr7ribfcjeo7tosrrotfxq

NOTE ON STURM'S THEOREM FOR A STRONGLY ELLIPTIC SYSTEMS

Janusz Wqsowski
1973 Demonstratio Mathematica  
Janusz Wqsowski NOTE ON STURM'S THEOREM FOR A STRONGLY ELLIPTIC SYSTEMS In this paper Sturm's Theorem (see [l] , [2] , [5] ) for a linear strongly elliptic systems of second order is given.  ... 
doi:10.1515/dema-1973-0303 fatcat:rkhjnvbrcjg3ta2efzmfa5depq

Sturm's theorem on zeros of linear combinations of eigenfunctions

Pierre Bérard, Bernard Helffer
2018 Expositiones mathematicae  
Motivated by recent questions about the extension of Courant's nodal domain theorem, we revisit a theorem published by C.  ...  Although not specialists in History of Sciences, we have tried to replace these theorems into the context of nineteenth century mathematics.  ...  We refer to [5, 30] for the relationships between Theorem 1.3 and Sturm's theorem on the number of real roots of real polynomials. Sturm's motivations.  ... 
doi:10.1016/j.exmath.2018.10.002 fatcat:72r7a2i3ifeb7d3djg6qfcscwa

An analogue of Sturm's theorem for Hilbert modular forms

Yuuki Takai
2013 Algebra & Number Theory  
We prove an analogue of Sturm's result for Hilbert modular forms associated to totally real number fields.  ...  An analogue of Sturm's theorem for Hilbert modular forms  ...  The reason why we call Theorem 1 an "analogue" of Sturm's theorem is that the constant κ includes a strange constant C.  ... 
doi:10.2140/ant.2013.7.1001 fatcat:5qnu4x3enfhjbbahgpr375ffcm

On an inverse spectral problem and a generalized Sturm's nodal theorem for nonlinear boundary value problems [article]

Y.Sh. Ilyasov, N.F. Valeev
2018 arXiv   pre-print
The key outcome of this relationship is a generalized Sturm's nodal theorem for the nonlinear boundary value problems.  ...  ., the proofs of assertions (2 o ), (3 o ) immediately follow from (2 o ), (3 o ) Theorem 1.  ...  We emphasize that this result is nothing more than a generalization of the well known Sturm's nodal theorem to the nonlinear problem.  ... 
arXiv:1809.00229v1 fatcat:igyk7buoh5ey5kttck3jyuix3u
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