Old and New on the Schwarzian Derivative

Osgood, Brad

doi:10.1007/978-1-4612-0605-7_16

Cited by 32 publications

(17 citation statements)

References 54 publications

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“…for the elements defined by Equations (14) to (19). Since we are primarily interested in y(x) = u 1 (x), the top row of Equation (21) shows the final desired solution for System Two:…”

Section: Of 24mentioning

confidence: 99%

“…The general solution to the differential equation describing System Two, Equation 22, shows the homogeneous solution with arbitrary constants in terms of the state-variable initial conditions and the particular response due to the driving function. For given a 11 (x) and a 22 (x), the resulting matrix elements are presented in equations (14) through (19) and summarized in the complete state transition matrix of Equation (20). Then, this general solution will also apply to System One if the appropriate equivalent coefficients a 11 (x) and a 22 (x) resulting from the given p(x) and q(x) of Equations (31) and (32) are incorporated in the Φ II 11 (x, x 0 ) and Φ II 12 (x, x 0 ) matrix element calculations of Equations (14) through (19).…”

Section: Application Of System Two Solutions To System Onementioning

confidence: 99%

“…are calculated via Equations (14) and (19) using the appropriately chosen functions a 11 (x) and a 22 (x) that result from the System One, System Two equalization process of Equations (28) to (32), they provide the System One counterpart matrix elements necessary and sufficient for solutions to that system. Upon completion of this process in what follows, y I (x) = y II (x) and Equation (35) solutions apply to either system.…”

Section: Application Of System Two Solutions To System Onementioning

confidence: 99%

“…The quantity f v (x) is the negative of the Schwarzian derivative for Equation 1and plays an important role in more generalized studies of differential equations [14].…”

Section: System One Solutions From the Riccati Equation Connectionmentioning

confidence: 99%

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Feedback Diagram Application for the Generation and Solution of Linear Differential Equations Solvable by Quadrature

Pascone

Callahan

2020

Axioms

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A novel method for generating and providing quadrature solutions to families of linear, second-order, ordinary differential equations is presented in this paper. It is based upon a comparison of control system feedback diagrams—one representing the system and equation under study and a second equalized to it and providing solutions. The resulting Riccati equation connection between them is utilized to generate and solve groups of equations parameterized by arbitrary functions and constants. This method also leads to a formal solution mechanism for all second-order linear differential equations involving an infinite series of integrals of each equation’s Schwarzian derivative. The practicality of this mechanism is strongly dependent on the series rates of and allowed regions for convergence. The feedback diagram method developed is shown to be equivalent to a comparable method based on the differential equation’s normal form and another relying upon the grouping of terms for a reduction of the equation order, but augmenting their results. Applications are also made to the Helmholtz equation.

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“…for the elements defined by Equations (14) to (19). Since we are primarily interested in y(x) = u 1 (x), the top row of Equation (21) shows the final desired solution for System Two:…”

Section: Of 24mentioning

confidence: 99%

Section: Application Of System Two Solutions To System Onementioning

confidence: 99%

Section: Application Of System Two Solutions To System Onementioning

confidence: 99%

“…The quantity f v (x) is the negative of the Schwarzian derivative for Equation 1and plays an important role in more generalized studies of differential equations [14].…”

Section: System One Solutions From the Riccati Equation Connectionmentioning

confidence: 99%

See 2 more Smart Citations

Feedback Diagram Application for the Generation and Solution of Linear Differential Equations Solvable by Quadrature

Pascone

Callahan

2020

Axioms

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“…(see e.g. [1,20,21]). Here the first right-hand summand is the action of f on a quadratic differential 16) in terms of the z-coordinate above, in turn leading to a transition function for the vector bundle K 2 Λ in an analogous way to the classical situation.…”

Section: Relationship With Projective Structures On Xmentioning

confidence: 99%

Differential Algebras with Banach-Algebra Coefficients II: The Operator Cross-Ratio Tau-Function and the Schwarzian Derivative

Dupré

Glazebrook

Превиато

2012

Complex Anal. Oper. Theory

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Several features of an analytic (infinite-dimensional) Grassmannian of (commensurable) subspaces of a Hilbert space were developed in the context of integrable PDEs (KP hierarchy). We extended some of those features when polarized separable Hilbert spaces are generalized to a class of polarized Hilbert modules, in particular the Baker and τ -functions, which become operator-valued. Following from Part I we produce a pre-determinant structure for a class of τ -functions defined in the setting of the similarity class of projections of a certain Banach *algebra. This structure is explicitly derived from the transition map of a corresponding principal bundle. The determinant of this map gives a generalized, operator-valued τ -function that takes values in a commutative C*-algebra. We extend to this setting the operator cross-ratio which had been used to produce the scalar-valued τ -function, as well as the associated notion of a Schwarzian derivative along curves inside the space of similarity classes. We link directly this cross-ratio with Fay's trisecant identity for the τ -function (equivalent to the KP hierarchy). By * Partial research support under grant NSF-DMS-0808708 is very gratefully acknowledged.

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2 × 2 Matrices: The Donoghue and Hansen–Tomiyama Theorems

Simon

2019

Grundlehren Der Mathematischen Wissenschaften

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Old and New on the Schwarzian Derivative

Cited by 32 publications

References 54 publications

Feedback Diagram Application for the Generation and Solution of Linear Differential Equations Solvable by Quadrature

Feedback Diagram Application for the Generation and Solution of Linear Differential Equations Solvable by Quadrature

Differential Algebras with Banach-Algebra Coefficients II: The Operator Cross-Ratio Tau-Function and the Schwarzian Derivative

2 × 2 Matrices: The Donoghue and Hansen–Tomiyama Theorems

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