Discontinuous, although “highly” differentiable, real functions and algebraic genericity

AbstractGiven a pair of real functions (k, f), we study the conditions they must satisfy for $$k+\lambda f$$ k + λ f to be the curvature in the arc-length of a closed planar curve for all real $$\lambda $$ λ . Several equivalent conditions are pointed out, certain periodic behaviours are shown as essential and a family of such pairs is explicitely constructed. The discrete counterpart of the problem is also studied.

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Coefficient inequalities related with typically real functions

Mathematica Slovaca ◽

10.1515/ms-2017-0396 ◽

2020 ◽

Vol 70 (4) ◽

pp. 829-838

Author(s):

Saqib Hussain ◽

Shahid Khan ◽

Khalida Inayat Noor ◽

Mohsan Raza

Keyword(s):

Unit Disk ◽

Real Functions ◽

Coefficient Inequalities ◽

Typically Real ◽

Class Of Functions ◽

Typically Real Functions

AbstractIn this paper, we are mainly interested to study the generalization of typically real functions in the unit disk. We study some coefficient inequalities concerning this class of functions. In particular, we find the Zalcman conjecture for generalized typically real functions.

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Preimages of Residual Sets of Continuous Functions under Operation of Superposition

Georgian Mathematical Journal ◽

10.1515/gmj.2005.763 ◽

2005 ◽

Vol 12 (4) ◽

pp. 763-768

Author(s):

Artur Wachowicz

Keyword(s):

Banach Space ◽

Continuous Functions ◽

Real Functions ◽

Second Category ◽

Residual Sets

Abstract Let 𝐶 = 𝐶[0, 1] denote the Banach space of continuous real functions on [0, 1] with the sup norm and let 𝐶* denote the topological subspace of 𝐶 consisting of functions with values in [0, 1]. We investigate the preimages of residual sets in 𝐶 under the operation of superposition Φ : 𝐶 × 𝐶* → 𝐶, Φ(𝑓, 𝑔) = 𝑓 ○ 𝑔. Their behaviour can be different. We show examples when the preimages of residual sets are nonresidual of second category, or even nowhere dense, and examples when the preimages of nontrivial residual sets are residual.

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Addition theorems for $${\mathcal {C}}^k$$ real functions and applications in ordinary differential equations

Aequationes Mathematicae ◽

10.1007/s00010-021-00822-w ◽

2021 ◽

Author(s):

Francisco Crespo ◽

Salomón Rebollo-Perdomo ◽

Jorge L. Zapata

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Addition Theorems ◽

Real Functions

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