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Smale's Conjecture on Mean Values of Polynomials and Electrostatics

Dimitrov, Dimitar (2007)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary 30C10, 30C15, 31B35.A challenging conjecture of Stephen Smale on geometry of polynomials is under discussion. We consider an interpretation which turns out to be an interesting problem on equilibrium of an electrostatic field that obeys the law of the logarithmic potential. This interplay allows us to study the quantities that appear in Smale’s conjecture for polynomials whose zeros belong to certain specific regions. A conjecture concerning the electrostatic equilibrium...

Smirnov domains

П.Л. Дьюрен (1989)

Zapiski naucnych seminarov Leningradskogo

Smooth quasiregular mappings with branching

Mario Bonk, Juha Heinonen (2004)

Publications Mathématiques de l'IHÉS

We give an example of a 𝒞 3 - ϵ -smooth quasiregular mapping in 3-space with nonempty branch set. Moreover, we show that the branch set of an arbitrary quasiregular mapping inn-space has Hausdorff dimension quantitatively bounded away from n. By using the second result, we establish a new, qualitatively sharp relation between smoothness and branching.

Smooth quasiregular maps with branching in 𝐑 n

Robert Kaufman, Jeremy T. Tyson, Jang-Mei Wu (2005)

Publications Mathématiques de l'IHÉS

According to a theorem of Martio, Rickman and Väisälä, all nonconstant Cn/(n-2)-smooth quasiregular maps in Rn, n≥3, are local homeomorphisms. Bonk and Heinonen proved that the order of smoothness is sharp in R3. We prove that the order of smoothness is sharp in R4. For each n≥5 we construct a C1+ε(n)-smooth quasiregular map in Rn with nonempty branch set.

Some applications of subordination theorems associated with fractional q -calculus operator

Wafaa Y. Kota, Rabha Mohamed El-Ashwah (2023)

Mathematica Bohemica

Using the operator 𝔇 q , ϱ m ( λ , l ) , we introduce the subclasses 𝔜 q , ϱ * m ( l , λ , γ ) and 𝔎 q , ϱ * m ( l , λ , γ ) of normalized analytic functions. Among the results investigated for each of these function classes, we derive some subordination results involving the Hadamard product of the associated functions. The interesting consequences of some of these subordination results are also discussed. Also, we derive integral means results for these classes.

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