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Charles A. Weibel - One of the best experts on this subject based on the ideXlab platform.

Christian Haesemeyer - One of the best experts on this subject based on the ideXlab platform.

Charles Fefferman - One of the best experts on this subject based on the ideXlab platform.

  • A sharp form of Whitney's extension theorem
    Annals of Mathematics, 2005
    Co-Authors: Charles Fefferman
    Abstract:

    3. Order relations involving multi-indices 4. Statement of two main lemmas 5. Plan of the proof 6. Starting the main induction 7. Nonmonotonic sets 8. A consequence of the main Inductive Assumption 9. Setup for the main induction 10. Applying Helly's theorem on convex sets 11. A Calder6n-Zygmund decomposition 12. Controlling auxiliary polynomials I 13. Controlling auxiliary polynomials II 14. Controlling the main polynomials 15. Proof of Lemmas 9.1 and 5.2 16. A rescaling lemma 17. Proof of Lemma 5.3 18. Proofs of the theorems 19. A bound for k# References

Spyros Alexakis - One of the best experts on this subject based on the ideXlab platform.

  • The Inductive Step of the Fundamental Proposition: The Simpler Cases
    The Decomposition of Global Conformal Invariants (AM-182), 2012
    Co-Authors: Spyros Alexakis
    Abstract:

    This chapter takes up the proof of Lemmas 4.16 and 4.19, which are easier than Lemma 4.24. The proof of these two lemmas relies on the study of the first conformal variation of the Assumption of Proposition 4.13. This study involves many complicated calculations and also the appropriate use of the Inductive Assumption of Proposition 4.13.

  • The Inductive Step of the Fundamental Proposition: The Hard Cases, Part I
    The Decomposition of Global Conformal Invariants (AM-182), 2012
    Co-Authors: Spyros Alexakis
    Abstract:

    This chapter takes up the proof of Lemma 4.24, which has two cases, A and B. The strategy goes as follows: It first repeats the ideas from Chapter 5 and derives a new local equation from the Assumption of Proposition 4.13. However, it finds that this new equation is very far from proving the claim of Lemma 4.24. It then has to return to the hypothesis of Proposition 4.13 and extract an entirely new equation from its conformal variation. It proceeds with a detailed study of this new equation (again using the Inductive Assumption of Proposition 4.13); the result is a second new local equation which again is very far from proving the claim of our lemma. Next, it formally manipulates this second new local equation and adds it to the first one, and observes certain miraculous cancellations, which yield new local equations that can be collectively called the grand conclusion. Lemma 4.24 in case A then immediately follows from the grand conclusion.

Jakub Šmerda - One of the best experts on this subject based on the ideXlab platform.

  • Environmental pressures on stomatal size may drive plant genome size evolution: evidence from a natural experiment with Cape geophytes.
    Annals of botany, 2020
    Co-Authors: Pavel Veselý, Petr Šmarda, Petr Bureš, Charles H. Stirton, A. Muthama Muasya, Ladislav Mucina, Lucie Horová, Kristýna Veselá, Alexandra Šilerová, Jakub Šmerda
    Abstract:

    Background and Aims The idea that genome (size) evolution in eukaryotes could be driven by environmental factors is still vigorously debated. In extant plants, genome size correlates positively with stomatal size, leading to the idea that conditions enabling the existence of large stomata in fossil plants also supported growth of their genome size. We test this Inductive Assumption in drought-adapted, prostrate-leaved Cape (South Africa) geophytes where, compared with their upright-leaved geophytic ancestors, stomata develop in a favourably humid microclimate formed underneath their leaves. Methods Stomatal parameters (leaf cuticle imprints) and genome size (flow cytometry) were measured in 16 closely related geophytic species pairs from seven plant families. In each pair, representing a different genus, we contrasted a prostrate-leaved species with its upright-leaved phylogenetic relative, the latter whose stomata are exposed to the ambient arid climate. Key Results Except for one, all prostrate-leaves species had larger stomata, and in 13 of 16 pairs they also had larger genomes than their upright-leaved relatives. Stomatal density and theoretical maximum conductance were less in prostrate-leaved species with small guard cells ( 1 pL). Giant stomata were observed in the prostrate-leaved Satyrium bicorne (89–137 µm long), despite its relatively small genome (2C = 9 Gbp). Conclusions Our results imply that climate, through selection on stomatal size, might be able to drive genome size evolution in plants. The data support the idea that plants from ‘greenhouse’ geological periods with large stomata might have generally had larger genome sizes when compared with extant plants, though this might not have been solely due to higher atmospheric CO2 in these periods but could also have been due to humid conditions prevailing at fossil deposit sites.