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| 2. |
- Degtyarev, Alexander, et al.
(författare)
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ON TOTAL REALITY OF MEROMORPHIC FUNCTIONS
- 2007
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Ingår i: Annales de l'Institut Fourier. - 0373-0956 .- 1777-5310. ; 57:6, s. 2015-2030
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Tidskriftsartikel (refereegranskat)abstract
- We show that if a meromorphic function of degree at most four on a real algebraic curve of an arbitrary genus has only real critical points then it is conjugate to a real meromorphic function after a suitable projective automorphism of the image.
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| 4. |
- Ekedahl, Torsten, et al.
(författare)
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First steps towards total reality of meromorphic functions.
- 2006
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Ingår i: Moscow Mathematical Journal. ; 6:1
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Tidskriftsartikel (refereegranskat)abstract
- It was earlier conjectured by the second and the third authors that any rational curve γ: CP1 → CPn such that the inverse images of all its flattening points lie on the real line RP1 ⊂ CP1 is real algebraic up to a Möbius transformation of the image CPn. (By a flattening point p on γ we mean a point at which the Frenet n-frame (γ′,γ″,…,γ(n)) is degenerate.) Below we extend this conjecture to the case of meromorphic functions on real algebraic curves of higher genera and settle it for meromorphic functions of degrees 2, 3 and several other cases.
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| 5. |
- Gravin, Nick, et al.
(författare)
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On moments of a polytope
- 2018
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Ingår i: Analysis and Mathematical Physics. - : Springer Science and Business Media LLC. - 1664-2368 .- 1664-235X. ; 8:2, s. 255-287
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Tidskriftsartikel (refereegranskat)abstract
- We show that the multivariate generating function of appropriately normalized moments of a measure with homogeneous polynomial density supported on a compact polytope P subset of R-d is a rational function. Its denominator is the product of linear forms dual to the vertices of P raised to the power equal to the degree of the density function. Using this, we solve the inverse moment problem for the set of, not necessarily convex, polytopes having a given set S of vertices. Under a weak non-degeneracy assumption we also show that the uniform measure supported on any such polytope is a linear combination of uniform measures supported on simplices with vertices in S.
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| 6. |
- Klionsky, Daniel J., et al.
(författare)
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Guidelines for the use and interpretation of assays for monitoring autophagy
- 2012
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Ingår i: Autophagy. - : Informa UK Limited. - 1554-8635 .- 1554-8627. ; 8:4, s. 445-544
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Forskningsöversikt (refereegranskat)abstract
- In 2008 we published the first set of guidelines for standardizing research in autophagy. Since then, research on this topic has continued to accelerate, and many new scientists have entered the field. Our knowledge base and relevant new technologies have also been expanding. Accordingly, it is important to update these guidelines for monitoring autophagy in different organisms. Various reviews have described the range of assays that have been used for this purpose. Nevertheless, there continues to be confusion regarding acceptable methods to measure autophagy, especially in multicellular eukaryotes. A key point that needs to be emphasized is that there is a difference between measurements that monitor the numbers or volume of autophagic elements (e.g., autophagosomes or autolysosomes) at any stage of the autophagic process vs. those that measure flux through the autophagy pathway (i.e., the complete process); thus, a block in macroautophagy that results in autophagosome accumulation needs to be differentiated from stimuli that result in increased autophagic activity, defined as increased autophagy induction coupled with increased delivery to, and degradation within, lysosomes (in most higher eukaryotes and some protists such as Dictyostelium) or the vacuole (in plants and fungi). In other words, it is especially important that investigators new to the field understand that the appearance of more autophagosomes does not necessarily equate with more autophagy. In fact, in many cases, autophagosomes accumulate because of a block in trafficking to lysosomes without a concomitant change in autophagosome biogenesis, whereas an increase in autolysosomes may reflect a reduction in degradative activity. Here, we present a set of guidelines for the selection and interpretation of methods for use by investigators who aim to examine macroautophagy and related processes, as well as for reviewers who need to provide realistic and reasonable critiques of papers that are focused on these processes. These guidelines are not meant to be a formulaic set of rules, because the appropriate assays depend in part on the question being asked and the system being used. In addition, we emphasize that no individual assay is guaranteed to be the most appropriate one in every situation, and we strongly recommend the use of multiple assays to monitor autophagy. In these guidelines, we consider these various methods of assessing autophagy and what information can, or cannot, be obtained from them. Finally, by discussing the merits and limits of particular autophagy assays, we hope to encourage technical innovation in the field.
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| 9. |
- Saldanha, Nicolau, et al.
(författare)
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Finiteness of rank for Grassmann convexity
- 2023
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Ingår i: Comptes rendus. Mathematique. - : Cellule MathDoc/CEDRAM. - 1631-073X .- 1778-3569. ; 361:1, s. 445-451
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Tidskriftsartikel (refereegranskat)abstract
- The Grassmann convexity conjecture, formulated in [8], gives a conjectural formula for the maximal total number of real zeroes of the consecutiveWronskians of an arbitrary fundamental solution to a disconjugate linear ordinary differential equation with real time. The conjecture can be reformulated in terms of convex curves in the nilpotent lower triangular group. The formula has already been shown to be a correct lower bound and to give a correct upper bound in several small dimensional cases. In this paper we obtain a general explicit upper bound.
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| 10. |
- Saldanha, Nicolau, et al.
(författare)
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Grassmann Convexity and Multiplicative Sturm Theory, Revisited
- 2021
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Ingår i: Moscow Mathematical Journal. - : National Research University, Higher School of Economics (HSE). - 1609-3321 .- 1609-4514. ; 21:3, s. 613-637
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we settle a special case of the Grassmann convexity conjecture formulated by the second and the third authors about a decade ago. We present a conjectural formula for the maximal total number of real zeros of the consecutive Wronskians of an arbitrary fundamental solution to a disconjugate linear ordinary differential equation with real time. We show that this formula gives the lower bound for the required total number of real zeros for equations of an arbitrary order and, using our results on the Grassmann convexity, we prove that the aforementioned formula is correct for equations of orders 4 and 5.
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