57 research outputs found
On a theory of the -function in positive characteristic
We present a theory of the -function (or Bernstein-Sato polynomial) in
positive characteristic. Let be a non-constant polynomial with coefficients
in a perfect field of characteristic Its -function is
defined to be an ideal of the algebra of continuous -valued functions on
The zero-locus of the -function is thus naturally
interpreted as a subset of which we call the set of roots of
We prove that has finitely many roots and that they are negative
rational numbers. Our construction builds on an earlier work of Musta\c{t}\u{a}
and is in terms of -modules, where is the ring of Grothendieck
differential operators. We use the Frobenius to obtain finiteness properties of
and relate it to the test ideals of Comment: Final versio
On D-modules related to the b-function and Hamiltonian flow
Let f be a quasi-homogeneous polynomial with an isolated singularity. We
compute the length of the D-modules generated by complex powers
of f in terms of the Hodge filtration on the top cohomology of the Milnor
fiber. For 1/f we obtain one more than the reduced genus of the singularity. We
conjecture that this holds without the quasi-homogeneous assumption. We also
deduce that the aforementioned quotient is nonzero when c is a root of the
b-function of f (which Saito recently showed fails to hold in the inhomogeneous
case). We obtain these results by comparing these D-modules to those defined by
Etingof and the second author which represent invariants under Hamiltonian
flow.Comment: 15 pages, final version. All comments welcom
Feynman integral relations from parametric annihilators
We study shift relations between Feynman integrals via the Mellin transform
through parametric annihilation operators. These contain the momentum space IBP
relations, which are well-known in the physics literature. Applying a result of
Loeser and Sabbah, we conclude that the number of master integrals is computed
by the Euler characteristic of the Lee-Pomeransky polynomial. We illustrate
techniques to compute this Euler characteristic in various examples and compare
it with numbers of master integrals obtained in previous works.Comment: v2: new section 3.1 added, several misprints corrected and additional
remark
Bernstein-Sato theory modulo
For fixed prime integer we develop a notion of Bernstein-Sato
polynomial for polynomials with -coefficients, compatible
with existing theory in the case . We show that the ``roots" of such
polynomials are rational and we show that the negative roots agree with those
of the mod- reduction. We give examples to show that, surprisingly, roots
may be positive in this context. Moreover, our construction allows us to define
a notion of ``strength" for roots by measuring -torsion, and we show that
``strong" roots give rise to roots in characteristic zero through mod-
reduction.Comment: Comments welcom
Altered splicing of the BIN1 muscle-specific exon in humans and dogs with highly progressive centronuclear myopathy
Amphiphysin 2, encoded by BIN1, is a key factor for membrane sensing and remodelling in different cell types. Homozygous BIN1 mutations in ubiquitously expressed exons are associated with autosomal recessive centronuclear myopathy (CNM), a mildly progressive muscle disorder typically showing abnormal nuclear centralization on biopsies. In addition, misregulation of BIN1 splicing partially accounts for the muscle defects in myotonic dystrophy (DM). However, the muscle-specific function of amphiphysin 2 and its pathogenicity in both muscle disorders are not well understood. In this study we identified and characterized the first mutation affecting the splicing of the muscle-specific BIN1 exon 11 in a consanguineous family with rapidly progressive and ultimately fatal centronuclear myopathy. In parallel, we discovered a mutation in the same BIN1 exon 11 acceptor splice site as the genetic cause of the canine Inherited Myopathy of Great Danes (IMGD). Analysis of RNA from patient muscle demonstrated complete skipping of exon 11 and BIN1 constructs without exon 11 were unable to promote membrane tubulation in differentiated myotubes. Comparative immunofluorescence and ultrastructural analyses of patient and canine biopsies revealed common structural defects, emphasizing the importance of amphiphysin 2 in membrane remodelling and maintenance of the skeletal muscle triad. Our data demonstrate that the alteration of the muscle-specific function of amphiphysin 2 is a common pathomechanism for centronuclear myopathy, myotonic dystrophy, and IMGD. The IMGD dog is the first faithful model for human BIN1-related CNM and represents a mammalian model available for preclinical trials of potential therapies
On the -supports of a holonomic -module
For a smooth variety over a perfect field of positive characteristic, the
sheaf of crystalline differential operators on (also called the sheaf
of -differential operators) is known to be an Azumaya algebra over
the cotangent space of the Frobenius twist of Thus to a
sheaf of modules over one can assign a closed subvariety of
called the -support, namely the support of seen as a sheaf
on We study here the family of -supports assigned to the
reductions modulo primes of a holonomic -module. We prove that
the Azumaya algebra of differential operators splits on the regular locus of
the -support and that the -support is a Lagrangian subvariety of the
cotangent space, for large enough. The latter was conjectured by
Kontsevich. Our approach also provides a new proof of the involutivity of the
singular support of a holonomic -module, by reduction modulo Comment: The article has been rewritten with much improved exposition as well
as some additional results, e.g. Corollary 6.3.1. This is the final version,
accepted for publication in Inventiones Mathematica
Guidelines for the use and interpretation of assays for monitoring autophagy (3rd edition)
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. For example, 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 versus those that measure fl ux through the autophagy pathway (i.e., the complete process including the amount and rate of cargo sequestered and degraded). In particular, a block in macroautophagy that results in autophagosome accumulation must be differentiated from stimuli that increase autophagic activity, defi ned as increased autophagy induction coupled with increased delivery to, and degradation within, lysosomes (inmost 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 fi eld 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. It is worth emphasizing here that lysosomal digestion is a stage of autophagy and evaluating its competence is a crucial part of the evaluation of autophagic flux, or complete autophagy. 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. Along these lines, because of the potential for pleiotropic effects due to blocking autophagy through genetic manipulation it is imperative to delete or knock down more than one autophagy-related gene. In addition, some individual Atg proteins, or groups of proteins, are involved in other cellular pathways so not all Atg proteins can be used as a specific marker for an autophagic process. 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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