Zeroes of polynomials on definable hypersurfaces: Pathologies exist, but they are rare

Given a sequence {Zd}d?N of smooth and compact hypersurfaces in Rn-1, we prove that (up to extracting subsequences) there exists a regular definable hypersurface ? RPn such that each manifold Zd is diffeomorphic to a component of the zero set on of some polynomial of degree d. (This is in sharp contrast with the case when is semialgebraic, where for example the homological complexity of the zero set of a polynomial p on is bounded by a polynomial in deg(p).) More precisely, given the above sequence of hypersurfaces, we construct a regular, compact, semianalytic hypersurface ? RPn containing a subset D homeomorphic to a disk, and a family of polynomials {pm}m?N of degree deg(pm) = dm such that (D, Z(pm)nD) ~ (Rn-1, Zdm ), i.e. the zero set of pm in D is isotopic to Zdm in Rn-1. This says that, up to extracting subsequences, the intersection of with a hypersurface of degree d can be as complicated as we want. We call these 'pathological examples'. In particular, we show that for every 0 = k = n - 2 and every sequence of natural numbers a = {ad}d?N there is a regular, compact semianalytic hypersurface ? RPn, a subsequence {adm }m?N and homogeneous polynomials {pm}m?N of degree deg(pm) = dm such that bk( n Z(pm)) = adm . (0.1) (Here bk denotes the kth Betti number.) This generalizes a result of Gwozdziewicz et al. [13]. On the other hand, for a given definable we show that the Fubini-Study measure, in the Gaussian probability space of polynomials of degree d, of the set dm,a, of polynomials verifying (0.1) is positive, but there exists a constant c such tha

Random fields and the enumerative geometry of lines on real and complex hypersurfaces

We introduce a probabilistic framework for the study of real and complex enumerative geometry of lines on hypersurfaces. This can be considered as a further step in the original Shub\u2013Smale program of studying the real zeros of random polynomial systems. Our technique is general, and it also applies, for example, to the case of the enumerative geometry of flats on complete intersections. We derive a formula expressing the average number En of real lines on a random hypersurface of degree 2 n- 3 in RP n in terms of the expected modulus of the determinant of a special random matrix. In the case n= 3 we prove that the average number of real lines on a random cubic surface in RP 3 equals: E3=62-3.This technique can also be applied to express the number Cn of complex lines on a generic hypersurface of degree 2 n- 3 in CP n in terms of the expectation of the square of the modulus of the determinant of a random Hermitian matrix. As a special case, we recover the classical statement C3= 27. We determine, at the logarithmic scale, the asymptotic of the quantity En, by relating it to Cn (whose asymptotic has been recently computed in [19]). Specifically we prove that: limn\u2192 1elogEnlogCn=12.Finally we show that this approach can be used to compute the number Rn= (2 n- 3) ! ! of real lines, counted with their intrinsic signs (as defined in [28]), on a generic real hypersurface of degree 2 n- 3 in RP n

Topologies of nodal sets of random band limited functions

It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In particular the results apply to random monochromatic waves and to random real algebraic hyper-surfaces in projective space.Comment: 62 pages. Major revision following referee repor

Interactions and scattering of quantum vortices in a polariton fluid

Quantum vortices, the quantized version of classical vortices, play a prominent role in superfluid and superconductor phase transitions. However, their exploration at a particle level in open quantum systems has gained considerable attention only recently. Here we study vortex pair interactions in a resonant polariton fluid created in a solid-state microcavity. By tracking the vortices on picosecond time scales, we reveal the role of nonlinearity, as well as of density and phase gradients, in driving their rotational dynamics. Such effects are also responsible for the split of composite spin-vortex molecules into elementary half-vortices, when seeding opposite vorticity between the two spinorial components. Remarkably, we also observe that vortices placed in close proximity experience a pull-push scenario leading to unusual scattering-like events that can be described by a tunable effective potential. Understanding vortex interactions can be useful in quantum hydrodynamics and in the development of vortex-based lattices, gyroscopes, and logic devices.Comment: 12 pages, 7 figures, Supplementary Material and 5 movies included in arXi

Evaluation of SHOX defects in the era of next‐generation sequencing

Short stature homeobox (SHOX) haploinsufficiency is a frequent cause of short stature. Despite advances in sequencing technologies, the identification of SHOX mutations continues to be performed using standard methods, including multiplex ligation‐dependent probe amplification (MLPA) followed by Sanger sequencing. We designed a targeted panel of genes associated with growth impairment, including SHOX genomic and enhancer regions, to improve the resolution of next‐generation sequencing for SHOX analysis. We used two software packages, CONTRA and Nexus Copy Number, in addition to visual analysis to investigate the presence of copy number variants (CNVs). We evaluated 15 patients with previously known SHOX defects, including point mutations, deletions and a duplication, and 77 patients with idiopathic short stature (ISS). The panel was able to confirm all known defects in the validation analysis. During the prospective evaluation, we identified two new partial SHOX deletions (one detected only by visual analysis), including an intragenic deletion not detected by MLPA. Additionally, we were able to determine the breakpoints in four cases. Our results show that the designed panel can be used for the molecular investigation of patients with ISS, and it may even detect CNVs in SHOX and its enhancers, which may be present in a significant fraction of patients.Copy number variants analyses and Sanger sequencing of breakpoint regions in Case 11, which has a heterozygous deletions involving exons 4, 5, and 6a of short stature homeobox (SHOX).Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/151254/1/cge13587.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/151254/2/CGE_13587-sup-0001-Supinfo.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/151254/3/cge13587_am.pd

Interactions and scattering of quantum vortices in a polariton fluid

Quantum vortices, the quantized version of classical vortices, play a prominent role in superfluid and superconductor phase transitions. However, their exploration at a particle level in open quantum systems has gained considerable attention only recently. Here we study vortex pair interactions in a resonant polariton fluid created in a solid-state microcavity. By tracking the vortices on picosecond time scales, we reveal the role of nonlinearity, as well as of density and phase gradients, in driving their rotational dynamics. Such effects are also responsible for the split of composite spin–vortex molecules into elementary half-vortices, when seeding opposite vorticity between the two spinorial components. Remarkably, we also observe that vortices placed in close proximity experience a pull–push scenario leading to unusual scattering-like events that can be described by a tunable effective potential. Understanding vortex interactions can be useful in quantum hydrodynamics and in the development of vortex-based lattices, gyroscopes, and logic devices.MAT2016- 79866-R project (AEI/FEDER, UE)

The Use of Social Media and Digital Devices Among Italian Neurologists

Background: Digital devices and online social networks are changing clinical practice. In this study, we explored attitudes, awareness, opinions, and experiences of neurologists toward social media and digital devices. Methods: Each member of the Italian Society of Neurology (SIN) participated in an online survey (January to May 2018) to collect information on their attitude toward digital health. Results: Four hundred and five neurologists participated in the study. At work, 95% of responders use the personal computer, 87% the smartphone, and 43.5% the tablet. These devices are used to obtain health information (91%), maintain contact with colleagues (71%), provide clinical information (59%), and receive updates (67%). Most participants (56%) use social media to communicate with patients, although 65% are against a friendship with them on social media. Most participants interact with patients on social media outside working hours (65.2%) and think that social media have improved (38.0%) or greatly improved (25.4%) the relationship with patients. Most responders (66.7%) have no wearable devices available in clinical practice. Conclusion: Italian neurologists have different practices and views regarding the doctor–patient relationship in social media. The availability of digital devices in daily practice is limited. The use of social networks and digital devices will increasingly permeate into everyday life, bringing a new dimension to health care. The danger is that advancement will not go hand in hand with a legal and cultural adaptation, thus creating ambiguity and risks for clinicians and patients. Neurologists will need to be able to face the opportunities and challenges of this new scenario