scholarly journals Linear projections and successive minima

2010 ◽  
Vol 197 ◽  
pp. 45-57
Author(s):  
Christophe Soulé
Keyword(s):  
Line Bundle ◽  
Lower Bounds ◽  
Generic Fiber ◽  
Arithmetic Surface ◽  
Successive Minima ◽  
Effective Version ◽  
Projective Curves ◽  
High Degree

LetXbe an arithmetic surface, and letLbe a line bundle onX. Choose a metrichon the lattice Λ of sections ofLoverX. When the degree of the generic fiber ofXis large enough, we get lower bounds for the successive minima of (Λ,h) in terms of the normalized height ofX. The proof uses an effective version (due to C. Voisin) of a theorem of Segre on linear projections and Morrison’s proof that smooth projective curves of high degree are Chow semistable.

scholarly journals Linear projections and successive minima

2010 ◽  
Vol 197 ◽  
pp. 45-57
Author(s):  
Christophe Soulé
Keyword(s):  
Line Bundle ◽  
Lower Bounds ◽  
Generic Fiber ◽  
Arithmetic Surface ◽  
Successive Minima ◽  
Effective Version ◽  
Projective Curves ◽  
High Degree

Let X be an arithmetic surface, and let L be a line bundle on X. Choose a metric h on the lattice Λ of sections of L over X. When the degree of the generic fiber of X is large enough, we get lower bounds for the successive minima of (Λ,h) in terms of the normalized height of X. The proof uses an effective version (due to C. Voisin) of a theorem of Segre on linear projections and Morrison’s proof that smooth projective curves of high degree are Chow semistable.

scholarly journals Isoperimetric Numbers of Regular Graphs of High Degree with Applications to Arithmetic Riemann Surfaces

10.37236/651 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Dominic Lanphier ◽  
Jason Rosenhouse
Keyword(s):  
Large Class ◽  
Lower Bounds ◽  
Riemann Surfaces ◽  
Upper And Lower Bounds ◽  
Regular Graphs ◽  
Eigenvalue Spectrum ◽  
Asymptotic Bounds ◽  
High Degree

We derive upper and lower bounds on the isoperimetric numbers and bisection widths of a large class of regular graphs of high degree. Our methods are combinatorial and do not require a knowledge of the eigenvalue spectrum. We apply these bounds to random regular graphs of high degree and the Platonic graphs over the rings $\mathbb{Z}_n$. In the latter case we show that these graphs are generally non-Ramanujan for composite $n$ and we also give sharp asymptotic bounds for the isoperimetric numbers. We conclude by giving bounds on the Cheeger constants of arithmetic Riemann surfaces. For a large class of these surfaces these bounds are an improvement over the known asymptotic bounds.

scholarly journals SLIDE REDUCTION, SUCCESSIVE MINIMA AND SEVERAL APPLICATIONS

2013 ◽  
Vol 88 (3) ◽  
pp. 390-406 ◽  
Author(s):  
JIANWEI LI ◽  
WEI WEI
Keyword(s):  
New York ◽  
Approximation Algorithm ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Reduced Basis ◽  
Successive Minima ◽  
Theory Of Computing

AbstractGama and Nguyen [‘Finding short lattice vectors within Mordell’s inequality’, in: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, New York, 2008, 257–278] have presented slide reduction which is currently the best SVP approximation algorithm in theory. In this paper, we prove the upper and lower bounds for the ratios $\Vert { \mathbf{b} }_{i}^{\ast } \Vert / {\lambda }_{i} (\mathbf{L} )$ and $\Vert {\mathbf{b} }_{i} \Vert / {\lambda }_{i} (\mathbf{L} )$, where ${\mathbf{b} }_{1} , \ldots , {\mathbf{b} }_{n} $ is a slide reduced basis and ${\lambda }_{1} (\mathbf{L} ), \ldots , {\lambda }_{n} (\mathbf{L} )$ denote the successive minima of the lattice $\mathbf{L} $. We define generalised slide reduction and use slide reduction to approximate $i$-SIVP, SMP and CVP. We also present a critical slide reduced basis for blocksize 2.

Variations of Minkowski's theorem on successive minima

2016 ◽  
Vol 28 (2) ◽  
Author(s):  
Martin Henk ◽  
Matthias Henze ◽  
María A. Hernández Cifre
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Geometry Of Numbers ◽  
Successive Minima

AbstractMinkowski's second theorem in the Geometry of Numbers provides optimal upper and lower bounds for the volume of an

scholarly journals Upper bounds for some Brill–Noether loci over a finite field

2018 ◽  
Vol 14 (03) ◽  
pp. 739-749 ◽  
Author(s):  
Kamal Khuri-Makdisi
Keyword(s):  
Line Bundle ◽  
Finite Field ◽  
Classical Result ◽  
Base Point ◽  
Upper Bounds ◽  
Line Bundles ◽  
Effective Version ◽  
General Line ◽  
Genus Formula ◽  
Explicit Constant

Let [Formula: see text] be a smooth projective algebraic curve of genus [Formula: see text], over the finite field [Formula: see text]. A classical result of H. Martens states that the Brill–Noether locus of line bundles [Formula: see text] in [Formula: see text] with [Formula: see text] and [Formula: see text] is of dimension at most [Formula: see text], under conditions that hold when such an [Formula: see text] is both effective and special. We show that the number of such [Formula: see text] that are rational over [Formula: see text] is bounded above by [Formula: see text], with an explicit constant [Formula: see text] that grows exponentially with [Formula: see text]. Our proof uses the Weil estimates for function fields, and is independent of Martens’ theorem. We apply this bound to give a precise lower bound of the form [Formula: see text] for the probability that a line bundle in [Formula: see text] is base point free. This gives an effective version over finite fields of the usual statement that a general line bundle of degree [Formula: see text] is base point free. This is applicable to the author’s work on fast Jacobian group arithmetic for typical divisors on curves.

Tissue section lifting for electron microscopy

1978 ◽  
Vol 36 (2) ◽  
pp. 86-87
Author(s):  
Adrian F. van Dellen
Keyword(s):  
Infectious Disease ◽  
Electron Microscopy ◽  
Tissue Culture ◽  
Organ System ◽  
Surrounding Tissue ◽  
Paraffin Sections ◽  
Surface Areas ◽  
Specific Determination ◽  
High Degree ◽  
Accuracy And Repeatability

The morphologic pathologist may require information on the ultrastructure of a non-specific lesion seen under the light microscope before he can make a specific determination. Such lesions, when caused by infectious disease agents, may be sparsely distributed in any organ system. Tissue culture systems, too, may only have widely dispersed foci suitable for ultrastructural study. In these situations, when only a few, small foci in large tissue areas are useful for electron microscopy, it is advantageous to employ a methodology which rapidly selects a single tissue focus that is expected to yield beneficial ultrastructural data from amongst the surrounding tissue. This is in essence what "LIFTING" accomplishes. We have developed LIFTING to a high degree of accuracy and repeatability utilizing the Microlift (Fig 1), and have successfully applied it to tissue culture monolayers, histologic paraffin sections, and tissue blocks with large surface areas that had been initially fixed for either light or electron microscopy.

Electron Microscopy in Molecular Biology

1967 ◽  
Vol 25 ◽  
pp. 2-3
Author(s):  
Cecil E. Hall
Keyword(s):  
Electron Microscopy ◽  
Molecular Biology ◽  
Noise Level ◽  
Molecular Structures ◽  
Material Structure ◽  
The Arts ◽  
Shadow Casting ◽  
Electron Image ◽  
High Degree ◽  
Very High

The visualization of organic macromolecules such as proteins, nucleic acids, viruses and virus components has reached its high degree of effectiveness owing to refinements and reliability of instruments and to the invention of methods for enhancing the structure of these materials within the electron image. The latter techniques have been most important because what can be seen depends upon the molecular and atomic character of the object as modified which is rarely evident in the pristine material. Structure may thus be displayed by the arts of positive and negative staining, shadow casting, replication and other techniques. Enhancement of contrast, which delineates bounds of isolated macromolecules has been effected progressively over the years as illustrated in Figs. 1, 2, 3 and 4 by these methods. We now look to the future wondering what other visions are waiting to be seen. The instrument designers will need to exact from the arts of fabrication the performance that theory has prescribed as well as methods for phase and interference contrast with explorations of the potentialities of very high and very low voltages. Chemistry must play an increasingly important part in future progress by providing specific stain molecules of high visibility, substrates of vanishing “noise” level and means for preservation of molecular structures that usually exist in a solvated condition.

A High Angle, Double Tilting Cold Stage for the AEI EM7

1974 ◽  
Vol 32 ◽  
pp. 450-451
Author(s):  
P.R. Swann ◽  
A.E. Lloyd
Keyword(s):  
Habit Plane ◽  
Temperature Control ◽  
Tilt Angle ◽  
Cooling System ◽  
Thermal Drift ◽  
High Angle ◽  
Cold Stage ◽  
High Degree ◽  
Better Than

Figure 1 shows the design of a specimen stage used for the in situ observation of phase transformations in the temperature range between ambient and −160°C. The design has the following features a high degree of specimen stability during tilting linear tilt actuation about two orthogonal axes for accurate control of tilt angle read-out high angle tilt range for stereo work and habit plane determination simple, robust construction temperature control of better than ±0.5°C minimum thermal drift and transmission of vibration from the cooling system.

New Aspects in the Design of Electron Optical Magnification Systems

1972 ◽  
Vol 30 ◽  
pp. 606-607
Author(s):  
Willem H.J. Andersen
Keyword(s):  
Electron Microscope ◽  
Imaging System ◽  
Chromatic Aberration ◽  
Research Tool ◽  
Energy Losses ◽  
Objective Lens ◽  
Theoretical Limit ◽  
Optical Magnification ◽  
Chromatic Aberrations ◽  
High Degree

Electron microscope design, and particularly the design of the imaging system, has reached a high degree of perfection. Present objective lenses perform up to their theoretical limit, while the whole imaging system, consisting of three or four lenses, provides very wide ranges of magnification and diffraction camera length with virtually no distortion of the image. Evolution of the electron microscope in to a routine research tool in which objects of steadily increasing thickness are investigated, has made it necessary for the designer to pay special attention to the chromatic aberrations of the magnification system (as distinct from the chromatic aberration of the objective lens). These chromatic aberrations cause edge un-sharpness of the image due to electrons which have suffered energy losses in the object.There exist two kinds of chromatic aberration of the magnification system; the chromatic change of magnification, characterized by the coefficient Cm, and the chromatic change of rotation given by Cp.

Cytoplasmic Organization in the Receptor Cells of the Crista Ampullaris

1972 ◽  
Vol 30 ◽  
pp. 60-61 ◽  
Author(s):  
Robert F. Dunn
Keyword(s):  
Fine Particles ◽  
Apical Cell ◽  
Apical Region ◽  
Receptor Cells ◽  
Cuticular Plate ◽  
Morphological Organization ◽  
Crista Ampullaris ◽  
Cytoplasmic Organization ◽  
High Degree ◽  
General Appearance

Receptor cells of the cristae in the vestibular labyrinth of the bullfrog, Rana catesbiana, show a high degree of morphological organization. Four specialized regions may be distinguished: the apical region, the supranuclear region, the paranuclear region, and the basilar region.The apical region includes a single kinocilium, approximately 40 stereocilia, and many small microvilli all projecting from the apical cell surface into the lumen of the ampulla. A cuticular plate, located at the base of the stereocilia, contains filamentous attachments of the stereocilia, and has the general appearance of a homogeneous aggregation of fine particles (Fig. 1). An accumulation of mitochondria is located within the cytoplasm basal to the cuticular plate.

Sign in / Sign up

Export Citation Format

Share Document