scholarly journals AND THE ARITHMETIC OF A FAMILY OF NON-HYPERELLIPTIC CURVES OF GENUS 3

2015 ◽  
Vol 3 ◽  
Author(s):  
JACK A. THORNE
Keyword(s):  
Rational Numbers ◽  
Hyperelliptic Curves ◽  
Simple Singularity ◽  
Integral Points ◽  
Average Size

We study the arithmetic of a family of non-hyperelliptic curves of genus 3 over the field$\mathbb{Q}$of rational numbers. These curves are the nearby fibers of the semi-universal deformation of a simple singularity of type$E_{6}$. We show that average size of the 2-Selmer sets of these curves is finite (if it exists). We use this to show that a positive proposition of these curves (when ordered by height) has integral points everywhere locally, but no integral points globally.

scholarly journals On the integer part of the reciprocal of the Riemann zeta function tail at certain rational numbers in the critical strip

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
WonTae Hwang ◽  
Kyunghwan Song
Keyword(s):  
Zeta Function ◽  
Rational Number ◽  
Riemann Zeta Function ◽  
Rational Numbers ◽  
Critical Strip ◽  
Integer Part ◽  
Integral Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove that the integer part of the reciprocal of the tail of $\zeta (s)$ ζ ( s ) at a rational number $s=\frac{1}{p}$ s = 1 p for any integer with $p \geq 5$ p ≥ 5 or $s=\frac{2}{p}$ s = 2 p for any odd integer with $p \geq 5$ p ≥ 5 can be described essentially as the integer part of an explicit quantity corresponding to it. To deal with the case when $s=\frac{2}{p}$ s = 2 p , we use a result on the finiteness of integral points of certain curves over $\mathbb{Q}$ Q .

scholarly journals Integral points on hyperelliptic curves

2008 ◽  
Vol 2 (8) ◽  
pp. 859-885 ◽  
Author(s):  
Yann Bugeaud ◽  
Maurice Mignotte ◽  
Samir Siksek ◽  
Michael Stoll ◽  
Szabolcs Tengely
Keyword(s):  
Hyperelliptic Curves ◽  
Integral Points

scholarly journals Computing integral points on hyperelliptic curves using quadratic Chabauty

2016 ◽  
Vol 86 (305) ◽  
pp. 1403-1434 ◽  
Author(s):  
Jennifer S. Balakrishnan ◽  
Amnon Besser ◽  
J. Steffen Müller
Keyword(s):  
Hyperelliptic Curves ◽  
Integral Points

Quadratic Chabauty: p-adic heights and integral points on hyperelliptic curves

2016 ◽  
Vol 2016 (720) ◽  
Author(s):  
Jennifer S. Balakrishnan ◽  
Amnon Besser ◽  
J. Steffen Müller
Keyword(s):  
Hyperelliptic Curves ◽  
Integral Points

AbstractWe give a formula for the component at

scholarly journals Rational points on hyperelliptic curves having a marked non-Weierstrass point

2017 ◽  
Vol 154 (1) ◽  
pp. 188-222
Author(s):  
Arul Shankar ◽  
Xiaoheng Wang
Keyword(s):  
Weierstrass Point ◽  
Hyperelliptic Curves ◽  
Rational Points ◽  
Selmer Groups ◽  
Automorphic Representations ◽  
The Family ◽  
Odd Degree ◽  
Fundamental Research ◽  
Average Size ◽  
Fixed Genus

In this paper, we consider the family of hyperelliptic curves over$\mathbb{Q}$having a fixed genus$n$and a marked rational non-Weierstrass point. We show that when$n\geqslant 9$, a positive proportion of these curves have exactly two rational points, and that this proportion tends to one as$n$tends to infinity. We study rational points on these curves by first obtaining results on the 2-Selmer groups of their Jacobians. In this direction, we prove that the average size of the 2-Selmer groups of the Jacobians of curves in our family is bounded above by 6, which implies a bound of$5/2$on the average rank of these Jacobians. Our results are natural extensions of Poonen and Stoll [Most odd degree hyperelliptic curves have only one rational point, Ann. of Math. (2)180(2014), 1137–1166] and Bhargava and Gross [The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point, inAutomorphic representations and$L$-functions, Tata Inst. Fundam. Res. Stud. Math., vol. 22 (Tata Institute of Fundamental Research, Mumbai, 2013), 23–91], where the analogous results are proved for the family of hyperelliptic curves with a marked rational Weierstrass point.

S-INTEGRAL POINTS ON HYPERELLIPTIC CURVES

2011 ◽  
Vol 07 (03) ◽  
pp. 803-824 ◽  
Author(s):  
HOMERO R. GALLEGOS-RUIZ
Keyword(s):  
Upper Bound ◽  
Hyperelliptic Curve ◽  
Rational Point ◽  
Hyperelliptic Curves ◽  
Integral Points ◽  
Finite Set ◽  
Genus 2 ◽  
The Right ◽  
Model C

Let C : Y2 = an Xn + ⋯ + a0 be a hyperelliptic curve with the ai rational integers, n ≥ 5, and the polynomial on the right irreducible. Let J be its Jacobian. Let S be a finite set of rational primes. We give a completely explicit upper bound for the size of the S-integral points on the model C, provided we know at least one rational point on C and a Mordell–Weil basis for J(ℚ). We use a refinement of the Mordell–Weil sieve which, combined with the upper bound, is capable of determining all the S-integral points. The method is illustrated by determining the S-integral points on the genus 2 hyperelliptic model Y2 - Y = X5 - X for the set S of the first 22 primes.

Grain Refinement in Titanium-Erbium Alloys

1974 ◽  
Vol 32 ◽  
pp. 522-523
Author(s):  
B. B. Rath ◽  
J. E. O'Neal ◽  
R. J. Lederich
Keyword(s):  
Electron Microscopy ◽  
Size Distribution ◽  
Grain Refinement ◽  
Grain Growth ◽  
Volume Fraction ◽  
Pure Titanium ◽  
Systematic Investigation ◽  
Second Phase ◽  
Titanium Matrix ◽  
Average Size

Addition of small amounts of erbium has a profound effect on recrystallization and grain growth in titanium. Erbium, because of its negligible solubility in titanium, precipitates in the titanium matrix as a finely dispersed second phase. The presence of this phase, depending on its average size, distribution, and volume fraction in titanium, strongly inhibits the migration of grain boundaries during recrystallization and grain growth, and thus produces ultimate grains of sub-micrometer dimensions. A systematic investigation has been conducted to study the isothermal grain growth in electrolytically pure titanium and titanium-erbium alloys (Er concentration ranging from 0-0.3 at.%) over the temperature range of 450 to 850°C by electron microscopy.

Identification by Electron Microdiffraction of Intermetallic Phases in a Duplex Stainless Steel

1990 ◽  
Vol 48 (2) ◽  
pp. 494-495
Author(s):  
A. Redjaïmia ◽  
J.P. Morniroli ◽  
G. Metauer ◽  
M. Gantois
Keyword(s):  
Stainless Steel ◽  
Duplex Stainless Steel ◽  
Convergent Beam Electron Diffraction ◽  
Intermetallic Phases ◽  
Small Particles ◽  
Parallel Beam ◽  
Diffraction Patterns ◽  
Average Size ◽  
Convergent Beam ◽  
Electron Microdiffraction

2D and especially 3D symmetry information required to determine the crystal structure of four intermetallic phases present as small particles (average size in the range 100-500nm) in a Fe.22Cr.5Ni.3Mo.0.03C duplex stainless steel is not present in most Convergent Beam Electron Diffraction (CBED) patterns. Nevertheless it is possible to deduce many crystal features and to identify unambiguously these four phases by means of microdiffraction patterns obtained with a nearly parallel beam focused on a very small area (50-100nm).From examinations of the whole pattern reduced (RS) and full (FS) symmetries the 7 crystal systems and the 11 Laue classes are distinguished without ambiguity (1). By considering the shifts and the periodicity differences between the ZOLZ and FOLZ reflection nets on specific Zone Axis Patterns (ZAP) which depend on the crystal system, the centering type of the cell and the glide planes are simultaneously identified (2). This identification is easily done by comparisons with the corresponding simulated diffraction patterns.

Depletion of coating color components in the blade coating process circulation

TAPPI Journal ◽  
2011 ◽  
Vol 10 (9) ◽  
pp. 17-23 ◽  
Author(s):  
ANNE RUTANEN ◽  
MARTTI TOIVAKKA
Keyword(s):  
Fine Particles ◽  
Particle Density ◽  
Stable Process ◽  
Size Fraction ◽  
Color Stability ◽  
Strong Interactions ◽  
Coating Process ◽  
Size Segregation ◽  
Coating Color ◽  
Average Size

Coating color stability, as defined by changes in its solid particle fraction, is important for runnability, quality, and costs of a paper coating operation. This study sought to determine whether the size or density of particles is important in size segregation in a pigment coating process. We used a laboratory coater to study changes in coating color composition during coating operations. The results suggest that size segregation occurs for high and low density particles. Regardless of the particle density, the fine particle size fraction (<0.2 μm) was the most prone for depletion, causing an increase in the average size of the particles. Strong interactions between the fine particles and other components also were associated with a low depletion tendency of fine particles. A stable process and improved efficiency of fine particles and binders can be achieved by controlling the depletion of fine particles.

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