Tamagawa Products of Elliptic Curves Over ℚ

We explicitly construct the Dirichlet series $$\begin{equation*}L_{\mathrm{Tam}}(s):=\sum_{m=1}^{\infty}\frac{P_{\mathrm{Tam}}(m)}{m^s},\end{equation*}$$ where $P_{\mathrm{Tam}}(m)$ is the proportion of elliptic curves $E/\mathbb{Q}$ in short Weierstrass form with Tamagawa product m. Although there are no $E/\mathbb{Q}$ with everywhere good reduction, we prove that the proportion with trivial Tamagawa product is $P_{\mathrm{Tam}}(1)={0.5053\dots}$. As a corollary, we find that $L_{\mathrm{Tam}}(-1)={1.8193\dots}$ is the average Tamagawa product for elliptic curves over $\mathbb{Q}$. We give an application of these results to canonical and Weil heights.

Mordell–Weil ranks and Tate–Shafarevich groups of elliptic curves with mixed-reduction type over cyclotomic extensions

Let [Formula: see text] be an elliptic curve defined over a number field [Formula: see text] where [Formula: see text] splits completely. Suppose that [Formula: see text] has good reduction at all primes above [Formula: see text]. Generalizing previous works of Kobayashi and Sprung, we define multiply signed Selmer groups over the cyclotomic [Formula: see text]-extension of a finite extension [Formula: see text] of [Formula: see text] where [Formula: see text] is unramified. Under the hypothesis that the Pontryagin duals of these Selmer groups are torsion over the corresponding Iwasawa algebra, we show that the Mordell–Weil ranks of [Formula: see text] over a subextension of the cyclotomic [Formula: see text]-extension are bounded. Furthermore, we derive an aysmptotic formula of the growth of the [Formula: see text]-parts of the Tate–Shafarevich groups of [Formula: see text] over these extensions.

Intramedullary Nailing of Intertrochanteric Fractures - Minimal Invasive Techniques for Reduction

Background: An intramedullary nail has become the implant of choice for intertrochanteric fractures. This paper introduced some minimal invasive techniques were used to improve quality of intertrochanteric fracture reduction. Methods: Of 119 intertrochanteric fractures treated from January 2014 to October 2019. All patients who received internal fixation on traction bed, and who could not achieve satisfactory closed reduction through the process of "external rotation, abduction, traction, adduction and internal rotation". Reductions were classified as good, acceptable, or poor. We had acceptable reduction in 83 cases and poor reduction in 37 cases though closed reduction. The displacement was reduced using some minimal invasive techniques. Results: After performing the relative techniques in these cases, no case had a poor result. 112(94.9%) cases were in a good reduction. Anatomical reduction should always be achieved in intertrochanteric fractures. Conclusion: The minimal invasive techniques could help the surgeon achieve satisfactory reduction in intertrochanteric fractures. This work had the potential to improve the cognition of reduction of intertrochanteric fractures for surgeons, especially beginners and juniors.

Large families of elliptic curves ordered by conductor

In this paper we study the family of elliptic curves $E/{{\mathbb {Q}}}$ , having good reduction at $2$ and $3$ , and whose $j$ -invariants are small. Within this set of elliptic curves, we consider the following two subfamilies: first, the set of elliptic curves $E$ such that the quotient $\Delta (E)/C(E)$ of the discriminant divided by the conductor is squarefree; and second, the set of elliptic curves $E$ such that the Szpiro quotient $\beta _E:=\log |\Delta (E)|/\log (C(E))$ is less than $7/4$ . Both these families are conjectured to contain a positive proportion of elliptic curves, when ordered by conductor. Our main results determine asymptotics for both these families, when ordered by conductor. Moreover, we prove that the average size of the $2$ -Selmer groups of elliptic curves in the first family, again when these curves are ordered by their conductors, is $3$ . The key new ingredients necessary for the proofs are ‘uniformity estimates’, namely upper bounds on the number of elliptic curves with bounded height, whose discriminants are divisible by high powers of primes.

Plant Fibers in Comparison with Other Fining Agents for the Reduction of Pesticide Residues and the Effect on the Volatile Profile of Austrian White and Red Wines

Pesticide residues in Austrian wines have so far been poorly documented. In 250 wines, 33 grape musts and 45 musts in fermentation, no limit values were exceeded, but in some cases high levels (>0.100 mg/L) of single residues were found, meaning that a reduction of these levels before bottling could make sense. In the course of this study, a white and a red wine were spiked with a mix of 23 pesticide residues from the group of fungicides (including botryticides), herbicides and insecticides. The influence of the following treatments on residue concentrations and volatile profiles were investigated: two activated charcoal products, a bentonite clay, two commercial mixed fining agents made of bentonite and charcoal, two yeast cell wall products, and a plant fiber-based novel filter additive. The results of this study show that all the agents tested reduced both residues and volatile compounds in wine, with activated charcoal having the strongest effect and bentonite the weakest. The mixed agents and yeast wall products showed less aroma losses than charcoal products, but also lower residue reduction. Plant fibers showed good reduction of pesticides with moderate aroma damage, but these results need to be confirmed under practical conditions.

FUNCTIONAL OUTCOME OF TIBIAL FRACTURES TREATED BY INTERLOCKING NAILING THROUGH SUPRAPATELLAR APPROACH – A PROSPECTIVE AND RETROSPECTIVE STUDY

The aim of this prospective study is to analyse the functional and radiological outcome of tibial fractures treated by intramedullary nailing through supra patellar approach.15 patients with tibial fractures were operated by intramedullary nailing through suprapatellar approach. Oxford knee scoring system was done to evaluate the functional outcome. Serial radiographs were taken to assess the fracture union at 2 weeks, 6 weeks and 12 weeks. Fifteen patients (male 11 female 4) with fracture both bones leg i Results : ncluded in our study. The most common cause was motor vehicle accidents (9 cases), self limiting fall (4 cases), assault (2 cases). The mean age was 45 years. All fractures united at a mean period of 3.5 months. To conclude, Suprapatellar tibial nailing has the benets of decreased operating time, good ouroscopy visualisation, minimal blood loss and ability to achieve and maintain good reduction throughout the procedure.

On the cohomology of Kobayashi’s plus/minus norm groups and applications

The plus and minus norm groups are constructed by Kobayashi as subgroups of the formal group of an elliptic curve with supersingular reduction, and they play an important role in Kobayashi’s definition of the signed Selmer groups. In this paper, we study the cohomology of these plus and minus norm groups. In particular, we show that these plus and minus norm groups are cohomologically trivial. As an application of our analysis, we establish certain (quasi-)projectivity properties of the non-primitive mixed signed Selmer groups of an elliptic curve with good reduction at all primes above p. We then build on these projectivity results to derive a Kida formula for the signed Selmer groups under a slight weakening of the usual µ = 0 assumption, and study the integrality property of the characteristic element attached to the signed Selmer groups.