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In this paper, we introduce an algorithm for computing p-adic integrals on bad reduction hyperelliptic curves. For bad reduction curves, there are two notions of p-adic integration: Berkovich-Coleman integrals which can be performed locally; and abelian integrals with desirable number-theoretic properties. By covering a bad reduction hyperelliptic curve by annuli and basic wide open sets, we reduce the computation of Berkovich-Coleman integrals to the known algorithms on good reduction hyperelliptic curves. These are due to Balakrishnan, Bradshaw, and Kedlaya, and to Balakrishnan and Besser for regular and meromorphic 1-forms on good reduction curves, respectively. We then employ tropical geometric techniques due to the first-named author with Rabinoff and Zureick-Brown to convert the Berkovich-Coleman integrals into abelian integrals. We provide examples of our algorithm, verifying that certain abelian integrals between torsion points vanish., Comment: Comments Welcome! figure taken from arxiv::1606.09618; v2 minor revisions

We show that for smooth and proper varieties over local fields with no non-trivial vector fields, good reduction descends over purely inseparable extensions. We use this to extend the Neron-Ogg-Shafarevich criterion for K3 surfaces to the equicharacteristic $p>0$ case., 15 pages, comments welcome!

We give good reduction criteria for hyperbolic polycurves, i.e., successive extensions of families of curves, under some assumptions. These criteria are higher dimensional versions of the good reduction criterion for hyperbolic curves given by Oda and Tamagawa.

Let $E$ be an elliptic curve defined over a number field $K$ where $p$ splits completely. Suppose that $E$ has good reduction at all primes above $p$. Generalizing previous works of Kobayashi and Sprung, we define multiply signed Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of a finite extension $F$ of $K$ where $p$ 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 $E$ over a subextension of the cyclotomic $\mathbb{Z}_p$-extension are bounded. Furthermore, we derive an aysmptotic formula of the growth of the $p$-parts of the Tate-Shafarevich groups of $E$ over these extensions., Comment: 20 pages

The goal of this manuscript is to provide a survey of good reduction criteria for hyperbolic polycurves. In particular, we give outlines of the proofs of the main theorems of the papers [19] and [20], which are details of the talk “Criteria for good reduction of hyperbolic polycurves” given at “Algebraic Number Theory and Related Topics 2018”. Also, this paper contains a proof of a specialization theorem of pro-L fundamental groups., Algebraic Number Theory and Related Topics 2018. November 26-30, 2018. edited by Takao Yamazaki and Shuji Yamamoto. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

We implement two-cover descent for plane quartics over Q with all 28 bitangents rational and show that on a significant collection of test cases, it resolves the existence of rational points. We also review a classical description of the relevant moduli space and use it to generate examples. We observe that local obstructions are quite rare for such curves, and only seem to occur in practice at primes of good reduction. In particular, having good reduction at 11 implies having no rational points. We also gather numerical data on two-Selmer ranks of Jacobians of these curves, which suggests that these often have non-trivial Tate-Shafarevich groups. We implement two-cover descent for plane quartics over Q with all 28 bitangents rational and show that on a significant collection of test cases, it resolves the existence of rational points. We also review a classical description of the relevant moduli space and use it to generate examples. We observe that local obstructions are quite rare for such curves and only seem to occur in practice at primes of good reduction. In particular, having good reduction at 11 implies having no rational points. We also gather numerical data on two-Selmer ranks of Jacobians of these curves, providing evidence these behave differently from those of general abelian varieties due to the frequent presence of an everywhere locally trivial torsor., 15 pages; Some minor improvements to algorithm and rank data analysis

This survey article is an outline of author's talk at the RIMS Workshop Algebraic Number Theory and Related Topics (2017). We study arithmetic properties of K3 surfaces with complex multiplication (CM) generalizing the results of Shimada for K3 surfaces with Picard number 20. Then, following Taelman's strategy and using Matsumoto's good reduction criterion for K3 surfaces with CM, we construct K3 surfaces over finite fields with given L-function, up to finite extensions of the base fields. We also prove the Tate conjecture for self-products of K3 surfaces over finite fields by CM lifts and the Hodge conjecture for self-products of K3 surfaces with CM proved by Mukai and Buskin., Algebraic Number Theory and Related Topics 2017. December 4-8, 2017. edited by Hiroshi Tsunogai, Takao Yamazaki and Yasushi Mizusawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

We define affinoids in the Lubin–Tate curve in even equal characteristic, and compute the reductions of them. Each reduction is isomorphic to a smooth affine curve with Artin–Schreier type equation. We expect that the cohomology of the reductions realizes the local Langlands correspondence and local Jacquet–Langlands correspondence for representations of conductor five.

Food sector produces million tons of waste. Out of which some is discarded as animal feed, as fertilizer or mostly treated as waste which further creates environmental hazards. The present study was done to utilize the waste pomace from vegetables – carrot and beetroot after juice extraction in the formation of Cellulose Nano Particles (CNPs). Pomaces of carrot and beetroot were treated with alkali treatment and bleaching treatment separately.So formed CNPs have shown good adsorption behavior and have shown good reduction in polluting parameters , specially colour was reduced upto 80 % in waste water.

Mandibular condylar fractures are among the most common facial fractures and some of the most difficult to manage. Opinions about the management of mandibular condylar fractures differ among surgeons. With the implementation of new technology, an increased understanding of fracture management, and better functional and morphological outcomes reported in the literature, open reduction and internal fixation is becoming many surgeons’ preferred choice for the treatment of condylar fractures. Because surgical treatment of such fractures is complex, certain factors must be considered to achieve satisfactory outcomes. In this article, we summarise six key points in the management of mandibular condylar fractures: virtual evaluation of condylar fracture, a suitable surgical approach, good reduction, stable internal fixation, repair of the articular disc, and restoration of the mandibular arch width. We believe that these points will help to improve the prognosis of mandibular condyle fractures. Keywords: Mandibular fractures, Condylar fractures, Template

Introduction: Today’s society is always more interested to the concept of aesthetics. The patients frequently ask to dentist to resolve unaesthetic problems of teeth, in particular that of the upper frontal group. The WSLs are enamel white alterations due to alteration during the demineralization and remineralization of enamel. This effect is caused by alteration of the pH in the oral cavity and buffer action of saliva. An alteration of this relationship leads to a progressive demineralization of enamel until the formation of a dental cavitation. Materials and Methods: For this study are selected 11 patients, of which 3 men and 8 women, with total of 17 WSLs. The inclusion criteria included WLSs with ICDAS = 2 and WLSs caused by hypomineralization of traumatic origin. These patients were subjected to treatment with infiltrating resin according to operative procedure. Discussion: The therapy with infiltrating resin gave great results in 12 lesions out of 17. In the lesions where there weren’t a complete remission we obtained a great aesthetic improvement and a good reduction of lesions. The follow up could improve the result after a better rehydration of hard tissue. Conclusions: With a correct selection of cases and good operative procedure, the use of the micro-infiltrative technique by low viscosity resin is a good procedure to resolve WSLs problems of non-orthodontic origins. Other studies with a larger sample are required to validate this clinical approach.

Let $p$ be an odd prime number, and $E$ an elliptic curve defined over a number field with good reduction at every prime of $F$ above $p$. In this short note, we compute the Euler characteristics of the signed Selmer groups of $E$ over the cyclotomic $\Zp$-extension. The novelty of our result is that we allow the elliptic curve to have mixed reduction types for primes above $p$ and that we allow mixed signs in the definition of the signed Selmer groups., 9 pages

We give a bound on the primes dividing the denominators of invariants of Picard curves of genus 3 with complex multiplication. Unlike earlier bounds in genus 2 and 3, our bound is based, not on bad reduction of curves, but on a very explicit type of good reduction. This approach simultaneously yields a simplification of the proof and much sharper bounds. In fact, unlike all previous bounds for genus 3, our bound is sharp enough for use in explicit constructions of Picard curves.

Objective: To determine percentages of types A (flat) and B (concave) of the distal tibiofibular joint in patients with ankle fractures or chronic ligament instabilities, with syndesmosis lesions; check the shape of the fixation and position of the fibula in this joint; to identify poor fibular reduction and its frequency in types A and B; patients according to the AOFAS criteria. Methods: 104 patients surgically treated with syndesmosis fixation underwent clinical evaluation using AOFAS functional criteria and tomographic exams to classify the distal tibiofibular joint in types A or B and evaluated the poor position of the fibula in this joint. Results: Distal tibiofibular joint type A was present in 27 ankles and type B in 77. Non-anatomical reduction of the fibula (17 ankles) was more frequent in type A than in type B and more frequent in fractures than in instabilities. The AOFAS score was 91.79 points in the 87 patients with good reduction and 86.76 points in the 17 patients with poor fibula reduction. Conclusion: Distal tibiofibular joint type B was more frequent than type A (p=0.00001); there was poor reduction of the fibula in this joint in 17 ankles (16.34%). Poor fibula reduction was more frequent in fractures than in instabilities (p=0.006). The poor reduction was more constant in type A than in type B, without statistical significance (p=0.34). The AOFAS score was 91.79 points in patients with good reduction and 86.76 points in patients with poor fibula reduction in the distal tibiofibular joint. Level of Evidence IV; Therapeutic Studies; Case Series.

This survey article is an outline of author's talk at the RIMS Workshop Algebraic Number Theory and Related Topics (2017). We study arithmetic properties of K3 surfaces with complex multiplication (CM) generalizing the results of Shimada for K3 surfaces with Picard number 20. Then, following Taelman's strategy and using Matsumoto's good reduction criterion for K3 surfaces with CM, we construct K3 surfaces over finite fields with given L-function, up to finite extensions of the base fields. We also prove the Tate conjecture for self-products of K3 surfaces over finite fields by CM lifts and the Hodge conjecture for self-products of K3 surfaces with CM proved by Mukai and Buskin.

This is a survey article on some recent developments in the arithmetic theory of linear algebraic groups over higher-dimensional fields, written for the Notices of the AMS., To appear in the Notices of the AMS. arXiv admin note: text overlap with arXiv:2005.05484

Introduction Proximal femoral nail (PFN) is a commonly used implant for intertrochanteric fractures which is designed according to western femoral measurements. However, anthropometry of proximal femur in Indian and in general, Asian, are smaller. So a modified short PFN with smaller dimensions was developed. This study analyses the radiological and functional outcome of treatment of intertrochanteric fractures with modified short PFN. Materials and methods A retrospective study analysed 120 adult patients operated between 2014-2017 using modified short PFN for intertrochanteric fractures, having a minimum follow-up of 12 months. Clinical and radiological parameters including tip-apex distance (TAD), position of tip of lag screw in femoral head, lateral slide of lag screw as well as length of anti-rotation screw were measured. Final functional outcome was assessed using Barthel's index and Kyle's criteria. Results Good reduction was achieved in 90.83% cases and 79.16% had ideal placement of lag screw in femoral head. Intra-operative difficulties were encountered in 13.33% (n=16). Mean TAD AP (anteroposterior) was 11.8mm, TAD LAT (lateral) was 11.0mm and mean TAD TOT was 22.8mm. Overall mean lateral slide was 3.20mm and it was more in unstable fracture. We had five mechanical failures, one patient with screw breakage without loss of reduction and two peri-implant fractures after union. 81.66% returned to pre-injury levels of activity with 88.33% good to excellent outcome as per Kyle's criteria. Conclusion Although, not devoid of complications, modified short PFN results in good functional recovery of patients with intertrochanteric fractures of femur.

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 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 latter projectivity results to derive a Kida formula for the signed Selmer groups, and study the integrality property of the characteristic element attached to the signed Selmer groups., 25 pages; corrected the proof of Lemma 5.1

We prove that a Kummer surface defined over a complete strictly Henselian discretely valued field $K$ of residue characteristic different from 2 admits a strict Kulikov model after finite base change. The Kulikov models we construct will be schemes, so our results imply that the semistable reduction conjecture is true for Kummer surfaces in this setup, even in the category of schemes. Our construction of Kulikov models is closely related to an earlier construction of K\"unnemann, which produces semistable models of Abelian varieties. It is well-known that the special fibre of a strict Kulikov model belongs to one of three types, and we shall prove that the type of the special fibre of a strict Kulikov model of a Kummer surface and the toric rank of a corresponding Abelian surface are determined by each other. We also study the relationship between this invariant and the Galois representation on the second $\ell$-adic cohomology of the Kummer surface. Finally, we apply our results, together with earlier work of Halle-Nicaise, to give a proof of the monodromy conjecture for Kummer surfaces in equal characteristic zero., Comment: 38 pages. Some typographical errors fixed. Small correction to the proof of Lemma 3.10

In this paper, we give a notion of the potentially good reduction locus of a Shimura variety. It consists of the points which should be related with motives having potentially good reductions in some sense. We show the existence of such locus for a Shimura variety of preabelian type. Further, we construct a partition of the adic space associated to a Shimura variety of preabelian type, which is expected to describe degenerations of motives. Using this partition, we prove that the cohomology of the potentially good reduction locus is isomorphic to the cohomology of a Shimura variety up to non-supercuspidal parts., 54 pages. This article subsumes arXiv:1109.4697. Minor modifications

We describe normal forms and minimal models of Picard curves, discussing various arithmetic aspects of these. We determine all so-called special Picard curves over $\mathbb{Q}$ with good reduction outside 2 and 3, and use this to determine the smallest possible conductor a special Picard curve may have. We also collect a database of Picard curves over $\mathbb{Q}$ of small conductor., 34 pages

Introduction: Proximal phalangeal fractures are common fractures of the hand. There is vicinity of two important joints and crossing of long tendons which make these fractures difficult to treat. The goal of this study was to evaluate the efficacy of nail traction technique in the management of proximal phalangeal fractures of the hand. Material and Methods: This is a prospective study of patients with proximal phalangeal fractures treated by nail traction. Assessment of all patients was done at the time of presentation and there was a standard protocol which was followed for recruiting patients. Assessment of the patients initially was done on 12th day after the application of nail traction. The outcome measures included radiographic evaluation post reduction and total active motion (TAM) in the finger at the final follow up. Follow up of all patients was done for a period of one year. Results: On x-ray evaluation post reduction, good reduction was seen in 33 cases, 8 cases had fair reduction and poor reduction in 2 cases. At final assessment, 35 patients had good, six had fair and two had poor TAM score. Complications were noted in two patients, which included pressure necrosis in palm and stiffness in proximal interphalangeal joint. Conclusions: The results of this study showed that nail traction seems to be simple, safe and effective technique for managing proximal phalangeal fractures.

Background: About 90 percent of all spinal injuries involve the thoracolumbar region. Unstable fractures need surgical treatment to achieve a painless, balanced and stable spine preserving or recovering neurological function, highest degree of spinal motion and to allow early patient mobilization.Objective: This study was carried out to evaluate radiological outcome of posterior instrumentation in thoracolumbar fractures.Methodology: A total of 110 patients with thoracolumbar fracture were included in the study, which was carried out at Kathmandu Medical College Teaching Hospital from December 2011 to December 2016. Unstable Arbeitsgemeinschaft fur Osteosynthesefragen type A and type B fractures were treated with short segment instrumentation and type C with long segment instrumentation. Radiological evaluation of postoperative correction of kyphotic angle and vertebral height was measured and was compared with immediate postoperative correction and loss of correction in two years final follow up.Results: Mean postoperative correction of vertebral kyphotic angle was 25° and loss of correction in final follow up was 5°. Mean postoperative vertebral height correction was 24% and its loss in final follow up was 2%. There was no significant difference in loss of correction of vertebral kyphosis and vertebral height in short segment and long segment fi xation in final follow up.Conclusion: Long segment posterior instrumentation results in good reduction and its maintenance for Arbeitsgemeinschaft fur Osteosynthesefragen type C thoracolumbar fractures whereas similar results can be achieved with short segment posterior instrumentation in type A and type B fractures. Journal of Kathmandu Medical College,Vol. 6, No. 4, Issue 22, Oct.-Dec., 2017, Page: 150-155

Let X be a smooth projective algebraic variety over a number field F , with an embedding τ : F ↪ C . The action of Gal ( F ¯ / F ) on l -adic cohomology groups H e t i ( X / F ¯ , Q l ) , induces Galois representations ρ l i : Gal ( F ¯ / F ) → GL ( H e t i ( X / F ¯ , Q l ) ) . Fix a non-archimedean valuation v on F , of residual characteristic p . Let F v be the completion of F at v and ′ W v be the Weil–Deligne group of F v . We establish new cases, for which the linear representations ρ l i _ of ′ W v , associated to ρ l i , form a compatible system of representations of ′ W v defined over Q . Under suitable hypotheses, we show that in some cases, these representations actually form a compatible system of representations of ′ W v , with values in the Mumford–Tate group of H B i ( τ X ( C ) , Q ) . When X has good reduction at v , we establish a motivic relationship between the compatibility of the system { ρ l i } l ≠ p and the conjugacy class of the crystalline Frobenius of the reduction of X at v .

OBJECTIVE. The aim of this study was to improve the results of the complex treatment of early periprosthetic infection in hip replacement by using a vacuum therapy. MATERIAL AND METHODS. The study included 82 patients with periprosthetic infection who had been hospitalized to the department of traumatology and orthopedics federal state budgetary educational Institution of higher education «north-Western state Medical University named after I. I. Mechnikov» from 2008 to 2016. We compared the results of the treatment in the retrospective group (without vacuum therapy) and in the prospective panel (using the local negative pressure method). RESULTS. The studies demonstrated the efficacy of the method of vacuum therapy in combination with standard methods of periprosthetic infection treatment. The clinical effects were: decrease of tissue edema, good reduction of wound exudate, decreasing the concentration of the microflora in the surgical wound area, granulation growth, stimulation of epithelialization, reducing of the time spent in the hospital. CONCLUSION. All these effects lead to the fact that the number of cases where we were able to maintain the implant increased from 52,9 % in retrospective groups to 78,8 % in prospective groups.

Given a map $\varphi:\mathbb{P}^1\rightarrow \mathbb{P}^1$ of degree greater than 1 defined over a number field $k$, one can define a map $\varphi_\mathfrak{p}:\mathbb{P}^1(\mathfrak{o}_k/\mathfrak{p})\rightarrow \mathbb{P}^1(\mathfrak{o}_k/\mathfrak{p})$ for each prime $\mathfrak{p}$ of good reduction, induced by reduction modulo $\mathfrak{p}$. It has been shown that for a typical $\varphi$ the proportion of periodic points of $\varphi_\mathfrak{p}$ should tend to $0$ as $|\mathbb{P}^1(\mathfrak{o}_k/\mathfrak{p})|$ grows. In this paper, we extend previous results to include a weaker set of sufficient conditions under which this property holds. We are also able to show that these conditions are necessary for certain families of functions, for example, functions of the form $\varphi(x)=x^d+c$, where $0$ is not a preperiodic point of this map. We study the proportion of periodic points by looking at the fixed point proportion of the Galois groups of certain extensions associated to iterates of the map., Comment: This article draws heavily from arXiv:1410.3378

Let K be a number field and let ϕ in K ( z ) be a rational function of degree d ≥ 2 . Let S be the set of places of bad reduction for ϕ (including the archimedean places). Let Per ( ϕ , K ) , PrePer ( ϕ , K ) , and Tail ( ϕ , K ) be the set of K-rational periodic, preperiodic, and purely preperiodic points of ϕ, respectively. The present paper presents two main results. The first result is a bound for | PrePer ( ϕ , K ) | in terms of the number of places of bad reduction | S | and the degree d of the rational function ϕ. This bound significantly improves a previous bound given by J. Canci and L. Paladino. For the second result, assuming that | Per ( ϕ , K ) | ≥ 4 (resp. | Tail ( ϕ , K ) | ≥ 3 ), we prove bounds for | Tail ( ϕ , K ) | (resp. | Per ( ϕ , K ) | ) that depend only on the number of places of bad reduction | S | (and not on the degree d). We show that the hypotheses of this result are sharp, giving counterexamples to any possible result of this form when | Per ( ϕ , K ) | 4 (resp. | Tail ( ϕ , K ) | 3 ).

Let $K$ be the field of fractions of a local Henselian DVR with perfect residue field. Assuming potential semi-stable reduction, we show that an unramified Galois-action on second $\ell$-adic cohomology of a K3 surface over $K$ implies that the surface has good reduction after a finite and unramified extension. We give examples where this unramified extension is really needed. Moreover, we give applications to good reduction after tame extensions and Kuga-Satake Abelian varieties. On our way, we settle existence and termination of certain semi-stable flops in mixed characteristic, and study group actions and their quotients on models of varieties., Comment: 40 pages, final version

Malleolar fractures are the most common ankle injuries treated by orthopedic surgeons. Objectives: The present study was undertaken to study the functional outcome of surgically managed medial malleolar fractures of ankle in adults. Material and Methods: A prospective study was conducted on 12 cases of medial malleolar fractures of ankle in adults, treated surgically by using various implants while 22 conservatively. This study was conducted for a period of 18 months and the results were assessed using criteria of Baird and Jackson’s ankle scoring system for evaluating the functional outcome. Results: This study shows the importance of pre-operative understanding of fracture mechanics for good reduction and internal fixation, in turn better functional outcome. Good functional results were obtained by surgical treatment of medial malleolar ankle fractures. There were no intra-operative complications observed in this study. Of the 12 patients with medial malleolar fractures treated surgically excellent to good results were achieved in 10 (83.3%) cases, fair and poor in 1 (8.33%) case each. Conclusion: Conservative treatment in selected fractures is justified. In our study the functional outcome and the results of operative treatment were found to be good as these operative methods restores the anatomy, biomechanics and contact loading characteristics of the ankle.

In deep-rod pumping wells, polished rod load tends to be very high and even exceeds the operating range of the pumping unit. To tackle the problem, mainly four rod load reduction techniques have been designed till now. They are side-flow pump, rod load reducer, fiberglass sucker rod and deep pumping with small-diameter pump. However, some of them still cause certain problems when put into use. That indicates the fundamental mechanisms of these techniques have not yet been fully studied. In this paper, on the basis of conventional polished rod load calculation models, the new calculation models for these techniques are respectively established. The rod load reduction effects are comparatively analyzed by calculation. The results indicate that the differences in mechanisms lead to different reduction effects. Side-flow pump and deep pumping with small-diameter pump can largely reduce the liquid pressure load, but its reduction effect for the polished rod load is very limited. On the contrary, fiberglass sucker rod can greatly reduce the polished rod load by decreasing the weight of the sucker rod. Rod load reducer does not change the previous load; instead, it can create an additional reduction force. The potential disadvantages are also discussed. When using a side-flow pump, the resistance on the plunger caused by the liquid pressure would increase. This resistance may cause the sucker rod to bend when it is moving downwards. Deep pumping with small-diameter pump needs to use very small pump in order to get a relatively good reduction effect, so its application is limited. These disadvantages should be considered in practical application.

We develop a theory of Hodge type Rapoport-Zink formal schemes, which uniformize certain formal completions of the canonical integral models of Shimura varieties of Hodge type at primes of good reduction. We then apply the general theory to the special case of Shimura varieties associated with groups of spinor similitudes, and, in the basic case, determine explicitly the reduced scheme underlying the Rapoport-Zink formal scheme., 66pp, final version, to appear in Compositio Math

Fix a number field $k$ and a rational prime $\ell$. We consider abelian varieties whose $\ell$-power torsion generates a pro-$\ell$ extension of $k(\mu_{\ell^\infty})$ which is unramified away from $\ell$. It is a necessary, but not generally sufficient, condition that such varieties have good reduction away from $\ell$. In the special case of $\ell = 2$, we demonstrate that for abelian surfaces $A/\mathbb{Q}$, good reduction away from $\ell$ does suffice. The result is extended to elliptic curves and abelian surfaces over certain number fields unramified away from $\{2,\infty\}$. An explicit example is constructed to demonstrate that good reduction is not sufficient, at $\ell = 2$, for abelian varieties of sufficiently high dimension., Comment: 9 pages, 1 table

Background: Bariatric surgery is currently the most effective method to ameliorate co-morbidities as consequence of morbidly obese patients with BMI over 35 kg/m2. Endoscopic techniques have been developed to treat patients with mild obesity and ameliorate comorbidities, but endoscopic skills are needed, beside the costs of the devices. Aim: To report a new technique for internal gastric plication using an intragastric single port device in an experimental swine model. Methods: Twenty experiments using fresh pig cadaver stomachs in a laparoscopic trainer were performed. The procedure was performed as follow in ten pigs: 1) volume measure; 2) insufflation of the stomach with CO2; 3) extroversion of the stomach through the simulator and installation of the single port device (Gelpoint Applied Mini) through a gastrotomy close to the pylorus; 4) performance of four intragastric handsewn 4-point sutures with Prolene 2-0, from the gastric fundus to the antrum; 5) after the performance, the residual volume was measured. Sleeve gastrectomy was also performed in further ten pigs and pre- and post-procedure gastric volume were measured. Results: The internal gastric plication technique was performed successfully in the ten swine experiments. The mean procedure time was 27±4 min. It produced a reduction of gastric volume of a mean of 51%, and sleeve gastrectomy, a mean of 90% in this swine model. Conclusion: The internal gastric plication technique using an intragastric single port device required few skills to perform, had low operative time and achieved good reduction (51%) of gastric volume in an in vitro experimental model. RESUMO Racional: A cirurgia bariátrica é atualmente o método mais efetivo para melhorar as co-morbidades decorrentes da obesidade mórbida com IMC acima de 35 kg/m2. Técnicas endoscópicas foram desenvolvidas para tratar pacientes com obesidade leve e melhorar as comorbidades, mas habilidades endoscópicas são necessárias, além dos custos. Objetivo: Relatar uma nova técnica para a plicatura gástrica interna utilizando um dispositivo intragástrico de portal único em modelo experimental de suínos. Métodos: Foram realizados 20 experimentos utilizando estômagos de cadáver de porco fresco em um instrutor laparoscópico. O procedimento foi realizado da seguinte forma em dez porcos: 1) medida de volume; 2) insuflação do estômago com CO2; 3) extroversão do estômago através do simulador e instalação do dispositivo de uma única via (Gelpoint Applied Mini) através de uma gastrotomia próxima ao piloro; 4) realização de quatro suturas de quatro pontos intra-gástricas com Prolene 2-0, desde o fundo gástrico até o antro; 5) medição do volume residual. A gastrectomia vertical foi também realizada em mais dez suínos e o volume gástrico pré e pós-procedimento foi medido. Resultados: A técnica de plicatura gástrica interna foi realizada com sucesso nos dez experimentos com suínos. O tempo médio do procedimento foi de 27±4 min. Produziu redução do volume gástrico em média de 51%, e a gastrectomia vertical em média de 90% neste modelo suíno. Conclusão: A técnica de plicatura gástrica interna, utilizando um dispositivo intragástrico de uma única via, exigiu poucas habilidades para ser realizada, teve baixo tempo operatório e obteve boa redução (51%) do volume gástrico em um modelo experimental in vitro.

Fix an elliptic curve E / Q . For each prime p of good reduction, let a p = p + 1 − # E ( F p ) . A well-known theorem of Hasse asserts that | a p | ≤ 2 p . We say that p is a champion prime for E if a p = − ⌊ 2 p ⌋ , that is, # E ( F p ) is as large as allowed by the Hasse bound. Analogously, we call p a trailing prime if a p = ⌊ 2 p ⌋ . In this note, we study the frequency of champion and trailing primes for CM elliptic curves. Our main theorem is that for CM curves, both the champion primes and trailing primes have counting functions ∼ 2 3 π x 3 / 4 / log ⁡ x , as x → ∞ . This confirms (in corrected form) a recent conjecture of James–Tran–Trinh–Wertheimer–Zantout.

Introduction: The proximal tibial fractures are associated with high energy trauma and present with difficulty in treatment due to poor skin condition. In our study, we have operated 32 patients with hybrid external fixator for definitive treatment of these fractures. This technique gives good reduction, adequate stability, early mobilization and less complications. Aim: To assess the functional and radiological outcome after application of hybrid external fixator in proximal tibial fractures. Material and Methods: Prospective analysis of 32 patients was done with proximal tibial fractures classified according to Schatzker classification in our institute who were treated by application of hybrid external fixator. Results: In our study, Road traffic accident was the most common mechanism of injury (n=23). There was a male preponderance (n=23) in our study. The mean age was 39.2±11.0 years. Functional and radiological outcome was assessed using Rasmussen functional and radiological criteria at final followup. 16 patients had an excellent functional outcome and 13 patients had excellent radiological outcome. Superficial pin tract infection was seen in 4 patients which healed uneventfully. Varus malalignment was seen in 2 patients. Conclusion: Hybrid external fixator is a simple way to treat proximal tibial fractures with a low complication rate and good clinical outcomes.

A rational map with good reduction in the field \begin{document} $\mathbb{Q}_p$ \end{document} of \begin{document} $p$ \end{document} -adic numbers defines a \begin{document} $1$ \end{document} -Lipschitz dynamical system on the projective line \begin{document} $\mathbb{P}^1(\mathbb{Q}_p)$ \end{document} over \begin{document} $\mathbb{Q}_p$ \end{document} . The dynamical structure of such a system is completely described by a minimal decomposition. That is to say, \begin{document} $\mathbb{P}^1(\mathbb{Q}_p)$ \end{document} is decomposed into three parts: finitely many periodic orbits; finite or countably many minimal subsystems each consisting of a finite union of balls; and the attracting basins of periodic orbits and minimal subsystems. For any prime \begin{document} $p$ \end{document} , a criterion of minimality for rational maps with good reduction is obtained. When \begin{document} $p=2$ \end{document} , a condition in terms of the coefficients of the rational map is proved to be necessary for the map being minimal and having good reduction, and sufficient for the map being minimal and \begin{document} $1$ \end{document} -Lipschitz. It is also proved that a rational map having good reduction of degrees \begin{document} $2$ \end{document} , \begin{document} $3$ \end{document} and \begin{document} $4$ \end{document} can never be minimal on the whole space \begin{document} $\mathbb{P}^1(\mathbb{Q}_2)$ \end{document} .

Let $E/\mathbb{Q}$ be a semistable elliptic curve such that $\mathrm{ord}_{s=1}L(E,s) = 1$. We prove the $p$-part of the Birch and Swinnerton-Dyer formula for $E/\mathbb{Q}$ for each prime $p \geq 5$ of good reduction such that $E[p]$ is irreducible: $$ \mathrm{ord}_p \left (\frac{L'(E,1)}{\Omega_E\cdot\mathrm{Reg}(E/\mathbb{Q})} \right ) = \mathrm{ord}_p \left (\#\mathrm{Sha}(E/\mathbb{Q})\prod_{\ell\leq \infty} c_\ell(E/\mathbb{Q}) \right ). $$ This formula also holds for $p=3$ provided $a_p(E)=0$ if $E$ has supersingular reduction at $p$.