Functions of class Ck without derivatives

We describe a general axiomatic way to define functions of class Ck, k ∈ N∪{∞} on topological abelian groups. In the category of Banach spaces, this definition coincides with the usual one. The advantage of this axiomatic approach is that one can dispense with the notion of norms and limit procedures. The disadvantage is that one looses the derivative, which is replaced by a local linearizing factor. As an application we use this approach to define C∞ functions in the setting of graded/super manifolds

On Weyl Quantization from geometric Quantization

A. Weinstein has conjectured a nice looking formula for a deformed product of functions on a hermitian symmetric space of non-compact type. We derive such a formula for symmetric symplectic spaces using ideas from geometric quantization and prequantization of symplectic groupoids. We compute the result explicitly for the natural 2-dimensional symplectic manifolds: the euclidean plane, the sphere and the hyperbolic plane. For the euclidean plane we obtain the well known Moyal-Weyl product. The other cases show that Weinstein's original idea should be interpreted with care. We conclude with comments on the status of our result.Comment: 11 pages. (v2: corrected a couple of typos

Symplectic connections and Fedosov's quantization on supermanifolds

A (biased and incomplete) review of the status of the theory of symplectic connections on supermanifolds is presented. Also, some comments regarding Fedosov's technique of quantization are made.Comment: Submitted to J. of Phys. Conf. Se

Projective Fourier Duality and Weyl Quantization

The Weyl-Wigner correspondence prescription, which makes large use of Fourier duality, is reexamined from the point of view of Kac algebras, the most general background for noncommutative Fourier analysis allowing for that property. It is shown how the standard Kac structure has to be extended in order to accommodate the physical requirements. An Abelian and a symmetric projective Kac algebras are shown to provide, in close parallel to the standard case, a new dual framework and a well-defined notion of projective Fourier duality for the group of translations on the plane. The Weyl formula arises naturally as an irreducible component of the duality mapping between these projective algebras.Comment: LaTeX 2.09 with NFSS or AMSLaTeX 1.1. 102Kb, 44 pages, no figures. requires subeqnarray.sty, amssymb.sty, amsfonts.sty. Final version with text improvements and crucial typos correction

Cotangent bundle quantization: Entangling of metric and magnetic field

For manifolds $\cal M$ of noncompact type endowed with an affine connection (for example, the Levi-Civita connection) and a closed 2-form (magnetic field) we define a Hilbert algebra structure in the space $L^2(T^*\cal M)$ and construct an irreducible representation of this algebra in $L^2(\cal M)$. This algebra is automatically extended to polynomial in momenta functions and distributions. Under some natural conditions this algebra is unique. The non-commutative product over $T^*\cal M$ is given by an explicit integral formula. This product is exact (not formal) and is expressed in invariant geometrical terms. Our analysis reveals this product has a front, which is described in terms of geodesic triangles in $\cal M$. The quantization of $\delta$-functions induces a family of symplectic reflections in $T^*\cal M$ and generates a magneto-geodesic connection $\Gamma$ on $T^*\cal M$. This symplectic connection entangles, on the phase space level, the original affine structure on $\cal M$ and the magnetic field. In the classical approximation, the $\hbar^2$-part of the quantum product contains the Ricci curvature of $\Gamma$ and a magneto-geodesic coupling tensor.Comment: Latex, 38 pages, 5 figures, minor correction

Obstruction Results in Quantization Theory

We define the quantization structures for Poisson algebras necessary to generalise Groenewold and Van Hove's result that there is no consistent quantization for the Poisson algebra of Euclidean phase space. Recently a similar obstruction was obtained for the sphere, though surprising enough there is no obstruction to the quantization of the torus. In this paper we want to analyze the circumstances under which such obstructions appear. In this context we review the known results for the Poisson algebras of Euclidean space, the sphere and the torus.Comment: 34 pages, Latex. To appear in J. Nonlinear Scienc

Comparison of laparoscopic versus robot-assisted versus transanal total mesorectal excision surgery for rectal cancer:a retrospective propensity score-matched cohort study of short-term outcomes

BACKGROUND: Laparoscopic total mesorectal excision (TME) surgery for rectal cancer has important technical limitations. Robot-assisted and transanal TME (TaTME) may overcome these limitations, potentially leading to lower conversion rates and reduced morbidity. However, comparative data between the three approaches are lacking. The aim of this study was to compare short-term outcomes for laparoscopic TME, robot-assisted TME and TaTME in expert centres. METHODS: Patients undergoing rectal cancer surgery between 2015 and 2017 in expert centres for laparoscopic, robot-assisted or TaTME were included. Outcomes for TME surgery performed by the specialized technique in the expert centres were compared after propensity score matching. The primary outcome was conversion rate. Secondary outcomes were morbidity and pathological outcomes. RESULTS: A total of 1078 patients were included. In rectal cancer surgery in general, the overall rate of primary anastomosis was 39.4, 61.9 and 61.9 per cent in laparoscopic, robot-assisted and TaTME centres respectively (P < 0.001). For specialized techniques in expert centres excluding abdominoperineal resection (APR), the rate of primary anastomosis was 66.7 per cent in laparoscopic, 89.8 per cent in robot-assisted and 84.3 per cent in TaTME (P < 0.001). Conversion rates were 3.7 , 4.6 and 1.9 per cent in laparoscopic, robot-assisted and TaTME respectively (P = 0.134). The number of incomplete specimens, circumferential resection margin involvement rate and morbidity rates did not differ. CONCLUSION: In the minimally invasive treatment of rectal cancer more primary anastomoses are created in robotic and TaTME expert centres

Long-term survival after hyperthermic intraperitoneal chemotherapy using mitomycin C or oxaliplatin in colorectal cancer patients with synchronous peritoneal metastases:A nationwide comparative study

Objectives: In the Netherlands, limited variability exists in performance of cytoreductive surgery and hyperthermic intraperitoneal chemotherapy (CRS-HIPEC) among centers treating colorectal peritoneal metastases (PM), except for the intraperitoneal drug administration. This offers a unique opportunity to investigate any disparities in survival between the two most frequently used HIPEC regimens worldwide: mitomycin C (MMC) and oxaliplatin.Methods: This was a comparative, population-based cohort study of all Dutch patients diagnosed with synchronous colorectal PM who underwent CRS-HIPEC between 2014 and 2017. They were retrieved from the Netherlands Cancer Registry. Main outcome was overall survival (OS). The effect of the intraperitoneal drug on OS was investigated using multivariable Cox regression analysis.Results: In total, 297 patients treated between 2014 and 2017 were included. Among them, 177 (59.6%) received MMC and 120 (40.4%) received oxaliplatin. Only primary tumor location was different between the two groups: more left-sided colon in the Oxaliplatin group (47.5% vs. 33.3%, respectively, p=0.048). The 1-, 2- and 3-year OS were 84.6% vs. 85.8%, 61.6% vs. 63.9% and 44.7% vs. 53.5% in patients treated with MMC and oxaliplatin, respectively. Median OS was 30.7 months in the MMC group vs. 46.6 months in the oxaliplatin group (p=0.181). In multivariable analysis, no influence of intraperitoneal drug on survival was observed (adjusted HR 0.77 [0.53-1.13]).Conclusions: Long-term survival between patients treated with either MMC or oxaliplatin during CRS-HIPEC was not significantly different. (C) 2020 Elsevier Ltd, BASO similar to The Association for Cancer Surgery, and the European Society of Surgical Oncology. All rights reserved.</p