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This is a survey of local and global classification results concerning Dupin hypersurfaces in Sn(or Rn) that have been obtained in the context of Lie sphere geometry. The emphasis is on results that relate Dupin hypersurfaces to isoparametric hypersurfaces in spheres. Along with these classification results, many important concepts from Lie sphere geometry, such as curvature spheres, Lie curvatures, and Legendre lifts of submanifolds of Sn(or Rn), are described in detail. The paper also contains several important constructions of Dupin hypersurfaces with certain special properties.

The complex conductivity of a superconducting thin film is related to the quasiparticle density, which depends on the physical temperature and can also be modified by external pair breaking with photons and phonons. This relationship forms the underlying operating principle of Kinetic Inductance Detectors (KIDs), where the detection threshold is governed by the superconducting energy gap. We investigate the electromagnetic properties of thin-film aluminum that is proximitized with either a normal metal layer of copper or a superconducting layer with a lower $T_{C}$, such as iridium, in order to extend the operating range of KIDs. Using the Usadel equations along with the Nam expressions for complex conductivity, we calculate the density of states and the complex conductivity of the resulting bilayers to understand the dependence of the pair breaking threshold, surface impedance, and intrinsic quality factor of superconducting bilayers on the relative film thicknesses. The calculations and analyses provide theoretical insights in designing aluminum-based bilayer kinetic inductance detectors for detection of microwave photons and athermal phonons at the frequencies well below the pair breaking threshold of a pure aluminum film.

IMPORTANCE: Pseudomyxoma peritoni, a rare condition characterized by mucinous ascites and peritoneal deposits, mainly originates from a ruptured mucinous appendix tumor and is considered an indolent disease but can progress and become fatal. Optimal treatment to improve cure and survival rates involves complete cytoreductive surgery (CCRS) and hyperthermic intraperitoneal chemotherapy (HIPEC). Accurate predictive models are useful in supporting and informing treatment strategies and stratifying patient follow-up. OBJECTIVE: To evaluate the prognostic significance of clinically important variables and generate validated nomograms to predict overall (OS) and disease-free survival (DFS) following CCRS and hyperthermic intraperitoneal HIPEC for pseudomyxoma peritonei (PMP) of appendiceal origin. DESIGN, SETTING, AND PARTICIPANTS: This retrospective study used prospectively collected data on patients who had cytoreductive surgery (CRS) and HIPEC in a single institution between 1994 and 2018. The cohort was randomly allocated into development (70%) and validation (30%) sets. Univariate and multivariate analyses were performed with Cox proportional hazards regression. MAIN OUTCOMES AND MEASURES: A prediction model was developed with significant prognostic factors identified by multivariate analysis. The model’s prognostic performance was evaluated with the concordance index (C index). The nomogram was calibrated by comparing the predicted and observed probabilities. RESULTS: Of 2637 CRS and HIPEC operations, 1102 patients (female, 64.4%; median age [IQR], 57.0 [48.0-66.0] years) (41.8%) had CCRS for PMP of appendiceal origin. Elevated tumor markers, peritoneal carcinomatosis index, gastrectomy, and tumor grade were independent predictive factors for DFS. Gender, age, elevated tumor makers, peritoneal carcinomatosis index, and tumor grade influenced OS. The nomograms were generated with respective prognostic factors. The nomograms showed good performance in predicting survival. Median OS of the cohort was 16.5 years (95% CI, 13.7-19.2) with a 5-year probability of survival of 80.2%. The median DFS was 10.3 years (95% CI, 7.2- 13.3) and the 5-year probability of recurrence-free survival was 60.5%. CONCLUSIONS AND RELEVANCE: Clinically important independent predictors for survival and recurrence were selected to develop the nomograms for OS and DFS. These 2 nomograms are user friendly and useful tools for patient management with clinical trial design applications.

PURPOSE: Most series report lymph node involvement as the main predictor for local recurrence. The principal lymphatic drainage of the rectum is to nodes in the mesorectum and then nodes along the superior rectal and inferior mesenteric arteries. If total mesorectal excision provides adequate block dissection of the lymphatics of the rectum, good local control with low rates of local recurrence should be achieved even in node-positive disease. METHODS: Prospective data on all rectal cancers have been collected since 1978; 170 patients with Dukes C rectal cancer have undergone anterior resection and total mesorectal excision. We did not perform any internal iliac node dissections. Follow-up data were analyzed for local recurrence and distant recurrence. RESULTS: The local recurrence rate was 2 percent for Dukes A cases, 4 percent for Dukes B, and 7.5 percent for Dukes C ( P = 0.0127). The systemic recurrence rate was 8 percent for Dukes A, 18 percent for Dukes B, and 37 percent for Dukes C ( P = 0.0001). CONCLUSIONS: If surgical priority is given to the difficult task of excision of the whole mesorectum, anterior resection with total mesorectal excision in node-positive rectal cancer, local recurrence rates of < 10 percent can be achieved.

Goblet cell carcinomas (GCC) are a rare, aggressive sub-type of appendiceal tumours with neuroendocrine features, and controversy exists with regards to therapeutic strategy. We undertook a retrospective review of GCC patients surgically treated at two tertiary referral centres. Clinical and histopathological data were extracted from a prospectively maintained database. Survival analyses utilised Kaplan–Meier methodology. Twenty-one patients were identified (9 females). Median age at diagnosis was 55 years (range 32–77). There were 3, 6 and 9 grade 1, 2 and 3 tumours, respectively. One, 10, 5 and 5 patients had stage I, II, III and IV disease at diagnosis, respectively. There were 8, 10 and 3 Tang class A, B and C tumours, respectively. Index operation was appendectomy (n= 12), right hemicolectomy (n= 6) or resections including appendix/right colon, omentum and the gynaecological system (n= 3). Eight patients underwent completion right hemicolectomy. Surgery for recurrence included small bowel resection (n= 2), debulking with peritonectomy and heated intraperitoneal chemotherapy, and hysterectomy and bilateral salpingo-oophorectomy (all n= 1). Median follow-up was 30 months (range 2.5–123). One-, 3- and 5-year OS was 79.4, 60 and 60%, respectively. Mean OS (1-, 3-, and 5-year OS) for Tang class A, B and C tumours were 73.1 months (85.7, 85.7, 51.4%), 83.7 months (all 66.7%) and 28.5 months (66.7, 66.7%, not reached), respectively. Chromogranin A/B and 68Ga-DOTATATE PET/CT were not useful in follow-up, but CEA, CA 19-9, CA 125 and 18F-FDG PET/CT identified tumour recurrence. GCC must be clearly discriminated from relatively indolent appendiceal neuroendocrine neoplasms. 18F-FDG PET/CT and CEA/CA19-9/CA 125 are useful in detecting recurrence of GCC.

Pseudomyxoma peritonei (PMP) is a complex disease with unique biological behavior that usually arises from appendiceal mucinous neoplasia. The classification of PMP and its primary appendiceal neoplasia is contentious, and an international modified Delphi consensus process was instigated to address terminology and definitions. A classification of mucinous appendiceal neoplasia was developed, and it was agreed that “mucinous adenocarcinoma” should be reserved for lesions with infiltrative invasion. The term “low-grade appendiceal mucinous neoplasm” was supported and it was agreed that “cystadenoma” should no longer be recommended. A new term of “high-grade appendiceal mucinous neoplasm” was proposed for lesions without infiltrative invasion but with high-grade cytologic atypia. Serrated polyp with or without dysplasia was preferred for tumors with serrated features confined to the mucosa with an intact muscularis mucosae. Consensus was achieved on the pathologic classification of PMP, defined as the intraperitoneal accumulation of mucus due to mucinous neoplasia characterized by the redistribution phenomenon. Three categories of PMP were agreed—low grade, high grade, and high grade with signet ring cells. Acellular mucin should be classified separately. It was agreed that low-grade and high-grade mucinous carcinoma peritonei should be considered synonymous with disseminated peritoneal adenomucinosis and peritoneal mucinous carcinomatosis, respectively. A checklist for the pathologic reporting of PMP and appendiceal mucinous neoplasms was also developed. By adopting the classifications and definitions that were agreed, different centers will be able to use uniform terminology that will allow meaningful comparison of their results.

Pseudomyxoma peritonei is a diffuse peritoneal malignancy that generally originates form a perforated appendiceal tumor. Optimal treatment requires extensive surgical resection to achieve complete cytoreduction combined with hyperthermic intraperitoneal chemotherapy. In a proportion of patients this is impossible, in particular, owing to extensive irresectable small-bowel involvement. There is ongoing debate as to the role of maximal tumor debulking in such cases.

It is well known in the industry that the technology nodes from 30nm and below will require model based SRAF / OPC for critical layers to meet production required process windows. Since the seminal paper by Saleh and Sayegh[1][2] thirty years ago, the idea of using inverse methods to solve mask layout problems has been receiving increasing attention as design sizes have been steadily shrinking. ILT in its present form represents an attempt to construct the inverse solution to a constrained problem where the constraints are all possible phenomena which can be simulated, including: DOF, sidelobes, MRC, MEEF, EL, shot-count, and other effects. Given current manufacturing constraints and process window requirements, inverse solutions must use all possible degrees of freedom to synthesize a mask. Various forms of inverse solutions differ greatly with respect to lithographic performance and mask complexity. Factors responsible for their differences include composition of the cost function that is minimized, constraints applied during optimization to ensure MRC compliance and limit complexity, and the data structure used to represent mask patterns. In this paper we describe the level set method to represent mask patterns, which allows the necessary degrees of freedom for required lithographic performance, and show how to derive Manhattan mask patterns from it, which can be manufactured with controllable complexity and limited shot-counts. We will demonstrate how full chip ILT masks can control e-beam write-time to the level comparable to traditional OPC masks, providing a solution with maximized lithographic performance and manageable cost of ownership that is vital to sub-30nm node IC manufacturing.

For low k1 lithography the resolution of critical patterns on large designs can require advanced resolution enhancement techniques for masks including scattering bars, complicated mask edge segmentation and placement, etc. Often only a portion of a large layout will need this sophisticated mask design (the hotspot), with the remainder of layout being relatively simple for OPC methods to correct. In this paper we show how inverse lithography technology (ILT) can be used to correct selected regions of a large design after standard OPC has been used to correct the simple portions of the layout. The hotspot approach allows a computationally intensive ILT to be used in a limited way to correct the most difficult portions of a design. We will discuss the most important issues such as: model matching between ILT and OPC corrections; transition region corrections near the ILT and OPC boundary region; mask complexity; total combined runtime. We will show both simulated and actual wafer lithographic improvements in the hotspot regions.

Using the method of moving frames, we prove that any irreducible Dupin hypersurface in S5with four distinct principal curvatures and constant Lie curvature is equivalent by Lie sphere transformation to an isoparametric hypersurface in S5.

The notions of focal point and support function are considered for a nondegenerate hypersurfaceMn in affine spaceRn+1 equipped with an equiaffine transversal field. IfMn is locally strictly convex, these two concepts are related via an Index theorem concerning the critical points of the support functions onMn. This is used to obtain characterizations of spheres and ellipsoids in terms of the critical point behavior of certain classes of affine support functions.

Pinkall's standard constructions for obtaining a Dupin hypersurface W in RN from a Dupin hypersurface M in Rn, N>n, are studied in the context of Lie sphere geometry. It is shown that a compact Dupin hypersurface W in RN with g distinct principal curvatures at each point is reducible to a compact Dupin hypersurface M in Rn if and only if g=2.

Abstract.: Using the method of moving frames, we prove that any locally irreducible Dupin hypersurface in S n with three distinct principal curvatures is equivalent by Lie sphere transformation to an isoparametric hypersurface in S n .

Let $ M$, and let $ J$ in terms of its second fundamental form. This takes a particularly tractable form when $ M$ is a principal vector for each unit normal $ \xi $. The rank of the focal map onto a sheet of the focal set of $ M$ is principal with corresponding principal curvature $ \mu $ has constant rank, then that sheet is a complex submanifold over which $ M$. First, there are no totally umbilic real hypersurfaces in $ C{P^m}$ Theorem 3. Let $ M$be a connected real hypersurface in $ C{P^m}$ $ m \geqslant 3$with at most two distinct principal curvatures at each point. Then $ M$is an open subset of a geodesic hypersphere. Secondly, we show that there are no Einstein real hypersurfaces in and characterize the geodesic hyperspheres and two other classes of hypersurfaces in terms of a slightly less stringent requirement on the Ricci tensor in Theorem 4.

For increasingly small and dense designs requiring adequate DOF, MEEF, and EL, numerous technologies have been employed to increase yield. Some techniques such as process optimization (i.e. SMO) are effective, but can be costly and time consuming, and are not easily modifiable once an initial choice is made. Design optimization can be done separately from knowledge of the fab's OPC correction, but for sub 32nm nodes the complexity and interaction of the design target shapes is becoming too complicated for predefined design rules to produce an acceptable result. In this paper we introduce a method called Lithographically Enhanced Edge Design (LEED) suited for IDMs. This joint target and mask optimization method takes into account the full OPC correction and process, and modifies the user's design in a controlled way so as to produce a new design with improved lithographic performance which can be used in place of the initial design. Control is given to the user so that inter-layer dependencies are not broken. Also, integrated target, mask, and source optimization is available in cases where target and mask optimization in not sufficient to produce adequate results. The use of ILT allows efficient target, mask, and source correction without extensive user OPC scripting and target modification sweeping. We show LEED results which enable production at 20x node.