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Dr. McKenzie is the Executive Director of the Office for Education Policy at the University of Arkansas. A teacher at heart, Sarah has taught students from pre-kindergarten to university level, has provided training and consulting to public school districts, and has presented nationally and internationally on educational statistics. Since her days as an elementary teacher, Sarah has been an advocate for valid and reliable assessment data for students. She is passionate about using high quality data with students, teachers and parents to inform their educational decisions. Dr. McKenzie received her B.S. in literature from Claremont McKenna College, M.A. in Early Childhood Education from Mills College, and Ph.D. in Education Statistics and Research Methods from the University of Arkansas. | {
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{"url":"https:\/\/iwaponline.com\/wst\/article\/82\/11\/2400\/77590\/Performance-analyses-of-existing-sanitary-sewers","text":"## Abstract\n\nThe widespread uptake of household water-saving systems (i.e. appliances, fittings, rainwater harvesting tanks, etc.) usually aims to reduce the gap between water demand and supply without considering the performances of downstream sanitary sewers (SSs). This paper presents an analysis approach that examines the lifespan interaction of water-saving schemes (WSSs) and operation of existing SSs. Examined are three probable ways of using (or not using) these water systems, including the conventional (baseline), full application and optimal selection of efficient WSSs. For optimality, a method that maximises the WSS potential efficiency (overall) and minimises the cost of WSSs including the associated savings across the entire existing SS subject to constraints at the end of the planning horizon has been formulated. The problem is solved using a non-dominated genetic algorithm to obtain optimal solutions. Decision variables include various water use (or saving) capacities of water-saving schemes at different inflow nodes (locations). The method was demonstrated on the subsystem of the Tsholofelo Extension SS. The results indicate impactful and revealing interactions between water use efficiency, instantaneous hydraulic performances and existing SS upgrade requirements due to different applications of WSSs. The impacts and revelations observed would inform decisions during lifespan operations and management of SSs.\n\n## HIGHLIGHTS\n\n\u2022 Conventional, optimal and maximum water savings have been analysed considering existing sanitary sewers.\n\n\u2022 Trade-offs between cost, cost savings and water efficiency have been analysed.\n\n\u2022 Water-saving related sanitary sewer (SS) upgrade postponement has been articulated.\n\n\u2022 Flows across the SS lifespan are quantified using the proposed velocity violation factor.\n\n\u2022 Spatial and temporal considerations would inform existing SS upgrades.\n\n## INTRODUCTION\n\nGlobally, the widespread manufacturing and uptake of water-saving schemes (WSSs) at household levels is currently on the rise. These developments reduce the gap between water demand and supply. Household water sources (rainwater harvesting and greywater recycling and\/or reuse systems) and water-saving products (appliances and fittings) all referred to as WSSs in this paper, are progressively being developed, promoted and used in households. In the context of water supply in general, as much as reducing water demand may have benefits (i.e. less wastewater generated), there are potential adverse effects on the performances of sanitary sewers (SSs) downstream. The adverse effects may arise because existing networks would be subjected to less amounts of wastewater than they were designed to dispose of.\n\nAccording to the sewer design principle, SS (pipe) flow velocities mainly depend on the pipe slopes, diameters and demands (or wastewater flows). Changing one or more of these parameters would prompt appropriate adjustment(s) of others to maintain the required levels of service in existing sewers. However, for existing networks, it is worth noting that slopes and conduit diameters remain the same (fixed) while wastewater inflows and velocities change, thus leading to questions that motivate the current study. This principle implies that a potential decrease in wastewater flows subjects existing sewers to potential risks such as those of blockages that are associated with the infringement of pipe flow self-cleansing velocities (see McGhee 1991; Mattsson et al. 2015). These velocities are generally used as design constraints (or objectives) that are necessary for obtaining the acceptable performances of sewers, i.e. SSs are required to meet certain hydraulic standards.\n\nThe aforementioned principles show the need to analyse WSSs (and their associated benefits) in the context of technical aspects of SSs. This analysis is necessitated by the link that exists between the use of WSSs and SSs (i.e. WSSs affect hydraulics of SSs), which suggests that the benefits (e.g. cost savings) of saving water in households may conflict with the hydraulic performances of existing SSs. WSSs also present challenges in terms of various lifespans that require cautious decision making in order to find more viable and cost-effective schemes. On the other hand, reduction of water supplied, and wastewater generated can contribute significantly to reduction in the capital and operation costs. Water saving decreases treatment by reducing the need for storage of effluent water, untreated waste overflows and providing for increased efficiency of the plant processes by reducing flow rates (Robinson et al. 1984). However, reduced flow rates can also impact wastewater treatment facilities negatively. In this regard, McKenna et al. (2018) predicted a significant increase in effluent total ammonia nitrogen, nitrates, nitrites and total inorganic nitrogen concentrations, and thus higher costs when the influent flow rates were reduced by more than 43% in the treatment plants they studied. Therefore, the modern trends of implementing WSSs require proactive management and integrated water system approaches to obtain sustainable solutions. Despite the principles discussed here, many studies (e.g., Fidar et al. 2010; Proenca et al. 2011; Campisano & Modica 2015) have considered water demand management separately without taking into account water network (e.g., SS) technical issues or vice versa (e.g., Austin et al. 2014; Duque et al. 2016).\n\nThe aim of this study is to develop and demonstrate a method that consists of analysis approaches, which examine, compare and reveal lifespan interactions of existing SS hydraulics and different levels of WSS uptake resulting from conventional, optimal and full applications of WSSs. This approach should provide efficiencies and SS upgrade requirements. The method would therefore support timely and appropriate decisions for system upgrades and promote incorporation of the widespread use of WSSs in the management of SSs because reduction of water consumption becomes the main priority both economically and environmentally. The method evaluates how the required hydraulic performance(s) of existing SSs may be violated by the selection of WSSs (i.e. certain level of water use efficiency) during their service life. In the uptake of WSSs, the existing SS operational performances would be basically tested against technical standards amidst the resultant benefits such as cost savings for SS wastewater treatment. However, water quality implications of WSSs are not considered. After the background of WSS applications, the proposed method is explained and applied to a subsystem of a SS network.\n\n## BACKGROUND\n\nIn the past, the predominant approach to water management has been supply development rather than integrated supply and demand management (Robinson et al. 1984). Household demand management increasingly play a key role in water supply sustainability. The demand management interventions considered include the use of water-saving appliances and fittings together with some measures, which provide alternative sources of water that promote water use efficiency. Trends in water research put emphasis on household WSSs because of their potential benefits to communities (e.g., Fidar et al. 2010; Basupi et al. 2014). Regarding the water source perspective, Campisano & Modica (2016) evaluated the potential of rainwater harvesting systems (RWHS) to mitigate peak roof runoff due to rainfall and found that a significant peak runoff reduction exists. In another study, GhaffarianHoseini et al. (2015) ascertain that accurate design and configuration, simulation, localisation and proper maintenance are expected to accomplish the goal of using RWHSs. Furthermore, Campisano et al. (2017) reviewed the practical, theoretical and social aspects of RWHSs to ascertain state-of-the-art after noticing that RWHSs have not been implemented with systems that maximise the benefits. They found out that the degree of RWHS implementation and technology solutions are strongly influenced by economic constraints and local regulations. In this study, the proposed method assumes social acceptance of all the WSSs. Therefore, the effects of RWHSs among other WSSs are also evaluated in the hydraulic performance and the related upgrade requirements of SSs in this study.\n\nOne of the most relevant studies to the current research was carried out by Penn et al. (2013b) who modelled the effects of on-site greywater reuse and low-flush toilets on municipal sewer systems. Later, Penn et al. (2013a) optimised types of greywater recycling systems that were combined with different flush volumes of toilets, but a full range of household WSSs and the ranges of product capacities that require attention as WSSs increasingly become popular were not examined. In the similar context, Basupi (2019) integrated WSSs in the design of SSs where new decisions of the SS components and\/or functions were made, together with the selection of WSSs. However, considering that most SSs have already been constructed (fixed) and WSSs are generally implemented in the vicinity of such existing SSs, a different scope, which include analyses that are most appropriate for existing SSs rather than SS designs is therefore considered in this paper. In terms of objectives considered, Basupi (2019) optimised cost (minimised) and cost savings (maximised) without the explicit inclusion of system-wide water efficiency (%), which would further enhance trade-offs (i.e. further influencing SSs) as it has been herein addressed accordingly. In terms of SS performance analyses, Basupi (2019) and other earlier studies did not articulate and use the novel timescale and hydraulic performance indicators associated with the implementation of WSSs such as the SS upgrade requirements and self-cleansing velocity violation factor. Therefore, consequent implications of WSSs on the proposed instantaneous SS hydraulics and SS upgrade requirements of SSs, during their entire service period and beyond, respectively, have not been quantified and\/or explored yet. Performance indicators that include self-cleansing velocity violation factor, upgrade postponement, and system wide WSS efficiency considering instantaneous demands in the interaction of existing SSs and different WSS applications are therefore articulated as shown in the following section and demonstrated in the case study.\n\n## METHODOLOGY\n\nThe integrated modelling and analysis approaches considered in this paper include fiscal, SS hydraulic and the related upgrade deferral performances of different water-saving interventions. These interventions include optimal solutions, conventional and full application of WSSs. The conventional approach is associated with options of standard intervention measures while the full application of WSSs refers to the use of the most efficient components of WSSs available. An informative methodology that captures the interaction effects of WSSs and SSs is articulated and explained in the following sections.\n\n### Performances of WSSs and SSs\n\nIn order to understand and reasonably compare the interactions of WSS interventions with existing SSs, different existing water-saving models (products) and their characteristics form part of the options evaluated. The WSSs consist of local water sources such as RWHSs (i.e. different tank sizes), water-saving appliances (washing machines, dishwashers) and fittings (toilets, kitchen taps, basin taps, baths, and showerheads). Further details of water-saving products are presented in the case study section.\n\n#### Economic cost and benefits\n\nAmong other performance indicators, WSSs are analysed in terms of costs (i.e. in any currency). The total cost includes the cost of WSSs, together with cost savings in the system. The cost is expressed as equivalent annual worth (EAW) (McGhee 1991) because costs of WSS investments have different lifespans, i.e. capital expenditures are basically related by lifetime to operational expenditures. In this cost method, the capital recovery factor is used to convert the capital costs of the interventions to comparable quantities. In other words, the EAW method allows rational comparisons of intervention options that do not have equal lifespans. This method may incorporate costs of transport, treatment and final disposal of wastewater. The annual total cost () of WSSs in any currency is therefore estimated as follows:\n(1)\n(2)\n(3)\nwhere is the total number of SS inflow nodes; is the total number of households and\/or any other water use entities in the catchments of the SS inflow nodes; is the total number of WSS components; is the capital recovery factor; r is the interest rate of the k-th WSS investment; nk is the lifespan of the k-th WSS component under consideration; and are the capital and annual operational costs of the k-th WSS component, respectively; and is the total cost saving. Note that all costs\/benefits are added regardless of cost ownership because the approach used is a holistic method, which considers the general cost to the society:\n(4)\nwhere and are the unit costs of energy () and billed water required by a particular WSS, respectively; and (used to determine Bss,op) are the volumes of water used by the k-th WSS component and the total (or system) volume, respectively. The energy required for heating water that is apportioned according to water use of fixtures and appliances is estimated in kWh using the equation below:\n(5)\nwhere m is the mass (kg) of water that requires heating. Well documented proportions of WSS components that use hot water are used to estimate m, and also ratios between cold and hot water use for components are required (e.g., equal use); c is the specific heat capacity of water (J\/kg\/oC); is the change in water temperature (oC); and is the efficiency of the heating system that is used. The constant in the equation is a factor that converts Joules to kWh. The benefits (cost savings) that are eventually subtracted from the total cost are calculated as functions of water-saving efficiencies of a specific WSS () and the overall system () as follows:\n(6)\n(7)\nwhere and correspond with standard products in this instance; and are the WSS component and SS operational savings, respectively, i.e. annual operational cost savings, which are generally equivalents of the differences between the costs and\/or losses that would be incurred assuming no water efficient measures applied () and the corresponding costs with efficient measures () as follows:\n(8)\nwhere is the general operational benefit of a water system. The total daily water consumption per household or other entities is required in the evaluation of water use efficiency.\n\n#### SS capacity, water use efficiency, SS upgrade postponement and hydraulics\n\nThe existing SS analysis presented here is based on the principle that existing SSs are designed for maximum inflows that satisfy flow velocities and flow depth requirements. In this regard, these performances would be achieved at the end of the planning horizon. However, SSs should also satisfy hydraulic requirements from the beginning when inflows are lower. Given these requirements, an approach that analyses the interactions of WSSs and SSs in the planning horizon is articulated.\n\nThe analysis proposed in this study establishes the intended initial and final demands at the beginning (time equals 0) and end of the design period of an existing SS by estimating the minimum and maximum demands that barely satisfy the required flow characteristics (velocities, flow depth). From known magnitudes of demands that are distributed across an existing and well-functioning SS (meeting design criteria), amounts of these demands can be decreased or increased proportionally until the minimum or maximum threshold demands that lead to acceptable hydraulics are obtained, respectively. These minimum and maximum demand thresholds correspond to the acceptable initial and final demands in the SS, respectively. The self-cleansing flow velocity requirement is necessary for obtaining both initial and final (max) demands while the flow depth ratio governs the maximum capacity of the SS at the end of the planning horizon. With initial () and final demands ), demands across the planning horizon (design period) can be determined by fitting an exponential curve of the general form , where represents an instantaneous system demand at time t; a and b are coefficients that are determined by formulating two equations and solving them simultaneously, using the known initial ( and final () points with their corresponding system water demands. In this notation, d is the design period of the SS. These simultaneous equations eventually simplify to a general curve as follows:\n(9)\n\nThe maximum flow depth ratio would not be exceeded with optimal WSSs obtained for the maximum demand, which corresponds to the maximum flow depth ratio. Varying over time would be costs, water demands, water use efficiencies and the system violation factor () of pipe flow self-cleansing velocity.\n\nThe efficiency of implementing a WSS is estimated relative to the known standard product. In other words, the maximum flow for any water use component corresponds with a standard product. When a lower flow product is used, the standard flow becomes the base used to measure the level of efficiency of such a product. This computation requires prior knowledge of water consumption and the typical proportions of water use components (see Alliance for Water Efficiency 2015), i.e. assuming standard products were used. The rainwater use efficiencies that are required in the evaluation of WSSs are obtained by simulating different sizes of rainwater tanks using hydrological models such as RainCycle (Roebuck 2007). Different sizes of rainwater tanks with their corresponding performances are subsequently used as opportunities towards water security. Therefore, the estimation of the overall system water-saving efficiency contributed by the WSSs in percentage (%) form is formulated as follows:\n(10)\nwhere is the corresponding nodal water demand at the i-th node;\n(11)\n(12)\n(13)\nwhere i,wss is the collective WSS efficiency (%) at a demand node. The formulation of the i,wss shown may also explicitly take into account greywater reuse. The RainCycle software used here can analyse RWHSs that accommodate the combined quantities of rainwater and greywater. Rainwater and\/or recycled water is used in the washing machines (WMs), flushing of the toilets (WCs) and any outdoor\/outside water activity\/event that contributes to water demand; and represent the efficiencies for RWHSs, showerheads (SHs), bathtubs (Bs), dishwashers (DWs), kitchen taps (KTs) and basin taps (BTs), respectively; and are the collective water use efficiencies of toilets and washing machines, respectively, in the case where rainwater is used for flushing the toilets and in washing machines (i.e. non-potable use) according to the assumptions made in the formulation. Other assumptions can be made and the same procedure can be followed. It should be noted that while RWHSs contribute to the overall water use efficiency in the network, they do not affect the sewer inflows because the water used would still be discharged; and are water-saving efficiencies for toilets (i.e. WC) and washing machines (i.e. WM), respectively. The resultant efficiencies of WSSs directly influence the amounts of sewer inflows. Therefore, the sewer nodal inflow () that is generated in households can be estimated as follows:\n(14)\nwhere is the effective system efficiency of WSSs that reduces inflow, i.e. less the effect of WSSs that do not reduce sewer inflows, although this is not the case in this study; is any other household water use event that reduces the inflows received in the sewer's i-th node.\nThe estimation of the effective service roof area () that is necessary for obtaining the potential potable water-saving efficiencies of RWHSs is based on the following assumptions as adapted from Ghisi et al. (2007):\n(15)\nwhere is the number of dwellings estimated from nodal and per capita\/day water demands. is multiplied by a weighted average roof area per dwelling; and are the percentages of houses and flats in the service area; is the number of people per dwelling; is the assumed area of a house and is the area per person in flats (e.g., 3.75 m2).\nThe energy consumption of RWHS local pumps is estimated in kWh as follows (adapted from Ward et al. 2012):\n(16)\n(17)\n(18)\nwhere is the total energy consumed (kWh); is the RWHS pump efficiency; is the RWHS pump power rating (kW), which is a function of head and flow. The head is considered to be equivalent to the minimum head required for water distribution and the flow is taken as the RWHS pump capacity, (m3\/h), which is the nodal non-potable water demand; is the start-up duration and is the operating duration; is the start-up energy factor; V is the daily (i.e. 24 h) volume of rainwater pumped to serve non-potable water use; , is the percentage of water that is pumped during operation; , is the percentage of water pumped on start-ups and is the number of RWHS pump start-ups; is the percentage of water pumped per start-up. With daily volume V, , and , hence and , the average flow rate pumped at the operating point and the average flow rate in the start-up can be estimated.\nWhen the existing SS capacity explained earlier is reached, there would be upgrade requirements. Reducing inflows would postpone SS upgrade requirements, which is derived as in the following formulation:\n(19)\n(20)\nwhere is the compound annual growth rate; and that correspond to and , respectively, are the initial and final times, with indicating the duration (if the initial time is zero) required by the SS to serve adequately until the next upgrade requirement. Taking logarithms on both sides of Equation (19) and solving for t yields:\n(21)\nIf is the maximum water demand that can be served by the SS, its upgrade postponement would be formulated as follows:\n(22)\nwhere is any lower demand managed by WSSs. At different ages of the SS, the velocity violation factor is obtained by averaging the accumulated pipe flow violations () across all the sewer pipes considered as follows:\n(23)\n(24)\nwhere is the number of pipes considered, which depends on the standard adopted; is the required pipe flow velocity in the i-th sewer conduit according to design criteria (i.e. regulation); must be reached (or exceeded) at least once a day; pipe flow velocities and flow depths are obtained by running the EPA Storm Water Management Model (SWMM) 5.0 hydraulic simulator\/solver (Rossman 2004) using the kinematic wave routing method or an alternative over a defined period (e.g., 24 h) while storing values at specified intervals. The maximum instantaneous velocity stored over the entire simulation for the i-th sewer conduit is referred to as .\n\nIt should be noted that the analyses proposed here are applicable on existing networks where appropriate conditions (e.g., diameters and slopes) would have been decided during the design phase, i.e. other design conditions would already have been met for a certain level of water demand. For analysis of existing SSs, slopes and conduit diameters remain the same (fixed) while wastewater inflows and velocities change, hence only the pipe flow self-cleansing velocity violations are the focus of this analysis. With fixed diameters and slopes, pipe flow velocities would mainly be affected by the changes in water demand, which is directly associated with WSSs.\n\n### Optimisation problem\n\nFor the optimal WSS application, the solutions sought are those with minimised total cost of WSSs (both capital and operational) in any currency, while household water-saving efficiencies (%) are maximised using the demand that can be served at the maximum capacity of the SS. The methodology utilises efficiencies of water-saving technologies that are available on the market. The decision variables consist of different water-saving appliances, fittings and local water sources with their corresponding range of water-saving capacities. As much as we want to save water, there is a cost associated with it that we need to minimise. The optimisation model is formulated with two objectives as follows:\n(25)\n(26)\nThe proposed approach ensures that the optimisation problem for selection of WSSs is subjected to decision variable constraints as follows:\n(27)\nwhere Di is the value of the i-th discrete decision variable; D is a discrete set of available WSS capacities\/variables; and is the number of decision variables. The approach used to derive the minimum and maximum water demands ensures that the maximum allowed flow depth ratio is not exceeded, while may be violated. Worth observing is that can be optimised without the term in Equation (25). The optimisation problem formulated in this study is solved by using the well-known non-dominated sorting genetic algorithm (NSGA-II) (Deb et al. 2002).\n\n## CASE STUDY\n\nThe methodology presented in this study has been demonstrated on a subsystem (Figure\u00a01) of the newly developed Tsholofelo Extension SS system whose components and physical characteristics were obtained from the \u2018As built\u2019 drawings provided by the Water Utilities Corporation of Botswana. Physical characteristics include fundamental features such as invert levels of the inflow nodes that also provide the sewer slopes. The description of the sewer system, data and the necessary assumptions required in the demonstration of the methodology discussed next.\n\nFigure 1\n\nA subsystem of the Tsholofelo Extension sanitary sewer.\n\nFigure 1\n\nA subsystem of the Tsholofelo Extension sanitary sewer.\n\n### Network and WSS descriptions\n\nThe Tsholofelo Extension SS is in Gaborone, which is the capital city of Botswana. For demonstration, an adequate portion of this sprawling development area is considered in this study. The sewer considered consists of 113 inflow nodes (manholes) and 138 links (i.e. 200, 110- and 160-mm uPVC pipes for conduits 33\u201336 that are adjacent to the outfall, conduits 113\u2013116 and the rest of conduits across the network, respectively) with a total length of approximately 4.5 km. The existing sewer system should serve about 271 dwellings and\/or properties. Household sewer pipes connect and discharge inflows into the immediate manholes of known characteristic levels\/elevations. In the evaluation of hydraulic performance, calibration of the sewer model could not be performed because the location is still developing. Moreover, with appropriate data assumptions, the hydraulic characteristics of the sewer lead to informative results, i.e. the focus of the study is the comparative differences that arise when implementing WSSs rather than actual values.\n\n### Data and assumptions\n\nWater demands were distributed according to \u2018As built\u2019 drawings, which show property plots that correspond with water consumption categories (WUC 2014) and manholes into which each property will discharge wastewater. Demands of about 3.61 and 4.51 litres\/second were derived as explained earlier for initial and final values, respectively. The water consumption pattern required for hydraulic simulations was obtained from WUC (2014). The wastewater generated from households was approximated at 90% of the household indoor water use. The technical aspects of the sewer used for demonstrating the proposed approach include a maximum pipe flow depth ratio of 0.5 and a pipe flow self-cleansing velocity of 0.6 m\/s (pipes 19\u201336; 67\u201371) at least once a day (Department of Sanitation and Waste Management 2003). The demands stated here lead to the performances that meet the criteria stipulated by the Botswana design manual for certain pipe sizes considered.\n\nThe study considered the WSS product capacities available on the market and typical costs as presented in Table\u00a01. Due to lack of local data and importation of most fixtures and appliances, typical values considered reasonable were adopted. The lifespans of fixtures and appliances were obtained from the International Association of Certified Home Inspectors, while RWHSs are expected to serve for 25 years, approximately. Standard capacities of household water use components such as WCs (6.06 litres), SHs (0.16 litres\/second), Bs (192.5 litres), WMs (628 litres\/cycle\/m3), DWs (18.9 litres\/cycle), KTs (0.14 litres\/second) and BTs (0.14 litres\/second) were assumed (Alliance for Water Efficiency 2014). It is also assumed that each household has two toilets and two basin taps. On the other hand, water use appliances and other fittings (shower head, bath, washing machine, dishwasher and a kitchen tap) are single in each household.\n\nTable\u00a01\n\nSummary of typical WSS capacities, water use proportion, lifespans and adapted costs used for analysis (National Renewable Energy Laboratory 2002; Alliance for Water Efficiency 2014; Alliance for Water Efficiency 2015; Plastic-mart 2017)\n\nWSS componentCapacitiesWater use proportion (%)Lifespan (years)Costs (US$) Rainwater harvesting system 2.5; 5.0; 10 (m3\u2013 25 495; 790; 1,360 Toilet 6.06; 4.85; 4.16 (litres\/flush) 26.7 100 356; 427; 467 Showerhead 0.16; 0.13; 0.09; 0.05 (litres\/second) 16.8 100 46; 55; 64; 78 Bath 192.5; 149 (litres) 1.8 100 1,030; 1,263 Washing machine 628; 495; 428; 374 (litres\/cycle\/m321.7 10 1,260; 1,401; 1,462; 1,482 Dishwasher 18.9; 16.1; 15.7; 13.2 (litres\/cycle) 1.4 1,037; 1,193; 1,213; 1,348 Kitchen tap 0.14; 0.08 (litres\/second) 9.8 17.5 489; 711 Basin tap 0.14; 0.09; 0.08; 0.06 (litres\/second) 5.9 17.5 489; 645; 711; 756 WSS componentCapacitiesWater use proportion (%)Lifespan (years)Costs (US$)\nRainwater harvesting system\u00a02.5; 5.0; 10 (m3\u2013\u00a025\u00a0495; 790; 1,360\nToilet\u00a06.06; 4.85; 4.16 (litres\/flush)\u00a026.7\u00a0100\u00a0356; 427; 467\nShowerhead\u00a00.16; 0.13; 0.09; 0.05 (litres\/second)\u00a016.8\u00a0100\u00a046; 55; 64; 78\nBath\u00a0192.5; 149 (litres)\u00a01.8\u00a0100\u00a01,030; 1,263\nWashing machine\u00a0628; 495; 428; 374 (litres\/cycle\/m321.7\u00a010\u00a01,260; 1,401; 1,462; 1,482\nDishwasher\u00a018.9; 16.1; 15.7; 13.2 (litres\/cycle)\u00a01.4\u00a01,037; 1,193; 1,213; 1,348\nKitchen tap\u00a00.14; 0.08 (litres\/second)\u00a09.8\u00a017.5\u00a0489; 711\nBasin tap\u00a00.14; 0.09; 0.08; 0.06 (litres\/second)\u00a05.9\u00a017.5\u00a0489; 645; 711; 756\n\nThe energy cost of US$0.12\/kWh (U.S. Department of Energy 2015) was used to calculate the cost of operation energy utilised by RWHS pumps (i.e. assuming a realistic efficiency of 0.65). This unit energy cost was also used to estimate energy cost savings of different WMs and DWs. The start-up energy factor of 0.6 (Ward et al. 2012) was used. The energy uses and the efficiencies of WMs and DWs were estimated using the cost-saving calculators obtained from the U.S. Department of Energy (2015). The energy required for water heating (geysers) and\/or savings were calculated assuming the specific heat capacity of water equal to 4,190 J\/kg\/\u00b0C and the temperature rise of 40\u00b0C. In the case of WSS components such as taps (basin and kitchen), baths and showerheads, hot and cold water uses were assumed to be equal (Fidar et al. 2010). RWHS pumps were expected to meet a conservative pressure head of 15 m, which is above the required water distribution system minimum pressure (WUC 2014). The cost of US$2.2\/m3 (U.S. Department of Energy 2015) paid by customers for both water and sewerage services was used. The energy required per unit of water (0.505 kWh\/m3) in WWTPs was obtained from EPRI (2002). This energy requirement assumes insignificant cost effects of wastewater concentrations on operation considering the relatively little maximum water efficiency (i.e. <30%) compared to the significant magnitude (>43%) observed by McKenna et al. (2018). The effects of wastewater concentrations would have even lesser effects on a centralised WWTP considered in this study.\n\nThe cost data for water-saving appliances were obtained from the National Renewable Energy Laboratory (2002). The costs of RWHSs were guided by Plastic-mart (2017). The discount rate of 5% was used for the cost of interventions. Note that all the costs were converted to a common year as shown in Table\u00a01 for sensible comparisons using the online inflation calculator (CoinNews Media Group LLC 2015).\n\nThe potential water-saving efficiencies of appliances were estimated using standard capacities according to the U.S. federal standards (Alliance for Water Efficiency 2014) and a variety of other commercially available water-saving models (see Alliance for Water Efficiency 2015). As for RWHSs, the efficiency of demand reduction is modelled by performing separate hydraulic analyses of each RWHS size (i.e. discrete sizes of 2.5, 5 and 10 m3) under each nodal catchment using the RainCycle Standard model (Roebuck 2007). An approximated average rainfall of 490 mm\/annum was expressed and input into the model in terms of mm\/day, together with the catchment area that differs for each network node. The effective rooftop catchment area for each node is estimated according to Equation (15). Additional key parameter inputs are catchment runoff coefficient (0.85), filter coefficient (0.9), recycled greywater and the daily non-potable water demand. Non-potable water demand differs for each network node. The water use components excluded from Table\u00a01 are outdoor water use (2.2%) and leakages (13.7%) (Alliance for Water Efficiency 2015).\n\nIt should be noted that more efficient WSSs with smaller proportions of water use can influence pipe flows more than those with larger proportions. For instance, the most efficient showerheads presented in Table\u00a01 would influence the pipe flows much more than the volumes of the toilet flushes despite toilets having a bigger share of water use. The total water use of taps was shared between the kitchen and basin taps according to the ratio of UK kitchen tap to basin tap water use (i.e. 5:3). The output of the RainCycle model is the overall water-saving efficiencies (%) of the non-potable water demand. Uncertainties that are associated with rainfall, demand and other parameters that influence efficiency of RWHSs are beyond the scope of the methodology presented here. In the case of optimisation, the NSGA-SWMM model was run with a population of 100 for 5,000 generations. Multiple (six) independent runs (with different initial seeds) were carried out to obtain and confirm the best convergence whose solutions are shown and analysed in this paper.\n\n## RESULTS AND DISCUSSION\n\nThe impacts of conventional and full application of WSS approaches on the SS are presented and compared against each other and the non-dominated solutions obtained from the formulated two-objective optimisation problem as shown in Figures\u00a025 and Table\u00a02. Table\u00a01 shows characteristics and costs of WSSs used. These approaches are herein referred to as conventional, full application of WSSs and optimal WSS solutions, respectively. They are all evaluated in the same model presented in this study to make comparable discussions. It should be noted that the full application of WSSs is also referred to as the maximum efficiency intervention, i.e. the application of maximum efficiency of each WSS component at every node in the network. Optimal WSS solutions form a trade-off curve, which suggests that there are many optimal solutions A \u2013 C that represent arbitrary low, medium and high efficiency WSSs were selected for comparative discussions.\n\nTable\u00a02\n\nCost breakdown, water use efficiencies and pipe hydraulic performances due to conventional, full application of WSSs and selected optimal interventions\n\nWSS interventions\nPerformance measureABCConventionalMax efficiency\nOverall cost (\u00d7 US$1056.080 6.030 6.022 7.018 6.286 RWHS capital cost (\u00d7 US$1034.686\u00a04.645\u00a04.666\u00a010.90\nRWHS pumping cost (\u00d7 US$1030.282 0.282 0.282 0.280 Total cost of fittings & appliances (\u00d7 US$1056.786\u00a06.729\u00a06.712\u00a07.018\u00a06.941\nFittings & appliance operational cost savings (\u00d7 US$1047.379 7.315 7.233 7.505 Wastewater treatment savings (\u00d7 US$1040.171\u00a00.170\u00a00.169\u00a00.171\nWater-saving efficiency (%)\u00a019.81\u00a019.68\u00a019.56\u00a00\u00a019.80\nSelf-cleansing velocity deficit (factor)\nWSS interventions\nPerformance measureABCConventionalMax efficiency\nOverall cost (\u00d7 US$1056.080 6.030 6.022 7.018 6.286 RWHS capital cost (\u00d7 US$1034.686\u00a04.645\u00a04.666\u00a010.90\nRWHS pumping cost (\u00d7 US$1030.282 0.282 0.282 0.280 Total cost of fittings & appliances (\u00d7 US$1056.786\u00a06.729\u00a06.712\u00a07.018\u00a06.941\nFittings & appliance operational cost savings (\u00d7 US$1047.379 7.315 7.233 7.505 Wastewater treatment savings (\u00d7 US$1040.171\u00a00.170\u00a00.169\u00a00.171\nWater-saving efficiency (%)\u00a019.81\u00a019.68\u00a019.56\u00a00\u00a019.80\nSelf-cleansing velocity deficit (factor)\nFigure 2\n\nComparative WSS solutions including low (solution C), medium (solution B) and high (solution A) water-saving interventions in terms of cost and water use efficiency.\n\nFigure 2\n\nComparative WSS solutions including low (solution C), medium (solution B) and high (solution A) water-saving interventions in terms of cost and water use efficiency.\n\nFigure 3\n\nComparative conventional, low (solution C), medium (solution B), high (solution A) and maximum water-saving interventions in terms of (a) water use efficiency and service duration (b) water demand and service duration, and (c) water use efficiency and water demand.\n\nFigure 3\n\nComparative conventional, low (solution C), medium (solution B), high (solution A) and maximum water-saving interventions in terms of (a) water use efficiency and service duration (b) water demand and service duration, and (c) water use efficiency and water demand.\n\nFigure 4\n\nComparative conventional, low (solution C), medium (solution B), high (solution A) and maximum water-saving interventions in terms of (a) velocity violation factor and service duration (b) water use efficiency and velocity violation factor, and (c) velocity violation factor and water demand.\n\nFigure 4\n\nComparative conventional, low (solution C), medium (solution B), high (solution A) and maximum water-saving interventions in terms of (a) velocity violation factor and service duration (b) water use efficiency and velocity violation factor, and (c) velocity violation factor and water demand.\n\nFigure 5\n\nComparative WSSs, including the selected low (solution C), medium (solution B) and high (solution A) water-saving solutions, together with the maximum water-saving interventions in terms of SS upgrade postponement and water use efficiency.\n\nFigure 5\n\nComparative WSSs, including the selected low (solution C), medium (solution B) and high (solution A) water-saving solutions, together with the maximum water-saving interventions in terms of SS upgrade postponement and water use efficiency.\n\nNear optimal WSS solutions are shown in Figure\u00a02 in terms of a graph of overall cost versus water use efficiency presented by a curve for WSSs that were obtained for the SS flows. The trade-off curve indicates that increasing water use efficiency in the selection of WSS is generally associated with increasing cost because water use efficiency is attained by adding more expensive WSSs. These curves also suggest that a compromise exists between the overall cost and water use efficiency in the selection of WSSs. It is worth observing that the performance of the conventional approach is not shown in Figure\u00a02 because it is extremely far from the full application of WSSs or WSS solutions, i.e. it is completely outcompeted by WSS interventions. For example, the conventional approach would cost US$7.018 \u00d7 105 with water use efficiency of 0% while the full application of WSSs would cost US$6.286 \u00d7 105 with water use efficiency of 19.8%. The use of maximum efficiency of each WSS component in all the nodes is clearly outperformed by optimal solutions with similar or equivalent water use efficiency in terms of cost (also see Table\u00a02). WSS solutions outperform the full application of WSSs because the latter do not maximise the cost savings or minimise the costs that are dependent on water demands, which should vary appropriately in time and space for optimality. The selected solutions that represent low, medium and high efficiency WSS solutions are indicated in Table\u00a02.\n\nThe nature of interventions, solutions and the in-depth analysis of factors that bring the differences between different approaches are displayed in Table\u00a02 (two indicators considered in system evaluation are shown in bold text). This table includes WSS overall cost and benefit breakdowns of extreme approaches. The extreme interventions considered in these analyses are the conventional and full applications of WSSs. The difference between their costs (about US$7.32 \u00d7 104) is the potential benefit revealed by considering water use efficiency. In addition, for interventions with equivalent water use efficiencies, the differences between the costs are the potential benefits revealed by the optimal selection of WSSs in the context of SSs. For example, the difference between US$6.286 \u00d7 105 (full application of WSSs) and US$6.07 \u00d7 105 (WSS solution in the trade-off curve) solutions that both have water use efficiencies of about 19.8% would signify such benefit. The improvement introduced by the WSS solutions is mainly attributed to low values in the total cost of fittings and appliances, which constitute the largest part of the annual cost. In addition, the costs of RWHSs in WSS solutions are much lower than those in the case of full application of WSSs. On the other hand, the conventional approach indicates the highest total cost of fittings and appliances (US$7.018 \u00d7 105) without any water use efficiency (i.e. 0%) and the associated cost savings.\n\nIn the analyses of the likely effects of WSSs on SSs during the entire SS design period, Figure\u00a03 illustrates the performances of WSSs in the lifespan (service duration) of the SS. The performances were analysed on three perspectives; water demand, water use efficiency and SS service duration. Figure\u00a03(a) reveals that as the SS age across the planning horizon, water use efficiencies of all the WSS interventions decrease with visible separate graphs except for the full application and Solution A. For example, the maximum application of WSS interventions leads to water use efficiencies of 20.47% and 19.8% at the beginning and the end of the planning horizon, respectively. Similarly, the least efficient WSS intervention (Solution C), efficiency drops from 20.18% down to 19.56% at 0 and 50th years, respectively. Figure\u00a03(b) shows that water demand increases over the planning horizon with demands associated with the conventional approach clearly differing with those of full application of WSS and WSS solutions, which are similar across the planning horizon. The increase in demands would increase with the population until the maximum SS capacity is reached. Furthermore, Figure\u00a03(c) shows the same trends shown in Figure\u00a03(a) for all the interventions with water efficiencies that drop as water demand increases across the planning horizon. The conventional approach is not plotted in this figure because of zero water use efficiency, which would diminish graphs of focal interventions.\n\nFurther analysis shown in Figure\u00a04 includes four perspectives: pipe flow velocity violation factor, water use efficiency, SS service duration and system water demand. Figure\u00a04(a) shows that the violation factors of system flow velocities decrease over the SS service duration when water demand increases (i.e. for full WSS application and WSS solutions) while the conventional (baseline) interventions generally do not violate the flow velocities. For water efficient interventions, the violation factors are similar across the planning horizon. The significance of this analysis is that, despite selecting WSSs that do not violate flow velocity requirements in SSs at the end of the planning horizon as shown in Table\u00a02, they may violate flow velocities for the larger part of the SS service life as opposed to the conventional approach.\n\nFigure\u00a04(b) reveals that the velocity violation factor increases less rapidly when water use efficiency is low towards the end of the SS operation life. Differences among interventions are distinct except for high efficiency (Solution A) and the full application of WSS interventions due to their close efficiencies. Figure\u00a04(c) confirms the velocity violations which happen due to low connections that result in lower flows in the presence of WSSs at the commissioning of the SS. Finally, an important perspective in the uptake of WSSs is the network upgrade postponement shown in Figure\u00a05. Results of this analysis reveal that upgrade requirements of the SS studied can be postponed by a significant magnitude of about 17.6\u201317.8 years depending on the respective water use efficiencies achieved. The results also confirm that the maximum use of WSSs does not necessarily guarantee the best results.\n\nDemonstrated in Figure\u00a06 are the temporal variations of flow velocities in selected pipes of the network. The selected pipes 26 and 67 shown in Figure\u00a01 are used to illustrate the significance of implementing WSSs in the existing SS. Considered are the extreme extents of WSS applications and the time perspective of the SS performance. Figure\u00a06(a) reveals that full application of WSSs in the SS introduces significant reduction of pipe flow velocities, which is demonstrated by a clear visible separation of 24-h flow velocity graphs for conventional and full applications of WSSs. The differences in pipe 26 are the results of reduced SS inflows. In Figure\u00a06(b), the time factor reveals that the application of WSS efficiency in the SS has the potential of negative effects on the hydraulics of SSs in terms of self-cleansing velocity. Even though WSSs would be satisfactory at the end (50th year) of the planning horizon, with a self-cleansing velocity of 6 m\/s being met in the presence of WSSs, the SS would violate the velocity requirements in the initial stages and most of the time of its operation. This violation happens despite the SS performing adequately under the conventional approach. The results obtained should inform decision makers in terms of prioritisation and timing of system operations and\/or upgrades for sustainable SSs.\n\nFigure 6\n\nEffects of (a) conventional and maximum WSSs on instantaneous flow velocities over 24 h in pipe 26 and (b) maximum water efficiency on instantaneous flow velocities over 24 h in pipe 67 at the start and the end of the SS operational life.\n\nFigure 6\n\nEffects of (a) conventional and maximum WSSs on instantaneous flow velocities over 24 h in pipe 26 and (b) maximum water efficiency on instantaneous flow velocities over 24 h in pipe 67 at the start and the end of the SS operational life.\n\n## CONCLUSIONS\n\nThis study articulated novel approaches that examine the implications of the conventional, full application of WSSs and optimal WSS efficiencies on existing SS hydraulics and SS upgrade postponement. The proposed analysis method was demonstrated on an existing real-life sewer network in Tsholofelo Extension, Gaborone. The results obtained and the conclusions that are based on the assumptions, data and cost models used in the case study presented in this paper are as follows:\n\n\u2022 The analysis approach revealed that cost, benefits and water use efficiency and the determining trade-off factors in different applications of WSSs in existing SSs. In this regard, the full application of WSSs (US$6.286 \u00d7 105) is more expensive than all WSS solutions while the conventional application would be the most expensive (US$7.018 \u00d7 105) due to lack of the benefits of water use efficiency, which would have reduced the overall cost of WSSs. Furthermore, water use efficiency of WSSs reduces over time when water demand increases. Despite no violation of self-cleansing velocities with a conventional approach in the entire planning horizon, WSS solutions and the full application of WSSs would lead to violation of pipe flow velocity. For instance, the full application of WSSs considered would have a maximum flow velocity violation factor of 0.012 at the beginning of the planning horizon, which could be addressed by incorporating WSSs in the design of SSs.\n\n\u2022 Despite the pipe flow violations that may be associated with water use efficiency, the uptake of WSSs also presents desirable impacts on SS upgrade requirements. In this view, the Tsholofelo SS would require upgrades after about 17.6\u201317.8 years beyond its design period, which is expected to differ with other SSs. Conversely, the maximum use of WSSs does not guarantee the best results in terms of both efficiency and SS upgrade postponement.\n\n\u2022 The impact of WSSs and existing SS interactions differ in terms of spatial and temporal variations as indicated by different pipe hydraulics. Therefore, considerable differences exist between the impacts of the conventional approach and the full use of WSSs in terms of temporal variations of pipe flow velocities in selected pipes of the SSs over 24 h (i.e. compliance to non-compliance). Similarly, the effects of WSSs on pipe flow velocities at the beginning and end of the planning horizon would differ.\n\nThe practical implications for considering the impacts of WSSs on SSs are the informed investments (rehabilitation and\/or redesign decisions) or operations of SSs that guide sustainable solutions. For completeness, the significance (or insignificance) levels of flow reductions and the water quality implications on wastewater treatment operations regarding any specific SS network would be integrated in this approach. Further studies that apply the proposed method of different SSs with different levels of complexities and uncertainties should be carried out before the findings of this study can be considered unique or general.\n\n## ACKNOWLEDGEMENTS\n\nThe author is grateful to the Water Utilities Corporation of Botswana for availing Tsholofelo SS data that supported this study. This research did not receive any specific grant from funding agencies in the public, commercial or not-for-profit sectors.\n\n## CONFLICT OF INTEREST\n\nThere is no conflict of interest associated with this study.\n\n## DATA AVAILABILITY STATEMENT\n\nAll relevant data are included in the paper or its Supplementary Information.\n\n## REFERENCES\n\nREFERENCES\nAlliance for Water Efficiency\n2014\nNational Efficiency Standards and Specifications for Residential and Commercial Water-Using Fixtures and Appliances\n.\nAlliance for Water Efficiency\/Koeller & Co\n.\nAlliance for Water Efficiency\n2015\n.\nAvailable from: http:\/\/www.home-water-works.org\/indoor-use (accessed 21 November 2015)\nAustin\nR. J.\nChen\nA. S.\nSavic\nD. 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M.\nHassan\nN. B.\n2015\nState of the art of rainwater harvesting systems towards promoting green built environments: a review\n.\nDesalination and Water Treatment\n.\ndoi:10.1080\/19443994.2015.1021097\n.\nMattsson\nJ.\nHedstroom\nA.\nAshley\nR. M.\nViklander\nM.\n2015\nImpacts and managerial implications for sewer systems due to recent changes to inputs in domestic wastewater \u2013 a review\n.\nJournal of Environmental Management\n161\n(\n2015\n),\n188\n197\n.\nMcGhee\nT. J.\n1991\nWater Supply and Sewerage\n, 6th edn.\nMcGraw-Hill series in Water Resources and Environmental Engineering\n,\nNew York\n.\nISBN 0-07-100823-3.\nMcKenna\nA.\nSilverstein\nJ.\nSharvelle\nS.\nHodgson\nB.\n2018\nModeled response of wastewater nutrient treatment to indoor water conservation\n.\nEnvironmental Engineering Science\n35\n,\n5\n.\ndoi:10.1089\/ees.2017.0161\n.\nNational Renewable Energy Laboratory\n2002\nFederal Energy Management Program; Domestic Water Conservation Technologies\n.\nU.S. Department of Energy, Office of Energy Efficiency and Renewable Energy, DOE\/EE-0264\n.\nPenn\nR.\nFriedler\nE.\nOstfeld\nA.\n2013a\nMulti-objective evolutionary optimization for greywater reuse in municipal sewer systems\n.\nWater Research\n47\n,\n5911\n5920\n.\nPenn\nR.\nSch\u00fctze\nM.\nFriedler\nE.\n2013b\nModelling the effects of on-site greywater reuse and low flush toilets on municipal sewer systems\n.\nJournal of Environmental Management\n114\n,\n72\n83\n.\nPlastic-mart\n2017\n.\nAvailable from: http:\/\/www.plastic-mart.com\/category\/232\/rainwater-tanks (accessed 22 February 2017)\nProenca\nL. C.\nGhisi\nE.\nTavares\nD. d.\nCoelho\nG. M.\n2011\nPotential for electricity savings by reducing potable water consumption in a city scale\n.\nResources, Conservation and Recycling\n55\n,\n960\n965\n.\nRobinson\nJ. E.\nFitzgibbon\nJ. E.\nBenninger\nB. A.\n1984\nIntegrating demand management of water\/wastewater systems: where do we go from here?\n9\n(\n4\n),\n29\n36\n.\ndoi:10.4296\/cwrj0904029\n.\nRoebuck\nR.\n2007\nThe RainCycle: Rainwater Harvesting Modelling Tool\n.\n,\nUK\n.\nRossman\nL. A.\n2004\nStormwater Management Model User's Manual\n.\nSWMM5.0. Environment Protection Agency\n.\nU.S. Department of Energy Office of Energy Efficiency & Renewable Energy\n2015\n.\nWard\nS.\nButler\nD.\nMemon\nF. A.\n2012\nBenchmarking energy consumption and CO2 emissions from rainwater-harvesting systems: an improved method by proxy\n.\nWater and Environment Journal\n26\n,\n184\n190\n.\ndoi:10.1111\/j.1747-6593.2011.00279.x\n.\nWUC\n2014\nGaborone Water Supply MasterPlan Update 2010; Pre-Investment Study, Final Report\n.\nBotswana Water Utilities Corporation\n.","date":"2021-04-20 07:56:42","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5295895934104919, \"perplexity\": 3139.425297390725}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618039379601.74\/warc\/CC-MAIN-20210420060507-20210420090507-00575.warc.gz\"}"} | null | null |
{"url":"https:\/\/blog.1a23.com\/2019\/09\/15\/translate-readme-files-in-restructuredtext-with-sphinx\/","text":"# Translate README files in reStructuredText with Sphinx\n\nreStructuredText (reST) is a markup language that is popular in the Python developers community. reST is the standard markup language for docutils, Sphinx documentation generator, and the Python Package Index (PyPI). However, reST by now is still not popular enough. Most translation platforms, including Crowdin which I\u2019m using now for EH Forwarder Bot, have no support to reST documents.\n\nSphinx has provided a plug-in sphinx-intl, which extract strings from reST documents and compile into GNU gettext message catalog template (.pot) files, and build new documents with translated strings in other languages. GNU gettext formats are widely accepted by translation platforms, making our life much easier. This would work out-of-box if you are generating HTML or PDF documentations, but not so simple if you want a reST output.\n\nTo utilize what we have in sphinx-intl, what we need is instead a document writer for Sphinx that outputs reST. There is a plug-in for called restbuilder that serves this very purpose, but it has not been updated for over a year. restbuilder is currently looking for maintainers. Unfortunately I don\u2019t really have much time to maintain such a complex project. What I did is just forked the project, included some other fixes from existing PRs, and fixed some more stuff myself.\n\n## Extract strings\n\nSince sphinx-build in general works with directories, we need to create a new temporary directory to isolate README.rst. We don\u2019t want to include write a conf.py file just for the translation either as everything can be indicated in command line.\n\nmkdir .build_readme\nrm -rf .build_readme\n\nFlags used on line 3:\n\n\u2022 -b gettext: Use gettext string extractor as builder\n\u2022 -C: Use no config file at all, only -D options\n\u2022 -D master_doc=README: Set the front page name as README (that\u2019s the only file we have here)\n\u2022 -D gettext_additional_targets=literal-block,image: Include code blocks and images (and captions) into the message catalog.\n\u2022 .build_readme: Source directory\n\u2022 .\/readme_translations\/locale\/: Destination directory\n\u2022 .build_readme\/README.rst: File to build\n\nThis will extract strings into .\/readme_translations\/locale\/README.pot. This file can then be uploaded onto any translation platform or directly passed over to translators.\n\nWhen you have got translated strings from your translators, you need to arrange it into a folder structure that sphinx-intl can recognize, namely {language_code}\/LC_MESSAGES\/README.(po|mo). I have arranged them in readme_translations\/locale\/{language_code}\/LC_MESSAGES\/ in my case.\n\nTo build translated README files, we need to install the restbuilder plugin for Sphinx. Here I\u2019ll use my fork as example.\n\ngit clone https:\/\/github.com\/blueset\/restbuilder.git\ncd restbuilder.git\npython3 setup.py\n\nThen, write a script to iterate through the locale folder and get a list of languages available, and use that to build a list of commands to run.\n\nimport glob\nfrom pathlib import Path\n\n# My language code is defined in a POSIX-like style. E.g. en_US\nlanguages = [i[i.rfind('\/')+1:] for i in glob.glob(\".\/readme_translations\/locale\/*_*\")]\n\n# Compile .po files to .mo\nsources = glob.glob(\".\/**\/*.po\", recursive=True)\ndests = [i[:-3] + \".mo\" for i in sources]\nactions = [[\"msgfmt\", sources[i], \"-o\", dests[i]] for i in range(len(sources))]\n\nlocale_dirs = (Path('.') \/ \"readme_translations\" \/ \"locale\").absolute()\nfor i in languages:\nactions.append([\"sphinx-build\", \"-b\", \"rst\", \"-C\",\n\"-D\", f\"language={i}\", \"-D\", f\"locale_dirs={locale_dirs}\",\n\"-D\", \"extensions=sphinxcontrib.restbuilder\",\nactions.append([\"rm\", \"-rf\", \".\/.build_readme\/source\"])\n\u2022 -b rst: Use reST as output format\n\u2022 -D language={language}: Indicate language code to use\n\u2022 -D extensions=sphinx.contrib.restbuilder: Load the restbuilder extension installed\nWith the commands above, you can build translated README files automatically with Sphinx and GNU gettext message catalogs. For full sample code with doit automation, visit the script in EFB Telegram Master Channel repository, and look for task_gettext and task_msgfmt methods.","date":"2021-07-26 00:26:43","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.301773339509964, \"perplexity\": 11690.347840346732}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-31\/segments\/1627046151972.40\/warc\/CC-MAIN-20210726000859-20210726030859-00571.warc.gz\"}"} | null | null |
By Shannon K. DeBra
Here are a few "quick hit" updates to stay informed about recent news affecting hospices.
Hospice audits
In late 2017, the Centers for Medicare and Medicaid Services (CMS) expanded its targeted probe and educate (TPE) audit program to include hospices. The TPE program focuses on providers with high improper payment rates and topics identified by the Medicare Administrative Contractors (MACs) through data analysis or billing patterns. During each "probe" round, the MAC reviews 20-40 claims and allows for one-on-one education to help the provider understand the cause of the errors. Any problems that fail to improve after three rounds of education sessions will be referred to CMS for next steps, which may include 100 percent prepay review, extrapolation, referral to a Recovery Auditor or other action. Hospices should familiarize themselves with the TPE process to avoid inadvertent denials and other consequences.
Electronic filing of Hospice Notice of Election
Effective January 1, 2018, hospice providers are now able to electronically submit the Notice of Election (NOE) to Medicare. While this is a welcome change for hospice providers, CMS notes that Medicare systems limitations could affect the timeliness of NOEs submitted electronically. Medicare Transmittal 3866 details the revised NOE policy and sets forth guidelines for seeking an exception to the timely filing requirement when delays due to Medicare system constraints are outside the hospice's control.
Physician assistants as attending physicians for hospice patients
On February 9, 2018, President Trump signed the Medicare Patient Access to Hospice Act into law, amending the Medicare statute to allow physician assistants to serve as the attending physician for hospice patients and perform other functions that are otherwise consistent with their scope of practice.
Pennsylvania hospice company and owner settle false claims lawsuits for $1.24 million
On February 8, 2018, the Department of Justice announced a $1.24 million settlement with Horizons Hospice, LLC (also known as 365 Hospice, LLC) and its owner/CEO to resolve allegations brought in two whistleblower lawsuits. The lawsuits alleged that Horizons submitted, or caused to be submitted, false claims to Medicare and Medicaid for patients who did not qualify for hospice, because they did not have life expectancy prognoses of less than six months, and that Horizons falsified records to support the false claims. 365 Hospice, LLC and the owner/CEO also entered into a five year corporate integrity agreement with the Office of Inspector General contemporaneously with the settlement.
Shannon K. DeBra Of Counsel
Cincinnati 513.870.6685 VCard | {
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Апопа () — місто та муніципалітет в центральній частині Сальвадору, на території департаменту Сан-Сальвадор.
Географія
Апопа примикає до міст Сояпанго і Сан-Сальвадор, будучи частиною великої агломерації Сан-Сальвадор, населення якої за даними на 2010 рік становить близько 1 900 000 осіб. Абсолютна висота — 404 метри над рівнем моря.
Населення
За даними на 2013 рік чисельність населення становить 145 705 осіб.
Примітки
Міста Сальвадору | {
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Today we had a Lovely time with Mitch Foy in his program Over The Rainbow at Bay FM radio station.
the intervie says it all!
Bay FM is realy great for us to spread the word about what we are actualy doing.
Shine your health coaching program gives you the real opportunity to lift your health, happiness and integrated wellbeing. With all the needed support you will do the program over a period of three months. Over this time you will experience improved immunity, detoxification, discovery of deeper meaning in your life, release of negativity and old patterns of thought and action. Your integrated health will the focus and you will learn all the little and big things that empower you to be the real master of your health.
12 Weekly Personal Consultation Sessions where you can get professional support and coaching to deal with any difficulty arising while in the process of elevating your body, mind and spirit. Personal adjustment to diet and exercises. Gestalt Counselling and Therapy.
Personalised Diet, Yoga Exercises and Natural Treatments.
A Monthly Day Workshop for 3 months, on Shine Your Health, enjoing together delightful juices and cooking classes; health and wellness nutritional informations; Gestalt group process and support; Yoga and fun!
Ongoing Facebook group interaction and support. You will become part of the Australia Wellness Yoga Detox Face-Book group to keep up with other members.
This integral detox coaching program is a natural and perfect follow up to the Detox retreat as it help you to put into practice in your daily life, the changes and the learning you gained from the retreat experience.
The program can also stand by itself and will support great changes in your life, improved health, detoxification, improved immunity, greater levels of energy, and a firm foundation to establish a meaningful and fulfilling lifestyle.
For a limited time, you can enrol in our Well Detox Retreat and Shine Your Health Program and save a massive 20%. Book now, and enter the coupon code Shine20 at the checkout to redeam your special discount.
This Retreat has been very small and intimate.
also looking at the practical applications of food preparations and home made probiotics, within easily accessible cooking classes.
" A wanderful nurturing and gentle team that allowed me to feel at easy.
Amazing 'living food' and the ability to see it being prepared.
The Diet, Health and Wellness presentations were invaluable.
An intimate group that allowed for lots of sharing on a wide range of subjects.
A beautiful 'rustic' retreat centre in a soothing, tranquil environment.
The ten days AMWellness Yoga Detox Retreat held last January in Mullumbimby, has been a great success! We had 13 participants that enjoyed the variety of programs. This unique program integrated the best practises for the complete wellbeing of body mind and spirit.
The program is uniquely synthesising Yoga Asanas, Kiirtan, Meditation, Alkaline diet, Juice Fasting, Gestalt group process, Classes on Health, Stimulation Treatments, and Therapies. All participants greatly enjoyed the program.
Everyone received a personalised set of recommendations and treatments to fit their own health and life situation, to bring home after the retreat ended.
The presence of our international teacher Dada Dharmavedanda was deeply inspiring and enlivening. Dada is a light hearted Yogi that has tremendous levels of energy and positivity, and everyone that spends time with him can take away something very valuable to enrich their life.
On the team were also Radha Cohen, Didi Ananda Devaniishta and Yogesh Alperovitch. Didi contributed with her very sweet, gentle and uplifting vibration that carried everyone swiftly over the detox. Thanks to Radha and Yogesh, one of the program's elements, unique to our Australian team, was the integration of Gestalt Group process. Everyone during the retreat noticed the added value and the deep impact that the gestalt process had on all participants, enabling to access group support to overcome life-long barriers in a matter of days.
The Wellness Detox Retreats can help people troubled by Stress, Obesity, Blood pressure, Diabetes, Arthritis and Body pain, Hyper-acidity, Digestive complications, IBS or Candida, Hepatitis, Menstrual problems and Hormonal imbalance, Migraine, Insomnia, Constipation, Skin problems, to mention a few. | {
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Home Page > Thematic > Funny and Moody: The Best of Aislin's Cartoons
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Drawing, cartoon
Le Dossier Lindros
Aislin (alias Terry Mosher)
1991, 20th century
Gift of Mr. Terry Mosher
M996.11.44
Keywords: Cartoon (19139) , Culture / Sports (256) , Drawing (18637) , drawing (18379) , Eric Lindros (3) , hockey (86) , Mass media (86) , People (413) , sports (154) , Sports personalities (50)
Keys to History
"It's an accident of history that cartoons appear on the editorial pages of newspapers. In the 1890s, some editors decided that their editorials might be more eye-catching if they were illustrated. But their initiative backfired because head-strong cartoonists began demanding the right to express their own points of view.
In my experience, earnest editorialists and oddball cartoonists make an effort to get along, even if they remain a puzzle to each other. Occasionally, though, editorial writers get emotionally worked up about a subject. That happened all over Quebec when hockey player Eric Lindros announced that he would not play for Les Nordiques in Quebec City. Well! Didn't that create a collective round of righteous indignation, as reflected in the generic characters in this cartoon."
Terry Mosher (alias Aislin)
The Lindros Affair, as it was called, hit news headlines in 1991. Which hockey team would Lindros play for? What conditions would he demand? Would he sign up to play for the Quebec Nordiques? Everyone had an opinion : journalists, columnists, editorial writers - even Prime Minister Mulroney. Here, the cartoonist reveals his surprise at seeing people get so worked up over the NHL hockey draft.
The Quebec Nordiques were never able to really lay down roots and win over fans. They were usually at the bottom of the NHL rankings. One year after the club was moved to Colorado, the Avalanche - the new name of the franchise - succeeded in doing something that the Nordiques had not been able to do in sixteen seasons in Quebec City, win the Stanley Cup. The Lindros Affair was a good illustration of the problems that plagued the Nordiques: how could a team caught in a dwindling market and never able to top its league meet all of the demands of a star hockey player?
The city of Quebec was home to its own National Hockey League (NHL) team from 1979 to 1995. During that time, the Nordiques, who wore the colours of the capital city of Quebec, rarely ranked among the leading teams. The salary increases awarded to players by the NHL, along with the small size of the Quebec City market, eventually sounded the death-knell of the Nordiques.
Born in 1973 in London, Ontario, Eric Lindros played for the Philadelphia Flyers from 1992 to 2000, and then for the New York Rangers. | {
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Catholics Instigate Anti-Catholicism
This might be a new one. We all know the Catholic Church is responsible for all the evils in the world but now Joel Connelly, a liberal columnist for the SeattlePi, wrote that Catholics who take Catholicism seriously are responsible instigating anti-Catholicism.
Connelly starts off his column pretty impressively with some obfuscation, smoke, and then quickly devolves into lies. And he does it all impressively in one line. Check it out:
When he should get time to reflect after his father's recent death, and to appreciate the life of his assassinated uncle, Rep. Patrick Kennedy, D-R.I., is being told — in effect — to get out of his church.
He should be given time to appreciate his uncle who died over 40 years ago? Look, how much time does this guy need? I think his therapist might be overcharging.
But you see, Patrick Kennedy unfortunately has done very very little with his own life so Joel Connelly was forced to raise the specter of Kennedy's father and uncle as a way to reflect some borrowed glory on Patrick Kennedy.
What is it about liberals and the name "Kennedy?" They all start acting like 13 year old girls who just saw the Jonas Brothers. Seriously, you want to see a liberal freak out just criticize a Kennedy. It's kinda' fun. Bishop Tobin knows this all too well by now.
So after Connelly's done dragging out every Kennedy but Bobby, he lies. Bishop Tobin obviously never told Patrick Kennedy to "get out of his church." Not even close.
But Connelly's rolling so we'll let him continue. He says that Bishop Tobin's words are instigating a rash of anti-Catholicism.
"Millions of American Catholics have gone somewhere else — in my case, long ago, to the Episcopal Church — or chosen to stay home, because of the actions and pronouncements of prelates like Bishop Tobin.
In recent years, the Vatican has seen fit to install bishops who show no respect for conscience, or for a U.S. Constitution that wisely separates Caesar from Peter.
They give orders, demand unquestioned obedience, and treat Communion wafers as political weapons..
The behavior of hard-line prelates elsewhere has stoked the fires of anti-religious and anti-Catholic prejudice. Just read certain "liberal" Seattle blog sites and in the printed word.
The bishops have made it difficult to begin a necessary dialogue on prevention of unwanted pregnancies. They've made it immeasurably tougher the job of articulating legitimate ethical objections to assisted suicide.
Nor have the faithful taken political counsel from purple hats. Barack Obama, the pro-choice presidential nominee, captured a majority of votes from American Catholics in the 2008 election.
Thomas Keefe, a Spokane lawyer (and expatriate Seattlite), Catholic University law grad and former nominee for Congress, reacted to the latest brouhaha by sending off a blistering letter to the Diocese of Providence.
"It should come as little wonder why the Catholic Church continues to lose appeal in our society when a fool like Bishop Tobin passes for a church leader," Keefe wrote.
So the loony anti-Catholics are out for blood and whose fault is it? Bishop Tobin's of course.
And the poor Vatican must be reeling because some guy who ran for Congress (AND LOST!) called Bishop Tobin a "fool." I'm sure the Vatican is quivering that some lawyer from Spokane dashed off a blistering letter.
The odd thing here is that Connelly admits that he left the Church. And so can Kennedy. So can anyone. It's a volunteer organization. The Church is the Church. Nobody's forced to join. But for some reason, people freak out all the time about the Church "forcing" people to do this or that. How? The Swiss Guard?
I know. I know. The Church is responsible for all the evils in the world. And now, according to Connelly the Church is also responsible for anti-Catholicism.
Oddly, I don't remember Connelly's column excoriating Muslim terrorists for instigating anti-Islamic behavior but maybe that's next.
anti-Catholic, bishop tobin, fr. kennedy
Subvet
"Oddly, I don't remember Connelly's column excoriating Muslim terrorists for instigating anti-Islamic behavior but maybe that's next."
Don't hold your breath.
canon1753
The key question in all of this is "Mr. C, where do you stand on abortion?" I'd be willing to bet he is pro-abortion. Because this Kennedy-Tobin thing is all about abortion funding. Kennedy threw the "He ordered me not to go to communion" line out there to turn it into an intrachurch fight over abortion and somehow discredit the bishops.
Hey, it doesn't matter that Bishop Tobin asked him not to go to communion, or really to search his conscience, which is a real difference from a precept to not go to communion. (does anyone actually know if Patrick Kennedy does go to Communion when in Providence?)
Of Course, now, as practicing Catholics, we are the enemy. We are the bad guys. We will be the ones on the "wrong side of history," and "standing in the way of progress." So inflexible that health care was wrecked because they wouldn't compromise a measly principle on abortion. Get ready for that to hit us at or after Christmas….
Ever wonder why so called liberals are almost universally rude, insulting and have trouble with the truth???
Our culture is doing so well without the Church, we really don't have a leg to stand on. Who doesn't know someone that settled down, got married, had a couple of kids, and then just started having tons of meaningless sex with women who got abortions because they didn't want to be "punished with a baby"?
Holla for doing whatever you want all the time!
1st Who is this guy?
2nd Who cares?
3rd He is Episcopalian, so see 2nd question.
Lastly, anyone who writes "They give orders, demand unquestioned obedience, and treat Communion wafers as political weapons.." is so unworthy of serious thought or attention it's almost scarey.
Dan Lower
This is the second money-commentary piece today from CMR, and I'm incredibly happy for that fact.
Seriously though, agreement with this is definite for me. People are acting all hot and bothered because bishops are actually exercising the authority that they've had for, oh, only a few hundred years now. Just because they took a break doesn't mean it disappeared.
The Bones
I know its just so wicked and cruel to refuse Communion to people who want to kill babies. Wicked and shameful. Shame on you Bishop Tobin…
mtmom
I want to know where the outrage is going to come in with this article or column? How dare this man start name calling!!! God bless Bishop Tobin and would that the Good LOrd grant us more of these "fools" to lead the church!
Bp. Tobin was really too kind and conciliatory. When someone is asked to stay away from the Eucharist, it is because that person is not only in a grave state of sin but is also willing to flaunt his wickedness and add sacrilege and scandal to his misdeeds. So, as the shepherd of souls, the good bishop was doing his job and was still trying to give Kennedy a chance to save face. That's is why he is a bishop and I am not because if Ted wants to jump off the barque of St. Peter, a part of me does not give a flying rats ass. | {
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Q: how to create activity indicator for blackberry 5.0.0? I am trying to prepare activity indicator for BLACKBERRY 5.0.0,but getting failed into it..
I am trying to create two thread classes, one with background functionality and one with activity indicator class which extends popup screen...but getting an error as "too many thread error...
please help me out
thanx in advance
A: This error will occur because blackberry supports only 63 threads. It gives "Too many thread exception", if count of threads exceeds by 63.
So try to use UiApplication.getUiApplication().invokeLater(new Runnable{}) for activity indicator. This will help to resolve the issue.
| {
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\section{Introduction}
Dense, neutral (\citealt{mea92}; \citealt{hug92}) knots with ionized
cometary tails are found in the central regions of planetary nebulae
(PNe); the most famous being those in the Helix nebula for, at a
distance of only 213 $+$30/$-$16 pc \citep{har07}, they were easily
detected and resolved by early ground--based observations (Baade and
reported by \citealt{vor68}). The incredibly clumpy nature of the
neutral material in the disk of the Helix nebula (when modelled as a
bi--polar PNe viewed along its axis, \citealt{mea98}; \citealt{mea05})
has now been revealed in spectacular fashion in the imagery in the
H$_{2}$ emission line by \citet{mat09}. Previously, \citet{mei05} had
estimated that there were 23,000 cometary knots and that inevitably
tail interactions must be occurring. \citet{mat09} show conclusively
that it is an inner region in the Helix disk, towards the central
star, where tails are observed from neutral globules surrounded by an
outer clumpy region free of such tails.
\citet{dys03} reviewed the two broad models for the creation of
the cometary tails; either they are shadowed from the ionizing
radiation of the central star, and their surfaces photo-ionized by
scattered Lyman photons in the nebula, or they are dynamically
produced as the particle winds from the central star swept past the
slowly expanding system of dense globules.
A critical distinction between these models arises if the cometary
tails can be seen to be flowing along their lengths away from their
parent globules. Only the `dynamic' model would have this effect. In
fact, a numerical simulation of the flow of a moderately supersonic
particle wind past the ionized head of a globule \citep{dys06}
predicted not only the creation of a cometary tail but a moderate
acceleration of ionized material parallel to the tail surface and away
from the globule. In this initial model the neutral material away from
the globule had a smooth density distribution.
The most comprehensive optical study of the kinematics of the Helix
system of globules remains that described in \citet{mea98}. Here,
spatially resolved \nii\ profiles, from 300 separate long-slit
(163\arcsec\ long) positions, were obtained with the Manchester
echelle spectrometer (MES -- \citealt{mea84} but now with a CCD as the
detector) on the Anglo--Australian telescope in exceptional `seeing'
conditions, over three separate `blocks' of globules and their tails
in the nebular core. Furthermore, a kinematical `case study' of the
globule which is apparently the 2nd closest to the nebular core (Knot
14), and a length of its tail, was carried out in a range of emission
lines. The principal outcome was to show that the system of central
globules is concentrated in a disk expanding itself at 14\kms. Flows
parallel to the surfaces of the cometary tails of two of the most
prominent globules (Knots 38 and 14) were also detected and even an
acceleration of this flow along the length of the longest tail (from
Knot 38) suggested. \citet{ode07} challenged these latter assertions
without making any further kinematical observations but simply because
they claim that more recent HST images reveal confusing minor globules
in the longest tail (Knot 38). With the dismissal of these kinematical
effects they proceeded to support the shadowing theories of the
creation of the tails.
The aim of the present paper is to reassess the strength of the
deductions from the \citet{mea98} kinematical data set in the light of
the subsequent HST and very latest H$_{2}$ imagery by \citet{mat09}.
In particular, to
consider if the evidence in \citet{mea98} for flows along the tail
surfaces of Knots 38 and 14, away from the central star, is now
invalidated by this HST imagery alone as suggested by
\citet{ode07}. Furthermore, the flow behind Knot 32, which was not
considered hitherto, is now presented to strengthen the original
suggestion that accelerating flows in the cometary tails could be
ubiquitous though in a clumpy medium.
\section[]{Knots 38 and 14}
Knot 38 has the longest (62\arcsec) cometary tail and is the
apparently closest knot to the central star of NGC 7293, while Knot 14
is the next closest. The heads of both have arcs of \oiii\ emission
facing this star which indicates that they protrude into the hard
radiation field of the central volume of the bi--polar,
ellipsoidally--shaped, nebula that produces the characteristic helical
appearance in emission lines of lower ionization species
(\citealt{mea98}; \citealt{mea05}). The HST archival images (PI NAME:
Meixner, PID:9700) of Knots 38 (HST ACS/WFC J8KR14040) and 14 (HST
ACS/WFC J8KR0840) are presented respectively in Figs. 1a and 2. These
should be compared with those taken with the New Technology Telescope
(NNT - Chile) and presented in Meaburn et al (1998). All are in the
light of the \ha\ plus \NII\ nebular emission lines.
The uniquely long ionized tail of Knot 38 in Fig. 1 is prominent
in both the HST and former \citep{mea98} NTT images and appears to be
a coherent structure in both i.e. it is not simply a consequence of
chance superposition of a large number of fore-- or background tails
along the same sightlines. Minor ionized knots appear along the tail
in the image in Fig. 1a but it is the H$_{2}$ 2.12$\mu$m
image \citep{mat09} shown in Fig. 1b that
emphasises that the tail of ionized gas, being so long, is formed
around an internal core composed of a large number of minor clumps of
neutral material.
With this as a starting point the kinematics along this tail should be
re--considered by examining the \nii\ line profiles presented in
\citet{mea98}. The centroids of these profiles are shown in fig. 13 of
that paper to change along the 62\arcsec\ length of this tail by a
radial velocity difference (from central knot to tail end) by
10\kms. The velocity images in figs. 4 and 12 of the same paper
confirm this systematic radial velocity change in a different
way. Unfortunately, the four images in different heliocentric radial
velocity (\vhel) ranges became jumbled in the production of fig. 12 in
the \citet{mea98} paper. These should be \vhel\ $= -$31 to $-$27 (top
right), $-$25 to $-$21 (top left), $-$20 to $-$15 (bottom right) and
$-$14 to $-$10 (bottom left) and all \kms. When these four images are
considered with this correction it is clear that the head of Knot 38
appears alone in the top right frame then progressively the tail
appears in the subsequent three frames towards more positive
velocities as the image of the head declines. The acceleration along
the tail length, if regarded as a coherent feature starting at Knot
38, is very clear and not dominated by the minor confusing knots
apparent in Fig. 1a. Those areas free of these along the tail length of
Knot 38 in Fig. 1 clearly show this radial velocity change. The appearance
of the neutral material in the tail of Knot 38 and seen in Fig. 1b
suggests that the ionized outflow is in a sheath around a clumpy neutral
core.
Similarly, the \nii\ line profiles up to 5\arcsec\ from the head of
Knot 14 (fig. 9 of \citealt{mea98}) show a systematic change of radial
velocity of their centroids of -6 \kms. This was not covered by the H$_{2}$
imagery of \citep{mat09}.
The tail from Knot 14 was modelled
kinematically as a flow parallel to the globule and tail
surfaces. The HST images in Fig. 2 show that there are no significant
minor ionized knots along this small length of the tail.
Again an
accelerating flow away from the central star is indicated. If tilted
at 25\degree\ to the plane of the sky this change of outflow velocity
amounts to 14 \kms\ compared to the apex of the head of the cometary
knot.
\section{Knot 32}
As shown in Figs. 1a \& b
Knot 32 has a more complex structure than that of
both Knots 38 and 14. Again, a faint tail of diffuse ionized gas
can be seen extending radially away from the central star though the
central knot is composed of several neutral clumps. This tail of course
could be an unrelated feature along the same sightline but its appearance
suggests that this is unlikely. The
kinematics of the Knot 32 tail have now been examined using the set of
profiles of the \nii\ emission line from 100 longslit positions. These
have an EW orientation and each is separated from its neighbour by
1\arcsec. The slit width is $\equiv$ 6\kms\ and 0.5\arcsec\ on the
sky. These observations and their analysis are described fully in
\citet{mea98} and the reader is referred there for detailed
information.
However, the \nii\ line profile from the very apex of this knot,
facing the ionizing star, is shown in Fig. 3a and that of the knot
head (1.5 \arcsec\ further from this star) in Fig. 3b. The \nii\
profiles from the faint material of the knot tail ( 14\arcsec\ and
16\arcsec\ away from the apex) are shown in Figs. 3c \& d
respectively. The positions where these line profiles were
obtained are marked a--d in the velocity image in Fig. 4c.
The \nii\ profiles from the host nebula have been subtracted in each
case leaving only the profiles of the \nii\ emission from Knot 32 and
its tail in those shown in Fig. 3a--d. This type of `velocity
imagery' is inevitably blurred in the NS dimension for the
adjacent longslit spectra, each orientated EW, that are used to generate
individual velocity images are separated by 1\arcsec\ which is then
convolved in the imagery with the angular slit width and the
0.8\arcsec\ seeing disk.
A distinct positive shift of around 8 \kms\
in radial velocity can be seen to be occurring between the apex of the
knot and the material in the extended tail.
The `velocity' images from the same data set in Figs. 4a--c show this
radial velocity shift in a different way. The head of the knot alone
appears in Fig. 4a (\vhel\ $= -$31 to $-$27\kms), the nearest part of
the tail in Fig. 4b (\vhel\ $= -$24 to $-$21\kms) and the furthest
extent of this tail in Fig. 4c (\vhel\ $= -$20 to $-$16\kms) all of
which is consistent with the profiles shown in Fig. 3a--d.
\section[]{Conclusions}
The 62\arcsec\ long tail from Knot 38 appears to be a coherent
structure in the HST, NTT and the previous
velocity imagery but formed around an
extremely clumpy neutral medium as shown by the H$_{2}$ imagery.
The evidence that there are accelerating flows along the walls of the
tails of the two most prominent cometary globules in NGC 7293 (Knots
38 and 14), away from the central star, is not substantially affected
by the higher resolution HST images or even the extremely clumpy
nature of the neutral material revealed in the H$_{2}$ imagery.
A similar flow away from the more complex Knot 32 is shown
to be occurring for the first time in the present paper.
It is valid to attempt to explain flows along the cometary tails in
the dynamical type of model in \citet{dys06} but now
modified to accommodate a very clumpy ambient medium similar
to that in \citet{pit05} which had been derived for
more general clumpy phenomena.
It is becoming very clear \citep{mat09} that the AGB particle wind
overflowing the clumpy neutral structure of the disk of NGC 7293, at
mildly supersonic velocity, can create and accelerate the cometary
tails. Previously, \citet{mea82} had shown that an inner
quasi--spherical volume of material shielded the system of cometary
knots from exposure to any subsequent fast stellar particle wind.
\section*{Acknowledgments}
We would like to thank the referee for constructive comments that have
improved the paper considerably. We also thank M. Matsuura who kindly
provided their molecular hydrogen image of Helix in FITS format. The
observations made with the NASA/ESA Hubble Space Telescope, obtained
from the data archive at the Space Telescope Institute. STScI is
operated by the association of Universities for Research in Astronomy,
Inc. under the NASA contract NAS 5-26555.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,954 |
\section{Introduction}
\setcounter{page}{1}
The (optimal) value function, that is, a function defined by optimizing an
objective function over one of multiple arguments, is used widely in economics,
finance, statistics and other fields. The statistical behavior
of estimated value functions appears, for instance, in the study of
distribution and quantile functions of treatment effects under partial
identification (see, e.g., \citet{FirpoRidder08, FirpoRidder19} and
\citet{FanPark10,FanPark12}). More recently, investigating counterfactual
sensitivity, \citet{ChristensenConnault19} construct interval-identified
counterfactual outcomes as the distribution of unobservables varies over a
nonparametric neighborhood of an assumed model specification. Statistical
inference for value functions can be analyzed at a point using knowledge of the
asymptotic distribution established in \citet{Roemisch06} or
\citet{CarcamoCuevasRodriguez19} and inference can be carried out using the
techniques of \citet{HongLi18} or \citet{FangSantos19}.
This paper extends these pointwise inference methods to uniform inference for
value functions that are estimated with nonparametric plug-in estimators.
Uniform inference has an important role for the analysis of statistical
models.\footnote{For example, \cite{AngristChernozhukovVal06} discuss the
importance of uniform inference \textit{vis-\`a-vis} pointwise inference in a
quantile regression framework. Uniform testing methods allow for multiple
comparisons across the entire distribution without compromising confidence
levels, and uniform confidence bands differ from corresponding pointwise
confidence intervals because they guarantee coverage over the family of
confidence intervals simultaneously.} However, uniform inference for the value
function can be quite challenging. In order to use standard statistical
inference procedures, one must show that the map from objective function to
value function is compactly differentiable as a map between two functional
spaces, which may require restrictive regularity conditions. As a result, the
delta method, as the conventional tool used to analyze the distribution of
nonlinear transformations of sample data, may not apply to marginal optimization
as a map between function spaces due to a lack of differentiability. We propose
a solution to this problem that allows for the use of the delta method to
conduct uniform inference for value functions.
Because in general the map of objective function to value function is not
differentiable, our solution is to analyze statistics that are applied to the
value function directly. This is in place of a more conventional analysis that
would first establish that the value function is well-behaved and as a second
step use the continuous mapping theorem to determine the distribution of test
statistics. In settings involving a chain of maps,
this can be seen as choosing how to define the ``links'' in the chain to which
the chain rule applies. The results are presented for $L_p$ functionals for $1
\leq p \leq \infty$, that are applied to a value function, which should cover
many cases of interest. In particular, this family of functionals includes
Kolmogorov-Smirnov and Cram\'er-von Mises tests.
By considering $L_p$ functionals of the value function, we may bypass the most
serious impediment to uniform inference. However, these $L_p$ functionals are
only Hadamard directionally differentiable. A directional generalization of
Hadamard (or compact) differentiability was developed for statistical
applications in \citet{Shapiro90, Duembgen93}, and there is recent growing
literature of applications using Hadamard directional differentiability~--- see,
e.g., \citet{SommerfeldMunk17}, \citet{ChetverikovSantosShaikh18},
\citet{ChoWhite18}, \citet{HongLi18}, \citet{FangSantos19},
\cite{MastenPoirier20}, \citet{CattaneoJanssonNagasawa20} and
\citet{ChristensenConnault19}. $L_p$ functionals for $1 \leq p < \infty$ and
the supremum norm are treated separately, as are one-sided variants used for
tests that have a sense of direction. As an intermediate step towards showing
the differentiability of supremum-norm statistics, we show directional
differentiability of a minimax map applied to functions that are not necessarily
convex-concave.
Directional differentiability of $L_p$ functionals provides a minimal amount of
regularity needed for feasible statistical inference. We use these results to
find the asymptotic distributions of test statistics estimated from sample data.
The distributions generally depend on features of the objective function, and
for practical inference, we turn to resampling to estimate them. We suggest the
use of a resampling strategy that was proposed by \citet{FangSantos19},
combining a standard bootstrap procedure with an estimate of the directional
derivative. This bootstrap technique can be tailored to impose a null hypothesis
and is simple to implement in practice. Moreover, we establish local size
control of the resampling procedure.
Monte Carlo simulations assess the finite-sample properties of the proposed
methods. The simulations suggest that Kolmogorov-Smirnov statistics used to
construct uniform confidence bands have accurate empirical size and power
against local alternatives. A Cram\'er-von Mises type statistic is used to test
a stochastic dominance hypothesis using bound functions, similar to the
breakdown frontier tests of \citet{MastenPoirier20}. This test has accurate
size at the breakdown frontier and power against alternatives that violate the
dominance relationship. All results are improved when the sample size increases
and are powerful using a modest number of bootstrap repetitions.
As a practical illustration of the proposed methods, we consider bounds for the
distribution function of a treatment effect. Treatment effects models have
provided a valuable analytic framework for the evaluation of causal effects in
program evaluation studies. When the treatment effect is heterogeneous, its
distribution function and related features, for example its quantile function,
are often of interest. Nevertheless, it is well known (e.g.
\citet{HeckmanSmithClements97, AbbringHeckman07}) that features of the
distribution of treatment effects beyond the average are not point identified
unless one imposes significant theoretical restrictions on the joint
distribution of potential outcomes. Therefore it has became common to use
partial identification of the distribution of such features, and to derive
bounds on their distribution without making any unwarranted assumptions about
the copula that connects the marginal outcome distributions.
Bounds for the CDF and quantile functions at one point in the support of the
treatment effects distribution were developed in \citet{Makarov82,
Rueschendorf82, FrankNelsenSchweizer87, WilliamsonDowns90}. These bounds are
minimal in the sense that they rely only on the marginal distribution functions
of the observed outcomes, not the joint distribution of (potential) outcomes.
We extend inference methods for these bounds at a point to uniform confidence
bands for the bound functions.
We illustrate the methods using job training program data set first analyzed by
\citet{LaLonde86} and subsequently by many others, including
\citet{HeckmanHotz89}, \citet{DehejiaWahba99}, \citet{SmithTodd01, SmithTodd05},
\citet{Imbens03}, and \citet{Firpo07}, without making any assumptions on
dependence between potential outcomes in the experiment. This experimental data
set has information from the National Supported Work Program used by
\citet{DehejiaWahba99}. We document strong heterogeneity of the job training
treatment across the distribution of earnings. The uniform confidence bands for
the treatment effect distribution function are imprecisely estimated at some
parts of the earnings distribution. These large uniform confidence bands may in
part be attributed to the large number of zeros contained in the data set, but
are also inherent to the fact that the distribution function is everywhere only
partially identified.
The remainder of the paper is organized as follows. Section \ref{sec:model}
defines the statistical model of interest for the value function. Section
\ref{sec:differentiability} establishes directional differentiability for the
(nonstochastic) maps of interest. Inference procedures are established in
Section \ref{sec:inference}. Section \ref{sec:bounds} provides an example for
the bounds on the treatment effect distribution function. It provides Monte
Carlo simulations, and an empirical application to job training is discussed in
Section \ref{sec:application}. Finally, Section \ref{sec:conclusion} concludes.
All proofs are relegated to the \hyperref[appn]{Appendix}.
\subsection*{Notation} For any set $T \subseteq \mathbb{R}^d$ let $\ell^\infty(T)$
denote the space of bounded functions $f: T \rightarrow \mathbb{R}$ and let $\mathcal{C}(T)$
denote the space of continuous functions $f: T \rightarrow \mathbb{R}$, both equipped
with the uniform norm $\| f \|_\infty = \sup_{t \in T} |f(t)|$. For measure
spaces, $\| \cdot \|_\infty$ is implicitly the essential supremum, and given a
sequence $\{f_n\}_n \subset \ell^\infty(T)$ and limiting random element $f$ we
write $f_n \leadsto f$ to denote weak convergence in $(\ell^\infty(T), \| \cdot
\|_\infty)$ in the sense of Hoffmann-J\o rgensen \citep{vanderVaartWellner96}.
Denote the positive-part map $[x]_+ = \max\{x, 0\}$. A set-valued map (or
correspondence) $S$ that maps elements of $X$ to the collection of subsets of
$Y$ is denoted $S: X \rightrightarrows Y$. For set-valued map $S: X
\rightrightarrows Y$, let $\mathrm{gr} S$ denote the graph of $S$ as a set in $X \times
Y$.
\section{The model}\label{sec:model}
Consider two sets $U \subseteq \mathbb{R}^{d_U}$ and $X \subseteq \mathbb{R}^{d_X}$, a
set-valued map $A: X \rightrightarrows U$ that serves as a choice set, and an
objective function $f \in \ell^\infty(\mathrm{gr} A)$.
The objective function and the value function are linked by a marginal
optimization step that maps one functional space to another. Let $\psi:
\ell^\infty(\mathrm{gr} A) \rightarrow \ell^\infty(X)$ map the objective function
to the value function obtained by marginally optimizing the objective $f$ with respect to
$u \in A(x)$ for each $x \in X$. Without loss of generality, we consider only
marginal maximization with respect to $u$ for each $x$:
\begin{equation}
\psi(f)(x) = \sup_{u \in A(x)} f(u, x). \label{m_def}
\end{equation}
The value function $\psi(f)$ is the object of statistical interest, for which we would like
to conduct uniform inference using a plug-in estimator $\psi(f_n)$.
In order to fix ideas, we introduce examples of the value function model in
equation \eqref{m_def}. To illustrate the results in this paper, we
provide a longer analysis of the first example in Sections
\ref{sec:bounds} and \ref{sec:application} below.
\begin{exm}\label{ex:bounds}
In our main application, the treatment effect of a policy is defined as the
difference $\Delta = X_1 - X_0$, where $(X_0, X_1)$ are potential outcomes
under control and treatment regimes. When no identification conditions are
placed on the dependence between the potential outcome variables, we may
attempt to bound the cumulative distribution function $F_\Delta$ by using the
most extreme possible dependence structures between $X_0$ and $X_1$. This
was developed in \citet{Makarov82} and \citet{Rueschendorf82}, and the bounds
of $F_\Delta(x)$ for any $x$ are
\begin{align}
L(x) &= \sup_{u \in \mathbb{R}} (F_1(u) - F_0(u - x)) \label{fbound_lo} \\
U(x) &= 1 + \inf_{u \in \mathbb{R}} (F_1(u) - F_0(u - x)), \label{fbound_hi}
\end{align}
where $F_0$ and $F_1$ are the CDFs of $X_0$ and $X_1$, which we assume are
identified. As a map from the pair of CDFs $F = (F_0, F_1)$, define $\Pi:
(\ell^\infty(\mathbb{R}))^2 \rightarrow \ell^\infty(\mathbb{R}^2)$ by
\begin{equation} \label{pi_def}
\Pi(f)(s, t) = f_1(s) - f_0(s - t).
\end{equation}
Then $L = \psi(\Pi(F))$ and $U = 1 - \psi(\Pi(F))$.
Similarly, letting $f^{-1}$ denote the generalized inverse of an increasing function, we may wish to bound the quantile function of $\Delta$, $F_\Delta^{-1}(\tau)$ for $0 < \tau < 1$. As discussed in \citet{FanPark10}, the functions in the relationship $L \leq F_\Delta \leq U$ may be inverted to find that for each quantile level $\tau$, $U^{-1}(\tau) \leq F^{-1}_\Delta(\tau) \leq L^{-1}(\tau)$. We may calculate them directly, based on the quantile functions of the control and treatment populations: these inverted bounds are \citep{WilliamsonDowns90}
\begin{align}
U^{-1}(\tau) &= \sup_{u \in (0, \tau)} \left\{ F_1^{-1}(u) - F_0^{-1}(u + (1 - \tau)) \right\} \label{qbound_lo}\\
L^{-1}(\tau) &= \inf_{u \in (\tau, 1)} \left\{ F_1^{-1}(u) - F_0^{-1}(u - \tau) \right\}. \label{qbound_hi}
\end{align}
The functions $U^{-1}$ and $L^{-1}$ are value functions where marginal optimization takes place over a set-valued map that varies with $\tau$.
It may be of interest to construct uniform confidence bands for these bound functions. Because the analysis is similar for all four of these functions, we focus below on uniform inference for $L$ in the sequel.
\end{exm}
\begin{exm}\label{ex:stoc_dominance}
Building upon the previous example, imagine testing for (first order)
stochastic dominance without assuming point identification. Let $F_A$ and
$F_B$ be the distribution functions of the treatment effects $\Delta_A = X_A -
X_0$ and $\Delta_B = X_B - X_0$. Without point identification we cannot test
the hypothesis $H_0: F_A\succeq_{FOSD} F_B$, which is equivalent to the
condition that $F_A(x) - F_B(x) \leq 0$ for all $x$. However, by using bounds, since
$L_A \leq F_A$ and $F_B \leq U_B$, a necessary condition of the dominance of
$F_A$ over $F_B$ is that $L_A(x) - U_B(x) \leq 0$ for all $x$. This holds
regardless of the correlations between treatment outcomes. Labeling the
underlying marginal CDFs from each group by $(G_0, G_A, G_B) \in
(\ell^\infty(\mathbb{R}))^3$, the function function used to indicate a violation of
dominance maps this triple into a function in $\ell^\infty(\mathbb{R})$: for each $x
\in \mathbb{R}$,
\begin{align}
L_A(x) - U_B(x) &= \sup_{u \in \mathbb{R}} (G_A(u) - G_0(u - x)) - 1
- \inf_{u \in \mathbb{R}} (G_B(u) - G_B(u - x)) \\
{} &= \psi(\Pi(G_A, G_0))(x) - 1 + \psi(-\Pi(G_B, G_0))(x).
\end{align}
We would like to test the hypothesis that $F_A$ dominates $F_B$, without
knowledge of the dependence between $X_0$, $X_A$ and $X_B$, by looking for
$x$ where $L_A(x) - U_B(x) > 0$. This is a case in which a one-sided
functional $f \mapsto \|[f]_+\|_p$ is preferred for testing.
\end{exm}
\begin{exm} \label{ex:quantile}
The quantile function of a random variable may be an object of interest;
see for example \citet{WeiHe06}. Suppose that the real-valued random variable
$X$ has distribution function $F$, and, following \citet{Kaji19}, consider the
quantile function of $X$ an integrable function on $\mathcal{T} = (0, 1)$. To frame
it in an optimization context, label the risk function $R: (\mathbb{R}, \mathcal{T})
\rightarrow \mathbb{R}$. This is not necessary for one-sample quantile estimates, of
course, but this form of the problem generalizes the search for quantiles to
quantile regression estimation \citep[equation 1.10]{Koenker05}. For each
$(q, \tau)$, let
\begin{equation} \label{quantile_obj}
R(q, \tau) = (\tau - 1) \int_{-\infty}^q (x - q) \textnormal{d} F(x) + \tau
\int_q^{\infty} (x - q) \textnormal{d} F(x).
\end{equation}
Then the $\tau$-th quantile of $X$ is the value function evaluated at $\tau$:
\begin{equation}
\psi(R)(\tau) = \min_{q \in \mathbb{R}} R(q, \tau),
\end{equation}
We would like to estimate a uniform confidence band for the quantile function
over $\mathcal{T}$.
\end{exm}
To analyze the asymptotic distributions of these examples, one would typically
rely on the delta method because $\psi$ is a nonlinear map from objective
function to value function. In the next section we discuss the difficulties we
encounter with this approach and a solution for the purposes of uniform
inference.
\section{Differentiability properties} \label{sec:differentiability}
The method of analysis that we employ to establish uniform inference methods for
value functions uses the delta method, a cornerstone of statistical analysis.
The delta method is applied to nonlinear maps of observable data and depends on
the notion of Hadamard (or compact) differentiability to linearize the map near
the population distribution (see, for example, \citet[Section
3.9.3]{vanderVaartWellner96}). This has the advantage of dividing the analysis
into a purely nonstochastic part and a straightforward statistical part.
Therefore this analysis starts with a section on Hadamard derivatives without
considering sample data, before explicitly considering the behavior of sample
statistics in the next section.
The appropriate notion of differentiability for application of the delta method
between two metric spaces is Hadamard differentiability
\citep[Section 3.9]{vanderVaartWellner96}. \citet{Shapiro90}, \citet{Duembgen93} and
\citet{FangSantos19} discuss Hadamard \emph{directional} differentiability and
show that this weaker notion also allows for the application of the delta method.
\begin{defn}[Hadamard differentiability]\label{def:diff} Let $\mathbb{D}$ and $\mathbb{E}$
be Banach spaces and consider a map $\phi: \mathbb{D}_\phi \subseteq \mathbb{D} \rightarrow
\mathbb{E}$.
\begin{enumerate}
\item $\phi$ is \emph{Hadamard differentiable} at $f \in \mathbb{D}_\phi$
tangentially to a set $\mathbb{D}_0 \subseteq \mathbb{D}$ if there is a continuous
linear map $\phi'_f: \mathbb{D}_0 \rightarrow \mathbb{E}$ such that \begin{equation*}
\lim_{n \rightarrow \infty} \left\| \frac{\phi(f + t_n h_n) -
\phi(f)}{t_n} - \phi'_f(h) \right\|_\mathbb{E} = 0 \end{equation*} for all
sequences $\{h_n\} \subset \mathbb{D}$ and $\{t_n\} \subset \mathbb{R}$ such that $h_n
\rightarrow h \in \mathbb{D}_0$ and $t_n \rightarrow 0$ as $n \rightarrow
\infty$ and $f + t_n h_n \in \mathbb{D}_\phi$ for all $n$.
\item $\phi$ is \emph{Hadamard directionally differentiable} at $f \in
\mathbb{D}_\phi$ tangentially to a set $\mathbb{D}_0 \subseteq \mathbb{D}$ if there is a
continuous map $\phi'_f: \mathbb{D}_0 \rightarrow \mathbb{E}$ such that
\begin{equation*} \lim_{n \rightarrow \infty} \left\| \frac{\phi(f + t_n
h_n) - \phi(f)}{t_n} - \phi_f'(h) \right\|_\mathbb{E} = 0 \end{equation*} for
all sequences $\{h_n\} \subset \mathbb{D}$ and $\{t_n\} \subset \mathbb{R}_+$ such that
$h_n \rightarrow h \in \mathbb{D}_0$ and $t_n \rightarrow 0^+$ as $n \rightarrow
\infty$ and $f + t_n h_n \in \mathbb{D}_\phi$ for all $n$.
\end{enumerate}
\end{defn}
In case $\phi$ is only directionally differentiable, the derivative map
$\phi'_f$ need not be linear although $\phi'_f$ is continuous
\citep[Proposition 3.1]{Shapiro90}. The Hadamard directional derivative is
also called a \emph{semiderivative} elsewhere \citep[Definition
7.20]{RockafellarWets98}. The derivative is linear if $\phi$ is fully
differentiable. The delta method and chain rule can be applied to maps that
are either Hadamard differentiable or Hadamard directionally differentiable.
To discuss the differentiability properties of $\psi$ in equation \eqref{m_def},
for any $\epsilon \geq 0$ let $U_f: X \rightrightarrows U$ define the
set-valued map of $\epsilon$-maximizers of $f(\cdot, x)$ in $u$ for each $x \in
X$:
\begin{equation} \label{marginal_epsmax_def}
U_f(x, \epsilon) = \left\{ u \in A(x): f(u, x) \geq \psi(f)(x) - \epsilon \right\}.
\end{equation}
The marginal optimization map $\psi$ has a long history in
optimization and statistics and results on its pointwise directional
derivatives date to \citet{Danskin67}. See \citet{MilgromSegal02} for an
introduction to this pointwise case. \citet[Theorem
2.1]{CarcamoCuevasRodriguez19} show that, in our notation, $\psi$ is
directionally differentiable in $\ell^\infty(A(x) \times \{x\})$, and for
directions $h(\cdot, x)$ for any $x \in X$,
\begin{equation*}
\psi'_f(h)(x) = \lim_{\epsilon \rightarrow 0} \sup_{u \in U_f(x, \epsilon)} h(u, x).
\end{equation*}
Assuming the stronger conditions that $A$ is continuous and compact-valued and
that $f$ is continuous on $\mathrm{gr} A$, the maximum theorem implies that $U_f$ is
non-empty, compact-valued and upper hemicontinuous \citep[Theorem
17.31]{AliprantisBorder06}, and \citet[Corollary 2.2]{CarcamoCuevasRodriguez19}
show that tangentially to $\mathcal{C}(U \times \{x\})$, the derivative of
$\psi(f)(x)$ simplifies to
\begin{equation*}
\psi'_f(h)(x) = \sup_{u \in U_f(x, 0)} h(u, x).
\end{equation*}
Further, when (for fixed $x$) $U_f(x, 0)$ is a singleton set, $\psi(f)(x)$ is
Hadamard differentiable, not just directionally so.
The pointwise directional differentiability of $\psi(f)(x)$ at each $x$ might
lead one to suspect that $\psi$ is differentiable more generally as a map from
$\ell^\infty(\mathrm{gr} A)$ to $\ell^\infty(X)$. However, this is not true. The
following example illustrates a case where $\psi$ is not Hadamard directionally
differentiable as a map from $\ell^\infty(\mathrm{gr} A)$ to $\ell^\infty(X)$, although
$f$ and $h$ are both continuous functions.
\begin{exm}
Let $U \times X = [0, 1] \times [-1, 1]$ and define $f: U \times X \rightarrow \mathbb{R}$ by
\begin{equation*}
f(u, x) = \begin{cases} 0 & x \in [-1, 0] \\ x(u^2 - u) & x \in (0, 1] \end{cases}.
\end{equation*}
Let $h(u, x) = -u^2 + u$. Then $\psi(f)(x) = 0$ for all $x$ and for any $t > 0$,
\begin{equation*}
\psi(f + th)(x) = \begin{cases} t / 4 & x \in [-1, 0] \\ (t - x) / 4 & x \in (0, t] \\ 0 & x \in (t, 1] \end{cases}.
\end{equation*}
Therefore for each $x$,
\begin{equation*}
\lim_{t \rightarrow 0^+} \frac{\psi(f + th)(x) - \psi(f)(x)}{t} = \frac{1}{4} I(x \leq 0).
\end{equation*}
However, for any $t > 0$,
\begin{equation*}
\sup_{x \in [-1, 1]} \left| (\psi(f + th)(x) - \psi(f)(x)) / t - (1/4) I(x \leq 0) \right| = 1 / 4.
\end{equation*}
This implies that no derivative exists as an element of $\ell^\infty([-1,
1])$. The functions $f$ and $h$ are well behaved, but because the (unique)
candidate derivative $I(x \leq 0) / 4$ is not continuous in $x$, uniform
convergence of the quotients to the candidate fails.
\end{exm}
\subsection{Directional differentiability of optimization maps} \label{sec:stats}
The lack of uniformity of the convergence to $\psi_f'(h)(\cdot)$ appears to be a
problem for the use of the delta method for uniform inference, because the
typical path of analysis would use the existence of a well-behaved limit of the
sequence $r_n(\psi(f_n) - \psi(f))$ and the continuous mapping theorem to
discuss the distribution of statistics applied to the limit. However, this
issue can be circumvented. In particular, because the tool used in analysis is
often a real-valued statistic, we examine the map that includes not only the
marginal optimization step but also a functional applied to the resulting value
function.
We consider the following uniform test statistics $\lambda_j: \ell^\infty(\mathrm{gr} A)
\rightarrow \mathbb{R}$: letting $m$ denote Lebesgue measure,
\begin{align}
\begin{aligned}
\lambda_1(f) &= \sup_{x \in X} \left| \sup_{u \in A(x)} f(u, x) \right|, &\lambda_2(f) &= \sup_{x \in X} \left[ \sup_{u \in A(x)} f(u, x) \right]_+, \\
\lambda_3(f) &= \left( \int_X \left| \sup_{u \in A(x)} f(u, x) \right|^p \textnormal{d}
m(x) \right)^{1/p}, &\lambda_4(f) &= \left( \int_X \left[ \sup_{u \in A(x)}
f(u, x) \right]_+^p \textnormal{d} m(x) \right)^{1/p}. \label{lambdas_abstract}
\end{aligned}
\end{align}
These maps are typical $L_p$ norms (for $1 \leq p \leq \infty$) applied to
$\psi(f)$ or $[\psi(f)]_+$. Below, we show that the statistics represented by the maps
in \eqref{lambdas_abstract} are directionally differentiable as maps from
$\ell^\infty(\mathrm{gr} A)$ to $\mathbb{R}$, and can be used to create tests as well as
inverted to construct uniform confidence bands.
The maps $\lambda_2$ and $\lambda_4$ are included because one-sided comparisons
may also be of interest, and the map $f \mapsto [f]_+$ is only pointwise
differentiable but not differentiable as a map from $\ell^\infty(X)$ to
$\ell^\infty(X)$, similarly to the previous example. The strategy of including
the positive-part map $[\cdot]_+$ in a compound map also works, and we illustrate
its use in a stochastic dominance example below.
We first define some functions that will be used extensively below and their
derivatives. First, let $\mu: \ell^\infty(\mathrm{gr} A) \rightarrow \mathbb{R}$ be
defined as the operation of finding the supremum of $f$ over the graph of $A$:
\begin{equation} \label{mu_def}
\mu(f) = \sup_{(u, x) \in \mathrm{gr} A} f(u, x).
\end{equation}
The derivative of supremum maps is well known, see for example
\citet{CarcamoCuevasRodriguez19}. Define for any $\epsilon \geq 0$ the set of
$\epsilon$-maximizers of $f$ in $\mathrm{gr} A$ by
\begin{equation} \label{epsmax_def}
A_f(\epsilon) = \left\{ (u, x) \in \mathrm{gr} A : f(u, x) \geq \mu(f) - \epsilon
\right\}.
\end{equation}
Then
\begin{equation} \label{muprime_def}
\mu'_f(h) = \lim_{\epsilon \rightarrow 0^+} \sup_{(u, x) \in A_f(\epsilon)}
h(u, x).
\end{equation}
In some cases detailed in Theorem~\ref{thm:supnorm_stats} below,
$A_f(\epsilon)$ is replaced with $U_f(X, \epsilon) = \bigcup_{x \in X} U_f(x,
\epsilon)$, which is a specialization related to the distribution of the test
statistics under a null hypothesis.
Next, we define the maximin map $\sigma: \ell^\infty(\mathrm{gr} A) \rightarrow \mathbb{R}$ by
\begin{equation} \label{sigma_def}
\sigma(f) = \sup_{x \in X} \inf_{u \in A(x)} f(u, x).
\end{equation}
Lemma~\ref{lem:saddle} in the \hyperref[appn]{Appendix} shows that this map is
Hadamard directionally differentiable for general objective functions $f$ and
that the derivative formula is the one shown in~\eqref{sigmaprime_def} below.
Theorem 5 of \citet{MilgromSegal02} discusses one form of differentiability for
saddle-point problems, the differentiability of a saddle value in a parameter
when the objective function is parameterized. We require differentiability with
respect to functional perturbations of the objective function, and we deal only
with the map from objective function to maximin value, without conditions that
ensure the existence of a saddle point (where the maximin equals the minimax
value). This is similar to Proposition~4 of \citet{Demyanov09} for $f \in
\ell^\infty(\mathrm{gr} A)$ and is similar to Lemma~4.4 of
\citet{ChristensenConnault19}.
The next theorem uses the set-valued map $U_{(-f)}(x, \epsilon)$ to denote the
$\epsilon$-minimizers of $f(\cdot, x)$, where $U_f(x, \epsilon)$ was defined
in~\eqref{marginal_epsmax_def}. Similar to the notation $U_f$, for a function
$f \in \ell^\infty(X)$, let
\begin{equation*}
X_f(\epsilon) = \left\{ x \in X : f(x) \geq \sup_{x \in X} f(x) - \epsilon
\right\}.
\end{equation*}
Then for any $\epsilon \geq 0$, the set of near-maximinimizers of $f$ is
\begin{align*}
\left\{ x \in X: \inf_{u \in A(x)} f(u, x) \geq \sigma(f) - \epsilon \right\}
&= \left\{ x \in X: -\psi(-f)(x) \geq \mu(-\psi(-f)) - \epsilon \right\} \\
{} &= X_{(-\psi(-f))}(\epsilon).
\end{align*}
This set appears in the derivative
\begin{equation}
\sigma_f'(h) = \lim_{\delta \rightarrow 0^+} \sup_{x \in
X_{(-\psi(-f))}(\delta)} \lim_{\epsilon \rightarrow 0^+} \inf_{u \in
U_{(-f)}(x, \epsilon)} h(u, x). \label{sigmaprime_def}
\end{equation}
The verification of derivative~\eqref{sigmaprime_def} is shown in the
\hyperref[appn]{Appendix}. Minimax problems often use convex-concave objective
functions. However, the derivative formula~\eqref{sigmaprime_def} holds for
functions that may not be concave-convex. This result is used in
Theorem~\ref{thm:supnorm_stats} below. There is a practical price to pay for
this theoretical generality, which is that finding minimax (or maximin, in our
case) points of a general function can be much harder to do than in the special
case of a convex-concave objective function, in which the set of saddle points
is a product set. Compare the result of \citet[Theorem 3.1]{Shapiro08} for the
convex-concave case with, for example, the practical difficulties that arise in
computation discussed in \citet{LinJinJordan19}. In our simulations we resort
to grid search conducted over a low-dimensional region and are aided by the
imposition of null hypotheses that simplify computation.
\subsection{Directional differentiability of norms of value functions}
In this section we show that under general conditions the compound map that
sends objective functions to $L_p$ norms of value functions are Hadamard
directionally differentiable. In particular, Theorem~\ref{thm:supnorm_stats}
verifies that the supremum-type maps $\lambda_1$ and $\lambda_2$ are Hadamard
directionally differentiable maps. The derivative formulas simplify when the
value function is identically zero, and these cases are separated from the
non-zero cases. Some care must be taken interpreting the theorem because the
derivatives are stated in terms of conditions on the objective function $f$,
while the special cases are dictated by properties of the value function
$\psi(f)$. The special cases are useful when imposing hypothesized
restrictions, as will be seen in an example below.
\begin{thm} \label{thm:supnorm_stats}
Suppose $f, h \in \ell^\infty(\mathrm{gr} A)$, where $A: X \rightrightarrows U$ is
non-empty-valued for all $x \in X$. Define $\mu$
by~\eqref{mu_def} and $\sigma$ by~\eqref{sigma_def}, and their derivatives
$\mu_f'$ and $\sigma_f'$ as in~\eqref{muprime_def} and~\eqref{sigmaprime_def}.
Then:
\begin{enumerate}
\item $\lambda_1(f)$ is Hadamard directionally differentiable and
\begin{equation*}
\lambda'_{1f}(h) = \begin{cases} \mu_f'(h), & \mu(f) > \sigma(-f) \\ \max
\left\{ \mu_f'(h), \sigma_{(-f)}'(-h) \right\}, & \mu(f) = \sigma(-f) \\
\sigma_{(-f)}'(-h), & \mu(f) < \sigma(-f) \end{cases}.
\end{equation*}
\item If $\|\psi(f)\|_\infty = 0$, then $\lambda_1(f)$ is Hadamard
directionally differentiable and
\begin{equation*}
\lambda'_{1f}(h) = \max \left\{ \lim_{\epsilon \rightarrow 0^+}
\sup_{(u, x) \in U_f(X, \epsilon)} h(u, x), \lim_{\epsilon \rightarrow
0^+} \sup_{x \in X} \inf_{u \in U_f(x, \epsilon)} (-h(u, x)) \right\}.
\end{equation*}
\item $\lambda_2(f)$ is Hadamard directionally differentiable and
\begin{equation*}
\lambda'_{2f}(h) = \begin{cases} \mu_f'(h), & \mu(f) > 0 \\ \left[
\mu_f'(h) \right]_+, & \mu(f) = 0 \\ 0 & \mu(f) < 0 \end{cases}.
\end{equation*}
\item If $\|[\psi(f)]_+\|_\infty = 0$, then $\lambda_2(f)$ is Hadamard
directionally differentiable and
\begin{equation*}
\lambda'_{2f}(h) = \begin{cases} \left[ \lim_{\epsilon
\rightarrow 0^+} \sup_{(u, x) \in A_f(\epsilon)} h(u, x)
\right]_+, & \mu(f) = 0 \\ 0, & \mu(f) < 0 \end{cases}.
\end{equation*}
\end{enumerate}
\end{thm}
The next result shows that the $L_p$ functionals for $1 \leq p < \infty$,
defined in~\eqref{lambdas_abstract}, are Hadamard directionally differentiable,
depending on whether $\lambda_j(f) = 0$ or $\lambda_j(f) \neq 0$.
In order to simplify notation, define
\begin{equation} \label{contact_def}
X_0 = \left\{ x \in X : \psi(f)(x) = 0 \right\}.
\end{equation}
\begin{thm} \label{thm:lpnorm_stats}
Suppose that $f, h \in \ell^\infty(\mathrm{gr} A)$ where $A: X \rightrightarrows U$ is
non-empty-valued for all $x \in X$ and $m(X) < \infty$. Then:
\begin{enumerate}
\item If $\|\psi(f)\|_p \neq 0$, then $\lambda_3$ is Hadamard directionally
differentiable and
\begin{equation*}
\lambda_{3f}'(h) = \| \psi(f) \|_p^{1-p} \int_X \mathrm{sgn}(\psi(f)(x))
|\psi(f)(x)|^{p-1} \psi_f'(h)(x) \textnormal{d} m(x).
\end{equation*}
\item If $\|\psi(f)\|_p = 0$, then $\lambda_3$ is Hadamard directionally
differentiable and
\begin{equation*}
\lambda_{3f}'(h) = \left( \int_X \left| \psi_f'(h)(x)
\right|^p \textnormal{d} m(x) \right)^{1/p}.
\end{equation*}
\item If $\|[\psi(f)]_+\|_p \neq 0$, then $\lambda_4$ is Hadamard directionally
differentiable and
\begin{equation*}
\lambda_{4f}'(h) = \| [\psi(f)]_+ \|_p^{1-p} \int_X [\psi(f)(x)]_+^{p-1}
\psi_f'(h)(x) \textnormal{d} m(x).
\end{equation*}
\item If $\|[\psi(f)]_+\|_p = 0$, then $\lambda_4$ is Hadamard directionally
differentiable and
\begin{equation*}
\lambda_{4f}'(h) = \left( \int_{X_0} \left[ \psi_f'(h)(x) \right]_+^p
\textnormal{d} m(x) \right)^{1/p}.
\end{equation*}
\end{enumerate}
\end{thm}
This is related to the results of \citet{ChenFang19}, who dealt with a squared
$L_2$ statistic. It is interesting to note here that using an $L_2$ statistic
instead of its square results in first-order (directional) differentiability of
the map, unlike the squared $L_2$ norm. Theorem~\ref{thm:lpnorm_stats} makes
use of the pointwise definition of $\psi'_f$. Although convergence to
$\psi'_f(h)$ may not be uniform, the dominated convergence theorem only requires
pointwise convergence and integrability.
Next, we revisit Examples \ref{ex:bounds} and \ref{ex:stoc_dominance} to
illustrate applications of Theorem \ref{thm:lpnorm_stats}. The next section will
extend the methods in Theorem \ref{thm:lpnorm_stats} for continuous functions and then return to Example \ref{ex:quantile}.
\begin{excont}{ex:bounds}
Suppose that $L$ is the true lower bound function based on the
population CDFs $F = (F_0, F_1)$. To construct a uniform confidence band for
$L$ we can invert a test for the hypothesis that $L = L_0$ for some
fixed $L_0$, specifically
\begin{equation}
\lambda(F) = \sup_{x \in \mathbb{R}} |\sup_{u \in \mathbb{R}} \Pi(F)(u, x) - L_0(x)|
= \sup_{x \in \mathbb{R}} |L(x) - L_0(x)|.
\end{equation}
This is analogous to the classical Kolmogorov-Smirnov statistic for empirical
distribution functions.
The map $L = \psi(\Pi(F))$ does not generally have a derivative in
$\ell^\infty(\mathbb{R})$. However, Theorem~\ref{thm:supnorm_stats} indicates that
$\lambda$ does have a directional derivative, using the function $f(u, x) =
F_1(u) - F_0(u - x) - L_0(x)$. Under the hypothesis that $L_0$ really is the
true lower bound, the derivative is, using the second part of
Theorem~\ref{thm:supnorm_stats},
\begin{equation} \label{cband_derivative}
\lambda'_F(h) = \max \left\{ \lim_{\epsilon \rightarrow 0^+}
\sup_{(u, x) \in U_{\Pi(F)}(X, \epsilon)} \Pi(h)(u, x), \lim_{\epsilon \rightarrow
0^+} \sup_{x \in X} \inf_{u \in U_{\Pi(F)}(x, \epsilon)} (-\Pi(h)(u, x)) \right\}.
\end{equation}
\end{excont}
\begin{excont}{ex:stoc_dominance}
Similarly to the previous example, the map $(G_0, G_A, G_B) \mapsto L_A - U_B$
does not have a derivative in $\ell^\infty(\mathbb{R})$. However, for testing this
necessary condition for stochastic dominance we can consider the test
statistic
\begin{equation}
\Lambda = \left( \int [ (L_{A}(x) - U_{B}(x))]_+^2 \textnormal{d} m(x) \right)^{1/2}.
\end{equation}
This test statistic is zero when $L_A \leq U_B$ and greater
than zero otherwise. This represents not so much a direct test of the hypothesis
that $F_A$ dominates $F_B$ as, using a phrase from \citet{MastenPoirier20}, a
frontier at which of our ability to entertain the notion that $F_A$ dominates
$F_B$ breaks down. Although the function may not have an
$\ell^\infty(\mathbb{R})$-valued derivative,
$\Lambda$ is directionally differentiable, and, extending
Theorem~\ref{thm:lpnorm_stats} to two functions inside the integral,
\begin{multline} \label{biglambdaprime_def}
\Lambda'_G(h) = \bigg( \int_{L_A = U_B} \Big[ \lim_{\epsilon \rightarrow 0^+}
\sup_{u \in U_{\Pi(G_0, G_A)}(\epsilon, x)} \Pi(h_A, h_0) (u, x) \\
- \lim_{\epsilon \rightarrow 0^+} \inf_{u \in U_{-\Pi(G_B, G_0)}(\epsilon,
x)} \Pi(h_B, h_0)(u, x) \Big]_+^2 \textnormal{d} m(x) \bigg)^{1/2}.
\end{multline}
\end{excont}
It is difficult to find a general condition that ensures uniformity of the
convergence of the scaled differences to the pointwise derivatives. For
example, if we assume that $U$ and $X$ are compact sets and $f$ and $h$ are
continuous functions, then it is necessary that $\max_{x \in U_f(x, 0)} h(u, x)$
be a continuous function for uniform convergence. One way of ensuring this is
through (strict) monotonicity or concavity conditions, as the next example
shows.
\begin{excont}{ex:quantile}
Taking a derivative of the objective function~\eqref{quantile_obj} with
respect to $q$ for any $\tau$ we have $\partial R(q, \tau) / \partial q = F(q)
- \tau$. From this expression it can be seen that for each $\tau$, since $F$
is increasing, the objective function is convex in $q$ and the argmin map is
single-valued wherever $F$ is strictly increasing in $q$ locally to the
minimizer. Therefore, when $F$ is everywhere strictly increasing, a Hadamard
derivative exists in $\mathcal{C}(\mathcal{T})$:
\begin{equation} \label{psiprime_quan_continuous}
\psi_f'(h)(\cdot) = h(F^{-1}(\cdot), \cdot)
\end{equation}
for all bounded integrable directions $h$. This derivative is
linear since $F^{-1}(\tau)$ is the only choice for the argument $q$. In
contrast, when there are point masses in the distribution of $X$, a (bounded,
integrable) derivative of the marginal optimization map does not exist.
Instead, convergence to a derivative is pointwise for each $\tau$:
\begin{equation} \label{psiprime_quan_general}
\psi_f'(h)(\tau) = \begin{cases} h(F^{-1}(\tau), \tau) & \text{when } F
\text{ is strictly increasing at } F^{-1}(\tau) \\ \min_{q \in
[\underline{Q}(\tau), \overline{Q}(\tau)]} h(q, \tau) & \text{when } F(q)
= \tau \text{ for } q \in [\underline{Q}(\tau), \overline{Q}(\tau)]
\end{cases}.
\end{equation}
This collection of pointwise derivatives cannot be considered a derivative of
the optimization map. However, in this general case we can find derivatives
of uniform test statistics applied to the quantile function, like in the
uniform confidence band example.
\end{excont}
\section{Inference}\label{sec:inference}
In this section we derive asymptotic distributions for uniform statistics
applied to value functions, and propose bootstrap estimators of their
distributions that can be used for practical inference.
\subsection{Asymptotic distributions}
Now the derivatives developed in the previous subsections can be used
in a functional delta method. We make a high-level assumption on the way that
observations are related to estimated functions, but conditions under which such
convergence holds are well-known, and examples will be shown below.
\begin{enumerate}[label=\textbf{A\arabic*.}, ref=\textbf{A\arabic*}]
\item \label{A:estimator} Assume that for each $n$ there is a random sample
$\{Z_i\}_{i=1}^n$ and a map $\{Z_i\}_{i=1}^n \mapsto f_n$ where $f_n \in
\ell^\infty(U \times X)$. In case $\lambda_3$ or $\lambda_4$
is used, also assume $m(X) < \infty$. Furthermore, for some
sequence $r_n \rightarrow \infty$, $r_n (f_n - f) \leadsto \mathcal{G}$, where
$\mathcal{G}$ is a tight random element of $\ell^\infty(\mathrm{gr} A)$.
\end{enumerate}
In case an $L_p$ statistic is used (with $p < \infty$), we restrict the
space of allowable functions. In order to ensure that $L_p$ statistics are
well-defined we make the assumption that the measure of $X$ is finite. While
stronger than the assumption used for the supremum statistic, it is sufficient
to ensure that the $L_p$ statistics of the value function are finite. This
assumption also has the advantage of providing an easy-to-verify condition based
on the objective function $f$, rather than the more direct, but less obvious
condition $\psi(f) \in L_p(m)$. Lifting this restriction would require some
other restriction on the objective function $f$ that is sufficient to ensure the
value function is $p$-integrable.\footnote{We attempted to show $p$-integrability by
assuming that $f$ is bounded and integrable (note that this would imply $f$ is
$p$-integrable in $\mathrm{gr} A$), but were unable to show that this implies that the value
function is integrable (in $X$). Note that if one were able to show that $f$ is
bounded and integrable, then the elegant results in \citet[Section 3]{Kaji19}
would apply for the purposes of verifying Assumption~\ref{A:estimator}.}
Given assumption~\ref{A:estimator} and the derivatives of the last section,
the asymptotic distribution of test statistics applied to the value function is
straightforward.
\begin{thm} \label{thm:asymptotic}
Under Assumption~\ref{A:estimator}, for $j \in \{1, \ldots 4\}$,
\begin{equation*}
r_n \left(\lambda_j(f_n) - \lambda_j(f) \right) \leadsto \lambda_{jf}'(\mathcal{G}).
\end{equation*}
\end{thm}
This theorem is abstract and hides the fact that the limiting distributions may
depend on features of $\lambda_j$ and $f$. Therefore it is only indirectly
useful for inference. A resampling scheme is the subject of the next section.
Our bootstrap technique ahead is tailored to resampling under the null
hypothesis that $\lambda_j(f) = 0$. We assume that when $j = 1$ or $j = 3$ (that
is, conventional two-sided statistics are used), the
statistics are used to test the null and alternative hypotheses
\begin{align}
H_0&: \psi(f)(x) = 0 \quad \text{for all } x \in X \label{eq_null} \\
H_1&: \psi(f)(x) \neq 0 \quad \text{for some } x \in X.
\end{align}
Meanwhile, when $j = 2$ or $j = 4$ (the one-sided cases) we assume the
hypotheses are
\begin{align}
H_0&: \psi(f)(x) \leq 0 \quad \text{for all } x \in X \label{ineq_null} \\
H_1&: \psi(f)(x) > 0 \quad \text{for some } x \in X.
\end{align}
Let $P$ denote the joint probability measure associated with the
observations. This measure is assumed to belong to a collection $\mathcal{P}$,
the set of probability measures allowed by the model. A few
subcollections of $\mathcal{P}$ serve to organize the asymptotic results below. When
$P$ is such that $\psi(f) \equiv 0$, we label $P \in \mathcal{P}_{00}^E$ (E for
equality). For functional inequalities, the behavior of test statistics under
the null is more complicated. If $P$ is such that $\psi(f) \leq 0$
everywhere, then label $P \in \mathcal{P}_0^I$. When $P \in \mathcal{P}_0^I$ makes
$\psi(f)(x) = 0$ for at least one $x \in X$, then we label $P \in \mathcal{P}_{00}^I$.
The following corollary combines the derivatives from
Theorem~\ref{thm:supnorm_stats} and Theorem~\ref{thm:lpnorm_stats} with the
result of Theorem~\ref{thm:asymptotic} for these distribution classes. Recall
that the set $X_0$, defined in~\eqref{contact_def}, represents the subset of $X$
where $\psi(f)$ is zero.
\begin{cor} \label{cor:null_asymptotics}
Under Assumption~\ref{A:estimator}, for $j \in \{1, \ldots 4\}$,
\begin{enumerate}
\item When $P \in \mathcal{P}_{00}^E$,
\begin{equation*}
r_n (\lambda_1(f_n) - \lambda_1(f)) \leadsto \max \left\{ \lim_{\epsilon
\rightarrow 0^+} \sup_{(u, x) \in U_f(X, \epsilon)} \mathcal{G}(u, x),
\lim_{\epsilon \rightarrow 0^+} \sup_{x \in X} \inf_{u \in U_f(x,
\epsilon)} (-\mathcal{G}(u, x)) \right\}.
\end{equation*}
\item
\begin{enumerate}
\item When $P \in \mathcal{P}_{00}^I$,
$r_n(\lambda_2(f_n) - \lambda_2(f)) \leadsto \lim_{\epsilon \rightarrow
0^+} \sup_{x \in A_f(\epsilon)} \left[ \mathcal{G}(u, x) \right]_+.$
\item When $P \in \mathcal{P}_0^I \backslash \mathcal{P}_{00}^I$,
$r_n(\lambda_2(f_n) - \lambda_2(f)) \stackrel{p}{\rightarrow} 0$.
\end{enumerate}
\item When $P \in \mathcal{P}_{00}^E$, $r_n(\lambda_3(f_n) -
\lambda_3(f)) \leadsto \left( \int_X \left| \psi_f'(\mathcal{G})(x) \right|^p
\textnormal{d} m(x) \right)^{1/p}$.
\item
\begin{enumerate}
\item When $P \in \mathcal{P}_{00}^I$, $r_n(\lambda_2(f_n) - \lambda_2(f)) \leadsto
\left( \int_{X_0} \left[ \psi_f'(\mathcal{G})(x) \right]_+^p \textnormal{d} m(x)
\right)^{1/p}$.
\item When $P \in \mathcal{P}_0^I \backslash \mathcal{P}_{00}^I$,
$r_n(\lambda_2(f_n) - \lambda_2(f)) \stackrel{p}{\rightarrow} 0$.
\end{enumerate}
\end{enumerate}
\end{cor}
The distributions described in the above corollary under the assumption that
$P \in \mathcal{P}_{00}^E$ or $\mathcal{P}_{00}^I$ are those that we emulate using
resampling methods in the next section.
\subsection{Resampling}
In the previous section we established asymptotic distributions for the test statistics of interest. However, for practical inference we turn to resampling
to estimate their distributions. This section suggests the use of a resampling strategy that was proposed by \citet{FangSantos19}, and it combines a standard bootstrap procedure with estimates of the directional derivatives $\lambda_{jf}'(\cdot)$.
The resampling scheme described below is designed to reproduce the null
distribution under the assumption that the statistic is equal to zero, or in
other terms, that $P \in \mathcal{P}_{00}^k$ for $k \in \{E, I\}$. This is achieved
by restricting the form of the estimates $\hat{\lambda}_{jn}'$, for $j \in \{1,
2, 3, 4\}$. The bootstrap routine is consistent under more general conditions,
as in Theorem 3.1 of \citet{FangSantos19}. However, we discuss behavior of
bootstrap-based tests under the assumption that one of the null conditions
described in the previous section holds. We ensure that the estimators hold
uniformly over all $P$ in the null region, in order to provide uniform size
control and avoid test bias, as emphasized in \citet{LintonSongWhang10}.
All the derivative formulas in Corollary~\ref{cor:null_asymptotics} require some
form of an estimate of the near-maximizers of $f$ in $u$, that is, of the set
$U_f(x, \epsilon)$ defined in~\eqref{marginal_epsmax_def} for various $x$.
The set estimators we use are similar to those used in
\citet{LintonSongWhang10}, \citet{ChernozhukovLeeRosen13} and
\citet{LeeSongWhang18} and depend on slowly decreasing sequences of constants to
estimate the relevant sets in the derivative formulas. For a sequence $a_n
\rightarrow 0^+$ we estimate $U_f(x, \epsilon)$ with the plug-in estimator
$U_{f_n}(x, a_n)$. The sequence $a_n$ should decrease more slowly than the rate
at which $f_n$ converges uniformly to $f$, an assumption which will be
formalized below.
Define the estimators of $\lambda_{1f}'$ and $\lambda_{2f}'$ as
\begin{equation} \label{est1_def}
\hat{\lambda}_{1n}'(h) = \max \left\{ \sup_{(u, x) \in U_{f_n}(X, a_n)} h(u,
x), \sup_{x \in X} \inf_{u \in U_{f_n}(x, a_n)} (-h(u, x)) \right\}
\end{equation}
and
\begin{equation}
\hat{\lambda}_{2n}'(h) = \left[ \sup_{(u, x) \in A_{f_n}(a_n)} h(u, x)
\right]_+.
\end{equation}
These estimates impose the condition that $P \in \mathcal{P}_{00}^E$ or
$\mathcal{P}_{00}^I$ on the behavior of the derivatives. When $P \in \mathcal{P}_{00}^E$
the imposition amounts to the assumption that $\psi(f)(x) = 0$ for each $x$, so
that the set of population $\epsilon$-maximizers in $\mathrm{gr} A$ is $U_f(X,
\epsilon)$, and the estimator $\hat{\lambda}_{1n}'$ emulates that. The
condition $P \in \mathcal{P}_{00}^I$ implies that for some $x^*$, $\psi(f)(x^*) = 0$.
The estimate $\hat{\lambda}'_{2n}$ uses the arguments that come close to
maximizing the function even if the true supremum is nonzero.
The estimator $\hat{\lambda}_{3n}'$ uses a near-maximizer set like
$\hat{\lambda}_{1n}'$: let
\begin{equation}
\hat{\lambda}_{3n}'(h) = \left( \int_X \left| \sup_{u \in U_{f_n}(x, a_n)}
h(u, x) \right|^p \textnormal{d} m(x) \right)^{1/p}.
\end{equation}
The estimator $\hat{\lambda}_{4n}'$ requires a second
estimate. For another sequence $b_n \rightarrow 0^+$, estimate the set $X_0$
defined in~\eqref{contact_def} with
\begin{equation*}
\hat{X}_0 = \{x \in X : |\psi(f_n)(x)| \leq b_n\}.
\end{equation*}
If $\hat{X}_0 = \varnothing$, then set $\hat{X}_0 = X$. This is a method of
enforcing the null hypothesis that $P \in \mathcal{P}_{00}^I$.
Then let
\begin{equation} \label{est4_def}
\hat{\lambda}_{4n}'(h) = \left( \int_{\hat{X}_0} \left[ \sup_{u \in U_{f_n}(x,
a_n)} h(u, x) \right]_+^p \textnormal{d} m(x) \right)^{1/p}.
\end{equation}
The estimate $\hat{X}_0$ makes $\hat{\lambda}_{4n}'$ a sort of combination of the
estimates $U_{f_n}(X, a_n)$ and $A_{f_n}(a_n)$: with $b_n = a_n$, assuming there
is some $x$ such that $|\psi(f_n)(x)| \leq a_n$, we have $A_{f_n}(a_n) =
U_{f_n}(\hat{X}_0, a_n)$, and then $\hat{\lambda}_{2n}'$ and
$\hat{\lambda}_{4n}'$ are evaluated over the same set. $\hat{\lambda}_{2n}'$
and $\hat{\lambda}_{4n}'$ are qualitatively different when
the initial set estimate $\hat{X}_0$ is empty, leading to the use of $\hat{X}_0
= X$, in which case the derivative is evaluated over the same set as
$\hat{\lambda}_{3n}'$, that is, $U_{f_n}(X, a_n)$.
We assume that an exchangeable bootstrap is used, which depends on a set of
weights $\{W_i\}_{i=1}^n$ that are independent of the observations and that put
probability mass $W_i$ at each observation $Z_i$. This type of bootstrap
describes many well-known bootstrap techniques \citep[Section
3.6.2]{vanderVaartWellner96}. We make the following assumptions to ensure the
bootstrap is well-behaved. To discuss bootstrap consistency in a Banach space
$\mathbb{D}$ we follow the precedent of considering functions $g \in BL_1(\mathbb{D})$,
which consists of $g$ with level and Lipschitz constant bounded by 1.
\begin{enumerate}[label=\textbf{A\arabic*.}, ref=\textbf{A\arabic*}]
\setcounter{enumi}{1}
\item \label{A:sequences} Assume that $r_n (f_n - f) \leadsto \mathcal{G}$ uniformly in
$\mathcal{P}$. Furthermore, $a_n, b_n \rightarrow 0^+$ and $a_n r_n \rightarrow
\infty$ and $b_n r_n \rightarrow \infty$.
\item \label{A:bootstrapf} Suppose that for each $n$, $W$ is independent of
the data $Z$ and there is a map $\{Z_i, W_i\}_{i=1}^n \mapsto f_n^*$ where
$f_n^* \in \ell^\infty(\mathrm{gr} A)$. If $\lambda_3$ or $\lambda_4$ is used, also
assume that $m(X) < \infty$. $r_n(f_n^* - f_n)$ is asymptotically
measurable, for all continuous and bounded $g$, $g(r_n(f_n^* - f_n))$ is a
measurable function of $\{W_i\}$ outer almost surely in $\{Z_i\}$, and
\begin{equation*}
\limsup_{n \rightarrow \infty} \sup_{P \in \mathcal{P}} \sup_{g \in
BL_1(\ell^\infty(\mathrm{gr} A))} \left| \ex{g(r_n(f_n^* - f_n)) | Z } -
\ex{g(\mathcal{G})} \right| = 0.
\end{equation*}
\end{enumerate}
Assumption~\ref{A:sequences} is used to ensure consistency of the
$\epsilon$-maximizer set estimators used in the bootstrap algorithm. It
requires that the weak limit of the scaled and centered estimated objective
function is stable enough that the $a_n$ and $b_n$ sequences provide estimators
under null and alternative distributions. This disallows limiting distributions
that are discontinuous in $P$. In the examples we use, objective functions are
built from uniform Donsker classes of functions, so this assumption is
satisfied.
Assumption~\ref{A:bootstrapf} ensures that a bootstrap version $f_n^*$ exists
and is well-behaved enough that $r_n(f_n^* - f_n)$ behaves asymptotically like
$r_n(f_n - f)$, uniformly in $\mathcal{P}$. Luckily, this condition is satisfied in
our examples, thanks to Lemma A.2 of \citet{LintonSongWhang10}, which states
that uniform Donsker classes are also bootstrap uniform Donsker classes. For our
examples, the underlying functions are one or a few distribution functions, and
the standard empirical process corresponds to the family of indicator functions
$\tilde{\mathcal{F}} = \{ I(X \leq x), x \in \mathbb{R}^p \}$, which is a uniform Donsker
class over all probability measures \citep[top of p.1995]{SheehyWellner92}, and
therefore $f_n$ and their bootstrap counterparts satisfy
Assumption~\ref{A:bootstrapf}. \citet{LintonSongWhang10} show conditions under
which observations that may be residuals from semiparametric estimation
constitute a uniform Donsker class.
\vspace{1em}
\noindent \textbf{Resampling routine to estimate the distribution of
$r_n(\lambda_j(f_n) - \lambda_j(f))$}
\begin{enumerate}
\item Estimate $\hat{\lambda}_{jn}'$ using sample data and
formulas~\eqref{est1_def}-\eqref{est4_def} above.
\end{enumerate}
Then repeat steps \ref{resample12}-\ref{resample22} for $r = 1, \ldots R$:
\begin{enumerate}
\setcounter{enumi}{1}
\item \label{resample12} Use an exchangeable bootstrap to construct $f_n^*$.
\item \label{resample22} Calculate the resampled test statistic $\lambda^*_r =
\hat{\lambda}_{jn}'(r_n(f_n^* - f_n))$
\end{enumerate}
Finally,
\begin{enumerate}
\setcounter{enumi}{3}
\item Let $\hat{q}_{\lambda^{*}}(1-\alpha)$ be the $(1-\alpha)$-th sample
quantile from the bootstrap distribution of $\{\lambda_r^*\}_{r=1}^R$, where
$\alpha \in (0, 1)$ is the nominal size of the test. Reject the null
hypothesis if $r_n \lambda_j(f_n)$ is larger than
$\hat{q}_{\lambda^{*}}(1-\alpha)$.
\end{enumerate}
The consistency of this resampling procedure under the null hypothesis is
summarized in the following theorem.
\begin{thm} \label{thm:bootstrap_consistency_teststats}
Under Assumptions~\ref{A:estimator}-\ref{A:bootstrapf}, if $j \in \{1, 3\}$,
\begin{equation*}
\limsup_{n \rightarrow \infty} \sup_{P \in \mathcal{P}_{00}^E} \sup_{g \in
BL_1(\mathbb{R})} \left| \ex{g \left( \hat{\lambda}_{jn}' \left( r_n (f_n^* - f_n)
\right) \right) \big| \{Z_i\}_{i=1}^n} - \ex{g \left( \lambda_{jf}'(\mathcal{G})
\right)} \right| = 0,
\end{equation*}
and if $j \in \{2, 4\}$,
\begin{equation*}
\limsup_{n \rightarrow \infty} \sup_{P \in \mathcal{P}_{00}^I} \sup_{g \in
BL_1(\mathbb{R})} \left| \ex{g \left( \hat{\lambda}_{jn}' \left( r_n (f_n^* - f_n)
\right) \right) \big| \{Z_i\}_{i=1}^n} - \ex{g \left( \lambda_{jf}'(\mathcal{G})
\right)} \right| = 0.
\end{equation*}
\end{thm}
We conclude this section by revisiting Examples \ref{ex:bounds}, \ref{ex:stoc_dominance}, and \ref{ex:quantile} and describing the resampling procedures in the corresponding cases.
\begin{excont}{ex:bounds}
Recall that the statistic $\lambda(F) = \sup_{x \in \mathbb{R}} |L(x) - L_0(x)|$
should be inverted to find a uniform confidence band for $L$ (where $L$ is the
value function $L(x) = \sup_u (F_1(u) - F_0(u - x))$ and the objective
function is a simple map of two marginal distribution functions). This
suggests the statistic
\begin{equation}
\lambda(\mathbb{F}) = \sqrt{n} \sup_{x \in \mathbb{R}} | \mathbb{L}_n(x) - L_0(x) |,
\end{equation}
which is the plug-in estimate using empirical CDFs, that is, denoting
$\mathbb{L}_n(x) = \sup_{u \in \mathbb{R}} (\mathbb{F}_{1n}(u) - \mathbb{F}_{0n}(u - x))$.
For each $c \geq 0$,
\begin{align*}
\prob{\sup_x \sqrt{n} | \mathbb{L}_n(x) - L_0(x) | \leq c} &= \prob{\sqrt{n} |
\mathbb{L}_n(x) - L_0(x) | \leq c \text{ for all } x} \\
{} &= \prob{\mathbb{L}_n(x) - c / \sqrt{n} \leq L_0(x) \leq \mathbb{L}_n(x) + c /
\sqrt{n} \text{ for all } x},
\end{align*}
implying that the appropriate quantile of the statistic provides us an
estimated uniform confidence band.
Standard conditions ensure that $\sqrt{n}(\mathbb{F}_n - F) \leadsto
\mathcal{G}_F$, where $\mathcal{G}_F$ is a Gaussian process. Under the null hypothesis,
\begin{align*}
\sqrt{n}\sup_x | \mathbb{L}_n(x) - L_0(x) | &\stackrel{H_0}{=} \sqrt{n} \left(
\sup_x | \mathbb{L}_n(x) - L_0(x) | - \sup_x | L(x) - L_0(x) | \right) \\
{} &= \sqrt{n} \left( \lambda(\mathbb{F}_n) - \lambda(F) \right) \\
{} &\leadsto \lambda'_F(\mathcal{G}_F) \\
{} &\sim \max \bigg\{ \lim_{\epsilon \rightarrow 0^+}
\sup_{(u, x) \in U_{\Pi(F)}(X, \epsilon)} \left( \mathcal{G}_{F_1}(u) -
\mathcal{G}_{F_0}(u - x) \right), \\
{} &\phantom{=} \qquad \qquad \qquad \lim_{\epsilon \rightarrow 0^+} \sup_{x
\in X} \inf_{u \in U_{\Pi(F)}(x, \epsilon)} \left( -\mathcal{G}_{F_1}(u) +
\mathcal{G}_{F_0}(u - x) \right) \bigg\},
\end{align*}
where the last line uses~\eqref{cband_derivative}. This derivative needs
to be estimated, and we estimate the set-valued map $U_{\Pi(F)}(x, \epsilon)$
with the plug-in estimate
\begin{equation}
U_{\Pi(\mathbb{F}_n)}(x, a_n) = \left\{ u \in \mathbb{R} : \mathbb{F}_{1n}(u) - \mathbb{F}_{0n}(u - x)
\geq \mathbb{L}_n(x) - a_n \right\},
\end{equation}
where $a_n$ is a sequence that converges slowly to zero.
Using Corollary~\ref{cor:null_asymptotics} and a slight modification of
Theorem~\ref{thm:bootstrap_consistency_teststats} for this problem to use the
estimate
\begin{equation}
\hat{\lambda}_F'(h) = \max \bigg\{ \sup_{(u, x) \in U_{\Pi(\mathbb{F}_n)}(X, a_n)}
\left( h_1(u) - h_0(u - x) \right), \sup_{x \in X} \inf_{u \in
U_{\Pi(\mathbb{F}_n)}(x, \epsilon)} \left( -h_1(u) + h_0(u - x) \right) \bigg\},
\end{equation}
we find
\begin{equation*}
\prob{ \sqrt{n} \hat{\lambda}_F' (\mathbb{F}_n^* - \mathbb{F}_n) \leq c \; \Big|
\{X_i\}_{i=1}^n } \rightarrow \prob{ \lambda_F' (\mathcal{G}_F) \leq c}.
\end{equation*}
We can find a critical value of the asymptotic distribution using the
bootstrap by estimating
\begin{equation*}
c^*_{1-\alpha} = \min \left\{ c : \prob{ \sqrt{n} \hat{\lambda}_F' (\mathbb{F}_n^* -
\mathbb{F}_n) \leq c \; \Big| \{X_i\}_{i=1}^n } \geq 1 - \alpha \right\},
\end{equation*}
simulating, for $r = 1, \ldots R$, $\lambda_r^* = \sqrt{n}
\hat{\lambda}_F'(\mathbb{F}_n - F)$ and finding the $(1-\alpha)$-th quantile of the
bootstrap sample $\{\lambda^*_r\}_{r=1}^R$. Simulation evidence presented in
the next section verifies that the coverage probability of these intervals is
accurate for a representative data generating process.
\end{excont}
\begin{excont}{ex:stoc_dominance}
Suppose that three independent iid samples
$\{X_{0i}\}_{i=1}^n$, $\{X_{Ai}\}_{i=1}^n$ and $\{X_{Bi}\}_{i=1}^n$ are
observed. Let $\mathbb{G}_n = (\mathbb{G}_{0n}, \mathbb{G}_{An}, \mathbb{G}_{Bn})$ be their empirical
distribution functions, and define the sample statistic
\begin{equation}
\sqrt{n} \hat{\Lambda} = \left( \int_\mathbb{R} \left[ \mathbb{L}_{An}(x) -
\mathbb{U}_{Bn}(x) \right]_+^2 \textnormal{d} m(x) \right)^{1/2},
\end{equation}
where $\mathbb{L}_{An}$ is the plug-in estimate of $L_A$ using $\mathbb{G}_{0n}$ and
$\mathbb{G}_{An}$ and $\mathbb{U}_{Bn}$ is the plug-in estimate of $U_B$ using $\mathbb{G}_{0n}$
and $\mathbb{G}_{Bn}$. Under standard conditions,
$\sqrt{n}(\mathbb{G}_n - G) \leadsto \mathcal{G}_G$, where $\mathcal{G}_G$ is a stochastic process in
$(\ell^\infty(\mathbb{R}))^3$. The theory above can be extended in a straightforward
way to show that
\begin{equation}
\sqrt{n} \hat{\Lambda} \leadsto \Lambda_G'(\mathcal{G}_G),
\end{equation}
where $\Lambda_G'(h)$ was defined in~\eqref{biglambdaprime_def}.
To estimate the distribution of $\Lambda_G'(\mathcal{G}_G)$, some estimates of the
derivative are required. Given sequences $\{a_n\}$ and $\{b_n\}$, let
\begin{equation*}
\hat{D}_0 = \{x \in \mathbb{R}: |\psi(\Pi(\mathbb{F}_0, \mathbb{F}_A))(x) -
\psi(\Pi(\mathbb{F}_0, \mathbb{F}_B))(x)| \leq b_n\},
\end{equation*}
and estimate the near-maximizers in $u$ for each $x$ as in the previous
example. Estimate the distribution by calculating resampled statistics, for
$r = 1, \ldots, R$,
\begin{multline*}
\Lambda^*_r = \sqrt{n} \Bigg( \int_{\hat{D}_0} \bigg[ \sup_{u \in
U_{\Pi(\mathbb{F}_{0n}, \mathbb{F}_{An})}(x, a_n)} \Pi(\mathbb{F}^*_{0n} - \mathbb{F}_{0n},
\mathbb{F}^*_{An} - \mathbb{F}_{An})(u, x) \\
+ \sup_{u \in U_{-\Pi(\mathbb{F}_{0n}, \mathbb{F}_{Bn})}(x, a_n)} (-\Pi)(\mathbb{F}_{0n}^*
- \mathbb{F}_{0n}, \mathbb{F}_{Bn}^* - \mathbb{F}_{Bn}) (u, x) \bigg]_+^2 \textnormal{d} m(x)
\Bigg)^{1/2}.
\end{multline*}
A test can be conducted by comparing $\sqrt{n}\hat{\Lambda}$ to the
$1-\alpha$-th quantile of the bootstrap distribution. A simulation in the
next subsection illustrates the accurate size and power of this testing
strategy.
\end{excont}
\begin{excont}{ex:quantile}
The derivatives~\eqref{psiprime_quan_continuous}
and~\eqref{psiprime_quan_general} imply some known facts about
bootstrapping quantile functions \citep[Section
3.9.4.2]{vanderVaartWellner96}. Equation~\eqref{psiprime_quan_continuous}
implies that if the distribution is continuous, standard resampling can be
used to estimate not just the sample quantiles at any $\tau$, but also as a
way of inferring features of the quantile function uniformly.
On the other hand, equation~\eqref{psiprime_quan_general} implies that the
although the standard bootstrap can be used to conduct inference on any
quantile where the distribution is continuous, for distributions
with point masses, alterations must be made.
\citet{ChernozhukovFernandezValMellyWuethrich16} propose an estimator that
addresses this concern.
\end{excont}
\subsection{Local size control}
It is also of interest to examine how these tests behave under sequences of
distributions local to distributions that satisfy the null hypothesis.
We consider sequences of local alternative distributions $\{P_n\}$ such that for
each $n$, $\{Z_i\}_{i=1}^n$ are distributed according to $P_n$, and $P_n$
converges towards a limit $P_0$ that satisfies the null hypothesis. To describe
this process, for $t \geq 0$ define a path $t \mapsto P_t$, where $P_t$ is an
element of the space of distribution functions $\mathcal{P}$, such that
\begin{equation} \label{local_sqrt}
\lim_{t \rightarrow 0} \int \left( ((\textnormal{d} P_t)^{1/2} - (\textnormal{d} P_0)^{1/2}) / t -
\frac{1}{2} h (\textnormal{d} P_0)^{1/2} \right)^2 \rightarrow 0,
\end{equation}
and $P_0$ corresponds to a distribution that satisfies the null hypothesis. The
direction that the sequence approaches the null is described asymptotically by
the score function $h \in L_2(F)$, which satisfies $\ex{h} = 0$. We assume that
by letting $t = c / r_n$ for $c \in \mathbb{R}$, we can parameterize distributions
that are local to $P_0$ and for $t \geq 0$ denote $f(P_t)$ as the function $f$
under distribution $f(P_t)$ so that the unmarked $f$ described above can be
rewritten $f = f(P_0)$. See, e.g., \citet[Section 3.10.1]{vanderVaartWellner96}
for more details.
The following assumption ensures that $f$ remains suitably regular under such
local perturbations to the null distribution.
\begin{enumerate}[label=\textbf{A\arabic*.}, ref=\textbf{A\arabic*}]
\setcounter{enumi}{3}
\item \label{A:local_alts} For all $c \in \mathbb{R}$,
\begin{enumerate}
\item There exists some $f'(\cdot) \in \ell^\infty(\mathrm{gr} A)$
such that $\|r_n (f(P_{c/r_n}) - f(P_0)) - f'(c)\|_\infty
\rightarrow 0$, where $P_{c/r_n}$ satisfy~\eqref{local_sqrt}.
\item $r_n(f_n - f(P_{c/r_n})) \leadsto \mathcal{G}$ in $\ell^\infty(\mathrm{gr} A)$,
where for each $n$, $\{X_i\}_{i=1}^n \sim P_{c/r_n}$.
\end{enumerate}
\end{enumerate}
Both parts of Assumption~\ref{A:local_alts} ensure that $f_n$ behaves regularly
as distributions drift towards the null region $\mathcal{P}_{00}^E$ or $\mathcal{P}_{00}^I$.
\begin{thm} \label{thm:size_control}
Make Assumptions~\ref{A:estimator}-\ref{A:local_alts}. If $j = 1$ or $3$,
$P \in \mathcal{P}_{00}^E$ and for each $n$, $P_n = P_{c / r_n} \in
\mathcal{P}_{00}^E$, then
\begin{equation*}
\limsup_{n \rightarrow \infty} P_n \left\{ r_n \hat{\lambda}_{jn}(f_n) >
\hat{q}_{\lambda^*}(1-\alpha) \right\} = \alpha.
\end{equation*}
If $j = 2$ or $4$, $P \in \mathcal{P}_0^I$ and $P_n = P_{c/r_n} \in
\mathcal{P}_{00}^I$ for all $n$, then
\begin{equation*}
\limsup_{n \rightarrow \infty} P_n \left\{ r_n \hat{\lambda}_{jn}(f_n) >
\hat{q}_{\lambda^*}(1-\alpha) \right\} \leq \alpha.
\end{equation*}
\end{thm}
Theorem~\ref{thm:size_control} shows that the size of tests can be controlled
locally to the null region, and the nominal rejection probability matches the
intended probability for tests of equality. The inequality in the second part
of this theorem results from the one-sidedness of the test statistics
$\lambda_2$ and $\lambda_4$. This is related to a literature in
econometrics on moment inequality testing. Tests may exhibit size that is
lower than nominal for local alternatives that are from the interior of the
null region. A few possible solutions to this problem have been proposed. For
example, one might evaluate the region where the moments appear to hold with equality,
which leads to contact set estimates like in the bootstrap routine described above
\citep{LintonSongWhang10}. Alternatively, we may alter the reference distribution by shifting
it in the regions where equality does not seem to hold \citep{AndrewsShi17}.
In the next section we illustrate the usefulness of our results by
returning to the example of Makarov bounds for treatment effect distributions
and providing details on the construction of uniform confidence bands around the
bound functions.
\section{Dependency bounds for treatment effect distributions} \label{sec:bounds}
In this section, we specialize the above theory to investigate uniform inference
for bounds on the distribution function of the treatment effects
distribution when the analyst has no knowledge of the dependence between
potential outcomes. In addition, we use this example to provide numerical simulations illustrating the empirical power function of the proposed tests.
Suppose a binary treatment is independent of two potential outcomes $(X_0,
X_1)$, where $X_0$ denotes outcomes under a control regime and $X_1$ denotes
outcomes under a treatment, and $X_0$ and $X_1$ have marginal distribution
functions $F_0$ and $F_1$ respectively. Suppose that interest is in the
distribution of the treatment effect $\Delta = X_1 - X_0$ but we are
unwilling to make any assumptions regarding the dependence between $X_0$ and
$X_1$. In this section we study the relationship between the identifiable
functions $F_0$ and $F_1$ and functions that bound the distribution function
of the unobservable random variable $\Delta$.
\subsection{The value functions}
As discussed above, the distribution function of interest is $F_\Delta(\cdot)$,
which is not point-identified because the full bivariate distribution
of $(X_0, X_1)$ is unidentified and the analyst has no knowledge of the
dependence between potential outcomes. However, $F_\Delta$ can be bounded.
The distribution bounds $L$ and $U$, defined in \eqref{fbound_lo} and \eqref{fbound_hi},
respectively, were derived independently by \citet{Makarov82},
\citet{Rueschendorf82}, \citet{FrankNelsenSchweizer87}, and extended by
\citet{WilliamsonDowns90} to the random variables defined by the four basic
binary arithmetic operators on $X_0$ and $X_1$. The map $(f_0, f_1) \mapsto
\inf_u (f_1(u) - f_0(u - x))$ is termed an \emph{epi-sum} or \emph{infimal
convolution} in other literatures \citep[Section 1-H]{RockafellarWets98}. The
bounds are derived by applying the Fr\'echet-Hoeffding copula bounds to the
joint distribution function $F_{X_0, X_1}$ \citep[p. 96-97]{WilliamsonDowns90}
to find the most extreme possible values of the distribution function of the
random variable $X_1 - X_0$. $L$ and $U$ satisfy $L(x) \leq
F_\Delta(x) \leq U(x)$ for each $x \in \mathbb{R}$, depend only on the
marginal distribution functions $F_0$ and $F_1$ and are pointwise sharp: for any
fixed $x_0$ there exist some $X_0^*$ and $X_1^*$ such that the resulting
$\Delta^* = X^*_1 - X^*_0$ has a distribution function $F^*$ such that
$F^*(x_0) = L(x_0)$ or $F^*(x_0) = U(x_0)$. Simple cases of equality occur
when one distribution is continuous and the other is degenerate at a
point~--- for example, if $F_0$ is continuous and $F_1(x) = I(x_0 \leq x)$ for
some $x_0 \in \mathbb{R}$, $L(x) = U(x) = F_\Delta(x) = 1 - F_0(x_0 - x)$ for all $x$,
and if $F_0(x) = I(x_0 \leq x)$ while $F_1$ is continuous, then $L(x) = U(x) =
F_\Delta(x) = F_1(x_0 + x)$. See Section 2 of \citet{FanPark10} for more
details and references.
\subsection{Estimation and uniform inference}
Suppose we observe samples $\{X_{ki}\}_{i=1}^{n_k}$ for $k \in \{0, 1\}$, and
recall that $\mathbb{L}_n$ and $\mathbb{U}_n$ are the nonparametric plug-in estimates of $L$ and $U$:
\begin{align}
\mathbb{L}_n(x) &= \sup_{u \in \mathbb{R}} \{ \mathbb{F}_{1n}(u) - \mathbb{F}_{0n}(u - x) \} \label{eq:SDLB}\\
\mathbb{U}_n(x) &= \inf_{u \in \mathbb{R}} \{ 1 + \mathbb{F}_{1n}(u) - \mathbb{F}_{0n}(u - x) \}. \label{eq:SDUB}
\end{align}
We make the following assumptions on the observed samples of treatment and control observations.
\begin{enumerate}[label=\textbf{B\arabic*.}, ref=\textbf{B\arabic*}]
\item \label{assume:first} The observations $\{X_{0i}\}_{i=1}^{n_0}$ and $\{X_{1i}\}_{i=1}^{n_1}$ are iid samples and independent of each other and are distributed with marginal distribution functions $F_0$ and $F_1$ respectively. Refer to the pair of distribution functions and their empirical distribution function estimates as $F = (F_0, F_1)$ and $\mathbb{F}_n = (\mathbb{F}_{0n}, \mathbb{F}_{1n})$.
\item \label{assume:last} The sample sizes $n_0$ and $n_1$ increase in such a way that $n_k / (n_0 + n_1) \rightarrow \nu_k$ as $n_0, n_1 \rightarrow \infty$, where $0 < \nu_k < 1$ for $k \in \{0, 1\}$. Define $n = n_0 + n_1$.
\end{enumerate}
Under Assumptions~\ref{assume:first}
and~\ref{assume:last}, it is a standard result \citep[Example
19.6]{vanderVaart98} that for $k \in \{0, 1\}$,
$\sqrt{n_k} (\mathbb{F}_{kn} - F_k) \leadsto \mathcal{G}_k$, where $\mathcal{G}_0$ and $\mathcal{G}_1$ are
independent $F_0$- and $F_1$-Brownian bridges, that is, mean-zero Gaussian
processes with covariance functions $\rho_k(x, y) = F_k(x \wedge y) - F_k(x)
F_k(y)$. This implies in turn that \( \sqrt{n} (\mathbb{F}_n - F) \leadsto \mathcal{G}_F = (
\mathcal{G}_0 / \sqrt{\nu_0}, \mathcal{G}_1 / \sqrt{\nu_1} )\), where $\mathcal{G}_F$ is a
mean-zero Gaussian process with covariance process $\rho_F(x, y) =
\textnormal{Diag}\{\rho_k(x, y) / \nu_k\}$.
Now we focus on the calculation of a uniform confidence band for $L$ only.
Because the bootstrap algorithm was described previously we only verify that the
regularity conditions for this
plan hold. Assumptions~\ref{assume:first} and~\ref{assume:last}, along with the
above discussion of the weak convergence of $\sqrt{n}(\mathbb{F}_n - F)$ imply that
Assumption~\ref{A:estimator} is satisfied. The class of functions $\{ (I(X \leq
x), I(Y \leq y), x,y \in \mathbb{R} \}$ is uniform Donsker and the sequences $a_n$ and
$b_n$ can be chosen to satisfy Assumption~\ref{A:sequences} by making them
converge to zero more slowly than $n^{-1/2}$. Assuming the weights are
independent of the observations, assumption~\ref{A:bootstrapf} is satisfied by
Lemma A.2 of \citet{LintonSongWhang10}, which implies that the bootstrap
algorithm described above is consistent, as described in
Theorem~\ref{thm:bootstrap_consistency_teststats}. It is also straightforward
to verify that, under the high-level assumption~\eqref{local_sqrt}, both parts
of Assumption~\ref{A:local_alts} are satisfied~--- in the language of the
assumption, $f(P_{c/\sqrt{n}})$ are the pair $(F_0^n, F_1^n)$ under the local
probability distribution $P_{c/\sqrt{n}}$, and $f' = (\int_{-\infty}^\cdot
h_0^c, \int_{-\infty}^\cdot h_1^c)$) for direction $h^c$ indexed by $c$ and
$\sqrt{n}(\mathbb{F}_n - F^n) \leadsto \mathcal{G}_F$ \citep[Theorem
3.10.12]{vanderVaartWellner96}.
\subsection{Computational details and simulation experiments}\label{sec:MC}
The bound functions were estimated using sample data and standard
empirical distribution functions. The bounds must be equal to zero for
small enough arguments and equal to one for large enough arguments, and are
monotonically increasing in between. However, bounds computed from two samples
could take unique values at all possible $X_{1i} - X_{0j}$ combinations, and
computing the function on all $n_1 \times n_0$ points could be computationally
prohibitive. Therefore we compute the bound functions on a reasonably
fine grid $\{x_k\}_{k=1}^K$.
In order to make calculations as efficient as possible, it is helpful
to know some features of the support of the bounds given sample data, where the
support of a bound is the region where it is strictly inside the unit interval. When
computing both bounds on the same grid, a grid of points over $\{\min_i X_{1i} -
\max_j X_{0j}, \max_i X_{1i} - \min_j X_{0j}\}$ is sufficient to capture the
supports of both bounds. Take the lower bound as an example of how to calculate
the support of one bound from two samples. The lower bound is the maximum of
the difference empirical process $\mathbb{F}_{1n}(\cdot) - \mathbb{F}_{0n}(\cdot - x)$,
which takes steps of size $+1/n_1$ at $X_{1i}$ observations and steps of size
$-1/n_0$ at $X_{0j} + x$ shifted observations. The upper endpoint of the
support of the lower bound is the smallest value for which it is equal to 1. If
for some (large) $x$, $X_{0j} + x \geq X_{1i}$
for all $i$ and $j$, then the shape of $\mathbb{F}_{1n}(u) - \mathbb{F}_{0n}(u - x)$ rises
monotonically to one, then falls, as $u$ increases. The $x$ that satisfy this
condition are $x \geq \max_{i,j} \{X_{1i} - X_{0j}\} = \max_i X_{1i} - \min_j
X_{0j}$. To find the minimum of the support, the maximum value of $x$ such
that $\mathbb{L}_n(x) = 0$, note that the function $\mathbb{F}_{1n}(u) -
\mathbb{F}_{0n}(u - x)$ is always equal to zero for some $u$, but zero is the maximum
only when $\mathbb{F}_{n1}$ first-order stochastically dominates $\mathbb{F}_{0n}(\cdot -
x)$. Therefore the smallest $x$ in the support of $\mathbb{L}_n$ is the smallest $x$
that shifts the $\mathbb{F}_{0n}$ distribution function enough so that it is
dominated by $\mathbb{F}_{n1}$. This is can be estimated by computing the minimum
vertical distance between quantile functions for both samples on a common set of
quantile levels, that is, $\min_k \{\hat{Q}_{1n}(\tau_k) -
\hat{Q}_{0n}(\tau_k)\}$. If an exact lower bound for the grid is not so
important, one could also bound the grid using the looser $\min_{i,j}\{X_{1i} -
X_{0j}\} = \min_i X_{1i} - \max_j X_{0j}$. Similarly, the support of the upper
bound is between $\min_i X_{1i} - \max_j X_{0j}$ and $\max_k
\{\hat{Q}_{1n}(\tau_k) - \hat{Q}_{0n}(\tau_k)\}$.
It is interesting to note that given two samples, one can always
reject the hypothesis of a constant treatment effect unless
one distribution is a location shift of the other. If $\Delta = x^*$
with probability one, then the value $x^*$ should pass through the bounds for
all quantile levels. However, $\mathbb{U}_n(x) < 1$ for all
$x < \max_k \{\hat{Q}_{1n}(\tau_k) - \hat{Q}_{0n}(\tau_k)\}$ (maximum
taken over the common set of quantile levels), while $\mathbb{L}_n(x) > 0$
for all $x > \min_k \{\hat{Q}_{1n}(\tau_k) - \hat{Q}_{0n}(\tau_k)\}$. This
means the hypothesis of a constant $\Delta$ is not rejected only when the
maximum and minimum quantile difference are equal to each other, that is, when
one is a vertical translate of the other (or the CDFs are horizontal
translates).
We illustrate the examples from the previous section with two simulation
experiments. The simulated observations in the first experiment are normally
distributed and we shift the location of one distribution to examine size and
power properties. We used this normal location experiment to choose a
sequence $\{a_n\}$ used to approximate the set of $\epsilon$-maximizers in the
estimation of the derivative $\psi'$: we decided on $a_n = 0.2
\log(\log(n)) / \sqrt{n}$ (where $n = n_0 + n_1$) using simulations that examined size
and power. For contact set estimation we chose the sequence $b_n = 3
\log(\log(n)) / \sqrt{n}$ based on the simulation results of
\citet{LintonSongWhang10}, who concentrated on estimating the contact set in
similar experiment.\footnote{They used sequences of the form $c
\log\log(\bar{n}) / \sqrt{\bar{n}}$, where $\bar{n} = (n_1 + n_2) / 2$, where
they suggested $c$ between 3 and 4. Using $c = 4$ but adjusting the formula to
depend on $n$, the sum of the two samples, we have $4 / \sqrt{2} \approx 3$.}
Figure~\ref{fig:powercurves_confidenceband} verifies that our proposed method
provides accurate coverage probability for uniform confidence bands and power
against local alternatives. It shows the results of an experiment in which we
test the null hypothesis that the lower bound of the treatment effect
distribution corresponds to the bound associated with two standard normal
marginal distributions, which is $L(x) = 2 \Phi(x / 2) - 1$ for $x > 0$ and zero
otherwise \citep[Section 4]{FrankNelsenSchweizer87}, where $\Phi$ is the CDF of
the standard normal distribution. The upper bound is symmetric so it is sufficient to examine only the lower bound test. We test size and power against local alternatives for samples of size 100, 500 and 1000 with respectively 499, 999 and 1999 bootstrap repetitions for each sample size, and 1,000 simulation repetitions. Alternatives are local location-shift alternatives with the mean of one distribution set to zero while the other ranges from $-5/\sqrt{n}$ to $5/\sqrt{n}$.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.75\columnwidth]{./fig/cband_powercurves.pdf}
\caption{Power curves for the first (two-sided test) experiment. The
horizontal axis shows local (not fixed) parameter values, where in all cases
the value zero corresponds to the null hypothesis and non-zero values are
local alternatives. Tests use two samples of size $n \in \{100, 500, 1000\}$
and bootstrap repetitions $R \in \{499, 999, 1999\}$ respectively, using the
bootstrap algorithm described in the text. 1,000 simulation repetitions were
used to find empirical rejection probabilities.} \label{fig:powercurves_confidenceband}
\end{center}
\end{figure}
Although all the power curves look very similar, it is important to remember
that they represent power against these local alternatives and not fixed
alternatives. The low point on these curves (the point closest to the tests'
nominal size) does not occur at $1$, but at $1/\sqrt{n}$. As discussed at the
end of Section~\ref{sec:inference}, asymptotic size is controlled locally with
this resampling procedure.
In a second experiment we simulate uniformly distributed samples and test for
the condition that implies stochastic dominance of $A$ over $B$. A control
sample is made up of uniform observations on the unit interval, as is treatment
$B$, while treatment $A$ is distributed uniformly on $[\mu, \mu + 1]$. We test
the hypothesis that $U_A - L_B \leq 0$ as in the theoretical example of the last
section. When $\mu = 1$, $U_A - L_B \equiv 0$ \citep[Section
4]{FrankNelsenSchweizer87} so that value of $\mu = 1$ represents a sort of
least-favorable null. When $\mu > 1$, the null hypothesis is satisfied with a
strict inequality and should not be rejected, while for $\mu < 1$ the null
should be rejected. We examine local alternatives around the central value $\mu
= 1$, which is normalized to zero in Figure~\ref{fig:powercurves_sd}. We test
size and power against local alternatives for samples of size 100, 500 and 1000
with respectively 499, 999 and 1999 bootstrap repetitions for each sample size.
Functions were evaluated on a grid with step size 0.02 and 1,000 simulation
repetitions were used for each sample size.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.75\columnwidth]{./fig/sd_powercurves.pdf}
\caption{Power curves for the second (stochastic dominance) experiment. The
horizontal axis shows local parameter values around the boundary of the null
region, which has been normalized to zero in this figure.
Tests use three samples of size $n \in \{100, 500, 1000\}$ and bootstrap
repetitions $R \in \{499, 999, 1999\}$ respectively, using the bootstrap
algorithm described in the text. 1,000 simulation repetitions were used to
find empirical rejection probabilities.} \label{fig:powercurves_sd}
\end{center}
\end{figure}
The results of this test are similar to what one would expect from the stochastic dominance tests in \citet{LintonSongWhang10}, except that here the test is applied to bounds and represents a breakdown frontier for the hypothesis that treatment $A$ dominates $B$, without any knowledge of how the treatments and control may be correlated. This shows how much information is lost by dropping the assumption of full treatment observability.
\section{The treatment effect distribution of job training on wages}\label{sec:application}
This section illustrates the inference methods with an evaluation of a job training program. We construct upper and lower bounds for both the distribution and quantile function of the treatment effects, confidence bands for these bound function estimates, and describe a few inference results. This application uses an experimental job training program data set from the National Supported Work (NSW) Program, which was first analyzed by \citet{LaLonde86} and later by many others, including \citet{HeckmanHotz89}, \citet{DehejiaWahba99}, \citet{SmithTodd01, SmithTodd05}, \citet{Imbens03}, and \citet{Firpo07}.
Recent studies in statistical inference for features of the treatment effects
distribution in the presence of partial identification include, among others,
\citet{FirpoRidder08, FirpoRidder19, FanPark09, FanPark10, FanWu10, FanPark12,
FanShermanShum14, FanGuerreZhu17}. These studies have concentrated on
distributions of finite-dimensional functionals of the distribution and quantile
functions, including these functions themselves evaluated at a point.
Additional work includes, among others, \citet{GautierHoderlein11},
\citet{ChernozhukovLeeRosen13}, \citet{Kim14} and \citet{ChesherRosen15}. Each
of these papers provides pointwise inference methods for bounds on the
distribution or quantile functions, and often for more complex objects. We
contribute to this literature by applying the general results in this paper to
provide uniform inference methods for the bounds developed by Makarov and
others, and hope that it may indicate the direction that pointwise inference for
bounds in more involved models may be extended to be uniformly valid.
The data set we use is described in detail in \citet{LaLonde86}. We use the
publicly available subset of the NSW study used by \citet{DehejiaWahba99}. The
program was designed as an experiment where applicants were randomly assigned
into treatment. The treatment was work experience in a wide range of possible
activities, such as learning to operating a restaurant or a child care center,
for a period not exceeding 12 months. Eligible participants were targeted from
recipients of Aid to Families With Dependent Children, former addicts, former
offenders, and young school dropouts. The NSW data set consists of information
on earnings and employment (outcome variables), whether treated or
not.\footnote{The data set also contains background characteristics, such as
education, ethnicity, age, and employment variables before treatment.
Nevertheless, since we only use the experimental part of the data we refrain
from using this portion of the data.} We consider male workers only and
focus on earnings in 1978 as the outcome variable of interest. There are a total
of 445 observations, where 260 are control observations and 185 are treatment
observations. Summary statistics for the two parts of the data are presented in
Table \ref{tb:summary}.
\begin{table}[!ht]
\linespread{1.5}
\par
\begin{center}
{\small \setlength{\tabcolsep}{4pt}
\begin{tabular}{ccccccccccc}
\cline{1-11}
\multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{4}{c}{Treatment Group} & \multicolumn{1}{c}{} & \multicolumn{4}{c}{Control Group}\\
\cline{3-6}\cline{8-11}
\multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{Mean} & \multicolumn{1}{c}{Median} & \multicolumn{1}{c}{Min.} & \multicolumn{1}{c}{Max.} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{Mean} & \multicolumn{1}{c}{Median} & \multicolumn{1}{c}{Min.} & \multicolumn{1}{c}{Max.}\\
\hline
Earnings (1978) & & 6,349.1 & 4,232.3 & 0.0 & 60,307.9 & & 4,554.8 & 3,138.8 & 0.0 & 39,483.5 \\
& & (7,867.4) & & & & & (5,483.8) & & & \\
\hline
\end{tabular}
}
\end{center}
\caption{Summary statistics for the experimental National Supported Work (NSW) program data.}
\label{tb:summary}
\end{table}
To provide a more complete overview of the data, we also compute the empirical
CDFs of the treatment and control groups in Figure~\ref{fig:lalonde_cdfs}. From
this figure we note that the empirical treatment CDF stochastically dominates
the empirical control CDF, and that there are a large number of zeros in each
sample. In particular, $\mathbb{F}_{n,tr}(0) \approx 0.24$ and $\mathbb{F}_{n,co}(0) \approx
0.35$.
\begin{figure}[!ht]
\begin{center}
\includegraphics[height=3in]{./fig/lalonde_cdfs.pdf}
\end{center}
\caption{Empirical CDFs of treatment and control observations in the
experimental NSW data. There are 260 control group observations and 185
treatment group observations, many outcomes equal to zero and the treatment group outcomes stochastically dominate the control group outcomes at first order.} \label{fig:lalonde_cdfs}
\end{figure}
The main objective is to provide uniform confidence bands for the CDF of the treatment effects distribution. We calculate the lower and upper bounds for the distribution function using the control and treatment samples as in equations \eqref{eq:SDLB}--\eqref{eq:SDUB}. The bounds are computed on a grid with increments of \$100 dollars along the range of the common support of the bounds, which is roughly from -\$40,000 to \$60,000. We focus on the region between -\$40,000 and \$40,000, which contains almost all the observations (there is a single \$60K observation in the treated sample). The results are presented in Figure~\ref{fig:uniform_bounds_lalonde} and given by the black solid lines in the picture. The most prominent feature is that, as expected, the upper bound for the CDF of treatment effects stochastically dominates the corresponding lower bound.
Next, we compute the uniform confidence bands as described in the text. They are shown in Figure~\ref{fig:uniform_bounds_lalonde} as the dashed lines around the corresponding solid lines.
\begin{figure}[!ht]
\begin{center}
\includegraphics[height=3in]{./fig/cdf_bounds_lalonde.pdf}
\end{center}
\caption{Bound functions and their uniform confidence bands. These confidence bands were constructed by inverting Kolmogorov-Smirnov-type test statistics as described in the text.} \label{fig:uniform_bounds_lalonde}
\end{figure}
Due to the large number of zero outcomes in both samples, these bounds have some
interesting features that we discuss further. First, we note that for any
$\epsilon$ greater than zero but smaller than the next smallest outcome (about
\$45), $\mathbb{F}_{n,tr}(0) - \mathbb{F}_{n,co}(-\epsilon) = 0.24$, which explains the jump
in the lower bound estimate near zero (it is really for a point in the grid just
above zero). Likewise, for the same $\epsilon$, $\mathbb{F}_{n,tr}(-\epsilon) -
\mathbb{F}_{n,co}(0) = -0.35$, which explains the jump in the upper bound just below
zero. Without these point masses at zero, both bounds would more smoothly tend
towards 0 or 1. Second, the point masses at zero imply another feature of the
bounds that can be discerned in the picture. The upper bound to the left of 0
is the same as $1 - \mathbb{F}_{n,co}(-x)$ and the lower bound to the right of zero is
the same as $\mathbb{F}_{n,tr}(x)$. Taking the lower bound as an example, for each $x
> 0$ find the closest observation from the control sample $y_{i^*,co}$, and set
$x^*(x) = X_{i^*,co} + \epsilon$, leading to the supremum $\mathbb{F}_{n,tr}(X_{i,co}
+ \epsilon)$ at every point where $X_{i,co} + \epsilon < X_{j,tr}$ for all $j$
in the treated sample. It is identical to the empirical treatment CDF for the
entire positive part of the support because the treatment first-order
stochastically dominates the control. The situation would be different if there
were a jump in the empirical control CDF at least as large as the jump in the
empirical treatment CDF at zero. Because the opposite is the case for the upper
bound, it does change slightly above the zero mark, tending from $1 -
\mathbb{F}_{n,tr}(0) + \mathbb{F}_{n,co}(0)$ to 1 as $x$ goes from 0 to the right.
We can also use the confidence bands for the bound functions to construct a confidence band for the true distribution function of treatment effects. This is shown in Figure~\ref{fig:cdf_confband_lalonde}. A $1 - \alpha$ confidence band can be constructed by using the upper $1 - \alpha / 2$ limit of the upper bound confidence band and the lower $\alpha / 2$ limit of the lower confidence band. This band is a uniform asymptotic confidence band for the true CDF, and uniform over correlation between the potential outcomes between samples. In other words, if $\mathcal{P}$ is the collection of bivariate distributions that have marginal distributions $F_{tr}$ and $F_{co}$, then
\begin{equation*}
\liminf_{n \rightarrow \infty} \inf_{\textnormal{P} \in \mathcal{P}}
\prob{F_\Delta(x) \in CB(x) \text{ for all } x} \geq 1 - \alpha.
\end{equation*}
This confidence band is likely conservative, since
\begin{align*}
\text{P} \big\{ \exists x : F_\Delta(x) &\not\in CB(x) \big\} \\
{} &= \prob{ \{\exists x : F_\Delta(x) < \mathbb{L}_n(x) - c^*_{L,
1-\alpha/2}/\sqrt{n} \} \cup \{\exists x : F_\Delta(x) > \mathbb{U}_n(x) + c^*_{U,
1-\alpha/2} / \sqrt{n}\} } \\
{} &\leq \prob{ \{\exists x : L(x) < \mathbb{L}_n(x) - c^*_{L, 1-\alpha/2}/\sqrt{n}
\} \cup \{\exists x : U(x) > \mathbb{U}_n(x) + c^*_{U, 1-\alpha/2}/\sqrt{n} \} }.
\end{align*}
See \citet{Kim14} and \citet{FirpoRidder19} for a more thorough discussion of
the sense in which these bounds are not uniformly sharp for the treatment effect
distribution function. We leave more sophisticated, potentially tighter
confidence bands for future research. Note that the technique of
\citet{ImbensManski04} cannot be used to tighten these bounds,
because the parameter, a function, could violate the null hypothesis at both
sides of the confidence band.
\begin{figure}[!ht]
\begin{center}
\includegraphics[height=3in]{./fig/cdf_confband_lalonde.pdf}
\end{center}
\caption{A uniform confidence band for the true treatment effect CDF. This bound is constructed by using the lower $\alpha/2$ limit of the lower bound and the upper $\alpha/2$ limit of the upper bound. A few other estimates of the treatment effect distribution are made: the vertical line at zero represents an informal hypothesis test that the effect is zero across the entire distribution, and is rejected (see the calculations at the beginning of this section about the nontrivial parts of the bounds to see why this is so). The vertical line to the right of zero is the average treatment effect, and it can be seen that this average ignores some variation in treatment effect outcomes. The dotted curve in between the bounds is the same as the (inverted) quantile treatment effect, which is equivalent to assuming rank invariance between potential treatment and control outcomes.} \label{fig:cdf_confband_lalonde}
\end{figure}
We plot some other features in this figure for context. First, the dotted
vertical line is positioned at $y = 0$, and it can be seen that we (just) reject
the null hypothesis $H_0: \prob{\Delta = 0} = 1$. This hypothesis is closest to
non-rejection, and it is clearer that one should reject the null that the
treatment effect distribution is degenerate at any other point besides zero.
This supports the notion that treatment effect heterogeneity is an important
feature of these observations, especially because this band is completely
agnostic about the form of the joint distribution. On the other hand, by examining the bands at horizontal levels, it can be seen that for the median effect and a wide interval in the center of the distribution, the hypothesis of zero treatment effect cannot be rejected (although these are uniform bounds and not tests of individual quantile levels).
The final feature in the figure is the dashed curve that represents the estimate that one would make under the assumption of comonotonicity (or rank invariance)~--- the assumption that, had an individual been moved from the treatment to the control group, their rank in the control would be the same as their observed rank. Under this strong assumption the quantile treatment effects are the quantiles of the treatment distribution and they can be inverted to make an estimate. Clearly, the estimate under this assumption is just one point-identified treatment effect distribution function of many.
\begin{figure}[!ht]
\begin{center}
\includegraphics[height=3in]{./fig/bounds_compare.pdf}
\end{center}
\caption{Uniform confidence bands for the CDF of treatment effects plotted with a collection of confidence intervals at treatment outcome levels. The confidence intervals are narrower because they represent confidence statements at each treatment effect individually, while the uniform bands give confidence statements about where the entire bound function lies. Pointwise confidence intervals were calculated using the method proposed in \citet{FanPark10}.} \label{fig:bounds_compare}
\end{figure}
Finally, to provide context for the uniform confidence band results within the
literature on inference for bounds, we compare the proposed uniform bands to the
pointwise confidence intervals suggested by \citet{FanPark10}. We used
\citet{BickelSakov08}'s automatic procedure to choose subsample size and
constructed confidence intervals for each individual point in the grid of the
treatment effect support. This collection of pointwise confidence intervals are
plotted along with uniform confidence bands in Figure~\ref{fig:bounds_compare}.
The results show that the uniform bands are farther from the bound estimates
than the set of pointwise confidence intervals.
\section{Conclusion}\label{sec:conclusion}
This paper develops uniform statistical inference methods for optimal value
functions, that is, functions constructed by optimizing an objective function in
one argument over all values of the remaining arguments. Value functions can be
seen as a nonlinear transformation of the objective function. The map from
objective function to value function is not compactly differentiabile, but
statistics used to conduct uniform inference are directionally differentiable.
We establish the asymptotic properties of nonparametric plug-in estimators of
these uniform test statistics and develop a resampling technique to conduct
practical inference. Examples involving dependency bounds for treatment effects
distributions are used for illustration. Finally, we provide an application to
the evaluation of a job training program, estimating a conservative uniform
confidence band for the distribution function of the program's treatment effect
without making any assumptions about the dependence between potential outcomes.
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 764 |
\section{Introduction}
Significant progress has been made on learning good representations for images, allowing impressive applications in image generation. However, it is still challenging to acquire accurate and complete 3D shapes and scenes due to shape geometry, surface material, and lighting conditions. However, their direct and naive extension from 2D images to 3D voxels introduces high memory costs and inefficient computation issues. For balancing computation and performance, it seems generating point cloud is widely used.
In this paper, we develop a generative neural network which outputs point clouds natively. In this challenge, Chamfer Distance (CD), one of the metrics, can compare two point sets while it does not consider the surface/mesh connectivity. For example, if we have a cube that only has eight vertices, the point cloud generated by our model closer to these eight vertices,
the small Chamfer Distance we will get. Another metric is F-score is also base on Chamfer distance. If we use implicit function models, it may generate high visual quality 3D models, generating a more smooth and continuous surface rather than points close to vertices. As a result, we choose AtlasNet as our baseline, which directly generates point clouds. What is more, In AtlasNet \cite{groueix2018papier} and IM-NET \cite{chen2019learning}, they all trained a 3{D} autoencoder (AE) first and then used it to help the single-view reconstruction training. This approach indicates that it is easy for networks to learn from 3{D} data due to single-view reconstruction, especially an ill-pose problem. Inspired by this, we replace the AtlasNet's decoder to FoldingNet \cite{yang2018foldingnet} decoder. FoldingNet mentioned a novel folding-base decoder deforms a canonical 2D grid onto the underlying 3D object surface of a point cloud, achieving low reconstruction errors even for objects with delicate structures.
\section{Related work}
Object-based single view reconstruction (SVR) calls for generating an object's 3D model given a single image. Recently, there are lots of work in learning and generative modeling 3D shapes.
It is still challenging to acquire accurate and complete 3D shapes due to the interference by shape geometry, surface material, lighting conditions, and the noise introduced in the capturing process.
Up to date, deep neural networks for shape reconstruction might be categorized into three categories:
voxel-based representations \cite{Choy20163DR2N2AU,liao2018deep}, point-based representations \cite{groueix2018papier,fan2017point}, and implicit fucntions representations \cite{mescheder2019occupancy,chen2020bsp,chen2019learning}.
\noindent\textbf{Voxel representations} are a straightforward generalization of pixels to 3D case. Unfortunately, the memory cost for voxel representations grows cubically with resolution, therefore, it can only generate $32^3$ or $64^3$ voxels. While it's possible to reduce memory cost by using special data structures such as octrees, this structure leads to complex algorithms and is still limited to relatively small $256^3$ voxel grids.
\noindent\textbf{Point-based representations} are widely used both in computer graphics and robotics. Pointnet \cite{qi2017pointnet} pioneered point clouds as a representation for discriminative deep learning tasks. They achieved permutation invariance by applying a global pooling operation.
As a result, a category of techniques pioneered by PointNet that express surfaces as point clouds, and techniques pioneered by Atlas that adopt 2D-to-3D
mapping process \cite{groueix2018papier}. But these methods do not guarantee watertight results and hardly scale beyond a hundred vertices
\noindent\textbf{Implicit functions representations} model shapes as learnable indicator functions rather than samples, as in the case of voxel methods. The resulting networks treat reconstruction as a classification problem and are universal approximators whose reconstruction precision is proportional to the network complexity. However, at inference time, these methods still require the execution of an expensive iso-surfacing operation whose performance effected by the desired resolution. Compared with other methods, implicit functions representations can achieve higher visual quality.
\begin{figure*}[h]
\centering
\includegraphics[width=\linewidth]{3D_Reconstruction_report.pdf}
%
%
\caption{\label{fig:arch} \noindent\textbf{Network Architecture. } Given an input image \textit{I}, we employ a ResNet to extract the latent vector.
The input of $Decoder2Dto3D$ and the $Decoder2Dto3D$ all need to concatenate the output of ResNet.
We further optimize the whole network by using Chamfer distance. \textcircled{+} means concatenate}
\end{figure*}
\section{Method}
\label{sec:model}
\noindent\textbf{Overview. }
Fig.~\ref{fig:arch} shows an overview of our method. For single-view 3D reconstruction,
Actually, we did not follow the idea from AtlaNet to first train an autoencoder by only using point cloud data,
then fix the parameters of the decoder when training SVR.
We replace the original AtlasNet's decoder with FoldingNet's decoder.
In our experiments, we adopted a more radical approach by only training the whole network to minimize the mean CD loss between the predicted and ground truth point clouds.
\noindent\textbf{Backbone. }We used ResNet \cite{he2016deep} as our image encoder to obtain 1024D features from $224 \times 224$ images.
In addition, all relu layers in ResNet were replaced with leakly-relu layers.
\noindent\textbf{Data pre-processing. }For data pre-processing,
since there are multi-view images and the scales of point clouds are different,
We use the official code to rotate the point cloud to the related image's view and then normalize the point cloud.
After normalization, a trade-off is between performance and computation is down-sampling the original point cloud to 2048 points.
Also, we add a small Gaussian noise to each point to improve the robustness of our network.
For image augmentation, we only keep resizing images to 224, which is different from AtlasNet.
\noindent\textbf{Decoder design. }To satisfy the submission requirements, the number of points in the predicted point cloud is 2048.
so that the primitives of the decoder are eight, and the number of each decoder's layer is 3.
The input latent vector of the $Decoder2Dto3D$ is obtained from the image encoder.
We replicate it $n$ times and concatenate the latent $n$-by-1024 matrix with an $n$-by-2 matrix that contains the n sampled points from $[0,1]^2$.
The result of the concatenation is a matrix of size $n$-by-514.
The matrix is processed row-wise by a 3-layer perception, and the output is a matrix of size$ n$-by-3 output and feeds it
into a 3-layer perception. For the $Decoder3Dto3D$, it is the same as the $Decoder2Dto3D$. The only difference is the input of its is an $n$-by-515 matrix.
\noindent\textbf{Sampling method. }During the training stage, random sample points from $[0,1]^2$ can improve the robustness of the whole network.
AtlasNet noted that sampling points regularly on a grid on the learned
parameterization yields better performance than sampling points randomly. For inference, all results used this regular sampling.
\noindent\textbf{End-to-end training. }
We briefly describe how the whole network is built on top of the AtlaNet and FoldingNet for end-to-end training.
Given a 2D input image \textit{I}, our goal is to learn making a prediction for the ground-truth $\emph{S}$.
Let the prediction of the final reconstruction output be $\widetilde{\emph{S}}$. Both of them are point sets.
The learning process tries to minimize the $Chamfer loss$ between the $\emph{S}$ and $\widetilde{\emph{S}}$:
\begin{equation}
L(\emph{S}, \widetilde{\emph{S}})={ \frac{1}{|S|}\sum_{x \in S} \min_{{y} \in \widetilde{S}} \|x-y \|_2}+{ \frac{1}{|\widetilde{S}|}\sum_{y \in \widetilde{S}} \min_{{x} \in S} \|x-y \|_2}
\end{equation}
the term $\min_{y \in \widetilde{S}}\|x-y\|$
enforces that any 3D point x in the ground-truth has a matching 3D point y in the reconstruction point cloud,
and the term $\min_{x \in S}\|x-y\|$ enforces the matching vice versa.
The $\frac{1}{|S|} $ and $\frac{1}{\widetilde{|S|}}$ are all mean operation to avoid too many points cause the huge loss.
In FoldingNet, they used max operation instead of mean operation but if there is an outlier, it will make loss huge and then, and then the next predicted point cloud still has an outlier so repeatedly. It is not easy to make the network converge.
\section{Experiment}
\label{sec:results}
This section shows qualitative and quantitative results on this task using our network and comparing them with AtlasNet.
To compare with the baseline, we first random sample points from the original point cloud to 2048 points. We use the dataset provided by \cite{fu20203dfuture},
We only use 5202 normalized models and their related 62424 multi-view images.
We evaluated our method on trackA and trackB server, which are provided by IJCAI-PRICAI 2020 3D AI Challenge:3D Object Reconstruction from A Single Image.
\noindent\textbf{Evaluation criteria. }
We evaluated our generated shape outputs by comparing them to ground truth shapes using two criteria.
First, we compared point sets of the output and ground-truth
shapes using CD. But CD only reflects the distance between the two point clouds as a whole.
Let the precision of a reconstructed model be the percentage of reconstructed points that have a ground truth point within distance $\tau$. So we introduce F-score to measure the accuracy of each point.
Let the recall of a reconstructed model be the percentage of ground-truth points
that have a reconstructed point withi distance $\tau$. For evaluating above two metrics, the overall score in trackA:
\begin{equation}
score={100\times\frac{2-CD}{2}\times0.5+{Fscore}\times0.5}
\end{equation}
In trackB:
\begin{equation}
score=100\times\frac{2-CD}{2}\times0.3+{Fscore}\times0.7
\end{equation}
\noindent\textbf{Ablative analysis. }
During the competition, several operations on the ground truth point clouds affect the final result.
\noindent\textbf{Normalization method. } In Table \ref{table:t3}, different normalization methods impact results.
Sphere normalization means to find the farthest point from the origin and then use its distance from the origin to normalize all points.
In this way, for compact objects such as desk or nightstand, the points on their surfaces are not very far away from a specific point,
which means all points could keep their relative position. However, for some specific objects like a chandelier,
it has at least one outlier point, which makes the normalized length is too considerable for the other points.
After normalization, the whole point cloud could not keep its original shape because of more points closer to the origin point.
Due to the essence of AtlasNet is to learn the mapping function from 2D points to 3D points,
the network is sensitive to the correspondence between 2D points and 3D points.
In summary, the key to getting more accurate results is to keep the whole point cloud shape no matter what to do any augmentation.
\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
Methods &CD$\downarrow$ & F-score$\uparrow$ &score$\uparrow$ \\
\hline
AtlasNet(baseline) & 4.2 &84.97 & 91.43\\
\hline
SQ & 1.49 & 96.59 & 97.92\\
\hline
\end{tabular}
\caption{Reconstruction metrics comparison on trackA.
baseline: use sphere normalization
SQ: use square normalization, default is unitball normalization;
Chamfer distance is reported, multiplied by $10^{2}$}
\label{table:t3}
\end{table}
\noindent\textbf{Projection. }In Table \ref{table:t7}, whether to use multi-view images to rotate original point clouds or not could bring a huge difference.
In Table \ref{table:t4}, different scale factors lead to distinguishing results.
Using multi-view images to rotate original point clouds is similar to calculate the relative projection matrix.
Using different scale factors is like to guess the focal length.
To sum up, it is not only lead by the reason from Table \ref{table:t3} but also relative to the projection transformation.
If the camera matrix and projection matrix are known, we can reconstruct point clouds from single RGB images.
As a result, our network also estimated the camera matrix and projection matrix implicitly.
\noindent\textbf{Sampling method. }
Fig. \ref{fig:sample} shows the point cloud before and after sampling mesh with lloyd's algorithm\cite{sample}.
After sampling, the density of point clouds increases. For the whole point cloud, their surfaces are more continuous.
It seems that it's not difficult to learn the mapping function.
However, Table \ref{table:t8} shows that the network trained by the sampled point cloud is not better than the network trained by the original point cloud.
In this way, we found that Chamfer distance is sensitive to the location of the given points.
In this challenge, the points in the ground-truth point cloud are all vertices,
which hopes all predicted points could be close to the edge or corner rather than distribute continuous surfaces.
For visualization purposes, the more point clouds distribute on the surface, the more comfortable people observe.
As a result, Chamfer distance maybe not enough to measure the quality of the generated point cloud.
It needs to introduce some criteria for visual quality or generated mesh.
\begin{figure*}[h]
\centering
\includegraphics[width=\linewidth]{lloyd_sample.pdf}
\caption{\label{fig:sample} Sample results, odd rows are original point cloud, even rows are point cloud after sampling mesh with Lloyd's algorithm. }
\end{figure*}
\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
Methods &CD$\downarrow$ & F-score$\uparrow$ &score$\uparrow$ \\
\hline
AtlasNet(w/o sampling) & 1.45 &96.85 & 98.06\\
\hline
AtlasNet(w sampling & 1.49 &85.06 & 97.92\\
\hline
\end{tabular}
\caption{Reconstruction metrics comparison on trackA. MV: use multi-view image to rotate original point clouds;
Chamfer distance is reported, multiplied by $10^{2}$}
\label{table:t8}
\end{table}
\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
Methods &CD$\downarrow$ & F-score$\uparrow$ &score$\uparrow$ \\
\hline
AtlasNet(baseline w/o MV) & 16.82 &52.92 & 72.26\\
\hline
AtlasNet(baseline w MV) & 4.2 &84.97 & 91.43\\
\hline
\end{tabular}
\caption{Reconstruction metrics comparison on trackA. MV: use multi-view image to rotate original point clouds;
Chamfer distance is reported, multiplied by $10^{2}$}
\label{table:t7}
\end{table}
\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
Methods &CD$\downarrow$ & F-score$\uparrow$ &score$\uparrow$ \\
\hline
$\times0.001$ & 4.1 &85.31 & 91.63\\
$\times0.01$ & 2.53 &91.52 & 95.13\\
$\times0.02$ & 1.47 &96.85 & 98.06\\
$\times0.1$ & 1.43 &96.79 & 98.03\\
$\times1$ & 1.46 &96.75 & 98.01\\
$\times10$ & 1.45 &96.84 & 98.06\\
$\times50$ & 1.43 &96.97 & 98.13\\
$\times100$ & 1.41 &96.98 & 98.08\\
\hline
\end{tabular}
\caption{Reconstruction metrics comparison on trackA.
$\times$n: means to scale the original point cloud by n, which means any point in ground-truth multiples n.
Chamfer distance is reported, multiplied by $10^{2}$}
\label{table:t4}
\end{table}
\noindent\textbf{Evaluation on trackA server. }
We report quantitative results for shape generation from single images in Table \ref{table:t5},
where each approach is trained on all categories and results are averaged over all categories.
According to the ablative analysis, we combine all tricks to get the best results.
\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
Methods &CD$\downarrow$ & F-score$\uparrow$ &score$\uparrow$ \\
\hline
AtlasNet(baseline w MV) & 4.2 &84.97 & 91.43\\
\hline
Our+SQ & 1.49 & 96.59 & 97.92\\
\hline
Our+SQ+FD &1.45&96.85&98.06\\
\hline
Our+SQ+FD+S &\textbf{1.41} &\textbf{96.98}&\textbf{98.13}\\
\hline
\end{tabular}
\caption{Reconstruction metrics comparison on trackA. SQ: use square normalization, default is unitball normalization;
FD: use decoder from FoldingNet; S: scale original point clouds, in our experiment, we use 50.
Chamfer distance is reported, multiplied by $10^{2}$}
\label{table:t5}
\end{table}
\section{Conclusion}
\label{sec:conclusion}
We have introduced our approach to the reconstruction point cloud from a single image. We have shown its benefits for 3{D} single-view reconstruction, out-performing existing baselines. Specifically, we use both a single-view reconstruction model and a 3{D} auto-encoder to yield robust and accurate reconstruction.
We also employ many empirical settings on the normalization, projection, and sampling trials to boost
performance.
Consequently, our proposed method achieved the 2nd place in the {\it IJCAI--PRICAI--20 3D AI Challenge: 3D Object Reconstruction from A Single Image}.
\noindent\textbf{Acknowledgements } Our work is supported by Em-Data Technology Co., Ltd.
We thank Jianfei Gao, Fengliang Qi, Yuan Gao, Changyong Shu, Yunjie Xu, Qi Liu, Yao Xiao for valuable discussions.
\bibliographystyle{named}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 321 |
Mat McLachlan is an Australian author, historian and television presenter. His first book, Walking With the Anzacs: A Guide to Australian Battlefields on the Western Front, was published by Hachette Australia in February 2007. It was reprinted in 2008, 2009, 2011, 2012, 2013 and 2014. A fully revised edition was published in 2015.
His second book, Gallipoli: The Battlefield Guide, was published in April 2010 and reprinted in 2015.
In 2007 McLachlan produced the documentary Lost in Flanders, which follows the investigation into the identities of five Australian soldiers uncovered on the First World War battlefields of Belgium. He appears as the program's key interviewee and is also Associate Producer. The documentary first screened on ABC TV 23 April 2009. McLachlan is also a regular presenter on The History Channel, a history commentator on Channel Seven's Sunrise program and regularly appears in print and on other television and radio programs.
In 2013 McLachlan hosted the 7-part National Geographic Series Australia: Life on the Edge, which was nominated for an ASTRA award for Best Factual Program. He has appeared in numerous other TV programs including Tony Robinsons World War One and Who Do You Think You Are.
McLachlan is also the founder of Mat McLachlan Battlefield Tours, a dedicated battlefield touring company that operates tours of the Western Front, Gallipoli, Vietnam, Guadalcanal and other destinations.
He was born in West Wyalong, NSW and currently lives in Sydney NSW
References
Living people
Year of birth missing (living people) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,485 |
Last week the "Writers for Democracy" gathered at the National Museum Auditorium, and there were writers of all races and faiths writing in Sinhala, Tamil and English. The organization was formed several months ago, because there was an urgent need for all writers to unite to save democracy in this country.
However, since the immediate threat for the system what we know as Democracy has been removed, it is time to think about what Democracy today has come to mean. A Bhutanese poet said "Democracy divides people" and it is very true, when we come to think of it. Even at this gathering of writers for democracy, there was division, some writers did not want to attend, for various reasons.
Thus it is perhaps time to name this organization as Writers for Humanity, because democracy has just become an empty word. What we need is for man to become humane once again, to be really human. Writers have a major role to play, and a major responsibility to regain the lost humanity. If we become really human once again, then all the evil on earth would disappear and we could live in peace and harmony with all races, creeds, castes and people speaking in different tongues.
At the gathering it was also discussed about the achievement towards Good Governance. We have been gloating over the achievement for two months already. We have been gloating over the victory over terrorism for over five years, but we have not been able to convert that war victory into a victory of humanity, into a country where all human beings could live together with trust and confidence with each other. It is time to start working as writers, to consolidate the victory for Good Governance.
Martin Wickramasinghe was known to some of the village folk at Koggala as 'Liyana Mahattaya'. That is one way of translating the word 'Writer'. It is a good term for a writer, because a genuine writer is a 'Mahattaya', a gentleman. That is what all writers should be, a gentleman or gentlewoman, or to please everyone, a gentleperson. Then we are on the path to be Writers for Humanity. Then we should be able to shed all our petty difference of party politics, personal interests, language or religious issues and even regional issues.
In his speech, Gamini Viyangoda pointed out that the meeting should not have started with observing Pansil (Five Precepts). This observance could have been the result of having a Buddhist monk as the Chief Guest. But it is also because we are still thinking inside the box, that any occasion where Buddhist monks are present, we should start with a Buddhist ritual. The Buddhist monks who attended the meeting came as writers and as the initiators of the "Writers for Democracy" movement. It was not a religious function, and if we are to have religious observances at the start of such secular functions, then we would not have time to conduct the meeting, after the Buddhist, Hindu, Christian and Islam religious observances are completed.
What I realized as we recited the Five Precepts was that if all of us observed these Five Precepts to the letter, then we already have humanity back with us, we already have democracy and good governance, and we do not even need to have a meeting to remind us or discuss the issue.
Almost all the writers who gathered as Writers for Democracy were all senior citizens of the country. There were very few young people in the auditorium, except perhaps from the media organizations. This is one major drawback for democracy or for humanity. It is the youth in this country, who have the vision, who have the strength to make this movement effective. They also have the latest technology, digital and electronic, and the knowledge to use this technology through their blog posts, social media sites and e-journals to take the message to other youth.
Another advantage with the youth is that their minds are not poisoned by racial, religious or language issues. They have no prejudices. It is the youth who should come forward, take the lead, show the country, and the rest of the world, that we in Sri Lanka are one nation, one people, and that we can resolve our issues by ourselves.
In order to achieve this, youth have a mighty weapon. The digital word, where they can use their most convenient, most handy, and most powerful weapons in the form of their computers, laptops, tablets and smartphones. They have already begun to use these weapons. They used them successfully in other countries, in Iran and in Egypt. They used these weapons here to save democracy.
Now they can use it to restore Humanity. For this we have to recognize and accept their writing medium. For they write in the clouds, they write in the electronic media, in e-books, e-journals, blogs and social media. The Ministry of Cultural Affairs should recognize e-writers as writers. They too have to think outside the box, that it is not necessary in the world today to publish something in the print medium to be accepted as a writer. The constitution of the Sri Lanka National Writers' Organization accepts only writers who have published (printed) a book, whatever the book may be. The Literary Panel of the Arts Council accepts only printed books for the State Literary Awards. It is time they accepted e-books too, to be considered for literary awards.
Why is it that young writers are not coming forward? Are they discouraged or prevented from coming forward by the seniors who want to retain their positions? Are they keeping away out of respect for the seniors?
Let us give youth the opportunity to win back democracy and humanity, because it is their future. | {
"redpajama_set_name": "RedPajamaC4"
} | 543 |
Q: Что дает наличие transform: translateZ(0) scale(1, 1) в body Не первый раз встречаю правило transform: translateZ(0) scale(1, 1), какое применяется к тегу <body>.
Вопрос, что дает данное правило?
Так как у меня от него следующий проблемы: border не всегда прорисовывается в таблице и иногда изображения не прогружаются (случай в Google Chrome).
Но! Когда отключаешь это правило, сайт сдвигается(пока еще не понял почему, ведь значения стандартные ИМХО).
На сайтах где-то прочитал, что это для того, чтобы мерцания не было ( но может я не правильно понял). Спасибо
A: Без минимального, самодостаточного и воспроизводимого примера вашей конкретной ошибки сложно сказать наверняка.
Обычно translateZ(0) прописывают для того, чтобы отрисовать элементы в GPU еще до того, как анимация началась, чтобы анимация была плавной, без дерганий. tranform не перерисовывает объект, он работают напрямую с GPU памятью, которая использует аппаратное ускорение.
scale(1, 1) увеличивает содержимое блока в 110% от нормы. Скорее всего именно поэтому вы не видите некоторые бордеры, границы содержимого. Ведь они больше ширины блока на 10% и скорее всего просто в него не влезли. Можете попробовать прописать принудительно overflow. Возможно у вас что-то наподобие вот этого:
div {
height: 50px;
margin: 20px;
border: 5px solid red;
transform: translateZ(0) scale(1.1);
}
.one {
overflow: hidden
}
.two {
overflow:auto
}
.three {
overflow: visible;
}
span {
height: 50px;
display: block;
border: 5px solid green;
}
<div class=one><span>1</span></div>
<div class=two><span>2</span></div>
<div class=three><span>3</span></div>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 782 |
package br.com.http.utils;
public class InvalidIPFormatException extends RuntimeException {
private static final long serialVersionUID = 1L;
public InvalidIPFormatException(String address, Throwable e) {
super("Invalid IP pattern : " + address, e);
}
public InvalidIPFormatException(String address) {
this("Invalid IP pattern : " + address, null);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,580 |
The sight of Erik Cole on Bell Centre ice has never been a welcome one for Habs fans. That's all about to change.
Peter Budaj's signing in Montreal may have surprised some this summer, yet the Habs' new netminder arrived in town knowing exactly what he was getting into.
A first name isn't the only thing Randy Cunneyworth and Randy Ladouceur have in common.
When it comes to hockey superstitions, goaltenders are in a league of their own. Gilbert Dionne found that out firsthand in the spring of 1993.
The arrival of Raphael Diaz arrival to the Canadiens organization is a breath of fresh air – both for him and for the team.
Running one of Montreal's hottest restaurants, Garde-Manger, is one thing; couple that with your own TV show, Chuck's Day Off, and a victory over the renowned Bobby Flay in front of millions of viewers on Iron Chef America and you have the recipe used by 34-year-old Saint-Sauveur native Chuck Hughes to become one of North America's premiere celebrity chefs. We caught up with the inked-up gourmand and lifelong Habs fan to see what's cooking in his future. | {
"redpajama_set_name": "RedPajamaC4"
} | 733 |
\section{Introduction}\label{s:intr}
\bekezdes
It is an exciting project to compare the topological and analytic invariants of a complex normal surface singularity
$(X,o)$.
The topological invariants are characterized by the link $M$
(an oriented closed 3--manifold), or by a dual resolution graph $\Gamma$
associated with a resolution $\tX$ of $(X,o)$. The analytic invariants usually are associated with the local ring,
e.g. by the structure of its ideals, or by a certain (multi-graded) filtration of it,
or they are defined using
different coherent sheaves of
$(X,o)$ or of the resolution $\tX$. The topological invariants are discrete, and usually (in principle) can be computed combinatorially from
$\Gamma$ (or, directly from the fundamental group of the link, or by certain topological constructions of $M$).
However, the analytic invariants might reflect essentially the variation of the analytic structure, they might even have
moduli. Their computation and identification is usually hard, even if they are discrete.
Therefore, usually, as a first step, we try to find topological bounds for them, or
redescribe them
topologically
whenever the analytic structure is `nice', special, (or, in the opposite case, if it is generic). In this procedure,
for several analytic invariants one tries to find some topological candidate,
which it is equal to for special analytic structures. But it can also happen, conversely, that for a known
topological object we wish to find its analytic counterpart or its analytic refinement.
In this way, we get `pairs' of invariants. The main advantage of these pairs is not just that they coincide for
certain analytic structures, but (if they are really `matching pairs') they also behave rather similarly in their own categories
(e.g. they satisfy similar types of identities, they behave similarly with respect to
surgeries or certain geometrical constructions, see a few examples below).
For such pairs and their interactions the reader might consult \cite{Pairs}.
Let us mention some of them (for more details see also \cite{Nfive,NJEMS,Adv,NGr,Pairs,Book}).
We fix a good resolution $\tX \to X$ (and in some of the cases below we need to assume that the link is a rational homology sphere).
\vspace{4mm}
\begin{tabular}{|l|l|} \hline
topological & analytic \\
\hline
Artin's fundamental cycle $Z_{min}$ & maximal ideal cycle $Z_{max}$\\
integral Lipman's monoid (cone) $\calS_{top}$ & monoid $\calS_{an}$ of divisors of function \\
multivariable topological Poincar\'e series $Z(\bt)$ & multivariable Poincar\'e series $P(\bt)$ of divisorial filtration\\
the canonical Seiberg-Witten invariant of $M$ & the geometric genus $p_g=h^1(\calO_{\tX})$\\
all the Seiberg--Witten invariants of $M$ & the equivariant geometric genera\\
topological lattice cohomology $\bH^*_{top}$& \hspace{2cm}????\\
topological graded root & \hspace{2cm}????\\
\hline\end{tabular}
\vspace{3mm}
Let us list also some matching pairs of statements regarding the above objects.
$\bullet$ $Z_{min}$ is the minimal nonzero element of $\calS_{top}$, $Z_{max}$ is the minimal nonzero element of $\calS_{an}$.
$\bullet$ $Z(\bt)$ is supported on $\calS_{top}$, $P(\bt)$ is supported on $\calS_{an}$.
$\bullet$
The additivity formula of Okuma for $p_g$ \cite{Opg} is the analogue of the additivity formula
from \cite{BN} valid for the Seiberg--Witten invariant (in the first case the correction term is the periodic constant
of a reduced series associated with $Z(\bt)$, while in the analytic case this $Z(\bt)$ is replaced by $P(\bt)$).
$\bullet$ The multivariable periodic constant of $Z(\bt)$ is the Seiberg--Witten invariant of the link,
the multivariable periodic constant of $P(\bt)$ is the geometric genus of $(X,o)$.
\vspace{1mm}
Let us mention also that on the topological side the theory and list of properties of the lattice cohomology and graded roots
run interweaved: the graded root is an `improvement' of
$\bH^0_{top}$. They were defined in \cite{NOSz,Nlattice,NGr} (from the topology of the link, whenever the link is a rational homology sphere, see below).
\bekezdes
The main motivation for the present article is the fact that in the literature there is no analytic analogue
of the topological lattice cohomology $\bH^*_{top}$, though $\bH^*_{top}$
is deeply connected to several other topological invariants that have well--defined matching pairs.
E.g., $\bH^*_{top}$ is defined as the cohomology of weighted cubes, and a particular sum of weights of cubes
provides $Z(\bt)$ (see \ref{rem:P0AN} here) --- \ and $Z(\bt)$ admits $P(\bt)$ as matching pair.
Or, the Euler characteristic of the cohomology theory
$\bH^*_{top}=\oplus_{q\geq 0}\bH^q_{top}$ is the Seiberg--Witten invariant (i.e. $\bH^*$ is a categorification of the Seiberg--Witten invariant, see Theorem \ref{th:ECharLC}) --- \ and the Seiberg--Witten invariant is matched by the (equivariant) geometric genus.
Hence, it is rather natural to ask: {\it is there any natural cohomology theory
(in this case, a graded $\Z[U]$--module, just like $\bH^*_{top}$ is), associated with the analytic structure $(X,o)$,
which is the categorification of the geometric genus $p_g$ ? Is there an analogue to the topological graded root?
If the answer is yes, is it able to effectively reflect the change of the analytic structures (on a fixed topological type)?}
The aim of this article is to give positive answers to these questions.
\bekezdes Let us assume that the link is a rational homology sphere.
Then the topological lattice cohomology of the link $M$, $\bH^*_{top}(M)$, is well--defined \cite{Nlattice}.
It has a rather different structure than any cohomology
theory associated with analytic spaces by complex analytic or algebraic geometry (e.g. like the various sheaf--cohomologies).
It has several gradings: first of all, it has a direct sum decomposition according to the spin$^c$--structures $\sigma$
of $M$ (${\rm Spin}^c(M)$ is an $H_1(M,\Z)$ torsor). Then each summand $\bH^*_{top}(M,\sigma)$ has a decomposition
$\oplus _{q\geq 0}\bH^q(M,\sigma)$, where each $\bH^q_{top}(M,\sigma)$ is a $\Z$--graded $\Z[U]$--module.
Probably the presence of this additional $U$--action is the most outstanding property compared with the usual cohomology theories.
Conjecturally (see \cite{Nlattice})
$\bH^*_{top}(M)$ is isomorphic to the Heegaard Floer cohomology $HF^+$ of Ozsv\'ath and Szab\'o (which is defined for any 3--manifold),
for $HF$--theory see their
long list of articles, e.g.
\cite{OSz,OSz7}. This conjecture was verified for several families of plumbed 3--manifolds (associated with
negative definite connected graphs), cf. \cite{NOSz,OSSz3},
but the general case is still open.
(In fact, Heegaard Floer theory is isomorphic with several other theories:
the Monopole Floer Homology of Kronheimer and Mrowka, or the Embedded Contact Homology of Hutchings.
They are based on different geometrical aspects of the 3--manifold $M$.)
$\bH^*_{top}$ is the categorification of the Seiberg--Witten invariant (similarly as $HF^+$ is).
For several properties of the lattice cohomology and applications in singularity theory see \cite{NOSz,NSurgd,NGr,Nexseq,NeLO}. For its connection with the classification of
projective rational plane cuspidal curves (via superisolated surface singularities) see \cite{NSurgd,BLMN2,BodNem,BodNem2,BCG,BL1}.
It provides sharp topological bounds for certain sheaf cohomologies (e.g. for $p_g$), see e.g. \cite{NSig,NSigNN}.
An improvement of $\bH^0_{top}$ is the set of graded roots parametrized by the spin$^c$--structures of $M$ \cite{NOSz,NGr}
(they have no analogues for general arbitrary 3--manifolds). The graded root is a special
tree with $\Z$--graded vertices, it provides
a very visual presentation of $\bH^0_{top}$ (e.g., the $U$--action is coded in the edges).
Hence, in particular it visualizes $HF^+$ too,
when the Heegaard Floer homology is known to be isomorphic to $\bH^0_{top}$ (see e.g. \cite{NOSz}). In such cases
the use of graded roots is significantly more convenient than any other method, see e.g.
\cite{DM,K1,K2,K3}.
\bekezdes Our goal is to define the corresponding analytic lattice cohomology associated with the analytic type of a normal surface singularity. The definition
and the development of the first key properties
(some of them under some topological assumptions, e.g that the link is a rational homology sphere)
are presented in this article.
In fact, in the present note
we present the analytic lattice cohomology associated with the {\it canonical spin$^c$--structure $\sigma_{can}$} only.
It will be denoted by $\bH^*_{an,0}(X,o)$.
The case of other spin$^c$--structures will be treated in \cite{AgNeIII}.
(For the general part,
under the assumption that the link is a rational homology sphere,
we need to generalize the present constructions to the level of the universal abelian covering of $(X,o)$ and we also to recall several technical
details of the theory of the `natural line bundles' of a
resolution. Therefore, we decided to separate these technical parts into \cite{AgNeIII}.)
Accordingly, in the present note, in the presentation of the topological lattice cohomology we
restrict ourself only to the module (summand)
which corresponds
to $\sigma_{can}$.
In this article, under the assumption that all the exceptional curves of a resolution are rational, we show that
the analytic lattice cohomology $\bH^*_{an,0}$ just constructed is a categorification of the
geometric genus. (This means that the Euler characteristic of the cohomology is the geometric genus.)
We also show that it admits a graded $\Z[U]$--module morphism $\bH^*_{an}(X,o)\to \bH^*_{top}(M, \sigma_{can})$ (which in some
`nice' cases is an isomorphism). Furthermore, through several examples we show that it
is sensitive to the change of the analytic structure. In fact, for several fixed topological types we even classify the
possible graded $\Z[U]$--modules $\bH^*_{an,0}$ associated with all the possible analytic structures supported on that topological type.
We are certain that the new theory will have similar power and applicability as the topological version
(or, as the $HF$--theory),
with the difference that in this case
its applications will be mostly in the analytic theory of singularities.
E.g. (as several examples
show already in this article, see also \cite{AgNeCurves}) it is deeply related with the deformation of singularities.
\bekezdes In order to define a lattice cohomology one needs a free $\Z$--module with a fixed basis and a weight
function (with certain properties) defined on the lattice points. In our case the lattice is that of
the divisors supported on the exceptional curve of a resolution. In the topological case the weight function is provided by the
Riemann--Roch expression. In the analytic case it is a combination of the coefficients of the Hilbert function associated with the divisorial filtration and the dimension of a (sheaf)
cohomology vectorspace. In both cases one proves that the output lattice cohomology is independent of the choice of the resolution.
In the Gorenstein case (in the presence of a certain symmetry) the analytic weight function can be deduced merely from the
coefficients of the Hilbert function of the divisorial filtration.
An important observation is that the general construction/definition of the lattice cohomology is very flexible: by providing different weight functions one
obtains different lattice cohomologies, and indeed, there are several possibilities to construct weight functions
with remarkable associated cohomologies. This is
suggested already in the present note by considering different filtrations (e.g. Newton filtration) and the associated weight functions.
The construction can be generalized to higher dimensional complex normal isolated singularities and also
to curve singularities (here the divisorial filtration is replaced by the valuative one), cf. \cite{AgNeCurves,AgNeHigh}.
In the higher dimensional
case it is the categorification of $h^{n-1}(\calO_{\tX})$
(as in the surface case), while in the curve case it is the categorification of the delta--invariant.
\bekezdes The structure of the article and some of the main results are the following:
Section \ref{s:Prem1} contains the general definition of the lattice cohomology and graded root
associated with a weight function, we follow \cite{NOSz,NGr,Nlattice}. Here we also recall certain basics regarding
the topological lattice cohomology, the key Reduction Theorem \cite{LN1}
(when we can reduce the rank of the lattice according to
`bad' vertices), and several needed technical statements regarding the case of `almost rational graphs'
\cite{NOSz}.
Section \ref{ss:NSSAnlattice} (after some analytic preliminaries, vanishing theorems, etc.) provides the definition of the
analytic lattice cohomology associated with a normal surface singularity via one of its resolutions. We prove
that it is independent of the resolution, and when
the link is a rational homology sphere we prove that
it is the categorification of
the geometric genus. We also prove an Analytic Reduction Theorem which allows us to compute it in
various cases,
e.g. for rational, weighted homogeneous, Gorenstein elliptic, and certain superisolated germs
(the special properties of almost rational graphs are used in this last case).
We also determine it for the generic analytic structure. On the other hand, by several examples we show how it indicates the
variation of the analytic structure. In \ref{bek:DEFAN} we connect it (by a Conjecture) with $p_g$--constant flat deformations.
Section \ref{ss:CombLattice} contains a combinatorial setup of the theory: several proofs which depend basically only on these
combinatorial properties are separated here. This part can and will be applied to several other weight functions in the forthcoming articles of the series.
In the body of the article we also
present several examples and problems/conjectures regarding the new theory.
\section{Preliminaries. Basic properties of lattice cohomology}\label{s:Prem1}
\subsection{The lattice cohomology associated with a weight function}\label{ss:latweight} \cite{NOSz,Nlattice}
\bekezdes
We consider a free $\Z$-module, with a fixed basis
$\{E_v\}_{v\in\calv}$, denoted by $\Z^s$, $s:=|\calv|$.
Additionally, we consider a {\it weight function} $w_0:\Z^s\to \Z$ with the property
\begin{equation}\label{9weight}
\mbox{for any integer $n\in\Z$, the set $w_0^{-1}(\,(-\infty,n]\,)$
is finite.}\end{equation}
\bekezdes\label{9complex} {\bf The weighted cubes.}
The space
$\Z^s\otimes \R$ has a natural cellular decomposition into cubes. The
set of zero-dimensional cubes is provided by the lattice points
$\Z^s$. Any $l\in \Z^s$ and subset $I\subset \calv$ of
cardinality $q$ defines a $q$-dimensional cube $\square_q=(l, I)$, which has its
vertices in the lattice points $(l+\sum_{v\in I'}E_v)_{I'}$, where
$I'$ runs over all subsets of $I$.
The set of $q$-dimensional cubes is denoted by $\calQ_q$ ($0\leq q\leq s$).
Using $w_0$ we define
$w_q:\calQ_q\to \Z$ ($0\leq q\leq s$) by
$w_q(\square_q):=\max\{w_0(l)\,:\, \mbox{$l$ is a vertex of $\square_q$}\}$.
(For a more general definition of a `system of weight functions' when the system $w_q(\square_q)$ is not
determined by $w_0$, see \cite{Nlattice}, that generality
will be not used here.)
For each $n\in \Z$ we
define $S_n=S_n(w)\subset \R^s$ as the union of all
the cubes $\square_q$ (of any dimension) with $w(\square_q)\leq
n$. Clearly, $S_n=\emptyset$, whenever $n<m_w:=\min\{w_0\}$. For any $q\geq 0$, set
$$\bH^q(\R^s,w):=\oplus_{n\geq m_w}\, H^q(S_n,\Z)\ \ \mbox{and}\ \
\bH^q_{red}(\R^s,w):=\oplus_{n\geq m_w}\, \widetilde{H}^q(S_n,\Z).$$
Then $\bH^q$ is $\Z$ (in fact, $2\Z$)-graded, the
$d=2n$-homogeneous elements
consist of $H^q(S_n,\Z)$.
Also, $\bH^q$ is a $\Z[U]$-module; the $U$-action is given by
the restriction map $r_{n+1}:H^q(S_{n+1},\Z)\to H^q(S_n,\Z)$.
Namely, $U*(\alpha_n)_n=(r_{n+1}\alpha_{n+1})_n$. The same is true for $\bH^*_{red}$.
Moreover, for
$q=0$, there exists an augmentation
(splitting)
$H^0(S_n,\Z)\simeq
\Z\oplus \widetilde{H}^0(S_n,\Z)$, hence an augmentation of the graded
$\Z[U]$-modules
$$\bH^0\simeq\calt^+_{2m_w}\oplus \bH^0_{red}=(\oplus_{n\geq m_w}\Z)\oplus (
\oplus_{n\geq m_w}\widetilde{H}^0(S_n,\Z))\ \ \mbox{and} \ \
\bH^*\simeq\calt^+_{2m_w}\oplus \bH^*_{red},$$
where $\calt_{2m}^+$ equals
$\Z\langle U^{-m}, U^{-m-1},\ldots\rangle$ as a $\Z$-module and it has the natural $U$--multiplication.
Though
$\bH^*_{red}(\R^s,w)$ has finite $\Z$-rank in any fixed
homogeneous degree, in general, without certain additional properties of $w_0$, it is not
finitely generated over $\Z$, in fact, not even over $\Z[U]$.
\bekezdes\label{9SSP} {\bf Restrictions.} Assume that $T\subset \R^s$ is a subspace
of $\R^s$ consisting of a union of some cubes (from $\calQ_*$). For any $q\geq 0$ define $\bH^q(T,w)$ as
$\oplus_{n\geq\min{w_0|T}} H^q(S_n\cap T,\Z)$. It has a natural graded $\Z[U]$-module
structure. The restriction map induces a natural graded (degree zero)
$\Z[U]$-module homomorphism
$r^*:\bH^*(\R^s,w)\to \bH^*(T,w)$.
In our applications to follow, $T$ --- besides the trivial $T=\R^s$ case --- will be one of the following:
(a) the first quadrant $(\R_{\geq o})^s$,
(b) the rectangle $[0,c]=\{x\in \R^s\,:\, 0\leq x\leq c\}$ for some lattice point $c\geq 0$,
(c) a path of composed edges in the lattice, cf. \ref{9PCl} and \ref{bek:path}.
\bekezdes \label{9F} {\bf The `Euler characteristic' of $\bH^*$.}
Fix $T$ as in \ref{9SSP} and we will assume that each $\bH^*_{red}(T,w)$ has finite $\Z$--rank.
The Euler characteristic of $\bH^*(T,w)$ is defined as
$$eu(\bH^*(T,w)):=-\min\{w(l)\,:\, l\in T\cap \Z^s\} +
\sum_q(-1)^q\rank_\Z(\bH^q_{red}(T,w)).$$
\begin{lemma}\cite{NJEMS}\label{bek:LCSW} If $T=[0,c]$ for a lattice point $c\geq 0$, then
\begin{equation}\label{eq:Ecal}
\sum_{\square_q\subset T} (-1)^{q+1}w_k(\square_q)=eu(\bH^*(T,w)).\end{equation}
\end{lemma}
\subsection{Path lattice cohomology}\label{9PCl}\cite{Nlattice}
\bekezdes \label{bek:pathlatticecoh}
Fix $\Z^s$ as in \ref{ss:latweight} and fix also the weight functions
$\{w_q\}_q$ as in \ref{9weight}.
Consider also a sequence $\gamma:=\{x_i\}_{i=0}^t$ so that $x_0=0$,
$x_i\not=x_j$ for $i\not=j$, and $x_{i+1}=x_i\pm E_{v(i)}$ for
$0\leq i<t$. We write $T$ for the
union of 0-cubes marked by the points $\{x_i\}_i$ and of the
segments of type $[x_i,x_{i+1}]$.
Then, by \ref{9SSP} we get a graded $\Z[U]$-module $\bH^*(T,w)$,
which is called the {\em
path lattice cohomology} associated with the `path' $\gamma$ and weights
$\{w_q\}_{q=0,1}$. It is denoted by $\bH^*(\gamma,w)$.
It has an augmentation with $\calt^+_{2m_\gamma}$,
where $m_\gamma:=\min_i\{w_0(x_i)\}$, hence one also gets the {\em reduced path lattice
cohomology} $\bH^0_{red}(\gamma,w)$ with
$$\bH^0(\gamma,w)\simeq \calt_{2m_\gamma}^+\oplus
\bH^0_{red}(\gamma,w).$$
It turns out that $\bH^q(\gamma,w)=0$ for $q\geq 1$ and its `Euler characteristic' can be defined as (cf. \ref{9F})
\begin{equation}\label{eq:euh0}
eu(\bH^*(\gamma,w)):=-m_\gamma+\rank_\Z\,(\bH^0_{red}(\gamma,w)).\end{equation}
\begin{lemma} \label{9PC2}
One has the following expression of $eu(\bH^*(\gamma,w))$ in terms of the values of $w_0$:
\begin{equation}\label{eq:pathweights}
eu(\bH^*(\gamma,w))=-w_0(0)+\sum_{i=0}^{t-1}\,
\max\{0, w_0(x_{i})-w_0(x_{i+1})\}.
\end{equation}
\end{lemma}
\begin{remark} It is convenient to
compare $\bH^*(T,w)$ and $\bH^0(\gamma,w)$ for certain (big) rectangles $T$, or for any $T$. In such cases it is convenient to
consider the following `truncated Euler characteristic' of $\bH^*(T,w)$ as well (as an analogue of (\ref{eq:euh0}),
even if $\bH^{\geq 1}(T,w)\not=0$):
$$eu(\bH^0(T,w)):=-\min\{w(l)\,:\, l\in T\cap \Z^s\} +
\rank_\Z(\bH^0_{red}(T,w)).$$
\end{remark}
\subsection{Graded roots and their cohomologies}\label{s:grgen} \cite{NOSz,NGr}
\begin{definition}\label{def:2.1} \
Let $\RR$ be an infinite tree with vertices $\calv$ and edges
$\cale$. We denote by $[u,v]$ the edge with
end-vertices $u$ and $v$. We say that $\RR$ is a {\em graded root}
with grading $\chic:\calv\to \Z$ if
(a) $\chic(u)-\chic(v)=\pm 1$ for any $[u,v]\in \cale$;
(b) $\chic(u)>\min\{\chic(v),\chic(w)\}$ for any $[u,v],\
[u,w]\in\cale$, $v\neq w$;
(c) $\chic$ is bounded below, $\chic^{-1}(n)$ is finite for any
$n\in\Z$, and $|\chic^{-1}(n)|=1$ if $n\gg 0$.
\vspace{1mm}
\noindent
An isomorphism of graded roots is a graph isomorphism, which preserves the gradings.
\end{definition}
\begin{examples}\label{ex:2.3e} \
(1) For any $n\in\Z$, let
$\RR_{(n)}$ be the tree with $\calv=\{v^{k} \}_{ k\geq n}$ and
$\cale=\{[v^{k},v^{k+1}]\}_{k\geq n}$. The grading is
$\chic (v^{k})=k$.
\vspace{2mm}
(2) Let $I$ be a finite index set. For each $i\in I$ fix an
integer $n_i\in \Z$; and for each pair $i,j\in I$ fix
$n_{ij}=n_{ji}\in\Z$ with the next properties:
(i) $n_{ii}=n_i$; \
(ii) $n_{ij}\geq \max\{n_i,n_j\}$; and
\
(iii) $n_{jk}\leq \max\{n_{ij},n_{ik}\}$ for any $ i,j,k\in I$.
For any $i\in I$ consider $\RR_i:=\RR_{(n_i)}$ with vertices
$\{v_i^{k}\}$ and edges $\{[v_i^{k},v_i^{k+1}]\}$, $(k\geq n_i)$.
In the disjoint union $\sqcup_i\RR_i$, for any pair $(i,j)$,
identify $v_i^{k}$ and $v_j^{k}$,
resp.\ $[v_i^{k},v_i^{k+1}]$ and $[v_j^{k},v_j^{k+1}]$, whenever
$k\geq n_{ij}$. Write $\bar{v}_i^{k}$ for the class of $v_i^k$.
Then $\sqcup_i\RR_i/_\sim$ is a graded root with
$\chic(\bar{v}_i^{k})=k$. It will be denoted by
$\RR=\RR(\{n_i\},\{n_{ij}\})$.
\vspace{2mm}
(3) Any map $\tau:\{0, 1,\ldots,T_0\}\to \Z$ produces a starting
data for construction (2). Indeed, set $I=\{0,\ldots,T_0\}$,
$n_i:=\tau(i)$ ($i\in I$), and $n_{ij}:=\max\{n_k\,:\, i\leq k\leq
j\}$ for $i\leq j$. Then $\sqcup_i\RR_i/_\sim $ constructed in (2)
using this data will be denoted by $(\RR_\tau,\chic_\tau)$.
This construction can be extended to the case of a map $\tau:\N\to \bZ$, whenever
$\tau$ has the property that there exists some $k_0\geq 0$ such that $\tau(k+1)\geq \tau(k)$ for any $k\geq k_0$.
In this case one can take any $T_0\geq k_0$ and construct the root associated with the
restriction of $\tau$ to $\{0,\ldots, T_0\}$.
It is independent of the choice of $T_0$ (since all
$\RR_k$ contributions for $k\geq k_0$ are superfluous). By definition, this is the root associated with $\tau$.
\end{examples}
\begin{definition}\label{9.2.6}{\bf The
$\Z[U]$-modules associated with a graded root.}
Let us identify a graded root $(\RR,\chic)$ with its topological realization provided by vertices (0--cubes)
and segments (1--cubes). Define $w_0(v)=\chic(v)$, and $w_1([u,v])=\max\{\chic(u),\chic(v)\}$ and let $S_n$ be the
union of all cubes with weight $\leq n$. Then we might set (as above) $\bH^*(\RR,\chi)=\oplus_{n\geq \min\chic}\
H^*(S_n,\Z)$. However, at this time $\bH^{\geq 1}(\RR,\chic)=0$; we set $\bH(\RR,\chic):=\bH^0(\RR,\chic)$.
Similarly, one defines $\bH_{red}(\RR,\chic)$ using the reduced cohomology, hence
$\bH(\RR,\chic)\simeq\calt_{2\min \chic}^+\oplus \bH_{red}(\RR,\chic)$.
\end{definition}
For a detailed concrete description of $\bH(\RR)$ in terms of the combinatorics of the root see \cite{NOSz}.
\begin{example}\label{9.2.9} (a) $\bH(\RR_n)=\calt_{2n}^+$.
(b) Let $(\RR_\tau,\chic_\tau)$ be
a graded root associated with some function $\tau:\N\to \Z$,
cf.\ \ref{ex:2.3e}(3). Then
$$\rank_\Z \bH_{red}(\RR_\tau,\chic_\tau)=-\tau(0)+\min_{i\geq 0}\tau(i)+\sum_{i\geq 0}\,
\max\{ \tau(i)-\tau(i+1),0\}.$$
\end{example}
\bekezdes\label{bek:GRootW}{\bf The graded root associated with a weight function.}
Fix a free $\Z$-module and a system of weights $\{w_q\}_q$.
Consider the sequence of topological spaces (finite cubical complexes) $\{S_n\}_{n\geq m_w}$
with $S_n\subset S_{n+1}$, cf. \ref{9complex}.
Let $\pi_0(S_n)=\{\calC_n^1,\ldots , \calC_n^{p_n}\}$ be the set of connected components of $S_n$.
Then we define the graded graph $(\RR_w,\chic_w)$ as follows. The
vertex set $\calv(\RR_w)$ is $\cup_{n\in \Z} \pi_0(S_n)$.
The grading $\chic_w:\calv(\RR_w)\to\Z$ is
$\chic_w(\calC_n^j)=n$, that is, $\chic_w|_{\pi_0(S_n)}=n$.
Furthermore, if $\calC_{n}^i\subset \calC_{n+1}^j$ for some $n$, $i$ and $j$,
then we introduce an edge $[\calC_n^i,\calC_{n+1}^j]$. All the edges
of $\RR_w$ are obtained in this way.
\begin{lemma}\label{lem:GRoot} $(\RR_w,\chic_w)$ satisfies all the required properties
of the definition of a graded root, except possibly the last one: $|\chic_w^{-1}(n)|=1$ whenever $n\gg 0$.
\end{lemma}
The property $|\chic_w^{-1}(n)|=1$ for $n\gg 0$ is not always satisfied.
However, the graded roots associated with connected negative definite plumbing graphs
(see below) satisfies this condition as well.
\begin{proposition}\label{th:HHzero} If $\RR$ is a graded root associated with $(T,w)$ and
$|\chic_w^{-1}(n)|=1$ for all $n\gg 0$ then $\bH(\RR)=\bH^0(T,w)$.
\end{proposition}
\section{Surface singularities and the topological lattice cohomology}\label{s:SSTLC}
\subsection{The combinatorics of the resolutions}\label{s:prel}
\cite{Nfive,NOSz,NGr}
\bekezdes
Let $(X,o)$ be the germ of a complex analytic normal surface singularity with link $M$.
Let $\phi:\widetilde{X}\to X$ be a good resolution of $(X,o)$ with
exceptional curve $E:=\phi^{-1}(0)$, and let $\cup_{v\in\calv}E_v$ be
the irreducible decomposition of $E$. (By good resolution we mean that each $E_v$ is smooth, and $E$ is a normal crossing divisor.)
Let $\Gamma$ denote the dual resolution graph of $\phi$. Note that $\partial \tX\simeq M$.
The lattice $L:=H_2(\widetilde{X},\mathbb{Z})$ is endowed
with the natural negative definite intersection form $(\,,\,)$. It is
freely generated by the classes of $\{E_v\}_{v\in\mathcal{V}}$.
The dual lattice is $L'={\rm Hom}_\Z(L,\Z) \simeq\{
l'\in L\otimes \Q\,:\, (l',L)\in\Z\}$.
$L$ is embedded in $L'$ with
$ L'/L\simeq {\rm Tors}( H_1(M,\mathbb{Z}))$, which is abridged by $H$.
The class of $l'$ in $H$ is denoted by $[l']$.
We define the Lipman cone as $\calS':=\{l'\in L'\,:\, (l', E_v)\leq 0 \ \mbox{for all $v$}\}$, and we also
set $\calS:=\calS'\cap L$. If $s'\in\calS'\setminus \{0\}$ then
all its $E_v$--coordinates are strictly positive.
There is a natural partial ordering of $L'$ and $L$: we write $l_1'\geq l_2'$ if
$l_1'-l_2'=\sum _v r_vE_v$ with every $r_v\geq 0$. We set $L_{\geq 0}=\{l\in L\,:\, l\geq 0\}$ and
$L_{>0}=L_{\geq 0}\setminus \{0\}$.
The support of a cycle $l=\sum n_vE_v$ is defined as $|l|=\cup_{n_v\not=0}E_v$.
If $H_1(M,\Q)=0$ then
each $E_v$ is rational, and the dual graph of any good resolution is a tree.
Artin's fundamental cycle $Z_{min}\in L_{>0}$ is defined as the smallest non-zero cycle in $\calS$
(with respect to $\geq $).
The {\it (anti)canonical cycle} $Z_K\in L'$ is defined by the
{\it adjunction formulae}
$(Z_K, E_v)=(E_v,E_v)+2-2g_v$ for all $v\in\mathcal{V}$, where
$g_v$ denotes the genus of $E_v$.
(The cycle $-Z_K$ is the first Chern class of the line bundle $\Omega^2_{\tX}$.)
We write $\chi:L'\to \Q$ for the (Riemann--Roch) expression $\chi(l'):= -(l', l'-Z_K)/2$.
The singularity (or, its topological type) is called numerically Gorenstein if $Z_K\in L$.
(Since $Z_K\in L$ if and only if the line bundle $\Omega^2_{X\setminus \{o\}}$ of holomorphic 2--forms
on $X\setminus \{o\}$ is topologically trivial, see e.g. \cite{Du}, the $Z_K\in L$ property
is independent of the resolution). $(X,o)$ is called Gorenstein if $Z_K\in L$ and
$\Omega^2_{\tX}$ (the sheaf of holomorphic 2--forms) is isomorphic to $ \calO_{\tX}(-Z_K)$ (or,
equivalently, if the line bundle $\Omega^2_{X\setminus \{o\}}$ is holomorphically trivial).
If $\tX$ is a minimal resolution then (by the adjunction formulae) $Z_K\in \calS'$.
\subsection{The topological lattice cohomology associated with $\phi:\tX\to X$}\label{s:latticeplgraphs}\cite{NOSz,Nlattice}
\bekezdes\label{9dEF1} We consider a good resolution $\phi$ as above
and we assume that the link $M$ is a
rational homology sphere. We write $s:=|\cV|$.
Then we automatically have a free $\Z$-module $L=\Z^s$ with
a fixed bases $\{E_v\}_v$. The Riemann--Roch expression $\chi$ defines a weight function
$w_0(l)=\chi(l)$, hence a set
of weight functions
$ w_q(\square_q)=\max\{\chi(v)\,:\, v\ \mbox{is a vertex of
$\square_q$}\}$.
\begin{definition}\label{9DEF} The $\Z[U]$-modules $\bH^*(\R^s,w)$ and
$\bH^*_{red}(\R^s,w)$ obtained in this way are
called the {\em topological lattice cohomologies} associated with the canonical spin$^c$--structure.
They are denoted by $\bH^*(\Gamma,-Z_K)$, respectively
$\bH^*_{red}(\Gamma,-Z_K)$ (or $\bH^*_{top,0}(M)$ and $\bH^*_{top,red,0}(M)$ respectively).
The same weight function defines a graded root $(\RR(\Gamma, -Z_K),\chic)=\RR_{top,0}(M)$ as well.
It is called the
{\it topological graded root} associated with the canonical spin$^c$--structure.
\end{definition}
\begin{proposition}\label{9STR2}
(a) $\bH^*_{red}(\Gamma,-Z_K)$ is finitely generated over $\Z$.
(b) The degree $d>0$ summands of \ $\bH^0_{red}(\Gamma,-Z_K)$ are zero.
(c) $\bH^*(\Gamma,-Z_K)$ and $(\RR(\Gamma, -Z_K),\chic)$
depend only on $M$ and they are independent of the choice of the good resolution $\phi$.
The root $(\RR(\Gamma, -Z_K),\chic)$ satisfies the property $|\chic^{-1}(n)|=1$ for $n\gg 0$.
(d) The restriction
$\bH^*(\Gamma,-Z_K)\to \bH^*((\R_{\geq 0})^s,-Z_K)$
induced by the inclusion $(\R_{\geq 0})^s\hookrightarrow \R^s$ is an isomorphism
of graded $\Z[U]$ modules.
\end{proposition}
\bekezdes {\bf The Euler characteristic and the Seiberg--Witten invariant.}\label{s:LCSW} \
The Seiberg--Witten invariant ${\rm Spin}^c(M)\to \Q$ associates a rational number
$\sw _{\sigma}(M)$ to each spin$^c$--structure $\sigma$ of the link.
\begin{theorem}\label{th:ECharLC} \cite{NJEMS} \
$eu(\bH^*(\Gamma,-Z_K))=\sw_{\sigma_{can}}(M)-(Z_K^2+|\cV|)/8$.
\end{theorem}
In other words, {\it
$\bH^*(\Gamma,-Z_K)$
is the categorification of $\sw_{\sigma_{can}}(M)$ (normalized by $(Z_K^2+|\cV|)/8$)}.
\bekezdes\label{bek:SWIC}
The geometric genus of any normal surface singularity $(X,o)$ is defined as
$h^1(\tX, \calO_{\tX})$, and it is denoted by $p_g$.
($h^1(\tX, \calO_{\tX})$ is independent of the choice of $\tX$.)
A priori (and usually) it is an analytic invariant.
We say, following \cite{NJEMS,NeNi,NeNiII},
that the singularity $(X,o)$ (with rational homology link)
satisfies the {\bf Seiberg--Witten Invariant Conjecture (SWIC) for the
canonical spin$^c$--structure} if $p_g(X,o)=eu(\bH^*(\Gamma,-Z_K))$, i.e., $p_g$ can be characterized topological by
the normalized Seiberg--Witten invariant.
The SWIC is satisfied in the following cases:
weighted homogeneous singularities, superisolated singularities associated with rational unicuspidal curves,
suspension singularities of type $f(x,y)+z^n$ with $f$ irreducible, splice quotient singularities (e.g.
rational or minimally elliptic singularities), see \cite{BN,Spany,BLMN2,NJEMS,NeNi,NeNiII,NeNiIII,NO2}.
\bekezdes\label{bek:pgbounds} {\bf Euler characteristic type bounds for $p_g$.}
Though the SWIC (for the canonical spin$^c$-structure) is not true in general, there are some generally valid
topological bounds provided by lattice cohomology.
Let ${\mathcal K}$ be the (topologically defined )
set of cycles $\lfloor Z_K\rfloor _++ L_{\geq 0}$. Here $\lfloor Z_K\rfloor _+$ is the effective part of the integral part
of $Z_K$.
By \ref{th:GR}, $h^1(\cO_l)=p_g$ for any $l\in {\mathcal K}$ and for any analytic structure supported by $\Gamma$, i.e.
$h^1(\cO_l)$ in this zone already `stabilizes'. The lattice point $\lfloor Z_K\rfloor _+$
is essential from topological point of view as well:
all the (path) lattice cohomologies can be computed already in the
rectangle $R(0, \lfloor Z_K\rfloor _+)$, cf. \cite{NOSz,Nlattice}.
\bekezdes \label{bek:path}
In the sequel we denote by $\cP$ the set of paths $\gamma=\{x_i\}_{i=0}^t$ with $x_0=0$ and arbitrary end-cycle
$x_t=c$ from ${\mathcal K}$.
A path is {\it increasing} if $\epsilon_i=+1$ for every $i$.
Let $\bH^0(\gamma,\Gamma, -Z_K)$ be the path lattice cohomology associated with $\gamma$ and the weight function
associated with $\chi$.
Let $eu(\bH^0(\gamma,\Gamma,-Z_K))$ be its Euler characteristic defined as in (\ref{eq:euh0}).
It turns out (see e.g. \cite{NSig,NSigNN}, or \cite{Book}) that
$\min_{\gamma \in \cP}\, eu(\bH^0(\gamma,\Gamma,-Z_K))$ is realized by an increasing path, and this expression
is independent of the choice of this $c$, whenever $c\in {\mathcal K}$.
Furthermore,
\begin{equation}\label{cor:INEQeu}
p_g\leq \min_{\gamma \in \cP}\, eu(\bH^0(\gamma,\Gamma,-Z_K))\leq
eu (\bH^0(\Gamma, -Z_K)).
\end{equation}
\begin{example}\label{ex:MINforwh}
The equality $p_g=\min_{\gamma \in \cP}\, eu(\bH^0(\gamma,\Gamma,-Z_K))$ is realized
(under the assumption that the link is a rational homology sphere)
by the following families:
(a) singularities which satisfies the
SWIC for the canonical spin$^c$ structure (cf. \ref{bek:SWIC})
and $\bH^{\geq 1}(\Gamma,-Z_K)=0$ (use (\ref{cor:INEQeu})). This includes e.g.
weighted homogeneous singularities,
rational singularities, or
maximally elliptic singularities (i.e. elliptic singularities which satisfy
$p_g=\mbox{length of the elliptic sequence}\ \ell_{seq}$).
(b) superisolated singularities, cf. \cite{NSig};
(c) local Weil divisors in affine toric 3-varieties with nondegenerate Newton principal part, cf.
\cite{NSigNN}.
\end{example}
\begin{remark}\label{rem:SPLIT}
Assume that $p_g=\min_{\gamma \in \cP}\, eu(\bH^0(\gamma,\Gamma,-Z_K))$, and that
$\min_{\gamma \in \cP}\, eu(\bH^0(\gamma,\Gamma,-Z_K))$ is realized by the increasing
path $\gamma=\{x_i\}_{i=0}^t$, $x_{i+1}=x_i+ E_{v(i)}$.
Then all the exact sequences $0\to \cO_{ E_{v(i)}}(-x_i)\to \cO_{x_{i+1}}\to \cO_{x_i}\to 0$ cohomologically must split
(for details see e.g. \cite{NSig,NSigNN}). In particular, $H^0(\cO_{x_{i+1}})\to H^0(\cO_{x_i})$ is surjective for every $i<t$.
\end{remark}
\subsection{Measure of non-rationality. `Bad' vertices}\label{ss:BadVer} \cite{NOSz,Book}\
Rational graphs play a distinguished role in this theory: they are the graphs with $\bH^*_{red}(\Gamma)=0$.
Usually, in the analysis of a graph $\Gamma$, we wish to understand how far is $\Gamma$ to be rational.
\begin{example}\label{ex:LatHomRat} {\bf Rational graphs.}
Recall that $(X,o)$ is called rational if $p_g=0$. By a result of Artin \cite{Artin62,Artin66}
$p_g=0$ if and only if
$\chi(l)\geq 1$ for all $ l\in L_{>0}$ (hence it is a topological property of $M$ readable from $\Gamma$, those graph which satisfy it are called rational).
The links of
any rational singularity is a
rational homology sphere.
The class of rational graphs is closed while taking subgraphs or/and
decreasing the Euler numbers $E_v^2$.
We have the following characterizations of the rationality in terms of topological lattice
cohomology (or graded roots):
$\Gamma$ is rational $\Leftrightarrow$
$\bH^0_{red}(\Gamma,-Z_K)=0$ $\Leftrightarrow$
$\bH^*_{red}(\Gamma,-Z_K)=0$ $\Leftrightarrow$ $\RR_{-Z_K}=\RR_{(0)}$, cf. \cite{NOSz}.
\end{example}
If $M$ is a $\Q HS^3$, then by
decreasing all the Euler numbers of $\Gamma$
we obtain a rational graph.
The next definition aims to
identify those vertices where such a decrease is really necessary.
\begin{definition}\label{def:SWrat} Let $\Gamma$ be a resolution graph such that $M$ is a rational homology sphere.
A subset of vertices
$\ocalj=\{v_1,\ldots, v_{\overline{s}}\}\subset \cV$ is called {\it B--set}, (set of `bad vertices')
if by replacing the Euler numbers
$e_v=E_v^2$ indexed by $v\in \ocalj$ by some more negative integers
$e'_v\leq e_v$ we get a rational graph.
A graph is called AR-graph (`almost rational graph') if it admits a B--set of cardinality $\leq 1$.
\end{definition}
\begin{example}\label{ex:*sets}
(a) A possible B--set can be chosen in many different ways, usually
it is not determined uniquely even if it is minimal with this property.
Usually we allow non-minimal $B$--sets as well.
(b) If $H_1(M,\Q)=0$ then the set of nodes is a B--set.
Hence any star-shaped graph (with $H_1(M,\Q)=0$) is AR.
Other AR families are: rational and elliptic graphs and graphs of superisolated singularities associated with a
rational unicuspidal curve \cite{NOSz,NGr}.
(c)
The class of AR graphs is closed while taking subgraphs or/and
decreasing the Euler numbers.
\end{example}
\bekezdes\label{bek:XI}
{\bf The definition of the lattice points $x(\bar{l})$.}
Assume that $\ocalj:=\{v_k\}_{k=1}^{\overline{s}}$ is any subset of $\calj$.
Then we split the set of vertices $\calj$ into the disjoint union $\overline{\calj}\sqcup\calj^*$.
Let $\{m_v(x)\}_v$ denote the coefficients of a cycle $x\in L\otimes \setQ$, that is $x=\sum_{v\in\calj}m_v(x)E_v$.
Our goal is to define some universal cycles $x(\bar{l})\in L$
associated with $\bar{l}\in L(\ocalj)$.
\begin{proposition}\label{lemF1} \ \cite[Lemma 7.6]{NOSz}, \cite{LN1}
For any
$\bar{l}:=\sum_{v\in \ocalj}\ell_v
E_v\in L(\ocalj)$
there exists a unique cycle
$x(\bar{l})\in L$
satisfying the next properties:
\begin{itemize}
\item[(a)] \ \ $m_{v}(x(\bar{l}))=\ell_v$ \ for any distinguished vertex $v\in\ocalj$;
\item[(b)] \ \ $(x(\bar{l}),E_v)\leq0$ \ for every `non-distinguished vertex' $v\in\calj^*$;
\item[(c)] \ \ $x(\bar{l})$ is minimal with the two previous properties (with respect to $\leq$).
\end{itemize}
\end{proposition}
\bekezdes
Let us fix a B--set $\ocalj\subset \calv$
as in \ref{def:SWrat} (with cardinality $\bar{s}$).
Our goal is to provide an equivalent description of the lattice cohomology using cubes in an $\bar{s}$--dimensional lattice.
Firs recall the isomorphism $\bH^*(\Gamma,-Z_K)\to \bH^*((\R_{\geq 0})^s,-Z_K)$ from
\ref{9STR2}{\it (d)}.
Having this in mind, for each $\bar{l}=\sum_{v\in \ocalj} \ell_vE_v\in L(\ocalj)$,
with every $\ell_v\geq 0$, we define the universal cycle $x(\bar{l})$ associated with
$\bar{l}$ as in \ref{lemF1}.
Then,
define the function
$\overline{w}_0:(\Z_{\geq 0})^{\bar{s}}\to \Z$ by
$\overline{w}_0(\bar{l}):=\chi(x(\bar{l}))$.
Then $\overline{w}_0$ defines a set $\{\overline{w}_q\}_{q=0}^{\bar{s}}$ of weight functions as in \ref{9dEF1} by
$\overline{w}_q(\square)=
\max\{\overline{w}_0(v)\,:\, v \ \mbox{ is a vertex of $\square$}\}$.
\begin{theorem}\label{th:red} {\bf (Topological Reduction Theorem)} \cite{LN1}
There exists a graded $\Z[U]$-module isomorphism
\begin{equation}\label{eq:reda}
\bH^*((\R_{\geq 0})^s,-Z_K)\cong\bH^*((\R_{\geq 0})^{\bar{s}},\overline{w}) \ \ \mbox{and} \
\ \RR((\R_{\geq 0})^s,-Z_K)\cong \RR((\R_{\geq 0})^{\bar{s}},\overline{w}).
\end{equation}
\end{theorem}
\subsection{Concatenated computation sequences of AR graphs}\label{ss:CCSAR} \cite{NOSz}
\bekezdes Assume that $\Gamma $ is an AR resolution graph. Let $\{v_0\}$ be an B--set.
Note that by Reduction Theorem \ref{th:red} $\bH^{\geq 1}(M(\Gamma), -Z_K)=0$.
The Reduction Theorem provides a formula for $\bH^{0}(\Gamma, -Z_K)$ too using a lattice cohomology associated with
$\Z_{\geq 0}\subset \R_{\geq 0}$ and weight function $\Z_{\geq 0}\ni \ell\mapsto \chi(x(\ell))$. In the next discussion we present another (equivalent) version,
which represents
$\bH^{0}(\Gamma, -Z_K)$ as the lattice cohomology of an increasing
path $T=\gamma=\{x_i\}_{i\geq 0} $ embedded in the lattice $L_{\geq 0}$.
The point is that $\gamma $ determines the 1-chain $C_\gamma:=
\cup_{i\geq 0}[x_i,x_{i+1}]$ of 1-cubes in
$L\otimes \R$ (without any loop), such that $C_\gamma\cap S_{n} \hookrightarrow S_{n}$ is a homotopy equivalence.
In particular, all the connected components of $S_n$ (whenever $S_n\not=\emptyset$) are contractible.
The construction of $\gamma$ runs as follows: it is
defined as a series of concatenated computation sequences.
It contains, as intermediate terms, all the universal cycles $\{x(\ell)\}_{\ell\geq 0}$ in an
increasing order.
The first term is $x_0=x(0)=0$.
The part of the sequence starting from $x(\ell)$ and ending with $x(\ell+1)$
starts with $x(\ell)$ and the next term is $x(\ell)+E_{v_0}$. Then, the continuation is a
generalized Laufer-type computation sequence connecting $x(\ell)+E_{v_0}$
with $x(\ell+1)$. Indeed, the multiplicity of $E_0$ in both $x(\ell)+E_{v_0}$
and $x(\ell+1)$ is $\ell+1$, and by the universal (minimality) property of $x(\ell)$ we get
that $x(\ell+1)\geq x(\ell)+E_{v_0}$.
Then, there is a (Laufer type) generalized computation sequence, or increasing path, $\gamma^{(\ell+1)}=\{x_i^{(\ell+1)}\}_i$,
which connects
$x(\ell)+E_{v_0}$ and $x(\ell+1)$ (see \cite{NOSz}).
Then we proceed inductively.
In general, it is not easy to concretely identify the cycles $x(\ell)$.
Fortunately, in several applications we only need
the values $\tau(\ell)=\chi(x(\ell))$. In
most of the cases they are computed inductively using \ref{lem:xellprops}{\it (d)}, hence
basically one needs only to know
$(x(\ell),E_{v_0})$ for any $\ell$.
\begin{proposition}\label{lem:xellprops}\cite{NOSz}
(a) The path $\{x_i\}_i$ is increasing: $x_{i+1}=x_i+E_{v(i)}$.
(b) For any $E_v$-coefficient one has $\lim_{\ell\to \infty} m_v(x(\ell))=\infty$ (where $v\in\calv$).
(c) $\chi$ along each part (subsequence) $\gamma^{(\ell)}$ is constant. This also implies that
$\bH^0((\R_{\geq 0})^{\bar{s}},\overline{w})$ from the reduction theorem equals the path lattice cohomology
$\bH^0(\gamma, -Z_K)$).
(d) Set $\tau(\ell)=\chi(x(\ell))$. Then $\tau(\ell+1)=\tau(\ell)+1-(x(\ell),E_{v_0})$ and
there exists $\ell_0$ such that $\tau(\ell+1)\geq \tau(\ell) $ for $\ell\geq \ell_0$.
(e) $eu(\bH^*(\Gamma,-Z_K) = eu(\bH^*(\gamma,-Z_K
=\sum _{\ell\geq 0}
\max\{\, \tau(\ell)-\tau(\ell+1), 0\,\}$.
\end{proposition}
\section{The analytic lattice cohomology of surface singularities}\label{ss:NSSAnlattice}
\setcounter{equation}{0}
\subsection{Review of some analytic properties}\label{ss:AnPrel}
\bekezdes\label{bek:PG}
Let $(X,o)$ be a normal surface singularity and we fix a good resolution $\phi$.
In this subsection we go over some statements that will help in the later discussion and proofs.
We start with a vanishing theorem.
\begin{theorem}\label{th:GR} {\bf Generalized Grauert--Riemenschneider Theorem.}
\cite{GrRie,Laufer72,Ram,Book}
Consider a line bundle $\cL\in {\rm Pic}(\widetilde{X})$ such that
$c_1(\cL(Z_K))\in \Delta -\cS_{\Q}$ for some
$\Delta\in L\otimes \Q$ with $\lfloor \Delta \rfloor =0$.
Then $h^1(l,\cL|_{l})=0$ for any $l\in L_{>0}$.
In particular, $h^1(\widetilde{X},\cL)=0$ too. (Here $\calS_\Q$ denotes the rational cone generated by $\calS$.)
\end{theorem}
In particular, if
$\calL\in {\rm Pic}(\tX)$ and $l\in L_{>0}$ satisfies $l\in c_1(\calL)+Z_K+\calS$,
then $h^1(\tX, \calL(-l))=0$, hence $H^1(\tX, \calL)=H^1(l, \calL|_l)$.
We denote by $\lfloor Z_K\rfloor $ the integral part of $Z_K$, and by
$\lfloor Z_K\rfloor_+ $ its effective part. The above statements imply the following.
If $\lfloor Z_K\rfloor_+=0$ then $p_g=0$.
If $\lfloor Z_K\rfloor_+>0$ then for
any $Z\geq \lfloor Z_K\rfloor_+$, $Z\in L$,
$p_g=h^1(\cO_{Z})$.
Furthermore, if $l\in \cS$ and $n\in\Z_{\geq 0}$ such that $nl+\lfloor Z_K\rfloor>0$ then
\begin{equation*}
\dim \frac{H^0(\cO_{\tX})}{H^0(\cO_{\tX}(-\lfloor Z_K\rfloor-nl))}=
\chi(\lfloor Z_K\rfloor+nl)+p_g.
\end{equation*}
This implies that for $l\in \calS\setminus \{0\}$ and $n\gg 0$,
\begin{equation}\label{eq:vanh0}
\dim \frac{H^0(\cO_{\tX})}{H^0(\cO_{\tX}(-nl))}=
-\frac{n^2l^2}{2}+\mbox{lower order terms in $n$}.
\end{equation}
For certain cycles the Grauert--Riemenschneider Theorem \ref{th:GR} can be improved.
\begin{proposition}\label{prop:VAN0}\ {\bf Lipman's Vanishing Theorem.} \cite[Theorem 11.1]{Lipman}, \cite{Book}\
Take $l\in L_{>0}$ with $h^1(\cO_l)=0$ and $\cL\in {\rm Pic}(\tX)$ for which
$(c_1\cL,E_v)\geq 0$ for any $E_v$ in the support of\, $l$. Then $h^1(l,\cL)=0$.
\ix{Vanishing Theorem!Lipman's|textbf}
\end{proposition}
\bekezdes
\label{rem:antmatroid} \cite[4.8]{MR} The set
$L_{p_g}:=\{l\in L_{>0}\, : \, h^1(\cO_l)=p_g\}$ has a unique minimal element, denoted by $Z_{coh}$, and called
the {\it cohomological cycle} of $\phi$. It has the property that
$h^1(\cO_l)<p_g$
for any $l\not\geq Z_{coh}$ ($l>0$).
By the consequences of Theorem \ref{th:GR} we obtain that $Z_{coh}\leq \lfloor Z_K\rfloor_+$.
In fact, the proof of the above statement in \cite{MR} shows the following.
Let $l_1, l_2\in L_{>0}$ be effective cycles, set
$l=\min\{l_1, l_2\}$ and $\overline{l}=\max\{l_1,l_2\}$. Then
$h^1(\calO_{\overline{l}})+h^1(\calO_l)\geq h^1(\calO_{l_1})+h^1(\calO_{l_2})$. We will refer
to this inequality as the {\it `opposite' matroid rank inequality} of $h^1$.
In particular, for any $l\in L_{>0}$ we have $h^1(\calO_l)=h^1(\calO_{\min \{l,Z_{coh}\}})$.
\bekezdes\label{bek:LauferDual} \cite{Laufer72},
\cite[p. 1281]{Laufer77}
Following Laufer, we can identify the dual space $H^1(\tX,\cO_{\tX})^*$ with the space of global holomorphic
2-forms on $\tX\setminus E$ up to the subspace of those forms which can be extended
holomorphically over $\tX$:
$H^1(\tX,\cO_{\tX})^*\simeq
H^0(\tX\setminus E,\Omega^2_{\tX})/ H^0(\tX,\Omega^2_{\tX})$.
Here $H^0(\tX\setminus E,\Omega^2_{\tX})$ can be replaced by
$H^0(\tX,\Omega^2_{\tX}(Z))$ for
any $Z>0$ with $h^1(\cO_Z)=p_g$. Indeed, for any $Z>0$, from the exact sequence of sheaves
$0\to\Omega^2_{\tX}\to \Omega^2_{\tX}(Z)\to \calO_{Z}(Z+K_{\tX})\to 0$ (where $\Omega^2_{\tX}=\cO_{\tX}(K_{\tX})$)
and from the vanishing
$h^1(\Omega^2_{\tX})=0$ and Serre duality
\begin{equation}\label{eq:duality}
H^0(\Omega^2_{\tX}(Z))/H^0(\Omega^2_{\tX})=H^0(\calO_Z(Z+K_{\tX}))\simeq H^1(\calO_Z)^*.
\end{equation}
If $H^1(\calO_Z)\simeq H^1(\calO_{\tX})$ then the inclusion
$H^0(\Omega^2_{\tX}(Z))/H^0( \Omega^2_{\tX})\hookrightarrow
H^0(\tX\setminus E, \Omega^2_{\tX})/H^0(\Omega^2_{\tX})$
is an isomorphism.
\bekezdes \label{bek:San}
For each holomorphic function $f\in \cO_{X,o}$ consider
${\rm div}_E(f)\in \cS$, that part of the divisor of $f\circ \phi$ which is supported on $E$.
Then define $\cS_{an}\subset \calS$, the {\it analytic semigroup of $\phi$}, as
$\{{\rm div}_E(f)\,:\, f\in \cO_{X,o}\}$.
The monoid $\calS_{an}\setminus \{0\}$ has a unique minimal element
$Z_{max}$ (${\rm div}_E$ of the generic element of the maximal ideal of $\cO_{X,o}$).
It is called {\it the maximal ideal cycle} of $\phi$. One has $Z_{max}\geq Z_{min}>0$.
\subsection{The analytic lattice cohomology, independence of the choice of the rectangle}\label{ss:anR}
\bekezdes
Next we construct the {\it analytic lattice cohomology } $\bH^*_{an,0}(X,o)$
of a normal surface singularity $(X,o)$.
(Though we will mention in the sequel nothing
about the spin$^c$--structures,
this module in fact corresponds to the canonical spin$^c$--structure of the link.
If the link is a rational homology sphere, the construction of the
analytic lattice cohomologies $\{\bH^*_{an, h}(X,o)\}_{h}$
associated with all spin$^c$--structures will be
presented in the forthcoming article \cite{AgNeIII}.)
Let us fix a good resolution $\phi$.
For any $c\in L$, $c\geq Z_{coh}$, we consider the rectangle $R(0,c)=\{l\in L\,:\,
0\leq l\leq c\}$.
Here we might consider the $c=\infty$ case too, in this case $R(0,c)=L_{\geq 0}$.
Then we consider the multivariable Hilbert function
$$\hh:R(0,c)\to\Z, \ \ \hh(l)=\dim H^0(\calO_{\tX})/H^0(\calO_{\tX}(-l))$$
associated with the
divisorial filtration of $\calO_{X,o}$ and the resolution $\phi$, cf. \cite{CDG,CHR,NJEMS}.
Clearly $\hh$ is increasing (that is, $\hh(l_1)\geq \hh(l_2)$ whenever $l_1\geq l_2$) and $\hh(0)=0$.
Next, for any $l\in R(0,c)$, we consider the function
$\hh^\circ(l)=p_g-h^1(\calO_l)$ too (where, by definition, $h^1(\calO_{l=0})=0$). Then $\hh^\circ$ is decreasing,
$\hh^\circ (0)=p_g$ and $\hh^\circ (c)=0$, cf. \ref{bek:PG}. (Compare with the notations of section \ref{ss:CombLattice}.)
We consider the natural cube-decomposition of $R(0,c)$
(where the 0-cubes are the lattice points) and
the set of cubes $\{\calQ_q\}_{q\geq 0}$ of $R(0,c)$ as in \ref{9complex}.
Then we define the weight function
\begin{equation}\label{eq:wean}
w_0:\calQ_0\to \Z, \ \ \ w_0(l)=\hh(l)+\hh^\circ (l)-p_g=\hh(l)-h^1(\calO_l).
\end{equation}
Clearly, $w_0(0)=0$.
This weight function has several useful properties.
First of all, note that $0\leq \hh^\circ(l)\leq p_g$ for every $l$, hence when $c=\infty$ then
$\hh$ and $w_0$ have comparable asymptotic behaviour for $l\gg 0$.
For any $l\in L$ let $s(l)\in L$ be the smallest element with $s(l)\geq l$ and $s(l)\in\calS$. Then
$\hh(l)=\hh(s(l))$. Using this fact, the monotonicity of $\hh$, and
(\ref{eq:vanh0})
a computation shows that $w_0$ satisfies the requirement \ref{9weight}(a), namely,
$w_0^{-1}((\-\infty, n]) \ \ \mbox{is finite for any $n\in\Z$}$.
Next, since $\hh$ is induced by a filtration, it satisfies the
matroid rank inequality
$\hh(l_1)+\hh(l_2)\geq \hh(\overline{l})+\hh(l)$,
where $l=\min\{l_1, l_2\}$ and $\overline{l}=\max\{l_1,l_2\}$.
On the other hand, $h^1$ satisfies the `opposite' matroid rank inequality, see
\ref{rem:antmatroid}. Therefore, $w_0$ itself satisfies the matroid rank inequality (where $l_1,l_2\geq 0$)
\begin{equation}\label{eq:matroidw00}
w_0(l_1)+w_0(l_2)\geq w_0(\overline{l})+w_0(l).
\end{equation}
(As a topological comparison: $\chi$, the topological weight function, also satisfies a similar inequality.)
Furthermore, similarly as in \ref{9dEF1}, we define
$w_q:\calQ_q\to \Z$ by $ w_q(\square_q)=\max\{w_0(l)\,:\, l \
\mbox{\,is any vertex of $\square_q$}\}$.
In the sequel we write $w$ for the system $\{w_q\}_q$ if there is no confusion.
These weight functions $\{w_q\}_q$ define the lattice cohomology $\bH^*(R(0,c),w)$ and the graded root
$\RR(R(0,c),w)$
associated with $R(0,c)$ and $w$.
\begin{lemma}\label{lem:INDEPAN}
$\bH^*(R(0,c),w)$ and $\RR(R(0,c),w)$ are independent of the choice of $c\geq Z_{coh}$.
\end{lemma}
\begin{proof}
Fix some $c\geq Z_{coh}$ and choose $E_v\subset |c- Z_{coh}|$.
Then for any $l\in R(0,c)$ with
$l_v=c_v$ we have $\min\{l, Z_{coh}\}=\min\{l-E_v, Z_{coh}\}$.
Therefore, by \ref{rem:antmatroid}, $h^1(\calO_{l-E_v})=h^1(\calO_l)$, thus
$w_0(l-E_v)\leq w_0(l)$. Then for any $n\in \Z$, a strong deformation retract in the direction $E_v$ realizes
a homotopy equivalence between the spaces $S_n\cap R(0,c)$ and $S_n\cap R(0,c-E_v)$.
A retract $r:S_n\cap R(0,c)\to S_n\cap R(0,c-E_v)$ can be defined as follows (for notation see \ref{9complex}).
If $\square =(l,I)$ belongs to $ S_n\cap R(0,c-E_v)$ then $r$ on $\square$ is defined as the identity.
If $(l,I)\cap R(0,c-E_v)=\emptyset$, then $l_v=c_v$, and we set $r(x)=x-E_v$. Else,
$\square =(l,I)$ satisfies $v\in I$ and $l_v=c_v-1$. Then we retract $(l,I)$ to $(l, I\setminus v)$ in the $v$--direction.
The strong deformation retract is defined similarly.
\end{proof}
\begin{corollary}\label{cor:veges}
(a) The graded root $\RR(R(0,c),w)$ satisfies $|\chic^{-1}(n)|=1$ for any $n\gg 0$.
(b) $\bH^*_{red}(R(0,c),w)$ is a finitely generated $\Z$-module (for any finite or infinite $c\geq Z_{coh}$).
\end{corollary}
\begin{proof}
For any $n\gg 0$ we have $R(0,c)=S_n$, hence $S_n$ is contractible for such $n$.
\end{proof}
\subsection{The analytic lattice cohomology of $(X,o)$, independence of $\phi$}\label{ss:anphi}
\bekezdes Let us abridge $\bH^*(R(0,c),w)$ as $\bH^*_{an}(\phi)$ and
$\RR(R(0,c),w)$ as $\RR_{an}(\phi)$.
\begin{theorem}\label{th:annlattinda} Assume that the resolution graph is a tree (a property independent of the
resolution). Then
$\bH^*_{an}(\phi)$ and $\RR_{an}(\phi)$ are independent of the choice of the resolution $\phi$.
\end{theorem}
(Later in \ref{bek:ADD} we will drop the assumption regarding the graph.)
\begin{proof}
Let us fix a resolution $\phi$, and denote the blow up of a point of $E_{v_0}\setminus \cup_{w\not= v_0}E_w$
by $\pi$, and set $\phi':= \phi\circ \pi$.
Let $\Gamma$ and $\Gamma'$ be the corresponding graphs, $L(\Gamma),\ L(\Gamma')$ the lattices
and $\, (\,,\,), \ (\,,\,)'$ the intersection forms.
We denote the new
$(-1)$-vertex of $\Gamma'$ by $E_{new}$. We use the same notation for $E_v\in L$ and for
its strict transform in $L'$. We have the natural morphisms:
$\pi_*:L(\Gamma')\to
L(\Gamma)$ defined by $\pi_*(\sum x_vE_v+x_{new}E_{new})=\sum
x_vE_v$, and $\pi^*:L(\Gamma)\to L(\Gamma') $ defined by
$\pi^*(\sum x_vE_v)=\sum x_vE_v +x_{v_0}E_{new}$. Then
$(\pi^*x,x')'=(x,\pi_*x')$. Thus
$(\pi^*x,\pi^*y)'=(x,y)$ and $(\pi^*x,E_{new})'=0$ for any
$x,y\in L(\Gamma)$.
Associated with $\phi$, let $\hh$ be the
Hilbert function, $w_0$ the analytic weight and
$S_n(\phi)=\cup\{\square\,:\, w(\square)\leq n\}$. We use similar notations $\hh'$, $w_0'$ and
$S_n(\phi')$ for $\phi'$.
Note that for any $x\in R$,
$H^0(\tX', \calO_{\tX'}(-\pi^*x))=H^0(\tX, \calO_{\tX}(-x))$.
Even more, for $a\leq 0$ and $x$ as above,
$H^0(\tX', \calO_{\tX'}(-\pi^*x-aE_{new}))=H^0(\tX', \calO_{\tX'}(-\pi^*x))$.
Indeed, take the exact sequence of sheaves
$$0\to \calO_{\tX'}(-\pi^*x)\to \calO_{\tX'}(-\pi^*x-aE_{new})\to \calO_{-aE_{new}}(-\pi^*x-aE_{new})\to 0$$
and use that $h^0(\calO_l(l)\otimes \calL)=0$ for any $l>0$ and line bundle $\calL$ with
$(c_1\calL,E_v)=0$ for any $E_v\in |l|$.
This last vanishing follows from the Grauert--Riemenschneider Theorem via Serre duality.
Therefore,
\begin{equation}\label{eq:MON1}
\hh'(\pi^*x+aE_{new}) \ \left\{ \begin{array}{l}
= \hh(x) \ \mbox{ for any $a\leq 0$} \\
\mbox{is increasing for $a\geq 0$}.\end{array}\right.
\end{equation}
Using the exact sequence $0\to \calO_{aE_{new}}(-\pi^*x)\to \calO_{\pi^*x+aE_{new}}\to \calO_{\pi^*x}\to 0$ and
Lipman's vanishing $h^1(\calO_{aE_{new}}(-\pi^*x))=0$ from \ref{prop:VAN0}, we get that
$h^1(\calO_{\pi^*x+aE_{new}})=h^1(\calO_{\pi^*x})$ for any $a\geq 0$. Furthermore, from
$0\to \calO_{E_{new}}(-\pi^*x +E_{new})\to \calO_{\pi^*x}\to \calO_{\pi^*x-E_{new}}\to 0$
we get that $h^1(\calO_{\pi^*x-E_{new}})=h^1(\calO_{\pi^*x})$ too. On the other hand (by Leray spectral sequence),
$h^1(\tX',\calO_{\pi^*x})=h^1(\tX,\calO_x)$. Therefore,
\begin{equation}\label{eq:MON2}
h^1(\calO_{\pi^*x+aE_{new}}) \ \left\{ \begin{array}{l}
\mbox{is increasing for $a\leq -1$}, \\
= h^1(\calO_x) \ \mbox{ for any $a\geq -1$}. \\
\end{array}\right.
\end{equation}
These combined provide
\begin{equation}\label{eq:HOM2a}
a\mapsto w_0'(\pi^*x+aE_{new}) \ \left\{ \begin{array}{l}
\mbox{is decreasing for $a\leq -1$},\\
= w_0(x) \ \mbox{ for $a= -1$ and $a=0$,} \\
\mbox{is increasing for $a\geq 0$}.\end{array}\right.
\end{equation}
Recall that we can compute $\bH^*_{an}(\phi)$ using the cube
$R(0,c)$ with $c\geq Z_{coh}(\phi)$, cf. \ref{lem:INDEPAN}. But then $\pi^*c\geq Z_{coh}(\phi')$,
hence $\bH^*_{an}(\phi')$ can be computed in $R(0, \pi^*c)$,
and $\pi^*$ sends the lattice points of $R(0,c)$ into $R(0, \pi^*c)$.
Furthermore, if $w_0'(\pi^*x+aE_{new})\leq n$, then
$w_0(x)\leq n$ too. In particular, the projection $\pi_{\R}$ in the direction of $E_{new}$
induces a well-defined map $\pi_{\R}:S_n(\phi')\to S_n(\phi)$.
We claim that (whenever $S_n(\phi)$ is non-empty)
$\pi_{\R}$ is a homotopy equivalence (with all fibers non-empty and contractible).
\bekezdes \label{bek:proof1}
We proceed in two steps.
First we prove that $\pi_{\R}:S_n(\phi')\to S_n(\phi)$ is onto.
Consider a zero dimensional cube (i.e. lattice point) $x\in S_n(\phi)$. Then $w_0(x)\leq n$. But then $w_0'(\pi^*x)
=w_0(x)\leq n$ too, hence $\pi^*(x)\in S_n(\phi')$ and $x=\pi_{\R}(\pi^*x)\in {\rm im}(\pi_{\R})$.
Next take a cube $(x,I)\subset S_n(\phi)$ ($I\subset \cV$). This means that
$w_0(x+E_{I'})\leq n$ for any $I'\subset I$. But then
\begin{equation}\label{eq:eps}
\pi^*(x+E_{I'})=\pi^*x+ E_{I'}+\epsilon\cdot E_{new},
\end{equation}
where $\epsilon=0$ if $v_0\not \in I'$ and $\epsilon =1$ otherwise. Hence
\begin{equation}\label{eq:eps2}
w_0'(\pi^*x+E_{I'}) =w_0'(\pi^*(x+E_{I'})-\epsilon E_{new})\stackrel {(\ref{eq:HOM2a})}{=}
w_0(x+E_{I'})\leq n.
\end{equation}
Therefore $(\pi^*x,I)\in S_n(\phi')$ and $\pi_{\R}$ projects $(\pi^*x, I)$ isomorphically onto $(x,I)$.
Next, we show that $\pi_{\R}$ is in fact a homotopy equivalence.
In order to prove this fact it is enough to verify that if
$\square\in S_{n}(\phi)$ and $\square ^\circ$ denotes its relative interior,
then $\pi_{\R}^{-1}(\square^\circ) \cap S_{n}(\phi')$ is contractible.
Let us start again with a lattice point $x\in S_n(\phi)$. Then $\pi_{\R}^{-1}(x)\cap S_n(\phi')$ is a real interval
(whose endpoints are lattice points, considered in the real line of the $E_{new}$ coordinate).
Let us denote it by $\cI(x)$. Now, if $\square=(x,I)$, then we have to show that
all the intervals $\cI(x+E_{I'})$ associated with all the subsets $I'\subset I$ have a common
lattice point. But this is exactly what we verified above: the $E_{new}$ coordinate of $\pi^*(x)$ is such a common
point. Therefore, $\pi_{\R}^{-1}(\square ^\circ)\cap S_n(\phi')$
has a strong deformation retraction (in the $E_{new}$ direction)
to the contractible space $(\pi^*x, I)^\circ$.
For any $l\in L$ let $N(l)\subset \R^s$ denote the union of all cubes which have $l$ as one of their vertices.
Let $U(l)$ be its interior. Write $U_n(l):=U(l)\cap S_n(\phi)$. If $l\in S_n(\phi)$ then $U_n(l)$ is a contractible
neighbourhood of $l$ in $S_n(\phi)$. Also, $S_n(\phi)$ is covered by $\{U_n(l)\}_l$.
Moreover, $\pi_{\R}^{-1}(U_n(l))$ has the homotopy type of $\pi_{\R}^{-1}(l)$, hence it is contractible.
More generally, for any cube $\square$,
$$\pi_{\R}^{-1}(\cap _{\mbox{$v$ vertex of $\square$}} U_n(l)) \sim \pi_{\R}^{-1}(\square ^\circ)$$
which is contractible by the above discussion. Since all the intersections of $U_n(l)$'s are of these type,
we get that the inverse image of any intersection is contractible. Hence by \v{C}ech covering
(or Leray spectral sequence) argument, $\pi_{\R}$
induces an isomorphism
$H^*(S_n(\phi'),\Z)=H^*(S_n(\phi),\Z)$. In fact, this already shows that
$\bH^*_{an}(\phi')=\bH^*_{an}(\phi)$ and $\RR_{an}(\phi')=\RR_{an}(\phi)$.
The fact that these identifications preserve the $U$--action follows from the natural inclusions of the spaces $S_n$.
In order to prove the homotopy equivalence,
one can use quasifibration, defined in \cite{DoldThom};
see also \cite{DadNem}, e.g. the relevant Theorem 6.1.5.
Since
$\pi_{\R}:S_{n}(\phi')\to S_{n}(\phi)$ is a quasifibration,
and all the fibers are contractible, the homotopy equivalence follows.
\bekezdes \label{bek:proof2}
The case when we blow up an intersection point $E_{v_0}\cap E_{v_1}$ starts very similarly, however at some point
there is a major difference, hence we need an additional argument.
Below we write $E_J:=\sum_{v\in J}E_v$ for any subset $J\subset \calv$.
With very similar notations, in this case we define $\pi^*(\sum_vx_vE_v)=\sum_v x_vE_v+(x_{v_0}+
x_{v_1})E_{new}$. Then all the statements till \ref{bek:proof1} (in fact, till (\ref{eq:eps}))
remain valid (including
the key (\ref{eq:HOM2a})).
However, the first part of \ref{bek:proof1} should be modified.
The first difference is in (\ref{eq:eps}). Indeed, in this case
\begin{equation}\label{eq:eps3}
\pi^*(x+E_{I'})=\pi^*x+ E_{I'}+\epsilon\cdot E_{new},
\end{equation}
where $\epsilon$ is the cardinality of $I'\cap \{v_0,v_1\}$. This can be 0, 1 or 2. Therefore, if
$\{v_0,v_1\}\not\subset I$, then $\epsilon \in \{0,1\} $ for any $I'$, hence for such cubes $(x,I)$
all the arguments of \ref{bek:proof1} work.
Assume in the sequel that $\{v_0, v_1\}\subset I$. Write $J=I\setminus \{v_0,v_1\}$.
There are two cube candidates of $L(\Gamma')\otimes \R$ which might
cover the cube $(x,I)\in S_n(\phi)$. One of them is $(\pi^*x,I)$ (as above). However, by (\ref{eq:HOM2a})
the lattice points $\pi^*(x+E_I)=\pi^*x+E_I+2E_{new}$ and $\pi^*(x+E_I)-E_{new}=\pi^*x+E_I+E_{new}$ are in
$S_n(\phi')$, but the vertex $\pi^*(x)+E_I$ of $(\pi^*x,I)$ might not be in $S_n(\phi')$.
Another candidate is $(\pi^*x+E_{new},I)$, but here again
$\pi^*x$ and $\pi^*x-E_{new}$ are in $S_n(\phi')$ but the vertex $\pi^*x+E_{new}$ of
$(\pi^*x+E_{new},I)$ might be not.
So both cubes a priori are obstructed if we apply (\ref{eq:HOM2a}) only.
Next we analyze these obstructions in more detail and we show that one of the candidate cubes works.
\bekezdes\label{bek:A} {\bf Case 1.}
Assume that $w_0'(\pi^*x)=w_0'(\pi^*x+E_{new})$. Then by (\ref{eq:MON1}) and (\ref{eq:MON2})
we obtain that $\hh'(\pi^*x)=\hh'(\pi^*x+E_{new})$. By the matroid inequality of $\hh'$
we get that $\hh'(\pi^*x+E_{J'})=\hh'(\pi^*x+E_{J'}+E_{new})$ for any $J'\subset J$.
This again via (\ref{eq:MON1}) and (\ref{eq:MON2})
shows that $w_0'(\pi^*x+E_{J'})=w_0'(\pi^*x+E_{J'}+E_{new})$.
In particular,
$$w_0'(\pi^*x+E_{J'}+E_{new})=w_0'(\pi^*x+E_{J'})=w_0'(\pi^*(x+E_{J'}))=w_0(x+E_{J'})\leq n.$$
That is, the vertices of type $\pi^*x+E_{J'}+E_{new}$ of $(\pi^*x+E_{new},I)$
are in $S_n(\phi')$. For all other vertices we already know this fact (use (\ref{eq:HOM2a})).
Hence $(\pi^*x+E_{new},I)$ is in $S_n(\phi')$ and it projects via $\pi_{\R}$ bijectively to $(x,I)$.
Furthermore, $\pi_{\R}^{-1}(x,I)^\circ \cap S_n(\phi')$ admits a deformation retract to
$(\pi^*x+E_{new},I)^\circ $, hence it is contractible.
\bekezdes \label{bek:B} {\bf Case 2.}
Assume that $w_0'(\pi^*x+E_I)=w_0'(\pi^*x+E_I+E_{new})$,
or $w_0'(\pi^*(x+E_I)-2E_{new})=w_0'(\pi^*(x+E_I)-E_{new})$.
Then by (\ref{eq:MON1}) and (\ref{eq:MON2})
we obtain that $h^1(\calO_{\pi^*x+E_I})=h^1(\calO_{\pi^*x+E_I+E_{new}})$. By the opposite
matroid inequality of $h^1$ and (\ref{eq:MON1}) and (\ref{eq:MON2}) again we obtain that
$w_0'(\pi^*x+E_I-E_{J'})=w_0'(\pi^*x+E_I-E_{J'}+E_{new})$.
In particular,
$$w_0'(\pi^*x+E_I-E_{J'})=w_0'(\pi^*x+E_I-E_{J'}+E_{new})=w_0'(\pi^*(x+E_I-E_{J'})-E_{new})=w_0(x+E_I-E_{J'})\leq n.$$
That is, the vertices of type $\pi^*x+E_I-E_{J'}$ of $(\pi^*x,I)$
are in $S_n(\phi')$. For all other vertices we already know this fact (use (\ref{eq:HOM2a})).
Hence $(\pi^*x,I)$ is in $S_n(\phi')$ and it projects via $\pi_{\R}$ bijectively to $(x,I)$.
Furthermore, $\pi_{\R}^{-1}(x,I)^\circ \cap S_n(\phi')$ admits a deformation retract to
$(\pi^*x,I)^\circ $, hence it is contractible.
\bekezdes \label{bek:C} {\bf Case 3.} Assume that the assumptions from {\bf Case 1} and {\bf Case 2} do not hold.
This means that
$$\left\{ \begin{array}{l}
\hh'(\pi^*x)<\hh'(\pi^*x+E_{new}), \ \mbox{and} \\
h^1(\calO_{\pi^*x+E_I})< h^1(\calO_{\pi^*x +E_I+E_{new}}).\end{array}\right.$$
This reads as follows (cf. (\ref{eq:duality})
$$\left\{ \begin{array}{l}
(a) \ \ H^0(\calO_{\tX'}(-\pi^*x-E_{new}) \subsetneq H^0(\calO_{\tX'}(-\pi^*x)), \ \mbox{and} \\
(b) \ \ H^0(\tX', \Omega^2_{\tX'}(\pi^* x +E_I)) \subsetneq
H^0(\tX', \Omega^2_{\tX'}(\pi^* x +E_I+E_{new})).
\end{array}\right.$$
Part {\it (a)} means the following: there exists a function $f\in H^0(\tX', \calO_{\tX'})$
such that ${\rm div}_{E'}(f)\geq \pi^*x$, and in this inequality the $E_{new}$--coordinate entries are equal.
By part {\it (b)}, there exists a global 2--form $\omega $ such that
${\rm div}_{E'}(\omega)\geq -\pi^*x-E_I-E_{new}$ and the $E_{new}$--coordinate entries are equal.
Therefore, the form $f\omega\in H^0(\tX'\setminus E', \Omega^2_{\tX'})$ has the property that
${\rm div}_{E'}(f\omega)\geq -E_I-E_{new}$ with equality at the $E_{new}$ coordinate.
In particular, again by duality (\ref{eq:duality}), we obtain that in $\tX'$ the following
strict inequality holds:
\begin{equation}\label{eq:ROSSZ}
h^1(\calO_{E_I+E_{new}})>h^1(\calO_{E_I}) \ \ (\cV'=\cV\cup\{new\}, \ I\subset \cV).\end{equation}
But if the graph is a tree then this strict inequality cannot happen.
\bekezdes In particular, for any $I\subset \cV$ either $\{v_0,v_1\}\not\subset I$, or in the opposite case
either {\bf Case 1} or {\bf Case 2} applies. Hence,
in any case,
$\pi_{\R}^{-1}(x,I)^\circ \cap S_n(\phi')$ is contractible. Therefore, $S_n(\phi)$ and $S_n(\phi')$
have the same homotopy type by the argument from the end of \ref{bek:proof1}.
\end{proof}
\bekezdes\label{bek:ADD} {\bf Addendum to Theorem \ref{th:annlattinda}.}
In the proof of Theorem \ref{th:annlattinda} the assumption (namely that $\Gamma$ should be a tree)
was used only to show that (\ref{eq:ROSSZ}) cannot hold.
However, the failure of (\ref{eq:ROSSZ}) can be guaranteed in other way as well.
Let $\Gamma$ be a resolution graph, which is not a tree. If $\lambda$ is a (simple) closed 1--cycle
in (the topological realization of) $\Gamma$ then let $\ell(\lambda)$ be its combinatorial length
(number of edges in it).
Furthermore, in general, if $V\subset \calv$ ($V\not=\emptyset$)
then $h^1(\calO_{E_V})=\sum_{v\in V}g_v+b_1(|E_V|)$, where $b_1(|E_V|)$ denotes the number of independent
closed 1--cycles of the support $|E_V|$.
Hence, (\ref{eq:ROSSZ}) can happen in $\Gamma'$ if and only if $I$ considered in $\calv$
contains a closed 1--cycle $\lambda$ in $\Gamma$, $\lambda$ has vertices $\{v_0,v_1\}\cup J$,
its lift $\lambda'$ to $\Gamma'$
has vertices $\{v_0,v_1\}\cup J\cup\{new\}$, hence this cycle $\lambda'$
is broken if we delete the new vertex $\{new\}$ of $\Gamma'$.
Thus, $|I|\geq \ell(\lambda)$.
Let $\ell_{min}(\phi)$ be the smallest $\ell(\lambda)$, where $\lambda$ is a closed 1--cycle of $\Gamma$.
Since $\phi$ is a good resolution, hence no $E_v$ has self-intersection, $\ell_{min}(\phi)\geq 2$.
The above discussion shows that for any $|I|<\ell_{min}(\phi)$ (\ref{eq:ROSSZ}) cannot hold.
Hence, any $q$--cube with $q<\ell_{min}(\phi)$ can be lifted (as in the proof), it is in the image of
$\pi_{\R}$, and its inverse image is contractible. Note also that the $ \pi_{\R}$--image of
any cube from $S_n(\phi')$ is in $S_n(\phi)$. Hence $S_n(\phi)$ is obtained from ${\rm im}(\pi_{\R})$
by adding cubes of dimension $\geq \ell_{min}(\phi)$ and
${\rm im}(\pi_{\R})$ has the homotopy type of
$S_n(\phi')$. In particular, $\bH^{\leq (\ell_{min}(\phi)-2)}_{an}(\phi)=\bH^{\leq (\ell_{min}(\phi)-2)}_{an}(\phi')$.
Since $\ell_{min}(\phi)\geq 2$ we obtain that in any case
$\bH^0_{an}(\phi)=\bH^0_{an}(\phi')$ and $\RR_{an}(\phi)=\RR_{an}(\phi)$; hence these objects are well
defined without any restriction regarding the link.
Now we concentrate on $\bH^q_{an}$ with fixed $q$. Let us consider two resolutions $\phi_1$ and $\phi_2$ such that
$\ell_{min}(\phi_1), \ell_{min}(\phi_2)\geq q+2$. Let $\phi_{12}$ be the resoltuion which dominates both $\phi_1$ and $\phi_2$. Then along the sequence of resolutions which connects $\phi_1$ and $\phi_2$
via $\phi_{12}$ the module $\bH^q_{an}$ is stable by the above discussion.
In particular, $\bH^q_{an}(\phi)$ is well--defined, whenever it is computed in a resolution
$\phi$ with $\ell_{min}(\phi)\geq q+2$.
\begin{definition} Assume that $\Gamma$ is a tree. Then $\bH^*_{an}(\phi)$ and $\RR_{an}(\phi)$
are independent of $\phi$.
In the sequel we will use the notations $\bH^*_{an,0}(X,o)$ and $\RR_{an,0}(X,o)$ for them.
(The index `zero' means `canonical spin$^c$--structure.)
They are called the {\it analytic lattice cohomology of $(X,o)$} and the {\it
analytic graded root of $(X,o)$} (associated with the canonical spin$^c$--structure).
\end{definition}
They are analytic invariant of the germ $(X,o)$. By Corollary \ref{cor:veges}
$\bH^*_{an,red,0}(X,o)$ has finite $\Z$--rank, hence the Euler characteristic $eu(\bH^*_{an,0}(X,o))$ is well--defined.
\subsection{The `Combinatorial Duality Property' of the pair $(\hh, \hh^\circ)$}\label{ss:anCDP}
\bekezdes
The following reinterpretation of $\hh^\circ:R(0,c)\to \Z$ will be helpful.
From \ref{bek:LauferDual} we have that
$\dim\, H^0(\calO_{\tX}(K_{\tX}+Z))/
H^0( \calO_{\tX}(K_{\tX}))=h^1(\calO_{Z})$
for any $Z>0 $.
On the other hand, by the opposite matroid rank inequality \ref{rem:antmatroid}
we also have $h^1(\cO_Z)=h^1(\cO_{\min\{Z,\lfloor Z_K\rfloor_+\}})$.
Hence, $\hh^\circ(l)=p_g-h^1(\calO_l)=h^1(\calO_c)-h^1(\calO_l)$ appears as
\begin{equation}\label{eq:reinter}
\hh^\circ(l):= \dim\, \frac{H^0(\tX, \calO_{\tX}(K_{\tX}+c))}{
H^0(\tX, \calO_{\tX}(K_{\tX}+l))}=
\dim\, \frac{H^0(\tX, \calO_{\tX}(K_{\tX}+\lfloor Z_K\rfloor_+))}{
H^0(\tX, \calO_{\tX}(K_{\tX}+\min\{l, \lfloor Z_K\rfloor_+\} ))}
.\end{equation}
(In (\ref{eq:reinter}) $\lfloor Z_{K}\rfloor _+$ can be replaced by $Z_{coh}$ as well.)
\begin{example} \label{ex:GorAN}
{\bf Gorenstein germs.}
Assume that $(X,o)$ is Gorenstein. Then $Z_K\in L$, and let us assume (for simplicity) that
$Z_K\geq 0$ (this happens e.g. in the good minimal resolution, cf. \cite{Book}).
Then for any $l\in R(0, Z_K)$, by (\ref{eq:reinter}) we have $\hh^\circ (l)=
\dim\, H^0(\calO_{\tX})/ H^0(\calO_{\tX}(-Z_K+l))=\hh(Z_K-l)$.
Therefore, $w_0(l)=\hh(l)+\hh(Z_K-l)-p_g$
is obtained as the {\it symmetrization} of $\hh$.
In particular $w_0(l)=w_0(Z_K-l)$. Note that this is true for the topological weight function too:
$\chi(l)=\chi(Z_K-l)$. However, in the analytic case, the symmetry might fail for non-Gorenstein germs
(even if we consider a numerically Gorenstein topological type).
\end{example}
The following property will be crucial in the Euler characteristic computation.
\begin{lemma}\label{lem:hsimult} {\bf (CDP)} Assume that $g_v=0$ for any $v\in\calv$. Then
there exists no $l\in L_{\geq 0}$ and $v\in\calv$ such that the differences
$\hh(l+E_v)-\hh(l)$ and $\hh^\circ (l)-\hh^\circ (l+E_v)$ are simultaneously strictly positive.
\end{lemma}
\begin{proof}
If $\hh(l+E_v)>\hh(l)$ then there exists a global function $f\in H^0(\calO_{\tX})$ with ${\rm div}_Ef\geq l$, where the
$E_v$-coordinate is $({\rm div}_Ef)_v = l_v$.
Similarly, if $\hh^\circ (l)>\hh^\circ(l+E_v)$ then there exists a
global 2-form $\omega$ with possible poles along $E$,
with ${\rm div}_E\omega \geq -l-E_v$ (i.e., the pole order is $\leq l+E_v$), and
$({\rm div}_E\omega )_v = -l_v-1$.
In particular, the form $f\omega$
satisfies
${\rm div}_E\ f\omega \geq -E_v$ and
$({\rm div}_E f \omega )_v = -1$. This implies $H^0(\Omega_{\tX}^2(E_v))/H^0(\Omega_{\tX}^2)\not=0$, or,
by (\ref{bek:LauferDual}), $h^1(\calO_{E_v})\not=0$.
\end{proof}
\bekezdes
In the next discussions we will assume that the link of $(X,o)$ is a rational homology sphere
(hence the graph in particular is a tree).
We show that $\bH^*_{an,0}(X,o)$ usually is non-trivial, its Euler characteristic is $p_g$,
it can be `compared' with the topological
lattice cohomology $\bH^*_{top,0}(M)$ of the link, and it really sees the variation of different
analytic structures supported on a fixed topological type.
It is worth to compare
several statements below with their topological analogues valid for
$\bH^*_{top,0}(M)$. (Recall that $\bH^*_{top,0}(M)$ was defined whenever the link is a $\Q HS^3$.)
\subsection{The Euler characteristic $eu(\bH^*_{an}(X,o))$}\label{ss:anEu}
\bekezdes Lemma \ref{lem:hsimult} will allow us to determine the Euler characteristic $eu(\bH^*_{an}(X,o))$
of the analytic lattice cohomology by a combinatorial argument. Surprisingly, this Euler characteristic automatically
equals the Euler characteristic of path cohomologies associated with any increasing path
(this equality definitely does not hold
in the topological versions of the corresponding lattice cohomologies).
First, let us fix the notations.
In the sequel
we will also consider
for any increasing path $\gamma$ connecting 0 and $c$
(that is, $\gamma=\{x_i\}_{i=0}^t$, $x_{i+1}=x_i+E_{v(i)}$, $x_0=0$ and $x_t=c$, $c\geq Z_{coh}$)
the path lattice cohomology $\bH^0(\gamma,w)$ as in \ref{bek:pathlatticecoh}.
Accordingly, we have the numerical
Euler characteristic
$eu(\bH^0(\gamma,w))$ as well.
The proof of the next theorem basically is combinatorial,
it is provided in the next subsection, where we separated certain combinatorial aspects of the
lattice cohomology, cf. Theorem \ref{th:comblattice}.
\begin{theorem}\label{th:euANLAT} Assume that the link is a $\Q HS^3$. Then
$eu(\bH^*_{an,0}(X,o))=p_g(X,o)$. Furthermore, for any increasing path $\gamma$ connecting 0 and $c$ (where
$c\geq Z_{coh}$)
we also have $eu(\bH^*(\gamma,w))=p_g$.
\end{theorem}
\begin{proof}
We claim that the assumptions of Theorem \ref{th:comblattice} are satisfied. Indeed,
the CDP was verified in \ref{lem:hsimult}, while the stability property of $\hh$ follows
since it is associated with a filtration.
\end{proof}
This in particular means that $\bH^*_{an,0}(X,o)$ is a {\it categorification of the geometric genus},
that is, it is a graded cohomology module whose Euler characteristic is $p_g$.
For more comments regarding Theorem \ref{th:euANLAT} see Remark \ref{rem:UJ}.
\subsection{Analytic reduction theorem}\label{ss:anRT}
\bekezdes
Our next goal is to prove a `Reduction Theorem', the analogue of the topological Theorem
\ref{th:red}.
Via such a result, the rectangle $R=R(0,c)$ can be replaced by another rectangle sitting in a
lattice of smaller rank. The procedure starts with identification of a set of `bad' vertices,
see \ref{ss:BadVer}. More precisely, we decompose $\calv$ as a disjoint union
$\overline{\calv}\sqcup \calv^*$, where the vertices $\overline {\calv}$ are the `essential' ones, the ones which dominate the others, and the coordinates $\calv^*$ are those which `can be eliminated'.
In the topological context the possible choice of $\overline{\calv}$ was dictated by combinatorial
properties of $\chi$ with a special focus on the topological characterization of rational germs.
In the present context we start with certain analytic properties of 2-forms (which reflects the dominance
of $\overline{\calv}$ over $\calv^*$).
(Note that $p_g=0$ if and only if $H^0(\tX\setminus E, \Omega^2_{\tX})=H^0(\tX, \Omega^2_{\tX})$.)
In this section we assume that the link is a rational homology sphere.
\begin{definition}\label{def:DOMAN}
We say that $\overline{\calv}$ is an B$_{an}$--set
if it satisfy the following
property: if
some differential form
$\omega\in H^0(\tX\setminus E, \Omega_{\tX}^2) $ satisfies
$({\rm div}_E\omega) |_{\overline{\calv}}\geq -E_{\overline{\calv}}$ \
then necessarily
$\omega\in H^0(\tX, \Omega_{\tX}^2)$.
By (\ref{bek:LauferDual}) this is equivalent with the vanishing $h^1(\calO_Z)=0$ for any
$Z=E_{\overline{\calv}}+l^*$, where $l^*\geq 0$ and it is supported on $\calv^*$.
\end{definition}
\begin{lemma}\label{lem:AnNodes}
If $\overline{\calv}$ is a B--set, then it is a B$_{an}$--set too.
(For the definition of B--sets see \ref{def:SWrat}.)
\end{lemma}
\begin{proof}
Let $\Gamma$ be the original graph, and let $\Gamma'$ be that rational graph which is obtained from $\Gamma$ be decreasing
the numbers $\{E_v^2\}_{v\in\overline{\calv}}$. Let $(\,,\,)'$ be its intersection form and $\chi'$ the RR expression.
Fix a cycle of type $Z=E_{\overline{\calv}}+l^*$ with $|l^*|\subset \calv^*$, $l^*\geq 0$. Then, regarding $\Gamma'$, we
claim the following: there exists a sequence $\{x_i\}_{i=0}^t$ such that
\begin{equation}\label{eq:SEQ}
x_0=0, \ x_t=Z, \ x_{i+1}=x_i+E_{v(i)}, \ (x_i, E_{v(i)})'\leq 1 \ \mbox{at every step $i$}.
\end{equation}
We construct the elements $x_i$ by decreasing order. Assume that $x_i$ is already constructed. Then, there exists at least one
$E_u\subset |x_i|$ such that $\chi'(x_i-E_u)\leq \chi'(x_i)$, equivalently $(\dag)$ $(x_i-E_u, E_u)'\leq 1$.
[Indeed, if not, then $(x_i, E_u)'\geq (Z'_K, E_u)'$ for every $E_u\subset |x_i|$, hence by summation
$\chi'(x_i)\leq 0$, a fact which contradicts the rationality of $\Gamma'$.] Then set $x_{i-1}:=x_i-E_u$. Note that
$(x_{i-1},E_u)'=(x_i-E_u,E_u)'\leq 1$ by ($\dag$). Hence the existence of $\{x_i\}_i$ follows.
Now, since $E_{\overline{\calv}}$ is reduced, and the self-intersections from the support of $l^*$ are not modified,
along these sequence $(x_i, E_{v(i)})'=(x_i, E_{v(i)})$. In particular, this sequence has the very same properties (\ref{eq:SEQ})
in $\Gamma$ too.
Then, using the exact sequences $0\to \calO_{E_{v(i)}}(-x_i)\to \calO_{x_{i+1}}\to \calO_{x_i}\to 0$
we get that $h^1(\calO_{x_{i+1}})=h^1(\calO_{x_i})$. Since $h^1(\calO_{x_0})=0$ by induction $h^1(\calO_{x_t})=0$ too.
\end{proof}
\begin{example}\label{ex:Ran} By the above lemma,
the set $\overline{\calv}={\mathcal N}$ of nodes is an B$_{an}$--set.
Moreover, if $\{\overline{v}\}$ is an B--set of an AR graph, then it is an B$_{an}$--set as well.
\end{example}
\bekezdes Associated with a disjoint decomposition $\calv=\overline{\calv}\sqcup \calv^*$, we write
any $l\in L$
as $\overline{l}+l^*$, or $(\overline{l}, l^*)$,
where $\overline {l}$ and $l^*$ are supported on $\overline{\calv}$ and
$\calv^*$ respectively. We also write $\overline{R}$ for the rectangle $R(0, \overline{c})$, the $\overline{\calv}$-projection of $R(0,c)$ with $c=Z_{coh}$.
For any $\overline {l}\in \overline {R}$ define the weight function
$$\overline{w}_0(\overline{l})=\hh(\overline{l})+\hh^\circ (\overline{l}+c^*)-p_g
=\hh(\overline{l})-h^1(\calO_{\overline{l}+c^*}).$$
Consider all the cubes of $\overline{R}$ and the weight function
$\overline{w}_q:\calQ_q(\overline{R})\to \Z$ by $ \overline{w}_q(\square_q)=\max\{w_0(\overline{l})\,:\, \overline{l} \
\mbox{\,is any vertex of $\square_q$}\}$.
\begin{theorem}\label{th:REDAN} {\bf Reduction theorem for the analytic lattice cohomology.}
If $\overline{\calv}$ is an B$_{an}$-set
then
$$\bH^*_{an}(R,w)=\bH^*_{an}(\overline{R}, \overline{w}).$$
\end{theorem}
\begin{proof}
For any $\cali\subset \calv$ write $c_\cali$ for the $\cali$-projection of $c=Z_{coh}$.
We proceed by induction, the proof will be given in $|\calv^*|$ steps.
For any $\overline{\calv}\subset \cali\subset \calv$ we create the inductive setup.
We write $\cali^*=\calv\setminus \cali$, and according to the disjoint union
$\cali\sqcup \cali^*=\calv$ we consider the coordinate decomposition
$l=(l_\cali,l_{\cali^*})$. We also set $ R_\cali=R(0, c_\cali)$ and the weight function
$$w_\cali(l_\cali)=\hh(l_\cali)+\hh^\circ(l_\cali+c_{\cali^*})-p_g. $$
Then for $\overline{\calv}\subset \cali\subset \cJ\subset \calv$, $\cJ=\cali\cup \{v_0\}$
($v_0\not\in\cali$),
we wish to prove that $\bH^*_{an}(R_\cali, w_\cali)=\bH^*_{an}(R_{\cJ}, w_{\cJ})$.
For this consider the projection $\pi_{\R}:R_{\cJ}\to R_{\cali}$.
For any fixed $y\in R_\cali$ consider the fiber $\{y+tE_{v_0}\}_{0\leq t\leq c_{v_0},\ t\in \Z}$.
Note that $t\mapsto \hh(y+tE_{v_0})$ is increasing. Let $t_0=t_0(y)$
be the smallest value $t$ for which
$\hh(y+tE_{v_0})< \hh(y+(t+1)E_{v_0})$.
If $t\mapsto \hh(y+tE_{v_0})$ is constant then we take $t_0=c_{v_0}$.
If $t_0<c_{v_0}$, then $t_0$ is characterized by the existence of a function
\begin{equation}\label{eq:1RED}
f\in H^0(\calO_{\tX}) \ \ \mbox{with} \ \
({\rm div}_Ef)|_\cali\geq y, \ \ \ ({\rm div}_Ef)_{v_0}=t_0.
\end{equation}
Symmetrically, $t\mapsto \hh^{\circ} (y+c_{\cJ^*}+ tE_{v_0})$ is decreasing. Let $t_0^\circ=t_0^\circ (y)$
be the smallest value $t$ for which
$\hh^{\circ} (y+c_{\cJ^*}+tE_{v_0})=\hh^{\circ} (y+c_{\cJ^*}+(t+1)E_{v_0})$. The value
$t_0^\circ$ is characterized by the existence of a form
\begin{equation}\label{eq:2RED}
\omega \in H^0(\tX\setminus E,\Omega_{\tX}^2) \ \ \mbox{with} \ \
({\rm div}_E\omega) |_\cali\geq - y, \ \ \ ({\rm div}_E\omega )_{v_0}=-t^{\circ}_0.
\end{equation}
This shows that there exist a form $f\omega\in H^0(\tX\setminus E, \Omega_{\tX}^2)$ such that
$({\rm div}_Ef\omega )|_\cali\geq 0$ and $({\rm div}_Ef\omega )_{v_0}=t_0-t^{\circ}_0$.
By the B$_{an}$ property we necessarily must have $t_0-t^{\circ}_0\geq 0$.
Therefore, the weight $t\mapsto w_{\cJ}(y+tE_{v_0})=\hh(y+tE_{v_0})+\hh^{\circ } (y+tE_{v_0}+c_{\cJ^*})-p_g$
is decreasing for $t\leq t_0^\circ$, is increasing for $t\geq t_0$. Moreover, for $t_0^\circ \leq t\leq t_0$
it takes the
constant value $\hh(y)+\hh^{\circ } (y+c_{v_0}E_{v_0}+c_{\cJ^*})-p_g=w_{\cali}(y)$.
Next we fix $y\in R_\cali$ and some $I\subset \cali$ (hence a cube $(y,I)$ in $R_{\cali}$).
We wish to compare the intervals
$[t_0^\circ (y+E_{I'}), t_0 (y+E_{I'})]$ for all subsets $I'\subset I$. We claim that they have at least one
common element (in fact, it turns out that $t_0(y)$ works).
Note that $\hh(y+tE_{v_0})=\hh(y+(t+1)E_{v_0})$ implies $\hh(y+tE_{v_0}+E_{I'})=\hh(y+(t+1)E_{v_0}+E_{I'})$
for any $I'$,
hence $t_0(y)\leq t_0(y+E_{I'})$. In particular, we need to prove that $t_0(y)\geq t_0^\circ (y+E_{I'})$.
Similarly as above, the value $t_0^\circ(y+E_{I'})$ is characterized by the existence of a form
\begin{equation*}
\omega_{I'} \in H^0(\tX\setminus E, \Omega_{\tX}^2) \ \ \mbox{with} \ \
({\rm div}_E\omega_{I'}) |_\cali\geq - y-E_{I'}, \ \ \ ({\rm div}_E\omega_{I'} )_{v_0}=-t^{\circ}_0(y+E_{I'}).
\end{equation*}
Hence the from $f\omega_{I'}\in H^0(\tX\setminus E, \Omega_{\tX}^2)$ satisfies
${\rm div}_Ef\omega_{I'} |_\cali\geq -E_{I'}$ and $({\rm div}_Ef\omega )_{v_0}=t_0(y)-t^{\circ}_0(y+E_{I'})$.
By the B$_{an}$ property we must have $t_0(y)-t^{\circ}_0(y+E_{I'})\geq 0$.
Set $S_{\cJ,n}$ and $S_{\cali,n}$ for the lattice spaces defined by $w_\cJ$ and $w_\cali$. If
$y+tE_{v_0}\in S_{\cJ,n}$ then $w_\cJ(y+tE_{v_0})\leq n$, hence by the above discussion $w_\cali(y)\leq n$ too.
In particular, the projection $\pi_{\R}:R_\cJ\to R_\cali$ induces a map $S_{\cJ,n}\to S_{\cali,n}$.
We claim that it is a homotopy equivalence. The argument is similar to the proof from
\ref{th:annlattinda} via the above preparations.
\end{proof}
\begin{corollary}\label{cor:AR} If $(X,o)$ admits a resolution $\phi$ with a B$_{an}$--set of cardinality $\overline{s}$,
then $\bH^{\geq \overline{s}}_{an,0}(X,o)=0$. In particular, if \ $\Gamma$ is almost rational (AR) then $\bH^{\geq 1}_{an,0}(X,o)=0$.
\end{corollary}
\begin{proposition}\label{prop:AnLatAR}
Assume that for $(X,o)$ and for its resolution $\phi$ with graph $\Gamma$ the following facts hold:
(i) $\Gamma$ is almost rational,
(ii) $p_g=\min _{\gamma} eu (\bH^0_{top}(\gamma,\Gamma, -Z_K))$.
\noindent
Then $\bH^*_{an,0} (X,o)=\bH^*_{top}(M, -Z_K)$. (Obviously, we have $\bH^{\geq 1}_{an,0} (X,o)=\bH^{\geq 1}_{top}(M, -Z_K)=0$, cf. \ref{cor:AR}.)
\end{proposition}
\begin{proof}
We can assume that $\phi$ is minimal good.
Then $Z_K\geq 0$ (see e.g. \cite[Example 6.3.4]{Book} or \cite{PPP,Veys}).
Write
$\calv$ as $\{v_0\}\sqcup \calv^*$, where $\{v_0\}$ is an B-set. By \ref{ex:Ran}
it is an B$_{an}$ set as well. In particular, $\bH^*_{an,0}(X,o)=\bH^*_{an}(\overline{R},\overline {w})$,
where $\overline{w}(\ell)=\hh(\ell E_{v_0})-h^1(\calO_{\ell E_{v_0}+\lfloor Z_K\rfloor^*})$, $0\leq \ell
\leq \lfloor Z_K\rfloor _{v_0}$.
Let $x(\ell)=x(\ell E_{v_0})$ be the universal cycle introduced in \ref{lemF1}. It is the smallest cycle whose
$E_{v_0}$-multiplicity is $\ell$ and
$(x(\ell), E_v)\leq 0$ for any $v\not=v_0$. If $s=s(\ell E_{v_0})$ is the smallest cycle in $\calS$ with $\ell E_{v_0}\leq s$
(for its existence see \cite{NOSz}), then by the universal properties $\ell E_{v_0}\leq x(\ell)\leq s$. Hence,
since $H^0(\cO_{\tX}(-\ell E_{v_0}))=H^0(\cO_{\tX}(-s))$, $\hh(\ell E_{v_0})=\hh(x(\ell))$.
Next, for any $l^*\in L_{>0}$ with $|l^*|\in \calv^*$ consider the exact sequence
$0\to \calO_{l^*}(-x(\ell))\to \calO_{x(\ell)+l^*}\to \calO_{x(\ell)}\to 0$. By the Lipman's vanishing
\ref{prop:VAN0} we have $h^1( \calO_{l^*}(-x(\ell)))=0$, hence
\begin{equation}\label{eq:daguj} h^1(\calO_{x(\ell)+l^*})= h^1(\calO_{x(\ell)}).\end{equation}
On the other hand, using $Z_{coh}\leq \lfloor Z_K\rfloor$ (cf. \ref{rem:antmatroid}), the opposite matroid rank inequality
from \ref{rem:antmatroid}
and (\ref{eq:daguj}) applied for $l^*$ sufficiently large, we obtain
$h^1(\calO_{x(\ell)+l^*})= h^1(\calO_{\min\{x(\ell)+l^*, \lfloor Z_K\rfloor\}})=
h^1(\calO_{\ell E_{v_0}+\lfloor Z_K\rfloor ^*})$.
Thus $\overline{w}(\ell)=\hh(x(\ell))-h^1(\calO_{x(\ell)})=w(x(\ell))$.
Finally, consider the exact sequence $0\to \calO_{\tX}(-x(\ell))\to \calO_{\tX}\to \calO_{x(\ell)}\to 0$ and
the morphism $r(\ell):H^0(\calO_{\tX})\to H^0(\calO_{x(\ell)})$. Then $\hh(x(\ell))+\dim\, {\rm coker} \,( r(\ell))=h^0(\calO_{x(\ell)})$, hence
$\overline{w}(\ell)=\chi(x(\ell))-\dim\,{\rm coker}\, (r(\ell))$.
The point in this identity is that in fact $r(\ell)$ is onto for all $\ell$. Indeed, this follows from
assumption {\it (ii)}
and from the fact that
$\min _{\gamma} eu (\bH^0_{top}(\gamma,\Gamma, -Z_K))$ is realized by the concatenated computation sequence $\gamma=\{x_i\}_{i=0}^t$
having as intermediate terms the universal cycles $x(\ell)$ (defined in \ref{ss:CCSAR}). In this case
$H^0(\calO_{x_{i+1}})\to H^0(\calO_{x_i})$ is necessarily onto, by the splitting of the cohomological exact sequence associated with consecutive members of the optimal path, cf. \ref{rem:SPLIT}.
Since $h^1(\cO_{x_t})=p_g=h^1(\cO_{\tX})$, we also conclude that
$H^0(\calO_{\tX})\to H^0(\calO_{x_i})$ is also surjective for all $x_i$, in particular for the intermediate terms
$x(\ell)$ as well.
In conclusion, $\overline{w}(\ell)=\chi(x(\ell))$. Then use the topological reduction theorem \ref{th:red}
for AR graphs.
\end{proof}
\begin{example}\label{ex:ARAN}
The two assumptions of Proposition \ref{prop:AnLatAR} are satisfied in the following cases,
cf. \ref{ex:MINforwh} (recall that
we assume that the link is a rational homology sphere):
$\bullet$ \ rational singularities,
$\bullet$ \ Gorenstein elliptic singularities, and, more generally,
non-necessarily numerically Gorenstein elliptic germs with $p_g=\mbox{length of the elliptic sequence}$
(combine \cite{weakly} and \cite{NNIIIb}),
$\bullet$ \ AR singularities for which the SWIC for the canonical spin$^c$ structure holds,
(in particular, weighted homogeneous germs and
superisolated germs associated with rational unicuspidal curves, or
splice quotient singularities associated with an AR graph $\Gamma$ \cite{NO2}).
\vspace{2mm}
Hence, for all these cases, $\bH^*_{an,0} (X,o)=\bH^*_{top}(M,-Z_K)$.
For concrete expressions of $\bH^*_{top}(M,-Z_K)$ in the above cases see \cite{NOSz,NGr,Nlattice,Book}.
\end{example}
\subsection{Comparison of $\bH^*_{an,0}(X,o)$ with $\bH^*_{top}(M,-Z_K)$ for any $(X,o)$ with $\Q HS^3$ link}\label{an:Comp}
\bekezdes The modules $\bH^*_{an,0}(X,o)$ with $\bH^*_{top}(M,-Z_K)$
can be compared even without the assumptions of Proposition \ref{prop:AnLatAR}.
Next, we define a morphism $\bH^*_{an,0}(X,o)\to \bH^*_{top}(M,-Z_K)$
of graded $\Z[U]$--modules, and we list some properties of the
analytic graded root associated with the analytic weight function $w_0$.
First, we compare the analytic and topological
weight functions.
From the exact sequence $0\to \calO_{\tX}(-l)\to \calO_{\tX}\to \calO_l\to 0$
we obtain:
\begin{lemma}\label{lem:weightTOPAN}
If $w_{an}$ denotes the analytic weight function, then $w_0(l)=\chi(l)-\dim \, {\rm coker}\, (r(l))$,
where $r(l):H^0(\calO_{\tX})\to H^0(\calO_l)$ is the natural morphism.
\end{lemma}
\bekezdes\label{bek:CONSEQW} {\bf Discussion.} Lemma \ref{lem:weightTOPAN} has several consequences.
Let us also write $w_{top}=\chi$.
(1) \ $w_{an}\leq \chi$, hence $\min \, w_{an}\leq \min\, \chi=\min\, w_{top}$.
(2) \ Set $S_{an,n}=\cup\{\square \,:\, w_{an}(\square)\leq n\}$ and
$S_{top,n}=\cup\{\square \,:\, w_{top}(\square)\leq n\}$. Then
$S_{top,n}\subset S_{an,n}$ for any $n\in\Z$.
(3) \ For any $q\geq 0$ there exists a {\it graded $\Z[U]$-module morphism}
$$\mathfrak{H}^q:\bH^q _{an,0}(X,o)\to
\bH^q_{top}(M,-Z_K).$$
This is an isomorphism in the cases when Proposition \ref{prop:AnLatAR} holds.
Similarly, the inclusion of the connected components induce a graded morphism of graph
$$\RR_{top,0}(M)\to \RR_{an,0}( X,o).$$
(4) The following (new) characterization of rational germs hold: $(X,o)$ is rational if and only if $w_{an}|_{L_{>0}}>0$.
Indeed, if $(X,o)$ is rational then $w_{an}(l)=\hh(l)>0$ for $l>0$.
Conversely use $\chi\geq w_{an}$ and Artin's criterion.
(5) If $\min\, w_{an}=0$, and $(X,o)$ is not rational, then $(X,o)$ is elliptic (use $\chi\geq w_{an}$). Conversely, if $(X,o)$ is
Gorenstein elliptic then $\min\, w_{an}=0$ (use \ref{ex:ARAN}).
Moreover, $\min\, w_{an}=0$ holds for any elliptic singularity with generic analytic structure as well (see
\ref{ex:GENERICANTYPE}). However, in general $\min\, w_{an}=0$ does not hold for any elliptic germ, see
\ref{ex:ELLipticAN}. For a complete discussion of the elliptic case see \cite{AgNeEll}.
(6) \ In the next discussions it is convenient to assume that $c$ is conveniently large or $c=\infty$.
Let $C$ be a component of $S_{an, n}$ such that for any
$l\in S_{an, n}$ we have $w_{an}(l)=n$ (that is, the class of $C$ represents a local minimum of the analytic weight in the
analytic graded root).
Assume that for some $s\in C$ we have $w_{an}(s+E_v)>w_{an}(s)$ for every $v\in\calv$ (that is, $s$
is a maximal element of $C$). Then $s\in \calS_{an}$.
(Indeed, $w_{an}(s+E_v)>w_{an}(s)$ implies $\hh(s+E_v)>\hh(s)$, cf. \ref{lem:hsimult}.)
In fact, using the matroid rank inequality of $w_{an}$ one can show that
$C$ has a unique maximal element.
This shows that the `ends' of the analytic graded root $\RR_{an} $ represent elements of the analytic semigroup $\calS_{an}$.
(7)
Assume that $m\in L_{>0}$ satisfies
$w_{an}(m-E_v)>w_{an}(m)$ for any $E_v\subset |m|$.
(E.g., it is a minimal element of a component $C$ as in (6).)
By \ref{lem:hsimult}
$h^1(\calO_{m-E_v})< h^1(\calO_{m})$. This can happen only if $h^1(\calO_{E_v}(-m+E_v))=h^0(\calO_{E_v}(m-Z_K))\not=0$,
which implies $(Z_K-m,E_v)\leq 0$ for every $E_v\subset |m|$. In particular, $\chi(m)\leq 0$.
(8) Assume that $n\geq 0$. We claim that any connected component of $S_{an, n}$ contains at least one
element of $S_{top,n}$.
Indeed, assume the opposite, and
let $C_n$ be a component of $S_{an, n}$ such that $\chi(l)>n$ for any $l\in C_n$.
Take a local $w_{an}$-minimum in $C_n$ with value $k\leq n$, consider the component $C$ of
$S_{an,k}$ which contains it, and let $m$ be
a minimal element of $C$. Then, by (7), $\chi(m)\leq 0$. But this is a contradiction since
$m\in C_n$, hence $\chi(m)>n\geq 0$.
(9) Recall that $S_{top, n}$ is connected for any $n\geq 1$, see Proposition \ref{9STR2}{\it (b)}.
Therefore, part (8) applied for $n\geq 1$ shows that
$S_{an, n}$ is connected too.
Furthermore, for $n=0$ it implies that at
the 0-graded level ${\mathfrak{H}}^0_{{\rm deg}=0}:\bH^0_{an,0}(X,o)_{{\rm deg}=0}\to \bH^0_{top,0}(M)_{{\rm deg}=0}$ is injective,
or, at the level of degree zero vertices of roots,
$\RR_{top,0}(M)_{{\rm deg}=0} \to \RR_{an,0}( X,o)_{{\rm deg}=0}$ is surjective.
\begin{problem}\label{prob:4.7.4}
{\it (a) For a fixed topological type find all the possible graded $\Z[U]$--modules $\{\bH^*_{an,0}\}_{an}$,
associated with all the possible analytic structures supported on that topological type. (For such a concrete classification see
Examples \ref{ex:ALLANTHESAME}, \ref{ex:CONTtwocuspsAN} and \ref{ex:ELLipticAN}.)
(b) For a fixed topological type (hence for a fixed $\bH^*_{top,0}(M)$) and analytic type $(X,o)$ supported on it
find special properties of
$\bH^*_{an,0}(X,o)$ (and of the morphism $\bH^*_{an,0}(X,o)\to \bH^*_{top,0}(M)$), which might characterize the
classification from part (a).}
\end{problem}
\noindent
Several examples support the following conjecture (see e.g. \ref{ex:CONTtwocuspsAN}), which makes Problem
\ref{prob:4.7.4} more precise.
\begin{conjecture}
Fix a topological type. Then for any analytic type supported on it
$\bH^*_{an,0}(X,o)\to \bH^*_{top,0}(M)$ is injective. At the graded roots level,
the graded graph-morphism $\RR_{top,0}(M)\to \RR_{an,0}( X,o)$ is surjective (at the level of vertices and edges).
Hence, Problem \ref{prob:4.7.4} reads as follows:
characterize those graded $\Z[U]$--submodules of $\bH^*_{top,0}(M)$ which appear as $\bH^*_{an,0}(X,o)$.
\end{conjecture}
Note that once $M$ is fixed, there are only finitely many graded $\Z[U]$--modules $\bH^*_{an,0}(X,o)$ and graded roots
$\RR_{an,0}( X,o)$ which satisfy this Conjecture.
\subsection{Examples}\label{ss:anEx}
\notation\label{9zu1}
Consider the graded $\Z[U]$-module
$\calt:=\Z[U,U^{-1}]$, and (following \cite{OSzP}) denote by
$\calt_0^+$ its quotient by the submodule $U\cdot \Z[U]$.
This has a grading in such a way that $\deg(U^{-d})=2d$ ($d\geq
0$). Similarly, for any $n\geq 1$, the quotient of $U^{-(n-1)}\cdot \Z[U]$
by $U\cdot \Z[U]$ (with the same grading) defines the graded module
$\calt_0(n)$. Hence, $\calt_0(n)$, as a $\Z$-module, is freely
generated by $1,U^{-1},\ldots,U^{-(n-1)}$, and has finite
$\Z$-rank $n$.
More generally, for any graded $\Z[U]$-module $P$ with
$d$-homogeneous elements $P_d$, and for any $r\in\Q$, we
denote by $P[r]$ the same module graded (by $\Q$) in such a way
that $P[r]_{d+r}=P_{d}$. Then set $\calt^+_r:=\calt^+_0[r]$ and
$\calt_r(n):=\calt_0(n)[r]$. Hence, for $m\in \Z$,
$\calt_{2m}^+=\Z\langle U^{-m}, U^{-m-1},\ldots\rangle$ as a $\Z$-module.
\begin{example}\label{ex:EATROOTAN} We claim that the following facts are equivalent:
(i){\bf $(X,o)$ is rational}, (ii) $\RR_{an,0}(X,o)=\RR_{(0)}$, (iii)
$\bH^*_{an,0}(X,o)=\calt^+_0$, (iv) $\bH^*_{an,0}(X,o)=\calt^+_{2m}$ for some $m$, (v)
$S_{an,0}$ is contractible.
Indeed, if $(X,o)$ is rational then $h^1(\calO_l)=0$ for $l>0$, hence $w_{an}$ is increasing.
In particular, any nonempty $S_{an,n}$ can be contracted to the origin, and the analytic root is $\RR_{(0)}$,
and $\bH^*_{an,0}=\calt^+_0$. (ii)$\Rightarrow$(iii)$\Rightarrow$(iv)$\Rightarrow$(v) are clear.
Assume that (v) holds.
Since $w_{an}(E_v)=\hh(E_v)-h^1(\cO_{E_v})=1$, the connectivity of $S_{an,0}$ implies that
$w_{an}|_{L_{>0}}>0$. Hence $\chi |_{L_{>0}}>0$ too, and $(X,o)$ is rational by Artin's criterion.
The above criterions are equivalent with the vanishing of the reduced cohomology $\bH^*_{an,red,0}(X,o)=0$ too.
\end{example}
\begin{example}\label{NNAN}
Consider the hypersurface singularity with nondegenerate principal part from Example \ref{ex:twonodesB}.
In this case $\bH^1_{an,0}$ is nonzero. In this Newton nondegenerate suspension case
$\hh$ is computed combinatorially using the weighted lattice points under the Newton diagram, and
$\hh^\circ (l)=\hh(Z_K-l)$.
This shows that $\bH^{>0}_{an,0}(X,o)\not=0$ can happen in the analytic case as well.
\end{example}
\begin{example}\label{ex:GENERICANTYPE} {\bf $\bH^*_{an,0}$ for the generic analytic structure.}
Let us fix a {\it non--rational} resolution graph $\Gamma$ with $M(\Gamma)$ a $\Q HS^3$.
Assume that $\tX$ is a resolution space of a singularity $(X,o)$ with dual graph $\Gamma$
and generic analytic structure in the sense of \cite{LauferDef,NNII}. In \cite{NNII} the following facts are proved:
(i) $p_g(X,o)=1-\min\,\chi$, \ (ii) $Z_{max}=\max\{ l\,:\, \chi(l)=\min\,\chi\}$, \
(ii) $Z_{coh}=\min\{ l\,:\, \chi(l)=\min\,\chi\}$. In particular, $Z_{coh}\leq Z_{max}$.
Since $\bH^*_{an,0}$ can be computed in $R(0,Z_{coh})$ (cf. \ref{lem:INDEPAN}), and
$\hh(0)=0$ and $\hh(l)=1$ for $0<l\leq Z_{max}$ --- hence similar identities hold in $R(0, Z_{coh})$ too ---,
the computation is immediate.
Note also that $h^1(\calO_{Z_{coh}})=p_g=1-\min\,\chi$.
Therefore,
$w_{an}(Z_{coh})=w_{an}(Z_{max})=1-p_g=\min\,\chi$. Furthermore,
$\bH^0_{an,0}(X,o)\simeq\calt^+_{2\,\min\,\chi}\oplus \calt_0(1)$ and $\bH^{\geq 1}_{an,0}(X,o)=0$. The analytic graded root is:
\begin{picture}(300,100)(20,330)
\put(77,380){\makebox(0,0){\small{$0$}}} \dashline{1}(100,350)(140,350)
\put(77,390){\makebox(0,0){\small{$1$}}}\put(70,350){\makebox(0,0){\small{$\min\,w_{an}$}}}
\dashline{1}(100,380)(140,380) \dashline{1}(100,390)(140,390)
\put(120,420){\makebox(0,0){$\vdots$}}
\put(120,365){\makebox(0,0){$\vdots$}}
\put(120,350){\circle*{3}}
\put(120,350){\line(0,1){5}}
\put(120,400){\circle*{3}}
\put(120,390){\circle*{3}} \put(110,380){\circle*{3}}
\put(120,380){\circle*{3}}
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\put(110,380){\line(1,1){10}}
\put(230,380){\makebox(0,0){\small{$(\RR_{gen}(m),\chic), \ \ \mbox{where}
\ m=\min\,w_{an}=\min\chi=1-\min\chic$}}}
\end{picture}
\noindent Note that in general $eu(\bH^0_{an,0})\geq 1-\min\,w_{an}$ (use (\ref{eq:euh0})). However, in the present case of the
generic analytic structure we have the sharp equality:
$eu(\bH^0_{an,0})=p_g=1-\min \, w_{an}=1-\min\,\chi=1-\min\, \chic$.
\end{example}
\begin{notation}\label{not:gen}
A graded root $(\RR,\chic)$ of the shape as in Example \ref{ex:GENERICANTYPE} is denoted by $\RR_{gen}(m)$, where
$m=\min\,\chic$.
\end{notation}
\begin{example}\label{ex:ALLANTHESAME}
Assume that $\Gamma$ is star--shaped with central vertex $-2$, with five legs, each of them
having one vertex with Euler number $-4$. Then the algorithm from \cite[\S11]{NOSz} (applied for the
canonical spin$^c$--structure) shows that the topological graded root is $\RR_{gen}(-1)$ with
$\min\,w_0=-1$ (cf. Notation \ref{not:gen}).
In fact, that algorithm also shows that that for {\it any} analytic structure $Z_{coh}=x(2)\leq x(3)=Z_{min}$, hence,
the analytic graded root is $R_{gen}(-1)$ independently of the analytic structure.
Note that from \ref{ex:ARAN}
we also obtain that $\bH^*_{an,0} (X,o)=\bH^*_{top}(M,-Z_K)$.
In particular, for this graph,
$\bH^*_{an,0}=\bH^*_{top,0}$ for {\it any} analytic structure supported on $\Gamma$.
\end{example}
\begin{example}\label{ex:CONTtwocuspsAN} Consider the topological type fixed by the following
resolution graph $\Gamma$.
\begin{picture}(300,45)(10,0)
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\put(175,25){\circle*{4}} \put(200,25){\circle*{4}}
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\put(200,5){\circle*{4}} \put(125,25){\line(1,0){100}}
\put(150,25){\line(0,-1){20}} \put(200,25){\line(0,-1){20}}
\put(125,35){\makebox(0,0){$-2$}}
\put(150,35){\makebox(0,0){$-1$}}
\put(175,35){\makebox(0,0){$-13$}}
\put(200,35){\makebox(0,0){$-1$}}
\put(225,35){\makebox(0,0){$-2$}} \put(160,5){\makebox(0,0){$-3$}}
\put(210,5){\makebox(0,0){$-3$}}
\end{picture}
\noindent In this case the link is an integral homology sphere and $\bH^1_{top,0}\not=0$, cf. \cite{Nlattice}.
\noindent The list of possible analytic structures
supported on $\Gamma$ is the following, see \cite{NO2cusps}:
(i) non-Gorenstein Kulikov analytic type with $p_g=3$ and $Z_{max}=Z_{min}$,
(ii) complete intersection (hence Gorenstein) splice quotient with $p_g=3$ and $Z_{max}=2Z_{min}$,
(iii) analytic structures with $p_g=2$ and $Z_{coh}\leq Z_{min}<2Z_{min}\leq Z_{max}$ (they are non--Gorenstein).
\noindent
In all these cases $\bH^{\geq 1}_{an,0}=0$, however the modules $\bH^0_{an,0}$ are all different. Below we give the graded roots
(and also the topological root, for comparison).
The needed information regarding $\hh$ and $\hh^\circ$ can be deduced e.g. from \cite{NO2cusps}.
\begin{picture}(300,90)(120,340)
\put(210,350){\makebox(0,0){\small{topological root}}}
\put(180,380){\makebox(0,0){\small{$0$}}} \put(177,370){\makebox(0,0){\small{$-1$}}}
\dashline{1}(200,370)(240,370)
\dashline{1}(200,380)(240,380)
\put(220,420){\makebox(0,0){$\vdots$}} \put(220,400){\circle*{3}}
\put(220,390){\circle*{3}} \put(210,380){\circle*{3}}
\put(230,380){\circle*{3}} \put(220,380){\circle*{3}}\put(210,370){\circle*{3}}
\put(230,370){\circle*{3}} \put(220,410){\line(0,-1){30}}
\put(210,380){\line(1,1){10}} \put(220,390){\line(1,-1){10}}
\put(210,370){\line(1,1){10}} \put(220,380){\line(1,-1){10}}
\dashline{3}(255,350)(255,420)
\put(300,350){\makebox(0,0){\small{Gorenstein type}}}
\dashline{1}(280,370)(320,370)
\dashline{1}(280,380)(320,380)
\put(300,420){\makebox(0,0){$\vdots$}} \put(300,400){\circle*{3}}
\put(300,390){\circle*{3}} \put(290,380){\circle*{3}}
\put(310,380){\circle*{3}} \put(300,380){\circle*{3}
\put(310,370){\circle*{3}}
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\put(300,380){\line(1,-1){10}}
\put(370,350){\makebox(0,0){\small{Kulikov type}}}
\dashline{1}(350,370)(390,370)
\dashline{1}(350,380)(390,380)
\put(370,420){\makebox(0,0){$\vdots$}} \put(370,400){\circle*{3}}
\put(370,390){\circle*{3}} \put(360,380){\circle*{3}}
\put(370,380){\circle*{3}}\put(360,370){\circle*{3}}
\put(380,370){\circle*{3}}
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\put(360,380){\line(1,1){10}}
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\put(440,350){\makebox(0,0){\small{$p_g=2$}}}
\dashline{1}(420,370)(460,370)
\dashline{1}(420,380)(460,380)
\put(440,420){\makebox(0,0){$\vdots$}} \put(440,400){\circle*{3}}
\put(440,390){\circle*{3}} \put(430,380){\circle*{3}}
\put(440,380){\circle*{3}}
\put(450,370){\circle*{3}}
\put(440,410){\line(0,-1){30}}
\put(430,380){\line(1,1){10}}
\put(440,380){\line(1,-1){10}}
\end{picture}
\noindent The local minima of the topological graded root reflect elements the topological semigroup (Lipman cone)$\calS$
in the rectangle $R(0, Z_K)$. On the other hand, the local minima
of each analytic graded root reflect the divisors of functions from the
analytic semigroup in the corresponding rectangle $R(0, Z_{coh})$
(of the corresponding
analytic type), cf. \ref{bek:CONSEQW}(6).
The above pictures indicate very intuitively which elements of the topological semigroup $\calS$
are not realized as divisors of functions (do not belong to $\calS_{an}$)
in different analytic structures: in the Goresntein case $Z_{min}$, in the non--Goresntein cases $Z_K$.
\end{example}
\begin{remark}
When we fix a topological type and we compare the possible graded roots (and analytic lattice cohomologies)
associated with different analytic structures then we realize that the `simplest' object is
provided by the generic analytic structure. However, it is very hard to identify those analytic structures
which provide the `most complicated' modules/roots. Of course, the candidate for the most complicated module/root
is provided by $\bH^*_{top,0}(M)$ and $\RR_{top,0}(M)$ respectively. However, they are not always
realized by a certain analytic structure, see Example \ref{ex:CONTtwocuspsAN}.
\end{remark}
\begin{problem} {\it
Which $\bH^*_{top,0}(M)$ and $\RR_{top,0}(M)$ is realized by a certain analytic structure?
If yes, for which (very special) analytic structure are they realized?
(Can this family be identified universally by certain analytic properties?) }
\end{problem}
\begin{remark}\label{rem:sq}
The Gorenstein case from Example \ref{ex:CONTtwocuspsAN} shows that even for {\bf splice quotient} analytic types
$\bH^q_{an,0}\not=\bH^q_{top,0}$ might happen. (In \ref{ex:CONTtwocuspsAN} they differ for both $q=0, 1$.)
However, the validity of SWIC guarantees that $eu(\bH^q_{an,0})=eu(\bH^q_{top,0})$ for any splice quotient.
\end{remark}
\begin{problem}{\it
Consider the equisingular family of splice quotient singularities associated with a (convenient) graph $\Gamma$.
Is it true that $\bH^*_{an,0}$ is constant along this family?
If yes (hence $\bH^*_{an,0}$ is determined by $\Gamma)$ describe it combinatorially from $\Gamma$.
(Note that usually it is \underline{not} $\bH^*_{top,0}(M(\Gamma))$, cf. Remark \ref{rem:sq}.)}
\end{problem}
\begin{example}\label{ex:ELLipticAN}
Assume that $(X,o)$ is a {\bf numerically Gorenstein elliptic
singularity} with rational homology sphere link.
In the next discussion we use the notations of \cite{weakly}.
In the Gorenstein case we already know (see \ref{ex:ARAN}) that $\bH^*_{an,0}(X,o)=\bH^*_{top,0}(M)$.
Next, consider a graph with $m=2$. In the Gorenstein case $p_g=3$, and $\bH^*_{an,0}$ is determined above.
Next, assume that $p_g=2$.
Since $(X_1,o_1)$ is Gorenstein, $h^1(\calO_{C'_1})=2$, hence $Z_{coh}\leq C'_1=C+Z_{B_1}$.
On the other hand, $H^0(\calO_{\tX}(-C_1))=H^0(\calO_{\tX}(-Z_{min}))$, hence $Z_{max}=C_1=Z_{min}+Z_{B_1}$.
Therefore, $Z_{coh}<Z_{max}$, and $w_{an}(Z_{coh})=1-p_g=-1$.
Since $eu=p_g=2$, this shows that the analytic graded root is necessarily
$R_{gen}(-1)$. The cohomology groups are $\bH^{\geq 1}_{an,0}=0$ and $\bH^0_{an,0}=\calt^+_{-2}\oplus \calt_0(1)$.
Surprisingly, even if $\Gamma$ is elliptic with $\min\,\chi=0$, for this analytic structure $\min\,w_{an}<0$.
For the discussion of the general elliptic case see \cite{AgNeEll}.
\end{example}
\begin{question}\label{question:AN}
Let us fix a non-rational topological type $\Gamma$ with $M$ a $\Q HS^3$.
What are the possible values of $\min\,w_{an}$ when we consider all the analytic types supported on $\Gamma$?
(Note that $1-p_g\leq \min\,w_{an}\leq \min\,\chi$.)
\end{question}
\subsection{$\bH^*_{an,0}$ and $p_g$--constant deformations} \label{bek:DEFAN}
\bekezdes
First we formulate the following conjecture.
\begin{conjecture}\label{conj:defAN}
$\bH^*_{an,0}$ is constant along flat $p_g$--constant deformations of normal surface singularities
(with $\Q HS^3$ links).
\end{conjecture}
\begin{example}\label{ex:CONJANNSIpelda}
Consider the superisolated singularity $(X,o)$
associated with a rational projective plane curve $C$ degree 5 with two cusps, both with
one Puiseux pair, namely (3,4) and (2,7) respectively (for notations see \cite{Spany}.
Denote the two singular points of $C$ by $p$ and $q$ respectively.
The curve $C$ is projective equivalent with
$y^3z^2+x^4z+x^3yz-x^2y^2z/2-x^5/4+x^4y/16$ \cite{Y4}. The resolution graph of $(X,o)$ is
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\put(125,35){\makebox(0,0){\small{$-2$}}}
\put(150,35){\makebox(0,0){\small {$-1$}}}
\put(175,35){\makebox(0,0){\small{$-31$}}}
\put(200,35){\makebox(0,0){\small{$-1$}}}
\put(225,35){\makebox(0,0){\small{$-3$}}} \put(160,5){\makebox(0,0){\small{$-4$}}}
\put(210,5){\makebox(0,0){\small{$-2$}}} \put(100,25){\circle*{4}}
\put(250,25){\circle*{4}} \put(275,25){\circle*{4}}
\put(100,35){\makebox(0,0){\small{$-2$}}}
\put(250,35){\makebox(0,0){\small{$-2$}}}
\put(275,35){\makebox(0,0){\small{$-2$}}}
\end{picture}
This is a numerically Gorenstein graph, and we can apply the topological reduction theorem for
the set of nodes. Then the topological lattice cohomology
computation is reduced to this 2-dimensional rectangle $R((0,0),(30,34))$. For the complete list of
$\chi$-weights see \cite{LThesis}. At $(0,0)$ and $(30,34)$
one has two symmetric local minima.
They will generate a summand $\calt_0(1)^2$ in $\bH^0_{top}(\Gamma,-Z_K)$.
The other generators and relations can be read from
the rectangle $R((12,14),(18,20))$ with the corresponding $\chi$--values:\\
{\small
\hspace*{3cm}$\begin{matrix}
{\bf -2} & {\bf -2} & {\bf -3} & -4 & -4 & -4 & \mathbf{-5} \\
{\bf -2} & -1 & {\bf -2} & -3 & -3 & -3 & -4\\
{\bf -3} & {\bf -2} & {\bf -2} & -3 & -3 & -3 & -4\\
-3 & -2 & -2 & -2 & -2 & -2 & -3\\
-4 & -3 & -3 & -3 & {\bf -2} & {\bf -2} & {\bf -3}\\
-4 & -3 & -3 & -3 & {\bf -2} & -1 & {\bf -2}\\
\mathbf{-5} & -4 & -4 & -4 & {\bf -3} & {\bf -2} & {\bf -2}\end{matrix}$ }\\
The two boldface $\mathbf{ -5}$'s are new generators of $\bH^0_{top}(M, -Z_K) $ of degree $-10$,
the middle $-2$ is a saddle point corresponding to the meeting of the two long legs of the graded root,
and the boldface loops generate $\bH^1_{top}(M, -Z_K) $.
Hence $\bH^0_{top}(\Gamma, -Z_K)=
\calt^+_{-10}\oplus \calt_{-10}(3)\oplus \calt_0(1)^2$ and
$\bH^1_{top}=\calt_{-2}(1)^2$ (see also
\cite{NSig}). The topological (canonical) graded root is
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\put(20,360){\makebox(0,0){\small{$-2$}}} \dashline{1}(40,370)(160,370)
\put(20,350){\makebox(0,0){\small{$-3$}}} \dashline{1}(40,360)(160,360)
\put(20,340){\makebox(0,0){\small{$-4$}}} \dashline{1}(40,350)(160,350)
\dashline{1}(40,340)(160,340) \dashline{1}(30,330)(160,330)
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\put(100,400){\circle*{3}}
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\put(20,410){\makebox(0,0)[t]{$\chi$}}
\end{picture}
The point is that there exists a $K_{min}^2$-- and $p_g$--constant deformation from $(X,o)$ to a singularity $(X_{5,5,6},o)$,
(equisingular to the Brieskorn hypersurface $x^5+y^5+z^6$), whose link is $\Sigma(5,5,6)$.
Indeed,
assume that the rational unicuspidal curve is given by $f_d(x,y,z)=0$ in $\C{\mathbb P}_2$.
We can fix the homogeneous coordinates in $\C{\mathbb P}_2$
in such a way that $z=0$ intersects $C$ generically.
A possible choice for the superisolated singularity
$f:(\C^3,0)\to(\C,0)$ (up to equisingularity) is $f=f_d+z^{d+1}$. Write $f_d$ as $\sum_{i=0}^d g_{d-i}(x,y)z^i$.
Then $g_d$ is a product of $d$ linear
factors corresponding to the points $C\cap \{z=0\}$, hence the germ
$g_d:(\C^2,0)\to (\C,0)$ is
equisingular with $(x,y)\mapsto x^d+y^d$.
Next, consider the following deformation $f_t:(\C^3,0)\to (\C,0)$ of isolated hypersurface germs, given by $f_t(x,y,z)=f_d(x,y,tz)+z^{d+1}=
\sum_i g_{d-i}(x,y)z^it^i+z^{d+1}$. For $t\not=0$ the deformation is
$\mu$-constant, the embedded topological type stays also constant, and it is equivalent
(up to such equivalences) to the type of $f$. However, for $t=0$ it is
equivalent (in similar sense) to the germ $x^d+y^d+z^{d+1}$.
The minimal good
resolution graph of $X_{d,d,d+1}$ is star-shaped with a $(-1)$ central vertex, and
it has $d$ identical legs, each consisting of one vertex with self-intersection number
$-(d+1)$. We invite the reader to verify that the deformation is $K_{min}^2$--constant (that means that $Z_K^2$ computed in the
minimal resolution is $t$--independent), and also $p_g$--constant.
Since $(X_{5,5,6},o)$ is AR and for it the SWIC holds, by \ref{ex:ARAN}
$\bH^*_{an,0}(X_{5,5,6},o)=\bH^*_{top,0}(\Sigma(5,5,6))$. That is, $\bH^{\geq 1}_{an,0}(X_{5,5,6},o)=0$ and the
{\it analytic} graded root of
$(X_{5,5,6},o)$ agrees with the topological graded root of $\Sigma(5,5,6)$.
On the other hand, one verifies (see e.g. \cite{Spany,BLMN2,Book}
that the topological lattice cohomologies and graded roots of $M(\Gamma)$ and $\Sigma(5,5,6)$ agree
(for the graded root see the root above). Hence, the conjecture predicts that the same facts are valid for
$(X,o)$ too, just like for $(X_{5,5,6},o)$. This, in particular means that $\bH^0_{an,0}(X,o)=\bH^0_{top,0}(M)$ and
$\bH^1_{an,0}(X,o)=0$. This is what we will verify next.
By the {\it analytic} Reduction Theorem \ref{th:REDAN} (applied for the two nodes)
we have to complete the $w_{an}$--table on $R((0,0),(30,34))$. Since $(X,o)$ is Gorenstein,
$\hh^\circ(l)=\hh(Z_K-l)$, cf. \ref{ex:GorAN},
hence we can concentrate only on $\hh$. Note also that $Z_{max}=Z_{min}$, and the generic line has multiplicities
$(12,14)$ along the nodes. Therefore, $\hh(l)=1$ for any $0<l\leq (12,14)$. Furthermore (by Laufer's algorithm \cite{Laufer72})
$h^1(\calO_{Z_{min}})=6$. Hence, basically it is enough to analyse
the symmetric rectangle $R((12,14), (18,20)) $, as in the above topological case.
The values of $\hh$ can be computed using computation sequences
via the following principles: (a) if $E_n$ is a node, $x(\ell)\in\calS_{an}$ and $(x(\ell),E_n)=0$
then ($\dag$) $\hh(x(\ell)+E_v)\geq \hh(x(\ell))+1$; (b)
$\hh$ is constant along a computation sequence which connects $x(\ell)+E_n$ with $s(x(\ell)+E_n)$.
In this rectangle we have the following elements of $\calS_{an}$, all with the above properties:
$x(12,14)$ being the divisor of the generic line, $x(15,14)$ of the generic line which contains $p$,
$x(16,14)$ of the tangent line at $p$, $x(12,16)$ of the generic line which contains $q$,
$x(15,16)$ of the line which contains both $p$ and $q$, and finally,
$x(12,18)$ of the tangent line at $q$.
(Note also that $\hh((18,20))=\hh(Z_K-Z_{min})=\hh^\circ(Z_{min})=p_g-h^1(\calO_{Z_{min}})=4$,
hence no other semigroup element can contribute and in $(\dag)$ we always have equality.)
These semigroup places are underlined in the next diagram from left,
which shows the $\hh$ values in
the rectangle $R((12,14),(18,20))$:\\
{\small
\hspace*{2cm}$\begin{matrix}
4 & \ \ 4 & \ \ 4 & \ \ 4 & \ \ 4 & \ \ 4 & \ \ 4 \\
4 & \ \ 4 & \ \ 4 & \ \ 4 & \ \ 4 & \ \ 4 & \ \ 4\\
\underline{3} & \ \ 4 & \ \ 4 & \ \ 4 & \ \ 4 & \ \ 4 & \ \ 4\\
3 & \ \ 4 & \ \ 4 & \ \ 4 & \ \ 4 & \ \ 4 & \ \ 4\\
\underline{2} & \ \ 3 & \ \ 3 & \ \ \underline{3} & \ \ 4 & \ \ 4& \ \ 4\\
2 & \ \ 3 & \ \ 3 & \ \ 3 & \ \ 4 & \ \ 4& \ \ 4\\
\underline{1} & \ \ 2 & \ \ 2 & \ \ \underline{2} & \ \ \underline{3} & \ \ 4 & \ \ 4\end{matrix}$
\hspace*{1cm}$\begin{matrix}
-2 & \ -2 & \ -3 & \ -4 & \ -4 & \ -4 & -5 \\
-2 & -2 & -2 & -3 & -3 & -3 & -4\\
-3 & -2 & -2 & -3 & -3 & -3 & -4\\
-3 & -2 & -2 & -2 & -2 & -2 & -3\\
-4 & -3 & -3 & -3 & -2 & -2 & -3\\
-4 & -3 & -3 & -3 & -2 & -2 & -2\\
-5 & -4 & -4 & -4 & -3 & -2 & -2\end{matrix}$ }\\
\noindent
The $w_{an}$--table is given on the right diagram.
Hence the wished statement can be read from it.
It is instructive to compare the above analytic $w_{an}$--table with the
$w_{top}=\chi$--table from above.
\vspace{2mm}
We note that this topological type (identified by the graph $\Gamma$) admits a splice quotient analytic structure as well.
WE expect that for this analytic structure
$\bH^*_{an,}(X,o)=\bH^*_{top,0}(M)$ and $\RR_{an,0}(X,o)=\RR_{top,0}(M)$.
\end{example}
\begin{example} The following deformation was communicated to us by Ignacio Luengo.
Let us consider the deformation of hypersurface singularities
$zy^3+x^5+z^{11}+tx^2y^2=0$, where the deformation parameter $t$ is small.
Along the deformation the following objects stay constant: the (non--degenerate)
Newton diagram, the Milnor number, the geometric genus,
the link (and even the embedded topological type). The stable resolution graph is
\begin{picture}(300,45)(20,0)
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\put(125,35){\makebox(0,0){\small{$-2$}}}
\put(150,35){\makebox(0,0){\small {$-1$}}}
\put(175,35){\makebox(0,0){\small{$-17$}}}
\put(200,35){\makebox(0,0){\small{$-3$}}}
\put(160,5){\makebox(0,0){\small{$-3$}}}
\end{picture}
Since the graph is AR, and $p_g=\min_{\gamma}\, eu \bH^0_{top}(\gamma, \Gamma, -Z_K)$, cf. \cite{NSig},
by \ref{ex:ARAN},
$\bH^*_{an,0}=\bH^*_{top,0}$ for any $t$. Since $\bH^*_{top,0}$ is $t$-stable, we obtain that
$\bH^*_{an,0}$ must stay $t$-stable as well.
In this example the point is that the deformation is not
$\mu^*$--stable ($\mu_1^{t=0}=11$ while $\mu_1^{t\not=0}=10$) (that is,
the deformation does not admit a strong simultaneous resolution). However, it still supports the above conjecture:
the stability of $\bH_{an,0}^*$.
\end{example}
\begin{example}\label{ex:Topjumps}
Consider the deformation of hypersurface singularities
$(X_t,o)=\{x^7+y^5+z^3+tx^4y^2=0\}$, $t\in (\C,0)$.
If $t=0$ then $(X_{t=0},0)$ is weighted homogeneous with the following minimal resolution graph, where the unmarked
vertices have $(-2)$ decorations.
\begin{picture}(300,45)(20,0)
\put(125,25){\circle*{4}} \put(150,25){\circle*{4}}
\put(175,25){\circle*{4}} \put(200,25){\circle*{4}}
\put(225,25){\circle*{4}}
\put(350,25){\circle*{4}} \put(250,25){\circle*{4}}
\put(275,25){\circle*{4}} \put(300,25){\circle*{4}}
\put(325,25){\circle*{4}}\put(375,25){\circle*{4}}
\put(275,5){\circle*{4}}
\put(125,25){\line(1,0){250}}
\put(275,25){\line(0,-1){20}}
\put(285,5){\makebox(0,0){\small{$-3$}}}
\end{picture}
For $t\not=0$ the germ $(X_t,0)$ has nondegenerate Newton principal part, its graph is
\begin{picture}(300,45)(-20,0)
\put(125,25){\circle*{4}} \put(150,25){\circle*{4}}
\put(175,25){\circle*{4}} \put(200,25){\circle*{4}}
\put(225,25){\circle*{4}}
\put(250,25){\circle*{4}}
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\put(225,5){\circle*{4}}
\put(150,5){\circle*{4}}
\put(125,25){\line(1,0){175}}
\put(150,25){\line(0,-1){20}}
\put(225,25){\line(0,-1){20}}
\put(235,5){\makebox(0,0){\small{$-3$}}}
\put(160,5){\makebox(0,0){\small {$-3$}}}
\end{picture}
Note that the deformation modifies the Newton diagram, and also the topological type.
However, along the deformation both $K_{\tX}^2$ and $p_g$ remain constant:
$K_{\tX}^2=-12$ and $p_g=4$.
(I.e., the deformation does not admit a weak simultaneous resolution, but it admits
a very weak simultaneous resolution, cf. \cite{LauferWeak}.)
Note also that both graphs are AR. Since both analytic types are Newton nondegenerate,
for both of them $p_g=\min_{\gamma}\, eu \bH^0_{top}(\gamma, \Gamma, -Z_K)$, cf. \cite{NSig}.
Hence, for any fixed $t$, by Proposition \ref{prop:AnLatAR}
the analytic and topological lattice cohomologies agree. In fact, by AR property
$\bH^{\geq 1}_{an,0}=0$ in all cases. On the other hand, for both $t=0$ and $t\not=0$,
the topological graded roots can be computed by the AR--algorithm and it turns out that they agree.
Hence $\bH^0_{an,0}$ and the analytic graded root
is independent of $t$ as well. The (common) graded root agrees with the topological graded root from the Example
\ref{ex:CONTtwocuspsAN}.
In particular, since $\bH^*_{an,0}$ is $t$--independent, the example supports the Conjecture, and it is also
compatible with Conjecture 11.3.51 from \cite{Book}, which predicts that along such deformations
the topological graded root stays constant.
\end{example}
\begin{problem} {\it
Consider the equisingular family of hypersurface singularities with Newton nondegenerate principal part associated with
a Newton diagram $N\Gamma$.
Is it true that $\bH^*_{an,0}$ is constant along this family?
If yes (hence $\bH^*_{an,0}$ is determined by $N\Gamma)$ describe it combinatorially from the Newton diagram $N\Gamma$.}
\end{problem}
\section{Combinatorial lattice cohomology} \label{ss:CombLattice}
In this section we prove several combinatorial statements regarding the lattice cohomology
associated with any weight function with certain combinatorial properties.
Here we also indicate that the geometric situations where some
analytic weight function can be defined (hence a lattice cohomology too) is very diverse and rich.
\subsection{The combinatorial setup}\label{ss:combsetup}
\bekezdes \label{bek:comblattice}
Fix $\Z^s$ with a fixed basis $\{E_v\}_{v\in\cV}$.
Write $E_I=\sum_{v\in I}E_v$ for $I\subset \cV$ and $E=E_{\cV}$.
Fix also an element
$c\in \Z^s$, $c\geq E$.
Consider the lattice points $R=R(0,c):=\{l\in\Z^s\,:\, 0\leq l\leq c\}$, and assume that
to each $l\in R$ we assign
(i) an integer $h(l)$ such that $h(0)=0$ and $h(l+E_v)\geq h(l)$
for any $v$,
(ii) an integer $h^\circ (l)$ such that $h^\circ (l+E_v)\leq h^\circ (l)$
for any $v$.
\noindent
Once $h$ is fixed with (i),
a possible choice for $h^\circ $ is
$h^{sym}$, where $h^{sym}(l)=h(c-l)$. Clearly, it depends on $c$.
We consider the set of cubes $\{\calQ_q\}_{q\geq 0}$ of $R$ as in \ref{9complex} and
the weight function
$$w_0:\calQ_0\to\Z\ \ \mbox{by} \ \ w_0(l):=h(l)+h^\circ (l)-h^\circ (0).$$
Clearly $w_0(0)=0$.
Furthermore, similarly as in \ref{9dEF1}, we define
$w_q:\calQ_q\to \Z$ by $ w_q(\square_q)=\max\{w_0(l)\,:\, l \
\mbox{\,is a vertex of $\square_q$}\}$. We will use the symbol $w$ for the
system $\{w_q\}_q$.
The compatible weight functions define the lattice cohomology $\bH^*(R,w)$.
Moreover, for any increasing path $\gamma$ connecting 0 and $c$
we also have
a path lattice cohomology $\bH^0(\gamma,w)$ as in \ref{bek:pathlatticecoh}. Accordingly, we have the numerical
Euler characteristics $eu(\bH^*(R,w))$, $eu(\bH^0(\gamma,w))$ and $\min_\gamma eu(\bH^0(\gamma,w))$ too.
\begin{lemma}\label{lem:comblat} We have $0\leq eu(\bH^0(\gamma,w))\leq h^\circ (0)-h^\circ (c)$
for any increasing path $\gamma$ connecting 0 to $c$.
The equality $eu(\bH^0(\gamma,w))=h^\circ (0)-h^\circ (c)$
holds if and only if for any $i$
the differences $h(x_{i+1})-h(x_i)$ and $h^\circ (x_{i})-h^\circ (x_{i+1})$ simultaneously are not nonzero.
\end{lemma}
\begin{proof} {\it (a)} Since $w_0(0)=0$ we have
$eu(\bH^0(\gamma,w))=-\min\, w_0+{\rm rank} \bH_{red}^0(\gamma,w)\geq 0$.
Next, by \ref{eq:pathweights} we have $eu(\bH^0(\gamma,w))=\sum_{i=0}^{t-1}
\max\{0,w(x_i)-w(x_{i+1})\}$.
On the other hand,
$w(c)+\sum_i
\max\{0,w(x_i)-w(x_{i+1})\}=-\sum_i
\min \{0,w(x_i)-w(x_{i+1})\}$, hence
$w(c)+ 2\cdot eu(\bH^0(\gamma,w))=\sum_i |w(x_i)-w(x_{i+1}|=
\sum_i |h(x_{i})+h^\circ (x_i)-h(x_{i+1})-h^\circ (x_{i+1})|\leq
\sum_i ( h(x_{i+1})-h(x_i))+\sum_i (h^\circ (x_i)-h^\circ (x_{i+1}))=h(c)+h^\circ (0)-h^\circ (c)$.
\end{proof}
\begin{remark}\label{rem:ketpelda}
The inequality $0\leq eu(\bH^*(R,w))$ is not true in general. Take e.g. the following table
($s=2$, $c=(2,2)$)
with $h^\circ =h^{sym}$:
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\put(140,20){\makebox(0,0){\small{$0$}}}
\put(110,30){\makebox(0,0){\small{$0$}}}
\put(125,30){\makebox(0,0){\small{$0$}}}
\put(140,30){\makebox(0,0){\small{$0$}}}
\put(-10,20){\makebox(0,0){\small{$h:$}}}
\put(90,20){\makebox(0,0){\small{$w_0:$}}}
\put(210,30){\makebox(0,0){\small{$\min w_0=0$}}}
\put(208,10){\makebox(0,0){\small{$eu=-1$}}}
\put(222,20){\makebox(0,0){\small{$\bH^0_{red}=0$,\ $\bH^1=\Z$}}}
\end{picture}
\noindent
The inequality $ eu(\bH^*(R,w))\leq h^\circ (0)-h^\circ (c)$ is not true either, see e.g.
\begin{picture}(100,40)(-40,0)
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\put(40,20){\makebox(0,0){\small{$1$}}}
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\put(40,30){\makebox(0,0){\small{$2$}}}
\put(110,10){\makebox(0,0){\small{$0$}}}
\put(125,10){\makebox(0,0){\small{$-1$}}}
\put(140,10){\makebox(0,0){\small{$-2$}}}
\put(110,20){\makebox(0,0){\small{$-1$}}}
\put(125,20){\makebox(0,0){\small{$0$}}}
\put(140,20){\makebox(0,0){\small{$-1$}}}
\put(110,30){\makebox(0,0){\small{$-2$}}}
\put(125,30){\makebox(0,0){\small{$-1$}}}
\put(140,30){\makebox(0,0){\small{$0$}}}
\put(-10,20){\makebox(0,0){\small{$h:$}}}
\put(90,20){\makebox(0,0){\small{$w_0:$}}}
\put(200,20){\makebox(0,0){\small{\mbox{(with $h^\circ =h^{sym}$)}}}}
\end{picture}
\noindent
Then $eu(\bH^0(\gamma,w))=2$, while $eu(\bH^0(R,w))=eu(\bH^*(R,w))=4$.
\end{remark}
\begin{example}\label{ex:latgrfilt} {\bf Graded and filtered vector spaces.}
In several geometrical constructions we face the following situation:
we have $\Z^s$ and $c$ as above, and a finite dimensional vector space $M$ with a $\Z^s$-grading $\{M_{{\bf a}}\}_{{\bf a}}$
such that $M_{{\bf a}}=0$ whenever either ${\bf a}\not\geq 0$ or ${\bf a}\geq c$. Let
$\hh$ be the Hilbert function
$\hh(l)=\sum_{{\bf a}\not \geq l}\,\dim M_{{\bf a}}$ and let $h$ be its restriction to $R(0,c)$.
Then $h(0)=0$ and $h(c)=\dim\, M$.
More generally, assume that $M$ is a finite dimensional vector space endowed with a decreasing $\Z^s$-filtration such that
$F(0)=M$ and $F(c)=0$ and define $\hh(l)=\dim (M/F(l))$ for any $l\geq 0$.
Again, define $h$ as the restriction of
$\hh$ to $R$.
The $h$--function associated with a filtration satisfies
the {\it `matroid rank inequality'}
\begin{equation}\label{eq:matroid}
h(l_1)+h(l_2)\geq h(\min\{l_1,l_2\})+h(\max\{l_1,l_2\}), \ \ l_1,l_2\in R.
\end{equation}
This implies the {\it `stability property'},
valid for any $\bar{l}\geq 0$ with $|\bar{l}|\not\ni E_v$
\begin{equation}\label{eq:stability}
h(l)=h(l+E_v)\ \ \Rightarrow\ \ h(l+\bar{l})=h(l+\bar{l}+E_v).
\end{equation}
Such a filtration might appear as follows. Take a ring $\cO$, endowed with $s$ valuations
$\frakv_i :\cO\to \Z_{\geq 0}\cup \{\infty\}$ and set for any $l=(l_1,\ldots, l_s)\in \Z^s$ the ideal
$F_{\cO}(l):=\{f\in\cO\,:\, \frakv_i(f)\geq l_i \ \mbox{for all $i$}\}$. Then, for a conveniently chosen
$c\geq E$ we take $M=\cO/F(c)$ and the filtration of $M$ given by
$F(l)=(F_{\cO}(l)+F_{\cO}(c))/F_{\cO}(c)$.
However, sometimes is preferable to keep the original setup $(\cO, \{F_{\cO}(l)\}_l)$ $(l\geq 0)$
since it can be connected directly to several classical invariants defined at the level of $\cO$.
This is possible since
$\dim ( M/F(l))=\dim \cO/F_{\cO}(l)$ for any $l\in R$.
For example, the {\it valuation semigroup} is defined as $\calS_{\cO}:=\{(\frakv_1,\ldots ,\frakv_s)(f)\ :\ f\in\cO\}$.
In our case (once $c$ is fixed) we will be interested only in its part $\{l\, :\ l\not\geq c\}$.
In the case of the divisorial filtration of surface singularities, or
in the case of the valuative filtration associated with plane curve singularities (via its normalization),
$\calS_{\cO}$ and the Hilbert function $\{\hh(l)\}_{l\geq 0}$ are related as follows:
\begin{equation}\label{eq:Sfromh}
\calS_{\cO}=\{l\ :\ \hh(l+E_v)>\hh(l)\ \mbox{for all $v$}\};
\end{equation}
and, for any $v\in\cV$,
\begin{equation}\label{eq:hfromS}
\hh(l+E_v)>\hh(l) \ \mbox{exactly when there exists $s\in \calS_{\cO}$ with $s\geq l$, $s_v=l_v$.}
\end{equation}
Additionally, in the presence of certain (e.g. Gorenstein) duality, the ring $\cO$ and the filtration $F_\cO$
might have the following properties:
\vspace{2mm}
\noindent {\it Combinatorial Duality Property of \ $\calS_\cO$}:
\begin{equation}\label{eq:CombGorS}
\mbox{there is no $v\in \cV$ and $s\in\calS_\cO$ so that $s_v=c_v-1$ and $s\geq c-E_v$;}
\end{equation}
\noindent {\it Combinatorial Duality Property of \ $\hh$}:
\begin{equation}\label{eq:CombGorh}
\left\{
\begin{array}{ll}\mbox{there is no $v\in \cV$ and $l\in R$ so that}\\
\mbox{$\hh(l+E_v)>\hh(l)$ and $\hh(c-l)>\hh(c-l-E_v)$.}\end{array}\right.
\end{equation}
Note that in the presence of
(\ref{eq:Sfromh}), (\ref{eq:hfromS}) the properties (\ref{eq:CombGorS}) and
(\ref{eq:CombGorh}) are equivalent (since \, $0\in\calS_{\cO}$).
\end{example}
\bekezdes\label{bek:LCgen}
In order to analyse more general cases, which are
not necessarily provided by filtrations, we wish to adjust the above
properties
in the language of an arbitrary system $(h,h^\circ,R)$ as in \ref{bek:comblattice}.
\begin{definition}\label{def:COMPGOR}
Fix $(h,h^\circ,R)$ as in \ref{bek:comblattice}.
We say that the pair $h$ and $h^\circ$ satisfy the `Combinatorial Duality Property' (CDP) if
$h(l+E_v)-h(l)$ and $h^\circ (l+E_v)-h^\circ (l)$ simultaneously cannot be nonzero
for $l,\, l+E_v\in R$. Furthermore,
we say that $h$ satisfies the CDP if
the pair $(h,h^{sym})$ satisfies it.
\end{definition}
\begin{example}\label{ex:CDP3pelda}
(1) If $\phi$ is the resolution of a normal surface singularity, $\hh$ is associated with the divisorial filtration and
$\hh^\circ(l)=p_g-h^1(\calO_l)$ then the pair $(\hh,\hh^\circ)$ satisfies
the CDP by
\ref{lem:hsimult}.
(2) If $\phi$ is the resolution of a normal surface singularity with $Z_K\in L$. If
$\hh$ is given by the divisorial filtration, and $\hh^{sym}(l)=\hh(Z_K-l)$, then
$\hh(l+E_v)-\hh(l)$ and $\hh^{sym} (l)-\hh^{sym} (l+E_v)$ cannot be simultaneously nonzero.
Indeed, otherwise there exists $s', s''\in\calS_{an}$ such that ${\rm div}_E(s')\geq l$, $s'_v=l_v$,
${\rm div}_E(s'')\geq Z_K-l- E_v$, $s''_v=(Z_K-l)_v$. Hence,
${\rm div}_E(s's'')\geq Z_K-E_v$ with equality at $E_v$--coordinate.
But this contradicts
$H^0(\calO_{E_v}(-Z_K+E_v)) =0$. (Compare with (\ref{eq:CombGorS})
applied for $s=s_1s_2\in \calS_{an}$ and $c=Z_K$.)
(3) Let $\hh$ be the Hilbert function associated with a Gorenstein curve singularity
$(C,o)$ (via valuations given by the normalization).
Let $c$ be the conductor. Then $\hh(l+E_v)-\hh(l)\in \{0,1\}$. Furthermore,
$\hh(l+E_v)-\hh(l)=1$ if and only if
$\hh^{sym} (l)-\hh^{sym} (l+E_v)=0$. This follows from the identity $h(c-l)-h(l)=\delta(C)-\sum_vl_v$, cf. \cite{cdk}.
For the general treatment of the analytic lattice cohomology associated with curve singularities see
\cite{AgNeCurves}.
\end{example}
\begin{example}\label{ex:rankone} {\bf Rank one case.} Take $R$ and $h$ as in \ref{bek:comblattice}.
Assume that $s=1$.
In this case, from the point of view of $h$,
we can assume that we are in the graded vector space case, cf. \ref{ex:latgrfilt}.
Indeed, in this case $h$ is associated with $M=\C^{h(c)}$ graded as
$M_l=\C^{h(l+1)-h(l)}$ for $l<c$ and $M_c=0$. Moreover, if $h^\circ=h^{sym}$, then
$$w_0(l)=\sum_{s<l} \,\dim M_s+ \sum_{s<c-l}\, \dim M_s \ -h(c).$$
If $h$ satisfies the CDP then
(cf. \ref{lem:comblat})
$$w_0(l+1)-w_0(l)=\left\{\begin{array}{ll}
\ \ \ \ \dim\, M_l & \mbox{if $h(l+1)>h(l)$}\\
-\dim\, M_{c-1-l} \ \ \ \ & \mbox{if \ $h(c-l)>h(c-l-1)$}\\
\ \ \ \ 0 & \mbox{otherwise.}
\end{array}\right.$$
Therefore, by a similar identity as in (\ref{eq:Ecal})
\begin{equation}\label{eq:eurankone}
eu ( \bH^*(R,h))=\sum_{l=0}^{c-1}
\big(\, w_1([l,l+1])-w_0(l)\,\big)=\sum_{l=0}^{c-1}\dim \, M_l=\dim\, M.
\end{equation}
\end{example}
\begin{definition}\label{def:comblat}
We say that the pair $(h, h^\circ) $ satisfy the
(a) {\it `path eu-coincidence'} if $eu(\bH^0(\gamma,w))=h^\circ (0)-h^\circ (c)$
for any increasing path $\gamma$.
(b) {\it `eu-coincidence'} if $eu(\bH^*(R,w))=h^\circ (0)-h^\circ (c)$.
\end{definition}
\begin{remark}\label{rem:UJ}
In the Example \ref{ex:twonodesB}
we present a $w_0$--rectangle $R=R((0,0),(14,14))$
can be realized as the $w_0$-table associated with a certain $h$ and $h^\circ=h^{sym}$
(provided by a
graded vector space). This diagram satisfies both the `path eu-coincidence' and `eu-coincidence'
properties, and it shows
the following two facts.
Even if $h$ satisfies the path eu-coincidence (and $h^\circ =h^{sym}$),
in general it is not true that $\bH^0(\gamma,w)$
is independent of the choice of the increasing path.
(This statement remains valid even if we consider only the symmetric increasing paths, where a
path $\gamma=\{x_i\}_{i=0}^t$ is symmetric if $x_{t-l}=c-x_l$ for any $l$.)
Even if $h$ satisfies both the path eu-coincidence and the eu-coincidence,
in general it is not true that $\bH^*(R,w)$ equals any of the path lattice cohomologies
$\bH^0(\gamma,w)$ associated with a certain increasing path.
Indeed, in the present mentioned case of \ref{ex:twonodesB} we have $\bH^1(R,w)\not=0$, a fact which does not hold for any
path lattice cohomology. However, amazingly, all the Euler characteristics agree.
\end{remark}
\begin{remark}\label{ex:WCGP} In the next discussion assume that $h^\circ =h^{sym}$.
(a)
The CDP of $h$ implies the `path eu-coincidence' of $h$, see
\ref{lem:comblat}.
However, the CDP of $h$ does not
imply the `eu-coincidence' of $h$.
As an example consider the second case from \ref{rem:ketpelda}.
Note that in this case the matroid or stabilization properties are not satisfied by $h$.
(b) On the other hand, an eu-coincidence type property cannot be hoped without
(some type of) CDP. Indeed, in the next example
$s=2$, $c=(2,2)$, and $h$ is associated with a
graded vector space of dimension 2 supported in $(0,1)$ and $(1,2)$.
In this case $h(c)=2$, $eu(\bH^0(R,w))=0$ and for any symmetric increasing path
$eu(\bH^0(\gamma,w))=0$ too, and for
any non-symmetric paths
$eu(\bH^0(\gamma,w))=1$.
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\put(125,30){\makebox(0,0){\small{$0$}}}
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\put(-10,20){\makebox(0,0){\small{$h:$}}}
\put(90,20){\makebox(0,0){\small{$w_0:$}}}
\end{picture}
\end{remark}
\subsection{The Euler characteristic formulae}\label{ss:combEU}
\begin{theorem}\label{th:comblattice}
Assume that $h$ satisfies the stability property, and the pair $(h,h^\circ)$
satisfies the Combinatorial Duality Property. Then $(h,h^\circ)$
satisfies both the path eu- and the eu-coincidence properties: for any increasing $\gamma$ we have
$$eu(\bH^*(\gamma,w))=eu(\bH^*(R,w))=h^\circ (0)-h^\circ (c).$$
\end{theorem}
\begin{proof}
The identity $eu(\bH^*(\gamma,w))=h^\circ (0)-h^\circ (c)$ follows from \ref{lem:comblat}. Next we focus on the second identity.
We claim that for any $I\subset \calv$ we have
\begin{equation}\label{eq:KEYIDEN}
w((l,I))-w(l)=h(l+E_I)-h(l).\end{equation}
We use induction over the cardinality $|I|$ of $I$. If $I=\{v\}$, then $w((l,I))-w(l)=\max\{ 0, w(l+E_v)-w(l)\}$. But if
$h(l+E_v)>h(l)$ then $h^\circ (l+E_v)=h^\circ (l)$
hence $w(l+E_v)-w(l)=h(l+E_v)-h(l)$. Otherwise $w(l+E_v)\leq w(l)$ and $w((l,I))=w(l)$.
Next, assume that $|I|>1$ and $h(l+E_v)=h(l)$ for every $v\in \calv$. Then by iterated use of the
stability property of $h$, $h(l+E_J)
=h(l)$ for any $J\subset I$. Moreover, $w((l,I))-w(l)=0=h(l+E_I)-h(l)$.
Finally, assume that $|I|>1$ and $h(l+E_v)>h(l)$ for a certain $v\in \calv$.
This means that $w(l+E_v)>w(l)$, hence $w((l,I))>w(l)$ ($\dag$).
Assume that $w((l,I))$ is realized for a certain $J\subset I$.
Let $J$ be minimal by this property. By ($\dag$) we know that $J\not=\emptyset$.
By minimality of $J$, $w(l+E_{J\setminus u})<w(l+E_J)$ for any $u\in J$. By CDP $h(l+E_{J\setminus u})<h(l+E_J)$ too,
hence by the stability of $h$ we also have $h(l)<h(l+E_u)$.
In particular we found $u\in I$ such that $h(l)<h(l+E_u)$
(hence $h^\circ (l)=h^\circ(l+E_u)$) and $w((l, I))=w((l+E_u, I^*))$, where $I^*:= I\setminus u$.
Now, we use induction applied for
the cube $(l+E_v, I^*)$. In particular,
$w((l,I))-w(l)=
w((l+E_v, I^*))-w(l)= w(l+E_v)+h(l+E_I)-h(l+E_v)-w(l)=
h(l+E_I)-h(l)$.
This ends the proof of the claim.
Let us denote by $\calQ$ the set of cubes of $R$.
For $eu(\bH^*(R,w))$ we use the following formula (with identical proof as in \ref{bek:LCSW})
$$eu(\bH^*(R,w))=\sum_{(l,I)\in\calQ}\, (-1)^{|I|+1} w((l,I)).$$
We will subdivide the lattice points of $R$ into the following disjoint subsets.
For any $0\leq t\leq s$ we denote by $R_t$ the set of those points $l\in R\cap L$ for which
the cardinality of $\{v\in\calv\,:\, l_v=c_v\}$ is $t$. If $l\in R_t$ set $I(l):=\{v\,:\, l_v<c_v\}$.
In particular, the set $\calQ$ is also a disjoint union
of the sets
$ \{(l, I)\,:\, l\in R_t, \ I\subset I(l)\}_t$. Then
$$eu(\bH^*(R,w))=\sum_t \, \sum_{l\in R_t}\, \sum_{I\subset I(l)}\, (-1)^{|I|+1} w((l,I)).$$
If $0\leq t<s$ and $l\in R_t$ then $I(l)\not=\emptyset$. In this case for any
$I$-independent (but maybe $l$-dependent) `constant' $a(l) $ we have $\sum _{I\subset I(l)}
(-1)^{|I|+1}a(l)=0$.
Therefore, by (\ref{eq:KEYIDEN}), for any such $l$,
\begin{equation}\label{eq:SUMt} \sum_{I\subset I(l)}\, (-1)^{|I|+1} w((l,I))=
\sum_{I\subset I(l)}\, (-1)^{|I|+1} h(l+E_I).\end{equation}
On the other hand, if $t=s$, then $R_t=\{c\}$, $I(c)=\emptyset$, hence
$\sum _{I\subset I(c)}
(-1)^{|I|+1}((w(c,I))=-w(c)$.
Hence, corresponding to $t=s$ (\ref{eq:SUMt}) fails, i.e. it must be corrected by
$-w(c)+h(c)=h^\circ (0)-h^\circ(c)$.
Therefore,
\begin{equation}\label{eq:SUMtt}
eu(\bH^*(R,w))=h^\circ (0)-h^\circ(c)\, +\, \sum_{(l,I)\in\calQ}\, (-1)^{|I|+1} h(l+E_I).
\end{equation}
For any fixed $\widetilde{l}\in R$, consider the following summation over $\{(l,I)\,:\,
l+E_I=\widetilde{l}\}$:
$$S(\widetilde{l}):=\sum \, (-1)^{|I|+1}h(l+E_I)=h(\widetilde{l})\cdot \sum \, (-1)^{|I|+1}.$$
Then, whenever the cardinality of $\{(l,I)\,:\,
l+E_I=\widetilde{l}\}$ is $>1$,
the above sum $\sum \, (-1)^{|I|+1}=0$.
Thus, $S(\widetilde{l})=0$ except when $\widetilde{l}=0$. But, for $\widetilde{l}=0$,
$S(0)=-h(0)=0$ too. Thus
$eu(\bH^*(R,w))=h^\circ (0)-h^\circ(c)+\sum_{\widetilde{l}}S(\widetilde{l})=h^\circ (0)-h^\circ(c)$.
\end{proof}
\begin{remark}\label{rem:P0AN}
The very same argument as in the previous proof provides the following fact.
Assume that instead of the rectangle $R(0,c)$ we take $L_{\geq 0}$ (i.e., $c=\infty$), and we also have
two functions $h, \, h^\circ:L_{\geq 0}\to \Z$ with the very same properties (i) and (ii) as in \ref{bek:comblattice}.
Furthermore, assume both that $h$ satisfies the stability property, and the pair $(h,h^\circ)$
satisfies the Combinatorial Duality Property (as in \ref{th:comblattice}). Then (\ref{eq:KEYIDEN}) implies
\begin{equation}\label{eq:sumsum}\sum_{l\geq 0}\,\sum _I\, (-1)^{|I|+1} w((l,I))\, \bt^{l}=
\sum_{l\geq 0}\,\sum _I\, (-1)^{|I|+1} h(l+E_I)\,\bt^{l}.\end{equation}
In the case of a normal surface singularity as in \ref{ss:anR}, the right hand side of (\ref{eq:sumsum})
is the $(h=0)$--component $P_0(\bt)$ of the multivariable Poincar\'e series (obtained from the Hilbert series),
cf. \cite{CDG,CHR,NJEMS}. Hence, the above identity recovers $P_0(\bt)$
in terms of the analytic weighted cubes.
This formula can be compared with its topological analogue:
there is an identical formula (of identical nature) for the topological multivariable series,
which is recovered in terms of the topological weighted cubes.
\end{remark}
\subsection{Examples}\label{ss:combExa}
\begin{example}\label{ex:NNGamma} {\bf Graded vector spaces determined by a Newton diagram.}
Assume that $N\Gamma$ is a convenient Newton diagram in $\R_{\geq 0}^{n+1}$ (see e.g. \cite{NSig}).
Let $\triangle^{(1)}$ denote the $n$-dimensional faces of $N\Gamma$, and for each $\sigma\in \triangle^{(1)}$
let $\ell_\sigma$ be the normal primitive integral vector of the corresponding face $F_\sigma$ (with all
entries positive). Then
$F_\sigma$ is on the affine $n$-plane $\langle \ell_\sigma,p\rangle=m_\sigma$.
Let $M$ be the $\C$--vector space generated by lattice points
$p\in \Z_{>0}^{n+1}$, which sit either on or below $N\Gamma$. This means that
$p\in \Z_{>0}^{n+1}$ should satisfy
$\langle \ell_\sigma,p\rangle\leq m_\sigma$ for at least one $\sigma$.
Then $M\not=0$ if and only if ${\bf 1}=(1,\ldots , 1)$ is such a point; in the sequel we will assume this fact.
We order $\triangle ^{(1)}$ as $\{\sigma_1,\ldots, \sigma_s\}$. We introduce a $\Z^s$-grading of $M$ by
$${\rm deg}(p)=(\, \langle \ell_{\sigma_1}, p-{\bf 1}\rangle, \ldots,
\langle \ell_{\sigma_s}, p-{\bf 1}\rangle \, )\in \Z^s.$$
Define $c=(c_1,\ldots , c_s)$ by
$c_i:= m_{\sigma_i}+1-\langle \ell_{\sigma_i}, {\bf 1}\rangle$.
Since $p\geq {\bf 1}$ for any such point,
we also have ${\rm deg}(p)\in \Z_{\geq 0}^s$. Moreover, for any $p$ there exists
$\sigma_i$ with $\langle \ell_{\sigma_i}, p\rangle\leq m_{\sigma_i}$, hence
$\langle \ell_{\sigma_i}, p-{\bf 1}\rangle <c_i$. Therefore, ${\rm deg}(p)\not\geq c$.
In particular, the conditions of \ref{ex:latgrfilt} are satisfied.
\end{example}
\begin{example}\label{ex:2552}
Consider the situation of \ref{ex:NNGamma} with $n=1$ and $N\Gamma$ generated by the lattice points
$(0,7), (2,2), (7,0)$. In fact, this is the Newton diagram of the Newton nondegenerate
plane curve singularity $f=(x^2+y^5)(y^2+x^5)$.
The normal vectors $(2,5)$ and $(5,2)$ of the two faces provide the degrees.
The next diagram shows the points $p$, which generate $M$ as base elements, and their degrees.
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\end{picture}
\noindent
The vector $c$ is $(8,8)$: this is exactly the conductor of the plane curve singularity $f$.
Furthermore, $\dim M=8$ is the delta-invariant $\delta(f)$ of $f$.
The next tables show the function $h$ and the weight function $w_0$ on $R$.
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\noindent
The circled points show the generators of $\bH^1(R,w)$, while the boxes the local minima of $w_0$.
Hence $\bH^1(R,w)=\calt_{-2}(1)^2$ and $\bH^0(R,w)=\calt^+_{-4}\oplus
\calt_{-4}(1)^6\oplus \calt_0(1)^2$.
The graded root is
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\noindent
$\min w_0=-2$, ${\rm rank}\bH^0_{red}(R,w)=8$, ${\rm rank}\bH^1(R,w)=2$, and $eu(\bH^*(R,w))=8=\dim M=h(c)$.
In particular, we constructed a set of cohomology groups $\bH^*(R, w)$ whose Euler characteristic is the
delta-invariant. This is a `categorification of the $\delta$--invariant'.
This case of curves will be fully exploited in \cite{AgNeCurves}.
From analytic point of view, one can show that
the above function $h$ is the Hilbert function associated with the
filtration of valuations associated with the normalization.
This can be verified e.g. by using the
formula of the analytic Hilbert function via the multivariable Alexander polynomials, see \cite{GorNem2015}.
In particular, in this case, the combinatorial lattice cohomology associated with
the lattice points below the diagram agrees with the analytic lattice cohomology
associated with the valuations given by the normalization.
(Nevertheless, we warn the reader
that for a more general $N\Gamma$, several filtrations --- projected, order, valuative ---
might all be different.)
\end{example}
\begin{example}\label{ex:twonodesB}
Consider the hypersurface $x^{13}+y^{13}+x^2y^2+z^3$ with Newton nondegenerate principal part.
Its dual resolution graph is
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Let $N\Gamma$ be its
Newton diagram. It has two faces with normal vectors
$(6,33,26)$ and $(33,6,26)$. The number of lattice point under the diagram is $p_g=5$, they are
$(1,1,1)$, $(2,1,1)$, $(3,1,1)$, $(1,2,1)$, $(1,3,1)$. Furthermore, $c$ (defined as in \ref{ex:NNGamma})
is exactly the restriction of $Z_K$ to the coordinates of the nodes (cf. \cite{NSig,NSigNN}).
Hence, in this case $R=R(0,Z_K|_{\tiny{\mbox{nodes}}})=R((0,0), (14,14))$. Then a computation shows that
the combinatorial weight function $w_0:R\to \Z$ associated with $(M, {\rm deg})$ from \ref{ex:NNGamma}
(that is, associated with the Newton diagram) is exactly the
table obtained by the procedure of the (topological) Reduction Theorem applied for the topological
RR-function $\chi$. Hence we have the coincidence of the corresponding lattice cohomologies as well:
$\bH^*(R,w)=\bH^*_{top}(M)$.
Furthermore, for this diagram, the weight function associated with the
divisorial filtration restricted to $R$ agrees with combinatorial weight function as well (see e.g. \cite{Lem,BaldurFiltr}).
These three common tables are the following:
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In particular, for this germ, $\bH^*_{an,0}(X,o)=\bH^*_{top,0}(M)$.
From the point of view of deformations, let us consider the following Newton nondegenerate germs:
$\{x^{13}+y^{13}+x^3y^2+x^2y^3+z^3=0\}$, or $\{x^{14}+y^{14}+x^2y^2+z^3=0\}$, or
$\{x^{13}+y^{9}z+x^2y^2+z^3=0\}$. All of them are $p_g$-constant deformations of
the germ $\{x^{13}+y^{13}+x^2y^2+z^3=0\}$ treated above. Then for any of them, the
corresponding $w_0$--table (determined by the Newton filtration)
in the rectangle given by $Z_K$ reduced to the nodes agree with the corresponding $\chi$--table
(though, the resolution graphs are rather different, and the size of $R(0,Z_K|_{\tiny{\mbox{nodes}}})$ is also changing). Furthermore, in all these cases,
the identity
$\bH^*_{an,0}(X,o)=\bH^*_{top,0}(M)$ holds, and this module stays stable under the corresponding deformations.
\end{example}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,383 |
El Odivelas Futebol Clube es un club de fútbol portugués de la ciudad de Odivelas. Fue fundado en 1939 y juega en la Tercera División de Portugal.
Jugadores
Plantilla 2009/10
Enlaces externos
Sitio web oficial
Odivelas
Odivelas | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,128 |
Vhils works in a variety of mediums but he is well known for his chiseled wall portraits of ordinary citizens. Through a combination of paint, chiseling and negative space, Vhils portraits come to life, projecting from the walls like a relief painting. Be sure to check out his personal website, alexandrefarto.com to see all of his fascinating artwork. There's also a video at the bottom of the post that documents the process he uses to create his chiseled wall portraits.
See More from Vhils on his Official Site! | {
"redpajama_set_name": "RedPajamaC4"
} | 5,155 |
TWITTER LOCAL: 5 Twitter Geolocation Features We Want
By Jennifer Van Grove 2009-08-20 17:17:56 UTC
Twitter's Geolocation API means that tweets are soon to be location-aware. Should you opt-in and share your geolocation with developers, your location in the physical world — ie. your longitude and latitude — will be associated with your tweets.
Geolocation and Twitter can make a beautiful pairing, as we've already seen with apps like Twinkle, Twitterfon, Ubertwitter, and Tweetmondo. These apps let you filter your timeline for tweets within a radius of your location, find nearby users, and view tweets on a map.
But there's still so much room to grow, because, with Twitter's Geolocation API, each tweet is like a check-in at your location, minus the extra step of checking in. So Twitter and Twitter app developers, you have a big task at hand. We think you're perfectly poised to build the dream location-aware apps we've always wanted, and we're going to point you toward our wishlist of features that we hope to see come about. This is just scratching surface, so if you've got a great location-based tweet idea, share it in the comments.
1. Trending Places
Twitter already has trending topics. We love the utility of knowing what's hot right now, and we especially love that some Twitter apps are built entirely around slicing and dicing tweets and memes around the trending topics. Now we want all of that goodness, but we want it in relation to places, and we want the power of Twitter search behind it too.
Perhaps we could see trending places worldwide, or filter by country, city, state, or town, but either way we think there's immediate value in seeing where we congregate. We think this could work by measuring the fluctuation in tweets per area, and we think it could be pretty cool.
2. Customizable DMs for Places
Tweetmondo has a "keep me updated" feature that DMs you whenever someone new is at a location of your interest. We like the idea, but we think it can be massively improved and rolled out to a global audience.
Here's our thinking. You tell Twitter, or a Twitter app, a location of interest — we're thinking a favorite coffee spot or bar — that location is saved, and you get a DM when someone tweets from the same spot (regardless of whether or not you're following each other). It's the location-based social connector with Twitter as the medium. In this model, you would be able to set the hours for when DMs are acceptable, save your places, and turn place DMs on or off (maybe with an SMS command).
Sounds nifty, right? Then think about all the great data one could harness from saved locations. This is powerful stuff we're playing with.
3. Audio Notifications for Nearby Friends
Tweetmondo has a "keep me updated" feature that DMs you whenever someone new is at a location of your interest. We think that's interesting, but we'd like the idea applied to friends and we'd like to see it in our desktop apps.
Here's what we want. TweetDeck, Seesmic Desktop, and other desktop applications should build in audio notifications for nearby friends. We want the apps to let us set our preferences by friend or by predefined column/group. This way we can have full control (and spying power) over these specialized notifications, giving us an even better system for tapping into a trusted network of friends and their physical location.
4. Push or SMS Notifications for Friends
This ties into to our previous wish, but with mobile devices in particular in mind. We want the same abilities to set preferences for audio notifications, but we want them in a mobile equivalent. We'd like to opt-in to push notifications, and maybe the occasional SMS update, from just specific friends or groups of friends.
5. Tweets as Check-ins
Plenty of us already enjoy the location-based social experience. Foursquare is certainly taking hyperlocal metros by storm, and Brightkite, Whirl, Loopt, and Latitude each have their own following, and each of them are positioned to win with location-aware tweets.
Think about it. Right now the physical act of checking in can be quite tedious, and we'd love to see the option to use our regular tweets as check-ins. This is likely more complicated than it sounds, but if we hand over the right permissions and customized settings, we could see it working out where your location-aware tweet prompts an approved service of choice to ask if we want to check-in there. The asking could happen via SMS, push notification, DM, email, IM, or any other distribution method of our choosing.
Topics: geolocation, location, opinion, Social Media, Twitter, Twitter Lists | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,411 |
"From The Miniature To The Monumental".
of the artist and sculptor Mark Richard Hall. Situated in the leafy hamlet of Cripplestyle, the Naughty Boy Studio is owned by and exclusively dedicated to Mark's work.
Mark works almost exclusively in bronze producing sculpture from the miniature to the monumental. Influences range from nature, childhood reflections and the everyday experiences seen from perhaps an alternate perspective.
Working as an artist for over twenty years Mark has developed an enthusiastic international following. Working hand in hand with his wonderful supporting gallery network, chances are there is someone close by that will be showing his work and will be able to help and assist you in acquiring a piece of his sculpture. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,808 |
class CreateTasks < ActiveRecord::Migration[5.0]
def change
create_table :tasks do |t|
t.string :name, :null => false
t.string :description
t.boolean :copmleted, :default => false
t.string :related_url
t.timestamps
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,088 |
agroforestry a land management system that uses trees or shrubs in cropland to improve sustainability.
agronomy the study and application of scientific methods of soil management and field crop production; scientific agriculture.
agt. abbreviation of "agent," a person with authority to perform certain actions on behalf of another.
ah used to express surprise, joy, pain, agreement, dislike, and other emotions or reactions according to the context.
aha an exclamation, usu. of surprise, mockery, or triumph.
a hard row to hoe a very difficult task or situation.
ahead of time before the time of some event.
ahem a sound as of clearing the throat, used to attract notice, express doubt, or fill in a pause.
ahimsa (Sanskrit) the principle of nonviolence based on the belief in the sacredness of all living creatures, as held by Buddhists, Hindus, and others.
A horizon the uppermost layer of soil in a geological soil profile; topsoil.
ahoy a greeting to attract attention or hail a ship.
ahtomp in Wampanoag language and culture, a weapon used for shooting arrows; bow.
Ahura Mazda in Zoroastrianism, the supreme deity and creator of the world; Ormazd.
ai1 used as an expression of distress, pity, surprise, or the like. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,132 |
{"url":"https:\/\/stats.stackexchange.com\/questions\/18707\/how-do-i-compare-bootstrapped-regression-slopes","text":"# How do I compare bootstrapped regression slopes?\n\nLet us assume I have two data sets with n observations of data pairs of independent variable x and dependent variable y each. Let us further assume I want to generate a distribution of regression slopes for each data set by bootstrapping the observations (with replacement) N times and calculating the regression y = a + bx each time. How do I compare the two distributions in order to say the slopes are significantly different? A U-test for testing the difference between the medians of the distributions would be heavily dependent on N, that is, the more often I repeat the bootstrapping the more significant will be the difference. How do I have to calculate the overlap between the distributions to determine a significant difference?\n\nBootstrapping is done to get a more robust picture of the sampling distribution than that which is assumed by large sample theory. When you bootstrap, there is effectively no limit to the number of bootsamples' you take; in fact you get a better approximation to the sampling distribution the more bootsamples you take. It is common to use $B=10,000$ bootsamples, although there is nothing magical about that number. Furthermore, you don't run a test on the bootsamples; you have an estimate of the sampling distribution--use it directly. Here's an algorithm:\n\n1. take a bootsample of one data set by sampling $n_1$ boot-observations with replacement. [Regarding the comments below, one relevant question is what constitutes a valid 'boot-observation' to use for your bootsample. In fact, there are several legitimate approaches; I will mention two that are robust and allow you to mirror the structure of your data: When you have observational data (i.e., the data were sampled on all dimensions, a boot-observation can be an ordered n-tuple (e.g., a row from your data set). For example, if you have one predictor variable and one response variable, you would sample $n_1$ $(x,y)$ ordered pairs. On the other hand, when working with experimental data, predictor variable values were not sampled, but experimental units were assigned to intended levels of each predictor variable. In a case like this, you can sample $n_{1j}$ $y$ values from within each of the $j$ levels of your predictor variable, then pair those $y$s with the corresponding value of that predictor level. In this manner, you would not sample over $X$.]\n2. fit your regression model and store the slope estimate (call it $\\hat\\beta_1$)\n3. take a bootsample of the other data set by sampling $n_2$ boot-observations with replacement\n4. fit the other regression model and store the slope estimate (call it $\\hat\\beta_2$)\n5. form a statistic from the two estimates (suggestion: use the slope difference $\\hat\\beta_1-\\hat\\beta_2$)\n6. store the statistic and dump the other info so as not to waste memory\n7. repeat steps 1 - 6, $B=10,000$ times\n8. sort the bootstrapped sampling distribution of slope differences\n9. compute the % of the bsd that overlaps 0 (whichever is smaller, the right tail % or the left tail %)\n10. multiply this percentage by 2\n\nThe logic of this algorithm as a statistical test is fundamentally similar to classical tests (e.g., t-tests) but you are not assuming the the data or the resulting sampling distributions have any particular distribution. (For example, you are not assuming normality.) The primary assumption you are making is that your data are representative of the population you sampled from \/ want to generalize to. That is, the sample distribution is similar to the population distribution. Note that, if your data are not related to the population you're interested in, you are flat out of luck.\n\nSome people worry about using, e.g., a regression model to determine the slope if you're not willing to assume normality. However, this concern is mistaken. The Gauss-Markov theorem tells us that the estimate is unbiased (i.e., centered on the true value), so it's fine. The lack of normality simply means that the true sampling distribution may be different from the theoretically posited one, and so the p-values are invalid. The bootstrapping procedure gives you a way to deal with this issue.\n\nTwo other issues regarding bootstrapping: If the classical assumptions are met, bootstrapping is less efficient (i.e., has less power) than a parametric test. Second, bootstrapping works best when you are exploring near the center of a distribution: means and medians are good, quartiles not so good, bootstrapping the min or max necessarily fail. Regarding the first point, you may not need to bootstrap in your situation; regarding the second point, bootstrapping the slope is perfectly fine.\n\n\u2022 Although I may well be wrong, I thought the bootstrap in regression had to be on the residuals rather than on the raw data, to be validated... Nov 22 '11 at 14:05\n\u2022 @Xi'an, I've been wrong before myself, but I don't understand why you think only bootstrapping residuals is valid. Efron & Tibshirani (1994) section 9.5 says \"Bootstrapping pairs is less sensitive to assumptions than bootstrapping residuals. The standard error obtained by bootstrapping pairs gives reasonable answers even if [the probability structure of the linear model] is completely wrong.\" The implication is that application of the bootstrap is more robust, although they imply it can be less efficient in some cases. Nov 25 '11 at 2:46\n\u2022 My worry with bootstrapping the pairs is that you also incorporate the distribution of the predictors, which is usually left outside of the picture in regular linear models. That's why I always teach my students to bootstrap only the residuals. Nov 25 '11 at 8:11\n\u2022 @Xi'an, that is a reasonable point, I suppose I was assuming an observational data structure. I have edited my answer to add more detail about these concerns. However, I do not see how that implies that bootstrapping pairs is necessarily invalid. Nov 25 '11 at 17:23\n\u2022 Forcing a pairing between two independent sets of data is artificial and inefficient. You can do much better than that!\n\u2013\u00a0whuber\nNov 25 '11 at 17:57\n\nYou can combine the two data sets into one regression. Let $s_i$ be an indicator for being in the first data set. Then run the regression $$\\begin{equation*}y_i = \\beta_0 + \\beta_1 x_i + \\beta_2 s_i + \\beta_3 s_i x_i + \\epsilon_i \\end{equation*}$$ Note that the interpretation of $\\beta_3$ is the difference in slopes from the separate regressions: \\begin{align*} \\text{E}[y_i \\mid x, s_i = 1] &= (\\beta_0 + \\beta_2) + (\\beta_1 + \\beta_3) x_i \\\\ \\text{E}[y_i \\mid x, s_i = 0] &= \\beta_0 + \\beta_1 x_i. \\end{align*} You can bootstrap the distribution of $\\beta_3$ if you want or just use standard testing procedures (normal\/t). If using analytical solutions, you need to either assume homoskedasticity across groups or correct for heteroskedasticity. For bootstrapping to be robust to this, you need to choose $n$ observations randomly among the first group and $n$ among the second, rather than $2n$ from the whole population.\n\nIf you have correlation among the error terms, you may need to alter this procedure a bit, so write back if that is the case.\n\nYou can generalize this approach to the seemingly unrelated regressions (SUR) framework. This approach still allows the coefficients for the intercept and the slope to be arbitrarily different in the two data sets.\n\n\u2022 It's a good idea. But doesn't this also assume the two regressions have iid errors?\n\u2013\u00a0whuber\nNov 21 '11 at 15:48\n\u2022 Good point. It requires that there not be different variances for the errors by group and that the errors not be correlated in the different groups. Nov 21 '11 at 17:23\n\nDoing everything in one regression is neat, and the assumption of independence is important. But calculating the point estimates in this way does not require constant variance. Try this R code;\n\nx <- rbinom(100, 1, 0.5)\nz <- rnorm(100)\ny <- rnorm(100)\ncoef(lm(y~x*z))\ncoef(lm(y~z, subset= x==1))[1] - coef(lm(y~z, subset= x==0))[1]\ncoef(lm(y~z, subset= x==1))[2] - coef(lm(y~z, subset= x==0))[2]\n`\n\nWe get the same point estimate either way. Estimates of standard error may require constant variance (depending on which one you use) but the bootstrapping considered here doesn't use estimated standard errors.\n\n\u2022 If you're going to test whether the difference of slopes is zero (as in @Charlie's reply, to which you seem to be following up), you need an accurate, valid estimate of standard errors. It doesn't matter whether you bootstrap that estimate or otherwise.\n\u2013\u00a0whuber\nNov 22 '11 at 3:45","date":"2021-09-19 11:05:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8558083176612854, \"perplexity\": 600.0818042908384}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780056856.4\/warc\/CC-MAIN-20210919095911-20210919125911-00687.warc.gz\"}"} | null | null |
MPEG-A is a group of standards for composing MPEG systems formally known as ISO/IEC 23000 - Multimedia Application Format, published since 2007.
MPEG-A consists of 20 parts, including:
MPEG-A Part 1: Purpose for multimedia application formats
MPEG-A Part 2: MPEG music player application format
MPEG-A Part 3: MPEG photo player application format
MPEG-A Part 4: Musical slide show application format
MPEG-A Part 5: Media streaming application format
MPEG-A Part 6: Professional archival application format
MPEG-A Part 7: Open access application format
MPEG-A Part 8: Portable video application format
MPEG-A Part 9: Digital Multimedia Broadcasting application format
MPEG-A Part 10: Surveillance application format
MPEG-A Part 11: Stereoscopic video application format
MPEG-A Part 12: Interactive music application format
MPEG-A Part 13: Augmented reality application format
MPEG-A Part 15: Multimedia preservation application format
MPEG-A Part 16: Publish/Subscribe Application Format
MPEG-A Part 17: Multiple sensorial media application format
MPEG-A Part 18: Media linking application format
MPEG-A Part 19: Common media application format (CMAF) for segmented media (MPEG CMAF), – a media application format for ABR (adaptive bitrate) media
See also
ISO/IEC JTC 1/SC 29
References
ISO/IEC standards
MPEG | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,668 |
Category Archives: Residential
Mosman 06
Completed /In Progress/
Boomey Hills
Completed /2012 - Present/
This terrace house is located in a distinct small heritage precinct of inner Sydney. Within the confines of being a Heritage Item, the house has been extensively remodeled. The design has responded to a way of living sustainably and efficiently in an increasingly hotter climate. The use of external shutters, glass louvres, ceiling fans, insulation and skylights, provide the means to tackle the discomfort of some summer days and contain warmth in winter.
The principles in remodeling this house within the confines of its heritage status were:
Stabilise structurally: The building was previously detached on the eastern side. By using this space on both levels, piles and underpinning were designed inserted to stabilise the existing building.
Provide a building that performs efficiently and effectively in summer and winter: Louvres are used in every possible location to facilitate cross -ventilation.
Design an integrated plan on the ground level to reduce the "boarding house" effect and introduce planting as integral to the building and visible from most of the house
Provide Light and Ai
The project has contributed greatly to the environmental factors now governing how we design such as:
Re using the existing structure.
Slotted timber shutter keep the house cool in summer, warm in winter.
The house is now well insulated.
Windows and doors are smart glazed.
Louvres and doors provide plentiful cross ventilation.
Ceiling fans in every room of the building assist in cooling and circulating the air.
Low energy, low cost heating with hydronic heating giving greater efficiency.
A solar consultant advised that the roof area did not receive enough direct sun light to effectively use solar panels. This is mainly due to the built up nature of the location.
Awarded AIA NSW chapter Architecture Award, 2009.
Winner Contribution to the Built Environment Award, Mosman Municipal Council's biannual Design Awards, April 2011.
The brief: Off-form concrete house for two adults, four children and a disabled grandparent.
The existing Edwardian basement walls set out the new building and children's courtyard. The new building is orientated with the site to the south, opening to the north and west.
The living-dining room is canted to allow high eastern windows drawing light into the children's bedrooms below. All windows hang within the walls of the building or from the roof minimizing heat transfer and allowing them to be open during rain.
Skylights have been used in both parts of the building to capture and draw down light through the immediate floor and to floors below.
Concrete has been used as blades, screens, overhangs and as the deep gills of the building. Concrete shades and ventilates the building. In summer there is significant cross ventilation and upwards drafting. In winter, aspect and thermal massing help with heating.
Awarded RAIA NSW chapter Commendation Award, 2005
Awarded People Choice prize in SMH Domain
The Brief: The original house was designed by architect Hugh Buhrich and was completed in 1961. When our work commenced, the house had been vacant for 2 years, it was riddled with concrete cancer, leaked badly in several places and had little or no north light internally at each level.
The main objective of this "renovation, restoration and rebuild" was to maintain the most dramatic elements of the original Buhrich building, the cantilevered living room and splayed terrace, with their slender concrete columns. This objective was to work in with accommodating a family of 4 and their visiting grandparents.The steel and zinc box slotted between the columns contains the main bedroom. It hangs from the rebuilt slab above. Steel haunches locate, separate and anchor the box to the existing columns. It is a deliberate intrusion that still allows the original cantilever and form to remain the hero.
The new central courtyard allows north light at each level to the internal spaces and provided.
Palm Beach Surf Club – Women's Showers
The Reformatory Caffeine Lab | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,860 |
Calliandra eriophylla est un arbuste pérenne de la famille des Fabaceae. Il est originaire du sud-ouest des États-Unis et du nord du Mexique. Comme d'autres espèces du genre Calliandra, il est parfois appelé « arbre aux houppettes ».
Description morphologique
Appareil végétatif
Calliandra eriophylla est un buisson bas non épineux, de 20 à 120 cm de hauteur, aux branches nombreuses. Les feuilles sont composées, alternes, bipennées en deux à quatre paires de groupes principaux, eux-mêmes divisés en 5 à 10 paires de folioles. Ces folioles, couverts d'un duvet grisâtre, ont une forme oblongue et mesurent environ 5 mm de long pour moins de 2 mm de largeur.
Appareil reproducteur
La floraison survient entre février et mai, ou entre avril et juillet.
La fleur à la forme d'une houpette d'un rose profond, de 5 cm de diamètre, constituée de nombreuses étamines (au moins 20) d'une longueur de 2 cm, dont la couleur va du blanc au rose violacé soutenu. Le calice et la corolle, constitués chacun de 5 pièces, sont minuscules (quelques millimètres) et rougeâtres. le style est rouge, un peu plus long que les étamines. Le fruit est une gousse plate, brune à maturité, de 5 cm de long environ. Ces gousses couvertes d'un fin duvet s'ouvrent en deux valves et contiennent d'une à plusieurs graines lisses.
La reproduction se fait par graines, mais il est possible de pratiquer le bouturage sur cette espèce, ou la repousse à partir de racines. Cette espèce possède 2 n = 16 chromosomes.
Répartition et habitat
Cette plante vit dans les steppes et déserts, du sud-ouest des États-Unis (sud de la Californie et de l'État du Nouveau-Mexique) au nord, jusqu'au nord du Mexique, au sud.
Elle préfère les sols à texture grossière et à pH basique.
C'est un arbuste résistant à la chaleur et à la sécheresse et appréciant les sols calcaires. Il nécessite beaucoup de lumière mais tolère un ombrage partiel.
Rôle écologique
Le feuillage de Calliandra eriophylla fournit une bonne source de nourriture pour les animaux herbivores sauvages. Les fleurs, productrices de nectar, attirent les insectes butineurs et les colibris ; les graines sont consommées par les colins de Californie. De plus, le système racinaire dense de cet arbuste lui confère un rôle important dans la limitation de l'érosion des sols
Systématique
Cette espèce a été décrite en 1844 par le botaniste britannique George Bentham dans le "London Journal of Botany". Elle s'est aussi vu attribuer les appellations Feuilleea eriophylla (Benth.) Kuntze en 1891 et Anneslia eriophylla (Benth.) Britton en 1894, mais ces deux appellations sont considérées comme non valides.
Selon ITIS, il existe deux sous-espèces:
Calliandra eriophylla var. chamaedrys Isely 1972
Calliandra eriophylla var. eriophylla Benth. 1844
Voir aussi
Arbre aux houppettes
Notes et références
Liens externes
Fabaceae
Mimosoideae
Espèce d'Angiospermes (nom scientifique)
Flore endémique d'Amérique du Nord | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,833 |
But then government bureaucrats declared their dads' deaths weren't heroic enough to be fully considered "in the line of duty."
"We've all been waiting at least five years just to get in the running to get the job. And now that we're here, they've taken away the legacy of our fathers," said Scott Barocas, whose father, Capt. Sheldon Barocas, died in 2011 from a 9/11-related cancer.
That meant Barocas lost so-called "legacy points," which vaulted him to the 20th spot in the long line of would-be firefighters.
They were near the top of the list thanks to the 10 extra bonus points added to their scores due to their father's tragic death following 27 years as a firefighter.
"My dad was the type of person who gave everything to his firehouse, the job was everything. I couldn't understand it. How could this be true?" said Michael Sullivan, 29.
His father listed on a plaque at FDNY headquarters honoring firefighters who died from illnesses linked to the "rescue and recovery operation" at the Trade Center site — as are the fathers of the other 12 applicants.
Yet that honor only extends so far. Without the points for his dad's service, Michael Sullivan dropped from 284th place to somewhere in the 11,000s.
John Sullivan Jr. said the legacy points didn't amount to preferential treatment, but fair treatment.
"There is a tragic disconnect between the observation of those who died that terrible day and those who have died every day since," said John Sullivan Jr., alluding to the agony caused by a 9/11-related illness.
State Sens. Martin Golden (R-Brooklyn) and Greg Ball (R-Carmel) spearheaded an amendment to civil service law that would define "killed in the line of duty" to include firefighters and cops who perished as a result of illnesses contracted during the cleanup.
"The bill is not on our desk at this time but it is under review," said Cuomo spokesman Rich Azzopardi.
"If the bill had been passed and signed into law earlier, they had a chance, but my understanding is now that the FDNY has started hiring, this list can't be changed," he said.
The FDNY usually only hires once every four years and may take anywhere from 4,000 to 12,000. Candidates who don't get selected can try again when the next test is given — as long as they're still 29 or younger.
"We've all got the same message: Please Gov. Cuomo, sign the bill," said Barocas.
Jennifer McNamara's son, Jack, was only 2 years old when his dad died in 2009 of colon cancer after logging 500 hours at the World Trade Center site post-9/11. She said Jack deserves the right to honor his father, John.
"It breaks my heart to see these kids who just want to get on the job like their dads. I understand the need to want to be like your father, especially when he's gone way too soon," Jennifer said.
"I don't know what Jack will want to do when he grows up, but to the extent that he wants to fulfill his dad's legacy, he should have every option to do it." | {
"redpajama_set_name": "RedPajamaC4"
} | 1,253 |
\section{Introduction}
The electronic structure of the negative nitrogen vacancy ($\mbox{NV}^{-}$)
center in diamond gives it properties that are attractive for aspects
of quantum information applications \cite{Jelezko2004b,Jelezko2004a,Childress2006,Wrachtrup2006,Hanson2006a,Gaebel2006,Santori2006a,Waldermann2007}.
One of the attractive features is the phenomenon of optically induced
spin polarization of the $S=1$ ground state \cite{Loubser1977,Reddy1987,Redman1991}.
It has been proposed that the polarization arises due to inter-system
crossing from the excited triplet state to singlet levels, and decay
back to the ground state with an overall change of spin. This process
was thought to be non-radiative due to the emission intensity dependence
on the level of spin polarisation \cite{Manson2006}, and there was
no direct knowledge of the singlets involved. However, in this work
infrared emission from the $\mbox{NV}^{-}$ centre is reported and
shown to be associated with the singlet levels. Spectral analysis
of this emission has provided information about the polarising decay
path and the electronic structure of the NV centre.
\section{Observations}
\subsection{Emission spectrum }
Two synthetic 1b diamond samples of different defect concentrations
were used. Both were 2 mm cubes that had been irradiated and annealed
to produce $\mbox{NV}^{-}$ centers with concentrations of about $3\times10^{18}\mbox{ cm}^{-3}$
({}``high'' concentration) and about $10^{17}\mbox{ cm}^{-3}$ ({}``low''
concentration). Each of the samples had $\langle110\rangle$, $\langle110\rangle$
and $\langle100\rangle$ faces. The samples were excited with a 532
nm laser at 100 mW and the emission at right angles was dispersed
by a monochromator and detected on an ADC model 403L cooled germanium
photodetector. A weak infrared emission band with a zero-phonon line
(ZPL) at 1046 nm was observed, and the spectrum is shown in Figures
\ref{fig:ir-spec-H-hot-cold} and \ref{fig:ir-spec-L-cold}. The characteristic
$\mbox{NV}^{-}$ emission has a higher energy ZPL at 637 nm and vibrational
sidebands that extend beyond 1000 nm, the extreme low energy tail
of which create the intensity baseline in Figure \ref{fig:ir-spec-H-hot-cold}.
Throughout this paper the emission from the 637 nm transition is referred
to as {}``visible'' in order to differentiate it from the infrared
emission band.
\begin{figure}
\hspace{71pt}\includegraphics[width=8.6cm]{spectrum_ir_high_conc}
\caption{Infrared emission band at (a) room and (b) liquid helium temperatures
for the high concentration sample with an $\mbox{NV}^{-}$ center
concentration of $3\times10^{18}\mbox{ cm}^{-3}$. Trace (b) from
the cryogenic measurement has been divided by a factor of 5 on this
graph in order to compensate for the enhanced ZPL intensity. The vibronic
tail of the characteristic $\mbox{NV}^{-}$ visible emission is apparent
under the infrared band. \label{fig:ir-spec-H-hot-cold}}
\end{figure}
\begin{figure}
\hspace{71pt}\includegraphics[width=8.6cm]{spectrum_ir_low_conc_cold}
\caption{Infrared emission band for low concentration ($10^{17}\mbox{ cm}^{-3}$)
sample at liquid helium temperature. The spectrum was measured with
and without an approximately 600 G magnetic field applied, and the
intensity with the field was 122\% of the intensity with no external
field. The sharp feature at 1064 nm common to both traces was due
to some scatter from the 532 nm laser. \label{fig:ir-spec-L-cold}}
\end{figure}
At room temperature the infrared ZPL at 1046 nm was clearly discernible
accompanied by a vibrational band, and at low temperature the features
were clearer and dominated by the zero-phonon line. The linewidth
was measured to be 0.3 nm (0.3 meV) in the lower concentrated sample,
but was broadened significantly to 4 nm in the higher concentration
sample. The vibrational sideband had similar integrated area to the
ZPL $(\mathcal{S}\mbox{-coefficient}\approx1)$ and was comprised
of peaks shifted by 42.6 meV (344 $\mbox{cm}^{-1}$), 133 meV (1070
$\mbox{cm}^{-1}$) and 221 meV (1780 $\mbox{cm}^{-1}$) from the ZPL.
\subsection{Magnetic field measurements}
The intensity of both infrared and visible emission bands was found
to vary with low magnetic fields, and the variation for the low concentration
sample is shown in Figure \ref{fig:mag_spectra} for fields between
0 and 1500 Gauss. The measurements were made at room temperature,
again involving 100 mW laser excitation at 532 nm. The visible emission
transmitted by a long pass filter at 615 nm was detected by a Si detector,
and the weakness of the infrared emission meant that this signal was
completely dominated by the $^{3}\! A_{2}\leftrightarrow{}^{3}\! E$
transition. The infrared signal was collected through a 1040 nm long
pass filter and detected on an InGaAs detector. In this case the infrared
band dominated, although a small contribution (10 - 15\%) remained
from the vibronic tail of the $^{3}\! A_{2}\leftrightarrow{}^{3}\! E$
transition. Spectrally isolating the emission bands gave a more reliable
measure of the change in emission magnitude with magnetic field, and
this is shown in Figures \ref{fig:ir-spec-L-cold} and \ref{fig:spectrum-vis-500G}.
\begin{figure}
\hspace{71pt}\includegraphics[width=8.6cm]{lac_spectra}
\caption{Emission intensity for varying external magnetic field. The field
was aligned to a $\langle111\rangle$ crystal axis to within $0.025^{\circ}$.
The magnetic field spectra are shown on the same scale, however the
infrared intensity was many orders of magnitude weaker than that of
the visible. \label{fig:mag_spectra}}
\end{figure}
\begin{figure}
\hspace{71pt}\includegraphics[width=8.6cm]{spectrum_vis}
\caption{Change in the visible emission intensity due to presence of an approximately
600 G external magnetic field. Across the entire band, the intensity
with the field was 87.7\% of the intensity with no magnetic field.
\label{fig:spectrum-vis-500G}}
\end{figure}
\begin{figure}
\hspace{71pt}\includegraphics[width=8.6cm]{ir_zpl_zeeman}
\caption{Infrared ZPL for 5 T magnetic field and zero field for sample at 4.2
K.\label{fig:ir-zpl-zeeman}}
\end{figure}
High field Zeeman measurements of the 1046 nm line were also undertaken.
These were made for the low concentration sample cooled to 4.2 K where
the linewidth was 0.3 nm. The sample was mounted within the core of
a super-conducting Helmholtz coil, and the magnetic field and 532
nm laser were directed along the $\langle110\rangle$ direction. The
infrared emission was detected at right angles along the $\langle100\rangle$
direction, and the ZPL spectrum for a 5 T field is shown in Figure
\ref{fig:ir-zpl-zeeman}. No Zeeman shift or splitting was observed,
although there was a small change in intensity consistent with the
room temperature measurements.
\subsection{Transient response}
The time dependence of the infrared and visible emission was investigated
for high intensity excitation pulses. Measurements were made with
the sample at room temperature being excited by a focused 532 nm laser
gated by an acousto-optic modulator. The emission was detected using
long pass filters at 615 nm and 1000 nm for the visible and infrared,
respectively, and to ensure consistency between measurements an InGaAs
detector was used in both cases. The pulse sequence consisted of excitation
for 700 ns, followed by a dark delay of 500 ns, and then a slightly
longer second pulse of 1700 ns. The measurements were repeated with
an approximately 600 Gauss field applied to the sample in a random
direction. The responses are shown in Figure \ref{fig:ir-dynamics-high-intensity}.
\begin{figure}
\hspace{71pt}\includegraphics[width=8.6cm]{dynamics_red_slow}
\hspace{71pt}\includegraphics[width=8.6cm]{dynamics_ir_slow}
\caption{Visible (a) and infrared (b) emission for a 700 ns, 1700 ns pulse
pair. Measured with and without a magnetic field of approximately
600 Gauss applied. \label{fig:ir-dynamics-high-intensity}}
\end{figure}
\subsection{Uniaxial stress}
The techniques of uniaxial stress spectroscopy \cite{Kaplyanskii1964,Kaplyanskii1964a,Hughes1967}
were used here to study the zero phonon line at 1046 nm. The low concentration
$2\mbox{ mm}$ cubic sample was held at a temperature of 4.2 K while
being compressed by a piston to pressures up to 0.7 GPa. Excitation
was at 532 nm in the vibronic band of the $^{3}\! A_{2}\leftrightarrow{}^{3}\! E$
transition with polarization parallel to the stress. The emission
at right angles was detected with the polarization in the $\pi$ (parallel
to axis of stress) and $\sigma$ (perpendicular to axis of stress)
directions. Spectra of the 1046 nm zero phonon line were recorded
for stresses along the $\langle110\rangle$ and $\langle100\rangle$
axes, and these are shown in Figures \ref{fig:ir-stress-splitting-spectra-110}
and \ref{fig:ir-stress-splitting-spectra-100}. Three distinct lines
were observed in the spectrum taken for the $\langle110\rangle$ stress
(although the lowest energy line could be two overlapping lines),
and two for the $\langle100\rangle$ stress. The splitting of the
637 nm ZPL was measured, but only used to calibrate the stress and
so is not shown here.
\begin{figure}
\hspace{71pt}\includegraphics[width=10.5cm]{ir_stress110_0and40psi_withdiagram}
\caption{(a) Emission spectra for approximately 0.3 GPa stress applied along
the $\langle110\rangle$ direction, measured along the $\langle110\rangle$
direction with polarization parallel $(\pi)$ and perpendicular $(\sigma)$
to stress. The zero stress ZPL is shown in grey for reference. (b)
Theoretical patterns for an $A\leftrightarrow E$ transition at a
site of $C_{3v}$ symmetry, showing the predicted relative intensities
of each line. The splittings are related to the stress coefficients
of \cite{Hughes1967} by $\Delta E_{\mbox{i}}=A_{1}+A_{2}+C-B$; $\Delta E_{\mbox{f}}=A_{1}+A_{2}-C+B$;
$\Delta E_{\mbox{h}}=A_{1}-A_{2}+C+B$; $\Delta E_{\mbox{g}}=A_{1}-A_{2}-C-B$.
Values for these coefficients were obtained from the measured splitting
of the infrared ZPL and are given in Table \ref{tab:Stress-coefficients}.
\label{fig:ir-stress-splitting-spectra-110}}
\end{figure}
\begin{figure}
\hspace{71pt}\includegraphics[width=10.5cm]{ir_stress100_0and50psi_thermal_withdiagram}
\caption{(a) Emission spectra for approximately 0.70 GPa stress along $\langle100\rangle$,
measured along the $\langle110\rangle$ direction with polarization
parallel $(\pi)$ and perpendicular $(\sigma)$ to stress. The zero
stress ZPL is shown in grey for reference. (b) Theoretical patterns
predicted for an $A\leftrightarrow E$ transition at a site of $C_{3v}$
symmetry, where the splittings are related to the stress coefficients
of \cite{Hughes1967} by $\Delta E_{\mbox{b}}=A_{1}+2B$ and $\Delta E_{\mbox{a}}=A_{1}-2B$.
\label{fig:ir-stress-splitting-spectra-100}}
\end{figure}
For stress along $\langle100\rangle$, measurements were repeated
at 43 K, and the results are included in Figure \ref{fig:ir-stress-splitting-spectra-100}.
The figure shows that there was no thermal variation in the emission
spectra over this temperature range.
\section{Discussion}
\subsection{Energy scheme for $C_{3v}$}
The low lying electronic states of the NV center have been considered
in previous publications \cite{Manson2006,Manson2007}. The levels
can be obtained by considering the one electron states at the vacancy
site adjacent to the nitrogen. In the notation for $C_{3v}$ point
group symmetry there are two one-electron orbits transforming as an
$A_{1}$ irreducible representation $(a_{1},a_{1}^{\prime})$ and
one transforming as an $E$ irreducible representation $(e)$, and
their energies are considered to be in that order \cite{Lenef1996}.
In the case of the NV$^{-}$ center there are six electrons occupying
these orbits and the lowest energy configuration is $a_{1}^{2}a_{1}^{\prime2}e^{2}$.
This configuration gives rise to $^{3}\! A_{2}$, $^{1}\! A_{1}$,
and $^{1}\! E$ states, whereas there is a higher energy excited triplet
$^{3}\! E$ and singlet $^{1}\! E$ associated with an $a_{1}^{2}a_{1}^{\prime}e^{3}$
configuration. The low lying states are, therefore, as shown in black
in Figure \ref{fig:levels-spin-orbit}.
\begin{figure}
\hspace{71pt}\includegraphics[width=8.6cm]{levels_spin-orbit}
\caption{The energy levels expected from consideration of the one-electron
states for the six $\mbox{NV}^{-}$ electrons in $C_{3v}$ symmetry
are shown in black (first and fourth columns). Diagonal spin-orbit
terms split the $^{3}\! E$ level as shown in the second column, and
spin-spin interactions give rise to the splittings in the third column.
Straight arrows indicate optical transitions, and the dashed lines
illustrate weakly allowed transitions that prevent perfect spin polarisation.
The curved lines show symmetry-allowed inter-system crossing transitions,
and the wavy line shows suspected vibronic decay between the singlets.
The states are labeled on the left and the symmetry transformation
properties of the spin-orbit wavefunctions are given on the right.
\label{fig:levels-spin-orbit}}
\end{figure}
The ground state is the $^{3}\! A_{2}$ and the characteristic strong
optical transition at 637 nm is from this ground state to the $^{3}\! E$
excited state. There is fine structure associated with both of the
triplet levels. In the ground state the spin levels are split by spin-spin
interaction into a non-degenerate $M_{s}=0$ level and a degenerate
$M_{s}=\pm1$ level. The excited state is split by diagonal spin-orbit
interaction $\lambda L_{z}S_{z}$ into three equally spaced doubly
degenerate levels and there are small displacements from the non-diagonal
spin-orbit interaction, $\lambda\left(L_{+}S_{-}+L_{-}S_{+}\right)$
\cite{Manson2006}. What is more important for this work is the effect
of spin-orbit interaction between the triplet and singlet levels.
The interaction causes mixing of states transforming as the same irreducible
representation, and states mixed in this way are indicated by curved
arrows in Figure \ref{fig:levels-spin-orbit}. The mixing can enable
radiative or non-radiative transfer between the triplet and singlet
levels, and calculation indicates that in the case of the excited
$^{3}\! E$ state the transfer will be predominantly out of $M_{s}=\pm1$
levels. This is significant as it provides an alternative decay path
to the visible emission. As a consequence the visible emission associated
with $M_{s}=\pm1$ spins is weaker than that for $M_{s}=0$.
\subsection{Magnetic field}
Optical excitation causes preferential population of the $M_{s}=0$
spin projection and the visible emission associated with this spin
is stronger than that for $M_{s}=\pm1$. Thus, as population is transferred
to the $M_{s}=0$ spin state the visible emission increases in intensity
and, conversely, if the spin polarization is reduced the emission
intensity will diminish. A static magnetic field mixes the ground
state spin levels and inhibits population transfer to $M_{s}=0$,
reducing the spin polarisation. Varying the field strength alters
the amount of mixing between levels which changes the intensity of
emission, and this effect is particularly noticeable for an axial
field of 1028 Gauss. At this field value there is a complete mixing
of $M_{s}=-1$ and $M_{s}=0$ states. The population will be equally
distributed between the two spins whereas it will be almost entirely
in $M_{s}=0$ at adjacent magnetic field values. Thus an axially aligned
magnetic field swept through 1028 Gauss causes a marked reduction
in population and noticeable drop in the visible emission intensity
as seen in trace (a) of Figure \ref{fig:mag_spectra}. With the reduction
of spin polarisation there is an increase in the $M_{s}=-1$ population,
which increases the transfer rate to the singlet levels and should
increase the emission intensity from any optical transitions within
the alternative decay path. Exactly such a rise is observed in the
infrared emission at 1028 Gauss.
All the other intensity variation of the visible emission in Figure
\ref{fig:mag_spectra} can be similarly explained by variation in
the spin polarization of the $\mbox{NV}^{-}$ centre. For example,
at the special cases of axial fields at 514 Gauss and 660 Gauss there
is cross relaxation between the $\mbox{NV}^{-}$ centre aligned with
the field and other spin systems in the crystal (single substitutional
nitrogen defects and non-aligned $\mbox{NV}^{-}$ centres, respectively)
\cite{Holliday1989,Epstein2005}. These other spin systems are not
spin polarised and the cross relaxation will hence reduce the polarisation
of the $\mbox{NV}^{-}$ centre, causing the visible emission to diminish.
It is immediately apparent from Figure \ref{fig:mag_spectra} that
the infrared emission contains the same features, and two conclusions
can be drawn. Firstly, the complete (anti-) correlation of this intensity
with that of the visible emission (which varies due to $\mbox{NV}^{-}$
spin polarisation) proves the new emission band is associated with
the $\mbox{NV}^{-}$ centre. Secondly, the fact that it is anti-correlated
shows the infrared emission is associated with the population involved
in the inter-system crossing.
High magnetic fields were experimentally found not to split the infrared
zero-phonon line. Although this rules out the possibility of the emission
arising from certain transitions, subtleties mean that it does not
conclusively identify the correct transition. A triplet-triplet cannot
immediately be eliminated, as it is already known that the $^{3}\! A_{2}\leftrightarrow{}^{3}\! E$
(637 nm) zero-phonon line is not split by a magnetic field \cite{Reddy1987,Hanzawa1993}.
The individual levels of the triplets are split, but the optical transitions
are between levels of like spin and so they remain degenerate, as
shown in Figure \ref{fig:Zeeman-splitting.}. The same could occur
for the infrared, but this option can be dismissed as there are no
other triplet levels in the $\mbox{NV}^{-}$ system. Unlike triplet
levels, spin-singlets are not split by a magnetic field, and thus
Zeeman splitting of the ZPL would be expected in general for a triplet-singlet
transition. A subtlety here is that no splitting would be observed
if the transition were restricted to a particular spin level of the
triplet state, as is indicated in Figure \ref{fig:Zeeman-splitting.}
for the transition that feeds back to the ground state. However this
diagram is accurate for an axially aligned field only, and the four
different $\mbox{NV}^{-}$ orientations in a bulk diamond sample makes
it impossible to achieve total alignment. Mixing between spin levels
in misaligned centres would lead to observable splitting.
\begin{figure}
\hspace{71pt}\includegraphics[width=8.6cm]{levels_zeeman}
\caption{Theoretical energy levels for a high (axial) magnetic field situation
are shown in the third column. The first two columns contain the spin-orbit
levels that are shown in detail in Figure \ref{fig:levels-spin-orbit}.
Straight arrows indicate optical transitions, curved lines show symmetry-allowed
inter-system crossing transitions, and the wavy line shows suspected
vibronic decay between the singlets. The wavefunctions in the presence
of a large field are given on the right. \label{fig:Zeeman-splitting.}}
\end{figure}
To first order, no splitting would occur for a singlet-singlet transition.
It is possible for orbitally degenerate states to separate and give
rise to some spectral broadening or splitting, but this can be expected
to be too small to result in a measurable splitting (as it is for
the $^{3}\! A_{2}\leftrightarrow{}^{3}\! E$ transition). The experimental
result in Figure \ref{fig:ir-zpl-zeeman} is most consistent with
the infrared emission arising from the $^{1}\! E\leftrightarrow{}^{1}\! A_{1}$
singlet-singlet transition.
\subsection{Transients\label{sub:Transients}}
The response of the visible emission to intense excitation pulses
has been interpreted previously \cite{Manson2006}. Initially the
$\mbox{NV}^{-}$ centres are evenly distributed between the three
spin projections and they are excited equally. With excitation, the
spin-selective inter-system crossing preferentially populates the
$M_{s}=0$ level and causes spin polarisation as has already been
discussed. This increase in spin polarisation typically increases
the visible emission intensity, however for intense excitation (as
used here) an equilibrium population is built up in a long-lived (300
ns) {}``storage'' state in the singlet system. This decreases the
population that contributes to the visible emission, causing the drop
in visible emission occurring over the first few hundred ns that is
prominent in Figure \ref{fig:ir-dynamics-high-intensity}. At the
start of the second pulse the population is still spin polarised and
so the inter-system crossing is slightly slower. This is observed
as a reduction in the rate that the emission drops to its equilibrium
intensity (ie the storage singlet builds up an equilibrium population).
The peak at the beginning of the second pulse is lower than that of
the first, as some population remains in the storage level after the
500 ns delay and some population is lost through photoionization \cite{Manson2005}.
The situation is changed by a weak magnetic field, which causes a
mixing of the ground state spin levels and prevents spin polarisation.
As a result, more population takes the alternative decay path through
the singlet levels and a larger equilibrium population is maintained
in the storage state. Thus the visible emission intensity is lower
than it was without the magnetic field. Since the difference between
the first and second pulses was explained by residual spin polarisation,
both pulses should be identical in the case of a magnetic field. In
the experiment, the observed difference that does occur between pulses
is due to imperfect quenching of spin polarisation and also to photoionization
\cite{Manson2005}.
Comparing the infrared and visible responses to excitation pulses
in Figure \ref{fig:ir-dynamics-high-intensity}, it is immediately
obvious that there is much similarity. The major difference is that
the magnetic field increases the infrared intensity whereas it decreases
the visible, which is consistent with the magnetic field spectra discussed
above. However, the drop in infrared emission intensity within the
first few hundred nanoseconds of each pulse (similar to that obtained
in the visible) indicates that the emitting level is not the {}``storage''
level. This confirms that the infrared band is emitted from the $^{1}\! E\rightarrow{}^{1}\! A_{1}$
singlet-singlet transition, and suggests that the lower singlet state
is the 300 ns storage level. Such a conclusion is plausible, as the
upper singlet could lie close to the excited triplet state to enable
efficient inter-system crossing and the lower singlet could then be
several hundred meV (1000s of $\mbox{cm}^{-1}$) above the ground
state with much slower inter-system crossing. Attempts were made to
re-pump the singlet-singlet transition by exciting in the infrared
to confirm this analysis, but they have not been successful. We were,
therefore, not able to establish where the singlet levels lie in relation
to the triplet levels.
The responses in Figure \ref{fig:ir-dynamics-high-intensity} provide
additional lifetime information. The rates of emission decay from
the initial intensity for each excitation pulse are similar for the
visible and infrared, indicating similar dynamics. The excited state
lifetime for the visible emission is known to be about 12 ns \cite{Collins1983,Batalov2008},
and the signal contrast ratio between spin polarised and unpolarised
(as can be seen in Figures \ref{fig:ir-spec-L-cold} and \ref{fig:spectrum-vis-500G})
indicates that population from the excited triplet crosses to the
singlet system at a similar rate. Thus the population must not spend
extra little time in the upper singlet level before giving the infrared
emission. However, a singlet-singlet transition is unlikely to appreciably
stronger than an allowed triplet-triplet transition and the most likely
explanation for this short lifetime is that there is also competing
non-radiative decay between the singlet levels as indicated by the
wavy line in Figures \ref{fig:levels-spin-orbit} and \ref{fig:Zeeman-splitting.}.
Significant non-radiative decay would account for both the short lifetime
and the weakness of the emission. This may be a general phenomenon
for transitions in the infrared as there are few reports of diamond
emitting at these wavelengths \cite{Zaitsev2001}. Such an increased
contribution of non-radiative decay at low-energy transitions in diamond
can be attributed to the strong electron phonon coupling and high
vibrational frequencies.
In summary, the following physical description is consistent with
the data. There is an almost 50\% branching of the population from
the $^{3}\! E$ to the singlets, and the upper singlet has a short
lifetime ($<1$ ns) mainly due to non-radiative decay. The lower singlet
level has a longer lifetime ($\approx300$ ns) identified previously
\cite{Manson2006}.
\subsection{Uniaxial stress measurements\label{sub:Uniaxial-stress-measurements}}
The splitting of the zero phonon line with uniaxial stress can be
used to determine the symmetry of the states involved in optical transitions.
A study of this type was undertaken by \cite{Davies1976} and they
showed the 637 nm zero-phonon line to be associated with an $A\leftrightarrow E$
transition at a site of trigonal symmetry. In the present work this
$A\leftrightarrow E$ transition was excited, but energy transferred
within the $\mbox{NV}^{-}$ system gives rise to the infrared transition
which was investigated.
For stress applied along a $\langle110\rangle$ direction, two pairs
of NV centres have equivalent orientations and both pairs are excited.
In all cases there is a component of strain perpendicular to the $\mbox{NV}^{-}$
axis and for an $A\leftrightarrow E$ transition a maximum of four
lines is therefore predicted \cite{Davies1976,Mohammed1982}. This
is consistent with the experimental observation shown in Figure \ref{fig:ir-stress-splitting-spectra-110},
and the polarization pattern for an $A\leftrightarrow E$ transition
shown below the experimental traces is also in plausible correspondence.
Some deviation of the polarisation pattern is likely to be due to
loss of polarization from scatter from the crystal faces, as they
were not optically polished.
Parameters for the stress splitting of an $A\leftrightarrow E$ transition
in trigonal symmetry were introduced by \cite{Hughes1967}, and their
values for the present transition were calculated from the energy
splittings of the four lines in Figure \ref{fig:ir-stress-splitting-spectra-110}
and are given in Table \ref{tab:Stress-coefficients}. They are of
the order of a factor 2.5 smaller than those for the 637 nm zero phonon
line determined for by \cite{Davies1976}. The small values for the
strain parameters and the small value for the inhomogeneous linewidth
are consistent with a transition such as $^{1}\! E(a_{1}^{2}a_{1}^{\prime2}e^{2})\leftrightarrow{}^{1}\! A_{1}(a_{1}^{2}a_{1}^{\prime2}e^{2})$,
which involves a spin change but no change of orbit between initial
and final state.
\begin{table}
\caption{Stress coefficients for the visible and infrared transitions. The
values for the visible are from \cite{Davies1976}, with the signs
of $B$ and $C$ adjusted to reflect the convention in \cite{Mohammed1982}
adopted here. \label{tab:Stress-coefficients}}
\hspace{71pt}\begin{tabular}{|c|c|c|}
\hline
& Visible & Infrared \tabularnewline
& $(\times10^{-12}\mbox{ eV Pa}^{-1})$ & $(\times10^{-12}\mbox{ eV Pa}^{-1})$\tabularnewline
\hline
\hline
$A_{1}$ & $1.47$ & $0.53$\tabularnewline
\hline
$A_{2}$ & $-3.85$ & $-1.44$\tabularnewline
\hline
$B$ & $-1.04$ & $-0.51$\tabularnewline
\hline
$C$ & $-1.69$ & $-0.58$\tabularnewline
\hline
\end{tabular}
\end{table}
For a stress along the $\langle100\rangle$ direction all four $\mbox{NV}^{-}$
orientations are at the same angle to the stress and the excitation
polarisation. Each orientation is thus equally excited, and for an
$E\rightarrow A$ transition the component of strain perpendicular
to the $\mbox{NV}^{-}$ axes would lift the degeneracy of the excited
$E$ state. The splitting would be the same for all orientations,
and so produce two lines. The spectra for $\langle100\rangle$ stress
shown in Figure \ref{fig:ir-stress-splitting-spectra-100} is consistent
with this description, and there is plausible correspondence with
the expected polarization pattern shown below the experimental traces.
Since the spin polarisation is to the $M_{s}=0$ level of the ground
state, and this spin level transforms with $A_{1}$ symmetry as indicated
in Figure \ref{fig:levels-spin-orbit}, the $^{1}\! A_{1}$ state
should be the lower of the singlet states \cite{Manson2006}. With
the $^{1}\! E$ as the upper level, the Boltzmann population distribution
of the split components might be expected to change with temperature.
This would cause a change in the relative intensities of the lines
in the spectrum, but no such change was observed between a temperature
of 4.2 and 43 K (Figure \ref{fig:ir-stress-splitting-spectra-100}).
In Section \ref{sub:Transients} it was argued that the lifetime of
this upper singlet is very short, and it is possible that there is
insufficient time to establish a Boltzmann distribution.
\section{Conclusion}
An emission band has been observed in the infrared, with a zero phonon
line at 1046 nm. Measurements have established that this infrared
emission is associated with the $\mbox{NV}^{-}$ defect centre, which
has previously been investigated through its well documented visible
transition at 637nm. From theoretical considerations, magnetic field
and uniaxial stress measurements the infrared emission is attributed
to a $^{1}\! A_{1}\leftrightarrow{}^{1}\! E$ transition where these
singlet levels lie between the ground and excited state triplets.
Although the results presented here are consistent with the $^{1}\! E$
being the higher of the singlets, the order of the levels has not
been conclusively established. Some contention over the order of these
singlet levels already exists due to previous numerical calculations
\cite{Goss1996,Gali2008}.
There are some further puzzling features about the infrared emission
that remain unresolved. The documented energies of the vibrational
sidebands associated with the visible transition (66 and 140 meV \cite{Davies1974})
are not matched by the infrared sideband energies (42.6, 133 and 221
meV). The first of these is very small and the last is large, well
outside the range of single phonons in diamond. The origin of these
frequencies is unclear but the small electron-phonon coupling, as
well as the small stress parameters and inhomogeneous line width,
are consistent with theoretical models of a transition involving only
a spin reorientation. Another strange aspect is the weakness of the
infrared emission. The inter-system crossing branching ratio is as
high as 50\% for the $M_{s}=\pm1$ spin state but yet the infrared
emission is orders of magnitude weaker than that of the visible emission.
The explanation that has been advanced is that there is also very
significant non-radiative decay for the same transition. Related to
this is that with emission arising almost exclusively from $M_{s}=\pm1$
spins then it is surprising that it only drops 15\% with spin polarization.
It suggests that the level of spin polarization attained in our the
samples was very small.
Despite these issues, the observation of the singlet to singlet transitions
adds significantly to our understanding of the electronic structure
of the NV centre. It provides a new avenue whereby the centre, and
in particular the process of spin polarisation, can be studied and
used for quantum information processing applications.
\ack{}{This work has been supported by Australian Research Council. }
\section*{References}{}
\bibliographystyle{unsrt}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 316 |
Features » July 11, 2011
Publicopoly Exposed
How ALEC, the Koch brothers and their corporate allies plan to privatize government.
BY Beau Hodai
ALEC openly advocates privatizing public education, transportation and the regulation of public health, consumer safety and environmental quality.
On February 25, 2011, Florida State Representative Chris Dorworth (R-Lake Mary) introduced HB 1021. The bill sought to curtail the political power of unions by prohibiting public employers from deducting any amount from an employee's pay for use by an employee organization (i.e., union dues) or for any political activity (i.e., the portion of union dues used for lobbying or for supporting candidates for office).
Furthermore, HB 1021 stated that, should a union seek to use any portion of dues independently collected from members for political activity, the union must obtain annual written authorization from each member.
In effect, this bill defunds public-sector unions–like AFSCME, SEIU, the American Federation of Teachers and the National Education Association–by making the collection of member dues an onerous, costly task. With public-sector unions denatured, they would no longer be able to stand in the way of radical free marketeers who plan to profit from the privatization of public services.
Given the similarities between HB 1021 and a rash of like-minded bills in states across the country, including Wisconsin, on March 30 a public records request was sent to Dorworth's office seeking copies of all documents pertaining to the writing of HB 1021, including copies of any pieces of model legislation the American Legislative Exchange Council (ALEC) may have provided.
Within an hour of submitting this request, Florida House Speaker Dean Cannon's (R-Winter Park) Communications Director Katherine Betta responded: "We received a note from Representative Dorworth's office regarding your request for records relating to the American Legislative Exchange Council and HB 1021. Please note that Mr. Dorworth's legislative offices did not receive any materials from ALEC relating to this bill or any 'model legislation' from other states."
But two weeks later Dorworth's office delivered 87 pages of documents, mostly bill drafts and emails, detailing the evolution of what was to become HB 1021. Buried at the bottom of the stack was an 11-page bundle of neatly typed material, labeled "Paycheck Protection," which consisted of three pieces of model legislation, with the words "Copyright, ALEC" at the end of each.
Dorworth legislative assistant Carolyn Johnson claims that, although Dorworth is an ALEC member, neither she nor her boss have any idea how the ALEC model legislation found its way into Dorworth's office. Dorworth could not be reached for comment.
Enter the Koch Brothers
Nov. 2, 2010 saw a radical cohort of Republicans swept into office in states across the country.
When the legislative sessions began in January, the American news-consuming public was shocked by the tenacity of this new breed of Grand Old Partier as it set to the task of breaking public employee unions, dismantling state government and privatizing civic services.
While battles still rage in the nation's legislatures and statehouses, mainstream media attention peaked in February and March with the culmination of the fight over Gov. Scott Walker's budget bill AB 11, which sought to curtail the collective bargaining rights of government employees and thus disempower Wisconsin's public sector unions.
When on February 23 the Buffalo Beast published recordings and transcripts of a prank call to Walker from a Beast reporter posing as billionaire GOP donor David Koch, it became apparent how intimately involved brothers David and Charles Koch were in Walker's efforts to break public sector unions.
Subsequently, bloggers and editorialists began batting around possible scenarios involving myriad right-wing public policy foundations funded by the Koch brothers and proceeds of Wichita, Kan.-based Koch Industries (and other Koch-controlled corporations). During such speculation, one name arose as the favorite villain behind the multitude of bills aimed squarely at public employee unions. That name was ALEC (see sidebar detailing the organization's Koch connections).
An exhaustive analysis of thousands of pages of documents obtained through public records requests from six states, as well as tax filings, lobby reports, legislative drafts and court records, reveal that these suddenly popular anti-public employee bills, while taking different forms from state to state, were indeed disseminated as "model legislation" by ALEC.
Not coincidentally, bills similar to those in Florida and Wisconsin have been introduced in Arizona, California, Illinois, Iowa, Indiana, Kansas, Maine, Maryland, Michigan, Minnesota, Missouri, North Carolina, New Hampshire, New Jersey, New Mexico, Ohio, Oklahoma, Rhode Island, Tennessee, Texas, Utah and Vermont.
The purported goal of this nationwide movement has been to reduce the budgetary burden posed by public employee salaries by limiting the right of public employees to collectively bargain for pay and other benefits. These restrictions, along with "paycheck protection" laws, curtail the political power of public employee unions by cutting off funds for political campaign and lobbying expenditures. These measures would effectively thwart attempts by public employee unions to resist privatization of government functions and to support candidates opposing elected officials who vote for corporate giveaways of public resources.
'Publicopoly' in play
ALEC contends that government agencies have an unfair monopoly on public goods and services. To change that situation, it has created a policy initiative to counter what it calls "Publicopoly." ALEC's stated aim is to provide "more effective, efficient government" via privatization–that is, the shifting of government functions to the private sector. ALEC lists its initiatives on its website (alec.org/publicopoly).
Though the specifics are secret and "restricted to members," ALEC openly advocates privatizing public education, transportation and the regulation of public health, consumer safety and environmental quality including bringing in corporations to administer:
• Foster care, adoption services and child support payment processing.
• School support services such as cafeteria meals, custodial staff and transportation.
• Highway systems, with toll roads presented as a shining example.
• Surveiling and detaining convicted criminals.
• Ensuring the quality of wastewater treatment, drinking water, and solid waste services and facilities. (After all, when someone mentions a safe and secure public water supply, the voter's next immediate thought is: "Only if it's cost-effective!")
To accomplish these initiatives, ALEC contends that "state governments can take an active role in determining which products and services should be privatized." ALEC advocates three reforms: creating a "Private Enterprise Advisory Committee" to review if government agencies unfairly compete with the private sector; creating a special council that would contract with private vendors if they can "reduce the cost of government"; and creating legislation that would require government agencies to demonstrate "compelling public interest" in order to continue as public agencies. (Who then oversees these committees to ensure the private sector doesn't unfairly profit by monopolizing public goods and services? One can only assume it is the same "Private Enterprise Advisory Committee.")
ALEC nuts and bolts
ALEC is a 501(c)(3) not-for-profit organization that in recent years has reported about $6.5 million in annual revenue. ALEC's members include corporations, trade associations, think tanks and nearly a third (about 2,000) of the nation's state legislators (virtually all Republican). According to the group's promotional material, ALEC's mission is to "advance the Jeffersonian principles of free markets, limited government, federalism, and individual liberty, through a nonpartisan public-private partnership of America's state legislators, members of the private sector, the federal government, and general public."
ALEC currently claims more than 250 corporations and special interest groups as private sector members. While the organization refuses to make a complete list of these private members available to the public, some known members include Exxon Mobil, the Corrections Corporation of America, AT&T, Pfizer Pharmaceuticals, Time Warner Cable, Comcast, Verizon, Wal-Mart, Phillip Morris International and Koch Industries, along with a host of right-wing think tanks and foundations.
ALEC is composed of nine task forces–(1) Public Safety and Elections, (2) Civil Justice, (3) Education, (4) Energy, Environment and Agriculture, (5) Commerce, Insurance and Economic Development, (6) Telecommunications and Information Technology, (7) Health and Human Services, (8) Tax and Fiscal Policy and (9) International Relations–each comprised of "Public Sector" members (legislators) and "Private Sector" members (corporations and interest groups).
Each of these task forces, which serve as the core of ALEC's operations, generate model legislation that is then passed on to member lawmakers for introduction in their home assemblies. According to ALEC promotional material, each year member lawmakers introduce an average of 1,000 of these pieces of legislation nationwide, 17 percent of which are enacted. For 2009, ALEC claimed a total of 826 pieces of introduced legislation nationwide, 115 of which were passed into law–slightly below the average at 14 percent. ALEC does not offer its model legislation for public inspection.
ALEC refused to comment on any aspect of the material covered here.
'Paycheck Protection'
The three pieces of model legislation contained in the ALEC "Paycheck Protection" bundle (archived at dbapress.com here) provided by Rep. Dorworth's office were titled "Employee Rights Reform Act," "Labor Organization Deductions Act" and "Political Funding Reform Act."
Employee Rights Reform Act (ERRA): This bill establishes limitations on fees that may be charged to nonunion public employees who are part of a collective bargaining unit represented by a union.
ERRA states that no nonunion public employee may have more than a proportionate share of collective bargaining union costs withheld from their pay by a public employer. Chargeable activities are defined as expenditures for purposes of collective bargaining, contract administration and grievance adjustment. ERRA states that whether or not a public employer can deduct funds from a public employees pay for political activity–union organizing campaigns, contributing to political campaigns of elected officials, lobbying on behalf of their members, or raising money from their members to pay for union organizing campaigns–is dependent on "controlling court decisions."
Labor Organizations Deductions Act (LODA): This is the only piece of the "Paycheck Protection" trilogy not aimed specifically at public employee unions (although the bill does name both the National Education Association and the American Federation of Teachers as entities that must comply with restrictions). LODA establishes a stringent set of criteria governing the means through which any labor organization may collect and use funds for political activity, such as lobbying, electoral and political activities, including contributions to any candidate, party or voter registration campaign.
LODA establishes criminal penalties for any labor organization found to have made a political contribution derived from dues or any other fee paid by union members. Further, LODA prohibits unions from soliciting funds for political use from any individual other than union members and their immediate family members.
Political Funding Reform Act (PFRA): While ERRA and LODA seek to significantly limit the amount and type of funds that may be deducted from employee pay–particularly as those funds may apply to union political activity–PFRA is designed to eliminate all withholding of public employee pay for use in any political activity. Simply put, under PFRA, unions would have to raise money for political purposes by directly fundraising to their members or other union supporters.
Florida: A case study
In the case of Florida's HB 1021, e-mails provided by Rep. Dorworth's office through a public records request reflect that the initial version of the bill had been drafted in January by then-Florida Chamber of Commerce (FCoC) Vice President of Government Affairs Adam Babington. A member of the FCoC Foundation's board of trustees, Cincy Marsiglio, the senior manager of public affairs and government relations in Florida for Wal-Mart, is the Florida ALEC "private sector" chair (see sidebar below for more on ALEC's public and private chairs). Babington's original draft (evidently based on ALEC "Paycheck Protection" model legislation) underwent a revision aimed at curtailing the political activity of public employee unions. This revision was made by Florida State Senate staff who were working with Babington to create a Senate companion version of the bill.
This companion bill, SB 830, was sponsored by Sen. John Thrasher (R-Jacksonville). Thrasher worked for the influential Tallahassee lobby firm of Southern Strategy Group, Inc., from 2002 through his election to the Florida Senate in 2009, where he represented several FCoC and ALEC member corporations, many with interests in the privatization of state governmental functions (particularly in the areas of mental health and healthcare service contracting).
The primary actor on the Senate end of HB 1021's formation was Andy Bardos, special counsel to Senate President Mike Haridopolos (R-Merrit Island). After a stringent anti-public employee union dues collecting provision was added by Bardos, Babington wrote in an e-mail to Dorworth and Johnson: "So, paycheck protection is about to go on steroids. Apparently the Senate wants to be more aggressive."
Bardos, prior to joining the office of Senate President Haridopolos in early 2011, had worked since 2005 for the Florida law firm of GrayRobinson as an attorney specializing in governmental affairs.
Bardos' former colleague, GrayRobinson attorney Fred Leonhardt, is currently on the board of directors of the FCoC, of which he was the former chair. Leonhardt is a member of Enterprise Florida, Inc., a "public-private partnership" that works as the economic development arm of the state.
Another director of Enterprise Florida is former Florida House Speaker Allan Bense (R-Panama City). Bense is the present chairman of FCoC, who derives a large portion of his annual income from a company he co-owns: GAC Contractors, Inc. As reported on his 2009 statement of financial interests (filed pursuant to his membership on the board of the quasi-public Enterprise Florida), Bense held nearly $5 million in GAC asssets, much of which was money earned from contracts to repair state and federal highways.
GAC is a prominent member of Associated Builders and Contractors, Inc. (ABC), which through its legislative efforts seeks to encourage the free flow of public-sector cash to nonunion private companies. ABC bills itself as being the nonunion "construction industry's voice within the legislative, executive and judicial branches" of government. The bundled "Paycheck Protection" package containing ERRA, LODA and PFRA in Dorworth's office had originated in ABC's 2010 "legislative handbook."
In addition to his FCoC, GAC and ABC connections, Bense is chair of the Florida-based, Koch-funded, ALEC-member public policy foundation, the James Madison Institute (JMI). FCoC baord member Leonhardt serves on the JMI board with Bense.
When asked why the FCoC was so deeply concerned with protecting the paychecks of public employees (to the point where FCoC top lobbyists were drafting legislation to such effect), FCoC Director of Public Affairs Edie Ousley declined to comment.
Both HB 1021 and SB 830 died in their respective chambers following pressure exerted on the FCoC by public employee union members.
According to materials obtained through a public records request, news of a large-scale opposition action made its way back to Dorworth in the form of an e-mail from Ousley, with the terse subject line "here's the issue." That e-mail contained a press release from a coalition of unions known as Floridians Outraged at the Chamber of Commerce's Attack on Workers, which read in part: "Wednesday, April 20…Workers respond to attacks from the Chamber of Commerce… Labor organizations and members withdrew close to $10 million in funds from the Chamber's largest banks." The press release went on to indicate that the group was prepared to issue further "wave(s) of withdrawals" and other actions.
Weeks later, on May 7, the bills' sponsors withdrew both bills from legislative hearings calendars.
Page 1 of 2 Continued »
Beau Hodai
Beau Hodai, a former In These Times Staff Writer, is the founder of DBA Press (dbapress.com), an online news publication and source materials archive.
Sneaks, snakes, snivelling nobodies...except, of course, they are cursed with the burden of more money than brains. So they tossed out bait and called up all the pusillanimous freaks who couldn't stand up for their hateful nihilism alone...like that faker, "Ayn Rand..." totally hooked on speed, using the old slavic trick of building a back-story that magnified her "Me, me,me..." refrain. Rand? What a loser. Not-too-bright people, insulated from most of modern life, banded together to make most of humanity miserable. Glued together by money they mostly did NOT earn, under a benevolent legal and political system that allows them to fester and spread their rot and pus by the most up-to-date electronics...a system that they fear and hate and will end as soon as they can stupefy themselves and more Americans...while picking their pockets by proxy...and their leaders, determined to uproot all hiumane advances since the Magna Carta? The short-order Kochers, staying as far away as they can from the heat...somebody once said something about those kind staying out of the kitchen...Occupy Wall Street will eventually show them how sausage is made, even...
Posted by Leslie Victor Piper on 2011-10-01 18:37:08
In my opinion, ALEC is doing very well. Education is the primary right of every person in the state since birth and if someone help in this regard, he/she must be appreciated. Resident of this world must take care of this world as their own home. We all must keep an eye on our environment quality. Good environment give good health to the livings. I would like to strongly vote to ALEC.
Posted by Philip First on 2011-09-24 02:52:29
It's not going to go away, but at this time The State, anywhere above the municipal level, where some examples of sane government still exist here and there, is so hopelessly befouled by corruption and incompetence that it is best to just stay out of its way, and depend on it for as little as possible. Only when the working class and underclass take responsibility for defending their own interests as vigorously as the elite do theirs will any real change come. The sooner we stop believing that the government can solve our problems, the sooner real solutions can be found. This is fascism, as defined by Mussolini, pure and simple, and the arcane "left vs. right" paradigm allows for a divided populace to be conquered and ruled. If we don't get serious about our rights, we won't even be allowed to have these online debates anymore.
Posted by Dan Mage on 2011-08-12 18:33:29
Interesting article. I will pass on this. Thanks for sharing information article. Laminate Flooring
Posted by davis4howard on 2011-08-02 06:03:00
That's all true terrible. But until we stop pretending that our elections are legitimate, how seriously can you expect me to take their complaints about thieves? bradblog.com good reports about our rigged elections, and is more important than partisan sniping. You are supporting a ruling, the hopelessly compromised system. Stop monitoring the truth. Testking 000-108 Fox is bad enough. Do not join them.
Posted by Jhon Breeno on 2011-07-28 22:20:19
The post is very informative. It is a pleasure reading it. I have also bookmarked you for checking out new posts. sell house quick
Posted by Jenny Hills on 2011-07-19 00:26:42
Every other organization asks their members of voluntarily contribute. Rotary or Elks or Shriners don't deduct dues off your paycheck. With electronic banking it is dead simple for union members to have their dues come out of their bank accounts like condo dues or mortgage payments or car loan payments. The union members fill out one form and its done. There is absolutely no reason for dues to come straight off a paycheck.
Posted by B Canuck on 2011-07-17 09:34:59
That's all real terrible stuff. But until you stop pretending our elections are legitimate, how seriously can you expect me to take your complaints about the usurpers? bradblog.com reports well on our fake elections, and is more important than partisan sniping. You're supporting a failed, hopelessly compromised system. Stop monitoring the truth. Fox is bad enough. Don't join them.
Posted by William Rigby on 2011-07-14 22:37:07
How could I have forgotten John Kasich? Piyush Jindal in Louisiana tried to sell off four state prisons to ostensibly raise funds for Medicare, but even the very Republican legislalture wasn't buying. Kasich tried to do him one better by selling off five state prisons to the corrupt for-profit operators. The Kochs also have propaganda factories in virtually every state, pretending to be "research" non-profits, churning out mendacious public relations. Calling themselves variously "institutes" or "foundations," they include Buckeye (OH), Allegheny and Commenwealth (PA), the Kansas Policy Foundation, Heartland Institute (IL), Mackinac Center (MI), Rio Grande (NM), Heritage (DC), Independent, Reason and Cato (CA), etc. Three are 85 in all, operated by Koch whores, and they are joined by various academics from around the U.S., either individually, or as departments or institutes within universities (i.e., the Mercatus Institute at George Mason U.).
Posted by Pancho on 2011-07-11 07:41:50
Another astounding investigative piece from Beau Hodai! Why can't the NY Times, the Wall Street Journal, the Washington Post do this sort of work? They certainly have the budget and the talent, but those mountains groan and bring forth mice. One of the Koch's prime objectives is to keep poor people, working class and lower middle class people, but particularly union members, away from the polls. If workers don't vote in their own interests, the Kochs can buy politicians so much more cheaply. Here's part of the list of their recent gubernatorial purchases, all with anti-union legislation, prison privatization, pension fund destabilizaton, deregulation of workplace and environmental safety, etc. Wisconsin: Scott Walker Florida: Rick Scott Michigan: Rick Snyder New Jersey: Chris Christie Pennsylvania: Tom Corbett Kansas: Sam Brownback South Carolina: Niki Haley Arizona: Jan Brewer Maine: Paul LePage Holdovers include: Louisiana: Piyush Jindal Mississippi: Haley Barbour Indiana: Mitch Daniels Essentially, anyplace you find more one of the key items on the Koch brothers "to do" list becoming a gubernatorial priority, you can find their direct and indirect contributions to campaigns: Anti-union (i.e., ending collective bargaining or dues checkoff) Anti-regulaltion (ending oversight on investments, environmental safeguards, etc.) Anti-tax (their bottom line) Anti-voter (i.e., "Real I.D. acts) Money comes from therm personally (David H. and/or Charles de Ganahl Koch and Richard Fink), from their corporations (Flint Hills refineries, Georgia Pacific paper and lumber, Koch Industries, etc.), and through their "independent" front groups, for media buys for candidates, ballot measures, etc. (Americans for Prosperity, Forward America, the Tea Party, Club for Growth, Citizens for a Sound Economy, etc.). | {
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Villa Maria College Celebrates Grads from 2020 and 2021 in an Outdoor Ceremony; Awards Given to Madelyn Jensen and José Colón
By Kristen SchoberJune 2, 2021Press Release
Home » News » Villa Maria College Celebrates Grads from 2020 and 2021 in an Outdoor Ceremony; Awards Given to Madelyn Jensen and José Colón
On Saturday, May 22, 2021, Villa Maria College celebrated its 56th Commencement ceremony.
In the name of safety, this year's event was held outside in front of the Main Building. The College's first outdoor ceremony since 1986 recognized 119 graduates from the class of 2021 and 84 graduates from the class of 2020. Candidates from both years earned degrees in arts, science, business administration, fine arts, and applied arts. Additionally, 5 students earned certificates for completing the College's Historic Preservation program.
Villa Maria College celebrated its 56th Commencement ceremony outside in front of the Main Building.
In keeping with the College's longstanding tradition, awards were given to two outstanding graduates.
The Martin Wanamaker Spirit Award is given annually by the Student Life Office to a graduate who has distinguished himself or herself in promoting spirit on campus. This year's Martin Wanamaker Spirit Award was given to José Colón.
José earned a B.S. in Music Industry while maintaining a 3.2 GPA. He truly embodies the Villa spirit, promoting everything that Villa Maria has to offer. He is well connected with his instructors and all resource offices on campus. Jose has served as an Orientation Leader, Student Ambassador, and is involved in the many events that occur on campus. José is the President of the Music Club and won the Outstanding Music Student Award in 2020. Throughout his time at Villa, he completed an internship at Blackrock Entertainment Production Studios, where he helped produce an entire album. He is also employed at Guitar Center, where he has gained extensive knowledge about instrument repair, lessons and sales. In the future, he hopes to apply this knowledge to continue to grow his own company, The Influence Entertainment Group. In his free time, José enjoys spending time with this family and friends., building race cars, and recording and producing music. He is currently a brown belt in karate and will continue to work towards earning his black belt.
José Colón, the winner of the 2020 Martin Wanamaker Spirit Award, addresses the Villa Maria College Community during commencement.
The Blessed Mary Angela Student Award is awarded annually at Commencement to the most outstanding graduate of Villa Maria College. Named after the foundress of the Felician Sisters, the award is selected by the President of the college and her cabinet, based on nominations from faculty and staff. Recipients embody the mission and values of Villa Maria College, demonstrated through outstanding achievement in academics, leadership, and service to both the college and local community. This year's award was given to Madelyn Jensen.
Jensen has completed a Business BBA, with highest honors, earning an impressive 4.0 GPA. Her coursework included a minor in marketing and a concentration in digital media marketing. As part of her programs, she became certified in Hubspot and Google Analytics. Madelyn is an exemplary student-athlete and a leader on campus, in and out of classroom. Besides captaining the College's bowling team for two years, she has served as an Orientation Leader, Student Ambassador, and is a member of the Leadership Development Program. This year, Madelyn was named that outstanding student in business in 2021 and is a member of the national scholastic honor society Delta Epsilon Sigma. Madelyn has participated in work-study, both at the front desk for admissions and as an administrative assistant. She gained internship experience with the YMCA's marketing department and ASTPS, a local tax firm. She also completed a full-year internship with the sales team at Lactalis. Outside of Villa, Madelyn serves the community as a volunteer firefighter.
2021 Blessed Mary Angela Student Award Winner Madelyn Jensen poses with the College's President, Dr. Giordano
Nyles Moore, 2020 winner of the Blessed Mary Angela Award, and Ashton Barrie, 2020 winner of the Martin Wanamaker Spirit Award, were present to be honored and receive their awards in person. To learn more about Nyles' an Ashton's contributions to the Villa Maria College community, click here.
Nyles Moore, winner of the 2020 Blessed Mary Angela Student Award, speaks during the 2021 commencement ceremony.
Ashton Barrie, winner of the 2020 Martin Wanamaker Spirit Award, poses with Dr. Giordano after accepting his award.
Additionally, the Founders Award was given to Gerard Place. David Zapfel, President and CEO of Gerard Place, accepted the award on behalf of the organization. Zapfel addressed the classes of 2021 and 2020, offering the graduates the advice to work hard, stay humble, and help others as they advance throughout life. To learn more about Gerard Place and Mr. Zapfel, click here.
Mr. David Zapfel accepts the 2021 Founders Award from the Chair of the Board of Trustees, Timothy Ryder.
To watch Commencement 2021 in its entirety, click here.
To view the photos from the ceremony, click here.
Congratulations to the Classes of 2021 and 2020!
Previous PostVilla Maria College Announces the Launch of its Community Health Baccalaureate Program; Opens Enrollment for an August 2021 Start
Next PostGraphic Design Students Earn First & Second Place in 2021 AAF Student Portfolio Review | {
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Perri Peltz is an Emmy-winning documentary filmmaker, journalist and public health advocate. Most recently, Perri created the documentary news series Axios on HBO with Matthew O'Neill. Perri & Matthew also co-directed and produced the 2019 HBO Documentary, Alternate Endings: Six New Ways to Die in America. Previously, Perri directed the HBO documentary, Warning: This Drug May Kill You, about the opioid addiction epidemic. She produced the HBO documentary Risky Drinking and co-directed A Conversation About Growing Up Black as part of the "Conversation on Race" series for The New York Times Op-Docs. Other films include HBO's Remembering the Artist: Robert De Niro, Sr. and Prison Dogs. Perri hosts "The Perri Peltz Show" on SiriusXM and is a doctoral candidate at Columbia University's School of Public Health. She was previously an award-winning broadcast journalist for NBC, ABC,and CNN.
Career
Peltz worked at WNBC from 1987 to 1996 where she co-anchored Weekend Today in New York with Ken Taylor, and weekend editions of News 4 New York at 6 and 11 with Ralph Penza.
Peltz joined Dateline NBC for two years. During that period, she often anchored live news coverage on NBC's 24-hour cable news television channel MSNBC.
She then worked for ABC's 20/20 for two years until she moved to CNN where she stayed until 2002.
Peltz left CNN to produce a feature film, Knights of the South Bronx starring Ted Danson. The film was based on the real-life story of a middle school chess team from the South Bronx that became national chess champions. The film aired on the A&E Network.
Peltz then went to work for the Robin Hood Foundation in New York City. Robin Hood is a non-profit organization dedicated to fighting poverty. While at Robin Hood, Peltz wanted to tell the stories of the people who were working on the front lines in the war against poverty.
In 2005 she rejoined WNBC after a nine-year absence to co-anchor Live at Five with Sue Simmons. She returned to WNBC to report on those people and the differences they were making. She also anchored Live at Five with Sue Simmons from 31 May 2005, until 12 March 2007, when she began hosting her own half-hour lifestyle broadcast titled News 4 You. The program was part of WNBC's attempt to boost ratings and features stories from the consumer, health and entertainment worlds.
On 10 September 2007, WNBC cancelled News 4 You. Peltz continued to report both for WNBC and for NBC Network on people who were making a difference.
Peltz co-produced and co-directed the documentary Prison Dogs, which premiered at the 2016 Tribeca Film Festival.
Education
Peltz graduated from The Dalton School in New York City, and then went to Brown University and then to Columbia for a Masters in Public Health. In 2008, she left WNBC to attend medical school.
Charitable work
References
External links
WNBC: Perri Peltz Returns To WNBC As Co-Anchor, Live At Five
1961 births
Living people
American television reporters and correspondents
Television anchors from New York City
New York (state) television reporters
American documentary film directors
American documentary film producers
Film producers from New York (state) | {
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package org.appng.core.templating;
import java.util.Collection;
import org.springframework.stereotype.Component;
import org.thymeleaf.context.ITemplateContext;
import org.thymeleaf.engine.AttributeName;
import org.thymeleaf.model.IProcessableElementTag;
import org.thymeleaf.processor.element.IElementTagStructureHandler;
import org.thymeleaf.standard.processor.StandardReplaceTagProcessor;
/**
* This interface has to be implemented by interceptors to intercept the call of the replace tag to determine if this
* call has to be replaced by another fragment. The interceptor can also add additional context variables. An
* interceptor also has to be available as bean in the spring context. Therefore the @{@link Component} annotation
* might be useful. <br/>
* <br/>
* Example:
*
* <pre>
* @Component
* public class CustomInterceptor extends ThymeleafReplaceInterceptorBase {
*
* public boolean intercept(ITemplateContext context, IProcessableElementTag tag, AttributeName attributeName,
* String attributeValue, IElementTagStructureHandler structureHandler,
* ThymeleafStandardReplaceTagProcessorCaller tagProcessor) {
*
* if (attributeValue.contains("::datacell")) {
* // we want to replace each call of datacell with our own fragment
* attributeValue = attributeValue.replace("::datacell", "customtemplate.html::datacell_custom");
* // proceed processing with our customized attribute value
* tagProcessor.callStandardDoProcess(context, tag, attributeName, attributeValue, structureHandler);
* // tell the replace tag processor that this intereptor already triggered the
* // processing
* return true;
* }
* // tell the replace tag processor that this interceptor did not intercept
* return false;
* }
*
* // add all needed template file names for this interceptor
* public Collection<String> getAdditionalTemplateResourceNames() {
* Set<String> resources = Sets.newHashSet();
* resources.add("personregister.html");
* return resources;
* }
*
* }
* </pre>
*
* @author Claus Stümke
*/
public interface ThymeleafReplaceInterceptor {
/**
* This method is called from the {@link ReplaceTagProcessor} before executing the ordinary replace logic as
* implemented by the {@link StandardReplaceTagProcessor}. With this method the interceptor can decide to overwrite
* the attribute value of the replace tag to call another target fragment instead. Probably the interceptor also has
* to add additional variables to the context if the other target fragment has parameters which does not exist in
* the given context If the interceptor wants to intercept, it has to modify the attributeValue and call the
* doProcess method of the given tagProcessor. It has to return true to indicate that processing was done within
* this interception to avoid another processing by the {@link ReplaceTagProcessor}
*
* @param context
* @param tag
* @param attributeName
* @param attributeValue
* @param structureHandler
* @param tagProcessor
*
* @return
*/
boolean intercept(final ITemplateContext context, final IProcessableElementTag tag,
final AttributeName attributeName, final String attributeValue,
final IElementTagStructureHandler structureHandler,
ThymeleafStandardReplaceTagProcessorCaller tagProcessor);
/**
* With this method the interceptor can define additional template files which should be added to the template
* engine that custom fragments are available during processing
*
* @return
*/
Collection<String> getAdditionalTemplateResourceNames();
/**
* In case that an application has different interceptors intercepting on the same replace call, the first
* interceptor will win. The order of interceptors can be affected by setting a high priority to interceptors which
* shall be processed first
*
* @return
*/
default int getPriority() {
return 0;
}
}
| {
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Q: Oracle Weblogic adding jars to the classpath? How do you add jars to the class path for Oracle 10.3.5...As I understood it, there is a bug (or incorrect info) with the documentation (readme) that states that any jars placed in the $DOMAIN_HOME/lib directory would be added to the classpath dynamically...but in the real documentation for 10.3.3 it states that these files dont get added to the classpath anymore...
so here I am trying to find out -- how do you add jars to the classpath...I have tried changing the commonEnv.sh and am currently looking for the setDomainEnv.sh (but cant find it as of yet) and none of these things have worked to add this jar to the classpath...
my whole problem is that i added datasources to my server...and I am trying to add the DB2 jar to the environment so that it can be used...funny thing is that after adding the jar in the $DOMAIN_HOME/lib I was able to get rid of a connection error in the admin console when trying to test the connection to the database...and that all seems to work but now im getting a class definition error...
]] Root cause of ServletException.
java.lang.NoClassDefFoundError: com/ibm/db2/jcc/DB2Connection
at java.lang.ClassLoader.defineClass1(Native Method)
at java.lang.ClassLoader.defineClassCond(ClassLoader.java:630)
at java.lang.ClassLoader.defineClass(ClassLoader.java:614)
at java.security.SecureClassLoader.defineClass(SecureClassLoader.java:141)
at weblogic.utils.classloaders.GenericClassLoader.defineClass(GenericClassLoader.java:343)
Truncated. see log file for complete stacktrace
Caused By: java.lang.ClassNotFoundException: com.ibm.db2.jcc.DB2Connection
at weblogic.utils.classloaders.GenericClassLoader.findLocalClass(GenericClassLoader.java:297)
at weblogic.utils.classloaders.GenericClassLoader.findClass(GenericClassLoader.java:270)
at java.lang.ClassLoader.loadClass(ClassLoader.java:305)
at java.lang.ClassLoader.loadClass(ClassLoader.java:246)
at weblogic.utils.classloaders.GenericClassLoader.loadClass(GenericClassLoader.java:179)
Truncated. see log file for complete stacktrace
idk what else to try - i searched for some answers but seemingly all of them are old and outdated...
A: $DOMAIN/lib should work fine, but not dynamically. You have to restart. However, handling JAR files for data source drivers is likely different.
Just curious - have you confirmed the jar file(s) contain he class in question?
Also try: http://docs.oracle.com/cd/E17904_01/web.1111/e13753/db2.htm
A: I ended up finding out the problem was that I was editing the commEnv.sh file on windows instead of the commEnv.cmd file...really dumb but editing that and adding the jar to the classpath there worked...bah!
| {
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\section{Introduction}
\subsection{Multiple zeta(-star) values, the cyclic sum formulas, and their generalizations}
For positive integers $k_{1},\dots,k_{r}$ with $k_{r}\ge2$,
the multiple zeta values (MZVs) and the multiple zeta-star values
(MZSVs) are defined by
\begin{align*}
\zeta(k_{1},\dots,k_{r})
&:=\sum_{0<n_{1}<\cdots<n_{r}}\frac{1}{n_{1}^{k_{1}}\cdots n_{r}^{k_{r}}} \in\mathbb{R},\\
\zeta^{\star}(k_{1},\dots,k_{r})
&:=\sum_{0<n_{1}\le\cdots\le n_{r}}\frac{1}{n_{1}^{k_{1}}\cdots n_{r}^{k_{r}}} \in\mathbb{R}.
\end{align*}
We say that an index $(k_{1},\dots,k_{r})\in\mathbb{Z}_{\ge1}^{r}$ is admissible if $k_{r}\ge2$.
It is known that there are a lot of linear relations over $\mathbb{Q}$ among MZVs.
The following relations obtained by Hoffman and Ohno \cite[eq.(1)]{HO03}, and Ohno and Wakabayashi \cite{OW06} are the clean-cut decompositions for the well-known sum formulas.
\begin{thm}[Cyclic sum formula for MZ(S)Vs; Hoffman--Ohno, Ohno--Wakabayashi] \label{cycsum}
For a non-empty index $(k_1,\dots,k_r)$ with $(k_1,\dots,k_r)\ne (\underbrace{1,\dots,1}_{r})$, we have
\begin{align*}
\sum_{l=1}^r \sum_{m=1}^{k_l-1} \zeta (m, k_{l+1}, \dots, k_r, k_1, \dots, k_{l-1}, k_l-m+1)
&=\sum_{l=1}^r \zeta (k_{l+1}, \ldots, k_r, k_1, \dots, k_{l-1}, k_l+1), \\
\sum_{l=1}^r \sum_{m=1}^{k_l-1} \zeta^\star (m, k_{l+1}, \dots, k_r, k_1, \dots, k_{l-1}, k_l-m+1)
&=k \zeta^\star (k+1),
\end{align*}
where we set $k:=k_1+\cdots+k_r$.
\end{thm}
\begin{rem}
The cyclic sum formulas for MZVs and MZSVs, i.e., the first and the second statements of Theorem \ref{cycsum}, are equivalent (see \cite[Section 4]{IKOO11} and \cite[Proposition 3.3]{TW10}).
\end{rem}
In \cite[Theorem 2]{HMM19}, Hirose, Murakami, and the author obtained the generalization of Theorem \ref{cycsum}
by considering the cyclic analogue of MZVs.
\begin{defn}[Cyclic analogue of MZVs]
Let $d$ and $r_1,\dots,r_d$ be positive integers.
For positive integers $n_{i,1},\dots,n_{i,r_i}$ and $k_{i,1},\ldots,k_{i,r_i} \ (i=1,\dots,d)$, we write
\begin{align*}
\boldsymbol{k}_i
&:=(k_{i,1},\dots,k_{i,r_i}), \\
\boldsymbol{n}_i^{\boldsymbol{k}_i}
&:=n_{i,1}^{k_{i,1}} \cdots n_{i,r_i}^{k_{i,r_i}}, \\
r
&:=r_1+\cdots+r_d
\end{align*}
and
\begin{align*}
\boldsymbol{k}
&:=[\boldsymbol{k}_{1},\dots,\boldsymbol{k}_{d}], \\
\boldsymbol{n}^{\boldsymbol{k}}
&:=\boldsymbol{n}_{1}^{\boldsymbol{k}_{1}}\cdots\boldsymbol{n}_{d}^{\boldsymbol{k}_{d}}.
\end{align*}
A multi-index $\boldsymbol{k}$ is called an admissible multi-index if
\begin{itemize}
\item for all $1\le i\le d$, the index $\boldsymbol{k}_i$ is admissible or equal to $(1)$,
\item there exists $1\le i\le d$ such that $\boldsymbol{k}_i \ne (1)$.
\end{itemize}
Then, for an admissible multi-index $\boldsymbol{k}$, the cyclic analogue of MZVs are defined by
\begin{align*}
\zeta^{\mathrm{cyc}} (\boldsymbol{k})
:=\sum_{S} \frac{ 1 }{ \boldsymbol{n}^{\boldsymbol{k}} },
\end{align*}
where
\begin{align*}
S&
:=\{(n_{1,1},\dots,n_{d,r_{d}})\in\mathbb{Z}_{\ge1}^{r}
\mid
n_{1,1}<\cdots<n_{1,r_{1}},\dots, n_{d,1}<\cdots<n_{d,r_{d}}, \\
&\qquad\qquad\qquad\qquad\qquad\qquad\,\,
n_{1,1} \le n_{2,r_{2}}, \dots, n_{d-1,1} \le n_{d,r_{d}}, n_{d,1} \le n_{1,r_{1}} \}.
\end{align*}
\end{defn}
\begin{rem}
When $d=1$, we have
$\zeta^{\mathrm{cyc}} ([(k_{1,1},\dots,k_{1,r_1})])=\zeta (k_{1,1},\dots,k_{1,r_1})$.
\end{rem}
We recall Hoffman's algebraic setup with a slightly different convention (see \cite{hoffman_97}).
Set $\mathfrak{H}:=\mathbb{Q}\langle x,y\rangle$.
We denote by $\mathfrak{H}^{\mathrm{cyc}}_{0}$ the subspace of $\oplus_{d=1}^{\infty}\mathfrak{H}^{\otimes d}$
spanned by
\[
\bigcup_{d=1}^{\infty}
\{w_{1}\otimes\cdots\otimes w_{d}\in\mathfrak{H}^{\otimes d}
\mid w_{1},\dots,w_{d}\in y\mathfrak{H}x \cup\{y\}\
\text{and there exists }i\ \text{such that }w_{i}\ne y\}.
\]
For a positive integer $k$, put $z_k:=yx^{k-1}$.
We define a $\mathbb{Q}$-linear map $Z^{\mathrm{cyc}}_{0}:\mathfrak{H}^\mathrm{cyc}\to\mathbb{R}$
by
\[
Z^\mathrm{cyc}(z_{k_{1,1}}\cdots z_{k_{1,r_{1}}}\otimes\cdots\otimes z_{k_{d,1}}\cdots z_{k_{d,r_{d}}})
:=\zeta^\mathrm{cyc}([(k_{1,1},\dots,k_{1,r_{1}}),\dots,(k_{d,1},\dots,k_{d,r_{d}})]).
\]
We define the shuffle product as the $\mathbb{Q}$-bilinear product $\sh:\mathfrak{H}\times\mathfrak{H}\to\mathfrak{H}$ given by
\begin{align*}
1\sh w&=w\sh 1=w, \\
wu\sh w'u'&=(w\sh w'u')u+(wu\sh w')u',
\end{align*}
where $w,w'\in\mathfrak{H}$ and $u,u'\in\{x,y\}$.
\begin{thm}[Cyclic relation; Hirose--Murahara--Murakami] \label{cycrel}
For $w_{1}\otimes\cdots\otimes w_{d}\in\mathfrak{H}^\mathrm{cyc}_{0}$, we have
\begin{align*}
&\sum_{i=1}^{d} Z^{\mathrm{cyc}}
(w_{1}\otimes\cdots\otimes w_{i-1}\otimes(y\shaub w_{i})\otimes w_{i+1}\otimes\cdots\otimes w_{d}) \\
&=\sum_{i=1}^{d} Z^{\mathrm{cyc}}
(w_{1}\otimes\cdots\otimes w_{i}\otimes y\otimes w_{i+1}\otimes\cdots\otimes w_{d}),
\end{align*}
where $y\shaub u_{i}=y\sh u_{i}-yu_{i}-u_{i}y$.
\end{thm}
\begin{rem}
Writing $w_i=z_{k_{i,1}}\cdots z_{k_{i,r_i}}$ for $i=1,\dots,d$, we find that the case $r_1=\cdots=r_d=1$ of Theorem \ref{cycrel} gives Theorem \ref{cycsum} (for details, see \cite[Section 5.1]{HMM19}).
\end{rem}
\begin{rem}
Recently, Onozuka and the author \cite{MO20} gave complex variable generalization of Theorem \ref{cycrel}.
\end{rem}
\subsection{Finite multiple zeta(-star) values and the cyclic sum formulas}
We set a $\mathbb{Q}$-algebra $\mathcal{A}$ by
\[
\mathcal{A}:=\biggl(\prod_{p}\mathbb{Z}/p\mathbb{Z}\biggr)\,\Big/\,\biggl(\bigoplus_{p}\mathbb{Z}/p\mathbb{Z\biggr)},
\]
where $p$ runs over all primes.
For positive integers $k_{1},\dots,k_{r}$,
the finite multiple zeta values (FMZVs) and the finite multiple zeta-star
values (FMZSVs) are defined by
\begin{align*}
\zeta_{\mathcal{A}}(k_{1},\dots,k_{r})
&:=\biggl(\sum_{0<n_{1}<\cdots<n_{r}<p} \frac{1}{n_{1}^{k_{1}}\cdots n_{r}^{k_{r}}}\bmod p \biggr)_{p} \in\mathcal{A}, \\
\zeta_{\mathcal{A}}^{\star}(k_{1},\dots,k_{r})
&:=\biggl(\sum_{0<n_{1}\le\cdots\le n_{r}<p} \frac{1}{n_{1}^{k_{1}}\cdots n_{r}^{k_{r}}}\bmod p \biggr)_{p} \in\mathcal{A}
\end{align*}
(for details of FMZVs, see \cite{Kan19,KZ20}).
Many families of relations among MZ(S)Vs have analogues for FMZ(S)Vs.
The counterparts of Theorem \ref{cycsum} for FMZ(S)Vs were obtained by Kawasaki and Oyama \cite{KO19}.
\begin{thm}[Cyclic sum formula for FMZ(S)Vs; Kawasaki--Oyama] \label{cycsumF}
For a non-empty index $(k_1,\dots,k_r)$ with $(k_1,\dots,k_r)\ne (\underbrace{1,\dots,1}_{r})$, we have
\begin{align*}
&\sum_{l=1}^{r} \sum_{m=1}^{k_l-1}
\zeta_\mathcal{A} (m,k_{l+1},\dots,k_r,k_1,\dots, k_{l-1}, k_{l}-m+1) \\
&=\sum_{l=1}^{r}
\bigl(\zeta_\mathcal{A} (k_{l+1},\dots,k_r,k_1,\dots,k_{l-1},k_{l}+1)
+\zeta_\mathcal{A} (k_{l+1}+1,k_{l+2},\dots,k_r,k_1,\dots,k_{l}) \\
&\qquad\quad +\zeta_\mathcal{A} (1,k_{l+1},\dots,k_r,k_1,\dots,k_{l}) \bigr), \\
&\sum_{l=1}^{r} \sum_{m=1}^{k_l-1}
\zeta_\mathcal{A}^\star (m,k_{l+1},\dots,k_r,k_1,\dots, k_{l-1}, k_{l}-m+1)
=\sum_{l=1}^{r} \zeta_\mathcal{A}^\star (1,k_{l+1},\dots,k_r,k_1,\dots,k_{l}).
\end{align*}
\end{thm}
\begin{rem}
Note that the cyclic sum formulas for FMZVs and FMZSVs are equivalent (see \cite[Section 2]{MOno19} and \cite[Section 6]{HMOno19}).
\end{rem}
\subsection{Main result}
To state our main theorem, we introduce the cyclic analogue of FMZVs (FCMZVs).
\begin{defn}[Cyclic analogue of FMZVs]
Let $d$ and $r_1,\dots,r_d$ be positive integers.
For a multi-index
$ \boldsymbol{k}=[\boldsymbol{k}_{1},\dots,\boldsymbol{k}_{d}]$
with
$\boldsymbol{k}_i=(k_{i,1},\dots,k_{i,r_i}) \in\mathbb{Z}_{\ge1}^{r_i} \;(i=1,\ldots,d)$, we define
\begin{align*}
\zeta^{\mathrm{cyc}}_{\mathcal{A}} (\boldsymbol{k})
:=\biggl( \sum_{S_p} \frac{ 1 }{ \boldsymbol{n}^{\boldsymbol{k}} } \bmod p \biggr)_{p} \in\mathcal{A},
\end{align*}
where we put
$
\boldsymbol{n}^{\boldsymbol{k}}
:=\boldsymbol{n}_{1}^{\boldsymbol{k}_{1}}\cdots\boldsymbol{n}_{d}^{\boldsymbol{k}_{d}}, \,
\boldsymbol{n}_i^{\boldsymbol{k}_i}
:=n_{i,1}^{k_{i,1}} \cdots n_{i,r_i}^{k_{i,r_i}},
$
and
\begin{align*}
S_p&
:=\{(n_{1,1},\dots,n_{d,r_{d}})\in\{ 1,\dots,p-1 \}^{r}
\mid
n_{1,1}<\cdots<n_{1,r_{1}},\dots, n_{d,1}<\cdots<n_{d,r_{d}}, \\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad
n_{1,1} \le n_{2,r_{2}}, \dots, n_{d-1,1} \le n_{d,r_{d}}, n_{d,1} \le n_{1,r_{1}} \}.
\end{align*}
\end{defn}
We denote by $\mathfrak{H}^{\mathrm{cyc}}$ the subspace of $\oplus_{d=1}^{\infty}\mathfrak{H}^{\otimes d}$
spanned by
\[
\bigcup_{d=1}^{\infty}
\{w_{1}\otimes\cdots\otimes w_{d}\in\mathfrak{H}^{\otimes d}
\mid w_{1},\dots,w_{d}\in y\mathfrak{H} \}.
\]
We define a $\mathbb{Q}$-linear map $Z_{\mathcal{A}}^{\mathrm{cyc}}:\mathfrak{H}^\mathrm{cyc}\to\mathcal{A}$
by
\[
Z_{\mathcal{A}}^\mathrm{cyc} (z_{k_{1,1}}\cdots z_{k_{1,r_{1}}}\otimes\cdots\otimes z_{k_{d,1}}\cdots z_{k_{d,r_{d}}})
:=\zeta_{\mathcal{A}}^\mathrm{cyc}([(k_{1,1},\dots,k_{1,r_{1}}),\dots,(k_{d,1},\dots,k_{d,r_{d}})]).
\]
\begin{thm}[Main theorem] \label{main}
For $w_{1}\otimes\cdots\otimes w_{d}\in\mathfrak{H}^\mathrm{cyc}$, we have
\begin{align*}
&\sum_{i=1}^{d} Z_{\mathcal{A}}^{\mathrm{cyc}}
(w_{1}\otimes\cdots\otimes w_{i-1}\otimes(y\sht w_{i})\otimes w_{i+1}\otimes\cdots\otimes w_{d}) \\
&=2\sum_{i=1}^{d} \sum_{j=1}^{r_i}
Z_{\mathcal{A}}^{\mathrm{cyc}}
(w_{1}\otimes\cdots\otimes w_{i-1}\otimes
z_{k_{i,1}} \cdots z_{k_{i,j-1}} z_{k_{i,j}+1} z_{k_{i,j+1}} \cdots z_{k_{i,r_i}}
\otimes w_{i+1}\otimes\cdots\otimes w_{d}),
\end{align*}
where we put
$w_i:=z_{k_{i,1}}\cdots z_{k_{i,r_i}}$
and
\begin{align*}
&y\sht z_{k}:=
\begin{cases}
-y^2 &\textrm{if }k=1, \\
0 &\textrm{if }k=2, \\
yx(y\sh x^{k-3})x &\textrm{if }k\ge3,
\end{cases} \\
&y\sht z_{k_1}\cdots z_{k_r}:=(y\sht z_{k_1})z_{k_2}\cdots z_{k_r}+\cdots+z_{k_1}\cdots z_{k_{r-1}}(y\sht z_{k_r}).
\end{align*}
\end{thm}
\begin{ex}
When $\boldsymbol{k}=[(1,2,3)]$, we have
\begin{align*}
2\zeta_{\mathcal{A}} (1,2,4) +2\zeta_{\mathcal{A}} (1,3,3) +2\zeta_{\mathcal{A}} (2,2,3)
+\zeta_{\mathcal{A}} (1,1,2,3) -\zeta_{\mathcal{A}} (1,2,2,2)=0.
\end{align*}
When $\boldsymbol{k}=[(1,3),(2)]$, we also have
\begin{align*}
&2\zeta^{\mathrm{cyc}}_{\mathcal{A}} ([(1,3), (3)]) +2\zeta^{\mathrm{cyc}}_{\mathcal{A}} ([(1,4), (2)])
+2\zeta^{\mathrm{cyc}}_{\mathcal{A}} ([(2,3), (2)]) \\
&+\zeta^{\mathrm{cyc}}_{\mathcal{A}} ([(1,1,3), (2)])
-\zeta^{\mathrm{cyc}}_{\mathcal{A}} ([(1,2,2), (2)])=0
\end{align*}
and
\begin{align*}
&4\zeta_{\mathcal{A}}(1, 6) +2\zeta_{\mathcal{A}}(2, 5) +2\zeta_{\mathcal{A}}(3, 4) +4\zeta_{\mathcal{A}}(4, 3)
+\zeta_{\mathcal{A}}(1, 1, 5) +\zeta_{\mathcal{A}}(1, 2, 4) \\
&+3\zeta_{\mathcal{A}}(1, 3, 3) -\zeta_{\mathcal{A}}(1, 4, 2) +2\zeta_{\mathcal{A}}(2, 2, 3)
+\zeta_{\mathcal{A}}(3, 1, 3) -\zeta_{\mathcal{A}}(3, 2, 2) \\
&+\zeta_{\mathcal{A}}(1, 1, 2, 3) +\zeta_{\mathcal{A}}(1, 2, 1, 3) -2\zeta_{\mathcal{A}}(1, 2, 2, 2)=0.
\end{align*}
\end{ex}
\begin{rem}
The case $r_1=\cdots=r_d=1$ of Theorem \ref{main} implies Theorem \ref{cycsumF}.
In fact, by the theorem, we have
\begin{align*}
&\sum_{i=1}^{d} Z_{\mathcal{A}}^{\mathrm{cyc}}
(z_{k_{1}}\otimes\cdots\otimes z_{k_{i-1}}\otimes(y\sht z_{k_{i}})\otimes z_{k_{i+1}}\otimes\cdots\otimes z_{k_{d}}) \\
&=2\sum_{i=1}^{d} \sum_{j=1}^{r_i}
Z_{\mathcal{A}}^{\mathrm{cyc}}
(z_{k_{1}}\otimes\cdots\otimes z_{k_{i-1}}\otimes
z_{k_{i}+1}
\otimes z_{k_{i+1}}\otimes\cdots\otimes z_{k_{d}}).
\end{align*}
Note that the R.H.S.\ of the above equality equals $0$ by the well-known identity of FMZVs $\zeta_{\mathcal{A}}(k)=0$.
Since $y\sht z_k=y(y\sh x^{k-2})x -y^2x^{k-1}=\sum_{m=1}^{k-1} z_m z_{k+1-m} -y^2x^{k-1}$ and by the definition of FCMZVs,
we get the second statement of Theorem \ref{cycsumF}.
\end{rem}
\section{Proof of Theorem \ref{main}}
Let
\begin{align*}
S_p^{(1)}&:=\{(n_{1,1},\dots,n_{d,r_{d}})\in\{ 1,\dots,p-1 \}^{r}
\mid
n_{1,1}<\cdots<n_{1,r_{1}}, \dots, n_{d,1}<\cdots<n_{d,r_{d}}, \\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad n_{1,1} \le n_{2,r_{2}}, \dots, n_{d-1,1} \le n_{d,r_{d}} \}, \\
S_p^{(i)}&:=\{(n_{1,1},\dots,n_{d,r_{d}})\in\{ 1,\dots,p-1 \}^{r}
\mid
n_{1,1}<\cdots<n_{1,r_{1}}, \dots, n_{d,1}<\cdots<n_{d,r_{d}}, \\
&\qquad n_{1,1} \le n_{2,r_{2}}, \dots, n_{i-2,1} \le n_{i-1,r_{i-1}},
n_{i,1} \le n_{i+1,r_{i+1}}, \dots, n_{d-1,1} \le n_{d,r_{d}}, n_{d,1} \le n_{1,r_{1}} \} \\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad (i\ne1).
\end{align*}
In the following, we understand $n_{0,j}=n_{d,j}$ and $n_{d+1,j}=n_{1,j}$.
\begin{lem} \label{lem1}
For a positive integer $i$ with $1\le i\le d$ and $w_{1}\otimes\cdots\otimes w_{d}\in\mathfrak{H}^\mathrm{cyc}$, we have
\begin{align*}
&Z_{\mathcal{A}}^{\mathrm{cyc}}
(w_{1}\otimes\cdots\otimes w_{i-1}\otimes(y\sht w_{i})\otimes w_{i+1}\otimes\cdots\otimes w_{d}) \\
&=\biggl(
\sum_{j=1}^{r_i-1} \sum_{\substack{ S_p \\ n_{i,j}<n<n_{i,j+1} }}
\frac{ 1 }{ \boldsymbol{n}^{\boldsymbol{k}} }
\biggl( \frac{ n_{i,j} }{ n(n-n_{i,j}) } -\frac{ n_{i,j}^{k_{i,j}-1} }{ n^{k_{i,j}-1} (n-n_{i,j}) } \biggr) \\
&\qquad +\sum_{\substack{ S_p^{(i)} \\ n_{i,r_i}<n<p \\ n_{i-1,1}\le n<p }}
\frac{ 1 }{ \boldsymbol{n}^{\boldsymbol{k}} }
\biggl( \frac{ n_{i,r_i} }{ n(n-n_{i,r_i}) } -\frac{ n_{i,r_i}^{k_{i,r_i}-1} }{ n^{k_{i,r_i}-1} (n-n_{i,r_i}) } \biggr)
\bmod p \biggr)_{p},
\end{align*}
where we understand $n_{i,r_i+1}=p$ for all $i$ with $1\le i\le d$.
\end{lem}
\begin{proof}
Let $w_i=z_{k_{i,1}}\cdots z_{k_{i,r_i}}$.
By definitions, we have
\begin{align*}
\textrm{L.H.S.}
=\biggl(
\sum_{j=1}^{r_i-1}
\sum_{\substack{ S_p \\ n_{i,j}<n<n_{i,j+1} }}
\frac{ N_{k_{i,j}} }{ \boldsymbol{n}^{\boldsymbol{k}} }
+\sum_{\substack{ S_p^{(i)} \\ n_{i,r_i}<n<p \\ n_{i-1,1}\le n<p }}
\frac{ N_{k_{i,r_i}} }{ \boldsymbol{n}^{\boldsymbol{k}} }
\quad \bmod p \biggr)_{p},
\end{align*}
where
\begin{align*}
N_{k_{i,j}}
=
\begin{cases}
\displaystyle{
-\frac{1}{n}
} & \textrm{if } k_{i,j}=1,
\\
\;\; 0 & \textrm{if } k_{i,j}=2,
\\
\displaystyle{
\sum_{m=2}^{k_{i,j}-1}
\frac{ n_{i,j}^{k_{i,j}} }{ n_{i,j}^m n^{k_{i,j}-m+1} }
} & \textrm{if } k_{i,j}\ge3
\end{cases}
\end{align*}
for $j=1,\ldots,r_i$.
Since
\[
N_{1}
=\frac{ n_{i,j} }{ n(n-n_{i,j}) } -\frac{ 1 }{ n-n_{i,j} }
\]
and
\begin{align*}
N_{k_{i,j}}
&=\frac{ n_{i,j}^{k_{i,j}-1} ( 1-(n/n_{i,j})^{k_{i,j}-2} ) }{ n^{k_{i,j}-1}(n_{i,j}-n) } \\
&=\frac{ n_{i,j} }{ n (n-n_{i,j}) }
-\frac{ n_{i,j}^{k_{i,j}-1} }{ n^{k_{i,j}-1} (n-n_{i,j}) }
\qquad (k_{i,j}\ge3),
\end{align*}
we obtain the result.
\end{proof}
For positive integers $p$ and $i$ with $1\le i\le d$, and a multi-index $\boldsymbol{k}$, let
\begin{align*}
A_i&:=A(p,i,\boldsymbol{k}) \\
&:=\sum_{ S_p }
\frac{1}{\boldsymbol{n}^{\boldsymbol{k}}} \left(
\biggl( \frac{ 1 }{ n_{i,1}+1 }+\cdots+\frac{ 1 }{ n_{i,2}-1 } \biggr) +\cdots+
\biggl( \frac{ 1 }{ n_{i,r_{i}-1}+1 }+\cdots+\frac{ 1 }{ n_{i,r_i}-1 } \biggr)
\right), \\
B_i&:=B(p,i,\boldsymbol{k}) \\
&:=\sum_{ S_p^{(i)} }
\displaystyle
\frac{1}{\boldsymbol{n}^{\boldsymbol{k}}} \biggl(
\biggl( \frac{1}{\max\{ 1, n_{i-1,1}-n_{i,r_i} \}}+\cdots+\frac{1}{ p-n_{i,r_i}-1 } \biggr) \\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad -
\biggl( \frac{1}{ \max\{ n_{i,r_i}+1, n_{i-1,1} \} } +\cdots+\frac{ 1 }{ p-1 } \biggr)
\biggr), \\
C_i&:=C(p,i,\boldsymbol{k}) \\
&:=\sum_{ S_p^{(i+1)} }
\displaystyle
\frac{1}{\boldsymbol{n}^{\boldsymbol{k}}} \biggl(
\biggl( \frac{1}{ \max\{ 1, n_{i,1}-n_{i+1,r_{i+1}} \} }+\cdots+\frac{1}{ n_{i,1}-1 } \biggr) \\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\,\, +
\biggl( 1+\cdots+\frac{1}{ \min\{ n_{i,1}-1, n_{i+1,r_{i+1}} \}} \biggr)
\biggr).
\end{align*}
\begin{lem} \label{lem2}
For positive integers $p$ and $i$ with $1\le i\le d$, and a multi-index $\boldsymbol{k}$, we have
\begin{align*}
\sum_{j=1}^{r_i-1} \sum_{\substack{ S_p \\ n_{i,j}<n<n_{i,j+1} }}
\frac{ 1 }{ \boldsymbol{n}^{\boldsymbol{k}} }
\biggl( \frac{ n_{i,j} }{ n(n-n_{i,j}) } -\frac{ n_{i,j}^{k_{i,j}-1} }{ n^{k_{i,j}-1} (n-n_{i,j}) } \biggr)& \\
+\sum_{\substack{ S_p^{(i)} \\ n_{i,r_i}<n<p \\ n_{i-1,1}\le n<p }}
\frac{ 1 }{ \boldsymbol{n}^{\boldsymbol{k}} }
\biggl( \frac{ n_{i,r_i} }{ n(n-n_{i,j}) } -\frac{ n_{i,r_i}^{k_{i,r_i}-1} }{ n^{k_{i,r_i}-1} (n-n_{i,r_i}) } \biggr)
&=-2A_i+B_i-C_i.
\end{align*}
\end{lem}
\begin{proof}
Let
\begin{align*}
D_{i,j}^{(1)}
&:=\sum_{\substack{ S_p \\ n_{i,j}<n<n_{i,j+1} }}
\frac{ 1 }{ \boldsymbol{n}^{\boldsymbol{k}} }
\frac{ n_{i,j} }{ n(n-n_{i,j}) }
\quad (j\ne r_i),
&D_{i,r_i}^{(1)}
&:=\sum_{\substack{ S_p^{(i)} \\ n_{i,r_i}<n<p \\ n_{i-1,1}\le n<p }}
\frac{ 1 }{ \boldsymbol{n}^{\boldsymbol{k}} }
\frac{ n_{i,r_i} }{ n(n-n_{i,r_i}) }, \\
D_{i,j}^{(2)}
&:=\sum_{\substack{ S_p \\ n_{i,j}<n<n_{i,j+1} }}
\frac{ 1 }{ \boldsymbol{n}^{\boldsymbol{k}} }
\frac{ n_{i,j}^{k_{i,j}-1} }{ n^{k_{i,j}-1} (n-n_{i,j}) }
\quad (j\ne r_i),
&D_{i,r_i}^{(2)}
&:=\sum_{\substack{ S_p^{(i)} \\ n_{i,r_i}<n<p \\ n_{i-1,1}\le n<p }}
\frac{ 1 }{ \boldsymbol{n}^{\boldsymbol{k}} }
\frac{ n_{i,r_i}^{k_{i,r_i}-1} }{ n^{k_{i,r_i}-1} (n-n_{i,r_i}) }.
\end{align*}
Then we have
\begin{align}
\begin{split} \label{eq1}
D_{i,j}^{(1)}
&=\sum_{\substack{ S_p \\ n_{i,j}<n<n_{i,j+1} }}
\frac{ 1 }{ \boldsymbol{n}^{\boldsymbol{k}} }
\biggl( \frac{ 1 }{ n-n_{i,j} } -\frac{ 1 }{ n } \biggr) \\
&=\sum_{S_p }
\frac{ 1 }{ \boldsymbol{n}^{\boldsymbol{k}} }
\biggl(
\biggl( 1+\cdots+ \frac{ 1 }{ n_{i,j+1}-n_{i,j}-1 } \biggr)
-\biggl( \frac{ 1 }{ n_{i,j}+1 }+\cdots+ \frac{ 1 }{ n_{i,j+1}-1 } \biggr)
\biggr)
\end{split}
\end{align}
for $j=1,\dots,r_i-1$, and
\begin{align*}
D_{i,r_i}^{(1)}
=\sum_{\substack{ S_p^{(i)} \\ n_{i,r_i}<n<p \\ n_{i-1,1}\le n<p }}
\frac{ 1 }{ \boldsymbol{n}^{\boldsymbol{k}} }
\biggl( \frac{ 1 }{ n-n_{i,r_i} } -\frac{ 1 }{ n } \biggr)
=B_i.
\end{align*}
We also have
\begin{align*}
D_{i,j}^{(2)}
&=\sum_{\substack{ S_p \\ n_{i,j}<n<n_{i,j+1} }}
\frac{ n_{i,j}^{k_{i,j}} }{ \boldsymbol{n}^{\boldsymbol{k}} }
\frac{ 1 }{ n_{i,j} n^{k_{i,j}-1} (n-n_{i,j}) } \\
&=\sum_{\substack{ S_p \\ n_{i,j-1}<n<n_{i,j} }}
\frac{ n_{i,j}^{k_{i,j}} }{ \boldsymbol{n}^{\boldsymbol{k}} }
\frac{ 1 }{ n_{i,j}^{k_{i,j}-1} n (n_{i,j}-n) }
\end{align*}
for $j=2,\dots,r_i-1$.
In the above equality, we interchanged $n$ and $n_{i,j}$.
Then we have
\begin{align}
\begin{split} \label{eq2}
D_{i,j}^{(2)}
&=\sum_{\substack{ S_p \\ n_{i,j-1}<n<n_{i,j} }}
\frac{ 1 }{ \boldsymbol{n}^{\boldsymbol{k}} }
\biggl( \frac{ 1 }{ n_{i,j}-n } +\frac{ 1 }{ n } \biggr) \\
&=\sum_{S_p }
\frac{ 1 }{ \boldsymbol{n}^{\boldsymbol{k}} }
\biggl(
\biggl( 1+\cdots+ \frac{ 1 }{ n_{i,j}-n_{i,j-1}-1 } \biggr)
+\biggl( \frac{ 1 }{ n_{i,j-1}+1 }+\cdots+ \frac{ 1 }{ n_{i,j}-1 } \biggr)
\biggr)
\end{split}
\end{align}
for $j=2,\dots,r_i-1$.
Similarly, we have
\begin{align*}
D_{i,1}^{(2)}
&=\sum_{\substack{ S_p^{(i+1)} \\ 1\le n<n_{i,1} \\ 1\le n\le n_{i+1,r_{i+1}} }}
\frac{ 1 }{ \boldsymbol{n}^{\boldsymbol{k}} }
\biggl( \frac{ 1 }{ n_{i,1}-n } +\frac{ 1 }{ n } \biggr)
=C_i
\end{align*}
and
\begin{align}
\begin{split} \label{eq3}
D_{i,r_i}^{(2)}
&=\sum_{S_p }
\frac{ 1 }{ \boldsymbol{n}^{\boldsymbol{k}} }
\biggl(
\biggl( 1+\cdots+ \frac{ 1 }{ n_{i,r_i}-n_{i,r_i-1}-1 } \biggr)
+\biggl( \frac{ 1 }{ n_{i,r_i-1}+1 }+\cdots+ \frac{ 1 }{ n_{i,r_i}-1 } \biggr)
\biggr).
\end{split}
\end{align}
From \eqref{eq1}, \eqref{eq2}, and \eqref{eq3}, we have
\begin{align*}
D_{i,j}^{(1)}-D_{i,j+1}^{(2)}
=-2\sum_{S_p }
\frac{ 1 }{ \boldsymbol{n}^{\boldsymbol{k}} }
\biggl( \frac{ 1 }{ n_{i,j}+1 }+\cdots+ \frac{ 1 }{ n_{i,j+1}-1 } \biggr)
\end{align*}
for $j=1,\dots,r_i-1$.
Hence we find the result.
\end{proof}
\begin{proof}[Proof of Theorem \ref{main}]
Note that
\begin{align*}
B_i
&=\sum_{ S_p }
\frac{1}{\boldsymbol{n}^{\boldsymbol{k}}} \biggl(
\biggl( 1+\cdots+\frac{1}{ p-n_{i,r_i}-1 } \biggr)
-\biggl( \frac{1}{ n_{i,r_i}+1 } +\cdots+\frac{ 1 }{ p-1 } \biggr)
\biggr), \\
&\quad +\sum_{\substack{ S_p^{(i)} \\ n_{i,r_i}<n_{i-1,1} }}
\frac{1}{\boldsymbol{n}^{\boldsymbol{k}}} \biggl(
\biggl( \frac{1}{ n_{i-1,1}-n_{i,r_i} }+\cdots+\frac{1}{ p-n_{i,r_i}-1 } \biggr)
-\biggl( \frac{1}{ n_{i-1,1} } +\cdots+\frac{ 1 }{ p-1 } \biggr)
\biggr)
\end{align*}
and
\begin{align*}
C_i
&=2\sum_{ S_p }
\frac{1}{\boldsymbol{n}^{\boldsymbol{k}}}
\biggl( 1+\cdots+\frac{1}{ n_{i,1}-1 } \biggr) \\
&\quad +\sum_{\substack{ S_p^{(i+1)} \\ n_{i+1,r_{i+1}}<n_{i,1} }}
\frac{1}{\boldsymbol{n}^{\boldsymbol{k}}} \biggl(
\biggl( \frac{1}{ n_{i,1}-n_{i+1,r_{i+1}} }+\cdots+\frac{1}{ n_{i,1}-1 } \biggr)
+\biggl( 1+\cdots+\frac{1}{ n_{i+1,r_{i+1}} } \biggr)
\biggr).
\end{align*}
Since
\begin{align*}
\biggl( \frac{ 1 }{ a }+\cdots+\frac{ 1 }{ p-a } \bmod{p} \biggr)_p =0,
\end{align*}
we have
\begin{align*}
&\biggl( \frac{1}{ n_{i-1,1}-n_{i,r_i} }+\cdots+\frac{1}{ p-n_{i,r_i}-1 } \bmod{p} \biggr)_p \\
&=\biggl( -\biggl( \frac{1}{ n_{i,r_i}+1 }+\cdots+\frac{1}{ n_{i-1,1}-n_{i,r_i}-1 } \biggr) \bmod{p} \biggr)_p.
\end{align*}
By Lemmas \ref{lem1} and \ref{lem2}, we have
\begin{align*}
&Z_{\mathcal{A}}^{\mathrm{cyc}}
(w_{1}\otimes\cdots\otimes w_{i-1}\otimes(y\sht w_{i})\otimes w_{i+1}\otimes\cdots\otimes w_{d}) \\
&=(-2A_i+B_i-C_i \bmod{p} )_p \\
&=\biggl( -2\sum_{ S_p }
\frac{1}{\boldsymbol{n}^{\boldsymbol{k}}}
\biggl( \sum_{m=1}^{p-1} \frac{ 1 }{ m } -\biggl( \frac{1}{ n_{i,1} } +\cdots+\frac{ 1 }{ n_{i,r_i} } \biggr) \biggr) \\
&\quad -\sum_{\substack{ S_p^{(i)} \\ n_{i,r_i}<n_{i-1,1} }}
\frac{1}{\boldsymbol{n}^{\boldsymbol{k}}}
\biggl(
\biggl( \frac{1}{ n_{i,r_i}+1 }+\cdots+\frac{1}{ n_{i-1,1}-n_{i,r_i}-1 } \biggr)
+\biggl( \frac{1}{ n_{i-1,1} } +\cdots+\frac{ 1 }{ p-1 } \biggr)
\biggr) \\
&\quad -\sum_{\substack{ S_p^{(i+1)} \\ n_{i+1,r_{i+1}}<n_{i,1} }}
\frac{1}{\boldsymbol{n}^{\boldsymbol{k}}} \biggl(
\biggl( \frac{1}{ n_{i,1}-n_{i+1,r_{i+1}} }+\cdots+\frac{1}{ n_{i,1}-1 } \biggr)
+\biggl( 1+\cdots+\frac{1}{ n_{i+1,r_{i+1}} } \biggr)
\biggr)
\bmod{p}\biggr)_p.
\end{align*}
Then we have
\begin{align*}
&\sum_{i=1}^{d} Z_{\mathcal{A}}^{\mathrm{cyc}}
(w_{1}\otimes\cdots\otimes w_{i-1}\otimes(y\sht w_{i})\otimes w_{i+1}\otimes\cdots\otimes w_{d}) \\
&=\biggl( 2\sum_{i=1}^{d} \sum_{j=1}^{r_i} \sum_{ S_p }
\frac{ 1 }{ \boldsymbol{n}^{\boldsymbol{k}} n_{i,j} }
\bmod{p} \biggr)_p.
\end{align*}
This finishes the proof.
\end{proof}
\begin{que}
It is known that the cyclic relation (Theorem \ref{cycrel}) includes the derivation relation for MZVs
obtained by Ihara, Kaneko, and Zagier \cite[Theorem 3]{IKZ06}).
However, Theorem \ref{main} does not include the derivation relation for FMZVs proved by the author \cite{Mur17}.
Are there any generalizations of Theorem \ref{main} that include this relation?
Let $R_x(w):=wx$ for $w\in\mathfrak{H}$.
Then, by Theorem \ref{cycrel} and the arguments in \cite[the last part of Section 2]{Mur17} and \cite[Section 5.3]{HMM19},
it can be proved
\begin{align*}
Z^{\mathrm{cyc}}_{\mathcal{A}}
(R_x^{-1} ((y\shaub w) \otimes \underbrace{y\otimes\cdots\otimes y}_{m}))
=(m+1) Z^{\mathrm{cyc}}_{\mathcal{A}}
(R_x^{-1} (w \otimes \underbrace{y\otimes\cdots\otimes y}_{m+1}))
\end{align*}
for non-negative integer $m$ and $w\in y\mathfrak{H}x$, which is essentially equivalent to the derivation relation for FMZVs.
\end{que}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,021 |
\section{Numerical solution of Fraunhofer diffraction pattern}
According to the Parseval's theorem \cite{Goodman2005}, the absolute square of Fourier spectrum presents the energy distribution of Fraunhofer diffraction pattern. So the intensity of images over the whole imaging domain can be expressed as
\begin{equation}
E(\omega)=\iint |U_{Q}(\rho\cos \phi,\rho\sin \phi)|^2 \mathrm {d}\rho \mathrm{d} \phi.
\end{equation}
We have proved that the phase shift generated by the different etch depth of grating determines the percentage of the incident light that is directed into each diffraction order \cite{Yan2013}. For our 3D imaging system based on QD grating, the working phase (thus the target etch depth) can lead to the desired intensity balance between multi-plane images in each diffraction order as well as the maximum total energy in those orders. Ideally, on the assumption that all the incident flux is focussed only on the first three diffraction orders, we set $E_0(\omega)=E_{\pm 1}(\omega)=1/3E(\omega),$ where $E_0(\omega)$ and $E_{\pm 1}(\omega)$ denote the image intensities of zeroth and first orders, respectively. Since the two images of first orders are identical, we only take one of them into account. Consequently, the working phase is the root of
\begin{equation}
C(\omega)=\frac{1}{3}E(\omega)-E_0(\omega).
\end{equation}
\begin{algorithm}
\caption{The bisection algorithm}\label{Alg:bisec}
\begin{algorithmic}[1]
\While{$|\omega_1-\omega_2|>\epsilon$ and $C(\omega)\neq 0, \text{where } \omega=\frac{\omega_1+\omega_2}{2},$ } \vspace{-0.1cm}
\begin{eqnarray*}
\omega_1&=&\omega, \text{ if } C(\omega_1)C(\omega)>0, \\ \vspace{-0.2cm}
\omega_2&=&\omega, \text{ otherwise,}
\end{eqnarray*}
\EndWhile \vspace{-0.3cm}
\State \textbf{return} $\omega.$
\end{algorithmic}
\end{algorithm}
Bisection algorithm \cite{Burden1985}, which is a root-finding method for a continuous function $C(\omega),$ typically works with two initial guesses, $\omega_1$ and $\omega_2$, such that $C(\omega_1)$ and $C(\omega_2)$ have opposite signs and at least one root can be bracketed within a subinterval of $[\omega_1,\omega_2]$ according to the intermediate value theorem. As Algorithm \ref{Alg:bisec} shown, the interval between $\omega_1$ and $\omega_2$ will become increasingly smaller, converging on the root of the function after a few iterations. Here the tolerance $\epsilon>0$ can reach up to $10^{-6}.$
A QD grating with moderate parameters (as Table \ref{Tbl:ParaQD} shown) is selected and applied in both our 2D model and the 1-D and period-fixed grating model \cite{Yan2013}, such that the two values of working phase obtained by both models should be close to each other.
\begin{table}[h]
\renewcommand\arraystretch{1.2}
\setlength{\belowcaptionskip}{1pt}
\centering
\caption{\bf Designed parameters of the 2D QD grating}
\begin{tabular}{cc}
\hline
Central Period ($d_0$) & $50\upmu \rm{m}$ \\
Radius ($R$) & $10$mm \\
$W_{20}$ & $50\lambda$\\
Wavelength ($\lambda$) & $532$nm\\
Number of Arcs &$801$ \\
\hline
\end{tabular}
\label{Tbl:ParaQD}
\end{table}
\\where $W_{20}$ is the standard coefficient of defocus and is equivalent to the extra path length introduced at the edge of the aperture. And the varying radii $r_j,$ for $ j\in J,$ can be obtained by \cite{Blanchard1999}
\begin{equation}
r_j=\left[ \frac{j\lambda R^2}{W_{20}}+\left(\frac{\lambda R^2}{2d_0 W_{20}}\right)^2 \right ]^{1/2}.
\end{equation}
\begin{figure}[htb]
\subfigure{
\hspace{0.5cm}
\includegraphics[width=0.4\textwidth]{Figure4a}
}\\[-0.35cm]
\subfigure{
\hspace{0.5cm}
\includegraphics[width=0.4\textwidth]{Figure4b}
}
\caption{Contour plot of normalized energy for Fraunhofer diffraction pattern of 2D QD phase gratings with working phases of (a) 1.99999rad; (b) 2.00777rad.}
\label{Fig:ContourQD}
\end{figure}
Since the rough 1D model gives a working phase of 2.00777rad \cite{Yan2013}, an initial interval of $[1.9, 2.1]$ is set in bisection algorithm. Then an optimized phase of 1.99999rad is obtained after 18 iterations. As Figure \ref{Fig:ContourQD} shown, being illuminated by a normally incident, unit-amplitude and monochromatic plane wave, the energy distributions across the QD grating at both phases look similar as anticipated. However, the energy difference between zeroth and first orders are $6\times 10^{-6}$ and $1.7\%$ at working phases of 1.99999rad and 2.00777rad, respectively. When applying the 1D model for the phase design of a QD grating with non-moderate parameters, especially if a big value of $W_{20}$ is selected (say $100\lambda$), the energy imbalance between first three orders will be even worse.
Given that the Fraunhofer diffraction at a circular sector and the 2D mathematical model of QD grating are first developed in this paper, our theories and algorithms should be verified in practice. Here a quasi-straight-line QD grating, which reserves the parameters shown in Table \ref{Tbl:ParaQD} but sets $W_{20}$ to be $0.5\lambda$, is applied in the 2D QD grating model and a working phase of 2.00831rad is obtained. The Fraunhofer diffraction pattern is demonstrated in Figure \ref{Fig:ContourStrQD}, in which the positions of the peaks are identical with those derived by classic theory of diffraction grating \cite{Born2001} and the energy distribution tends to be the same with that of straight-line and period-fixed grating.
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{Figure5}
\caption{Contour plot of normalized energy for Fraunhofer diffraction pattern of the quasi-straight-line QD grating.}
\label{Fig:ContourStrQD}
\end{figure}
In conclusion, we have established an elaborate 2D analytic model of QD grating and obtained the Fraunhofer diffraction pattern. This model can be extended to the design of crossed QD grating for simultaneous 9-plane imaging. Beyond the design of grating, it can also be utilized in the design and optimization of simultaneous multi-plane imaging system. An updated model involved with a chromatic correction scheme using grisms \cite{Yan2013OE} is in progress and high order aberrations, i.e. spherical aberration and coma, will be considered in the near future.\\
\noindent \textbf{\Large{Funding.}} This work at HWU was funded by the Science and Technology Facilities Council (STFC). YF was funded by SUPA Prize Studentship and research grant of Prof. Xiaohong Fang of ICCAS. YF is currently funded by the France BioImaging (FBI) infrastructure ANR-10-INBS-04 (by Dr. Nadine Peyri\'{e}ras).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,337 |
«Gitano» -«Gitana», o sus plurales - puede referirse:
El pueblo gitano es una comunidad o etnia originaria de la India.
Obras
Gitano
Gitano (1997), álbum de Los Chichos.
Gitano (1949), película dirigida por Manuel Silos.
Gitano (1970), película argentina con el actor y cantante Sandro.
Gitano (2000), película española con el bailaor [oaquín Cortés.
Gitana
Gitana tenías que ser (1953) es una película mexicana dirigida por Rafael Baledón.
Gitana (1965), película dirigida por Joaquín Bollo Muro.
Gitana (1988), álbum de Daniela Romo.
Gitana (1984), canción de Willie Colón.
Gitana (1990), telenovela colombiana.
Gitana (2010), canción de Shakira.
Gitanas
Gitanas (2004), telenovela mexicana. | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,856 |
Vera Aleksandrovna Sokolova (Russisch: Вера Александровна Соколова) (Solianoy (Tsjoevasjië), 8 juni 1987) is een Russische atlete, die zich heeft gespecialiseerd in het snelwandelen.
Loopbaan
Wereldkampioene bij de B-junioren
Amper zestien jaar oud was Sokolova, toen zij in 2002 voor het eerst deelnam aan een internationaal kampioenschaptoernooi, de wereldkampioenschappen voor junioren in het Jamaicaanse Kingston. Ze werd er negende op de 10.000 m snelwandelen. Een jaar later veroverde ze haar eerste grote titel, toen zij in het Canadese Sherbrooke op de WK voor B-junioren de 5000 m snelwandelen won, waarna ze in 2004, opnieuw op de WK voor junioren, ditmaal in het Italiaanse Grosseto, brons won op de 10.000 m snelwandelen.
Europees juniorkampioene in recordtijd
In 2005 greep de Russische, nog steeds juniore, haar eerste Europese titel. Op de Europese kampioenschappen voor junioren in Kaunas won zij de 10.000 m snelwandelen in 43.11,34, wat een wereldrecord voor junioren betekende. Dit record zou zes jaar stand houden.
In haar laatste juniorenjaar was Sokolova opnieuw present op de WK voor junioren, die ditmaal in Peking plaatsvonden. Op de 10.000 m snelwandelen ging ze aanvankelijk op kop en leek zij op weg naar weer een medaille. Toen ze echter halverwege de wedstrijd al haar tweede waarschuwing wegens onreglementair lopen te pakken had, kon ze haar tempo niet volhouden en zakte zij terug. Winnares werd nu de Chinese Liu Hong in 45.12,84, voor Sokolova's landgenote Tatjana Sjemjakina in 45.34,41 en de Roemeense Anamaria Greceanu in 46.58,21). Sokolova finishte als vierde in 46.58,21 en keerde dus met lege handen huiswaarts.
Brons op EK bij de senioren
Eenmaal senior richtte Vera Sokolova zich vooral op het snelwandelen op de weg, maar het duurde enkele jaren, voordat zij ook hier aansprekende resultaten boekte. Bleef zij op de wereldkampioenschappen in 2009 op de 20 km snelwandelen nog steken op een bescheiden veertiende plaats in 1:34.55, een jaar later wist zij zich tijdens de Europese kampioenschappen in Barcelona op dezelfde afstand op de derde plaats te nestelen in een aanzienlijk snellere tijd, 1:29.32, dan in Berlijn.
In 2011, op de WK in Daegu, stelde Sokolova echter weer teleur. In 1:32.13 eindigde zij op de 20 km snelwandelen op een tegenvallende negende plaats. Deze prestatie was des te teleurstellender, omdat de Russische aan het begin van het jaar, tijdens de winterkampioenschappen snelwandelen in Sotsji, de klokken op dit onderdeel nog had laten stilstaan op 1:25.08, een verbetering van het wereldrecord van haar landgenote Olimpiada Ivanova, gevestigd op de WK in 2005, met ruim een halve minuut.
DQ op universiade en WK
In 2013 nam Sokolova, weer op de 20 km snelwandelen, deel aan de universiade in eigen land, maar daar deed zij de wrange ervaring op dat, terwijl haar landgenotes als team de gouden plak veroverden, zijzelf werd gediskwalificeerd. Ditzelfde overkwam haar, opnieuw in eigen land, kort hierna op de WK in Moskou.
Schorsing
Wrang van een geheel andere orde was de bekendmaking van het Hof van Arbitrage voor Sport (CAS) op 13 oktober 2016, dat Vera Sokolova tot de vijf Russische snelwandelaars behoorde die in juni van het vorige jaar waren betrapt op het gebruik van epo. Als gevolg hiervan werd Sokolova door het CAS voor vier jaar geschorst.
Titels
Russisch kampioene 10 km snelwandelen – 2005
Russisch kampioene 20 km snelwandelen - 2011
Europees juniorenkampioene 10.000 m snelwandelen – 2005
Wereldkampioene B-junioren 5000 m snelwandelen - 2003
Persoonlijke records
Baan
Weg
Indoor
Palmares
5000 m snelwandelen
2003: WK voor B-junioren – 22.50,23
10.000 m snelwandelen
2002: 9e WJK - 47.59,14
2004: WJK – 46.53,02
2005: EJK – 43.11,34 (WJR)
2006: 4e WJK – 46.58,21
10 km snelwandelen
2005: Russische kamp. – 44.25
2005: Europa Cup te Miskolc – 44.09
2006: Russische kamp. – 43.49
20 km snelwandelen
2009: Russische kamp. 1:25.26
2009: 10e Europa Cup te Metz – 1:36.43
2009: 13e WK - 1:34.55 (na DQ van Olga Kaniskina)
2010: Russische kamp. – 1:25.35
2010: 4e World Cup te Chihuahua – 1:33.54
2010: EK – 1:29.32
2011: Russische kamp. – 1:25.08 (WR)
2011: Europa Cup te Olhão – 1:30.02
2011: 9e WK – 1:32.13
2013: DQ Universiade
2013: DQ WK
Russisch atleet
Snelwandelaar | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,097 |
Carrie Underwood arrives for 9th season finale of 'American Idol'
Country music star and former 'American Idol' winner Carrie Underwood arrives for the 9th season finale of 'American Idol' in Los Angeles May 26, 2010
Country music star and former 'American Idol' winner Carrie Underwood arrives for the 9th season finale of 'American Idol' in Los Angeles May 26, 2010. [Agencies]
Celebrity Gossips, Hot celebrities | | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,656 |
Q: Update database values using javascript I have a php application for showing some pictures stored in a database. I want to use javascript to change photos order. Index.php page looks like:
$photos = array();
$photos_query = "SELECT * FROM `photos` WHERE `id`=".$_SESSION['id']."ORDER BY `order`";
$data = mysqli_query($dbc, $photos_query);
while ($photos_row= mysqli_fetch_assoc($data)){
$photos[] = array(
'photo_id' => $photos_row['photo_id'],
'id' => $photos_row['id'],
'photo_name' => $photos_row['photo_name'],
'order' => $photos_row['photo_order']
);
} ...
<div id="message"></div>
<div id="showphotos">
if (empty($photos)) {
echo 'No photos';
} else {
echo '<ul>';
foreach ($photos as $photo) {
echo '<li id="photoid_'.$photo['photo_id'].'">
<img src="uploads/', $photo['id'], '/', $photo['photo_id'],'"' .$photo['photo_name']. 'height="150" width="150">
... </li>';
} echo '</ul>';
}
</div>
This page calls move.js once user tries to change image's position:
$(document).ready(function(){
function slideOut(){
setTimeout(function(){
$("#message").slideUp("slow", function () {});
}, 2000);
};
$("#message").hide();
$(function() {
$('ul').sortable({
cursor: 'move',
opacity: 0.7,
revert: false,
update: function(){
var neworder = $(this).sortable("serialize") + '&update=update';
$.post("change_image_position.php", neworder, function (themessage) {
$("#message").html(themessage);
$("#message").slideDown('slow');
slideOut();
});
}
});
});
});
Finally change_image_position.php contains:
$array = $_POST['photoid'];
if ($_POST['update'] == "update"){
$count = 1;
foreach ($array as $idval) {
$query = "UPDATE photos SET order = " . $count . " WHERE photo_id = " . $idval;
mysqli_query($query) or die('Error, insert query failed');
$count ++;
}
echo 'All saved! refresh the page to see the changes';
}
I am facing two problems... First, order by statement(index.php) is not working and second, query for altering photos through database query(change_image_position.php) does not return any result. Thanks in advance for any help.
A: Your mySQL statement has a missing space -
$photos_query = "SELECT * FROM `photos` WHERE `id`=".$_SESSION['id']."ORDER BY `order`";
Should be
$photos_query = "SELECT * FROM `photos` WHERE `id`=".$_SESSION['id']." ORDER BY `order`";
I think the way you are currently doing the photo saving is failing on the neworder portion.
You currently pass in $update and then call $_POST['update'] which would be looking for just update=update.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 101 |
\section{Introduction and statement of main results}\label{sec1}
We consider the following 1D system of pressureless Euler equations
with nonlocal interaction forces
\begin{align}
&\pa_t \rho + \pa_x (\rho u) = 0,\quad x \in \R, \quad t > 0, \label{eq:mainrho} \\
&\,\,\,\, \pa_t u + u\pa_x u = \int_{\R} \psi(x-y)(u(y,t) - u(x,t))\rho(y,t)dy
- \pa_x K \star \rho,\label{eq:mainu}
\end{align}
subject to initial density and velocity
\begin{equation}\label{eq:maininitial}
(\rho(\cdot,t),u(\cdot,t))|_{t=0}= (\rho_0, u_0).
\end{equation}
The term on the right hand side of \eqref{eq:mainu} consists of two
parts: an \emph{alignment} interaction with communication weight
$\psi$, and an \emph{attraction-repulsion} interaction through a
potential $K$.
\subsection{Self-organized dynamics with three-zone
interactions}\label{subsec:intro:3zone}
System \eqref{eq:mainrho}-\eqref{eq:mainu} arises from many contexts
in mathematical physics and biology. In particular, it serves as a
macroscopic system in modeling collective behaviors of complex
biological systems. The corresponding agent-based model has the form
\begin{equation}\label{eq:ABM}
\dot{x}_i=v_i, \quad
m\dot{v}_i=\frac{1}{N}\sum_{j=1}^N\psi(x_i-x_j)(v_j-v_i)-
\frac{1}{N}\sum_{j=1}^N\nabla_{x_i}K(x_i-x_j),
\end{equation}
where $(x_i, v_i)_{i=1}^N$ represent the position and velocity of agent
$i$. The dynamics is governed by Newton's second law, with the
interaction force modeled under a celebrated ``three-zone'' framework
proposed in \cite{reynolds1987flocks}, including
long-range attraction, short-range repulsion, and mid-range alignment.
The first part of the force describes the alignment interaction, where
$\psi$ characterizes the strength of the velocity alignment between
two agents. Naturally, it is a decreasing function of the distance
between agents. Such alignment force has been proposed by Cucker and Smale
in \cite{cucker2007emergent}. The corresponding dynamics enjoys the
flocking property \cite{ha2009simple}, which is a common phenomenon
observed in animal groups.
The second part of the force represents the attraction-repulsion
interaction. The sign of the force $-\nabla K$ determines whether the
interaction is attractive or repulsive. This type of potential driven
interaction force is widely considered in many physical and biological
models, e.g. \cite{d2006self,mogilner1999non}.
Starting from the agent-based model \eqref{eq:ABM}, one can derive a
kinetic representation of the system that describes the mean-field
behavior as $N\to\infty$, see \cite{carrillo2010asymptotic, ha2008particle,
tan2017discontinuous}. Then, a variety of hydrodynamics limits can be
obtained that capture the macroscopic behaviors in different regimes
\cite{fetecau2016first, karper2015hydrodynamic, poyato2017euler}. In
particular, if we consider the mono-kinetic regime, the corresponding
macroscopic system becomes \eqref{eq:mainrho}-\eqref{eq:mainu}.
\subsection{Global regularity versus finite time blowup}
We are interested in the global existence and regularity for the
solution of the system \eqref{eq:mainrho}-\eqref{eq:mainu}.
Let us start with the case with no interaction forces, namely
$\psi=K\equiv0$. The system can be recognized as the pressureless Euler system. In
particular, \eqref{eq:mainu} becomes the classical inviscid Burgers
equation, where smooth data forms shock discontinuity in finite
time due to nonlinear convection $u\pa_xu$.
Together with \eqref{eq:mainrho}, it is well-known that the
solution generates singular shocks in finite time:
$\rho(x,t)\to\infty$ at the position and time when shock occurs.
With alignment force $\psi\geq0$ and $K\equiv0$, the system is called
the \emph{Euler-Alignment system}. When $\psi$ is Lipschitz, the system has been
studied in \cite{carrillo2016critical, tadmor2014critical}, where it is
discovered that the alignment force tends to regularize the solution
and prevent finite time blowup, but only for some initial data. This is so
called \emph{critical threshold phenomenon}: for subcritical initial
data, the alignment force beats the nonlinear convection, and the
solution is globally regular; while for supercritical initial data,
the convection wins and the solution admits a finite time blowup.
Another interesting and natural setting is when $\psi$ is singular, taking the
form
\begin{equation}\label{eq:singularpsi}
\psi(x)=\frac{c_\alpha}{|x|^{1+\alpha}},\quad\alpha>0,
\end{equation}
with $c_\alpha$ be a positive constant. The range $0 <\alpha \leq 2$ is most natural,
and the case $0 < \alpha <1$ is most interesting for the reasons explained later in this sub-section.
The Euler-Alignment system corresponding to the choice \eqref{eq:singularpsi} is studied in
\cite{do2017global} for the periodic case.
Without loss of generality, we set the scale and let $\T=[-1/2,1/2]$ be the periodic domain
of size 1. The singular alignment force can be equivalently expressed as
\begin{equation}\label{eq:singularforce}
\int_\T\psi_\alpha(y)(u(x+y,t)-u(x,t))\rho(x+y,t)dy,
\end{equation}
with the periodic influence function $\psi_\alpha$ defined as
\begin{equation}\label{eq:psip}
\psi_\alpha(x)=\sum_{m\in\mathbb{Z}}\frac{c_\alpha}{|x+m|^{1+\alpha}},
\quad\forall~x\in\T\backslash\{0\}.
\end{equation}
Clearly, $\psi_\alpha$ is singular at $x=0$. Moreover, it has a positive
lower bound
\begin{equation}\label{eq:psim}
\psi_m=\psi_m(\alpha):=\min_{x\in\T}\psi_\alpha(x)=\psi_\alpha\left(\frac{1}{2}\right)>0.
\end{equation}
This leads to the following fractional Euler-Alignment system
\begin{align}
&\pa_t \rho + \pa_x (\rho u) = 0,\qquad x\in\T,\label{eq:fEArho}\\
&\pa_t u + u\pa_x u=\int_\T\psi_\alpha(y)(u(x+y,t)-u(x,t))\rho(x+y,t)dy.
\label{eq:fEAu}
\end{align}
It is shown in \cite{do2017global} that system
\eqref{eq:fEArho}-\eqref{eq:fEAu} has a global smooth solution for all
smooth initial data with $\rho_0>0$. This result is most interesting for the case $0<\alpha<1:$
if we set $\rho \equiv 1$ in \eqref{eq:fEAu}, we get Burgers equation with fractional dissipation.
It is well-known that in this case, there exist initial data leading to finite time blow up (when
$0<\alpha <1;$ the $1 \leq \alpha \leq 2$ range leads to global regularity). However, it turns out
that in the nonlinear
disspation/allignment case described by \eqref{eq:fEAu},
the singularity in the influence function and density modulation
dominate the nonlinaer convection, for all initial data. This also contrasts with the case of Lischitz regular influence function
$\psi,$ where one has critical threshold in the phase space separating initial data leading to finite time blow up
and to global regularity.
\subsection{Euler-Poisson-Alignment system}\label{subsec:intro:EAP}
Now, we take into account the attraction-repulsion force, namely
$K\not\equiv0$. We shall begin with a particular potential
\begin{equation}\label{eq:NewtonianR}
\mathcal{N}(x)=\frac{k|x|}{2}.
\end{equation}
The potential is the \emph{1D Newtonian potential}, and it is the
kernel for the 1D Poisson equation, namely
\[\pa_x^2\mathcal{N}\star\rho=k\rho.\]
When $k>0$, the Newtonian force
$\pa_x^2 \mathcal{N}\star\rho$ is attractive, and when $k<0$, the Newtonian force
is repulsive.
We call the corresponding system Euler-Poisson-Alignment (EPA) system. It
has the form
\begin{align}
&\pa_t \rho + \pa_x (\rho u) = 0,\label{eq:EPArho}\\
&\pa_t u + u \pa_x u = -\pa_x \phi + \int_{\R} \psi(x-y)(u(y,t) - u(x,t))
\rho(y,t) dy, \label{eq:EPAu}
\end{align}
where the stream function $\phi=\mathcal{N}\star\rho$ satisfies the Poisson equation
\begin{equation}\label{eq:EPAphi}
\pa_x^2\phi =k\rho.
\end{equation}
When there is no alignment force $\psi\equiv0$, the system coincides
with the 1D pressureless Euler-Poisson equation, which has been
extensively studied in \cite{engelberg2001critical}. The result is as follows:
when $k>0$, the attraction force together with convection
drives the solution of Euler-Poisson equation to a finite time
blowup for all smooth initial data; when $k<0$, the repulsive force
competes with the convection, and there exists a critical
threshold on initial conditions which separates global regularity and
finite time blowup.
The EPA system \eqref{eq:EPArho}-\eqref{eq:EPAphi} is studied in
\cite{carrillo2016critical}, in the case when $\psi$ is Lipschitz .
When $k<0$, a larger subcritical region of initial
data is obtained that ensures global regularity. This implies that the
alignment force helps repulsive potential to compete with the
convection. However, it is also shown that when $k>0$, the alignment
force is too weak to compete with convection and attractive potential,
so all smooth initial data lead to finite time blow up.
Our first result concerns EPA system with singular alignment force,
where the influence function has the form \eqref{eq:singularpsi}. The
main goal is to understand whether the singular alignment can
still regularize the solution when the Newtonian force is present.
We shall study the system in the periodic setting.
The 1D periodic Newtonian potential reads
\begin{equation}\label{eq:Newtonian}
\mathcal{N}(x)=-\frac{k}{2}\left(\frac{1}{2}-|x|\right)^2,\quad\forall~x\in\T.
\end{equation}
It is the kernel of the Poisson equation with background, namely
\[\partial_x^2(\mathcal{N}\ast\rho)=k(\rho-\bar{\rho}),\]
where $\bar{\rho}$ is the average density
\begin{equation}\label{eq:rhobar}
\bar{\rho}=\frac{1}{|\T|}\int_\T\rho(x,t)dx=\int_\T\rho_0(x)dx.
\end{equation}
Note that $\bar{\rho}$ is conserved in time due to conservation of mass by evolution.
The stream function $\phi$ in \eqref{eq:EPAu} satisfies the Poisson
equation with constant background
\begin{equation}\label{eq:EPAphiP}
\pa_x^2\phi =k(\rho-\bar{\rho}).
\end{equation}
The presence of the background $\bar{\rho}$ could change the
behavior of the solution.
For Euler-Poisson equation in periodic domain, namely
\eqref{eq:EPArho}-\eqref{eq:EPAu},\eqref{eq:EPAphiP} with $\psi\equiv0$, it is pointed out in
\cite{engelberg2001critical} that the
background has the tendency to balance both the
convection and attractive forces. So for the attractive case $k>0$,
instead of finite time blowup for all initial data, a critical
threshold is obtained.
Though similar techniques in \cite{carrillo2016critical}, one can
derive critical thresholds for EPA system
\eqref{eq:EPArho}-\eqref{eq:EPAu},\eqref{eq:EPAphiP} with bounded Lipschitz
influence function $\psi$.
The EPA system with singular alignment force \eqref{eq:singularforce} and potential
\eqref{eq:Newtonian} reads
\begin{align}
&\pa_t \rho + \pa_x (\rho u) = 0,\label{eq:EPASrho}\\
&\pa_t u + u \pa_x u = -\pa_x \phi + \int_\T\psi_\alpha(y)(u(x+y,t)-u(x,t))\rho(x+y,t)dy, \,\,\pa_x^2\phi =k(\rho-\bar{\rho}).\label{eq:EPASu}
\end{align}
The following theorem shows that the singular alignment force
dominates the Poisson force, and global regularity is obtained for all
initial data.
\begin{theorem}\label{thm:EAP}
For $\alpha\in(0,1)$, the fractional EPA system
\eqref{eq:EPASrho}-\eqref{eq:EPASu} with smooth periodic initial data
$(\rho_0, u_0)$ such that $\rho_0>0$ has a unique smooth
solution.
\end{theorem}
\begin{remark} The proof can be easily extended to the range $\alpha \geq 1$ with more straightforward arguments
for $\alpha >1;$ see also \cite{shvydkoy2017eulerian} for a different approach.
We focus on the $0 < \alpha <1$ case in the rest of the paper.
\end{remark}
We note that the proof of global regularity in \cite{do2017global} is based, in particular, on rather precise algebraic
structures that we will discuss below. Even though the interaction force we are adding is formally sub-critical,
it is far from obvious that the fairly intricate arguments of \cite{do2017global} survive such perturbation.
\subsection{Euler dynamics with general three-zone interactions}
The results on EPA system can be extended to systems with more
general interaction forces.
In \cite{carrillo2016critical}, critical thresholds are obtained for
the system \eqref{eq:mainrho}-\eqref{eq:mainu}, with
Lipschitz influence function $\psi$, and regular potential $K\in W^{2,\infty}$.
In this paper, we will also consider the case of more general singular influence function $\psi$.
More precisely, we assume that $\psi\geq\psi_m>0$, and can
be decomposed into
two parts
\begin{equation}\label{eq:psidecomp}
\psi=c\psi_\alpha+\psi_L,
\end{equation}
where $c>0$, $\psi_\alpha$ is defined in \eqref{eq:psip}, and $\psi_L$ is a bounded
Lipschitz function.
\begin{theorem}\label{thm:3zone}
Consider system
\eqref{eq:mainrho}-\eqref{eq:mainu} in the periodic setup
\begin{align}
&\pa_t \rho + \pa_x (\rho u) = 0,\quad x \in \T, \quad t > 0, \label{eq:mainrhop} \\
&\,\,\,\, \pa_t u + u\pa_x u = \int_\T\psi(y)(u(x+y,t)-u(x,t))\rho(x+y,t)dy
- \pa_x K \star \rho,\label{eq:mainup}
\end{align}
with smooth initial data $(\rho_0, u_0)$ such that $\rho_0>0$. Assume
$\psi$ is singular in the sense of \eqref{eq:psidecomp}, and $K$ is
a linear combination of Newtonian potential \eqref{eq:Newtonian}
and regular $W^{2,\infty}(\T)$ potential.
Then, the system has a unique global smooth solution.
\end{theorem}
We summarize the global behaviors of Euler equations with nonlocal
interaction \eqref{eq:mainrho}-\eqref{eq:mainu} under different
choices of interaction forces.
\begin{table}[h]
{\def\arraystretch{1.5}
\begin{tabular}{|c|c|c|c|l|}
\hline
Potential&Alignment
&Name&Domain&Behaviors\\ \hline
No&No&Euler& $\R$ or $\T$&Finite time blow up \\ \cline{2-5}
&Lipshitz&Euler-Alignment&$\R$ or $\T$& Critical threshold \cite{carrillo2016critical,tadmor2014critical}\\ \cline{2-5}
&Singular&Fractional EA&$\T$& Global regularity \cite{do2017global}\\ \hline
Newtonian&No&Euler-Poisson&$\R$& Finite time blow up \cite{engelberg2001critical}
\\ \cline{4-5}
&&&$\T$& Critical threshold \cite{engelberg2001critical}
\\ \cline{2-5}
&Lipschitz&EPA&$\R$& Finite time blow up (attractive)\\
&&&& Critical threshold (repulsive) \cite{carrillo2016critical}\\ \cline{4-5}
&&&$\T$& Critical threshold\\ \cline{2-5}
&Singular&Fractional EPA&$\T$& Global regularity (Theorem \ref{thm:EAP})\\ \hline
General&Lipshitz&Euler-3Zone& $\R$ or $\T$& Critical thresholds \cite{carrillo2016critical}\\ \cline{2-5}
&Singular&Singular 3Zone& $\T$& Global regularity (Theorem \ref{thm:3zone})\\ \hline
\end{tabular}
}
\end{table}
\section{Euler-Poisson-Alignment system}\label{sec:EAP}
In this section, we consider Euler-Poisson-Alignment system
\eqref{eq:EPASrho}-\eqref{eq:EPASu}
with singular alignment force \eqref{eq:singularforce}.
Following the idea in \cite{do2017global}, we let
\begin{equation}\label{eq:G}
G=\pa_xu-\Lambda^\alpha\rho,
\end{equation}
and calculate the dynamics of $G$ using \eqref{eq:EPASrho} and
\eqref{eq:EPASu}:
\begin{align*}
\pa_tG=&~\pa_t\pa_xu-\pa_t\Lambda^\alpha\rho
=-\pa_x(u\pa_xu)-k(\rho-\bar{\rho})+\pa_x\left(-\Lambda^\alpha(\rho
u)+u\Lambda^\alpha\rho\right)+\Lambda^\alpha\pa_x(\rho u)\\
=&-u\pa_x(\pa_xu-\Lambda^\alpha\rho)-\pa_xu(\pa_xu-\Lambda^\alpha\rho)-k(\rho-\bar{\rho})
=-\pa_x(Gu)-k(\rho-\bar{\rho}).
\end{align*}
So, we can rewrite the dynamics in terms of $(\rho,G)$ as
\begin{align}
&\pa_t \rho + \pa_x (\rho u) = 0,\label{eq:EPASrho2}\\
&\pa_t G + \pa_x (Gu) = -k(\rho-\bar{\rho}),\label{eq:EPASG}\\
&\pa_xu=\Lambda^\alpha\rho+G.\label{eq:EPASux}
\end{align}
The velocity $u$ can be recovered as
\begin{equation}\label{eq:urec}
u(x,t)=\Lambda^\alpha\partial_x^{-1}(\rho(x,t)-\bar{\rho})+\partial_x^{-1}G(x,t)+I_0(t),
\end{equation}
where $I_0$ can be determined by conservation of momentum.
\begin{equation}\label{eq:mom}
\int_\T\rho(x,t) u(x,t)dx=\int_\T\rho_0(x) u_0(x)dx.
\end{equation}
See \cite{do2017global} for detailed discussion.
\subsection{A priori bounds}
We first show an upper and lower bounds on density $\rho$ for all
finite times. For $k=0$, a uniform in time
bound is obtained in \cite{do2017global}.
With the Newtonian potential, especially when $k>0$, the
attractive force definitely helps density concentration. Hence, the upper bound
on $\rho$ can be expected to grow in time. However, the bound we obtain in this section indicates
that there is no finite time singular concentration on density, thanks to the
singular alignment force.
Let $F=G/\rho$. We can rewrite \eqref{eq:EPASrho2} as
\begin{equation}\label{eq:rhody}
(\pa_t+u\pa_x)\rho=-\rho\Lambda^\alpha\rho-\rho^2F.
\end{equation}
The first step is to obtain a bound on $F$. We calculate
\[\pa_tF=\frac{\rho\pa_tG-G\pa_t\rho}{\rho^2}
=\frac{\rho(-\pa_x(Gu)-k(\rho-\bar{\rho}))-G(-\pa_x(\rho u))}{\rho^2}
=-u\pa_xF-\frac{k(\rho-\bar{\rho})}{\rho}.
\]
This implies that
\begin{equation}\label{eq:Fcha}
(\pa_t+u\pa_x) F=-k\left(1-\frac{\bar{\rho}}{\rho}\right).
\end{equation}
Denote $X(x,t)$ the trajectory of the characterstic path starting
at $x$, namely
\begin{equation}\label{eq:path}
\frac{d}{dt}X(x,t)=u(X(x,t),t),\quad X(x,0)=x.
\end{equation}
Then, we can solve for $F$ along the characteristic path
\begin{equation}\label{eq:F}
F(X(x,t),t)=F_0(x)-kt+\int_0^t\frac{k\bar{\rho}}{\rho(X(x,s),s)}ds.
\end{equation}
Define $\rho_m(t)$ as the lower bound of $\rho$ on time interval $[0,t]$
\begin{equation}\label{eq:rhom}
\rho_m(t)=\min_{s\in[0,T]}\min_{x\in\T}\rho(x,s).
\end{equation}
Then, we get a bound on $F$ from \eqref{eq:F}:
\begin{equation}\label{eq:Fbound}
\|F(\cdot,t)\|_{L^\infty}\leq\|F_0\|_{L^\infty}+|k|t+|k|\bar{\rho}\int_0^t\frac{1}{\rho_m(s)}ds.
\end{equation}
Therefore, in order to control $F$ in $L^\infty$, we need a lower bound estimate on
the density.
\begin{theorem}[Lower bound on density]\label{thm:lower}
Let $(\rho, u)$ be a strong solution to EPA system
\eqref{eq:EPASrho}\eqref{eq:EPASu} with smooth periodic initial
conditions $(\rho_0, u_0)$ such that $\rho_m(0)>0$.
Then, there exist two positive constants $A_m$ and $C_m$,
depending only on the initial conditions, such that
for any $t\geq 0$,
\begin{equation}\label{eq:lowerbound}
\rho_m(t)\geq C_me^{-A_mt}.
\end{equation}
\end{theorem}
\begin{proof}
We depart from \eqref{eq:rhody} and estimate $\Lambda^\alpha\rho$ and $F$.
For a fixed time $t$, denote $\underline{x}$ be a point where $\rho$
attains its minimum. Note that $\underline{x}$ depends on $t$ and it
is not necessarily unique. The estimates below apply at any such point. We have
\begin{equation}\label{eq:lb1}
\begin{aligned}
-\Lambda^\alpha\rho(\underline{x},t)=&~c_\alpha\int_{-\infty}^\infty
\frac{\rho(\underline{x}+y,t)-\rho(\underline{x},t)}{|y|^{1+\alpha}}dy
=\int_\T\psi_\alpha(y)\big(\rho(\underline{x}+y,t)-\rho(\underline{x},t)\big)dy
\\
\geq&~
\psi_m\int_\T\big(\rho(\underline{x}+y,t)-\rho(\underline{x},t)\big)dy
=\psi_m\big(\bar{\rho}-\rho(\underline{x},t)\big).
\end{aligned}
\end{equation}
Here, we recall that $\psi_m$ is the positive lower bound of
$\psi_\alpha$ defined in \eqref{eq:psim}.
Combining \eqref{eq:Fbound} and \eqref{eq:lb1}, we obtain
\begin{equation}\label{eq:rhombound1}
\pa_t\rho(\underline{x},t)\geq~
\big(\psi_m\bar{\rho}\big)\rho(\underline{x},t)-\left[\psi_m+
\|F_0\|_{L^\infty}+|k|t+|k|\bar{\rho}\int_0^t\frac{1}{\rho_m(s)}ds\right]\rho(\underline{x},t)^2.
\end{equation}
We prove \eqref{eq:lowerbound} by contradiction.
For $t=0$, the bound \eqref{eq:lowerbound} holds if we let
$C_m\leq\rho_m(0)$. Suppose \eqref{eq:lowerbound} does not hold for
all $t\geq0$. Then, there exists a finite time $t_0>0$ so that the
inequality is violated for the first time at $t=t_0+$.
Pick any $\underline{x}=\underline{x}(t_0)$. Due to continuity of
$\rho$, we know
\begin{equation}\label{eq:rhomb}
\rho_m(t_0)=\rho(\underline{x},t_0)=C_me^{-A_mt_0}.
\end{equation}
Plug in \eqref{eq:rhomb} to \eqref{eq:rhombound1} and use the fact
that \eqref{eq:lowerbound} holds for all $t\in[0,t_0]$. We get
\begin{align*}
\pa_t\rho(\underline{x},t_0)\geq&~\rho_m(t_0)\left[
\big(\psi_m\bar{\rho}\big)-\left(\psi_m+
\|F_0\|_{L^\infty}+|k|t+|k|\bar{\rho}\int_0^{t_0}\frac{1}{\rho_m(s)}ds\right)\rho_m(t_0)
\right]\\
\geq&~\rho_m(t_0)\left[
\big(\psi_m\bar{\rho}\big)-\left(\psi_m+
\|F_0\|_{L^\infty}+|k|t_0+\frac{|k|\bar{\rho}}{A_mC_m}(e^{A_mt_0}-1)\right)C_me^{-A_mt_0}
\right]\\
\geq&~\rho_m(t_0)\left[
\left(\psi_m\bar{\rho}-\frac{|k|\bar{\rho}}{A_m}\right)-\left(\psi_m+
\|F_0\|_{L^\infty}+|k|t_0-\frac{|k|\bar{\rho}}{A_mC_m}\right)C_me^{-A_mt_0}
\right]\\
\geq&~\rho_m(t_0)\left[
\left(\psi_m\bar{\rho}-\frac{|k|\bar{\rho}}{A_m}-\frac{|k|C_m}{eA_m}\right)
+\left(\frac{|k|\bar{\rho}}{A_m}-C_m(\psi_m+\|F_0\|_{L^\infty})\right)e^{-A_mt_0}
\right].
\end{align*}
The right hand side is positive if we pick $A_m$ large enough and
$C_m$ small enough. For instance, we can pick
\begin{equation}\label{eq:AC}
A_m=\frac{|k|}{\psi_m}(1+\epsilon),\quad
C_m=\min\{\rho_m(0), \epsilon e\bar{\rho}\},
\end{equation}
for any $\epsilon\in(0,\epsilon_*)$, where $\epsilon_*=\frac{1}{2}\left(\sqrt{1+\frac{4\psi_m}{e(\psi_m+\|F_0\|_{L^\infty})}}-1\right)$.
With this choice of $A_m$ and $C_m$, we get
$\pa_t\rho(\underline{x},t_0)> 0$.
Now we obtain that $\rho(\underline{x}) < C_m e^{-A_m t_0} < C_m e^{-A_m t}$ for some $t < t_0.$
This contradicts our choice of $t_0.$
\end{proof}
\begin{remark}
The bound \eqref{eq:lowerbound} with decay rate \eqref{eq:AC}
is not necessarily sharp, but is enough for our purpose, as it
eliminates the possibility of finite time creation of vacuum.
One important observation is that for $k=0$, we get $A_m=0$.
In this case, the lower bound is uniform in time.
\end{remark}
Applying the lower bound \eqref{eq:lowerbound} to \eqref{eq:Fbound}, we
immediately derive a bound on $F$
\begin{equation}\label{eq:Fbound2}
\|F(\cdot,t)\|_{L^\infty}\leq \|F_0\|_{L^\infty} +
|k|t+\frac{|k|\bar{\rho}}{A_mC_m}e^{A_mt}
=:F_M(t).
\end{equation}
Now, we are ready to obtain an upper bound on density $\rho$.
\begin{theorem}[Upper bound on density]\label{thm:upper}
Let $(\rho, u)$ be a strong solution to EPA system
\eqref{eq:EPASrho}\eqref{eq:EPASu} with smooth periodic initial
conditions $(\rho_0, u_0)$ such that $\rho_m(0)>0$.
Then, there exist two positive constants $A_M$ and $C_M$,
depending only on the initial conditions, such that
for any $t\geq 0$ and $x\in\T$,
\begin{equation}\label{eq:upperbound}
\rho(x,t)\leq \rho_M(t):=C_Me^{A_Mt}.
\end{equation}
\end{theorem}
\begin{proof}
We again depart from \eqref{eq:rhody} and start with a lower bound
estimate on $\Lambda^\alpha\rho$.
For a fixed time $t$, denote $\bar{x}$ be a point where $\rho$
attains its maximum. Applying
nonlinear maximum principle by Constantin and Vicol
\cite{constantin2012nonlinear}, one can estimate
\begin{equation}\label{eq:CVbound}
\Lambda^\alpha\rho(\bar{x},t)\geq C_1\rho(\bar{x},t)^{1+\alpha},
\end{equation}
if $\rho(\bar{x},t)\geq 3\bar{\rho}$. The constant $C_1$ only depends
on initial conditions. One can consult \cite{do2017global} for more
details of the estimate.
Plugging the estimates \eqref{eq:Fbound2} and \eqref{eq:CVbound} into
\eqref{eq:rhody}, we obtain
\begin{equation}\label{eq:upperbound1}
\pa_t\rho(\bar{x},t)\leq -C_1\rho(\bar{x},t)^{2+\alpha}+
F_M(t)\rho(\bar{x},t)^2.
\end{equation}
It follows that $\pa_t\rho(\bar{x},t)<0$ if
$\rho(\bar{x},t)>(F_M/C_1)^{1/\alpha}$. Therefore,
\begin{equation}\label{eq:rhomaxb}
\rho(x,t)\leq\rho(\bar{x},t)\leq\max\left\{\|\rho_0\|_{L^\infty},
3\bar{\rho}, \left(\frac{F_M(t)}{C_1}\right)^{1/\alpha}\right\},
\end{equation}
and \eqref{eq:upperbound} holds with
\[A_M=\frac{A_m}{\alpha},\quad
C_M=\max\left\{\max_{x\in\T}\rho_0(x), ~ 3\bar{\rho}, ~
\left[\frac{1}{C_1}\left(\|F_0\|_{L^\infty}+\frac{|k|}{eA_m}+\frac{|k|\bar{\rho}}{A_mC_m}\right)\right]^{1/\alpha}\right\}.\]
\end{proof}
\subsection{Local wellposedness}
With the apriori bounds, we state a local wellposedness result for the fractional
EPA system~(\ref{eq:EPASrho})-(\ref{eq:EPASu}), as well as a
Beale-Kato-Majda type necessary and sufficient condition to guarantee global
wellposedness. The local wellposedness theory has been presented in detail in
\cite{do2017global} for fractional Euler-Alignment system.
We will show that presence of the Poisson force does not seriously affect the
argument, no matter whether it is attractive or repulsive. We will only sketch the
proof, indicating changes necessary.
\begin{theorem}[Local wellposedness]\label{thm:local}
Consider EPA system~(\ref{eq:EPASrho})-(\ref{eq:EPASu}) with initial
conditions $\rho_0$ and $u_0$ that satisfy
\begin{equation}\label{eq:smoothinit}
\rho_0\in H^s(\T),\quad \min_{x\in\T}\rho_0(x)>0,\quad
\partial_xu_0-\Lambda^\alpha\rho_0\in H^{s-\frac{\alpha}{2}}(\T),
\end{equation}
with a sufficiently large even integer $s>0$. Then, there exists $T_0>0$
such that the EPA system
has a unique strong solution $\rho(x,t), u(x,t)$ on $[0,T_0]$, with
\begin{equation}\label{eq:smoothsol}
\rho\in C([0,T_0], H^s(\T))\cap L^2([0,T_0], H^{s+\frac{\alpha}{2}}(\T)),\quad
u\in C([0,T_0], H^{s+1-\alpha}(\T)).
\end{equation}
Moreover, a necessary and sufficient condition for the solution to
exist on a time interval~$[0,T]$ is
\begin{equation}\label{eq:BKM}
\int_0^T\|\partial_x\rho(\cdot,t)\|_{L^\infty}^2 dt<\infty.
\end{equation}
\end{theorem}
\begin{proof}
We follow the proof in \cite{do2017global} and rewrite the equations
\eqref{eq:EPASrho2} and \eqref{eq:EPASG} in terms of
$(\theta, G)$ where $\theta=\rho-\bar{\rho}$.
\begin{align}
\pa_t\theta+\pa_x(\theta u)&=-\bar{\rho}\pa_xu,\label{eq:EAPtheta}\\
\pa_tG+\pa_x(Gu)&=-k\theta,\label{eq:EAPG2}
\end{align}
The velocity $u$ is defined in \eqref{eq:urec}.
Given any $T>0$, we will obtain a differential inequality on
\begin{equation}\label{eq:Y}
Y(t):=1+\|\theta(\cdot,t)\|_{H^s}^2+\|G(\cdot,t)\|_{H^{s-\frac{\alpha}{2}}}^2,
\end{equation}
for all $t\in[0,T]$.
Through a commutator estimate \cite[equation (3.23)]{do2017global}, one can get
\begin{equation}\label{eq:thetaHs}
\frac{1}{2}\frac{d}{dt}\|\theta\|_{H^s}^2\leq
C\left(1+\frac{1}{\rho_m}\right)
(1+\|\partial_x\theta\|_{L^\infty}^2+\|G\|_{L^\infty})
Y(t)-
\frac{\rho_m}{3}\|\theta\|_{H^{s+\frac{\alpha}{2}}}^2,
\end{equation}
where $\rho_m(t)$ has a positive lower bound for $t\in[0,T]$ due to
Theorem \ref{thm:lower}. Also, $\|G(\cdot,t)\|_{L^\infty}$ is bounded
for $t\in[0,T]$ as $G=F\rho$ and both $F$ and $\rho$ are bounded, see
\eqref{eq:Fbound2} and \eqref{eq:upperbound} respectively.
We also compute
\begin{equation}\label{eq:GHs}
\frac{1}{2}\frac{d}{dt}\|G\|_{\dot{H}^{s-\frac{\alpha}{2}}}^2
=-\int_\T(\Lambda^{s-\frac{\alpha}{2}}
G)\cdot(\Lambda^{s-\frac{\alpha}{2}}\partial_x (Gu))dx
-k\int_\T(\Lambda^{s-\frac{\alpha}{2}}
G)\cdot (\Lambda^{s-\frac{\alpha}{2}}\theta)dx=I+II.
\end{equation}
The first term can be controlled by the following estimate
\cite[equation (3.25)]{do2017global}
\begin{equation}\label{eq:GHsI}
|I|\leq
\frac{\rho_m}{6}\|\theta\|_{H^{s+\frac{\alpha}{2}}}^2 +C
\left(1+\frac{1}{\rho_m}\|G\|_{L^\infty}^2+\|\partial_x\theta\|_{L^\infty}+\|G\|_{L^\infty})\right) \|G\|_{H^{s-\frac{\alpha}{2}}}^2.
\end{equation}
The $II$ term encodes the contribution of the attractive-repulsive
potential. We have the following estimate
\begin{equation}\label{eq:GHsII}
|II|\leq
|k|\|G\|_{\dot{H}^{s-\frac{\alpha}{2}}}\|\theta\|_{\dot{H}^{s-\frac{\alpha}{2}}}
\leq C|k|\|G\|_{\dot{H}^{s-\frac{\alpha}{2}}}\|\theta\|_{H^s}\leq C|k|Y(t).
\end{equation}
Combine \eqref{eq:thetaHs}, \eqref{eq:GHsI} and \eqref{eq:GHsII}, we
get
\begin{equation}\label{eq:YHs}
\frac{d}{dt}Y(t)\leq C(1+\|\pa_x\theta(\cdot,t)\|_{L^\infty}^2)Y(t)-\frac{\rho_m(t)}{6}\|\theta\|_{H^{s+\frac{\alpha}{2}}}^2,
\end{equation}
where $C$ is a positive constant which might depend on $T$.
Applying Gronwall's inequality, we get
\begin{equation}\label{eq:gronwall}
Y(t)+\frac{1}{6}\min_{t\in[0,T]}\rho_m(t)\|\theta\|_{L^2([0,T];H^{s+\frac{\alpha}{2}}(\T))}^2\leq
Y(0)\exp\left[C(T)\int_0^T(1+\|\pa_x\theta(\cdot,s)\|_{L^\infty}^2)ds\right],
\end{equation}
for all $t\in[0,T]$.
The right hand side is bounded as long as condition \eqref{eq:BKM} is
satisfied. Therefore,
\[\theta\in C([0,T], H^s(\T))\cap L^2([0,T], H^{s+\frac{\alpha}{2}}(\T)),\quad
G\in C([0,T], H^{s-\frac{\alpha}{2}}(\T)).\]
This directly implies the regularity conditions on $\rho$ in
\eqref{eq:smoothsol}. The regularity conditions on $u$ can also be easily
obtained from \eqref{eq:urec}.
\end{proof}
\subsection{Global wellposedness}\label{sec:global}
In this section, we prove that the Beale-Kato-Majda type condition
\eqref{eq:BKM} holds for any finite time $T$. This will imply global
wellposedness of the fractional EPA system and hence finish the proof
of Theorem \ref{thm:EAP}. Throughout the section, we fix a time $T>0$
(which is arbitrary).
To derive a uniform $L^\infty$ bound on $\pa_x\rho$, we argue that
$\rho(\cdot,t)$ will obey certain modulus of continuity for
$t\in[0,T]$. Such method has been successfully used to obtain global
regularity for 2D quasi-geostrophic equation with critical dissipation
\cite{kiselev2007global}, fractal Burgers equation
\cite{kiselev2008blow}, as well as fractional Euler-Alignment system
\cite{do2017global}. In all these examples, the solution has a certain
scaling invariance property. Unfortunately, such property is not
available for the fractional EPA system
\eqref{eq:EPASrho}-\eqref{eq:EPASu}. We note that the modulus method
has been applied to subcritical perturbations destroying scaling before
(e.g. \cite{kiselev2010global}). The argument in \cite{kiselev2010global}, however, relies on the specific
structure of the perturbation, and cannot be readily ported to other settings.
A novel feature compared to both \cite{kiselev2010global} and \cite{do2017global} will be
dependence of the modulus on time. This feature is linked to the possible decay
of $\rho_m$ and growth of $\|\rho\|_{L^\infty},$ and appears to be an intrinsic
property of the problem.
We use the same family of moduli of continuity as in \cite{do2017global},
\begin{equation}\label{eq:mocf}
\omega(\xi)=\begin{cases}\xi-\xi^{1+{\alpha}/{2}},&0\le\xi<\delta \leq 1\\
\gamma\log(\xi/\delta)+\delta-\delta^{1+\alpha/2},&\xi\geq\delta,
\end{cases}
\end{equation}
where $\gamma, \delta$ are small constants to be determined. Set
$\omega_B(\xi)=\omega(B\xi)$, where $B$ is a large constant to be
determined as well. Due to lack of scaling invariance, we
will work directly on $\omega_B$.
\begin{equation}\label{eq:omegaB}
\omega_B(\xi)=\begin{cases}B\xi-(B\xi)^{1+{\alpha}/{2}},&0\le\xi<B^{-1}\delta\\
\gamma\log\frac{B\xi}{\delta}+\delta-\delta^{1+\alpha/2},&\xi\geq B^{-1}\delta,
\end{cases}
\end{equation}
We say that a function $f$ obeys modulus of continuity $\omega$ if
\begin{equation}\label{eq:defmod}
|f(x)-f(y)| < \omega(|x-y|),\quad\forall~x,y\in\T.
\end{equation}
Our plan is to find a $\omega_B$ such that $\rho(\cdot,t)$ obeys
$\omega_B$ for all $t\in[0,T]$. To construct $\omega_B$, we will first
choose $\delta$ and $\gamma$ which
depend on initial conditions and $T$, but not on $B$.
Then, we will choose $B$ that depend on $T,\delta,\gamma$ as well as initial conditions.
First, we would like to make sure that $\rho_0$ obeys $\omega_B$.
\begin{lemma}\label{lem:initmod}
Let $\rho_0\in C^1(\T)$. Then, $\rho_0$ obeys $\omega_B$ if
\begin{equation}\label{eq:deltagammaBinit}
\delta<\frac{2\|\rho_0\|_{L^\infty}}{\|\pa_x\rho_0\|_{L^\infty}},\quad
B>\frac{\delta\|\pa_x\rho_0\|_{L^\infty}}{2\|\rho_0\|_{L^\infty}}\exp\left(\frac{2\|\rho_0\|_{L^\infty}}{\gamma}\right).
\end{equation}
\end{lemma}
\begin{proof}
We start with an elementary inequality
\begin{equation}\label{eq:rhoele}
|\rho_0(x)-\rho_0(y)|\leq\min\{2\|\rho_0\|_{L^\infty},
\|\pa_x\rho_0\|_{L^\infty}|x-y|\}.
\end{equation}
As $\omega_B$ is concave and monotone increasing, the right hand side
of \eqref{eq:rhoele} is bounded by $\omega_B(|x-y|)$ if
\begin{equation}\label{eq:modinit}
\omega_B\left(\frac{2\|\rho_0\|_{L^\infty}}{\|\pa_x\rho_0\|_{L^\infty}}\right)
>2\|\rho_0\|_{L^\infty}.
\end{equation}
Since $\omega_B(\xi)\to+\infty$ as $B\to+\infty$, \eqref{eq:modinit}
is satisfied by taking $B$ large enough. Indeed, if $\delta$ and $B$ satisfy
\eqref{eq:deltagammaBinit}, then
\[\omega_B\left(\frac{2\|\rho_0\|_{L^\infty}}{\|\pa_x\rho_0\|_{L^\infty}}\right)
>\gamma\log\left(\frac{2B\|\rho_0\|_{L^\infty}}{\delta\|\pa_x\rho_0\|_{L^\infty}}\right)>
2\|\rho_0\|_{L^\infty}.\]
\end{proof}
The following lemma describes the only possible breakthrough scenario
for the modulus.
\begin{lemma}\label{lem:breakthrough}
Suppose $\rho_0$ obeys a modulus of continuity $\omega_B$ as in \eqref{eq:omegaB}.
If the solution~$\rho(x,t)$ violates
$\omega_B$ at some positive time, then there must exist $t_1>0$ and
$x_1\neq y_1$ such that
\begin{equation}\label{eq:breakthrough}
\rho(x_1,t_1)-\rho(y_1,t_1)=\omega_B(|x_1-y_1|),
\hbox{ and $\rho(\cdot,t)$ obeys $\omega_B$ for every $0\leq t<t_1$.}
\end{equation}
\end{lemma}
The main point of the lemma is the existence of two distinct points where the solution touches the
modulus (as opposed to a single point $x$ with $|\nabla \rho(x)| = \omega'_B(0)=B$).
This property is a consequence of $\omega''_B(0) = -\infty;$ see \cite{kiselev2007global} for more
details.
We will show that in the breakthrough scenario as above,
\begin{equation}\label{eq:nobreak}
\pa_t(\rho(x_1,t_1)-\rho(y_1,t_1))<0,\quad\forall~ t_1\in(0,T],
\end{equation}
achieving a contradiction with the choice of time $t_1$ - and thus showing
that the modulus $\omega_B$ cannot be broken.
Together
with Lemma \ref{lem:initmod} this implies that
$\rho(\cdot,t)$ obeys
$\omega_B$ for all $t\in[0,T]$. Therefore,
\begin{equation}\label{eq:rhoxbound}
\|\pa_x\rho(\cdot,t)\|_{L^\infty}\leq\omega_B'(0)=B,\quad\forall~t\in[0,T].
\end{equation}
This proves global regularity of the fractional EPA system and ends the proof of Theorem \ref{thm:EAP}.
The rest of the section is devoted to proof of \eqref{eq:nobreak}.
We fix $t_1$ and drop the time variable for simplicity.
Let $\xi=|x_1-y_1|$. Then
\begin{equation}\label{eq:rhoxi}
\begin{split}
\partial_t&(\rho(x_1)-\rho(y_1))=
-\partial_x(\rho(x_1)u(x_1))+\partial_x(\rho(y_1)u(y_1))\\
&=-\big(u(x_1)\partial_x\rho(x_1)-u(y_1)\partial_x\rho(y_1)\big)
-\big(\rho(x_1)-\rho(y_1)\big)\partial_xu(x_1)
-\rho(y_1)\big(\partial_xu(x_1)-\partial_xu(y_1)\big)\\
&=I+II+III.
\end{split}
\end{equation}
Decompose $u$ into two parts $u=u_1+u_2$ where
\begin{equation}\label{eq:u1u2}
u_1(x)=\Lambda^\alpha\pa_x^{-1}(\rho(x)-\bar{\rho}),\quad
u_2(x)=\pa_x^{-1}G(x)+I_0.
\end{equation}
Then, we can write \eqref{eq:rhoxi} as
\begin{equation}\label{eq:rhoxi2}
\partial_t(\rho(x_1)-\rho(y_1))=I_1+II_1+III_1+I_2+II_2+III_2,
\end{equation}
where $I_1, II_1, III_1$ represent the contributions from $u_1$, and
$I_2, II_2, III_2$ represent the contribution from $u_2$.
For $I_1, II_1, III_1$, we proceed with an argument parallel to
\cite{do2017global}. Let us recall the result.
The following quantities play a role in the proof:
\[ \Omega(\xi) = c_{1,\alpha} \left( \int_0^\xi \frac{\omega(\eta)}{\eta^\alpha} \,d\eta +
\xi \int_{\xi}^\infty \frac{\omega(\eta)}{\eta^{1+\alpha}}\,d\eta \right); \]
\[ A(\xi) = c_{2,\alpha} \int_{\R} \frac{\omega(\xi) - \omega(|\xi-\eta|)}{|\eta|^{1+\alpha}}\,d\eta; \]
{\small\[ D(\xi) = c_{3,\alpha} \left( \int_0^{\xi/2} \frac{2\omega(\xi) - \omega(\xi+2\eta)-\omega(\xi-2\eta)}{\eta^{1+\alpha}}\,d\eta
+\int_{\xi/2}^\infty \frac{2\omega(\xi) - \omega(\xi+2\eta)+\omega(2\eta-\xi)}{\eta^{1+\alpha}}\,d\eta \right). \]}
\begin{lemma}[{\cite[Lemma 4.4 and 4.5]{do2017global}}]\label{lem:homopart1}
Let $\rho(\cdot,t)$ obey the modulus of continuity $\omega$ as in
\eqref{eq:mocf} for $0\leq t<t_1\leq T$, and let $x_1$, $y_1$
be the breakthrough points at the first breakthrough time $t_1$.
Suppose $\delta$ and $\gamma$ are small constants such that
\begin{equation}\label{eq:deltagamma}
\delta<1,\quad\gamma\leq\frac{\delta-\delta^{1+\alpha/2}}{2\log2}.
\end{equation}
Then, there exist positive constants $C_I$, $C_{II}$ and $C_{III}$, which may
only depend on $\alpha$, such that
\begin{align}
|I_1|\leq&~\omega'(\xi)\Omega(\xi),\quad\text{where }
\Omega(\xi)\leq
\begin{cases}
C_I\xi,&0<\xi<\delta,\\
C_I\xi^{1-\alpha}\omega(\xi),&\xi\geq\delta.
\end{cases}\label{eq:I1homo}\\
II_1\leq&~\omega(\xi)A(\xi),\quad\text{where }
A(\xi) \leq
\begin{cases}
C_{II},&0<\xi<\delta,\\
C_{II}\gamma\xi^{-\alpha},&\xi\geq\delta.
\end{cases}\label{eq:I2homo}\\
III_1\leq&-\rho_mD(\xi),\quad\text{where }
D(\xi) \geq
\begin{cases}
C_{III}\xi^{1-\alpha/2},&0<\xi<\delta,\\
C_{III}\omega(\xi)\xi^{-\alpha},&\xi\geq\delta.
\end{cases}\label{eq:I3homo}
\end{align}
\end{lemma}
Applying the proof of Lemma \ref{lem:homopart1} to the modulus of
continuity $\omega_B$, we get the following estimates.
\begin{lemma}\label{lem:homopart}
Let $\rho(\cdot,t)$ obey the modulus of continuity $\omega_B$ as in
\eqref{eq:omegaB} for $0\leq t<t_1\leq T$, and let $x_1$, $y_1$
be the breakthrough points at the first breakthrough time $t_1$, as in
\eqref{eq:breakthrough}.
Suppose $\delta$ and $\gamma$ are small constants satisfying
\eqref{eq:deltagamma}.
Then there exist positive constants $C_2$ and $C_3$, which may only
depend on $\alpha$, such that
\begin{equation}\label{eq:I1I2}
|I_1|, ~II_1\leq\begin{cases}C_2B^{1+\alpha}\xi,&0<\xi<B^{-1}\delta,\\
C_2\gamma\omega_B(\xi)\xi^{-\alpha},&\xi\geq B^{-1}\delta,
\end{cases}
\end{equation}
and
\begin{equation}\label{eq:I3}
III_1\leq -\rho_m D_B(\xi),~~
D_B(\xi):=\begin{cases}C_3B^{1+{\alpha}/{2}}\xi^{1-{\alpha}/{2}},&0<\xi< B^{-1}\delta,\\
C_3{\omega_B(\xi)}{\xi^{-\alpha}},&\xi\geq B^{-1}\delta.
\end{cases}
\end{equation}
\end{lemma}
\begin{proof}
Through the same proof of Lemma \ref{lem:homopart1} and replacing
$\omega$ by $\omega_B$, one can obtain the following estimates similar to
\eqref{eq:I1homo}, \eqref{eq:I2homo} and \eqref{eq:I3homo}:
\[|I_1|\leq\omega_B'(\xi)\Omega_B(\xi),\quad
II_1\leq\omega_B(\xi)A_B(\xi),\quad
III_1\leq-\rho_mD_B(\xi).\]
Here $\omega_B$ is defined in \eqref{eq:omegaB}, and
\[\Omega_B(\xi)=B^{\alpha-1}\Omega(B\xi),\quad
A_B(\xi)=B^\alpha A(B\xi),\quad
D_B(\xi)=B^\alpha D(B\xi).\]
This directly implies \eqref{eq:I1I2} and \eqref{eq:I3} with
$C_2=\max\{C_I, C_{II}\}$ and $C_3=C_{III}$.
\end{proof}
If we pick $\delta$ small enough so that
\begin{equation}\label{eq:delta}
\delta<\left(\frac{C_3}{4C_2}\rho_m(T)\right)^{2/\alpha},
\end{equation}
then
\[
C_2B^{1+\alpha}\xi\leq C_2
B^{1+\alpha/2}\xi^{1-\alpha/2}\delta^{\alpha/2}\leq\frac{1}{4}\rho_mD_B(\xi),\quad
\forall~\xi\in(0,B^{-1}\delta).
\]
Also, pick $\gamma$ small enough so that
\begin{equation}\label{eq:gamma}
\gamma<\frac{C_3}{4C_2}\rho_m(T),
\end{equation}
then
\[C_2\gamma\omega_B(\xi)\xi^{-\alpha}\leq\frac{1}{4}\rho_mD_B(\xi),\quad
\forall~\xi\geq B^{-1}\delta.\]
Therefore, we have
\begin{equation}\label{eq:est1}
I_1+II_1+III_1\leq -\frac{1}{2}\rho_mD_B(\xi).
\end{equation}
It remains to control $I_2, II_2$ and $III_3$. We start with the
estimate on $I_2$.
\begin{lemma}\label{lem:drift}
Let $\rho(\cdot,t)$ obey the modulus of continuity $\omega_B$ as in
\eqref{eq:omegaB} for $0\leq t<t_1\leq T$, and let $x_1$, $y_1$
be the breakthrough points at the first breakthrough time $t_1$, as in
\eqref{eq:breakthrough}.
Suppose $\delta$ and $\gamma$ satisfy \eqref{eq:deltagamma}, and in addition
\begin{equation}\label{eq:deltagamma2}
\delta<
\left(\frac{\rho_m(T)C_3}{6\rho_M(T)F_M(T)}\right)^{2/\alpha},
\gamma<\alpha(\delta-\delta^{1+\alpha/2}),
\text{and }
B>\max\left\{1, 2\delta \exp\left(\frac{6\rho_M(T)F_M(T)}{C_3\rho_m(T)}\right)\right\}.
\end{equation}
Then,
\begin{equation}\label{eq:II1}
|I_2|\leq\frac{1}{6}\rho_mD_B(\xi).
\end{equation}
\end{lemma}
\begin{proof}
We start with the estimate
\[|I_2|\leq\|\pa_xu_2\|_{L^\infty}\xi\omega_B'(\xi)=\|G\|_{L^\infty}\xi\omega_B'(\xi)
\leq \rho_M(T)F_M(T) \xi\omega_B'(\xi).\]
For $\xi\in(0,B^{-1}\delta)$, $\omega_B'(\xi)<B$. So,
\begin{equation}\label{eq:II1a}
|I_2|\leq \rho_M(T)F_M(T)B\xi\leq\frac{1}{6}\rho_mD_B(\xi),
\end{equation}
provided that $\delta$ is small enough, satisfying
\eqref{eq:deltagamma2}, and $B>1$.
For $\xi\geq B^{-1}\delta$, since $\rho$ is periodic and $\omega_B$ is
increasing, the breakthrough can not happen first at $\xi>1/2$. So we
only need to consider $\xi\in(B^{-1}\delta, 1/2]$.
As $\omega_B'(\xi)=\frac{\gamma}{\xi}$ in this range, we get
\begin{equation}\label{eq:II1b}
|I_2|\leq \rho_M(T)F_M(T)\gamma.
\end{equation}
On the other hand, compute
\[\frac{d}{d\xi}D_B(\xi)=C_3\xi^{-\alpha-1}(-\alpha\omega_B(\xi)+\gamma)\leq C_3\xi^{-\alpha-1}(-\alpha(\delta-\delta^{1+\alpha/2})+\gamma)<0,
\]
for all $\xi\geq B^{-1}\delta$, provided that $\gamma$ is small
enough, satisfying \eqref{eq:deltagamma2}.
Therefore,
\begin{equation}\label{eq:minDBxi}
\min_{B^{-1}\delta\leq\xi\leq
1/2}D_B(\xi)=D_B(1/2)\geq C_3\gamma\log\left(\frac{B}{2\delta}\right).
\end{equation}
Combining \eqref{eq:II1b}, \eqref{eq:minDBxi} and the assumption
on $B$ in \eqref{eq:deltagamma2}, we conclude
\begin{equation}\label{eq:II1b2}
|I_2|\leq
\rho_M(T)F_M(T)\gamma\leq\frac{C_3}{6}\gamma\rho_m(T)\log\left(\frac{B}{2\delta}\right)
\leq\frac{1}{6}\rho_mD_B(\xi).
\end{equation}
\end{proof}
The estimates on $II_2$ and $III_2$ are more subtle.
To proceed, it is convinient to decompose $II_2+III_2$ in an
alternative way
\begin{equation}\label{eq:GtoF}
\begin{split}
II_2+III_2=&-\big(\rho(x_1)\pa_xu_2(x_1)-\rho(y_1)\pa_xu_2(y_1)\big)
=-\big(\rho(x_1)^2F(x_1)-\rho(y_1)^2F(y_1)\big)\\
=&-\big(\rho(x_1)^2-\rho(y_1)^2\big)F(x_1)-\rho(y_1)^2\big(F(x_1)-F(y_1)\big)
=IV+V.
\end{split}
\end{equation}
We first consider the case when $\xi< B^{-1}\delta$.
For $IV$, the estimate is similar to \eqref{eq:II1a}
\begin{equation}\label{eq:estIV}
|IV|=\omega_B(\xi)(\rho(x_1)+\rho(y_1))|F(x_1)|\leq
2\rho_MF_MB\xi\leq\frac{1}{6}\rho_mD_B(\xi),
\end{equation}
where the last inequality holds if
$\delta$ is picked to be small enough, satisfying
\begin{equation}\label{eq:delta3}
\delta<\left(\frac{C_3\rho_m(T)}{12\rho_M(T)F_M(T)}\right)^{2/\alpha}.
\end{equation}
For $V$, we need the following lemma.
\begin{lemma}\label{lem:FxFy}
Let $\rho(\cdot,t)$ obey the modulus of continuity $\omega_B$ with
any $B>1$ as in
\eqref{eq:omegaB} for $0\leq t<t_1\leq T$.
Then, there exists a constant $C_F=C_F(T)$ such that
\begin{equation}\label{eq:FxFy}
|F(x,t)-F(y,t)|\leq C_F(T)B|x-y|,\quad
\forall~x,y\in\T,\quad \forall~ t\in[0,t_1].
\end{equation}
\end{lemma}
\begin{proof}
Recall the dynamics of $F$
\begin{equation}\label{eq:F2}
\pa_tF+u\pa_xF=-k\left(1-\frac{\bar{\rho}}{\rho}\right).
\end{equation}
Let $f=\pa_xF$. Differentiate \eqref{eq:F2} with respect to $x$ and
get
\begin{equation}\label{eq:Fx}
\partial_tf+\pa_x(uf)=-k\bar{\rho}\frac{\pa_x\rho}{\rho^2}.
\end{equation}
Let $q=f/\rho$. Using \eqref{eq:EPASrho} and \eqref{eq:Fx}, we obtain
\begin{equation}\label{eq:q}
\partial_tq+u\pa_xq=-k\bar{\rho}\frac{\pa_x\rho}{\rho^3}.
\end{equation}
It follows that
\[q(X(x,t),t)=q_0(x)-k\bar{\rho}\int_0^t
\frac{\pa_x\rho(X(x,s),s)}{\rho(X(x,s),s)^3}ds,\]
where $X$ is the trajectory of the characteristic path defined in \eqref{eq:path}.
Then, since for $t\leq t_1$, $\rho(\cdot,t)$ obeys $\omega_B$, we
obtain the following estimate
\begin{equation}\label{eq:qmax}
\|q(\cdot,t)\|_{L^\infty}\leq\|q_0\|_{L^\infty}+|k|\bar{\rho}\int_0^t\frac{B}{\rho_m(s)^3}ds
\leq C'(T)B,
\end{equation}
where the finite constant $C'$ depends on $T$ and initial
data. This implies
\[
|F(x)-F(y)|\leq\|f\|_{L^\infty}|x-y|\leq\rho_M(T) C'(T)B\xi=:C_F(T)B|x-y|.
\]
\end{proof}
Applying the estimate \eqref{eq:FxFy} at the breakthrough points and using
the upper bound on $\rho$ \eqref{eq:upperbound}, we get
\begin{equation}\label{eq:estV}
|V|\leq \rho_M(T)^2C_F(T)B\xi< \frac{1}{6}\rho_mD_B(\xi),
\end{equation}
where the second inequality holds by picking sufficiently small
$\delta$, satisifying
\begin{equation}\label{eq:delta4}
\delta<\left(\frac{C_3\rho_m(T)}{6\rho_M(T)^2C_F(T)}\right)^{2/\alpha},
\end{equation}
similar to the estimate in \eqref{eq:II1a}.
Combining \eqref{eq:est1}, \eqref{eq:II1a}, \eqref{eq:estIV} and
\eqref{eq:estV}, we conclude that
\[\pa_t(\rho(x_1)-\rho(x_2))<0, \quad
\forall~\xi=|x_1-x_2|<B^{-1}\delta.\]
Finally, we estimate $II_2+III_2$ for $\xi\in[B^{-1}\delta,1/2]$.
As $\rho$ and $F$ are bounded, it is clear that
\begin{equation}\label{eq:estIVV}
|II_2+III_2|\leq 2\rho_M(T)^2 F_M(T)<\frac{1}{3}\rho_mD_B(\xi).
\end{equation}
The second inequality holds by picking $B$ large enough. This is due
to the fact that $D_B(\xi)$ is an increasing in $B$ with
$\lim_{B\to\infty}D_B(\xi)=\infty$. More precisely,
using the bound \eqref{eq:minDBxi}, it suffices to pick
\begin{equation}\label{eq:B2}
B>2\delta\exp\left(\frac{6\rho_M(T)^2 F_M(T)}{C_3\gamma\rho_m(T)}\right).
\end{equation}
Combining \eqref{eq:est1}, \eqref{eq:II1b2} and
\eqref{eq:estIVV}, we conclude that
\[\pa_t(\rho(x_1)-\rho(x_2))<0, \quad
\forall~\xi=|x_1-x_2|\in[B^{-1}\delta,1/2].\]
Let us summerize the procedure on the construction of the modulus of
continuity $\omega_B$. First, we fix a time $T$.
Then, we pick a small parameter $\delta$
satisfying \eqref{eq:deltagammaBinit}, \eqref{eq:deltagamma},
\eqref{eq:delta}, \eqref{eq:deltagamma2}, \eqref{eq:delta3} and \eqref{eq:delta4}:
\begin{equation}\label{eq:deltaall}
\delta<\min\left\{1,
\frac{2\|\rho_0\|_{L^\infty}}{\|\pa_x\rho_0\|_{L^\infty}},
\left(\frac{C_3\rho_m(T)}{\max\{4C_2,
12\rho_M(T)F_M(T), 6\rho_M(T)^2C_F(T)\}}\right)^{2/\alpha}\right\}.
\end{equation}
Next, we pick a small parameter $\gamma$ satisfying
\eqref{eq:deltagamma}, \eqref{eq:gamma} and \eqref{eq:deltagamma2}:
\begin{equation}\label{eq:gammaall}
\gamma<\min\left\{
\frac{C_3}{4C_2}\rho_m(T),~\min\left(\frac{1}{2\log2},\alpha\right)\cdot
(\delta-\delta^{1+\alpha/2})
\right\}.
\end{equation}
Finally, we pick a large parameter $B$ satisfying
\eqref{eq:deltagammaBinit}, \eqref{eq:deltagamma2} and \eqref{eq:B2}:
\begin{equation}\label{eq:Ball}
B>\max\left\{1,
\frac{\delta\|\pa_x\rho_0\|_{L^\infty}}{2\|\rho_0\|_{L^\infty}}\exp\left(\frac{2\|\rho_0\|_{L^\infty}}{\gamma}\right),
2\delta\exp\left(\frac{6\rho_M(T)^2 F_M(T)}{C_3\gamma\rho_m(T)}\right)\right\}.
\end{equation}
Here we assume, without loss of generality, that $\gamma \leq 1$ and $\rho_M(T) \geq 1$ to simplify
the expression.
With this choice of $\omega_B$, we have shown that $\rho(\cdot, t)$
obeys $\omega_B$ for all $t\in[0,T]$. Hence,
\[\|\pa_x\rho(\cdot, t)\|_{L^\infty}\leq B,\quad\forall~t\in[0,T].\]
Therefore, condition \eqref{eq:BKM} is satisfied, and we obtain global
regularity of the system.
We end this section by the following remark.
\begin{remark}
When $k=0$, all the quantities $\rho_m, \rho_M, F_M$ and $C_F$ do not
depend on $T$. As a consequence, $\delta, \gamma$
and $B$ do not depend on $T$ as well. Therefore,
$\|\pa_x\rho(\cdot, t)\|_{L^\infty}\leq B$ for any $t\geq0$. This
estimate improves the result obtained in \cite{do2017global},
where the bound on $\pa_x\rho$ could grow in time. We note that
stationary in time bound on $\pa_x\rho$ for the Euler-Alignment model
(without Possion forcing) has been derived in \cite{shvydkoy2017eulerian3} by a different
argument.
For $k\neq 0$, with the singular attractive or repulsive force, our
estimate on $\rho_m$ and $\rho_M$ is not uniform in time. We are able
to obtain time-dependent bounds \eqref{eq:lowerbound} and
\eqref{eq:upperbound}, where $\rho_m$ can decay exponentially in time,
and $\rho_M$ (and $F_M, C_F$) can grow exponentially in time. From
\eqref{eq:deltaall} and \eqref{eq:gammaall}, we see that $\delta$ and
$\gamma$ decay exponentially in time. Finally, from
\eqref{eq:Ball}, $B$ grows double exponentially in time. Therefore,
we obtain a double exponential in time bound on
$\|\pa_x\rho(\cdot,t)\|_{L^\infty}$. It is not clear whether such
bound is optimal. We will leave it for future investigation.
\end{remark}
\section{Euler dynamics with general three-zone interactions}
In this section, we extend our global regularity result for EPA system
to more general Euler dynamics with three-zone interactions. Recall
the \emph{Euler-3Zone system} under periodic setup
\begin{align}
&\pa_t \rho + \pa_x (\rho u) = 0, \quad x\in\T,\,\,t>0,\label{eq:grho} \\
&\,\,\,\, \pa_t u + u\pa_x u = \int_{\T} \psi(y)(u(x+y,t) - u(x,t))\rho(y,t)dy
- \pa_x K \star \rho.\label{eq:gu}
\end{align}
We will discuss the global wellposedness of the system with more general
singular influence function $\psi$ and interaction potential $K$.
\subsection{General singular influence function}
Consider a general influence function $\psi$ which is positive
\begin{equation}\label{eq:psimg}
\psi_m:=\min_{x\in\T}\psi(x)>0,
\end{equation}
and singular at origin.
Recall the decomposition \eqref{eq:psidecomp}:
we will consider the class of functions where one
can decompose $\psi$ into two parts
\begin{equation}\label{eq:psidecomp2}
\psi=c\psi_\alpha+\psi_L.
\end{equation}
Here $\psi_\alpha$ is the singular power defined in \eqref{eq:psip}, and $\psi_L$ is
bounded and Lipschitz.
In this case, let
\begin{equation}\label{eq:gg}
G=\pa_xu-c\Lambda^\alpha\rho+\psi_L\star\rho.
\end{equation}
Then, the dynamics of $G$ reads:
\begin{align*}
\pa_tG=&~\pa_t\pa_xu-c\pa_t\Lambda^\alpha\rho+\psi\star\pa_t\rho\\
=&-\pa_x(u\pa_xu)+c\pa_x\big(-\Lambda^\alpha(\rho
u)+u\Lambda^\alpha\rho\big)-\pa_x\big(\psi_L\star(\rho
u)-u(\psi_L\star\rho)\big)-\pa_{xx}^2K\star\rho\\
&+c\Lambda^\alpha\pa_x(\rho u)-\psi_L\star\pa_x(\rho u)\\
=&-\pa_x\big(u(\pa_xu-c\Lambda^\alpha\rho+\psi_L\star\rho)\big)-\pa_{xx}^2K\star\rho=-\pa_x(Gu)-\pa_{xx}^2K\star\rho.
\end{align*}
Therefore, $(\rho,G)$ still satisfy \eqref{eq:EPASrho2} and
\eqref{eq:EPASG},
\begin{equation}\label{eq:rhoandg}
\pa_t\rho+\pa_x(\rho u) = 0,\quad
\pa_tG+\pa_x(G u) = -\pa_{xx}^2K\star\rho,
\end{equation}
with a different relation
\begin{equation}\label{eq:uxg}
\pa_xu=\Lambda^\alpha\rho+G-\psi_L\star\rho.
\end{equation}
Then the velocity field $u$ can be recovered as
\begin{equation}\label{eq:urecg}
u(x,t)=\Lambda^\alpha\partial_x^{-1}(\rho(x,t)-\bar{\rho})+
\partial_x^{-1}\big(G(x,t)-\psi_L\star\rho(x,t)\big)+I_0(t),
\end{equation}
where $I_0(t)$ can be determined by conservation of momentum
\eqref{eq:mom}. The second term on the right hand side is well-defined since
\[\int_\T \big(G(x,t)-\psi_L\star\rho(x,t)\big)dx=
\int_\T\big(\pa_xu(x,t)-\Lambda^\alpha\rho(x,t)\big)dx=
0,\quad\forall~t\geq0.\]
We can decompose $u$ into two parts $u=u_S+u_L$, where $u_S$ is the
singular part
\begin{equation}\label{eq:using}
\begin{split}
u_S(x,t)=&\Lambda^\alpha\partial_x^{-1}(\rho(x,t)-\bar{\rho})+
\partial_x^{-1}\left(G(x,t)-\int_\T G(x,0)dx\right),\\
\pa_xu_S=&\Lambda^\alpha\rho+G-\int_\T G(x,0)dx,
\end{split}
\end{equation}
and $u_L$ is the Lipschitz part
\begin{equation}\label{eq:uLip}
\begin{split}
u_L(x,t)=&-
\partial_x^{-1}\left(\psi_L\star\rho(x,t)-\int_\T
G(x,0)dx\right)+I_0(t),\\
\pa_xu_L=&-\psi_L\star\rho+\int_\T G(x,0)dx.
\end{split}
\end{equation}
Now, we follow the same procedure as fractional EPA system to show
global regularity of system \eqref{eq:rhoandg}, \eqref{eq:urecg}. We
first take the Newtonian potential \eqref{eq:Newtonian}. General interaction
potentials will be discussed in the next section. The arguments below follow
the same outline, so we focus on indicating changes.
\subsubsection*{Step 1: Apriori lower bound on $\rho$}
The statement and proof are identical to Theorem \ref{thm:lower},
except that estimate \eqref{eq:lb1} is replaced by
\begin{equation}\label{eq:lbg}
\begin{aligned}
-c\Lambda^\alpha\rho(\underline{x},t)+&\psi_L\star\rho(\underline{x},t)\\
=&~\int_\T\big(c\psi_\alpha(y)+\psi_L(y)\big)\big(\rho(\underline{x}-y,t)-\rho(\underline{x},t)\big)dy+\rho(\underline{x},t)\int_\T\psi_L(y)dy\\
\geq&~
\psi_m\int_\T\big(\rho(\underline{x}-y,t)-\rho(\underline{x},t)\big)dy
-\rho(\underline{x},t)\|\psi_L\|_{L^\infty}\\
=&~\psi_m\bar{\rho}-(\psi_m+\|\psi_L\|_{L^\infty})\rho(\underline{x},t).
\end{aligned}
\end{equation}
Hence, estimate \eqref{eq:rhombound1} becomes
\begin{equation}\label{eq:rhomboundg}
\pa_t\rho(\underline{x},t)\geq~
\big(\psi_m\bar{\rho}\big)\rho(\underline{x},t)-\left[\psi_m+\|\psi_L\|_{L^\infty}+
\|F_0\|_{L^\infty}+|k|t+|k|\bar{\rho}\int_0^t\frac{1}{\rho_m(s)}ds\right]\rho(\underline{x},t)^2,
\end{equation}
where the only extra term
$\|\psi_L\|_{L^\infty}\rho(\underline{x},t)^2$ is quadratic in
$\rho$, and can be
controlled by the linear term
$\big(\psi_m\bar{\rho}\big)\rho(\underline{x},t)$ if $\rho_m$ is small
enough.
Following the same proof, we obtain the lower bound \eqref{eq:lowerbound}
with coefficient $A_m, C_m$ satisfying \eqref{eq:AC} for any
$\epsilon\in(0,\epsilon_*)$, where $\epsilon_*=\frac{1}{2}\left(\sqrt{1+\frac{4\psi_m}{e(\psi_m+\|\psi_L\|_{L^\infty}+\|F_0\|_{L^\infty})}}-1\right)$.
\subsubsection*{Step 2: Apriori upper bound on $\rho$}
We follow the proof of Theorem \ref{thm:upper}. The estimate
\eqref{eq:upperbound1} becomes
\begin{align*}
\frac{d}{dt}\rho(\bar{x},t)\leq& -C_1\rho(\bar{x},t)^{2+\alpha}+
F_M(t)\rho(\bar{x},t)^2+\rho(\bar{x},t)\cdot\psi_L\star\rho(\bar{x},t)\\
\leq& -C_1\rho(\bar{x},t)^{2+\alpha}+
F_M(t)\rho(\bar{x},t)^2+\|\psi_L\|_{L^\infty}\bar{\rho}\rho(\bar{x},t).
\end{align*}
Both second and third terms are dominated by the first term if
$\rho(\bar{x},t)$ is big enough. In particular $\pa_t\rho(\bar{x},t)<0$ if
$\rho(\bar{x},t)>\max\{(2F_M/C_1)^{1/\alpha},
(2\|\psi_L\|_{L^\infty}\bar{\rho}/C_1)^{1/(1+\alpha)}\}$
. Therefore,
\begin{equation}\label{eq:rhomaxbg}
\rho(x,t)\leq\rho(\bar{x},t)\leq\max\left\{\|\rho_0\|_{L^\infty},
3\bar{\rho}, \left(\frac{2F_M(t)}{C_1}\right)^{1/\alpha},
\left(\frac{2\|\psi_L\|_{L^\infty}\bar{\rho}}{C_1}\right)^{1/(1+\alpha)}\right\},
\end{equation}
and \eqref{eq:upperbound} holds with $A_M=A_m/\alpha$ and
\[C_M=\max\left\{\max_{x\in\T}\rho_0(x), ~ 3\bar{\rho}, ~
\left[\frac{2}{C_1}\left(\|F_0\|_{L^\infty}+\frac{|k|}{eA_m}+\frac{|k|\bar{\rho}}{A_mC_m}\right)\right]^{\frac{1}{\alpha}},
\left(\frac{2\|\psi_L\|_{L^\infty}\bar{\rho}}{C_1}\right)^{\frac{1}{1+\alpha}}\right\}.\]
\subsubsection*{Step 3: Local wellposedness}
We write the system \eqref{eq:rhoandg} \eqref{eq:urecg} in terms of
$\theta=\rho-\bar{\rho}$ and $G$ as follows
\begin{align}
\pa_t\theta+\pa_x(\theta u_S)+\pa_x(\theta u_L)&=-\bar{\rho}\pa_xu_S-\bar{\rho}\pa_xu_L,\label{eq:thetag}\\
\pa_tG+\pa_x(Gu_S)+ \pa_x(G u_L)&=-k\theta,\label{eq:localgg}
\end{align}
where $u_S$ and $u_L$ are defined in \eqref{eq:using} and
\eqref{eq:uLip} respectively.
We proceed with a Gronwall estimate on the quantity $Y$ in
\eqref{eq:Y}.
The estimates in Theorem \ref{thm:local} can be applied directly to
the $u_S$ part. We will focus on the Lipschitz part $u_L$. The
procedure is similar to \cite[Theorem Appendix A.1]{carrillo2016critical}. We
will summarize in below.
For the term $\pa_x(\theta u_L)$, we have
\[
\int_\T\Lambda^s\theta\cdot\Lambda^s\pa_x(\theta u_L)dx
=\int_\T\Lambda^s\theta\cdot\Lambda^s\pa_x\theta\cdot u_Ldx
+\int_\T\Lambda^s\theta\cdot[\Lambda^s\pa_x, u_L]\theta~dx
=:L_1+L_2.\]
We estimate the two terms one by one. For $L_1$,
\begin{equation}\label{eq:l1est}
|L_1|=\left|\int_\T\pa_x\left(\frac{(\Lambda^s\theta)^2}{2}\right)
u_Ldx\right|\leq
\frac{1}{2}\int_\T(\Lambda^s\theta)^2|\pa_xu_L| dx
\leq\frac{1}{2}\|\psi_L\|_{L^\infty}\bar{\rho}\|\theta\|_{H^s}^2.
\end{equation}
For $L_2$, we apply commutator estimate (e.g. \cite[Lemma Appendix
A.1]{carrillo2016critical}) and get
\begin{align}
|L_2|\leq&~\|\theta\|_{H^s}\big\|[\Lambda^s\pa_x,
u_L]\theta\big\|_{L^2}
\lesssim\|\theta\|_{H^s}\left(\|\pa_xu_L\|_{L^\infty}\|\theta\|_{H^s}+
\|\pa_xu_L\|_{H^{s}}\|\theta\|_{L^\infty}\right)\nonumber\\
\leq&~\|\theta\|_{H^s}\left(\|\psi_L\|_{L^\infty}\bar{\rho}\|\theta\|_{H^s}+
\|\psi_L\|_{L^\infty}\|\rho\|_{H^{s}}\|\theta\|_{L^\infty}\right)\nonumber\\
\leq&~\|\psi_L\|_{L^\infty}(2\bar{\rho}+\|\theta\|_{L^\infty})\|\theta\|_{H^s}^2
+\frac{1}{4}\|\psi_L\|_{L^\infty}\bar{\rho}.\label{eq:l2est}
\end{align}
Note that for the last inequality, we have used
$\|\rho\|_{H^s}\leq \|\theta\|_{H^s}+\|\bar{\rho}\|_{H^s}=
\|\theta\|_{H^s}+\bar{\rho}$.
For the term $-\bar{\rho}\pa_xu_L$,
\begin{equation}\label{eq:l3est}
\left|-\bar{\rho}\int_\T\Lambda^s\theta\cdot\Lambda^s\pa_x u_Ldx\right|
\leq
\bar{\rho}\|\theta\|_{H^s}\|(\pa_x\psi_L)\star(\Lambda^s\rho)\|_{L^2}
\leq
\bar{\rho}\|\pa_x\psi_L\|_{L^\infty}\|\theta\|_{H^s}(\|\theta\|_{H^s}
+\bar{\rho}).
\end{equation}
For the term $\pa_x(Gu_L)$, the estimate is similar to the term
$\pa_x(\theta u_L)$.
\begin{align*}
\int_\T\Lambda^{s-\frac{\alpha}{2}}G&\cdot\Lambda^{s-\frac{\alpha}{2}}
\pa_x (Gu_L)dx\\
=&\int_\T\Lambda^{s-\frac{\alpha}{2}}G\cdot\Lambda^{s-\frac{\alpha}{2}}
\pa_x G\cdot u_Ldx
+\int_\T\Lambda^{s-\frac{\alpha}{2}}G\cdot
[\Lambda^{s-\frac{\alpha}{2}}\pa_x,u_L]G~dx
=:L_4+L_5.
\end{align*}
where
\begin{equation}\label{eq:l4est}
|L_4|=\left|\int_\T\pa_x\left(\frac{(\Lambda^{s-\frac{\alpha}{2}}G)^2}{2}\right)
u_Ldx\right|\leq
\frac{1}{2}\int_\T(\Lambda^{s-\frac{\alpha}{2}}G)^2|\pa_xu_L| dx
\leq\frac{1}{2}\|\psi_L\|_{L^\infty}\bar{\rho}\|G\|_{H^{s-\frac{\alpha}{2}}}^2,
\end{equation}
and
\begin{align}
|L_5|\leq&~\|G\|_{H^{s-\frac{\alpha}{2}}}\big\|[\Lambda^{s-\frac{\alpha}{2}}\pa_x,
u_L]G\big\|_{L^2}
\lesssim\|G\|_{H^{s-\frac{\alpha}{2}}}\left(\|\pa_xu_L\|_{L^\infty}
\|G\|_{H^{s-\frac{\alpha}{2}}}+
\|\pa_xu_L\|_{H^{s}}\|G\|_{L^\infty}\right)\nonumber\\
\leq&~\|\psi_L\|_{L^\infty}\bar{\rho}\|G\|_{H^{s-\frac{\alpha}{2}}}^2
+\|\psi_L\|_{L^\infty}\bar{\rho}(\|\theta\|_{H^s}+\bar{\rho})
\|G\|_{H^{s-\frac{\alpha}{2}}}\nonumber\\
\leq&~\|\psi_L\|_{L^\infty}\bar{\rho}\left[2\|G\|_{H^{s-\frac{\alpha}{2}}}^2
+\frac12\|\theta\|_{H^{s-\frac{\alpha}{2}}}^2+\frac{\bar{\rho}^2}{2}
\right].\label{eq:l5est}
\end{align}
Combining \eqref{eq:YHs}, \eqref{eq:l1est}, \eqref{eq:l2est}, \eqref{eq:l3est},
\eqref{eq:l4est}, \eqref{eq:l5est} and the fact that
$\|G(\cdot,t)\|_{L^\infty}$ is controlled from above by a finite (growing in time) bound, we obtain that for all $t\in[0,T]$,
\begin{equation}\label{eq:YHsg}
\frac{d}{dt}Y(t)\leq C(T)(1+\|\pa_x\theta(\cdot,t)\|_{L^\infty}^2)Y(t)-\frac{\rho_m(t)}{6}\|\theta\|_{H^{s+\frac{\alpha}{2}}}^2,
\end{equation}
where the constant $C$ depends on initial data and $T$.
The same Gronwall's inequality yields local wellposedness as well as
BKM-type blowup condition \eqref{eq:BKM}.
\subsubsection*{Step 4: Global wellposedness}
To check the condition \eqref{eq:BKM}, we will use the procedure identical
to that in section \ref{sec:global}. Let us decompose $u$ as in
\eqref{eq:urecg}, $u = u_1+u_2$ where
\begin{equation}\label{eq:u1u2g}
u_1(x,t)=\Lambda^\alpha\partial_x^{-1}(\rho(x,t)-\bar{\rho}),\quad
u_2(x,t)=\partial_x^{-1}\big(G(x,t)-\psi_L\star\rho(x,t)\big)+I_0(t).
\end{equation}
The only difference between our system \eqref{eq:rhoandg}
\eqref{eq:urecg} and the EPA system is that there is an extra term in
$u_2$.
Throughout the proof in section \ref{sec:global}, the only property of
$u_2$ we have used is that $\pa_xu_2$ is bounded, namely
\[\|\pa_xu_2(\cdot,t)\|_{L^\infty}\leq \rho_M(T)F_M(T)<\infty,\quad\forall~t\in[0,T].\]
For our $u_2$ defined in \eqref{eq:u1u2g}, we also have a bound on
$\pa_xu_2$:
\[\|\pa_xu_2(\cdot,t)\|_{L^\infty}=\|G(\cdot,t)-\psi_L\star\rho(\cdot,t)\|_{L^\infty}\leq
\rho_M(T)F_M(T)+\|\psi_L\|_{L^\infty}\bar\rho<\infty,\quad\forall~t\in[0,T].\]
Hence, global regularity follows from the same procedure by
controlling the modulus of continuity.
\subsection{General interaction potential}
In this part, we consider system \eqref{eq:grho}-\eqref{eq:gu} with a
general interaction potential $K\in W^{2,\infty}(\T)$. This class of
potentials is more regular than the Newtonian potential $\mathcal{N}$
defined in \eqref{eq:Newtonian}, as
$\pa^2_{xx}\mathcal{N}=k(\delta_0-1)\not\in L^\infty$, where
$\delta_0$ is the Dirac delta at $x=0$. We will show global
wellposedness of Euler-3Zone system with $W^{2,\infty}$
potentials. The result automatically extends to systems with
potentials that can be decomposed into a sum of a Newtonian potential
and a $W^{2,\infty}$ potential.
Now, let us assume $K\in W^{2,\infty}(\T)$. After the transformation,
the dynamics for $(\rho,G)$ becomes \eqref{eq:rhoandg}, with velocity
field $u$ defined as \eqref{eq:urecg}. We shall run through the same
procedure and point out the differences.
\subsubsection*{Step 1: Apriori lower bound on $\rho$}
Due to the change of the potential, the dynamics of $F$
\eqref{eq:Fcha} becomes
\begin{equation}\label{eq:Fchag}
(\pa_t+u\pa_x)F=-\frac{\pa_{xx}^2K\star\rho}{\rho}.
\end{equation}
Therefore, we get
\begin{equation}\label{eq:Fg}
F(X(x,t),t)=F_0(x)-\int_0^t\frac{\pa_{xx}^2K\star\rho(X(x,s),s)}{\rho(X(x,s),s)}ds,
\end{equation}
where $X(x,t)$ is the characteristic path defined in \eqref{eq:path}.
Then, we obtain a bound
\begin{equation}\label{eq:Fboundg}
\|F(\cdot,t)\|_{L^\infty}\leq~\|F_0\|_{L^\infty}+\|\pa_{xx}K\|_{L^\infty}\bar{\rho}
\int_0^t\frac{1}{\rho_m(s)}ds,
\end{equation}
which is similar as \eqref{eq:Fbound}. In fact, it is a simpler
bound as the right hand side does not contain a linear term on $t$.
The lower bound \eqref{eq:lowerbound} follows then by the same
argument, with
\[A_m=\frac{\|\pa_{xx}^2K\|_{L^\infty}}{\psi_m}, \quad
C_m=\min\left\{\rho_m(0),\frac{\psi_m\bar{\rho}}{\psi_m+\|\psi_L\|_{L^\infty}+\|F_0\|_{L^\infty}}\right\}.\]
\subsubsection*{Step 2: Apriori upper bound on $\rho$}
The upper bound estimate \eqref{eq:rhomaxbg} can be obtained without any
additional difficulties. Since we have
\[F_M(t)=\|F_0\|_{L^\infty}+\frac{\|\pa_{xx}^2K\|_{L^\infty}\bar{\rho}}{A_mC_m}e^{A_mt}\]
by \eqref{eq:Fboundg} and the lower bound estimate on $\rho$,
the upper bound \eqref{eq:upperbound} holds with
$A_M=A_m/\alpha$ and
\[C_M=\max\left\{\max_{x\in\T}\rho_0(x), ~ 3\bar{\rho}, ~
\left[\frac{2}{C_1}\left(\|F_0\|_{L^\infty}+\frac{\|\pa_{xx}^2K\|_{L^\infty}\bar{\rho}}{A_mC_m}\right)\right]^{\frac{1}{\alpha}},
\left(\frac{2\|\psi_L\|_{L^\infty}\bar{\rho}}{C_1}\right)^{\frac{1}{1+\alpha}}\right\}.\]
\subsubsection*{Step 3: Local wellposedness}
Since the potential only enters the dynamics of $G$ equation, so the system in
terms of $(\theta, G)$ is identical to
\eqref{eq:thetag}-\eqref{eq:localgg}, except the right hand side of
\eqref{eq:localgg}
is replaced by $-\pa_{xx}^2K\star\rho$. Hence, we only need to
estimate this extra term.
\begin{align*}
\left|\int_\T\Lambda^{s-\frac{\alpha}{2}}G\right.&\left.\cdot~\Lambda^{s-\frac{\alpha}{2}}(\pa_{xx}^2K\star\rho)~dx\right|
=\left|\int_\T\Lambda^{s-\frac{\alpha}{2}}G\cdot(\pa_{xx}^2K\star\Lambda^{s-\frac{\alpha}{2}}\rho)~dx\right|\\
\lesssim&~\|\pa_{xx}^2K\|_{L^\infty}\|G\|_{H^{s-\frac{\alpha}{2}}}\|\theta\|_{H^s}
\leq \frac{1}{2}\|\pa_{xx}^2K\|_{L^\infty}Y(t).
\end{align*}
The local wellposedness and BKM-type blowup condition \eqref{eq:BKM}
follow by applying the same Gronwall's inequality on $Y$.
\subsubsection*{Step 4: Global wellposedness}
The argument for EPA system can be directly applied to the general system as
the $\rho$ equations in both cases are the same. The different
potential does change the estimates on $\rho_m, \rho_M, F_M, C_F$,
which are needed to construct the modulus $\omega_B$. Since $\rho_m,
\rho_M$ and $F_M$ have been treated in the previous steps, we are left
with estimating $C_F$, namely proving Lemma \ref{lem:FxFy} for the
general system.
\begin{proof}[Proof of Lemma \ref{lem:FxFy}]
Let $f=\pa_xF$. Differentiate \eqref{eq:Fchag} with respect to $x$ and
get
\begin{equation}\label{eq:Fxg}
\partial_tf+\pa_x(uf)=\frac{-(\pa_{xxx}^3K\star\rho)\rho+(\pa_{xx}^2K\star\rho)\pa_x\rho}{\rho^2}.
\end{equation}
Let $q=f/\rho$. Using \eqref{eq:EPASrho} and \eqref{eq:Fxg}, we obtain
\begin{equation}\label{eq:qg}
\partial_tq+u\pa_xq=\frac{-(\pa_{xxx}^3K\star\rho)\rho+(\pa_{xx}^2K\star\rho)\pa_x\rho}{\rho^3}.
\end{equation}
For $t\leq t_1$, $\rho(\cdot,t)$ obeys $\omega_B$. Then
$\|\pa_x\rho(\cdot,t)\|_{L^\infty}\leq\omega_B'(0)=B$. Therefore, we
can bound the right hand side of \eqref{eq:qg} as follows:
\begin{align*}
\left|\frac{-(\pa_{xxx}^3K\star\rho)\rho+(\pa_{xx}^2K\star\rho)\pa_x\rho}{\rho^3}\right|
\leq&~\frac{\|\pa_{xx}^2K\|_{L^\infty}\|\pa_x\rho\|_{L^1}}{\rho_m(t)^2}
+\frac{\|\pa_{xx}^2K\|_{L^\infty}\bar{\rho}\|\pa_x\rho\|_{L^\infty}}{\rho_m(t)^3}\\
\leq&~B\|\pa_{xx}^2K\|_{L^\infty}\left(\frac{1}{\rho_m(t)^2}+\frac{\bar{\rho}}{\rho_m(t)^3}\right).
\end{align*}
Then, we obtain the bound on $q$ for all $0\leq t\leq t_1<T$,
\begin{equation}\label{eq:qmaxg}
\|q(\cdot,t)\|_{L^\infty}\leq\|q_0\|_{L^\infty}+B\|\pa_{xx}^2K\|_{L^\infty}\int_0^t\left(\frac{1}{\rho_m(t)^2}+\frac{\bar{\rho}}{\rho_m(t)^3}\right)ds
\leq C'(T)B,
\end{equation}
where the finite constant $C'$ depends on $\|\pa_{xx}^2K\|_{L^\infty},
T$ and initial
data. This implies
\[
|F(x)-F(y)|\leq\|f\|_{L^\infty}|x-y|\leq\rho_M(T) C'(T)B\xi=:C_F(T)B|x-y|.
\]
\end{proof}
\textbf{Acknowledgment.} This work has been partially supported by the NSF grants DMS 1412023 and DMS 1712294.
\bibliographystyle{plain}
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Unique and Magical: 14 Serbian Landscapes of Outstanding Features
In Magazine, Public diplomacy, Relax for Digital Diplomacy, Serbia in my heart
Imagine a place adorned not only with an unforgettable appearance but with unbelievable natural, biological, aesthetic, cultural and historical value? Does such a place even exist and how does it look like? We'll answer you straight away – it does. And Serbia has 14 such destinations! Welcome to the magical world of outstanding natural landscapes!
Paradise has been found! And you will come to the same conclusion once you look at the photos we have prepared for you. But, of course, if you wish to feel the magic that courses through these landscapes, you'll have to visit them yourself. So, we suggest you don't hesitate too much – just look at the photos and choose your first destination!
1. Mount Kosmaj
Even though that it's the second lowest mountain in Šumadija after Avala, Kosmaj is a mountain with a lot to offer. From surreal nature and golden sunsets to fascinating history and a rich culture.
Photo: www.portalmladi.com
In the rich forests of Kosmaj a number of beautiful Serbian sanctities has been tucked away, and on the top of this mountain there is an impressive monument dedicated to the soldiers who have fallen in WWII.
2. The Great War Island
On the very confluence of the Danube and the Sava rivers lies a true natural oasis and an ecological point. Of course, we're talking about the Great War Island – the main jewel of the watery heart of Belgrade.
Photo: www.flickr.com, whl.travel
You can enjoy a phenomenal view of Zemun from the Lido beach on the north of the island, and due to the large number of diverse bird species the island is protected by the state as a nature reserve.
3. The Gradac river gorge
Did you know that one of the cleanest rivers in Europe is in Serbia? If you didn't, it's high time you met the Gradac river, its fascinating canyon and the magnificent nature that surrounds it!
Photo: www.panoramio.com
The magical mix of highly preserved, remarkable natural features and valuable cultural and historical landmarks makes Gradac a unique place for hiking, relaxing, recreation and enjoying nature!
4. The mountains of Vršac
A unique position, diversity of flora, rich forest ecosystems, beautiful landscapes and fascinating lookouts brought the mountains of Vršac, located in Banat, under the protection of the state as a landscape of outstanding features. If you ever get the chance to visit this place, do so and you'll find out why!
The thing you mustn't miss is a visit to the main tourist centre and picnic area called the "Vršački breg". You also have to visit the famous Vršac vineyards, the Vršac tower and numerous sanctities in the vicinity.
5. The Subotica or Telečka sands
Unique ecological features, different types of soul, a special regime of subterranean waters, moist habitats, highly diverse flora and rich fauna are only some of the attributes of this unique landscape of great value – the Subotica sands.
Photo: www.lazarus.rs
Near this unusual yet outstanding natural landscape there are also beautiful lakes that have to have their place on your itinerary. The Palić, Ludaš, Krvavo (Bloody) or Slano (Salt) lake – it's up to you to choose!
6. The Vlasina lake
Be it for its unique floating islands, untouched nature, clean air or the special energy that flows here, once you visit the Vlasina lake – you'll want to come back to it!
And although the things we said are quite enough, we have to add another attribute to the magical Vlasina lake – a whole kingdom of medicinal plants and diverse river fish. Visit it and witness its magic!
7. The Ovčar-Kablar gorge
Prepare yourself for enjoying impressive landscapes, for falling in love with the unusual meandres of West Morava, sightseeing of many monasteries and relaxing in beautiful and intact nature. Because all that and more awaits you in the surreal Ovčar-Kablar gorge!
Here, nature showed all her artistic skills and created one of her greatest artworks. It bestowed upon this whole area near Čačak rich flora and fauna, diverse landscapes and numerous mineral sources. Look inside the fairytale world of one of the most scenic gorges in Serbia!
8. Mount Avala
Even Prince Miloš realized the value and beauty of Avala in 1859 and decided to bring it under state protection. So, this small mountain near Belgrade was considered a landscape of outstanding features since the distant past, as well as an oasis of diverse plant and animal life and one of the favourite picnic places of Belgraders nowadays.
Dense coniferous forests and fresh air are the ideal combination for relaxation, and the Avala Tower, the Monument to the Unknown Hero and other cultural landmarks are perfect for sightseeing.
9. Lepterija-Sokograd
They don't call this place the ''green heart of Serbia'' for nothing. There are many reasons to do so – beautiful, intact nature, fresh air, a soothing climate and the attractive "Lepterija" picnic spot.
And in its vicinity are the famous Sokobanja spa and the medieval town of Sokograd which is a cultural monument of great importance. Now, do you realize why you have to visit this part of Serbia?
10. Zaovine
It's nearly impossible to pick one place in Serbia and call it the most beautiful and most valuable. But, when it comes to Zaovine one thing's for sure – it's on the top of the list. Green hills, glades covered in flowers, rivers, lakes and so on and so forth. A picture is worth a thousand words…
Photo: www.flickr.com, Irene Becker
Because of the beauty of all these scenic landscapes, natural habitats and extremely diverse animal and plant life, Zaovine has been proclaimed a landscape of outstanding features.
11. Mount Radan
The south of Serbia is adorned by a mountain which is perhaps best described as a "fascinating combination of nature, mystery and legends". Mt. Radan is also the home of many protected plant and animal species.
This mysterious mountain is famous for the stories of a river which flows from its base to the top of the mountain and cars which start to go uphill once the engines are shut down. Although these stories remain unexplained, the thing that's for certain is that near this mountain you can find many interesting locations such as the Devil's Town, the Prolom Banja spa, the Justiniana Prima, Ivan's Tower, etc.
12. The Đetinja river gorge
On the south-western slopes of mt. Tara springs the river Đetinja and so becomes serious competition when it comes to its beauty. The reasons for it being proclaimed a landscape with outstanding features are many: from dynamic morphology with numerous caves, karst sinkholes and pits to many important heritage monuments and rich flora and fauna…
Photo: www.flickr.com, Neonci
The only problem with this magnificent river's canyon located near Užice and Mokra Gora is that it's out-of-the-ordinary beauty can't be described but only experienced in person! So come and see it!
13. The Pčinja valley
On the very southeast of Serbia is another landscape of outstanding features and a national natural heritage site – the valley of Pčinja. Just look at the photo and you'll understand why this valley is so special. Exactly! Preserved natural resources, unspoiled soil, the wealth it holds in forest land, the fauna and small rural settlements are only some of them!
Photo: www.anterija.rs
We witnessed the beauty of Pčinja, which is near Vranje, for ourselves. Now it's your turn! And don't forget to tell us how you liked it!
14. Kamena Gora
You must think there can be nothing more beautiful than what we've shown you so far? We'll have to say that's not exactly true. And Kamena Gora is proof. Are you ready for a stunning landscape of unbelievable beauty? Here you have it!
Photo: www.commons.wikimedia.org
We're sure you realize now how Kamena Gora earned its place on the list of destinations with outstanding features, but we'll make it even more easy for you: the Holy pine which is estimated to be 500 years old, many rare plant and animal species and a mountain region authentic for its deep gorges and mosaic woodland and meadow areas…
Serbia.com
Serbia in my heart
Leaving the EU: Impact on UK-US Relations
ce.gatech.edu
image (not entry) from
Wondering what to expect from Brexit now that the UK Parliament has upheld its confidence in Prime Minister Theresa May's government but voted against her Brexit deal? Join us on Feb 6 for British Consul General Andrew Staunton's talk "Leaving the EU: Impact on UK-US Relations." 3-4:15pm, 1116 West Seminar Room Klaus Advanced Computing Building.
More about Consul General Andrew Straunton:
Andrew Staunton took up his appointment in Atlanta as Her Majesty's Consul General in June 2018. Andrew is the senior UK government representative in the Southeast United States responsible for relations with the states of Georgia, Tennessee, North Carolina, South Carolina, Mississippi and Alabama. He leads a team which works to promote UK-US trade and investment, support British nationals, conduct public diplomacy [JB emphasis] on key issues, and build scientific and research co-operation. He also sits on the Marshall Scholarship selection committee.
Royce leaves lasting legacy on development
devex.com; original article contains links
Image and caption from article: Representative Ed Royce, outgoing chairman of the House Foreign Affairs Committee. Photo by: CSUF / CC BY-NC-SA
Excerpt:His father's experience had a profound impact on Royce, who is retiring from U.S. Congress after serving as a Republican representative from California for 26 years, the last six of which he was chairman of the House Foreign Affairs Committee.
"The pictures of that concentration camp, the images there, the lesson of what happens when the U.S. is not engaged, the lesson of what happens during a period of isolationism, as was our policy at the time, and then all hell breaks loose," Royce said recently at a U.S. Global Leadership Coalition event.
"The lesson of the cost of that in human lives — 6 million Jews in those camps and 50 million others around this globe who lost their lives — teaches you the importance of a focus on human rights, on human freedom, on the support for the infrastructure of..
New Documents on the 1965 Dominican Republic Invasion
Greg Weeks, weeksnotice.blogspot
The State Department just published a new Foreign Relations of the United States volume: Public Diplomacy, 1964-1968. Another word for "public diplomacy" is actually "propaganda." So, for example, in May 1965 the Director of the United States Information Agency wrote a memo to LBJ about the invasion of the Dominican Republic. He noted how difficult it was to get support in the region. We need to convince non-Communist governments of our good intentions.If we are to succeed in making other Latin American nations believe that our actions are vital to their safety and freedom, it is of utmost importance that we get some members of the OAS, and perhaps non-OAS neighbors of the Dominican Republic like Jamaica, to speak out about the Communist involvement in the Dominican Republic, and to offer troops or other support to our efforts to end the bloodshed.That did not work out too well. The OAS did eventually send people, including some troops, but this was not..
Executive Assistant to the Ambassador
ethiojobs.net
Executive Assistant to the AmbassadorJob by Australian Embassy, Addis Ababa(Job Id: 188162 | 265 Views)Posted13NovCategory:Admin, Secretarial and Clerical, Business and Administration, ManagementLocation:
Addis Ababa Career Level:
Senior Level (5+ years experience)Employment Type:
Full timeSalary:
View Jobs by this companyJob DescriptionAgency: Department of Foreign Affairs and Trade (DFAT)Place of Work: Australian Embassy, Addis AbabaPosition Title: Executive Assistant to the Ambassador (P/N 11032)Classification: LE3Section: CorporateReporting to: Ambassador
About the Australian Government's Department of Foreign Affairs and Trade (DFAT)The role of the Department of Foreign Affairs and Trade (DFAT) is to advance the interests of Australia and Australians internationally. This involves strengthening Australia's security, enhancing Australia's prosperity, delivering an effective and high quality overseas aid program and helping Australian travellers and Australians overse..
CFP: 27th Iamhist Conference on "Power and the Media". July 16-19, 2019 @ Northumbria University (UK). Deadline Jan 14, 2019.
cstonline.net
XXVII IAMHIST Conference
POWER AND THE MEDIA (Northumbria University, Newcastle upon Tyne, 16-19 July 2019)Confirmed keynote speakers include:James Curran (Goldsmiths, University of London)Jennifer Smyth (University of Warwick)Papers and panels are invited for the 2019 conference of the International Association for Media and History. The conference theme this year is POWER AND THE MEDIA. Scholars of media history have not just been concerned with analysis of the individuals, institutions and elites exerting control, but also with how the media has represented, perpetuated or challenged power structures. Taking place in the immediate aftermath of Britain's planned exit from the European Union, the conference invites scholars and practitioners from all relevant disciplines to take part in a timely conversation about the relationship between power and the media, from the film and broadcasting industries and the press, to new media, social media and advertising. I.. | {
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{"url":"https:\/\/labs.tib.eu\/arxiv\/?author=A.%20A.%20Zijlstra","text":"\u2022 ### Opening PANDORA's box: APEX observations of CO in PNe(1803.08446)\n\nMarch 28, 2018 astro-ph.GA, astro-ph.SR\nContext. Observations of molecular gas have played a key role in developing the current understanding of the late stages of stellar evolution. Aims. The survey Planetary nebulae AND their cO Reservoir with APEX (PANDORA) was designed to study the circumstellar shells of evolved stars with the aim to estimate their physical parameters. Methods. Millimetre carbon monoxide (CO) emission is the most useful probe of the warm molecular component ejected by low- to intermediate-mass stars. CO is the second-most abundant molecule in the Universe, and the millimeter transitions are easily excited, thus making it particularly useful to study the mass, structure, and kinematics of the molecular gas. We present a large survey of the CO (J = 3 - 2) line using the Atacama Pathfinder EXperiment (APEX) telescope in a sample of 93 proto-planetary nebulae and planetary nebulae. Results. CO (J = 3 - 2) was detected in 21 of the 93 objects. Conclusions. CO (J = 3 - 2) was detected in all 4 observed pPNe (100%), 15 of the 75 PNe (20%), one of the 4 wide binaries (25%), and in 1 of the 10 close binaries (10%). Using the CO (J = 3 - 2) line, we estimated the column density and mass of each source.\n\u2022 ### The Coordinated Radio and Infrared Survey for High-Mass Star Formation III. A catalogue of northern ultra-compact H II regions(1803.09334)\n\nMarch 25, 2018 astro-ph.GA\nA catalogue of 239 ultra-compact HII regions (UCHIIs) found in the CORNISH survey at 5 GHz and 1.5\" resolution in the region $10^{\\circ} < l < 65^{\\circ}, ~|b| < 1^{\\circ}$ is presented. This is the largest complete and well-selected sample of UCHIIs to date and provides the opportunity to explore the global and individual properties of this key state in massive star formation at multiple wavelengths. The nature of the candidates was validated, based on observational properties and calculated spectral indices, and the analysis is presented in this work. The physical sizes, luminosities and other physical properties were computed by utilising literature distances or calculating the distances whenever a value was not available. The near- and mid-infrared extended source fluxes were measured and the extinctions towards the UCHIIs were computed. The new results were combined with available data at longer wavelengths and the spectral energy distributions (SEDs) were reconstructed for 177 UCHIIs. The bolometric luminosities obtained from SED fitting are presented. By comparing the radio flux densities to previous observational epochs, we find about 5% of the sources appear to be time variable. This first high-resolution area survey of the Galactic plane shows that the total number of UCHIIs in the Galaxy is ~ 750 - a factor of 3-4 fewer than found in previous large area radio surveys. It will form the basis for future tests of models of massive star formation.\n\u2022 ### An Infrared Census of DUST in Nearby Galaxies with Spitzer (DUSTiNGS). IV. Discovery of High-Redshift AGB Analogs(1711.02129)\n\nNov. 6, 2017 astro-ph.SR\nThe survey for DUST in Nearby Galaxies with Spitzer (DUSTiNGS) identified several candidate Asymptotic Giant Branch (AGB) stars in nearby dwarf galaxies and showed that dust can form even in very metal-poor systems (Z ~ 0.008 $Z_\\odot$). Here, we present a follow-up survey with WFC3\/IR on the Hubble Space Telescope (HST), using filters that are capable of distinguishing carbon-rich (C-type) stars from oxygen-rich (M-type) stars: F127M, F139M, and F153M. We include six star-forming DUSTiNGS galaxies (NGC 147, IC 10, Pegasus dIrr, Sextans B, Sextans A, and Sag DIG), all more metal-poor than the Magellanic Clouds and spanning 1 dex in metallicity. We double the number of dusty AGB stars known in these galaxies and find that most are carbon rich. We also find 26 dusty M-type stars, mostly in IC 10. Given the large dust excess and tight spatial distribution of these M-type stars, they are most likely on the upper end of the AGB mass range (stars undergoing Hot Bottom Burning). Theoretical models do not predict significant dust production in metal-poor M-type stars, but we see evidence for dust excess around M-type stars even in the most metal-poor galaxies in our sample (12+log(O\/H) = 7.26-7.50). The low metallicities and inferred high stellar masses (up to ~10 $M_\\odot$) suggest that AGB stars can produce dust very early in the evolution of galaxies (~30 Myr after they form), and may contribute significantly to the dust reservoirs seen in high-redshift galaxies.\n\u2022 ### The SAGE-Spec Spitzer Legacy program: The life-cycle of dust and gas in the Large Magellanic Cloud. Point source classification III(1705.02709)\n\nMay 7, 2017 astro-ph.GA, astro-ph.SR\nThe Infrared Spectrograph (IRS) on the {\\em Spitzer Space Telescope} observed nearly 800 point sources in the Large Magellanic Cloud (LMC), taking over 1,000 spectra. 197 of these targets were observed as part of the Sage-Spec Spitzer Legacy program; the remainder are from a variety of different calibration, guaranteed time and open time projects. We classify these point sources into types according to their infrared spectral features, continuum and spectral energy distribution shape, bolometric luminosity, cluster membership, and variability information, using a decision-tree classification method. We then refine the classification using supplementary information from the astrophysical literature. We find that our IRS sample is comprised substantially of YSO and H\\,{\\sc ii} regions, post-Main Sequence low-mass stars: (post-)AGB stars and planetary nebulae and massive stars including several rare evolutionary types. Two supernova remnants, a nova and several background galaxies were also observed. We use these classifications to improve our understanding of the stellar populations in the Large Magellanic Cloud, study the composition and characteristics of dust species in a variety of LMC objects, and to verify the photometric classification methods used by mid-IR surveys. We discover that some widely-used catalogues of objects contain considerable contamination and others are missing sources in our sample.\n\u2022 ### The very fast evolution of Sakurai's object(1701.06804)\n\nJan. 24, 2017 astro-ph.SR\nV4334 Sgr (a.k.a. Sakurai's object) is the central star of an old planetary nebula that underwent a very late thermal pulse a few years before its discovery in 1996. We have been monitoring the evolution of the optical emission line spectrum since 2001. The goal is to improve the evolutionary models by constraining them with the temporal evolution of the central star temperature. In addition the high resolution spectral observations obtained by X-shooter and ALMA show the temporal evolution of the different morphological components.\n\u2022 ### Early Science with the Large Millimetre Telescope: Molecules in the Extreme Outflow of a proto-Planetary Nebula(1701.01179)\n\nJan. 4, 2017 astro-ph.GA\nExtremely high velocity emission likely related to jets is known to occur in some proto-Planetary Nebulae. However, the molecular complexity of this kinematic component is largely unknown. We observed the known extreme outflow from the proto-Planetary Nebula IRAS 16342-3814, a prototype water fountain, in the full frequency range from 73 to 111 GHz with the RSR receiver on the Large Millimetre Telescope. We detected the molecules SiO, HCN, SO, and $^{13}$CO. All molecular transitions, with the exception of the latter are detected for the first time in this source, and all present emission with velocities up to a few hundred km s$^{-1}$. IRAS 16342-3814 is therefore the only source of this kind presenting extreme outflow activity simultaneously in all these molecules, with SO and SiO emission showing the highest velocities found of these species in proto-Planetary Nebulae. To be confirmed is a tentative weak SO component with a FWHM $\\sim$ 700 km s$^{-1}$. The extreme outflow gas consists of dense gas (n$_{\\rm H_2} >$ 10$^{4.8}$--10$^{5.7}$ cm$^{-3}$), with a mass larger than $\\sim$ 0.02--0.15 M$_{\\odot}$. The relatively high abundances of SiO and SO may be an indication of an oxygen-rich extreme high velocity gas.\n\u2022 ### (sub)Millimeter Emission Lines of Molecules in Born-again Stars(1612.07037)\n\nDec. 21, 2016 astro-ph.GA, astro-ph.SR\nThe detection and study of molecular gas in born-again stars would be of great importance to understand their composition and chemical evolution. In addition, the molecular emission would be an invaluable tool to explore the physical conditions, kinematics and formation of asymmetric structures in the circumstellar envelopes of these evolved stars. However, until now, all attempts to detect molecular emission from the cool material around born-again stars have failed. We carried out observations using the APEX and IRAM 30m telescopes to search for molecular emission toward four well studied born-again stars, V4334 Sgr, V605 Aql, A30 and A78, that are thought to represent an evolutionary sequence. We detected for the first time emission from HCN and H$^{13}$CN molecules toward V4334 Sgr, and CO emission in V605 Aql. No molecular emission was detected above the noise level toward A30 and A78. A first estimate of the H$^{12}$CN\/H$^{13}$CN abundance ratio in the circumstellar environment of V4334 Sgr is $\\approx$3, which is similar to the value of the $^{12}$C\/$^{13}$C ratio measured from other observations. We derived a rotational temperature of $T_{\\rm rot}$=13$\\pm1$ K, and a total column density of $N_{{\\rm HCN}}$=1.6$\\pm0.1\\times$10$^{16}$ cm$^{-2}$ for V4334 Sgr. This result sets a lower limit on the amount of hydrogen that was ejected into the wind during the born-again event of this source. For V605 Aql, we obtained a lower limit for the integrated line intensities $I_{^{12}\\rm C}$\/$I_{^{13}\\rm C}$>4.\n\u2022 ### The ALMA detection of CO rotational line emission in AGB stars in the Large Magellanic Cloud(1609.09647)\n\nSept. 30, 2016 astro-ph.GA, astro-ph.SR\nContext: Low- and intermediate-mass stars lose most of their stellar mass at the end of their lives on the asymptotic giant branch (AGB). Determining gas and dust mass-loss rates (MLRs) is important in quantifying the contribution of evolved stars to the enrichment of the interstellar medium. Aims: Attempt to, for the first time, spectrally resolve CO thermal line emission in a small sample of AGB stars in the Large Magellanic Cloud. Methods: ALMA was used to observe 2 OH\/IR stars and 4 carbon stars in the LMC in the CO J= 2-1 line. Results: We present the first measurement of expansion velocities in extragalactic carbon stars. All four C-stars are detected and wind expansion velocities and stellar velocities are directly measured. Mass-loss rates are derived from modelling the spectral energy distribution and Spitzer\/IRS spectrum with the DUSTY code. Gas-to-dust ratios are derived that make the predicted velocities agree with the observed ones. The expansion velocities and MLRs are compared to a Galactic sample of well-studied relatively low MLRs stars supplemented with \"extreme\" C-stars that have properties more similar to the LMC targets. Gas MLRs derived from a simple formula are significantly smaller than derived from the dust modelling, indicating an order of magnitude underestimate of the estimated CO abundance, time-variable mass loss, or that the CO intensities in LMC stars are lower than predicted by the formula derived for Galactic objects. This could be related to a stronger interstellar radiation field in the LMC. Conclusions: Although the LMC sample is small and the comparison to Galactic stars is non-trivial because of uncertainties in their distances it appears that for C stars the wind expansion velocities in the LMC are lower than in the solar neighbourhood, while the MLRs appear similar. This is in agreement with dynamical dust-driven wind models.\n\u2022 ### Post-AGB evolution much faster than previously thought(1609.08680)\n\nSept. 27, 2016 astro-ph.GA, astro-ph.SR\nFor 32 central stars of PNe we present their parameters interpolated among the new evolutionary sequences. The derived stellar final masses are confined between 0.53 and 0.58 $M_\\odot$ in good agreement with the peak in the white dwarf mass distribution. Consequently, the inferred star formation history of the Galactic bulge is well restricted between 3 and 11 Gyr and is compatible with other published studies. The new evolutionary tracks proved a very good as a tool for analysis of late stages of stars life. The result provide a compelling confirmation of the accelerated post-AGB evolution.\n\u2022 ### First Detection of $^3$He$^+$ in the Planetary Nebula IC$\\,$418(1604.02679)\n\nApril 10, 2016 astro-ph.GA, astro-ph.SR\nThe $^3$He isotope is important to many fields of astrophysics, including stellar evolution, chemical evolution, and cosmology. The isotope is produced in low-mass stars which evolve through the planetary nebula (PN) phase. $^3$He abundances in PNe can help test models of the chemical evolution of the Galaxy. We present the detection of the $^3$He$^+$ emission line using the single dish Deep Space Station 63, towards the PN IC$\\,$418. We derived a $^3$He\/H abundance in the range 1.74$\\pm$0.8$\\times$10$^{-3}$ to 5.8$\\pm$1.7$\\times$10$^{-3}$, depending on whether part of the line arises in an outer ionized halo. The lower value for $^3$He\/H ratio approaches values predicted by stellar models which include thermohaline mixing, but requires that large amounts of $^3$He are produced inside low-mass stars which enrich the interstellar medium (ISM). However, this over-predicts the $^3$He abundance in HII regions, the ISM, and proto-solar grains, which is known to be of the order of 10$^{-5}$. This discrepancy questions our understanding of the evolution of the $^3$He, from circumstellar environments to the ISM.\n\u2022 ### EU Del: exploring the onset of pulsation-driven winds in giant stars(1512.04695)\n\nDec. 15, 2015 astro-ph.SR\nWe explore the wind-driving mechanism of giant stars through the nearby (117 pc), intermediate-luminosity ($L \\approx 1600$ L$_\\odot$) star EU Del (HIP 101810, HD 196610). APEX observations of the CO (3--2) and (2--1) transitions are used to derive a wind velocity of 9.51 $\\pm$ 0.02 km s$^{-1}$, a $^{12}$C\/$^{13}$C ratio of 14 $^{+9}_{-4}$, and a mass-loss rate of a few $\\times$ 10$^{-8}$ M$_\\odot$ yr$^{-1}$. From published spectra, we estimate that the star has a metallicity of [Fe\/H] = --0.27 $\\pm$ $\\sim$0.30 dex. The star's dusty envelope lacks a clear 10-$\\mu$m silicate feature, despite the star's oxygen-rich nature. Radiative transfer modelling cannot fit a wind acceleration model which relies solely on radiation pressure on condensing dust. We compare our results to VY Leo (HIP 53449), a star with similar temperature and luminosity, but different pulsation properties. We suggest the much stronger mass loss from EU Del may be driven by long-period stellar pulsations, due to its potentially lower mass. We explore the implications for the mass-loss rate and wind velocities of other stars.\n\u2022 ### 3D pyCloudy modelling of bipolar planetary nebulae: evidence for fast fading of the lobes(1509.08017)\n\nSept. 26, 2015 astro-ph.SR\nWe apply an axially symmetric pseudo-3D photoionization model, pyCloudy, to derive the structures of 6 bipolar nebulae and 2 suggested post-bipolars in a quest to constrain the bipolar planetary nebulae evolution. HST images and VLT\/UVES spectroscopy are used for the modelling. The targets are located in the direction of the Galactic bulge. A 3D model structure is used as input to the photoionization code, so as to fit the HST images. Line profiles of different ions constrain the velocity field. The model and associated velocity fields allow us to derive masses, velocities, and ages. The 3D models find much lower ionized masses than required in 1D models: ionized masses are reduced by factors of 2-7. The selected bi-lobed planetary nebulae show a narrow range of ages: the averaged radii and velocities result in values between 1300 and 2000 yr. The lobes are fitted well with velocities linearly increasing with radius. These Hubble-type flows have been found before, and suggest that the lobes form at a defined point in time. The lobes appear to be slightly younger than the main (host) nebulae, by ~500 yr, they seem to form at an early phase of PN evolution, and fade after 1-2 kyr. We find that 30-35% of bulge PNe pass through a bipolar phase.\n\u2022 ### Spitzer Infrared Spectrographic point source classification in the Small Magellanic Cloud(1505.04499)\n\nMay 31, 2015 astro-ph.SR\nThe Magellanic clouds are uniquely placed to study the stellar contribution to dust emission. Individual stars can be resolved in these systems even in the mid-infrared, and they are close enough to allow detection of infrared excess caused by dust.We have searched the Spitzer Space Telescope data archive for all Infrared Spectrograph (IRS) staring-mode observations of the Small Magellanic Cloud (SMC) and found that 209 Infrared Array Camera (IRAC) point sources within the footprint of the Surveying the Agents of Galaxy Evolution in the Small Magellanic Cloud (SAGE-SMC) Spitzer Legacy programme were targeted, within a total of 311 staring mode observations. We classify these point sources using a decision tree method of object classification, based on infrared spectral features, continuum and spectral energy distribution shape, bolometric luminosity, cluster membership and variability information. We find 58 asymptotic giant branch (AGB) stars, 51 young stellar objects (YSOs), 4 post-AGB objects, 22 Red Supergiants (RSGs), 27 stars (of which 23 are dusty OB stars), 24 planetary nebulae (PNe), 10Wolf-Rayet (WR) stars, 3 Hii regions, 3 R Coronae Borealis (R CrB) stars, 1 Blue Supergiant and 6 other objects, including 2 foreground AGB stars. We use these classifications to evaluate the success of photometric classification methods reported in the literature.\n\u2022 ### Witnessing the Emergence of a Carbon Star(1504.03349)\n\nApril 13, 2015 astro-ph.SR\nDuring the late stages of their evolution, Sun-like stars bring the products of nuclear burning to the surface. Most of the carbon in the Universe is believed to originate from stars with masses up to a few solar masses. Although there is a chemical dichotomy between oxygen-rich and carbon-rich evolved stars, the dredge-up itself has never been directly observed. In the last three decades, however, a few stars have been shown to display both carbon- and oxygen-rich material in their circumstellar envelopes. Two models have been proposed to explain this dual chemistry: one postulates that a recent dredge-up of carbon produced by nucleosynthesis inside the star during the Asymptotic Giant Branch changed the surface chemistry of the star. The other model postulates that oxygen-rich material exists in stable keplerian rotation around the central star. The two models make contradictory, testable, predictions on the location of the oxygen-rich material, either located further from the star than the carbon-rich gas, or very close to the star in a stable disk. Using the Faint Object InfraRed CAmera (FORCAST) instrument on board the Stratospheric Observatory for Infrared Astronomy (SOFIA) Telescope, we obtained images of the carbon-rich planetary nebula (PN) BD+30 3639 which trace both carbon-rich polycyclic aromatic hydrocarbons (PAHs) and oxygen-rich silicate dust. With the superior spectral coverage of SOFIA, and using a 3D photoionisation and dust radiative transfer model we prove that the O-rich material is distributed in a shell in the outer parts of the nebula, while the C-rich material is located in the inner parts of the nebula. These observations combined with the model, suggest a recent change in stellar surface composition for the double chemistry in this object. This is evidence for dredge-up occurring ~1000yr ago.\n\u2022 ### Investigating the nature of the Fried Egg nebula: CO mm-line and optical spectroscopy of IRAS 17163-3907(1501.03362)\n\nJan. 14, 2015 astro-ph.SR\nThrough CO mm-line and optical spectroscopy, we investigate the properties of the Fried Egg nebula IRAS 17163-3907, which has recently been proposed to be one of the rare members of the yellow hypergiant class. The CO J=2-1 and J=3-2 emission arises from a region within 20\" of the star and is clearly associated with the circumstellar material. The CO lines show a multi-component asymmetrical profile, and an unexpected velocity gradient is resolved in the east-west direction, suggesting a bipolar outflow. This is in contrast with the apparent symmetry of the dust envelope as observed in the infrared. The optical spectrum of IRAS 17163-3907 between 5100 and 9000 {\\AA} was compared with that of the archetypal yellow hypergiant IRC+10420 and was found to be very similar. These results build on previous evidence that IRAS 17163-3907 is a yellow hypergiant.\n\u2022 ### The relationship between polycyclic aromatic hydrocarbon emission and far-infrared dust emission from NGC 2403 and M83(1412.7344)\n\nDec. 23, 2014 astro-ph.GA\nWe examine the relation between polycyclic aromatic hydrocarbon (PAH) emission at 8 microns and far-infrared emission from hot dust grains at 24 microns and from large dust grains at 160 and 250 microns in the nearby spiral galaxies NGC 2403 and M83 using data from the Spitzer Space Telescope and Herschel Space Observatory. We find that the PAH emission in NGC 2403 is better correlated with emission at 250 microns from dust heated by the diffuse interstellar radiation field (ISRF) and that the 8\/250 micron surface brightness ratio is well-correlated with the stellar surface brightness as measured at 3.6 microns. This implies that the PAHs in NGC 2403 are intermixed with cold large dust grains in the diffuse interstellar medium (ISM) and that the PAHs are excited by the diffuse ISRF. In M83, the PAH emission appears more strongly correlated with 160 micron emission originating from large dust grains heated by star forming regions. However, the PAH emission in M83 is low where the 24 micron emission peaks within star forming regions, and enhancements in the 8\/160 micron surface brightness ratios appear offset relative to the dust and the star forming regions within the spiral arms. This suggests that the PAHs observed in the 8 micron band are not excited locally within star forming regions but either by light escaping non-axisymmetrically from star forming regions or locally by young, non-photoionising stars that have migrated downstream from the spiral density waves. The results from just these two galaxies show that PAHs may be excited by different stellar populations in different spiral galaxies.\n\u2022 ### A Spitzer Space Telescope survey of extreme Asymptotic Giant Branch stars in M32(1410.4504)\n\nOct. 16, 2014 astro-ph.GA, astro-ph.SR\nWe investigate the population of cool, evolved stars in the Local Group dwarf elliptical galaxy M32, using Infrared Array Camera observations from the Spitzer Space Telescope. We construct deep mid-infrared colour-magnitude diagrams for the resolved stellar populations within 3.5 arcmin of M32's centre, and identify those stars that exhibit infrared excess. Our data is dominated by a population of luminous, dust-producing stars on the asymptotic giant branch (AGB) and extend to approximately 3 mag below the AGB tip. We detect for the first time a sizeable population of `extreme' AGB stars, highly enshrouded by circumstellar dust and likely completely obscured at optical wavelengths. The total dust-injection rate from the extreme AGB candidates is measured to be $7.5 \\times 10^{-7}$ ${\\rm M}_{\\odot} \\, {\\rm yr}^{-1}$, corresponding to a gas mass-loss rate of $1.5 \\times 10^{-4}$ ${\\rm M}_{\\odot} \\, {\\rm yr}^{-1}$. These extreme stars may be indicative of an extended star-formation epoch between 0.2 and 5 Gyr ago.\n\u2022 ### The Second Data Release of the INT Photometric H-Alpha Survey of the Northern Galactic Plane (IPHAS DR2)(1406.4862)\n\nAug. 12, 2014 astro-ph.SR, astro-ph.IM\nThe INT\/WFC Photometric H-Alpha Survey of the Northern Galactic Plane (IPHAS) is a 1800 square degrees imaging survey covering Galactic latitudes |b| < 5 deg and longitudes l = 30 to 215 deg in the r, i and H-alpha filters using the Wide Field Camera (WFC) on the 2.5-metre Isaac Newton Telescope (INT) in La Palma. We present the first quality-controlled and globally-calibrated source catalogue derived from the survey, providing single-epoch photometry for 219 million unique sources across 92% of the footprint. The observations were carried out between 2003 and 2012 at a median seeing of 1.1 arcsec (sampled at 0.33 arcsec\/pixel) and to a mean 5\\sigma-depth of 21.2 (r), 20.0 (i) and 20.3 (H-alpha) in the Vega magnitude system. We explain the data reduction and quality control procedures, describe and test the global re-calibration, and detail the construction of the new catalogue. We show that the new calibration is accurate to 0.03 mag (rms) and recommend a series of quality criteria to select the most reliable data from the catalogue. Finally, we demonstrate the ability of the catalogue's unique (r-Halpha, r-i) diagram to (1) characterise stellar populations and extinction regimes towards different Galactic sightlines and (2) select H-alpha emission-line objects. IPHAS is the first survey to offer comprehensive CCD photometry of point sources across the Galactic Plane at visible wavelengths, providing the much-needed counterpart to recent infrared surveys.\n\u2022 ### A study of rotating globular clusters - the case of the old, metal-poor globular cluster NGC 4372(1406.1552)\n\nJune 6, 2014 astro-ph.GA\nAims: We present the first in-depth study of the kinematic properties and derive the structural parameters of NGC 4372 based on the fit of a Plummer profile and a rotating, physical model. We explore the link between internal rotation to different cluster properties and together with similar studies of more GCs, we put these in the context of globular cluster formation and evolution. Methods: We present radial velocities for 131 cluster member stars measured from high-resolution FLAMES\/GIRAFFE observations. Their membership to the GC is additionally confirmed from precise metallicity estimates. Using this kinematic data set we build a velocity dispersion profile and a systemic rotation curve. Additionally, we obtain an elliptical number density profile of NGC 4372 based on optical images using a MCMC fitting algorithm. From this we derive the cluster's half-light radius and ellipticity as r_h=3.4'+\/-0.04' and e=0.08+\/-0.01. Finally, we give a physical interpretation of the observed morphological and kinematic properties of this GC by fitting an axisymmetric, differentially rotating, dynamical model. Results: Our results show that NGC 4372 has an unusually high ratio of rotation amplitude to velocity dispersion (1.2 vs. 4.5 km\/s) for its metallicity. This, however, puts it in line with two other exceptional, very metal-poor GCs - M 15 and NGC 4590. We also find a mild flattening of NGC 4372 in the direction of its rotation. Given its old age, this suggests that the flattening is indeed caused by the systemic rotation rather than tidal interactions with the Galaxy. Additionally, we estimate the dynamical mass of the GC M_dyn=2.0+\/-0.5 x 10^5 M_Sun based on the dynamical model, which constrains the mass-to-light ratio of NGC 4372 between 1.4 and 2.3 M_Sun\/L_Sun, representative of an old, purely stellar population.\n\u2022 ### Herschel Planetary Nebula Survey (HerPlaNS) - First Detection of OH+ in Planetary Nebulae(1404.2431)\n\nMay 14, 2014 astro-ph.GA\nWe report the first detections of OH$^+$ emission in planetary nebulae (PNe). As part of an imaging and spectroscopy survey of 11 PNe in the far-IR using the PACS and SPIRE instruments aboard the Herschel Space Observatory, we performed a line survey in these PNe over the entire spectral range between 51 and 672$\\mu$m to look for new detections. OH$^+$ rotational emission lines at 152.99, 290.20, 308.48, and 329.77$\\mu$m were detected in the spectra of three planetary nebulae: NGC 6445, NGC 6720, and NGC 6781. Excitation temperatures and column densities derived from these lines are in the range of 27 to 47 K and 2$\\times$10$^{10}$ to 4 $\\times$10$^{11}$ cm$^{-2}$, respectively. In PNe, the OH+ rotational line emission appears to be produced in the photodissociation region (PDR) in these objects. The emission of OH+ is observed only in PNe with hot central stars (T$_{eff}$ > 100000 K), suggesting that high-energy photons may play a role in the OH+ formation and its line excitation in these objects, as it seems to be the case for ultraluminous galaxies.\n\u2022 ### The evolving spectrum of the planetary nebula Hen 2-260(1405.0800)\n\nMay 5, 2014 astro-ph.SR\nWe analysed the planetary nebula Hen 2-260 using optical spectroscopy and photometry. We compared our observations with the data from literature to search for evolutionary changes. The nebular line fluxes were modelled with the Cloudy photoionization code to derive the stellar and nebular parameters. The planetary nebula shows a complex structure and possibly a bipolar outflow. The nebula is relatively dense and young. The central star is just starting $\\rm O^+$ ionization ($\\rm T_{eff} \\approx 30,000 \\, K$). Comparison of our observations with literature data indicates a 50% increase of the [OIII] 5007 \\AA\\ line flux between 2001 and 2012. We interpret it as the result of the progression of the ionization of $\\rm O^{+}$. The central star evolves to higher temperatures at a rate of $\\rm 45 \\pm 7\\,K\\, yr^{-1}$. The heating rate is consistent with a final mass of $\\rm 0.626 ^{+0.003}_{-0.005} \\, M_{\\odot}$ or $\\rm 0.645 ^{+0.008}_{-0.008} \\, M_{\\odot}$ for two different sets of post-AGB evolutionary tracks from literature. The photometric monitoring of Hen 2-260 revealed variations on a timescale of hours or days. The variability may be caused by pulsations of the star. The temperature evolution of the central star can be traced using spectroscopic observations of the surrounding planetary nebula spanning a timescale of roughly a decade. This allows us to precisely determine the stellar mass, since the pace of the temperature evolution depends critically on the core mass. The kinematical age of the nebula is consistent with the age obtained from the evolutionary track. The final mass of the central star is close to the mass distribution peak for central stars of planetary nebulae found in other studies. The object belongs to a group of young central stars of planetary nebulae showing photometric variability.\n\u2022 ### Accelerated post-AGB evolution, initial-final mass relations, and the star-formation history of the Galactic bulge(1404.6353)\n\nApril 25, 2014 astro-ph.SR\nWe study the star-formation history of the Galactic bulge, as derived from the age distribution of the central stars of planetary nebulae that belong to this stellar population. The high resolution imaging and spectroscopic observations of 31 compact planetary nebulae are used to derive their central star masses. The Bloecker tracks with the cluster IFMR result in ages, which are unexpectedly young. We find that the Bloecker post-AGB tracks need to be accelerated by a factor of three to fit the local white dwarf masses. This acceleration extends the age distribution. We adjust the IFMR as a free parameter to map the central star ages on the full age range of bulge stellar populations. This fit requires a steeper IFMR than the cluster relation. We find a star-formation rate in the Galactic bulge, which is approximately constant between 3 and 10 Gyr ago. The result indicates that planetary nebulae are mainly associated with the younger and more metal-rich bulge populations. The constant rate of star-formation between 3 and 10 Gyr agrees with suggestions that the metal-rich component of the bulge is formed during an extended process, such as a bar interaction.\n\u2022 ### PAH Formation in O-rich Planetary Nebulae(1403.1856)\n\nMarch 7, 2014 astro-ph.SR\nPolycyclic aromatic hydrocarbons (PAHs) have been observed in O-rich planetary nebulae towards the Galactic Bulge. This combination of oxygen-rich and carbon-rich material, known as dual-dust or mixed chemistry, is not expected to be seen around such objects. We recently proposed that PAHs could be formed from the photodissociation of CO in dense tori. In this work, using VISIR\/VLT, we spatially resolved the emission of the PAH bands and ionised emission from the [SIV] line, confirming the presence of dense central tori in all the observed O-rich objects. Furthermore, we show that for most of the objects, PAHs are located at the outer edge of these dense\/compact tori, while the ionised material is mostly present in the inner parts of these tori, consistent with our hypothesis for the formation of PAHs in these systems. The presence of a dense torus has been strongly associated with the action of a central binary star and, as such, the rich chemistry seen in these regions may also be related to the formation of exoplanets in post-common-envelope binary systems.\n\u2022 ### The VST Photometric Halpha Survey of the Southern Galactic Plane and Bulge (VPHAS+)(1402.7024)\n\nMarch 2, 2014 astro-ph.GA\nThe VST Photometric Halpha Survey of the Southern Galactic Plane and Bulge (VPHAS+) is surveying the southern Milky Way in u, g, r, i and Halpha at 1 arcsec angular resolution. Its footprint spans the Galactic latitude range -5 < b < +5 at all longitudes south of the celestial equator. Extensions around the Galactic Centre to Galactic latitudes +\/-10 bring in much of the Galactic Bulge. This ESO public survey, begun on 28th December 2011, reaches down to 20th magnitude (10-sigma) and will provide single-epoch digital optical photometry for around 300 million stars. The observing strategy and data pipelining is described, and an appraisal of the segmented narrowband Halpha filter in use is presented. Using model atmospheres and library spectra, we compute main-sequence (u - g), (g - r), (r - i) and (r - Halpha) stellar colours in the Vega system. We report on a preliminary validation of the photometry using test data obtained from two pointings overlapping the Sloan Digital Sky Survey. An example of the (u - g, g - r) and (r - Halpha, r - i) diagrams for a full VPHAS+ survey field is given. Attention is drawn to the opportunities for studies of compact nebulae and nebular morphologies that arise from the image quality being achieved. The value of the u band as the means to identify planetary-nebula central stars is demonstrated by the discovery of the central star of NGC 2899 in survey data. Thanks to its excellent imaging performance, the VST\/OmegaCam combination used by this survey is a perfect vehicle for automated searches for reddened early-type stars, and will allow the discovery and analysis of compact binaries, white dwarfs and transient sources.\n\u2022 ### A new HCN maser in IRAS 15082-4808(1402.2895)\n\nFeb. 13, 2014 astro-ph.GA, astro-ph.SR\nWe have identified a new vibrational HCN maser at 89.087 GHz in the asymptotic giant branch (AGB) star IRAS 15082-4808, a maser which is thought to trace the innermost region of an AGB envelope. The observations of this maser at three epochs are presented: two positive detections and one null detection. The line profile has varied between the positive detections, as has the intensity of the maser. The major component of the maser is found to be offset by -2.0+\/-0.9 km\/s with respect to the systemic velocity of the envelope, as derived from the 88.631 GHz transition of HCN. Similar blueshifts are measured in the other 9 sources where this maser has been detected. Maser variability with pulsation phase has been investigated for the first time using the 10 stars now available. Comparisons with AGB model atmospheres constrain the position of the formation region of the maser to the region between the pulsation shocks and the onset of dust acceleration, between 2 and 4 stellar radii.","date":"2021-02-26 09:56:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6150469779968262, \"perplexity\": 2715.4044894840913}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178356456.56\/warc\/CC-MAIN-20210226085543-20210226115543-00181.warc.gz\"}"} | null | null |
Q: How to change text in an EditText before and after typing I have an EditText that by default has this string in it "[ ]".
What I want is that when the user selects the EditText the brackets and space all disappear. Then after they are done editing I would like to place the brackets around their edit.
So, for example, if they typed "123" then when they are done typing the EditText will show "[123]".
I tried following the answer here, EditText not updated after text changed in the TextWatcher, by doing setText on the EditText from the afterTextChanged, but that throws an error.
Thanks for any help.
Here's my code:
I just added the xml that creates the first line of the program since that's the line I'm having problems with.
activity_main.xml
<?xml version="1.0" encoding="utf-8"?>
<LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
android:layout_width="match_parent"
android:layout_height="match_parent"
android:orientation="vertical" >
<!-- Top Information -->
<LinearLayout
android:layout_width="match_parent"
android:layout_height="wrap_content"
android:orientation="horizontal" >
<LinearLayout
android:layout_width="0dp"
android:layout_height="wrap_content"
android:layout_weight=".70"
android:orientation="vertical" >
<LinearLayout
android:layout_width="match_parent"
android:layout_height="wrap_content"
android:orientation="horizontal" >
<EditText
android:id="@+id/main_character_name_edit"
android:layout_width="0dp"
android:layout_height="wrap_content"
android:layout_weight=".25"
android:hint="@string/main_character_name_edit_hint" />
<EditText
android:id="@+id/main_player_name_edit"
android:layout_width="0dp"
android:layout_height="wrap_content"
android:layout_weight=".25"
android:hint="@string/main_player_name_edit_hint" />
<TextView
android:layout_width="0dp"
android:layout_height="wrap_content"
android:layout_weight=".05"
android:gravity="right"
android:paddingRight="7dp"
android:text="@string/main_tl_view" />
<EditText
android:id="@+id/main_tl_edit"
android:layout_width="0dp"
android:layout_height="wrap_content"
android:layout_weight=".10"
android:gravity="center_horizontal"
android:inputType="number" />
<EditText
android:id="@+id/main_tl_cost_edit"
android:layout_width="0dp"
android:layout_height="wrap_content"
android:layout_weight=".05"
android:gravity="center_horizontal"
android:inputType="numberSigned"
android:text="@string/cost_edit" />
</LinearLayout>
<EditText
android:id="@+id/main_character_description_edit"
android:layout_width="match_parent"
android:layout_height="wrap_content"
android:gravity="top|left"
android:hint="@string/main_character_description_edit"
android:lines="2"
android:maxLines="10"
android:minLines="2"
android:scrollbars="vertical" />
</LinearLayout>
<!-- Points Box -->
<LinearLayout
android:layout_width="0dp"
android:layout_height="wrap_content"
android:layout_weight=".30"
android:background="@drawable/border"
android:orientation="vertical"
android:padding="10dp" >
<TextView
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:layout_gravity="center_horizontal"
android:text="@string/main_points_view" />
<LinearLayout
android:layout_width="match_parent"
android:layout_height="wrap_content"
android:orientation="horizontal" >
<TextView
android:layout_width="0dp"
android:layout_height="wrap_content"
android:layout_weight=".25"
android:text="@string/main_points_total_view" />
<EditText
android:id="@+id/main_points_total_edit"
android:layout_width="0dp"
android:layout_height="wrap_content"
android:layout_weight=".75"
android:gravity="center_horizontal"
android:inputType="number" />
</LinearLayout>
<LinearLayout
android:layout_width="match_parent"
android:layout_height="wrap_content"
android:orientation="horizontal" >
<TextView
android:layout_width="0dp"
android:layout_height="wrap_content"
android:layout_weight=".25"
android:text="@string/main_points_unspent_view" />
<EditText
android:id="@+id/main_points_unspent_edit"
android:layout_width="0dp"
android:layout_height="wrap_content"
android:layout_weight=".75"
android:gravity="center_horizontal"
android:inputType="number" />
</LinearLayout>
</LinearLayout>
</LinearLayout>
</LinearLayout>
MainActivity.java
package com.gmail.james.grider.gurpscharactersheet;
import android.app.Activity;
import android.content.Intent;
import android.os.Bundle;
import android.text.Editable;
import android.text.TextWatcher;
import android.view.LayoutInflater;
import android.view.Menu;
import android.view.MenuItem;
import android.widget.ArrayAdapter;
import android.widget.EditText;
import android.widget.Spinner;
import android.widget.TableLayout;
import android.widget.TableRow;
public class MainActivity extends Activity {
/* -- Top Info -- */
private EditText mMainCharacterEdit, mMainPlayerEdit, mMainTlEdit,
mMainTlCostEdit, mMainDescriptionEdit;
/* -- Points Box -- */
private EditText mMainPointsTotalEdit, mMainPointsUnspentEdit;
/* -- Stats Derived Box -- */
private EditText mMainFirstStatsDerivedHpTotalEdit,
mMainFirstStatsDerivedHpCostEdit,
mMainFirstStatsDerivedHpCurrentEdit,
mMainFirstStatsDerivedFpTotalEdit,
mMainFirstStatsDerivedFpCostEdit,
mMainFirstStatsDerivedFpCurrentEdit,
mMainFirstStatsDerivedWillEdit, mMainFirstStatsDerivedWillCostEdit,
mMainFirstStatsDerivedPerEdit, mMainFirstStatsDerivedPerCostEdit;
/* -- Stats Boxes -- */
private EditText mMainSecondStStEdit, mMainSecondStCostEdit,
mMainSecondStLastSkillEdit, mMainSecondStLastLvlEdit,
mMainSecondStLastRelEdit, mMainSecondStLastCostEdit;
private TableLayout mStatStSkillTable;
private EditText mMainSecondDxDxEdit, mMainSecondDxCostEdit,
mMainSecondDxLastSkillEdit, mMainSecondDxLastLvlEdit,
mMainSecondDxLastRelEdit, mMainSecondDxLastCostEdit;
private TableLayout mStatDxSkillTable;
private EditText mMainSecondIqIqEdit, mMainSecondIqCostEdit,
mMainSecondIqLastSkillEdit, mMainSecondIqLastLvlEdit,
mMainSecondIqLastRelEdit, mMainSecondIqLastCostEdit;
private TableLayout mStatIqSkillTable;
private EditText mMainSecondHtHtEdit, mMainSecondHtCostEdit,
mMainSecondHtLastSkillEdit, mMainSecondHtLastLvlEdit,
mMainSecondHtLastRelEdit, mMainSecondHtLastCostEdit;
private TableLayout mStatHtSkillTable;
/* -- Secondary Stats Box -- */
private EditText mMainThirdSecondaryLiftEdit, mMainThirdSecondarySpeedEdit,
mMainThirdSecondarySpeedCostEdit, mMainThirdSecondaryBasicMoveEdit,
mMainThirdSecondaryBasicMoveCostEdit, mMainThirdSecondaryMoveEdit;
private Spinner mMainThirdSecondaryDamageThrustSpinner,
mMainThirdSecondaryDamageSwingSpinner,
mMainThirdSecondaryEncumbranceSpinner;
/* -- Defense Box -- */
private EditText mMainFourthDefenseDamageReductionEdit,
mMainFourthDefenseDodgeEdit, mMainFourthDefenseParryEdit,
mMainFourthDefenseBlockEdit;
private void updateSkillRow(int stat, boolean action) {
TableRow row;
if (action) {
switch (stat) {
case Info.ST:
row = (TableRow) mStatStSkillTable.getChildAt(mStatStSkillTable
.getChildCount() - 1);
mMainSecondStLastSkillEdit = (EditText) row.getChildAt(0);
mMainSecondStLastLvlEdit = (EditText) row.getChildAt(1);
mMainSecondStLastRelEdit = (EditText) row.getChildAt(2);
mMainSecondStLastCostEdit = (EditText) row.getChildAt(3);
break;
case Info.DX:
break;
case Info.IQ:
break;
case Info.HT:
}
}
}
private void addSkillRow(int stat) {
int skillCount;
LayoutInflater inflater = LayoutInflater.from(MainActivity.this);
TableRow row = (TableRow) inflater.inflate(
R.layout.activity_main_skill_row, null);
switch (stat) {
case Info.ST:
skillCount = mStatStSkillTable.getChildCount();
if (skillCount > 0)
mStatStSkillTable.addView(row, skillCount - 1);
else
mStatStSkillTable.addView(row);
break;
case Info.DX:
skillCount = mStatDxSkillTable.getChildCount();
if (skillCount > 0)
mStatDxSkillTable.addView(row, skillCount - 1);
else
mStatDxSkillTable.addView(row);
break;
case Info.IQ:
skillCount = mStatIqSkillTable.getChildCount();
if (skillCount > 0)
mStatIqSkillTable.addView(row, skillCount - 1);
else
mStatIqSkillTable.addView(row);
break;
case Info.HT:
skillCount = mStatHtSkillTable.getChildCount();
if (skillCount > 0)
mStatHtSkillTable.addView(row, skillCount - 1);
else
mStatHtSkillTable.addView(row);
}
updateSkillRow(stat, Info.ADD);
}
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.activity_main);
mMainTlCostEdit = (EditText) findViewById(R.id.main_tl_cost_edit);
mMainTlCostEdit.addTextChangedListener(new TextWatcher() {
@Override
public void afterTextChanged(Editable s) {
String text = mMainTlCostEdit.getText().toString();
text = "[" + text + "]";
mMainTlCostEdit.setText(text);
}
@Override
public void beforeTextChanged(CharSequence s, int start, int count,
int after) {
}
@Override
public void onTextChanged(CharSequence s, int start, int before,
int count) {
// TODO Auto-generated method stub
}
});
// mMainTlCostEdit.setOnClickListener(new View.OnClickListener() {
//
// @Override
// public void onClick(View v) {
// String text = mMainTlCostEdit.getText().toString();
// text.replace("[", "");
// text.replace("]", "");
// mMainTlCostEdit.setText(text);
// }
// });
mStatStSkillTable = (TableLayout) findViewById(R.id.main_second_st_skills_tablelayout);
addSkillRow(Info.ST);
mMainSecondStLastSkillEdit.addTextChangedListener(new TextWatcher() {
@Override
public void afterTextChanged(Editable s) {
// TODO Auto-generated method stub
}
@Override
public void beforeTextChanged(CharSequence s, int start, int count,
int after) {
// TODO Auto-generated method stub
}
@Override
public void onTextChanged(CharSequence s, int start, int before,
int count) {
String text = mMainSecondStLastSkillEdit.getText().toString();
if (text == "") {
} else {
addSkillRow(Info.ST);
}
}
});
mStatDxSkillTable = (TableLayout) findViewById(R.id.main_second_dx_skills_tablelayout);
mStatIqSkillTable = (TableLayout) findViewById(R.id.main_second_iq_skills_tablelayout);
mStatHtSkillTable = (TableLayout) findViewById(R.id.main_second_ht_skills_tablelayout);
addSkillRow(Info.DX);
addSkillRow(Info.IQ);
addSkillRow(Info.HT);
ArrayAdapter<CharSequence> adapter = ArrayAdapter.createFromResource(
this, R.array.damage_table_array, R.layout.my_spinner_item);
adapter.setDropDownViewResource(android.R.layout.simple_spinner_dropdown_item);
mMainThirdSecondaryDamageThrustSpinner = (Spinner) findViewById(R.id.main_third_secondary_damage_thrust_spinner);
mMainThirdSecondaryDamageThrustSpinner.setAdapter(adapter);
mMainThirdSecondaryDamageThrustSpinner.setSelection(4);
mMainThirdSecondaryDamageSwingSpinner = (Spinner) findViewById(R.id.main_third_secondary_damage_swing_spinner);
mMainThirdSecondaryDamageSwingSpinner.setAdapter(adapter);
mMainThirdSecondaryDamageSwingSpinner.setSelection(6);
mMainThirdSecondaryEncumbranceSpinner = (Spinner) findViewById(R.id.main_third_secondary_enchumbrance_spinner);
adapter = ArrayAdapter.createFromResource(this,
R.array.encumbrance_level_array, R.layout.my_spinner_item);
adapter.setDropDownViewResource(android.R.layout.simple_spinner_dropdown_item);
mMainThirdSecondaryEncumbranceSpinner.setAdapter(adapter);
}
@Override
public boolean onCreateOptionsMenu(Menu menu) {
// Inflate the menu; this adds items to the action bar if it is present.
getMenuInflater().inflate(R.menu.main, menu);
return true;
}
@Override
public boolean onOptionsItemSelected(MenuItem item) {
switch (item.getItemId()) {
case R.id.menu_item_extras:
Intent extrasIntent = new Intent(MainActivity.this,
ExtraActivity.class);
startActivity(extrasIntent);
return true;
case R.id.menu_item_equipment:
Intent equipmentIntent = new Intent(MainActivity.this,
EquipmentActivity.class);
startActivity(equipmentIntent);
return true;
case R.id.menu_item_items:
Intent itemsIntent = new Intent(MainActivity.this,
ItemActivity.class);
startActivity(itemsIntent);
return true;
case R.id.menu_item_options:
Intent optionsIntent = new Intent(MainActivity.this,
OptionListActivity.class);
startActivity(optionsIntent);
return true;
default:
return super.onOptionsItemSelected(item);
}
}
}
A: you can do somthing like:
String temp = EditText.getText().toString();
temp="[temp]";
EditText="[]";
or do you want it to be showed inside the EditText?
you can use OnKeyListener and make the EditText look like you want after some key is pressed..
e.g "ENTER"
EditText=.setText("[temp]");
A: Use a View.OnFocusChangeListener on the EditText. It contains the onFocusChange method which is called when the focus state on the view is changed. For this case, you can set the braces surrounding the text once the EditText goes out of focus and add those back upon the EditText gaining focus.
mMainTlCostEdit.setOnFocusChangeListener(new OnFocusChangeListener() {
@Override
public void onFocusChange(View v, boolean hasFocus) {
if (hasFocus) {
String text = mMainTlCostEdit.getText().toString();
mMainTlCostEdit.setText(text.substring(1, text.length() - 1));
} else {
mMainTlCostEdit.setText(String.format("[%s]", mMainTlCostEdit.getText().toString()));
}
}
});
For more reference, check out this documentation on the Android Developers page -
https://developer.android.com/reference/android/view/View.OnFocusChangeListener.html
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,238 |
Q: How to Add a table in Presentation using ONLYOFFICE API? We are trying to add table using the ONLYOFFICE API from the presentation editor plugin. We could easily do that with the document editor but are unable to find a way for a presentation.
We tried inserting a table using attached scripts but it's not working. We tried the same in Document Builder also but faced an error while creating a document.
We tried the following:
oDocContent.InsertContent([oTable]);
and
oDocContent.Push(oTable);
With the following scripts we tried to insert a table in the presentation editor.
Script 1:
builder.CreateFile("pptx");
var oPresentation = Api.GetPresentation();
var oParagraphCell, oTable, oTableStyle, oCell, oTableRow, oParagraph;
var oSlide = oPresentation.GetSlideByIndex(0);
oSlide.RemoveAllObjects();
var oShape = Api.CreateShape("rect", 300 * 36000, 130 * 36000, oFill, oStroke);
oShape.SetPosition(608400, 1267200);
oDocContent = oShape.GetDocContent();
oParagraph = oDocContent.GetElement(0);
oParagraph.AddText("This is an example of the ppt.");
oParagraph = Api.CreateParagraph();
oDocContent.InsertContent([oParagraph]);
oTableStyle = oDocContent.CreateStyle("CustomTableStyle", "table");
oTableStyle.SetBasedOn(oDocument.GetStyle("Bordered - Accent 5"));
oTable = Api.CreateTable(3, 3);
oTable.SetWidth("percent", 100);
oTableRow = oTable.GetRow(0);
oTableRow.SetHeight("atLeast", 1440);
oCell = oTable.GetRow(0).GetCell(0);
oCell.SetVerticalAlign("top");
oParagraphCell = oCell.GetContent().GetElement(0);
oParagraphCell.AddText("Align top");
oCell = oTable.GetRow(0).GetCell(1);
oCell.SetVerticalAlign("center");
oParagraphCell = oCell.GetContent().GetElement(0);
oParagraphCell.AddText("Align center");
oCell = oTable.GetRow(0).GetCell(2);
oCell.SetVerticalAlign("bottom");
oParagraphCell = oCell.GetContent().GetElement(0);
oParagraphCell.AddText("Align bottom");
oTable.SetStyle(oTableStyle);
oDocContent.InsertContent([oTable]);
oSlide.AddObject(oShape);
builder.SaveFile("pptx", "SampleText.pptx");
builder.CloseFile();
Script 2:
builder.CreateFile("pptx");
var oPresentation = Api.GetPresentation();
var oParagraphCell, oTable, oTableStyle, oCell, oTableRow, oParagraph;
var oSlide = oPresentation.GetSlideByIndex(0);
oSlide.RemoveAllObjects();
oFill = Api.CreateSolidFill(Api.CreateRGBColor(61, 74, 107));
oStroke = Api.CreateStroke(0, Api.CreateNoFill());
var oShape = Api.CreateShape("rect", 300 * 36000, 130 * 36000, oFill, oStroke);
oShape.SetPosition(608400, 1267200);
oDocContent = oShape.GetDocContent();
oParagraph = oDocContent.GetElement(0);
oParagraph.AddText("This is an example of the ppt.");
oTableStyle = oDocContent.CreateStyle("CustomTableStyle", "table");
oTableStyle.SetBasedOn(oDocContent.GetStyle("Bordered - Accent 5"));
oTable = Api.CreateTable(3, 3);
oTable.SetWidth("percent", 100);
oTableRow = oTable.GetRow(0);
oTableRow.SetHeight("atLeast", 1440);
oCell = oTable.GetRow(0).GetCell(0);
oCell.SetVerticalAlign("top");
oParagraphCell = oCell.GetContent().GetElement(0);
oParagraphCell.AddText("Align top");
oCell = oTable.GetRow(0).GetCell(1);
oCell.SetVerticalAlign("center");
oParagraphCell = oCell.GetContent().GetElement(0);
oParagraphCell.AddText("Align center");
oCell = oTable.GetRow(0).GetCell(2);
oCell.SetVerticalAlign("bottom");
oParagraphCell = oCell.GetContent().GetElement(0);
oParagraphCell.AddText("Align bottom");
oTable.SetStyle(oTableStyle);
oDocContent.Push(oTable);
oSlide.AddObject(oShape);
builder.SaveFile("pptx", "SampleText.pptx");
builder.CloseFile();
How to add a table in Presentation using ONLYOFFICE API?
A:
We could easily do that with the document editor but are unable to
find a way for a presentation.
Soon you also will be able to add the tables in ONLYOFFICE Presentations.
It is planned in the next versions.
Keep in touch with ONLYOFFICE support team, they will let you know.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,844 |
Q: How to configure a Zepplin notebook to use only X cores in my local Spark Cluster? Config - Ubuntu; Apache Zepplin (7.3.0); Spark 2.2.0; Hadoop 2.6;
A cluster of 6 machines with 14gb ram & 4 cores each. Need to split these for two notebooks. Please advice.
A: In case of 7.3.0, check-in Zepplin Interpreter settings.
A dropdown option is available select the SparkContext as Global or per Note.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,480 |
Example for Half Subtractor class
---------------------------------
.. code:: python
# Imports
from __future__ import print_function
from BinPy.combinational.combinational import *
.. code:: python
# Initializing the HalfSubtractor class
hs = HalfSubtractor(0, 1)
# Output of HalfSubtractor
print (hs.output())
.. parsed-literal::
[1, 1]
.. code:: python
# The output is of the form [DIFFERENCE, BORROW]
# Input changes
# Input at index 1 is changed to 0
hs.set_input(1, 0)
# New Output of the HalfSubtractor
print (hs.output())
.. parsed-literal::
[0, 0]
.. code:: python
# Changing the number of inputs
# No need to set the number, just change the inputs
# Input length must be two
hs.set_inputs(1, 1)
.. code:: python
# New output of HalfSubtractor
print (hs.output())
.. parsed-literal::
[0, 0]
.. code:: python
# Using Connectors as the input lines
# Take a Connector
conn = Connector()
# Set Output at index to Connector conn
hs.set_output(0, conn)
# Put this connector as the input to gate1
gate1 = AND(conn, 0)
# Output of the gate1
print (gate1.output())
.. parsed-literal::
0
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,671 |
Q: Laravel Stop Registration from admin panel I building settings page where the admin can change some setting in the website
I want to add an option to stop registration
for example:
if registration is disabled and a user trying to go to register page he will automatically redirect to 404
Setting Table:
Name: The Name of option or setting
Value: The value (if this value = 0 that means this option is disabled and if it's 1 that means it's enabled)
I already add a column in setting table "stop_register"
what I want is when the value of this column is 0 then registration is off and when it's 1 then the registration is on
A: You can use middleware to do this task,
To create a new middleware, use the make:middleware Artisan command:
php artisan make:middleware CheckRegistration
The above command will create CheckRegistration class in app/Http/Middleware directory.
In this middleware, you can apply your logic to allow registration route or not depending on the value
Middleware:
<?php
namespace App\Http\Middleware;
use Closure;
use App/Registration;
class CheckRegistration
{
/**
* Handle an incoming request.
*
* @param \Illuminate\Http\Request $request
* @param \Closure $next
* @return mixed
*/
public function handle($request, Closure $next)
{
$value = Registration::select("value"); // assuming value is either 0 or 1
if ($value == 0) {
return redirect('404'); // view with 404 display error
}
return $next($request);
}
}
As in the above code, it will redirect to 404 error view if value is 0 otherwise, the request will be passed further into the application.
Code is not tested.
Reference here
A: try this version:
public function Register()
{
$stop_reg = DB::table('settingstable')->value('stop_register');
if( $stop_reg==1 )
{
return view('register_page');
}
elseif( $stop_reg==0 )
{
return view('404_page');
}
}
A: try something like this on your blade. though i have not tested the code.
@php $stop_reg = DB::table('settingstable')->value('stop_register'); @endphp
@if($stop_reg==1)
<a class="nav-link" href="{{ route('register') }}" style="font-size:11px">Register</a>
@elseif($stop_reg==0)
<a class="nav-link" href="{{ route('stop_register') }}" style="font-size:11px">Register</a>
@endif
on the above code snippet, declare two routes; one for registration page and the second for 404 page. On the controller, have two functions, for registration page and 404 page.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,374 |
I would like to report a user for an 'inappropriate build'. The username is 'IGamingJB'. If you could please check this out, that would be great. Thank you!
Apoligies for the delay in reply - I've not been on here for a while (*makes forum home page*).
I have removed the inapropriate builds off the plot, and have placed a warning on the players profile to make other staff aware of this incident if anything similar is to happen again. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,123 |
Gender-dependent expression of alpha and beta estrogen receptors in human nontumor and tumor lung tissue
Michael J. Fasco, Gregory J. Hurteau, Simon D. Spivack
Estrogen receptor (ER) expression in human lung has been understudied, particularly in light of its potential biological importance in the female lung cancer epidemic. Reverse transcription-polymerase chain reaction was used to probe mRNA expression of wild-type ERα and ERβ and their splice variants in human bronchogenic tumor and adjacent nontumor specimens. In tumor tissue from 13 women and 13 men, ERα was expressed in 85% of women versus 15% in men [P=0.001]. ERβ was expressed equally in tumors from women versus men [92% vs. 69%, P=ns]. Both ERα and β forms were expressed simultaneously in the lung tumors of 77% of women versus 15% of men [P=0.005]. Among adjacent nontumor lung specimens, 31% of the women expressed ERα mRNA versus 0% of men [P=0.101], and 39% of women expressed ERβ mRNA versus 31% of men [P=ns]; only one woman and no men expressed both ERα and β in nontumor tissue. Females expressed ERα [P=0.017], ERβ [P=0.013], and ERα+β [P=0.002] more frequently in tumor versus nontumor tissue, whereas in males expression of ERα, β and both α+β was not clearly different for tumor versus nontumor tissue. In specimens expressing ERα mRNA, the transcript lacking exon 7 (Δ7) was the major splice variant with varying contributions from the transcripts Δ4, Δ3+4, Δ5 and others unidentified. Alternative splicing of ERβ mRNA was observed, but not to as great an extent as for ERα mRNA. ERα promoter usage in tumors varied among individuals. When the ER receptors were co-expressed in tumors, ERα was quantitatively more abundant in the majority of cases than ERβ. Within this small group of 26 patients, no correlation was found between age, smoking history, plasma nicotine, cotinine, estradiol concentrations or histopathologic type with tumor or nontumor estrogen receptor status of any type. However, several positive correlations imply that: (1) ERα expression occurs more often in the lungs of women than men; (2) ERβ is expressed with approximately equal frequency in the lungs of both genders; and (3) tumors display a higher frequency of both receptor types than nontumors in women. We hypothesize that these putative gender-dependent differences in ERα and ERβ expression could contribute unique phenotypic characteristics to lung cancer development or progression in women.
Molecular and Cellular Endocrinology
Estrogen Receptor beta
Estrogen Receptor alpha
Alternatively spliced estrogen receptor mRNAs
Estrogen receptor mRNA analysis
Estrogen receptor transcriptional regulation
Fasco, M. J., Hurteau, G. J., & Spivack, S. D. (2002). Gender-dependent expression of alpha and beta estrogen receptors in human nontumor and tumor lung tissue. Molecular and Cellular Endocrinology, 188(1-2), 125-140.
Gender-dependent expression of alpha and beta estrogen receptors in human nontumor and tumor lung tissue. / Fasco, Michael J.; Hurteau, Gregory J.; Spivack, Simon D.
In: Molecular and Cellular Endocrinology, Vol. 188, No. 1-2, 25.02.2002, p. 125-140.
Fasco, MJ, Hurteau, GJ & Spivack, SD 2002, 'Gender-dependent expression of alpha and beta estrogen receptors in human nontumor and tumor lung tissue', Molecular and Cellular Endocrinology, vol. 188, no. 1-2, pp. 125-140.
Fasco MJ, Hurteau GJ, Spivack SD. Gender-dependent expression of alpha and beta estrogen receptors in human nontumor and tumor lung tissue. Molecular and Cellular Endocrinology. 2002 Feb 25;188(1-2):125-140.
Fasco, Michael J. ; Hurteau, Gregory J. ; Spivack, Simon D. / Gender-dependent expression of alpha and beta estrogen receptors in human nontumor and tumor lung tissue. In: Molecular and Cellular Endocrinology. 2002 ; Vol. 188, No. 1-2. pp. 125-140.
@article{8f20d897e96e4eccbf7bbbdb45edd9a8,
title = "Gender-dependent expression of alpha and beta estrogen receptors in human nontumor and tumor lung tissue",
abstract = "Estrogen receptor (ER) expression in human lung has been understudied, particularly in light of its potential biological importance in the female lung cancer epidemic. Reverse transcription-polymerase chain reaction was used to probe mRNA expression of wild-type ERα and ERβ and their splice variants in human bronchogenic tumor and adjacent nontumor specimens. In tumor tissue from 13 women and 13 men, ERα was expressed in 85{\%} of women versus 15{\%} in men [P=0.001]. ERβ was expressed equally in tumors from women versus men [92{\%} vs. 69{\%}, P=ns]. Both ERα and β forms were expressed simultaneously in the lung tumors of 77{\%} of women versus 15{\%} of men [P=0.005]. Among adjacent nontumor lung specimens, 31{\%} of the women expressed ERα mRNA versus 0{\%} of men [P=0.101], and 39{\%} of women expressed ERβ mRNA versus 31{\%} of men [P=ns]; only one woman and no men expressed both ERα and β in nontumor tissue. Females expressed ERα [P=0.017], ERβ [P=0.013], and ERα+β [P=0.002] more frequently in tumor versus nontumor tissue, whereas in males expression of ERα, β and both α+β was not clearly different for tumor versus nontumor tissue. In specimens expressing ERα mRNA, the transcript lacking exon 7 (Δ7) was the major splice variant with varying contributions from the transcripts Δ4, Δ3+4, Δ5 and others unidentified. Alternative splicing of ERβ mRNA was observed, but not to as great an extent as for ERα mRNA. ERα promoter usage in tumors varied among individuals. When the ER receptors were co-expressed in tumors, ERα was quantitatively more abundant in the majority of cases than ERβ. Within this small group of 26 patients, no correlation was found between age, smoking history, plasma nicotine, cotinine, estradiol concentrations or histopathologic type with tumor or nontumor estrogen receptor status of any type. However, several positive correlations imply that: (1) ERα expression occurs more often in the lungs of women than men; (2) ERβ is expressed with approximately equal frequency in the lungs of both genders; and (3) tumors display a higher frequency of both receptor types than nontumors in women. We hypothesize that these putative gender-dependent differences in ERα and ERβ expression could contribute unique phenotypic characteristics to lung cancer development or progression in women.",
keywords = "Alternatively spliced estrogen receptor mRNAs, Estrogen receptor mRNA analysis, Estrogen receptor transcriptional regulation, Gender, Lung cancer",
author = "Fasco, {Michael J.} and Hurteau, {Gregory J.} and Spivack, {Simon D.}",
journal = "Molecular and Cellular Endocrinology",
publisher = "Elsevier Ireland Ltd",
T1 - Gender-dependent expression of alpha and beta estrogen receptors in human nontumor and tumor lung tissue
AU - Fasco, Michael J.
AU - Hurteau, Gregory J.
AU - Spivack, Simon D.
N2 - Estrogen receptor (ER) expression in human lung has been understudied, particularly in light of its potential biological importance in the female lung cancer epidemic. Reverse transcription-polymerase chain reaction was used to probe mRNA expression of wild-type ERα and ERβ and their splice variants in human bronchogenic tumor and adjacent nontumor specimens. In tumor tissue from 13 women and 13 men, ERα was expressed in 85% of women versus 15% in men [P=0.001]. ERβ was expressed equally in tumors from women versus men [92% vs. 69%, P=ns]. Both ERα and β forms were expressed simultaneously in the lung tumors of 77% of women versus 15% of men [P=0.005]. Among adjacent nontumor lung specimens, 31% of the women expressed ERα mRNA versus 0% of men [P=0.101], and 39% of women expressed ERβ mRNA versus 31% of men [P=ns]; only one woman and no men expressed both ERα and β in nontumor tissue. Females expressed ERα [P=0.017], ERβ [P=0.013], and ERα+β [P=0.002] more frequently in tumor versus nontumor tissue, whereas in males expression of ERα, β and both α+β was not clearly different for tumor versus nontumor tissue. In specimens expressing ERα mRNA, the transcript lacking exon 7 (Δ7) was the major splice variant with varying contributions from the transcripts Δ4, Δ3+4, Δ5 and others unidentified. Alternative splicing of ERβ mRNA was observed, but not to as great an extent as for ERα mRNA. ERα promoter usage in tumors varied among individuals. When the ER receptors were co-expressed in tumors, ERα was quantitatively more abundant in the majority of cases than ERβ. Within this small group of 26 patients, no correlation was found between age, smoking history, plasma nicotine, cotinine, estradiol concentrations or histopathologic type with tumor or nontumor estrogen receptor status of any type. However, several positive correlations imply that: (1) ERα expression occurs more often in the lungs of women than men; (2) ERβ is expressed with approximately equal frequency in the lungs of both genders; and (3) tumors display a higher frequency of both receptor types than nontumors in women. We hypothesize that these putative gender-dependent differences in ERα and ERβ expression could contribute unique phenotypic characteristics to lung cancer development or progression in women.
AB - Estrogen receptor (ER) expression in human lung has been understudied, particularly in light of its potential biological importance in the female lung cancer epidemic. Reverse transcription-polymerase chain reaction was used to probe mRNA expression of wild-type ERα and ERβ and their splice variants in human bronchogenic tumor and adjacent nontumor specimens. In tumor tissue from 13 women and 13 men, ERα was expressed in 85% of women versus 15% in men [P=0.001]. ERβ was expressed equally in tumors from women versus men [92% vs. 69%, P=ns]. Both ERα and β forms were expressed simultaneously in the lung tumors of 77% of women versus 15% of men [P=0.005]. Among adjacent nontumor lung specimens, 31% of the women expressed ERα mRNA versus 0% of men [P=0.101], and 39% of women expressed ERβ mRNA versus 31% of men [P=ns]; only one woman and no men expressed both ERα and β in nontumor tissue. Females expressed ERα [P=0.017], ERβ [P=0.013], and ERα+β [P=0.002] more frequently in tumor versus nontumor tissue, whereas in males expression of ERα, β and both α+β was not clearly different for tumor versus nontumor tissue. In specimens expressing ERα mRNA, the transcript lacking exon 7 (Δ7) was the major splice variant with varying contributions from the transcripts Δ4, Δ3+4, Δ5 and others unidentified. Alternative splicing of ERβ mRNA was observed, but not to as great an extent as for ERα mRNA. ERα promoter usage in tumors varied among individuals. When the ER receptors were co-expressed in tumors, ERα was quantitatively more abundant in the majority of cases than ERβ. Within this small group of 26 patients, no correlation was found between age, smoking history, plasma nicotine, cotinine, estradiol concentrations or histopathologic type with tumor or nontumor estrogen receptor status of any type. However, several positive correlations imply that: (1) ERα expression occurs more often in the lungs of women than men; (2) ERβ is expressed with approximately equal frequency in the lungs of both genders; and (3) tumors display a higher frequency of both receptor types than nontumors in women. We hypothesize that these putative gender-dependent differences in ERα and ERβ expression could contribute unique phenotypic characteristics to lung cancer development or progression in women.
KW - Alternatively spliced estrogen receptor mRNAs
KW - Estrogen receptor mRNA analysis
KW - Estrogen receptor transcriptional regulation
KW - Gender
KW - Lung cancer
JO - Molecular and Cellular Endocrinology
JF - Molecular and Cellular Endocrinology | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,877 |
{"url":"https:\/\/www.projecteuclid.org\/euclid.aoms\/1177692707","text":"## The Annals of Mathematical Statistics\n\n### Maximum Likelihood Estimation of a Translation Parameter of a Truncated Distribution\n\nMichael Woodroofe\n\n#### Abstract\n\nLet $f_\\theta(x) = f(x - \\theta), \\theta, x\\in R$, where $f(x) = 0$ for $x \\leqq 0$ and let $\\hat{\\theta}_n$ be the maximum likelihood estimate (MLE) of $\\theta$ based on a sample of size $n$. If $\\alpha = \\lim f'(x)$ exists as $x \\rightarrow 0$, and $0 < \\alpha < \\infty$, then under some regularity conditions, it is shown that $\\alpha_n(\\hat{\\theta}_n - \\theta)$ has an asymptotic standard normal distribution where $2\\alpha_n^2 = \\alpha n \\log n$ and that if $\\theta$ is regarded as a random variable with a prior density, then the posterior distribution of $\\alpha_n(\\theta - \\hat{\\theta}_n)$ converges to normality in probability.\n\n#### Article information\n\nSource\nAnn. Math. Statist., Volume 43, Number 1 (1972), 113-122.\n\nDates\nFirst available in Project Euclid: 27 April 2007\n\nhttps:\/\/projecteuclid.org\/euclid.aoms\/1177692707\n\nDigital Object Identifier\ndoi:10.1214\/aoms\/1177692707\n\nMathematical Reviews number (MathSciNet)\nMR298817\n\nZentralblatt MATH identifier\n0251.62018\n\nJSTOR","date":"2019-10-19 21:07:16","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8971444368362427, \"perplexity\": 321.5902764814496}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986697760.44\/warc\/CC-MAIN-20191019191828-20191019215328-00131.warc.gz\"}"} | null | null |
"use strict";
angular
.module('app')
.run([
'StatusService',
function(StatusService) {
StatusService.start();
}
]); | {
"redpajama_set_name": "RedPajamaGithub"
} | 2,515 |
After making a name for himself through releasing a host of well-received EPs and remixes, Montreal-bred producer Jacques Greene has unveiled his first full-length project in Feel Infinite, and you can hear the entire thing right here.
Arriving through Arts & Crafts/LuckyMe today (March 10), the record finds the producer "trying to find new approaches of pushing the Jacques Greene sound forward" over 11 tracks. As he tells Exclaim!, "I also wanted to do something that was almost like a culmination of all these previous records too."
Previously released singles "Afterglow" and "To Say" appear on the disc, in addition to a collaboration with How to Dress Well titled "True." A calling card of his production style, Greene doubles down on pitch-shifted vocal snippets to great effect. Our review of the record concludes that this is vintage Jacques Greene, but you're never left feeling like you've heard it all before.
"If you didn't like it three records ago, you probably won't like this album," he explains to Exclaim! "And I'd like to think that if you liked my records three years ago, you'll like this finely tuned, slightly matured, well-rounded 45-minute album-world version of it now. So, this is me sticking to my guns in a way that I'm really happy about."
Take in Feel Infinite below. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,030 |
class CloudFile < ActiveRecord::Base
attr_accessible :name, :asset
mount_uploader :asset, CloudFileUploader
validates :name, :asset, :presence => true
before_create :save_metadata
private
def save_metadata
self.content_type = asset.file.content_type
self.file_size = asset.file.size
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,794 |
{"url":"https:\/\/americans-world.org\/docs\/friyy.php?700a45=gpm-to-m%2Fs","text":"1 You are currently converting Volumetric flow rate units from British gallon per minute to cubic meter per second 1 gpm = 0.0000757683211 m 3 \/s British gallon per minute cubic meter per second 1 gallon per minute(US) = 3.785411784 liters per minute. To link to this flow rate gallon US per minute to cubic meters per second online converter simply cut and paste the following. My best, Please attribute www.kylesconverter.com when using the work, thank you! It 5 gallons per minute to cubic metre\/hour = 1.36383 cubic metre\/hour. Calculate age in future. Metric units. 1 gallon per minute(US) = 3.785411784 liters per minute. Do a quick conversion: 1 gallons\/minute = 63.090196666667 milliliters\/second using the online calculator for metric conversions. implies that the speed of the object is also equal to $$r \\times \\omega$$. With the above mentioned two-units calculating service it provides, this flow rate converter proved to be useful also as a teaching tool: 1. in practicing gallons US per minute and cubic meters per second ( gal\/min vs. m3\/sec ) measures exchange. The angular velocity of the wheel US gallons per minute or cubic metre\/second The SI derived unit for volume flow rate is the cubic meter\/second. This tool converts gallons per minute to liters per minute (gal\/m to lt\/m) and vice versa. Many \u2026 inch, 100 kg, US fluid ounce, 6'3\", 10 stone 4, cubic cm, CONVERT : \u00a0 between other flow rate measuring units - complete list. It is the EQUAL flow rate value of 1 cubic meter per second but in the gallons US per minute flow rate unit alternative. Before using any of the provided tools or data you must check with a competent authority to validate its correctness. The gallons US per minute unit number 15,850.32 gal\/min converts to 1 m3\/sec, one cubic meter per second.\n\nThe cubic meters per second unit number 0.000063 m3\/sec converts to 1 gal\/min, one gallon US per minute. Gallons per Minute(gpm) to Velocity, in feet per second (fps) Internal Pipe Diameter (i.d. 1 cubic meter\/second is equal to 13198.154897945 gallons per minute, or 3600 cubic metre\/hour. Assume an object (or point) attached to a rotating wheel. Unit Conversions | Convert flow rate of gallon US per minute (gal\/min) and kilograms (water mass) per second (kg\/sec) units in reverse from kilograms (water mass) per second into gallons US per minute.\n\nPrivacy policy | Terms of Use & Disclaimer | Contact | Advertise | Site map \u00a9 2019 www.traditionaloven.com, from cubic meters per second into gallons US per minute, cubic meters per second into gallons US per minute, flow rate from gallon US per minute (gal\/min) to cubic meters per second (m3\/sec).\n\nConvert 1 m3\/sec into gallon US per minute and cubic meters per second to gal\/min.\n\nPrivacy policy | Terms of Use & Disclaimer | Contact | Advertise | Site map \u00a9 2019 www.traditionaloven.com, from kilograms (water mass) per second into gallons US per minute, kilograms (water mass) per second into gallons US per minute, flow rate from gallon US per minute (gal\/min) to kilograms (water mass) per second (kg\/sec). l\/min cubic meter per second . A gallon per minute (us gallon, gal\/m) is unit of volume flow rate equal to a gallon flow per minute. Formula gallons per minute in liters per minute (gal\/m in lt\/m). To link to this flow rate gallon US per minute to kilograms (water mass) per second online converter simply cut and paste the following. gallons per minute = However, the accuracy cannot be guaranteed.\n\nsymbols, abbreviations, or full names for units of length, It measures the number of meters traveled in a second. First unit: gallon US per minute (gal\/min) is used for measuring flow rate. For a whole set of multiple units for volume and mass flow on one page, try the Multi-Unit converter tool which has built in all flowing rate unit-variations. With the above mentioned two-units calculating service it provides, this flow rate converter proved to be useful also as a teaching tool: 1. in practicing gallons US per minute and kilograms (water mass) per second ( gal\/min vs. kg\/sec ) measures exchange. 1 l\/min = 0.000016666666666667 m 3 \/s. Fill one of the following fields, values will be converted and updated automatically. The cubic meters per second unit number 0.000063 m3\/sec converts to 1 gal\/min, one gallon US per minute. Second: cubic meter per second (m3\/sec) is unit of flow rate. How many cubic meters per second are in 1 gallon US per minute? Switch units Starting unit.\n\nThe 40 gallons per minute to cubic metre\/hour = 10.91062 cubic metre\/hour How many kilograms (water mass) per second are in 1 gallon US per minute? The other way around, how many kilograms (water mass) per second - kg\/sec are in one gallon US per minute - gal\/min unit? Converting gallon US per minute to kilograms (water mass) per second value in the flow rate units scale. The conversions on this site will not be accurate enough for all applications. Second: kilogram (water mass) per second (kg\/sec) is unit of flow rate. Convert flow rate of gallon US per minute (gal\/min) and cubic meters per second (m3\/sec) units in reverse from cubic meters per second into gallons US per minute. Between m3\/sec and gal\/min measurements conversion chart page. UnitJuggler offers free and easy unit conversions. Calculators | There are 0.001 m\/s \u2026 can be converted to $$rad.s^{-1}$$ thanks to the following formula: $$\\omega_{ (rad.s^{-1}) } = \\frac {2\\pi}{60}.N_{(rpm)}$$. Page with flow rate by mass unit pairs exchange. TOGGLE : \u00a0 from kilograms (water mass) per second into gallons US per minute in the other way around. ): inches: Gallons per Minute (gpm): gpm: Note : This formula is used for converting gallons per minute (gpm) to velocity, in feet per second (fps) within a given pipe diameter. 2. for conversion factors between unit pairs. This unit-to-unit calculator is based on conversion for one pair of two flow rate units. You can find metric conversion tables for SI units, as well A gallon per minute (us gallon, gal\/m) is unit of volume flow rate equal to a gallon flow per minute. 3. work with flow rate's values and properties.\n\nThe symbol for meters per second is m\/s and the International spelling for this unit is metres per second. QUESTION: 15 gal\/min = ? Convert gallon [US]\/minute to cubic metre\/second, cubic metre\/second to US gallons per minute, US gallons per minute to cubic meter\/minute, US gallons per minute to cubic yard\/second, US gallons per minute to cubic decimeter\/minute, US gallons per minute to cubic mile\/minute, US gallons per minute to thousand cubic foot\/hour, US gallons per minute to million cubic foot\/day, US gallons per minute to hectare meter\/hour, US gallons per minute to cubic foot\/second. The link to this tool will appear as: flow rate from gallon US per minute (gal\/min) to cubic meters per second (m3\/sec) conversion. Meters per second is a unit of Speed or Velocity in the Metric System. Contact | conversion can be done thanks to the following formula: $$v_{(m.s^{-1})} = r \\times \\omega_{(rad.s^{-1})} = r \\times \\frac {2 \\pi }{60}.N_{(rpm)}$$, $$N_{(rpm)} = \\frac {60} { 2 \\pi \\times r} v_{(m.s^{-1})}$$, converted to $$rad.s^{-1}$$ thanks to the following formula, Convert from binary to decimal and vice-versa, Convert newton-metre [N.m] to kilogramme-centimeter [Kg.cm] and vice-versa, Convert newton-metre [N.m] to millinewton-metre [mN.m] and vice-versa, Convert inches [in] to centimeters [cm] and vice-versa, Convert meters per second [m\/s] to kilometers per hour [km\/h] and vice-versa, Convert meters [m] to millimeters [mm] and vice-versa, Convert newton-meter [N.m] to newton [N] and vice-versa, Convert kilometers per hour [km\/h] to radians per second [rad\/s] and vice-versa, Convert radians per second [rad\/s] to meters per second [m\/s] and vice-versa, Convert revolutions per minute [rpm] to radians per second [rad\/s] and vice-versa, Convert revolutions per second [rps] to radians per second [rad\/s] and vice-versa, Convert radians [rad] to degrees [\u00b0] and vice-versa, Convert revolutions per minute [rpm] to kilometers per hour [km\/h] and vice-versa, Convert revolutions per minute [rpm] to meters per second [m\/s] and vice-versa.\n\nThis also means that the wheel rotates from $$\\omega$$ radians during one second. 2. for conversion factors between unit pairs.\n\n* Whole numbers, decimals or fractions (ie: 6, 5.33, 17 3\/8)* Precision is how many digits after decimal point (1 - 9). 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To learn how we use any data we collect about you see our privacy policy.\n\n.\n\nWhere Are You Now Meaning, Stratified Squamous Epithelium Keratinized, Storks 2 Release Date, Comic Vine General, Ephesians 2:8-10 Nkjv, Siasat Definition In Urdu, Tater Tot Calories Baked, Light Blue And Coral Bedroom, How Many Abortions Before Roe V Wade, First In Importance - Crossword Clue, Lakanto Monk Fruit Sweetener Keto, Shielding Meaning In Tamil, Hear Music Record Label, All Our Yesterdays Theme, Ricotta Cheese Mixture For Lasagna, Technicolor Router Reset, Black Plastic Chair With Wooden Legs, Another Word For Says In An Essay, Morrisville, Nc Weather, Best Coffee In The World, Best Hunter Bow Ac Origins, Johnnie Walker Black Label 2 Litre Price, Bed Terraria Recipe, Vfwax Vs Vtiax, The Mastery Of Love Review, Tuna Pesto Pasta Clara Ole, Martha Bakes Cookies, Best Rendang Paste Singapore, Anu Hasan Siblings, Monoterpenes Essential Oils, Splendor Plus I3s Bs6, Lemon Blueberry Oatmeal Muffins No Flour, Calories In Half And Half, Forever Together Images, Jonathan Nelson Medley, Samsung Galaxy A10e Price, Pink Midi Dress Casual, Unit Of Energy Density, 5 Lines On Milk For Class 1, What Is Cloud Cotton Made Of, Chest Pain After Drinking Coffee, Assam Legislative Council, Slimming World Healthy Extras Explained, Emerald Green Color Palette, Low Loft Bed Frame, Church Movies List, Legitimate Power Meaning In Tamil, Philadelphia No Bake Cheesecake Filling Recipes, Ac Origins Best Hunter Bow, Chicken Leek Soup, 3-methyl-2-butanone 1h Nmr, Thrustmaster T Flight Hotas One Compatible Games, National Physique Committee 2020, Hanging Real Estate Signs, Assassins Creed Brotherhood Statue, Amazing Fruits Images, Jonathan Nelson Medley, Silver Crest Electric Grater, Web Design Course Details, Swift Meaning In Urdu, Toni Braxton Height, Aesthetic Cherry Png, Best Graphics Card For Fortnite, Bone Marrow Cancer How Long To Live, Redmi 8 Price In Bangladesh, Collagen + C Pomegranate Liquid Side Effects, Strawberry Curd For Macarons, Ethyl Benzoate Formula, Chocolate Orange Cheesecake, Wizard Of Oz Trick, Toe-nailing Pergola Rafters, Gender Roles Essay Conclusion, Mandarin Orange In Tamil, Kyoto Restaurants Orlando, Jailhouse Recipes Pdf, Mandarin Orange In Tamil, Mifi Router 4g,","date":"2021-04-13 04:09:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 2, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.20027987658977509, \"perplexity\": 7758.8318748426145}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038072082.26\/warc\/CC-MAIN-20210413031741-20210413061741-00213.warc.gz\"}"} | null | null |
Your letter is getting so strong. And I know you can do this! And meanwhile, your CP is in POS revision hell, BUT she has an amazing idea to add the fun back to her story (you know I have to have fun)… I will reveal all later!!
Hugs! You will get this done and congrats on the fab contest results. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,924 |
Q: "These" with uncountable nouns I know that "These" can only be followed by plural nouns.
ex: These flowers are beautiful.
My questions are:
*
*Can I use multiple uncountable nouns after "These"?
ex: These milk, butter, and bread are beyond their sell-by dates.
*Can I mention uncountable nouns together with countable ones after "These"?
ex: These evidence and witnesses prove him guilty.
A: If you're using demonstrative adjectives with a list, you should use one for each item in the list. See more on "demonstrative adjectives" on this page.
This milk, this butter, and this bread are beyond their sell-by dates.
Even though those nouns are uncountable, you use "this" and not "these," because uncountable nouns are generally treated as singular. See here for more information on that subject.
In your second example, one item is singular (uncountable and treated as singular) and one is plural, so you would use different demonstrative adjectives for each:
This evidence and these witnesses prove him guilty.
Using a demonstrative adjective is only necessary to point to which objects you're talking about. In most cases, you could avoid this problem by simply using an article like "the."
The milk, butter, and bread are beyond their sell-by dates.
The evidence and witnesses prove him guilty.
A: I don't think people would say or write this. I would expect something like:
These things, the milk, butter, and bread are beyond their sell-by dates.
Your other example might simply be:
These points, the evidence and witnesses prove him guilty.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,696 |
Q: Using KNSemiModal or MJPopUpView with storyboard Whenever I try and dismiss the semi modal view, I am left with a crashed app. I think I've properly set up the dismiss modal view but it doesn't seem to be working. Here's what it looks like in the demo app using XIB files:
MJDetailViewController *detailViewController = [[MJDetailViewController alloc] initWithNibName:@"MJDetailViewController" bundle:nil];
[self presentPopupViewController:detailViewController animationType:MJPopupViewAnimationSlideBottomBottom];
Here's what I am trying to replace it with:
[self presentPopupViewController:[self.storyboard instantiateViewCOntrollerWithIdentifier:@"example"] animationType:MJPopupViewAnimationSlideBottomBottom];
The [self.storyboard instantiateViewControllerWithIdentifier:NSString] doesn't seem to be replacing the instWithNib (or whatever function that is) properly.
Any ideas why?
A: I did something similar.
I would just delete all the init methods in MJDetailViewController and also added "example" in the 'Storyboard ID' attribute of identity inspector then called
[self presentPopupViewController:[self.storyboard instantiateViewCOntrollerWithIdentifier:@"example"] animationType:MJPopupViewAnimationSlideBottomBottom];
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 952 |
Q: Site Actions menu in SharePoint is WSS-like, whereas I have installed MOSS I had WSS installed on my VM and then uninstalled in order to install MOSS Enterprise
I can see a lot of the MOSS stuff in the Central Administration pages but when I create a new web application and site collection, my Site Actions dropdown is showing the WSS version, i.e. Create, Edit Page and Site Settings as opposed to the MOSS menu which has View All Site Content, View Reports, Manage Content and Structure etc.
I just tried deleting my web app and creating it all again but it's come back with the same thing again.
Is somewhere remembering that I used to have WSS installed? (yes I know MOSS is built on top of WSS but you know what I mean!)
A: Got it - had to enable "Office SharePoint Server Publishing Infrastructure" in Site Collection features and then activate "Office SharePoint Server Publishing" from the site in question.
Now appears fine
A: If you want the menus like View Reports, Manage Content and Structure etc.. you need to activate "Office SharePoint Server Publishing Infrastructure" feature from the site collection featire.
When you create new site using published site template, the "Office SharePoint Server Publishing Infrastructure" feature is activated defaultly.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,779 |
\section{Introduction}
The energy and power density spectra of low-mass X-ray binaries
(LMXBs) change with time in a correlated way, generally following
changes of the source luminosity, supporting the scenario in which
these changes are a function of mass accretion rate in the system
\citep[e.g., ][]{Wijnands97, Mendez99, GierlinskiDone02}. Evolution of the broad-band energy spectrum
in low-luminosity systems is thought to reflect changes in the configuration
of the accretion-disc flow \citep[see review by][and references therein]{Done07}.
\citet{GierlinskiDone02} find a strong correlation in the LMXB 4U 1608--52 between
the position of the source in the colour-colour diagram and the truncation radius of the
inner accretion disc, which is likely driven by the average
mass accretion rate through the disc. At low luminosity, the spectrum is consistent with emission
from an accretion disc truncated far from the neutron star; as the luminosity increases the
spectrum softens and the inner radius of the accretion disc moves inwards.
\noindent
In a very similar way, changes of the power density spectra appear to be driven by
mass accretion rate. At low luminosity, when the energy spectrum of the source is hard, all timing
components in the power spectrum have relatively low characteristic frequencies. These
frequencies increase as the energy spectrum softens and the inferred mass accretion rate through the disc increases \citep[see e.g.,][]{vdKlis97,
Mendez97,Mendez99,Mendez01, Homan02,Straaten02,Straaten03,Straaten05,Altamirano05,
Altamirano08a, Altamirano08b, Linares05,Linares07}. The fact that fits to
the energy spectra suggest that the accretion disc moves closer to the NS, and that the characteristic
frequencies in the power density spectra increase as the luminosity increases,
supports the idea that those frequencies are set by the dynamical frequencies in the accretion disc. The kilohertz quasi-periodic oscillations (kHz QPOs) are
especially interesting because of the close correspondence between their frequencies and the Keplerian frequency
at the inner edge of the accretion disc \citep[e.g.,][]{MillerLambPsaltis98,Stella98}. On short time-scales (within a day or less),
the frequency of the kHz QPOs increases monotonically as the source brightens, and the inferred mass accretion
rate increases. However, on longer time-scales this correlation breaks down and the intensity-frequency
diagram shows the so-called ``parallel tracks'' \citep{Mendez99}.
Broad asymmetric iron (Fe) lines have been often observed in accreting systems with the compact object
spanning a large range of masses, from supermassive black holes in AGNs \citep[see][for an extensive review]{Fabian00}
to stellar-mass black holes \citep[e.g.,][]{Miller02,Miller04} and neutron star systems \citep{Bhattacharyya07}.
The Fe K-$\alpha$ emission line at 6--7 keV is an important feature of the spectrum that
emerges from the accretion disc as a result of reflection of the corona and the NS surface/boundary layer photons
off the accretion disc.
The mechanism responsible for the broad asymmetric profile of the line is still under discussion.
\citet{Fabian89} proposed that the line is broadened by Doppler and relativistic effects due to motion of the matter in the accretion disc. \citet{diSalvo05} and \citet{Bhattacharyya07} discovered broad iron lines in the NS LMXBs
4U 1705--44 and Serpens X-1, respectively. \citet{Cackett08} confirmed \citet{Bhattacharyya07} results using
independent observations, and also discovered broad, asymmetric, Fe K-$\alpha$ emission lines
in the LMXBs 4U 1820--30 and GX 349+2. All these authors interpreted the broadening of the line
as due to relativistic effects. Relativistic Fe lines have been observed at least in a dozen NS binary systems
in the last decade \citep{diSalvo05,Bhattacharyya07,Cackett08, Pandel08, Cackett09b, Papitto09,diSalvo09,
Reis09, Iaria09,DAi09,Cackett10,DAi10,Egron11,Cackett12,Sanna13}. A different interpretation for the broadening of
the line has been suggested by \citet{Ng10} who re-analysed the data of several NS systems showing Fe K-$\alpha$
emission lines. \citet{Ng10} claim that for most of the lines there is no need to invoke special and general
relativity to explain the broad profile, and that Compton broadening is enough.
If the relativistic interpretation of the Fe line is correct, we can directly test
accretion disc models by studying the line properties, as the shape of the profile depends on the inner
and the outer disc radius. We expect then the iron line to be broader in the soft state -- when the inner radius
is smaller and the relativistic effects stronger -- than in the hard state.
Accretion disc models can also be tested using simultaneous measurements of kHz QPOs and iron lines
\citep{Piraino00,Cackett10}. If both the Keplerian interpretation of the kHz QPOs frequency and the
broadening mechanism (Doppler/relativistic) of the Fe line are correct, these two observables
should provide consistent information about the accretion disc. Furthermore, if the changes in the spectral continuum also
reflect changes of the inner edge of the accretion disc, the Fe line should vary in correlation with the
frequency of the kHz QPOs \citep[see, e.g.,][]{Bhattacharyya07}. Understanding the relation between kHz QPOs,
Fe emission line and spectral states may have an impact beyond accretion disc physics. As discussed by \citet{Piraino00},
\citet{Bhattacharyya07}, and \citet{Cackett08}, measurements of the line could also help constraining the mass and radius of the neutron star \citep{Piraino00}, parameters needed to determine the neutron star equation of state. \citet{Cackett10}\footnote{At about the same time, Altamirano et al. (2010) presented a similar analysis in a manuscript that was eventually never published (c.f. \S 5 in \citet{Cackett10})} tested this idea using three observations of 4U 1636--53.
With all of this in mind, in
this paper we investigate the correlation between the iron line, kHz QPOs and spectral states in the
LMXB 4U 1636--53, with the aim of understanding whether the existing interpretations of these phenomena
are consistent.
The fact that 4U 1636--53 is well sampled with \textit{XMM-Newton}, and \textit{Rossi X-ray Timing Explorer (RXTE)} observations, is one
of the most prolific sources of kHz QPOs, shows strong Fe-K$\alpha$ lines, and shows regular
hard-to-soft-to-hard state transitions on time scales of weeks, makes this source an excellent target for this study.
\section{Observations}
\label{sec:obs}
The NS LMXB 4U 1636--53 has been observed with \textit{XMM-Newton} nine times between 2000 and
2009. The first 2 observations (in 2000 and 2001) were performed in imaging-mode, and suffered from severe
pile-up and data loss due to telemetry drop outs; we therefore excluded these two observations from the analysis. In \citet{Sanna13}
we reanalysed the 2005, 2007 and 2008 observations \citep{Pandel08,Cackett10} and we analysed for the first time four new observations taken in 2009.
The observation taken on March 14th, 2009 had a flaring high-energy background during the full
$\sim$40 ks exposure, and we therefore excluded this observation from the analysis \citep[see][for more details
on the analysis of the \textit{XMM-Newton} observations]{Sanna13}. In this paper we make use of the results reported there.
Since 1996, 4U 1636--53 was observed $\sim$1300 times with \textit{RXTE}; from March 2005 the source was regularly observed
for $\sim$2 ks every other day (except for periods with solar viewing occultation). The first results of this monitoring
campaign have been reported by \citet{Belloni07}, \citet{Zhang12}, and \citet{Sanna12a}. We took all the information related to the kHz QPOs used for this paper from \citet[][]{Sanna12a}. Following \citet{Sanna13}, we refer to the 2005, 2006, 2007,
25th March 2009, 5th September 2009, and 11th September 2009 \textit{XMM-Newton} observations as Obs.~1--6,
respectively. We will use the same labelling (Obs.~1, etc.) for the simultaneous \textit{RXTE} observations used to study the
X-ray variability. When these names are used in the context of power spectral
components, spectral hardness, intensities and colours, we refer to the \textit{RXTE} observations.
\section{Data analysis}
\subsection{Timing analysis}
We used data from 1280 \textit{RXTE} observations with the Proportional Counter Array \citep[PCA,][]{Zhang93}, covering
14 years of data since 1996. We produced light curves and colours using the Standard 2 data as in
\citet{Altamirano08a}: we used 16-s time-resolution Standard 2 data to calculate the hard and soft colours, defined
as the 9.7--16.0 keV / 6.0--9.7 keV and 3.5--6.0 keV / 2.0--3.5 keV count rate ratio, respectively. We measured the intensity in
the 2.0--16.0 keV range. We normalised the colours and the intensity by the corresponding Crab Nebula
values \citep[closest in time and within the same PCA gain epoch; e.g.,][]{Kuulkers94,Straaten03}.
We produced Fourier power density spectra (PDS) using the 2--60 keV data from the $\sim$125 $\mu$s (1/8192 s)
time resolution Event mode. We created Leahy-normalised PDS for each 16-s data segment with time bins of
1/8192 s, such that the lowest available frequency is 1/16 Hz, and the Nyquist frequency is 4096 Hz. We fitted the PDS
in the 50--4000 Hz range using a combination of one or two Lorentzians to fit the kHz QPOs, a constant for the
Poissonian noise, and if needed, an extra Lorentzian to fit the broad-band noise at frequencies between 50 Hz and the
frequency range spanned by the kHz QPOs.
\begin{figure*}
\resizebox{2\columnwidth}{!}{\rotatebox{0}{\includegraphics[clip]{figure1.ps}}}
\caption{Colour-colour, hard colour vs. intensity and soft colour vs. intensity
diagrams (upper, lower left and lower right panels, respectively) of
4U~1636--53 for all available \textit{RXTE} pointed observations (grey
circles).
Colours and intensities during \textit{XMM-Newton} observations
computed from pointed
\textit{RXTE} observations (see Section~\ref{sec:obs})
are marked with different symbols following the legend at the upper-left corner. The position of the source on the colour-colour diagram is parametrized by the length of the black solid line that represents the coordinate $S_a$.
For each observation we report the frequency of the upper kHz QPO (when detected)
and the flux of the line (Fe Flux) in units of $10^{-3}$ photons cm$^{-2}$ s$^{-1}$ for fits with a \textsc{kyrline} model with $a_*=0.27$ \citep[see][for more details]{Sanna13}.
}
\label{fig:cc}
\end{figure*}
\subsection{Spectral analysis}
\label{models}
For this work we use the results of the spectral analysis by \citet{Sanna13}, that we briefly summarised hereunder.\\
The energy spectrum of 4U 1636-53 can be well fitted with a multicolour disc blackbody (\textsc{DISKBB}) plus a single-temperature blackbody (\textsc{BBODY}) and a thermally comptonized component (\textsc{NTHCOMP}) to account for the thermal emission from the accretion disc, the thermal emission from the NS surface/boundary layer, and the high-energy emission from the corona-like region surrounding the systems, respectively. The soft seed photons of the comptonised component were assumed to come from the accretion disc. Besides the continuum emission, the data require a emission-like feature to model prominent residuals in the energy range 4-9 keV, around the Fe-K$\alpha$ emission line region. \citet{Sanna13} fitted this feature with a set of so-called phenomenological models (\textsc{gaussian, diskline, laor}, and \textsc{kyrline}), that only model the Fe emission line, and with two reflection models (\textsc{rfxconv}, and \textsc{bbrefl}), which self-consistently model the whole reflection spectrum, of which the Fe-K$\alpha$ emission line is the strongest feature. \citet{Sanna13} showed that the Fe-K$\alpha$ emission line is well fitted by a symmetric gaussian profile characterised, however, by a large breadth, which is difficult to explain with mechanisms other than relativistic broadening.
\section{Results}
\subsection{Long term spectral behaviour of 4U 1636--53}
In Figure~\ref{fig:cc} we show the colour-colour (top), hard colour vs. intensity (bottom left), and the soft colour vs. intensity (bottom right)
diagrams for all the available \textit{RXTE} pointed observations. We use different symbols and colours to mark the location of the source
during the \textit{XMM-Newton} observations as estimated by the simultaneous \textit{RXTE} observations (see Section~\ref{sec:obs}).
Although 4U 1636--53 is a persistent X-ray source, it shows variations in intensity up to a factor of 6, following a narrow track
in the colour-colour and colour-intensity diagrams \citep{Belloni07,Altamirano08a}. Using data from the all-sky monitor (ASM)
onboard \textit{RXTE}, \citet{Shih05} found a long-term modulation with a period of 30--40 days, which corresponds
to the regular transition between the hard and soft states \citep{Belloni07}.
The \textit{XMM-Newton} observations sample different parts of the diagrams shown in Figure~\ref{fig:cc}. During Obs.~1 and 6, the intensity was low and the spectrum was hard (see Table 4 and 5 in \citealt{Sanna13} for more details on the spectral properties). Observations 2 to 5 were all done when the source was bright, and the source spectrum was relatively soft. The shape of the colour-colour diagram in the top panel of Figure~\ref{fig:cc} shows that 4U 1636--53 belongs to the so-called Atoll class \citep{Hasinger89}. The upper right corner of the diagram represents times when the source is in the so-called transitional state. As the mass accretion rate increases, the source first moves down to the left of the diagram, and then to right to the soft state \citep[see, e.g.,][]{vdKlis06}. The \textit{XMM-Newton} observations did not uniformly sample the full range of source colours and intensities, but covered two small regions of the diagram. Obs.~1 and 6 sampled the source in the transitional state, while Obs.~2--5 covered the soft state. The position of the source on the colour-colour diagram is parameterised by the length of the coordinate $S_a$ \citep{Mendez99}, which approximates the shape of the diagram with a curve. The length of $S_a$ is arbitrarily normalised to the distance between $S_a=1$ at the top right corner, and $S_a=2.5$ at the bottom right corner, via $S_a=2$ at the bottom left vertex of the colour-colour diagram (see Figure~\ref{fig:cc}). Similar to the $S_z$ coordinate in the Z sources \citep{Vrtilek90}, $S_a$ is considered to map mass accretion rate \citep[see e.g.,][]{Hasinger89,Kuulkers94, Mendez99}. As reported in \citet{Sanna13}, the order of the six \textit{XMM-Newton/RXTE} observations, going from the transitional state to the soft state, according with their $S_a$ values is: 1--6--2--3--5--4.
\begin{comment}
\begin{table*}
\caption{Inner radius inferred from iron line measurements (from \citealt{Sanna13})}
\begin{tabular}{|c|c|c|c|c|c|c|}
& Obs.~1 & Obs.~2 & Obs.~3 & Obs.~4 & Obs.~5 & Obs.~6 \\
& $R_{in}$(GM/c$^2$) & $R_{in}$(GM/c$^2$)& $R_{in}$(GM/c$^2$) & $R_{in}$(GM/c$^2$)&$R_{in}$(GM/c$^2$) & $R_{in}$(GM/c$^2$)\\
\hline
\textsc{diskline} & $10.6^{+1.5}_{-2.6}$ & $10.7^{+4.5}_{-2.4}$ & $8.4^{+0.7}_{-1.5}$ & $6.0^{+2.8}_{-0.0*}$ & $6.5^{+0.6}_{-0.5*}$ & $8.0^{+6.3}_{-2.0*}$ \\\\
\textsc{laor} &$10.8^{+0.6}_{-2.9}$ & $4.0^{+5.6}_{-0.8}$ & $2.3^{+0.2}_{-0.5}$ & $2.8^{+1.2}_{-0.8}$ & $2.0^{+0.4}_{-0.2}$& $6.2\pm1.9$ \\\\
\textsc{kyrline} a$_*$=0 & $10.8^{+2.0}_{-1.3}$ & $6.3^{+1.1}_{-0.3*}$ & $13.1^{+1.2}_{-1.4}$ & $6.0^{+1.9}_{-0.0*}$& $6.2^{+0.4}_{-0.2*}$& $12.2^{+1.9}_{-2.6}$ \\\\
\textsc{kyrline} a$_*$=0.27 & $10.6^{+1.9}_{-1.2}$ & $12.5^{+2.9}_{-2.0}$ & $13.1^{+1.3}_{-1.5}$ & $5.7^{+2.0}_{-0.6*}$& $5.9^{+0.4}_{-0.8*}$& $12.2^{+1.7}_{-2.6}$ \\\\
\textsc{kyrline} a$_*$=1 & $9.9^{+1.9}_{-1.1}$ & $11.8^{+3.2}_{-1.7}$ & $12.9^{+3.2}_{-1.7}$ & $5.3^{+1.4}_{-1.9}$& $2.6\pm0.1$&$12.1^{+2.0}_{-2.1}$ \\\\
\textsc{reflection} & $12.6^{+1.5}_{-1.8}$ & $7.8^{+3.1}_{-2.7}$ & $15.4\pm2.7$ & $5.6^{+2.2}_{-0.3}$& $5.4\pm0.1$& $19.1^{+7.6}_{-10.8}$ \\
\hline
\end{tabular} \\
\flushleft Notes: A * means that the radius pegged at the hard limit of the range.
\label{tab:line_radius}
\end{table*}
\end{comment}
\begin{table*}
\caption{Inner radius inferred from the iron line measurements sorted as a function of the $S_a$ parameter (from \citealt{Sanna13})}
\begin{tabular}{|c|c|c|c|c|c|c|}
& Obs.~1 & Obs.~6 & Obs.~2 & Obs.~3 & Obs.~5 & Obs.~4 \\
& $R_{in}$(GM/c$^2$) & $R_{in}$(GM/c$^2$)& $R_{in}$(GM/c$^2$) & $R_{in}$(GM/c$^2$)&$R_{in}$(GM/c$^2$) & $R_{in}$(GM/c$^2$)\\
\hline
\textsc{diskline} & $10.6^{+1.5}_{-2.6}$ & $8.0^{+6.3}_{-2.0*}$ & $10.7^{+4.5}_{-2.4}$ & $8.4^{+0.7}_{-1.5}$ & $6.5^{+0.6}_{-0.5*}$ & $6.0^{+2.8}_{-0.0*}$ \\\\
\textsc{laor} &$10.8^{+0.6}_{-2.9}$ & $6.2\pm1.9$ & $4.0^{+5.6}_{-0.8}$ & $2.3^{+0.2}_{-0.5}$ & $2.0\pm{0.3}$&$2.8^{+1.2}_{-0.8}$ \\\\
\textsc{kyrline} a$_*$=0 & $10.8^{+2.0}_{-1.3}$ & $12.2^{+1.9}_{-2.6}$ &$6.3^{+1.1}_{-0.3*}$ & $13.1\pm{1.3}$ &$6.2^{+0.4}_{-0.2*}$& $6.0^{+1.9}_{-0.0*}$ \\\\
\textsc{kyrline} a$_*$=0.27 & $10.6^{+1.9}_{-1.2}$ &$12.2^{+1.7}_{-2.6}$ & $12.5^{+2.9}_{-2.0}$ & $13.1\pm{1.4}$ & $5.9^{+0.4}_{-0.8*}$ & $5.7^{+2.0}_{-0.6*}$ \\\\
\textsc{kyrline} a$_*$=1 & $9.9^{+1.9}_{-1.1}$ &$12.1\pm{2.1}$ & $11.8^{+3.2}_{-1.7}$ & $12.9^{+3.2}_{-1.7}$ & $2.6\pm0.1$&$5.3^{+1.4}_{-1.9}$ \\\\
\textsc{reflection} & $12.6\pm{1.7}$ &$19.1^{+7.6}_{-10.8}$ & $7.8^{+3.1}_{-2.7}$ & $15.4\pm2.7$ & $5.4\pm0.1$&$5.6^{+2.2}_{-0.3}$ \\
\hline
\end{tabular} \\
\flushleft Notes: A * means that the radius pegged at the hard limit of the range.
\label{tab:line_radius}
\end{table*}
\subsection{Iron line and measurements of the inner accretion radius}
In Table~\ref{tab:line_radius} we report the values
of the inner disc radius for the different models used by \citet{Sanna13} to fit the line. The observations in Table~\ref{tab:line_radius}
are sorted according to their $S_a$ values. As already noted in \citet{Sanna13}, the inner disc radius
inferred from the relativistic profile of the iron line in 4U 1636--53 does not change in correlation with the position in
the colour-colour diagram contrary to what is predicted by the standard accretion disc model \citep[see e.g.,][and references therein]{Done07}.
\citet{Sanna13} also found that other parameters used to fit the line, such as line energy, source inclination and equivalent
width, do not show any clear correlation with the source state \citep[see Figure~6 in][]{Sanna13}.
\subsection{kHz QPOs and measurements of the inner accretion radius}
\label{sec:qpo_analysis}
We detected one or two kHz QPOs in the average PDS of each \textit{RXTE} observation that
was performed simultaneously with an \textit{XMM-Newton} observation. In Obs.~1 and 6 we detected a
strong broad-band noise component extending up to few hundred Hz plus a single kHz QPO at
$\sim$480 and $\sim$540 Hz, respectively. The kHz QPOs in Obs.~1 and 6 have rms
amplitudes of $\sim$12\% and $\sim$14\%, respectively. The overall power-spectral
shape in both cases (not shown) is similar to those previously observed in the transitional state of 4U 1636--53
\citep[e.g., intervals A--C in][]{Altamirano08a} and other sources \citep[e.g.,][]{Straaten02,Straaten03}.
In Obs.~4 we detected a single kHz QPO at $\sim$920 Hz, with an rms amplitude of about 7\%.
Twin kHz QPOs at $\sim$600 and $\sim$905 Hz are present in Obs.~2 with rms amplitudes of $\sim$7\% and $\sim$10\%,
respectively, and at around 700 and $\sim$1020 Hz in Obs.~3, both with rms amplitudes of $\sim$6\%. In Obs.~5 we
detected two peaks with a frequency separation significantly lower than the average frequency difference
between the lower and the upper kHz QPOs previously reported for this source \citep{Mendez98,Jonker02,Mendez07,Altamirano08a}. We investigated the evolution of the QPO frequency with time in this observation, and we found that the QPO
signal appears sporadically during the long observation, and when it is detectable the frequency changes with
time between $\sim$750 Hz and $880$ Hz. It is likely that Obs.~5 showed only one kHz QPO that
moved in frequency, we therefore applied the shift-and-add
method \citep{Mendez98} to recover the properties of the QPO. Note that we did not find the upper kHz QPO (see Figure~\ref{fig:qpo}), and that
the frequency reported for the kHz QPO in Obs.~5 is the weighed average of the frequencies spanned by the QPO during the observation.
In Figure~\ref{fig:qpo} we show the Leahy normalised power density spectra at high frequencies for the six observations, and in Table~\ref{tab:kHz}
we give the best-fitting parameters of the kHz QPOs.
\begin{figure*}
\begin{center}$
\begin{array}{ccc}
\includegraphics[scale=0.41]{figure2.ps} &
\includegraphics[scale=0.41]{figure3.ps}&
\includegraphics[scale=0.41]{figure4.ps}\\
\includegraphics[scale=0.41]{figure5.ps} &
\includegraphics[scale=0.41]{figure6.ps}&
\includegraphics[scale=0.41]{figure7.ps}\\
\end{array}$
\end{center}
\caption{Leahy normalised power density spectra for Obs.~1--6 of 4U 1636--53, calculated from the \textit{RXTE} observations.
The power density spectra were fitted with a model consisting of a constant, one or two Lorentzians to fit the kHz QPOs, and
(if required) a Lorentzian to model the residual broad band noise at low frequencies. For Obs.~5 we show the kHz QPO after we applied the shift-and-add method \citep[see][for more details]{Mendez98}. }
\label{fig:qpo}
\end{figure*}
In those observations in which we detected two simultaneous QPOs we can readily identify the lower and the upper kHz QPOs.
For the other observations we used the frequency vs. hard colour digram that, as shown in \citet[][see also \citealt{Belloni07}]{Sanna12a}, can be
used to identify the lower and upper kHz QPOs in 4U 1636--53.
In Figure~\ref{fig:freq-hard} we show the centroid frequency of the kHz QPOs detected in 4U 1636--53 as a function
of hard colour, with different symbols for the lower (grey filled bullets) and the upper (grey empty bullets) kHz QPOs.
On top of that we show the kHz QPOs detected in Obs.~1--6. As expected, Figure~\ref{fig:freq-hard} confirms
that the two simultaneous QPOs detected in Obs.~2 and 3, are indeed the lower and the upper kHz QPOs.
Figure~\ref{fig:freq-hard} also shows that the QPOs detected in Obs.~1 and 6 are both upper kHz QPOs,
while the QPOs in Obs.~4 and 5 are both the lower kHz QPOs.
\begin{figure}
\resizebox{1\columnwidth}{!}{\rotatebox{0}{\includegraphics[clip]{figure8.ps}}}
\caption{Frequency of the kHz QPOs in 4U 1636--53 as a function of hard colour. Grey filled
and empty bullets represent the lower and the upper kHz QPOs, respectively \citep[see][]{Sanna12a}. With large symbols we
represent the kHz QPOs detected in Obs.~1--6. The symbols have the same meaning as in Figure~\ref{fig:cc}.}
\label{fig:freq-hard}
\end{figure}
\begin{comment}
\begin{table*}
\caption{kHz QPOs parameters}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
& & Obs.~6& Obs.~1 & Obs.~2 & Obs.~3 & Obs.~5 & Obs.~4 \\
\hline
\multirow{3}{*}{$L_{\ell}$} & $\nu$ (Hz) &$597\pm13$ & -- & $700\pm5$ & $917\pm3$ & $795.1\pm0.5$ & -- \\
& FWHM (Hz) & $115^{+42}_{-29}$ &-- & $99\pm14$ & $15\pm6$ & $11.5\pm1.3$& -- \\
& rms (\%) & $6.7\pm0.8$ &-- & $8.5\pm0.4$ & $6.8\pm0.9$ & $11.7\pm0.3$& -- \\
\hline
\multirow{3}{*}{$L_{u}$} & $\nu$ (Hz) &$906\pm5$ & $482\pm11$ & $1020\pm14$ & -- & --&$537\pm18$ \\
& FWHM (Hz) &$99\pm13$ & $178\pm45$ & $140^{+48}_{-37}$ & -- & -- & $247^{+72}_{-58}$\\
& rms (\%) & $9.8\pm0.5$ &$11.5\pm1.5$ & $6.3\pm0.7$ & -- & -- &$14.1\pm1.4$\\
\hline
\end{tabular} \\
\flushleft Notes: $L_{\ell}$ and $L_{u}$ stand for the lower and the upper kHz QPO, respectively. .
\label{tab:kHz}
\end{table*}
\end{comment}
\begin{table*}
\caption{Parameters of the kHz QPOs in 4U 1636--53 sorted as a function of the $S_a$ parameter}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
& & Obs.~1 & Obs.~6 & Obs.~2 & Obs.~3 & Obs.~5 & Obs.~4 \\
\hline
\multirow{3}{*}{$L_{\ell}$} & $\nu$ (Hz) & -- &--& $597\pm13$ & $700\pm5$ & $795.1\pm0.5$ & $917\pm3$ \\
& FWHM (Hz) & -- & -- & $115^{+42}_{-29}$ & $99\pm14$ & $11.5\pm1.3$ &$15\pm6$ \\
& rms (\%) & -- & -- & $6.7\pm0.8$ & $8.5\pm0.4$ & $11.7\pm0.3$ & $6.8\pm0.9$ \\
\hline
\multirow{3}{*}{$L_{u}$} & $\nu$ (Hz) &$482\pm11$ &$537\pm18$ & $906\pm5$ & $1020\pm14$ & -- & -- \\
& FWHM (Hz) & $178\pm45$& $247^{+72}_{-58}$ & $99\pm13$ & $140^{+48}_{-37}$ & -- & -- \\
& rms (\%) & $11.5\pm1.5$&$14.1\pm1.4$ & $9.8\pm0.5$ & $6.3\pm0.7$ & -- & -- \\
\hline
\end{tabular} \\
\flushleft Notes: $L_{\ell}$ and $L_{u}$ stand for the lower and the upper kHz QPO, respectively. For Obs.~5 we report the kHz QPO parameters after applying the shift-and-add method \citep{Mendez98}.
\label{tab:kHz}
\end{table*}
In order to investigate whether the frequency of the detected kHz QPOs remains approximately constant in time,
we studied the dynamical power spectra \citep[e.g.,][]{Berger96} using different frequency and time binning factors.
We were unable to follow the time evolution of the upper kHz QPO in any of the observations. We were able to trace the frequency of the lower kHz QPO only in two cases: During Obs.~3 the QPO frequency varied between $\sim$620 Hz and
$\sim$810 Hz while, as already mentioned, during Obs.~5 the QPO moved between $\sim$750 Hz and $\sim$880 Hz.
In accordance with the scenario of an accretion disc truncated at a larger radii in the
hard than in the soft state, and a disc extending very close to the NS surface in the soft state \citep{GierlinskiDone02, Done07},
the frequency of the upper kHz QPO is lower (larger inner radius) in Obs.~6 and 1 (hard state) than in Obs.~2 and 3 (soft state).
\subsection{Iron lines and kHz QPOs as tracers of the inner radius of the accretion disc}
Both kHz QPOs and relativistically-broadened iron lines likely reflect properties of
the accretion flow in the inner edge of the accretion disc \citep[e.g.,][]{Fabian89,MillerLambPsaltis98}. To investigate whether the two interpretations match,
we compared the inner radius estimated from the frequency of the kHz QPO and the profile of the iron line
when both were detected simultaneously.
Most models predict that the upper kHz QPO frequency in LMXBs represents the orbital
frequency at the inner edge of the accretion disc \citep[e.g.,][]{MillerLambPsaltis98,Stella98}. The expression
for the orbital frequency in the space time outside a slowly and uniformly rotating NS is
$\nu_{\phi}=\nu_k(1+a_*(R_g/r)^{3/2})^{-1}$ where $\nu_k = (1/2\pi)\sqrt{GM/r^3}$ is the Keplerian frequency,
$a_*=Jc/GM^2$ is the angular momentum parameter, $R_g=GM/c^2$ is the gravitational radius,
\textit{G} is the gravitational constant, \textit{M} the mass of the NS star, \textit{r} the radial
distance from the center of the NS star, and \textit{c} is the speed of light.
In order to estimate the inner radius we need both the angular momentum parameter and the NS mass. As
reported in \citet{Sanna13}, taking into account the spin frequency of 581 Hz of the NS in 4U 1636--53 \citep{Zhang97,Giles02,Strohmayer02},
we estimated $a_*$ to be $~0.27$. On the other hand, the mass of the NS in this system is unknown, therefore
we calculated the inner radius of the accretion disc, $R_{in}$, as a function of the NS mass. The lack of information on the NS
mass does not allow us to directly compare $R_{in}$ inferred from the kHz QPO and the iron line simultaneously,
however we can compare the trend of $R_{in}$ as a function of other properties of the source, e.g., position in the
colour-colour diagram or intensity, and test whether there is a single value of the mass of the NS for which the two
different estimates of the inner disc radius agree.
\begin{figure}
\centering
\caption{}
\begin{subfigure}[b]{1\columnwidth}
\centering
\includegraphics[width=0.75\textwidth]{figure9.eps}
\caption{Probability distribution function of the inner radius of the accretion disc as a function NS mass in 4U 1636--53, inferred from the upper kHz QPO in Obs.~1. Colour density represents the confidence level.}
\label{fig:radius_qpo}
\end{subfigure}\\
\begin{subfigure}[b]{1\columnwidth}
\centering
\includegraphics[width=0.75\textwidth]{figure10.eps}
\caption{Probability distribution function of the inner radius of the accretion disc as a function of NS mass in 4U 1636--53, inferred from the fit of the iron emission line in Obs.~1 using the relativistic line model \textsc{kyrline} with $a_*=0.27$. Colour density represents the confidence level. }
\label{fig:radius_line}
\end{subfigure}\\
~
\begin{subfigure}[b]{1\columnwidth}
\centering
\includegraphics[width=\textwidth]{figure11.eps}
\caption{Joint probability distribution function (top-right panel), marginal probability distribution functions for the inner disc radius (top-left panel) and the NS mass (bottom panel) for 4U 1636--53, calculated combining the probability density distribution functions from Figures~\ref{fig:radius_qpo} and \ref{fig:radius_line}. }
\label{fig:joint_prop}
\end{subfigure} \label{fig:qpo_lines}
\end{figure}
To explain in more detail how we compared the simultaneous timing and spectral information, in Figure~\ref{fig:qpo_lines}
we show a step-by-step example for the case of Obs.~1. As reported in Table~\ref{tab:kHz}, Obs.~1 showed
an upper kHz QPO at a frequency of $\sim$480 Hz. In Figure~\ref{fig:radius_qpo} we show the inner radius of the accretion disc
in units of $R_g$ versus the NS mass inferred from the relation between $\nu_{\phi}$, $M$ and $R_{in}$.
Colour density represents the probability density distribution (PDF) of the
inner radius taking into account the uncertainties from the fit of the QPO frequency. In Figure~\ref{fig:radius_line} we show
the inner radius in units of $R_g$ inferred from the iron emission line when we fitted it with the relativistic line model \textsc{kyrline} with $a_*=0.27$ \citep{Sanna13}.
Similar to Figure~\ref{fig:radius_qpo}, the PDF of the inner radius is shown in colour density.
Note that the radius derived from the iron line does not depend upon the NS mass; however, for practical purposes, we
plotted the PDF of the inner radius using a similar layout as for the kHz QPO.
In Figure~\ref{fig:joint_prop} (central panel) we show the joint probability distribution function of the inner radius and the NS mass derived from
the iron line profile and the frequency of the kHz QPO. In Figure~\ref{fig:joint_prop} we also show the marginal distributions for the
NS mass and the inner disc radius in units of $R_g$, calculated by integrating the joint PDF
over the radius and the mass, respectively.
From this example, we find that for this observation of 4U 1636--53 the kHz QPO frequency
and the iron line modelling are consistent for a NS mass of $\sim$1.9 M$_\odot$, and an
accretion disc extending down to $\sim$11 R$_g$; the NS mass is slightly high,
but still consistent with most NS equations of state \citep[see, e.g.,][]{Lattimer07}. The fact that the NS mass estimate is consistent
with theoretical expectations suggests that, for this particular case, the kHz QPO and iron line interpretations both hold.
Since we have observations mapping different positions in the colour-colour diagram, we can test whether the kHz QPOs and
the Fe line give a consistent value of the NS mass for all accretion states. To do that we used the four observations,
two in the transitional state (Obs.~1 and 6) and two in the soft state (Obs.~2 and 3), where
we detected both the upper kHz QPOs and the broad iron line simultaneously. For completeness, we used the inner radius
estimated from fits of the iron line profile with different line models \citep[see Table~\ref{tab:line_radius} for the list of models; see][for a discussion
of the line models]{Sanna13}.
In Figure~\ref{fig:masses} we show, for different line models, the marginal probability distribution functions of the NS mass for the four
observations with both the upper kHz QPOs and the iron emission line. The mass values derived
from different observations do not yield consistent results, regardless of the iron line model. We further notice
that the NS mass values in 4U 1636--53 derived from this method span a range between $\sim$0.5 M$_\odot$ and $\sim$3.0 M$_\odot$,
with the exception of the mass inferred from the fits of the line with the \textsc{laor} model, which spans a wider range (1.2--10 M$_\odot$).
\section{discussion}
We detected kHz QPOs in all the \textit{RXTE} observations simultaneous with the six \textit{XMM-Newton} observations of the NS LMXB 4U 1636--53 for which \citet{Sanna13} studied the broad iron line in the X-ray spectrum. Combing the measurements of the frequency of the kHz QPOs and the parameters of the iron lines in 4U 1636--53 we investigated the hypothesis that both the iron line and the kHz QPOs originate at (or very close to) the inner radius of the accretion disc in this system.
From these observations we found that the inner disc radius, deduced from the upper kHz QPO frequency,
decreases as the spectrum of the source softens, and the inferred mass accretion rate increases.
On the other hand, the inner radius estimated from the modelling of the relativistically-broadened iron line
did not show any clear correlation with the source state, except for the line model \textsc{laor} for which the inferred
inner disc radius consistently decreases going from the transitional state to the soft state (see Table~\ref{tab:line_radius}).
Combining the disc radius inferred from the frequency of the upper kHz QPO and the iron line profile,
we found that the mass of the NS in 4U 1636--53 deduced from the four observations are inconsistent with being the same. A similar conclusion was drawn by \citet{Cackett10} from the first three observations in the sample that we studied here.
The latter result implies that either the upper kHz QPO frequency does not reflect the orbital frequency at the inner edge of the disc, the Fe line profile is not (only) shaped by relativistic effects, the models used to fit the iron line are incorrect, or the the kHz QPOs and the Fe line are not produced in the same region of the accretion disc.
We assumed that the upper kHz QPO is the one which reflects the orbital frequency at the inner edge of
the accretion disc \citep[e.g.,][]{MillerLambPsaltis98, Stella98}. There are, however, alternative models that associate
instead the lower kHz QPO to the orbital disc frequency \citep[e.g.,][]{Meheut09}. If this is the case, then the radius profile showed in
Figure~\ref{fig:radius_qpo} would shift to higher values of $R_{in}$, and therefore, the mass for which $R_{in}$ from kHz QPOs and
iron lines would match will also shift to higher values. Since the difference in frequency between upper and lower kHz QPOs in 4U 1636--53 is more
or less constant across the colour-colour diagram \citep[e.g.,][]{Mendez98, Jonker02}, using the lower kHz QPOs would lead to similar results as those shown in Figure~\ref{fig:masses}, with NS masses shifted toward higher values.
Under the assumption that the kHz QPOs are generated in the accretion disc, and considering circular orbits in the equatorial plane for Kerr space-time, the only characteristic frequencies (other than the orbital frequency) that match the observed kHz QPO frequency range are the periastron precession and the vertical epicyclic frequencies. Interpreting the upper kHz QPO as the vertical epicyclic frequency and combining the inner radius estimates with the iron line findings we found results consistent with the ones reported above. On the other hands, interpreting the upper kHz QPO as the periastron precession frequency led to meaningless NS mass values (lower than 0.1 M$_\odot$).
The kHz QPOs may still reflect the orbital (quasi-Keplerian) frequency at a radius far from the inner edge of the disc.
A possible scenario to reconcile this idea, for instance, could be a mechanism that amplifies the orbital frequencies of matter orbiting within a narrow ring in the disc to produce the QPO. The process could be similar to the lamp-post model by \citet{Matt91}. Such mechanism, however, must be able to pick a narrow range of radii in order to reproduce the observed high QPO coherence values \citep[e.g.,][]{Barret05b}. For instance, for a 1.8 solar mass neutron star with a QPO at 800 Hz, if this is the Keplerian frequency in the disc, the putative mechanism should pick a ring of $\sim$600 m to produce a QPO with $Q = 200$. Besides generating the kHz QPOs, this mechanism should also affect other properties of the disc, such as the emissivity index or the ionisation balance, which would in turn affect the properties of the iron emission line.
From the behaviour of the time derivative of the frequency of the lower kHz QPO, \citet{Sanna12a} found that the kHz QPOs (both the lower and the upper) in 4U 1636--53 are consistent with the orbital frequency at the sonic radius in the accretion disc. We also note that
the frequency of the upper kHz QPO increases monotonically across the colour-colour diagram. All this lends
support to the interpretation of the kHz QPO reflecting the orbital frequency at the inner edge of the accretion disc.
\begin{figure*}
\begin{center}$
\begin{array}{cc}
\includegraphics[scale=0.23]{figure12.ps} &
\includegraphics[scale=0.23]{figure13.ps}\\
\includegraphics[scale=0.23]{figure14.ps}&
\includegraphics[scale=0.23]{figure15.ps} \\
\includegraphics[scale=0.23]{figure16.ps}&
\includegraphics[scale=0.23]{figure17.ps}\\
\end{array}$
\end{center}
\caption{Marginal probability distribution functions of the NS mass in 4U 1636--53 inferred from simultaneous
measurements of the upper kHz QPO and the iron emission line, for four different observations represented with different colours. Different panels represent
different models used to fit the iron line profile. }
\label{fig:masses}
\end{figure*}
Besides Doppler and relativistic effects, the iron emission line can be broadened by other processes.
For example, the broadening may be ( partially) due to Compton
scattering in a disc corona (\citealt{Misra98}; \citealt{Misra99};
see also \citealt{Reynolds00}; \citealt{Ruszkowski00}; \citealt{Turner02}, and \citealt{Ng10}).
However, \citet{Sanna13} showed that Compton broadening alone cannot explain the broad profile of the iron emission line in 4U 1636--53. \citet{Titarchuk03} argued that the red
wing of the Fe-K$\alpha$ lines is not due to Doppler/relativistic
effects, but to relativistic, optically-thick,
wide-angle (or quasi-spherical) outflows (\citealt{Laming04}; \citealt{Laurent07}, see however
\citealt{Miller04}; \citealt{Miller07}, and \citealt{Pandel08}). As explained by \citet{Titarchuk09}, in this scenario the
red wing of the iron line is formed in a strong extended wind
illuminated by the radiation emanating from the innermost part of the
accreting material.
One of the main predictions of this model is that all high-frequency
variability should be strongly suppressed.
The fact that we detected kHz QPOs and broad iron lines simultaneously
in 4U~1636--53 casts doubt on this interpretation.
Although our findings contradict this scenario, the model under
discussion has been developed for black holes, so it is not clear how
the boundary layer or the neutron star surface could change these
predictions.
Compared to other sources, the iron line in 4U 1636--53 shows unusual properties; for instance, the best-fitting inclination is $i \gtrsim 80^\circ$ \citep{Pandel08,Cackett10,Sanna13}, which is at odds with the lack of dips or eclipse in the light curve. \citet{Pandel08} proposed that the line profile could be the blend of
two (or more) lines, for example, formed at different radii in the disc, or
due to separate regions with different ionisation balance. If this is correct,
the total line profile would be the result of iron lines at different energies.
To proceed further with this idea would require to solve the
ionisation balance in the accretion disc where the line is formed.
\citet{Sanna13} investigated this scenario by fitting the
reflection spectrum with a self-consistent ionised reflection model, but
they did not find any supporting evidence for this idea \citep[see also][]{Cackett10}.
The fact that in Obs.~3 and 5 the kHz QPO frequency significantly
varied within the 20-30~ksec required to detect the iron line
suggests that the iron line profile we model may be affected by
changes of the disc during those 20-30~ksec.
If the kHz QPO frequency depends upon the inner disc radius,
the iron line profile we observe would be the average of different line
profiles, one for each value of the inner disc radius. This is independent of whether
the kHz QPO frequency reflects the orbital disc frequency, or whether
the relation between frequency and inner radius is more complicated.
The line energy or the disc emissivity could also vary if the inner disc radius changes.
To proceed further, detailed simulations (assuming scenarios in which
only the $R_{in}$ changes with time, as well as scenarios in which all
line parameters change) are needed to test to what extent
changes in the accretion flow on timescales of $\sim$30~ksec
(approximately the time needed with present instruments
to fit the line accurately) can affect the final line profile.
A similar consideration applies to the inner radius inferred from
the kHz QPO frequency. As mentioned in Section~\ref{sec:qpo_analysis},
in Obs.~3 and 5 the frequency of the lower kHz QPO spanned a frequency range of $\sim$200 Hz during the $\sim$25 ks observation. Although, we did not directly see
the upper kHz QPO changing frequency with time, it is likely that the upper kHz QPO followed the
lower one. If this was the case, then the full width at half maximum (FWHM)
of the upper kHz QPO observed contains information on the frequency range
covered by the QPO during the observation. To bring this information into the inner disc
radius estimates, we should use the QPO FWHM instead of the frequency error (which is relatively small)
to calculate the probability distribution function of the inner radius of the accretion disc. In Figure~\ref{fig:diskline_fwhm} we show the marginal probability distribution functions
of the NS mass inferred from the four observations combining the upper kHz QPOs and the iron lines
modelled with \textsc{diskline}. Solid and dotted lines represent the marginal probability distribution functions using the
error in the QPO frequency and the half-width half-maximum (HWHM) as error, respectively.
In the latter case the marginal probability distribution functions of the NS mass show a broader profile, and
the range of mass values where they are consistent increases (although the overlapping area is still small).
By combining the marginal probability distribution functions we get the mass profile (joint probability) for which,
in the 4 observations, kHz QPO and iron line estimates of the inner disc radius are consistent.
This is shown in the inset of Figure~\ref{fig:diskline_fwhm}. The most likely value of the NS mass,
for this specific case, ranges between $\sim$1.1 and $\sim$1.5 M$_\odot$, which is consistent with theoretical NS mass predictions \citep[e.g.,][]{Lattimer07}.
However, it should be
noticed that in Figure~\ref{fig:diskline_fwhm}, two out of the four PDFs (Obs.~1 and 2) marginally overlap, therefore the final joint probability function is likely not fully representative of all observations. Statistically speaking, the overlapping area between the intersecting marginal distributions in Figure~\ref{fig:diskline_fwhm} represents the likelihood of measuring 4 values $M_i$ of the NS mass $M_0$ (assuming the mass is always the same), and the hypothesis $H$ that one of the kHz QPOs is Keplerian, the Fe line is relativistic, and both phenomena arise from the same region of the accretion disc is valid. In Table~\ref{tab:probability} we report the the NS mass values and the likelihood for the different models used to fit the Fe line profile. It is interesting to notice that the highest likelihood of the data given the model is obtained when the Fe-K$\alpha$ emission line is modelled with \textsc{diskline}.
\begin{table}
\resizebox{0.9\columnwidth}{!}{\begin{minipage}{\columnwidth}
\begin{tabular}{lcccc}\hline
\multicolumn{1}{c}{} &
\multicolumn{2}{c}{$\delta \nu$} &
\multicolumn{2}{c}{HWHM}\\\hline
\multicolumn{1}{c}{Fe line model} &
\multicolumn{1}{c}{$M_0(M_\odot)$} &
\multicolumn{1}{c}{$\mathcal{L}$} &
\multicolumn{1}{c}{$M_0(M_\odot)$} &
\multicolumn{1}{c}{$\mathcal{L}$} \\\hline
\textsc{Diskline} & 1.4$\pm$0.2 & 8.2\e{-3} &1.3$\pm$0.2 &1.4\e{-1} \\
\textsc{Laor} &2.6$\pm$0.6& \textless\, 1\e{-9} &3.0$\pm$1.2& 1.8\e{-6} \\
\textsc{Kyrline} a$_{*}$=0 &0.9$\pm$0.3 &\textless\, 1\e{-9} &0.9$\pm$0.3& \textless\, 1\e{-9}\\
\textsc{Kyrline} a$_{*}$=0.27&0.8$\pm$0.2 & \textless\, 1\e{-9} &0.7$\pm$0.1& 3.1\e{-3}\\
\textsc{Kyrline} a$_{*}$=1&0.8$\pm$0.2 & \textless\, 1\e{-9}&0.8$\pm$0.1 &1.9\e{-3}\\
\textsc{Reflection} &0.6$\pm$0.1 & 3.7\e{-3} &0.6$\pm$0.1& 2.5\e{-2}\\\hline
\end{tabular}
\end{minipage}}
\caption{Likelihood ($\mathcal{L}$) values of measuring the 4 values NS $M_i$ if the NS mass $M_0$ is always the same and under the hypothesis $H$, for different models of the Fe emission line. The two columns represent the likelihood measured from the marginal probability distribution functions of the NS mass in 4U 1636--53 calculated using the error in the QPO frequency ($\delta \nu$) and the half-width half-maximum of the upper kHz QPOs (HWHM), respectively.}
\label{tab:probability}
\end{table}
The frequency of the kHz QPO and the source intensity (likely the mass accretion rate) are degenerate on long (longer than $\sim$ a day) time-scales \citep[``parallel tracks'',][]{Mendez99}: The same QPO frequency may appear at very different source intensities. It remains to be seen whether this phenomenon can affect some properties of the disc, such as the emissivity index or the ionisation balance, which would affect the profile of the iron line, and hence the inferred value of the inner radius of the accretion disc.
\begin{figure}
\resizebox{1\columnwidth}{!}{\rotatebox{0}{\includegraphics[clip]{figure18.ps}}}
\caption{Marginal probability distribution functions of the NS mass in 4U 1636--53 using the iron line model \textsc{Diskline}. Solid and dotted
lines are calculated using the error in the QPO frequency and the half-width half-maximum of the
upper kHz QPOs, respectively. Colours are as in Figure~\ref{fig:masses}. The inset shows the joint probability distribution function for the four observations with Fe line and upper kHz QPO, using the half-width half-maximum as the error in the frequency. The area under the probability function shown in the inset represents the likelihood of measuring the masses $M_i$ of the NS mass $M_0$ under the hypothesis $H$ that one of the kHz QPOs is Keplerian, the Fe line is relativistic, and both phenomena arise from the same region of the accretion disc is valid.}
\label{fig:diskline_fwhm}
\end{figure}
\section*{Acknowledgments}
AS wish to thank Cole Miller for interesting discussions. TB acknowledges support from ASI- INAF grant I/009/10/0 and from INAF PRIN 2012-6. AS, MM, TB and DA wish to thank ISSI for their hospitality. DA acknowledges support from the Royal Society.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,168 |
{"url":"https:\/\/socratic.org\/questions\/5991353d7c0149668c0ec36a","text":"# Question #ec36a\n\nAug 14, 2017\n\n(a) 5.00 mol; (b) 50.0 u; (c) 10.0 g; (d) 25.0 g.\n\n#### Explanation:\n\n(a) Mass of $\\text{A\"_2\"B}$\n\nYour equation is not in its lowest terms. It should be\n\n${M}_{\\textrm{r}} : \\textcolor{w h i t e}{m} 40.0 \\textcolor{w h i t e}{m l} 30.0 \\textcolor{w h i t e}{m m l} 50.0$\n$\\textcolor{w h i t e}{m m m l l} \\text{A\"_2 + \"2AB\" \u2192 \"2A\"_2\"B}$\n\nStep 2. Calculate the moles of $\\text{A\"_2\"B}$.\n\n$\\text{Moles of A\"_2\"B\" = 2.50 color(red)(cancel(color(black)(\"mol A\"_2))) \u00d7 (\"2 mol A\"_2\"B\")\/(1 color(red)(cancel(color(black)(\"mol A\"_2)))) = \"5.00 mol A\"_2\"B}$\n\n(b) Molecular mass of $\\text{A\"_2\"B}$\n\nIf $\\text{Molec. mass of A\"_2 = \"40.0 u}$, then $\\text{at. mass of A\" = \"20.0 u}$\n\n$\\text{Molec. mass of AB\" = \"30.0 u\" = \"20.0 u + at. mass of B}$\n\n$\\text{At. mass of B\" = \"30.0 u - 20.0 u\" = \"10.0 u}$\n\n$\\text{Molec. mass of A\"_2\"B\" = \"2 \u00d7 20.0 u + 10.0 u\" = \"40.0 u + 10.0 u\" = \"50.0 u}$\n\n(c) Mass of ${\\text{A}}_{2}$ required\n\nStep 1. Calculate the moles of $\\text{AB}$\n\n$\\text{Moles of AB\" = 15.0 color(red)(cancel(color(black)(\"g AB\"))) \u00d7 (1 \"mol AB\")\/(30.0color(red)(cancel(color(black)(\"g AB\")))) = \"0.500 mol AB}$\n\nStep 2. Calculate the moles of of ${\\text{A}}_{2}$\n\n${\\text{Moles of A\"_2 = 0.500 color(red)(cancel(color(black)(\"mol AB\"))) \u00d7 \"1 mol A\"_2\/(2 color(red)(cancel(color(black)(\"mol AB\")))) = \"0.250 mol A}}_{2}$\n\nStep 3. Calculate the mass of ${\\text{A}}_{2}$\n\n${\\text{Mass of A\"_2 = 0.250 color(red)(cancel(color(black)(\"mol A\"_2))) \u00d7 \"40.0 g A\"_2\/(1 color(red)(cancel(color(black)(\"mol A\"_2)))) = \"10.0 g A}}_{2}$\n\n(d) Mass of $\\text{A\"_2\"B}$ formed\n\nStep 1. Calculate the moles of $\\text{A\"_2\"B}$ formed\n\n$\\text{Moles of A\"_2\"B\" = 0.500 color(red)(cancel(color(black)(\"mol AB\"))) \u00d7 (\"2 mol A\"_2\"B\")\/(2 color(red)(cancel(color(black)(\"mol AB\")))) = \"0.500 mol A\"_2\"B}$\n\nStep 2. Calculate the mass of $\\text{A\"_2\"B}$\n\n$\\text{Mass of A\"_2\"B\" = 0.500 color(red)(cancel(color(black)(\"mol A\"_2\"B\"))) \u00d7 (\"50.0 g A\"_2\"B\")\/(1 color(red)(cancel(color(black)(\"mol A\"_2\"B\")))) = \"25.0 g A\"_2\"B}$","date":"2020-01-20 04:55:11","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 23, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8424064517021179, \"perplexity\": 4818.653214446106}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579250597230.18\/warc\/CC-MAIN-20200120023523-20200120051523-00559.warc.gz\"}"} | null | null |
package org.ovirt.engine.ui.webadmin.section.main.view.tab.quota;
import javax.inject.Inject;
import org.ovirt.engine.core.common.businessentities.Quota;
import org.ovirt.engine.core.common.businessentities.QuotaVdsGroup;
import org.ovirt.engine.ui.common.idhandler.ElementIdHandler;
import org.ovirt.engine.ui.common.uicommon.model.SearchableDetailModelProvider;
import org.ovirt.engine.ui.common.widget.table.column.TextColumnWithTooltip;
import org.ovirt.engine.ui.uicommonweb.models.qouta.QuotaClusterListModel;
import org.ovirt.engine.ui.uicommonweb.models.qouta.QuotaListModel;
import org.ovirt.engine.ui.webadmin.section.main.presenter.tab.quota.SubTabQuotaClusterPresenter;
import org.ovirt.engine.ui.webadmin.section.main.view.AbstractSubTabTableView;
import com.google.gwt.core.client.GWT;
public class SubTabQuotaClusterView extends AbstractSubTabTableView<Quota, QuotaVdsGroup, QuotaListModel, QuotaClusterListModel>
implements SubTabQuotaClusterPresenter.ViewDef {
interface ViewIdHandler extends ElementIdHandler<SubTabQuotaClusterView> {
ViewIdHandler idHandler = GWT.create(ViewIdHandler.class);
}
@Inject
public SubTabQuotaClusterView(SearchableDetailModelProvider<QuotaVdsGroup, QuotaListModel, QuotaClusterListModel> modelProvider) {
super(modelProvider);
ViewIdHandler.idHandler.generateAndSetIds(this);
initTable();
initWidget(getTable());
}
private void initTable() {
getTable().addColumn(new TextColumnWithTooltip<QuotaVdsGroup>() {
@Override
public String getValue(QuotaVdsGroup object) {
return object.getVdsGroupName();
}
}, "Name");
getTable().addColumn(new TextColumnWithTooltip<QuotaVdsGroup>() {
@Override
public String getValue(QuotaVdsGroup object) {
return (object.getMemSizeMBUsage() == null ? "0" : object.getMemSizeMBUsage().toString()) + "/"
+ (object.getMemSizeMB() == null ? "*" : object.getMemSizeMB().toString()) + " GB";
}
},
"Used Memory/Total");
getTable().addColumn(new TextColumnWithTooltip<QuotaVdsGroup>() {
@Override
public String getValue(QuotaVdsGroup object) {
return (object.getVirtualCpuUsage() == null ? "0" : object.getVirtualCpuUsage().toString()) + "/"
+ (object.getVirtualCpu() == null ? "*" : object.getVirtualCpu().toString());
}
},
"Running CPU/Total");
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 795 |
I have well water with a manifold inside the house that has a faucet, pressure tank, temperature sensor (for pump) and pressure relief valve. My current pump pressure is set to 60 which effectively goes from about 53 to 67.
I would like to bump up the pressure another 10 psi to improve reverse osmosis system pressure on the 2nd floor as well as the general improvement on the 3rd floor. I get about 10 psi drop every floor now. The pressure relief valve says it's set to 100 psi (at least from manufacturer), but it starts leaking around 70 psi.
First of all, is it advisable to go up another 10 psi? I assume the likelihood of leaks goes up and maybe the pump life is decreased. I'll probably also need to increase the pressure in the expansion tank. ~63-77 psi on the first floor (and lower on the above floors) doesn't seem to be extremely high to me, but I'm no expert.
Second, is there a tool to adjust the valve? It has two notches with a hole in the middle. A 5/8" or 11/16" slotted bit should fit, but those are hard to find and something like $20 online. How is this usually adjusted on the field?
Most (or the ones I've met) well system pressure reliefs are non-adjustable. So if it leaks 30 PSI below rated pressure, I'd replace it (for less cost than your proposed tool purchase, as far as I recall.) Odds are that parts to rebuild yours will be more than simply buying a new one.
Toilet valves get wonky (or may) above 80 PSI or so, and any increase in well system pressure will reduce the effective capacity of the pressure tank (and should also be accompanied by adjusting the pressure in the pressure tank to a few PSI below the new low setpoint.) If your pump does not run at least a minute to refill the pressure tank (with nothing else drawing water) you should increase the size of the tank or add an additional tank.
Not the answer you're looking for? Browse other questions tagged water-pressure well-pump or ask your own question.
How does one know if a well pump is "Short Cycling" or pressure tank is water logged?
Max pressure I can set my well to? | {
"redpajama_set_name": "RedPajamaC4"
} | 1,509 |
module Spree
OrderPopulator.class_eval do
def populate(*args)
attempt_cart_add(*args) if args.any?
valid?
end
private
def attempt_cart_add(order_params)
# 2,147,483,647 is crazy.
# See issue #2695.
quantity = order_params[:quantity].to_i
if quantity > 2_147_483_647
errors.add(:base, Spree.t(:please_enter_reasonable_quantity, :scope => :order_populator))
return false
end
variant = get_variant(order_params)
if quantity > 0
if option_value_ids = order_params[:attached_options_ids]
line_item = @order.contents.add_with_option_values(variant, option_value_ids, quantity, currency)
else
line_item = @order.contents.add(variant, quantity, currency)
end
unless line_item.valid?
errors.add(:base, line_item.errors.messages.values.join(" "))
false
end
end
end
def get_variant(order_params)
Variant.find order_params[:variant_id]
end
end # end OrderPopulator
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,369 |
\section*{\arabic{chapter} $\ $ #1}
\newcommand{\par\bigskip\par}{\par\bigskip\par}
\newcommand{\mathbb{P}}{\mathbb{P}}
\newcommand{\mathbb{A}}{\mathbb{A}}
\newcommand{\mathbb{N}}{\mathbb{N}}
\newcommand{\mathbb{Z}}{\mathbb{Z}}
\newcommand{\mathbb{Q}}{\mathbb{Q}}
\newcommand{\mathbb{R}}{\mathbb{R}}
\newcommand{\mathbb{C}}{\mathbb{C}}
\newcommand{\mathbb{F}}{\mathbb{F}}
\newcommand{\mathbb{G}}{\mathbb{G}}
\newcommand{{{\mathbb{G}}_m}}{{{\mathbb{G}}_m}}
\newcommand{\mathbb{S}}{\mathbb{S}}
\newcommand{\mathcal{D}}{\mathcal{D}}
\newcommand{\varepsilon}{\varepsilon}
\newtheorem{Lemma}{Lemma}
\newtheorem{Folgerung}{Folgerung}
\newtheorem{Satz}{Satz}
\newtheorem{Theorem}{Theorem}
\newtheorem{Korollar}{Korollar}
\DeclareMathSizes{12}{11}{8}{5}
\begin{document}
\title{Torelli's theorem from the topological point of view}
\maketitle
\bigskip
Torelli's theorem states, that the isomorphism class of a smooth
projective curve of genus $g\geq 2$ over an algebraically closed
field $k$ is uniquely determined by the isomorphism class of the
associated pair $(X,\Theta)$, where $X$ is the Jacobian variety of
$C$ and $\Theta$ is the canonical theta divisor. The aim of this
note is to give a \lq{topological\rq}\ proof of this theorem.
Although Torelli's theorem is not a topological statement the
proof to be presented gives a characterization of $C$ in terms of
perverse sheaves on the Jacobian variety $X$, which are attached
to the theta divisor by a \lq{topological}\rq\ construction.
\bigskip
For complexes $K,L\in D_c^b(X,\overline\mathbb{Q}_l)$ define $K*L\in D_c^b(X,\overline\mathbb{Q}_l)$ by the
direct image complex $Ra_*(K\boxtimes L)$, where $a:X\times X\to X$ is the addition law of $X$.
Let $K^0_*(X)$ be the tensor product of the Grothendieck group of perverse sheaves on $X$ with
the polynomial ring $\mathbb{Z}[t^{1/2},t^{-1/2}]$. $K^0_*(X)$ is a commutative ring with ring
structure defined by the convolution product, hence also the quotient ring $K_*(X)$ obtained
by dividing the principal ideal generated by the constant perverse sheaf $\delta_X$ on $X$.
Both rings $K_*^0(X)$ and $K_*(X)$ resemble properties of the homology ring of $X$ endowed with
the $*$-product, but have a much richer structure. A sheaf complex $L\in
D_c^b(X,\overline\mathbb{Q}_l)$ defines a class in $K_*(X)$ by the perverse Euler characteristic
$\sum_\nu (-1)^\nu\cdot {}^p\! H^\nu(L)\cdot t^{\nu/2}$ . Similar to the homology ring every
irreducible closed subvariety $Y$ has a class in $K_*(X)$ defined as the class of the perverse
intersection cohomology sheaf $\delta_Y$ of $Y$. For details see [W]. This allows to consider
the product
$$ \delta_\Theta * \delta_\Theta \in K_*(X) \ .$$
Whereas the corresponding product in the homology ring of $X$ is
zero, this product turns out to be nonzero. $\delta_\Theta
* \delta_\Theta$ is of the form $\sum_{\nu,\mu} A_{\nu,\mu}
t^{\mu/2}$. Recall, the coefficients are irreducible perverse
sheaves $A_{\nu,\mu}$ on $X$. For a perverse sheaf $A$ on $X$,
which is a sheaf complex on $X$, let ${\cal H}^i(A)$ denote the
associated cohomology sheaves for $i\in \mathbb{Z}$. Let $\kappa\in X(k)$
be the Riemann constant defined by $\Theta = \kappa - \Theta$. It
depends on the choice of the Abel-Jacobi map $C\to X$.
\bigskip
{\bf Theorem}: {\it Let $C$ be a curve of genus $g\geq 3$. There
exists a unique irreducible perverse sheaf $A=A_{\nu,0}$, among
the coefficients of $\delta_\Theta*\delta_\Theta$, characterized
by one of the following equivalent properties
\begin{enumerate} \item ${\cal H}^{-1}(A)$ is nonzero, but
not a constant sheaf on $X$.
\item ${\cal H}^{-1}(A)$ is the skyscraper sheaf $H^1(C)\otimes \delta_{\{\kappa\}}$ with
support in the point $\kappa\in X$.
\end{enumerate}
Furthermore the support of the perverse sheaf $A$ is
$\kappa+C-C\subseteq X$.}
\bigskip
Taking this for granted, Torelli's theorem is an immediate
consequence. In fact the subvariety $C-C$ uniquely determines the
curve $C$. This is well known. For instance, $C$ is the unique
one-dimensional fiber of the minimal desingularization of $C-C$.
Since the theorem stated above allows to recover the sheaf complex
$A$ from $\Theta$, this determines $C$ from the data $(X,\Theta)$
via the support $\kappa+C-C$ of $A$ for $g\geq 3$. The cases
$g=1,2$ are trivial.
\bigskip
\underbar{Remark}: $A$ is a direct summand of the complex
$\delta_{C}*\delta_{\kappa -C}$. If $C$ is not hyperelliptic, it
splits into two irreducible perverse summands $\delta_{\{\kappa\}}
\oplus A$. If $C$ is hyperelliptic, then it splits into the three
irreducible perverse summands $\delta_{\{\kappa\}} \oplus
\delta_{\kappa+C-C}\oplus A$.
\bigskip
We now give a sketch of the theorem in the non-hyperelliptic case.
For all integers $r\geq 0$ let $\delta_r\in K_*(X)$ be the class
of the direct image complex $Rp_{r,*}\delta_{C^{(r)}}$, where
$p_r:C^{(r)}=C^r/\Sigma_r \to X$ are the higher Abel-Jacobi maps
from the symmetric quotient of $C^r$ to $X$. For $r\leq g-1$ the
image of $p_r$ is the Brill-Noether subvariety $W_r=C+\cdots + C$
($r$ copies) of $X$. If $C$ is not hyperelliptic, then
$$ \delta_{W_r} =\delta_r \ ,$$ since $p_r$ is a small morphism
by the theorem of Martens [M] for $r\leq g-1$. In particular
$\delta_\Theta=\delta_{g-1}$, which will be used in the proof. (In
the hyperelliptic case $\delta_\Theta=\delta_{g-1}-\delta_{g-3}$.
For this and further details we refer to [W]).
\bigskip
\underbar{Proof of the theorem}: Suppose $C$ is not hyperelliptic.
\bigskip
1) Since the canonical morphism
$$\tau:C^{(i)}\times C^{(j)}\to C^{(i+j)}$$ is a finite ramified
covering map, the direct image $R\tau_* \delta_{C^{(i)}\times C^{(j)}}$ decomposes into a
direct sum of etale sheaves $ \bigoplus_{\nu} m(i,j,\nu)\cdot {\cal F}_{i+j-\nu,\nu}$ by
keeping track of the underlying action of the symmetric group $\Sigma_{i+j}$ for the map
$C^{i+j}\to C^{(i+j)}$ (see [W].4.1). If we apply $Rp_{i+j,*}$, this gives a formula for
$\delta_i* \delta_j$. From $p_{i+j}\circ \tau = a\circ (p_i\times p_j)$, where $a:X\times X\to
X$ is the addition law of $X$, one obtains for $i\geq j$ that the convolution $\delta_i
* \delta_j$ is $ \delta_{i+j} \oplus \delta_{i+j-1,1} \oplus \cdots
\oplus \delta_{i-j,j} $, where $\delta_{r,s}= Rp_{i+j,*}({\cal
F}_{r,s})$. A special case is
$$ \delta_\Theta * \delta_\Theta= \delta_{g-1}*\delta_{g-1} = \delta_{2g-2} \oplus \delta_{2g-3,1}
\oplus \cdots \oplus \delta_{g-1,g-1} \ .$$ Another case is $ \
\delta_1
* \delta_{2g-3} = \delta_{2g-2} \oplus \delta_{2g-3,1}$, and
together this implies $$ \fbox{$
\delta_{2g-3}*\delta_1\hookrightarrow \delta_\Theta*\delta_\Theta
$} \ .$$
\bigskip
2) The morphism $f:C\times C\to \kappa+C-C \subseteq X$, defined
by $(x,y)\mapsto \kappa+x-y$, is semi-small. If $C$ is not
hyperelliptic, then $f$ is a birational map, which blows down the
diagonal to the point $\kappa$, and is an isomorphism otherwise.
Hence the direct image $Rf_*(\delta_C\boxtimes\delta_C)$ is
perverse on $X$, and necessarily decomposes
$Rf_*(\delta_C\boxtimes\delta_C) =\delta_C* \delta_{\kappa-C} =
\delta_{\{\kappa\}} \oplus \delta_{\kappa+C-C}$ such that
$$ \fbox{$ {\cal H}^{-1}(\delta_{\kappa+C-C}) \cong H^1(C)\otimes \delta_{\{\kappa\}} \
$} \ .$$
\bigskip
3) We claim $\delta_{2g-3} \equiv \delta_{\kappa-C}$ and $\delta_{2g-2} \equiv
\delta_{\{\kappa\}}$ in $K_*(X) $ (ignoring Tate twists). These are the simplest cases of the
duality theorem [W] 5.3. This implies
$$ \fbox{$ \delta_{2g-3}*\delta_1 \ \equiv\ \ \ \delta_{\{\kappa\}} + \delta_{\kappa+C-C} $} \ ,$$
in $K_*(X)$ using step 2.
\bigskip
\underbar{Proof of the claim}: By the theorem of Riemann-Roch $
C^{(2g-3)} \overset{p}{\to} X $ is a $\mathbb{P}^{g-2}$-bundle over
$\kappa-C$ and a $\mathbb{P}^{g-3}$-bundle over the open complement
$X\setminus \ (\kappa-C)$. Hence $Rp_*\delta_{C^{(2g-3)}}$ is a
direct sum of $ \delta_{\kappa - C}$ and a sum of translates of
constant sheaves on $X$. Similarly
$pr_{2g-2}^{-1}(\{\kappa\})=\mathbb{P}^{g-1}$, and $pr_{2g-2}$ is a
$\mathbb{P}^{g-2}$-bundle over the open complement $X\setminus \
\{\kappa\}$. Hence $\delta_{2g-2}\equiv \delta_{\{\kappa\}} $ in
$K_*(X)$.
\bigskip
4) $\Theta=\kappa-\Theta$ and the definition of the convolution
product
implies
$$ {\cal H}^{-1}(\delta_\Theta * \delta_\Theta) \ \cong \
IH^{2g-1}(\Theta)\otimes \delta_{\{\kappa\}} \ $$ for an arbitrary principally polarized
abelian varieties $(X,\Theta)$, where $IH^{2g-1}(\Theta)$ denotes the intersection cohomology
group of $\Theta$. If the singularities of $\Theta$ have codimension $\geq 3$ in $\Theta$, then
ignoring Tate twists this implies (see [W] {2.9} and [W2])
$$ \fbox{$ {\cal H}^{-1}(\delta_\Theta * \delta_\Theta) \ \cong \
H^1(X)\otimes \delta_{\{\kappa\}} $} \ .$$
\bigskip In fact
by the Hard Lefschetz theorem $IH^{2g-3}(\Theta)$ and $H^1(X)$
have the same dimensions. A more elementary argument proves this
for Jacobians including the case of hyperelliptic curves: For
example for non-hyperelliptic curves we have $IH^{\bullet}(W_d) =
H^{\bullet}(X,Rp_{d,*}\delta_{C^{(d)}}[-d])=H^{\bullet}(C^{(d)})=(\bigotimes^d
H^\bullet(C))^{\Sigma_d}$, since $p_{d}$ is a small morphism. Thus
$$IH^{d+\bullet}(W_d)\ \cong\ \bigoplus_{a+b=d}\ Sym^a\Bigl(H^0(C)[1]\oplus
H^2(C)[-1]\Bigr) \otimes \Lambda^b(H^1(C))\ .$$ For $IH^{2d-1}(W_d)$ only $a=d-1$ contributes,
hence $IH^{2d-1}(W_d)\!\cong\! H^1(C)\!\cong\! H^1(X)$. (For the hyperelliptic case see [W]
4.2).
\bigskip
\underbar{Conclusion}: For curves $C$, which are not
hyperelliptic, the perverse sheaf $A$ defined by
$\delta_{\kappa+C-C}$ satisfies all the assertions of the theorem.
$\delta_{\kappa+C-C}$ is a direct summand of
$\delta_\Theta*\delta_\Theta$ by step 1 and 3. By step 2 and 4 we
obtain modulo constant sheaves on $X$
$${\cal H}^{-1}(\delta_\Theta*\delta_\Theta) \equiv H^1(X)\otimes
\delta_{\{\kappa\}} \equiv H^1(C)\otimes \delta_{\{\kappa\}}
\equiv {\cal H}^{-1}(\delta_{\kappa+C-C}) \ .$$ Since $K_*(X)$ is
a quotient of $K_*^0(X)$, the last identity only holds modulo
constant sheaves on $X$. But this suffices to imply the theorem.
\bigskip
\underbar{Remark}: In [W] we constructed a $\overline\mathbb{Q}_l$-linear
Tannakian category ${\cal BN}$ attached to $C$ equivalent to the
category of finite dimensional $\overline\mathbb{Q}_l$-representations
$Rep(G)$, where $G$ is $Sp(2g-2,\overline\mathbb{Q}_l)$ or
$Sl(2g-2,\overline\mathbb{Q}_l)$ depending on whether $C$ is hyperelliptic
or not. In this category $\delta_\Theta$ corresponds to the
alternating power $\Lambda^{g-1}(st)$ of the standard
representation, and $A$ corresponds to the adjoint representation.
\bigskip\noindent
{\bf Bibliography}
\bigskip\noindent
[M] Martens H.H, On the variety of special divisors on a curve,
Crelle 227 (1967), p.111 -- 120.
\bigskip\noindent
[W] Weissauer R., Brill-Noether sheaves (preprint)
\bigskip\noindent
[W2] Weissauer R., Inner cohomology (preprint)
\end{document}
| {
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Home > JETLAW > Vol. 21 > Iss. 2 (2020)
Investor-State Arbitration and Human Rights
Timothy Feighery
After decades of growth and popularity, the international investor-state dispute settlement (ISDS) regime has come under intense criticism recently-particularly concerning the perceived chilling effect the regime imposes on states' ability to regulate in the public interest. This Article seeks to contextualize this criticism by examining the historical antecedent of ISDS in international law: the law of diplomatic protection. It proceeds to focus on the flexibility of ISDS as a critical advance over diplomatic protection, and shows how ISDS has evolved over time-particularly as developed states have moved from approaching the regime from a predominantly investment-exporting perspective to a more balanced perspective that accounts for inbound foreign investments. In concrete terms, the inherent flexibility of ISDS has permitted it increasingly to protect states' interests in regulating in the public interest, while at the same time protecting foreign investment against inappropriate governmental interference. The Article ultimately argues that the ISDS system should be permitted to continue to evolve to arrive at the appropriate equilibrium for its time
Timothy Feighery, Investor-State Arbitration and Human Rights, 21 Vanderbilt Journal of Entertainment and Technology Law 417 (2020)
Available at: https://scholarship.law.vanderbilt.edu/jetlaw/vol21/iss2/2
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Śródmieście (lit. Centro seniūnija) – śródmiejska dzielnica administracyjna Kowna, położona na prawym brzegu Niemna, u ujścia Wilii; obejmuje Stare Miasto i Nowe Miasto; pełni funkcje mieszkaniowo-usługowe; szkoły wyższe, teatry, muzea.
Osią dzielnicy jest aleja Wolności, łącząca Stare i Nowe Miasto.
Przypisy
Linki zewnętrzne
Dzielnice Kowna | {
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Q: Grub2 Function parameters Grub2 allows for powerful scripting capabilities. if, while, function, etc all mean the language is pretty powerful (Turing complete?).
However, I cannot figure out how to pass parameters in grub2.
grub> function hello {
> echo hello $1
> }
grub>
grub> hello world
hello
grub> # i'd expect to see 'hello world' here
grub> # instead in only get 'hello'
grub>
grub> # this works however
grub> 1=world
grub> hello
hello world
grub>
Does Grub2 allow passing parameters to user defined functions?
A: Function parameters only appeared in Grub in May 2010. At the moment the last grub release is 1.98 from March 2010, so if you want them, you have to get Grub from the Bazaar repository.
If you hope to do serious programming in Grub, though, you'll have to go all the way to lua support.
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Justine Henin
Justine Henin's main goal to win Wimbledon, her coach worked out tactics
World number one Justine Henin has always claimed that she would prefer another Roland Garros title over winning Wimbledon. However, her opinion has changed, and when asked "A fifth Roland Garros or a first Wimbledon?" she replied "Wimbledon!" without hesitation.
Henin's coach, Carlos Rodriguez, has prepared a plan, some tactical and psychological changes, which will enable the Belgian to win the only Grand Slam title she misses.
Justine's father Jose said: "Carlos is right about the way to go for a better Justine next year. He is going to make her more and more positive, stronger in the head. She will be going to the net much more so that she can feel more comfortable making volleys. Wimbledon is the goal."
Even though Henin is the best female tennis player the world has at the moment, she feels there is room for improvement. "I think I can still improve a lot, in my attitude especially," said the 25-year-old Belgian. "I have to keep improving my aggressive game, moving forward and taking my opportunities." (sources: Daily Mail, Reuters)
Previous articleAmelie Mauresmo hasn't disappeared
Next articleJustine Henin supports life bans for match-fixing
ATP fines LTA for banning Russian, Belarusian players
Wimbledon relaxes white clothing rule to accommodate WTA players on their period
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Lennox Lewis - the famous boxer, professional. British boxer born in 1965 in Vestheme. Athlete Climbing started an amateur boxer, like many of his predecessors. Lewis boxing's most prestigious weight category - heavy.
Amateur boxing
Lennox Lewis was a member of two Olympic Games: in 1984 and 1988. At that time he was a member of the Canadian national team, and therefore, it is represented that country.
In the first game Lennox failed to achieve any results: the boxer failed to even pass the qualifying round. But the second time Lewis was able to rehabilitate. Boxer could win a major achievement in his career. While many did not consider winning Lennox unconditional, because in those games danger is Cuban team. In the final battle, the British boxer managed to beat Riddick Bowe - boxer whose fate brought Lewis back in the professional ring.
The absolute champion of
In autumn 1992, there was an open tournament in which his skills show strong global heavyweights. The tournament was held for the purpose of awarding the title of absolute champion of his weight.
First Lennox Lewis easily defeated Radokoma Donovan. The second semi-final boxing fans remember the victory of the British over Evander Holyfield. But known Riddick Bowe izyavil no desire to fight with Lennox, for which he paid his championship belt. It was rare to see a boxing Lennox Lewis could impress the judges and the audience.
The dramatic confrontation
Lennox Lewis fights are always notable for their dramatic. For example, in 1994 he was defeated in a duel with Oliver McCall. But after 3 years Lennox managed in a rematch to win the same McCauley and regain the title.
Lennox Lewis met in the ring, even with such eminent heavyweight Mike Tyson in 2002. By the way, this fight he won in the 8th round knockout strike through. In combat Vitali Klychko (Ukrainian boxer) defeated in Round 6. The bout ended by TKO. Klitschko was badly cracked eyebrow.
All boxing fans in 1999, watching the spectacular fights between Evander Hollyfild and Lennox Lewis. The first fight was a draw, but in the second part of the Lewis won a spectacular victory and won the title of invincible.
Call Klitschko
After his retirement from professional boxing on one of the press conferences, the British champion told everyone that he was ready to return to the ring for his fight with Klitschko, but with some of the brothers is not explained.
In addition, he said the boxer is very interesting news. It turns out that during his visit to the Russian capital, he was offered $ 50 million for the fact that he will fight with any of the Klitschko brothers.
But the Briton did not agree on the amount of voiced, it is much more satisfied with a reward of 100 million. And it seems that Lewis says it's not a joke. He said that the Russian side is thinking over the offer.
British champion said he would be ready to fight in six months. Boxer diligently preparing for this battle.
Life Brit
In 2002, all the American media are full of information about what the absolute boxing champion met his other half and is going to marry. Many said that Lennox wants to connect his life with the singer Aisha Max. But gossip was not destined to be realized.
3 years after the fight with Tyson, Lewis married a Jamaican beauty Violet Chang. The couple had four lovely daughters. Wife of boxer insisted that Lewis ended his professional career. The question was acutely athlete appeared before a choice: either sports or family. Of course, Lennox has chosen his wife and daughters.
CHudinov Fedor: ups and downs
Boxer Floyd Patterson: biography and career
Daniel Cormier. Biography. best matches | {
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\section{Introduction.}
Microgels are polymeric particles whose networks are swollen in a good solvent. When dispersed, microgels take up the given solvent to a multitude of their dry volume. Changing the solvent quality, \emph{i.e.}, the microgel-solvent interactions, leads to a (partial) expulsion of the solvent. This influences the properties of the dispersed microgels, such as size, structure, colloidal stability, and interaction potential. The most studied microgels are composed of poly-\emph{N}-\-iso\-pro\-pyl\-acryl\-amide (pNIPAM), which readily disperses in aqueous solution and is thermo-responsive. \cite{Pel00} Typically, their polymer network is cross-linked by \emph{N,N'}-\-methyl\-ene\-bis\-acryl\-am\-ide (BIS) which is consumed faster during the synthesis. \cite{Kro17} For that reason, the microgels have a dense-core-fuzzy-shell structure with a decreasing polymer content and cross-linker concentration from the center to the periphery. \cite{Sti04FF}
Responsive microgels are used for many applications in which the material properties have to be varied \emph{in situ}. For example, the swelling and deswelling of the microgels can be exploited for targeted release of drugs \cite{Swi20, Gue17,Bus17}, responsive micro-reactors \cite{Bor18}, adaptive viscosity modifiers and lubricants \cite{And19}, or sensors \cite{Dan19, Wei19}. In fact, microgels have another remarkable property: a high interfacial activity. pNIPAM-based microgels efficiently lower the surface tension \cite{Zha99,Ngai05} and stabilise emulsions, \cite{Bru09,Ngai05,Ngai06,Fuj05} foams \cite{Dup08,Hor18,Wud18,Fam15}, or coat solids \cite{Ish99,Kes18,MFS19}. Their responsiveness is imparted to the interface, which leads to emulsions and foams that can be broken on-demand. \cite{Ngai05,Fuj05,Dup08,Fam15}
Although the formation and use of responsive emulsions using microgels is reported regularly, the answer to one of the most crucial question remains: Why do microgel-stabilised emulsions become unstable when changing the environmental conditions, such as temperature or pH? Indeed, this question is not trivial. There might be more than one answer, due to the complexity of the emulsion (de-) stabilization \cite{Tad13book} and the many different chemical compositions \cite{Kee07,Ima10}, architectures \cite{Men09, Nic19}, and stimuli-responsivenesses \cite{Chi86,Tan82,Mer15} of microgels. So far, many explanations have been proposed in the literature. One of the first theories includes the desorption of microgels from the interface into the oil-phase due to an increase in their hydrophobicity and a consequent loss of interfacial activity. \cite{Ngai05,Ngai06} However, it has been shown that microgels are surface active at temperatures above the volume phase transition temperature (VPTT) or at different pH values. \cite{Mon10, Gei14} Another explanation attributes the destabilization to the size change of adsorbed microgels within the interface: deswelling in the lateral and vertical directions reduces the microgel-covered interfacial area, which reduces the stability of the emulsion.\cite{Ngai05,Ngai06} Experiments and simulations have provided arguments for and against this theory. \cite{Gei14, Har19, Sch20, Boc19, Boc20, Mae18, Rav17} Moreover, the change of the visco-elastic properties of the microgel monolayers due to the change in the environment has been proposed as a reason for the demulsification. \cite{Bru08,Des14} Indeed, in solution, the swelling state of the microgels has a pivotal influence on their microgel-microgel interaction and the resulting flow properties. \cite{Sen99,Dal08,Sco20}
In general, the stability of emulsions is determined by the interfacial tension, as well as the mechanical properties and the repulsive electrostatic and/or steric interactions between the films consisting of the emulsifying agents at the interfaces. The aforementioned explanations are focused on the mechanical properties of the monolayers and the decrease in the surface tension.
Recently, Harrer \emph{et al.} discussed that the collapse of the dangling polymer chains of the microgel fraction in the aqueous phase may cause the steric repulsion between the microgel-covered emulsion droplets to be minimized. The swelling state of the microgels normal to the interfaces may also contribute to the destabilization mechanism.
In this work, we aimed to elucidate the effect of the swelling state and compression on the interaction forces between microgel-covered interfaces. We used the simplified model system of a microgel monolayer at an air-water interface and a rigid colloid in the aqueous phase. In order to measure the interactions, we used the Monolayer Particle Interaction Apparatus (MPIA), which is based on the technique of atomic force microscopy. \cite{Gil05} The Langmuir-trough part of the MPIA allows the surface pressure and temperature to be controlled. The MPIA has already been successfully used to measure the interaction between different probes in aqueous solutions and films of surfactants, soft particles, or rigid particles at air-aqueous interfaces as a function of pH, salt concentration and lateral compression. \cite{Mcn10b,Mcn11,Mcn16, Mcn10a, Mcn18}
We investigated two pNIPAM-based microgels with different volume phase transition temperatures (VPTT). The different VPTTs were achieved by copolymerizing \emph{N,N}-\-di\-ethyl\-acryl\-amide (DEAAM) and NIPAM in one of the microgels. \cite{Kee07} Both microgels were slightly positively charged and had the same size and structure in solution. Compression isotherms and depositions showed that the microgels had the same two-dimensional phase behavior. We exploited the different VPTT to relate the experimental results to the swelling state of the microgels.
Although air-water and oil-water interfaces are different environments, it has been shown that microgels show a similar behavior at the two interfaces. \cite{Boc20} Therefore, we discuss our result in the context of emulsion stabilization. The results give strong evidence that the deswelling of the microgels fractions in the aqueous phase has a major influence on the interaction of microgel-covered interfaces. For the mechanism of demulsification of microgel-stabilized emulsions, not only the strength of the interfacial microgel layer, \cite{ Gei14, Sch20, Boc19, Boc20, Mae18, Rav17, Bru08, Des14} but also the interaction forces between the microgel-covered droplets need to be taken into account.
\section{Experimental.}
\subsection{Materials.}
\emph{N}-\-iso\-pro\-pyl\-acryl\-amide (Acros Orga\-nics, Belgium),
\emph{N,N}-\-Di\-ethyl\-acryl\-amide (Polysciences Inc., USA),
\emph{N,N'}-\-methyl\-ene\-bis\-acryl\-am\-ide (Alfa Aesar, USA),
\emph{N}-(3-aminopropyl) meth\-acryl\-amide hydro\-chloride (Polysciences Inc., USA),
2,2'-azobis(2-methylpropionamidine) dihydrochloride (V50) (Sigma-Aldrich, USA), chloroform (CHCl$_{3}$, 99.0~$\%$ purity, Wako Pure Chemical Industries, Japan), methanol (MeOH, HPLC grade, Fisher Scientific, Germany), ethanol (EtOH, JIS Special Grade, Wako Pure Chemical Industries, Japan), aqueous ammonia solution (NH3(aq), 25 wt$\%$, analytic grade, WTL Laborbedarf GmbH, Kastellaun, Germany), and N-trimethoxysilylpropyl-N,N,N -trimethylammonium chloride (hydrophilpos, 50~$\%$ in
methanol, ABCR, Germany), and
cetyltrimethylammonium bromide (CTAB) (Fluka Biochemica, Switzerland)
were used as received.
For all the interface experiments ultra pure water (Astacus$^2$, membraPure GmbH, Germany) with a resistivity of 18.2 MOhm$\cdot$cm was used as the sub-phase. To facilitate spreading, propan-2-ol (Merck KGaA, Germany) was used.
\subsection{Synthesis.}
pNIPAM-co-DEAAM and pNIPAM-based microgels were synthesized by precipitation polymerization using the following feed parameters and instructions. For pNIPAM-co-DEAAM microgels the mono\-mers, NIPAM (2.0069~g), DEAAM (3.4854), BIS (0.3748~g), and APMH (0.1384~g) were dissolved in 330~mL double-distilled water. For pNIPAM microgels the mono\-mers, NIPAM (5.4546~g), BIS (0.3398~g), and APMH (0.1474~g), were dissolved in 330~mL double-distilled water. Both microgels have small amount of the comonomer APMH ($\approx$ 2~mol\%), which allows for post modification of the microgels, such as covalent labeling with fluorescent dyes.\cite{Lyo11, Gel16} The primary amine is positively charged at neutral pH. Therefore, the cationic initiator (V50) and cationic surfactant (CTAB) were used to prevent aggregation. The monomer solution was heated to 65$^\circ$\,C and purged with nitrogen under constant stirring (270~rpm). Simultaneously, the initiator and CTAB (for pNIPAM-co-DEAAM microgels 0.2151~g and 0.0251~g, respectively; for pNIPAM microgels 0.2253~g and 0.0340~g, respectively) were each dissolved in 20~mL water in separated vessels and degassed for one hour. The surfactant-solution was injected into the reaction vessel and stirred for 30 more minutes to equilibrate. The polymerization was initiated by adding the initiator solution in one shot to the reaction flask. The reaction was carried out for 4~h at 65$^\circ$\,C under constant nitrogen flow and stirring. The obtained microgels were purified by threefold ultra-centrifugation at 30~000~rpm (70~000 RCF) and subsequent re-dispersion in fresh, double-distilled water. For the pNIPAM-co-DEAAM microgels, the centrifugation was carried out at T~=~10$^\circ$\,C, in order to assure the microgels were fully swollen. Lyophilization was applied for storage.
\subsection{Monolayer Particle Interaction Apparatus.}
A micro-manipulator (Model MMO-202D, Narishige) and a light microscope (BX51, Olympus) were used to attach a silica particle (nominal diameter = 6.8~$\mu$m, Bangs Laboratory, Fishers, USA) to a gold-plated Si$_3$N$_4$ cantilever (V-shaped, nominal spring constant k = 0.15 N/m, OTR8-PS-W, Olympus) using an epoxy resin (Araldite Rapid). The silica probe was next modified with hydrophilpos using the method reported elsewhere. \cite{Mcn17}
The forces between the microgel films at the air-water interface and the colloidal probe in water were measured using the MPIA. The MPIA combines a Langmuir trough (Riegler \& Kirstein GmbH, Potsdam, Germany) and a force measurement unit. Detailed information about the MPIA can be found in the literature. \cite{Mcn11}
The Langmuir trough was cleaned using CHCl$_{3}$ and EtOH. The colloidal probe was then attached to the cantilever holder, water added to the trough, and the water surface suctioned cleaned. The temperature of the water sub-phase was controlled by circulating thermostated water through the base of the trough using a circulation system (C25P, ThermoHaake, Karlsruhe, Germany). Next, a clean mica substrate was placed across the edges of the Langmuir trough filled with water, and force curves were measured between the probe in the aqueous phase and the mica. The calibration factor (CF$_{mica}$), which was needed to convert raw force curves (force [V] versus piezo position [nm] curves) to calibrated force curves (force [nN] versus piezo position [nm] curves), was calculated from the linear contact region of the approach force curves. After the calibration, the water in the trough was removed and new water added, a minimum of 30 min allowed for water to reach the desired temperature, and then the water surface suctioned cleaned. The microgel monolayer was spread at the air-water interface, 10 min allowed for the spreading solvent to evaporate, and the films compressed to the desired surface pressure. The forces were then measured between that film and the same probe that was used to measure CF$_{mica}$ at an approach/retract velocity of 33 $\mu$m s$^{-1}$ (scan rate: 0.66 Hz, scan size (in z-direction): 25 $\mu$m). At least 50 force curves were measured at each surface pressure.
CF$_{mica}$ was used to convert the raw force curves measured between the particle film and the probe to calibrated force curves. Zero force was defined at large cantilever-film separations, where no surface forces acted on the cantilever. Zero distance was defined as the intersection of the slope of the linear contact region (the area where the probe was in contact with the mica substrate or with the air-aqueous interface in the absence or presence of the microgel monolayer) to the zero force line.
The adhesion ($F_{ad}$) between the probe and the air-water interface in the absence or presence of a film of microgels at a given surface pressure was taken as the average of the adhesive force measured in all the retract force curves. The effective stiffness ($S_N$) of the air-water interface in the absence or presence of a microgel monolayer was determined from $S_N$ = $S_1$/ $S_2$. Here, $S_1$ is the averaged slope of the linear contact region recorded for all the approach force curves measured between the probe and the air-water interface. $S_2$ is the average of the slope of the linear contact region recorded for all the approach force curves between the probe and the mica substrate, which were used to determine the calibration factor CF$_{mica}$.
\subsection{Compression Isotherms and Depositions.}
Combined compression isotherms and depositions, \emph{i.e.}, gradient Langmuir-Blodgett depositions, were conducted at the air-water interface in a customized liquid-liquid Langmuir-Blodgett trough (KSV NIMA, Biolin Scientific Oy, Finland) made of poly(oxymethylene) glycol. The surface pressure ($\Pi$) was probed with a highly porous platinum Wilhelmy plate (perimeter = 39.24 mm, KSV NIMA, Biolin Scientific Oy, Finland). The plate was attached to an electronic film balance (KSV NIMA, Biolin Scientific Oy, Finland) and placed parallel to the barriers.
Before each measurement, the trough was cleaned and a fresh air-water interface was created. For temperature control, the trough was connected to an external water bath. Water was added to the trough and the trough was tempered to the desired temperature. A plasma-cleaned rectangular piece of ultra-flat silicon wafer ($\approx$ 1.1 x 6.0 cm, P\{100\}, NanoAndMore GmbH, Germany) was mounted to the substrate holder with an angle of $\approx25^\circ$ with respect to the interface. The substrate was immersed in the water and the water surface suctioned cleaned. After temperature equilibration, the microgel solution (5~mg~mL$^{-1}$ in 50~$\%$ v/v water-propan-2-ol) was added to the interface. The barriers were closed ($v$ =~6.48 cm$^2$ min$^{-1}$) in order to increase the interfacial concentration of the microgels. Simultaneously, the substrate was raised ($v$ = 0.15~$\pm$~0.004~mm min$^{-1}$) through the air-water interface. Compression isotherms and depositions were conducted at $(10.0\pm0.5)~^\circ$C, $(20.0\pm0.5)~^\circ$C and $(40.0\pm0.5)~^\circ$C. A detailed description of gradient Langmuir-Blodgett depositions can be found elsewhere.\cite{Rey16}
\subsection{Atomic Force Microscopy.}
Atomic force microscopy (AFM) measurements of the microgels in the dry state, \emph{i.e.}, at the solid-air interface, were performed using a Dimension Icon with a closed loop (Veeco Instruments Inc., USA, Software: Nanoscope 9.4, Bruker Co., USA). The images were recorded in tapping mode using OTESPA tips with a resonance frequency of 300~kHz, a nominal spring constant of 26 N~m$^{-1}$ of the cantilever and a nominal tip radius of $<$ 7~nm (NanoAndMore GmbH, Germany).
\subsection{Image Analysis.}
The open-source analysis software \textit{Gwyddion} 2.54 was used to process the AFM images. All images were flattened to remove the tilt and the minimum value was fixed to zero height. The processed AFM micrographs were analyzed with a custom-written MATLAB script\cite{Boc19}. A Delaunay triangulation and Voronoi tesselation were used to compute their nearest neighbor connections. A more detail description can be found in Ref.\cite{Boc19}.
\subsection{Dynamic Light Scattering.}
The hydrodynamic radius of the microgels was measured \emph{via}
Dynamic light scattering (DLS). A laser a with vacuum wavelength of $\lambda_0 = 633$~nm was used.
The diluted suspensions of the microgels in water have a refractive index of $n({\lambda_0}) = 1.33$. For pNIPAM-co-DEAAM microgels the temperature increment was between T =~(6.0$\pm$0.1)$^\circ$\,C to T = (40.0~$\pm$~0.1)$^\circ$\,C and for pNIPAM microgels between T =~$(15.0\pm0.1)~^\circ$C to T = (49.0~$\pm$~0.1)$^\circ$\,C. The measurements were conducted in 2$^\circ$\,C steps. A thermal bath was filled with toluene to match the refractive index of the glass cuvettes. The scattering angle ($\theta_s$) was varied between 30$^\circ$ and 130$^\circ$, in steps of 20$^\circ$, in order to change the scattering vector $q = 4\pi n/\lambda_0\sin(\theta_s/2)$.
$R_h$ was computed from the diffusion coefficient using the Einstein-Stokes equation. We determined the VPTT from the inflection point of a logistic ``S" shape function fitted to $R_h$ \emph{versus} T.
\subsection{Small-Angle Neutron Scattering.}
Small-angle neutron scattering (SANS) experiments were performed at the KWS-2 instrument (Heinz Maier-Leibnitz Zentrum, Garching, Germany). The scattering vector ($q = 4\pi/ \lambda\sin(\theta_s/2)$, with $\theta_s$ being the scattering angle) was varied by using a wavelength of $\lambda$ = 0.5 and 1~nm for the neutron beam and three sample-detector distances: 20, 8 and 2~m.
The detector was a 2D-$^3$He tubes array with a pixel size of 0.75 cm and a resolution of $\Delta\lambda/\lambda = 10\%$. Data were corrected with sample transmittance and dark count (B$_4$C used). The background, heavy water, was subtracted from all data. The data were acquired at T = 20 and 40$^\circ$\,C and fitted with the Fuzzy-Sphere model.
\subsection{Electrophoretic Mobility and Zeta-Potential.}
A NanoZS Zetasizer (Malvern Instruments Ltd., England) was used to measure the electrophoretic mobility ($\mu_{el}$) and dynamic light scattering at a scattering angle of 173$^\circ$ (backscatter angle). The temperature range for the pNIPAM-co-DEAAM microgels was between T = 4 to 40$^\circ$\,C, and for the pNIPAM microgels between T = 10 to 50$^\circ$\,C. The measurements were conducted in 2$^\circ$\,C steps for both microgels. A laser with a vacuum wavelength of $\lambda_0$ = 633~nm was used. The Smoluchowski approximation was employed as a model to calculate the zeta potential ($\zeta_{pot}$). The values of $\zeta_{pot}$ are meant to show simply a qualitative trend of the data are shown for comparison with the literature. The temperature of the electrokinetic transition \citep{Lop06} is calculated by fitting a logistic ``S" shape function to the data.
\section{Results and Discussion.}
The two microgels studied in this work have different VPTTs. The different VPTT are achieved by varying of the chemical composition of the microgels. We used the monomers of NIPAM and DEAAM, which have the same backbone (acryl\-amide), but are differently \emph{N}-substituted. In solution, statistical copolymer microgels of NIPAM ($\approx$ 40~mol\%) and DEAAM ($\approx$ 60~mol\%) have a VPTT of $\approx$ 20$^\circ$\,C, compared to $\approx$ 32$^\circ$\,C of pNIPAM microgels. \cite{Kee07}
The temperature-dependent swelling of the pNIPAM-co-DEAAM and pNIPAM microgels is shown in Figure~\ref{fig:figure_Solution}A. The results confirm the expected VPTT \cite{Kee07}. In the swollen (T $<$ VPTT) and deswollen state (T $>$ VPTT), the microgels have nearly the same $R_h$ of 160 and 80~nm, respectively. The microgels display the typical inhomogeneous internal structure \cite{Sti04FF} with a decreasing amount of polymer and cross-linker from the center to the periphery (Fig. S1 and Ref.~\cite{Boc19}). Both microgels are monodisperse with size polydispersities calculated from SANS data of (8 $\pm$ 1)~\% for the pNIPAM-co-DEAAM and (7 $\pm$ 1)~\% for the pNIAPM microgels.
\begin{figure}[ht!]
\includegraphics[width=0.5\textwidth]{_Figures/_Figure_SB103_104c_DLS_Mob.pdf}
\caption{Properties of pNIPAM-co-DEAAM and pNIPAM microgels in solutions. (A) Temperature-dependent swelling of the microgels measured with multi-angle dynamic light scattering. (B) Electrophoretic mobility in pure Milli-Q water (0~mM KCl) and with 1~mM KCl as background salt at temperatures below and above the VPTT of the microgels.}
\label{fig:figure_Solution}
\end{figure}
In gerneral, microgels are colloidally stable in solution due to steric and electrostatic repulsions. Below the VPTT, steric repulsions between dangling polymer chains of microgels leads to their colloidal stability. At T $>$ VPTT, electrostatic repulsions between the charged moities
provides the stability. \cite{Pel00} The electrokinetic characteristics of the pNIPAM-co-DEAAM and pNIPAM microgels were studied with electrophoretic light scattering (ELS). The microgels were synthesized using a cationic initiator (V50) and a small amount of primary amine (APMH). As a consequence, the microgels were positively charged in pure water, as shown by the positive electrophoretic mobility (Fig.~\ref{fig:figure_Solution}B). At elevated temperatures, the microgels displayed a higher $\mu_{el}$ in purified water.
Analogous to the VPTT, we determined the temperature of the electrokinetic transition from the electrophoretic mobility curves in pure water (Fig.~S2A and B). This transition takes place $\approx$ 2\,K above the VPTT for both microgels. This difference indicates that the strong increase in $\mu_{el}$ with temperature is not only associated with a reduction in the frictional coefficient due to the deswelling of the microgels, but is also due to an increase in the local charge concentration on the microgels' surface caused by the reorganization of the charged moities. \cite{Lop06,Dal00}
In the case of the pNIPAM and pNIPAM-co-DEAAM microgels, the addition of 1~mM potassium chloride (KCl) leads to a strong decrease in $\mu_{el}$ below the VPTT and a negative mobility above the VPTT (Fig.~\ref{fig:figure_Solution}B). The electrophoretic mobility of the microgels as a function of the temperature in different electrolyte solutions is presented in the ESI (Figure~S2). DLS measurements in 1~mM KCl show an increase of the microgel radius at elevated temperatures (Fig.~S3).
Although electrostatic repulsion stabilizes the pNIPAM and pNIPAM-co-DEAAM microgels in purified water, the presence of 1~mM salt is sufficient to screen the charges. The microgels become colloidally instable and aggregate above their VPTT. This result highlights that the microgels contain only a very small proportion of charged groups. The characterization of the two microgel systems in bulk shows that the biggest difference between the two microgel systems is their VPTT in aqueous solution.
\begin{figure}[ht!]
\includegraphics[width=\textwidth]{_Figures/_Figure_Structure_Monolayers_V3.jpeg}
\caption{$\Pi$ $\emph{versus}$ $Area/Mass$. (A) Monolayers of pNIPAM and pNIPAM-co-DEAAM microgels below their VPTT. (B) Monolayers of pNIPAM and pNIPAM-co-DEAAM microgels above their VPTT. (C) AFM images of the deposited microgel monolayers in dried state. The positions of the images (d-o) on the compression isotherms are highlighted in (A) and (B). Dashed vertical lines represent the onset point of the isostructural phase transition. Gray dashed and dash-dotted horizontal lines highlight the sharp increases in the compression isotherms. Scale bars are 2 $\mu$m.}
\label{fig:figure_structure_MM}
\end{figure}
The compression isotherms at the air-water interface of the monolayers of both microgels are shown in Figure~\ref{fig:figure_structure_MM}A and B.
$\Pi$ is plotted as a function of the area normalized by the amount of microgel initially added to the interface, \emph{i.e.}, $Area/Mass$.
We cannot exclude the possibility that a small fraction of the microgels will disperse into the aqueous phase when the microgel solution is spread. Incomplete adsorption may be a problem for microgels with a large amount of charged moities, but is reportedly not a problem for low- or uncharged microgels. \cite{Sch20}
Furthermore, once adsorbed, microgels are considered irreversibly confined at the interfaces due to their high adsorption energy ($\approx$~10\textsuperscript{6}~$k_{\text{B}}T$), which is comparable to solid particles. \cite{Mont14} Hence, the desorption of microgels into the sub-phase during compression is very unlikely.
The microstructures of the monolayers were investigated using atomic force microscopy images (Figure~\ref{fig:figure_structure_MM}C) after a gradient Langmuir-Blodgett deposition \cite{Rey16} to a silicon wafer. %
Figure~\ref{fig:figure_structure_MM}C shows height images of the dried monolayers at the surface pressures used in the MPIA measurements. The images show that closely packed monolayers were formed in all cases.
The images were analysed using a custom-written Matlab script to obtain quantitative information about the monolayers \cite{Boc19}. The values of the number concentrations per area ($N_{area}$) and the center-to-center distances in the first and second crystalline phase ($NND_{1st}$ and $NND_{2nd}$) are summarized in Table~\ref{tab:tab_NND}.
\begin{table}[ht!]
\centering
\resizebox{\textwidth}{!}{\begin{tabular}{cccccc}
\toprule
Microgel & Temperature & $\Pi$ & $N_{area}$ & $NND_{1st}$ & $NND_{2nd}$\\
- & ($^\circ$\,C) & (mN~m$^{-1}$) & (nm) & (nm) \\
\midrule
& 10 & 15 & 4.4 & (507 $\pm$~28) & - \\
& & 24 & 5.9 & (438 $\pm$~26) & - \\
pNIPAM-co-DEAAM & & 26 & 8.3 & (372 $\pm$~54) & (183 $\pm$~22) \\
& 40 & 15 & 4.6 & (490 $\pm$~30) & - \\
& & 24 & 6.0 & (431 $\pm$~26) & - \\
& & 29 & 14.8 & (296 $\pm$~26) & (191 $\pm$~33) \\
\midrule
& 20 & 15 & 4.7 & (489 $\pm$~30) & - \\
& & 24 & 6.3 & (421 $\pm$~24) & - \\
pNIPAM & & 26 & 8.1 & (380 $\pm$~35) & (220 $\pm$~20)\\
& 40 & 15 & 4.8 & (481 $\pm$~21) & - \\
& & 24 & 6.2 & (431 $\pm$~23) & - \\
& & 29 & 15.0 & (275 $\pm$~37) & (195 $\pm$~32) \\
\bottomrule
\caption{Number concentration per area ($N_{area}$) and center-to-center distance for the first crystalline phase ($NND_{1st}$) the second crystalline phase ($NND_{2nd}$) of pNIPAM and pNIPAM-co-DEAAM microgels at different compressions and temperatures.}
\label{tab:tab_NND}
\end{tabular}}
\end{table}
The two-dimensional phase behavior of microgel monolayers has been extensively discussed in the literature, both below \cite{Gei14,Rey16} and above the VPTT. \cite{Boc19, Boc20} Briefly, the softness of the microgels allows their deformation at the interface. Upon adsorption, the contact area of the microgel with the interface is maximised in order to reduce as many unfavourable air- (or oil-) water contacts as possible and to lower the surface tension. The deformation is limited by the cross-links within the microgels' polymer network, which decrease in amount from their center to the periphery \cite{Sti04FF, Kro17}. The resulting shape of the adsorbed microgels is described as a core-corona or ``fried-egg''-like structure. \cite{Bru09,Des11,Gei14,Rey16,Boc19,Fab19} The corona represents the portion of the microgel at and near the interface and the core represents the portion that is still in the aqueous phase.
At sufficiently large concentrations, the microgel coronae contact one another and a densely packed monolayer is formed. The compression isotherms below and above the VPTT show a sharp first increase in the surface pressure (Figures~\ref{fig:figure_structure_MM}A and B, between gray dashed horizontal lines) starting at $\approx$~3000 and 2500~cm$^2$ mg$^{-1}$ for the pNIPAM-co-DEAAM and pNIPAM microgels, respectively. The microgels in the corona-corona contact region have a hexagonal order as shown in Figure~\ref{fig:figure_structure_MM} C, (d), (g), (j), and (m). Lateral compression of the monolayers decreases the lattice constant, \emph{i.e.}, the center-to-center distance, causing the surface pressure to increase. \cite{Boc19}
At higher lateral compressions, an isostructural phase transition takes place, where two the crystalline phases coexist (AFM images (f), (i), (l), and (o) in Fig.~\ref{fig:figure_structure_MM}). \cite{Rey16, Boc20} The onset of the transition is highlighted by the dashed vertical lines in Figures~\ref{fig:figure_structure_MM}A and B. As the microgels progress into the second crystalline phase, the compression isotherms show a pseudo-plateau where the $\Pi$ becomes nearly independent of $Area/Mass$. The influence of temperature on the microgel monolayers becomes visible at even higher lateral compressions, when all the microgels are in the second crystalline phase. \cite{Boc19, Boc20} Below the VPTT, the second crystalline phase is compressible and the surface pressure increases once more (Fig.~\ref{fig:figure_structure_MM}A, between the gray dash-dotted horizontal lines) before the monolayer collapses. In contrast, at T $>$ VPTT, the second crystalline phase is incompressible and the monolayer collapses without a further strong increase in $\Pi$ (Fig.~\ref{fig:figure_structure_MM}B). The compression isotherms of the microgel monolayers change from two sharp increases in the surface pressure at T $<$ VPTT (Fig.~\ref{fig:figure_structure_MM}A dashed and dash-dotted horizontal lines) to a single increase above the VPTT (Fig.~\ref{fig:figure_structure_MM}B dashed horizontal lines).
\begin{figure}[ht!]
\includegraphics[width=0.8\textwidth ]{_Figures/_Figure_Sketch_Microgels.jpeg}
\caption{Sketches of microgel monolayers at air-water interfaces and AFM images. AFM images show microgel monolayers in dried state (top view). The small drawings show microgel monolayers from the aqueous phase looking at the air-water interface (bottom view). Large sketches display the monolayer from the side (side view). To clarify the size relationship, the probe is sketched at the bottom edge. Microgel monolayers are shown below (A and B) and above (C and D) the VPTT and at low (A and C) and high (B and D) lateral compression.
}
\label{fig:figure_sketch}
\end{figure}
For the interpretation and discussion of the MPIA experiments below, we also briefly summarize the results for swelling/deswelling of microgels at liquid interfaces. \cite{Mae18, Har19, Boc19, Boc20} For illustration, the microgel monolayers at different temperatures and compressions are sketched in Figure~\ref{fig:figure_sketch} together with the colloidal probe. The crucial difference between the microgels in solution and those at the air-water interface is their altered volume-phase transition when the microgels are adsorbed. \cite{Har19, Boc19, Boc20} The size of the directly adsorbed fractions of the microgels, \emph{i.e.}, the corona, depends on the surface tension and is nearly temperature-independent. In contrast, the cores of the adsorbed microgels have similar properties as microgels in the aqueous bulk phase, \emph{e.g.}, they can still deswell as a function of temperature. \cite{Boc19,Har19,Boc20} The microgel cores are therefore strongly swollen by water and have dangling polymer chains below the VPTT (Figures~\ref{fig:figure_sketch}A and B, side view). At T $>$ VPTT, the cores are deswollen and their dangling polymer chains are collapsed (Figures~\ref{fig:figure_sketch}C and D, side view). \cite{Har19, Boc19} Compared to the swollen state, the deswollen cores not only have a a reduced thickness, but also a smaller (lateral) diameter and a larger polymer volume fraction compared to the swollen state. \cite{Boc19, Mae18} The bottom view in Figure~\ref{fig:figure_sketch}C also shows that by deswelling the cores, more of the firmly adsorbed polymer layer, \emph{i.e.}, the corona, is effectively exposed and accessible from the aqueous sub-phase at the same center-to-center distance. At the onset of the isostructural phase transition, the microgel cores contact one another (Figures~\ref{fig:figure_sketch}B and D, bottom view). \cite{Boc19} As a result, the monolayers are less undulated below and above the VPTT (Figures~\ref{fig:figure_sketch}B and D, side view).
We used the MPIA \cite{Gil05} to measure force-distance curves between microgel monolayers at flat air-water interfaces
and a colloidal probe. The colloidal probe was funtionalized with hydrophilpos, in order to obtain a hydrophilic, positively charged surface of the probe. \cite{Mcn09} This reduces the attractive electrostatic interactions that existed between the originally negatively charged silica particle and the cationic microgel monolayer and prevented the deposition of the microgels to the probe. Force distance curves were measured between the probe in the Milli-Q water sub-phase and the monolayer at the water surface. The raw data were converted into force (nN) \emph{versus} distance (nm) and shifted to zero distance (for details see the experimental part). The interactions were measured as a function of temperature (10, 20, 33 and 40$^\circ$\,C) and monolayer surface pressure (15, 24, and 26 or 29~mN~m$^{-1}$).
\begin{figure}[ht!]
\includegraphics[width=0.7\textwidth]{_Figures/_Figure_FD_curves_SB104.pdf}
\caption{Examples of force-distance curves of pNIPAM microgel monolayers at the air-water interface measured with the probe (diameter of 6.84 $\mu$m) from the aqueous phase. Approach (left) and retraction (right) force-distance curves. (A,B) Force distance curves as a function of temperature. (C,D) Force distance curves as a function of compression for T = 20~$^\circ$\,C. (E,F) force distance curves as a function of compression for T = 40 $^\circ$\,C.}
\label{fig:figure_FD_SB104c}
\end{figure}
Examples of the force-distance curves are shown in Figure~\ref{fig:figure_FD_SB104c} for the pNIPAM microgels and in Figure~S4 for pNIPAM-co-DEAAM microgels. The approach curves of both microgels show long-range repulsive interactions at low temperatures (T $<$ VPTT), expressed by a gradual increase in the force. Temperatures close to the microgels' VPTT caused the distance of the onset of the long-range repulsion to significantly decrease; it fully vanished above the VPTT. Increasing the lateral compression had no effect on the long-range repulsion (Figures~\ref{fig:figure_FD_SB104c}E,G and S4E,G). In comparison, the force distance approach curves in Figure~S5 showed an attraction between the probe and the bare air-water interfaces.
The air-aqueous interface is negatively charged. \cite{tak05,Cha09,Pol20} Therefore, the attraction is explained by electrostatic interactions between the negatively charged bare air-water surface and positively charged probe.
We explain the long-range repulsion between the monolayers and the probe by steric repulsions resulting from the swelling state of microgels' fractions in the aqueous phase. As illustrated in Figure~\ref{fig:figure_sketch}, below the VPTT, the cores of the microgels are hydrated and posses dangling chains that extend into the aqueous phase. The dangling polymer chains and the hydrated cores are compressed normal to the interface during the approach of the probe.
At elevated temperatures, the cores and dangling chains are collapsed (Fig.~\ref{fig:figure_sketch}C and D). In this case, the probe cannot compress the deswollen cores and the microgel film is deformed or bent. The incompressibility of the deswollen microgel cores was also observed upon lateral compression, \emph{i.e.}, in the compression isotherms (Fig.~\ref{fig:figure_structure_MM}A and B).
The approach curves (Figures~\ref{fig:figure_FD_SB104c} and~S4) showed a linear contact region at negative distances.
In this constant compliance region, the probe and monolayer cannot come closer together, \emph{i.e.}, they have a constant separation. It is expected that the movement of the colloidal probe deforms the fluid interface, which is the microgel film in this case. \cite{Pit02,Dav14,Ana16,But17} Thus, the slope of the region of constant compliance is proportional to the stiffness of the interface and the spring constant of cantilever. \cite{Duc94, Har99} The slope of the linear contact region decreases across the VPTT of the microgels and with lateral compression of the pNIPAM microgel monolayer.
Examples of the retract curves are shown in Figures~\ref{fig:figure_FD_SB104c}B, D, F, H and S4B, D, F, H. The influence of temperature on the retract force-distance curves is predominately observed across the VPTT of the microgels in solution. Increasing the temperature causes the colloidal probe to adhere stronger to the monolayer. Larger distance and forces are required to separate the probe form the microgel film (Fig.~\ref{fig:figure_FD_SB104c}B, D). Lateral compression of the monolayers yields the opposing result. At higher concentrations of microgels at the interface, \emph{i.e.}, larger $\Pi$-values, the probe can be separated from the monolayer by a weaker force. The strongest reduction was observed between 15~mN~m$^{-1}$ and 24~mN~m$^{-1}$ for all the temperatures (Fig~\ref{fig:figure_FD_SB104c}F, H).
We determined the effective stiffness ($S_N$) and the force of adhesion ($F_{ad}$) of the monolayers from the force distance curves. Figure~\ref{fig:figure_SN_Fad} shows $S_N$ and $F_{ad}$ for pNIPAM-co-DEAAM and pNIPAM microgel monolayers as a function of the temperature (Fig.~\ref{fig:figure_SN_Fad}A and C) and $\Pi$ (Fig.~\ref{fig:figure_SN_Fad}B and D).
$S_N$ was calculated from the region of constant compliance of the approach force distance curves. $S_N$ is the ratio of the slope of the region of constant compliance between the probe and the microgel monolayers and between the probe and a mica substrate (for details see the experimental part). The constant compliance region gives a linear force-distance region. This region was fit by a linear line to give $S_N$. Examples of the fits are given in Figures~\ref{fig:figure_FD_SB104c}A,C,E,G and ~S4A,C,E,G. $F_{ad}$ between the probe and the monolayer was quantified as the difference between the minimum and zero force in the retract force distance curves. Both parameters were averaged over at least 50 force curves. The error bars are a consequence of the variation in the force curves for the same temperature and lateral compression.
\begin{figure}[ht!]
\includegraphics[width=0.7\textwidth ]{_Figures/_Figure_N_stiff_F_ad.pdf}
\caption{Effective stiffness ($S_N$) and adhesion force ($F_{ad}$) of the pNIPAM-co-DEAAM and pNIPAM microgel monolayers. (A) $S_N$ as a function of temperature. The black circle displays the bare water interface. (B) $S_N$ as a function of lateral compression. The black circle displays the bare water interface. (C) $F_{ad}$ as a function of temperature. (D) $F_{ad}$ as a function of lateral compression. In C and D, $F_{ad}$ of the bare water interfaces is out of scale with $F_{ad}$ = (620 $\pm$ 25)~nN. All values are averaged over at least 50 force curves, the standard deviation is given as error.
}
\label{fig:figure_SN_Fad}
\end{figure}
$S_N$ describes the deformability of the interface. \cite{Cha01} For bare air-water interfaces, the surface tension is thought to affect the $S_N$, as the effective stiffness of an air-aqueous interface has been shown to decrease with a decreasing surface tension. \cite{Cha01} Figure~\ref{fig:figure_SN_Fad}A shows that the microgel monolayers were stiffer than the bare air-water interface at T = 20~$^\circ$\,C, even though the surface tension was greatly reduced. Thus, additional factors need to be considered for the effective stiffness when a film of particles or surfactants is present.
Increasing the lateral attraction between particles within the film, \emph{e.g.}, by enhancing capillary forces \cite{Mcn16} or hydrophobic interactions \cite{Mcn14, Mcn11}, leads to stiffer monolayers. In general, $S_N$ of the film is higher for films of particles with a lower lateral mobility. This is because the particles in such a film are moved less by the incoming probe, causing the film to remain more intact rather than a film whose particles have a high lateral mobility. Films with a low lateral mobility may result from films with a high viscosity or films whose particles show strong inter-particle attractions. The higher stiffness of the microgel monolayers compared to the clean air-water interface is in line with expectations from interfacial rheological measurements. Both dilatational and shear rheology have shown that microgel monolayers strongly affect the visco-elastic properties of fluid interfaces. \cite{Zif14,Rey16,Hua17,Mae18, Bru10} This increased viscosity of the microgel-laden interface also impacts the deformability of the monolayer normal to the interface.
Heating the microgel monolayer tended to decrease $S_N$ (Fig.~\ref{fig:figure_SN_Fad}A). The center-to-center distances of the microgels are virtually the same below and above the VPTT, but the microgel cores deswell in vertical and lateral directions. \cite{Boc19,Mae18,Har19} The decreased thickness of the monolayer, the collapse of the cores, and the dangling polymer chains in the aqueous side lead to larger areas of the interfaces being covered by the thin coronae of the microgels (Fig.~\ref{fig:figure_sketch}A and C, bottom view). This may allow the microgels film to deform or bend easier.
Similarly, the stiffness of surfactant monolayers has been found to decrease with the chain length of surfactants, \emph{i.e.}, a decrease of the film thickness. \cite{Mcn10a} Moreover, Particle Tracking Microrheology has shown that increasing the temperature leads to a transition from visco-elastic to a viscous fluid, \cite{Mae18} displaying the enhanced mobility of the adsorbed microgels.
Below and above the VPTT (Fig.~\ref{fig:figure_SN_Fad}B), lateral compression also tends to decrease $S_N$. If we consider our above explanation for the effect of temperature on the monolayer stiffness, the decrease of $S_N$ with $\Pi$ is surprising. The image analysis of the deposited microgels (Table~\ref{tab:tab_NND}) show that the microgels are pushed laterally into each other. This should jam the microgels as illustrated in Figure~\ref{fig:figure_sketch} and drastically decrease their mobility, which should increase $S_N$. In contrast, shear rheology measurements \cite{Rey16} and calculations of the surface elasticity of microgel monolayers \cite{Pin14} show a maximum around 15-20~mN~m$^{-1}$. A similar trend has been observed for other adsorbed species, such as TiO$_2$ particles \cite{Mcn16}, polystyrene particles \cite{Mcn14, Mcn17}, and surfactants \cite{Mcn12,Mcn10a}. The decrease of $S_N$ is explained by the decreased surface tension, that is an increase in surface pressure. At small surface pressures, \emph{i.e.}, at larger center-to-center distance, more force is needed for the deformation. This is because the Laplace pressure of the system is higher, counter acting the increase of the surface area. \cite{Cha01} Consequently, the stiffness is reduced for higher $\Pi$.
The presence of microgels at the interface significantly reduces $F_{ad}$ between the probe and the interface. For the bare water interface, $F_{ad}$ is (620 $\pm$ 25) nN. A microgel monolayer at 15~mN~m$^{-1}$ and T = 10~$^\circ$\,C has an $F_{ad}$ of roughly 250~nN (Fig.~\ref{fig:figure_SN_Fad}C). This strong adhesion observed for bare water can be explained by a three-phase contact formation between the probe and the interface. A strong capillary force needs to be overcome, when the probe is removed from the air-water interface. Scheludko $\&$ Nikolov \cite{Sch75} have calculated the force required to pull a sphere out of a liquid ($F_{adh}$). When a particle is moved from air into a liquid,
\begin{equation}
F_{adh}~=~2 \pi R \gamma sin^2 \left(\frac{\theta}{2}\right) \notag \,,
\label{eq:Scheludko}
\end{equation}
where $R$, $\gamma$, and $\theta$ are the radius of the sphere, the interfacial tension of the air-liquid interface, and the advancing contact angle of the sphere with the air-water interface, respectively. Inserting $R$ =3.4~$\mu$m, $F_{ad}$ = 620 nN for $F_{adh}$, and $\gamma$~= 0.072~N~m$^{-1}$ leads to $\theta$~=~79$^\circ$. The reduction in $F_{ad}$ by the presence of the microgels at the air-water interface is explained by i) the microgels prevent the probe from forming a three-phase contact line and ii) there is an electrostatic repulsion between the microgel and the probe, as both are positively charged.
The adhesion between the microgel monolayers and the probe decreases with increasing temperature (Figure~\ref{fig:figure_SN_Fad}C).
For particle monolayers it was shown that the $F_{ad}$ decreases as a result of: (i) electrostatic repulsion between the monolayer and the probe (if the charges of both have the same sign); and (ii) inhibition of direct contact between the probe and the negatively charged sites of the air-water interface, \emph{i.e.}, the monolayer blocks the attractive electrostatic interactions. \cite{Mcn17} Collapsed thermo-responsive microgels are known to be colloidally stabilized in solution by charged moieties originating, for example, from the ionic initiator fragments. \cite{Pel00} Both pNIPAM and pNIPAM-co-DEAAM microgels showed the expected increase in electrophoretic mobility with temperature in Milli-Q water (Fig~\ref{fig:figure_Solution}B). As discussed above, the cores of adsorbed microgels and the microgels in solution have similar properties. Following this assumption, the aqueous side of the microgels monolayers is expected to become more positively charged with increasing temperatures. Since the probe is also positively charged, $F_{ad}$ should decrease with temperature due to electrostatic repulsion. However, in Figure~\ref{fig:figure_SN_Fad}C the opposite was found. Electrostatic repulsion between the like-charged microgels and the probe therefore seems be of minor importance. Moreover, (dominant) repulsive electrostatic forces caused a long-range repulsion in the approach curves, \cite{Mcn12} which was not observed in Figures~\ref{fig:figure_FD_SB104c}A,C and S4A,C. Therefore, the reduced probe-microgel monolayer adhesion with increasing temperature is explained by the blockage of the negatively charged sites of the air-water interface by the presence of the pNIPAM or pNIPAM-co-DEAAM microgels. The deswelling of the microgel cores leads to larger areas of the interfaces being covered by the thin coronae of the microgels (Fig.~\ref{fig:figure_sketch}A and C, bottom view). This results in a poorer blockage of the negatively charged sites at higher temperatures and an increasing adhesion.
Additionally, hydrophobic interactions between the probe and the monolayer contribute to the adhesion. The contact angle of the hydrophilpos probe calculated from above (79$^\circ$) shows that the hydrophilpos probe is not perfectly hydrophilic. When the probe is retracted from the microgel monolayer, parts of the polymer network that were adsorbed to the probe need to be rehydrated. At elevated temperatures, rehydration is more unfavorable than at low temperatures, due to the increased hydrophobicity of the pNIPAM and pNIAPM-co-DEAAM polymer. In the literature, increasing $F_{ad}$ across the VPTT has been reported for adhesion measurements of pNIPAM coated solid substrates (both brush and microgel systems) in aqueous environment. \cite{Sch10,Kes10}
The decrease in adhesion with increasing $\Pi$ that is shown in Figure~\ref{fig:figure_SN_Fad}D can be attributed to the lateral compression of the microgels. During lateral compression, the center-to-center distance of the microgels decreases (Table~\ref{tab:tab_NND}) and the monolayer becomes denser (Fig.~\ref{fig:figure_sketch}). This causes the attractive electrostatic interaction between the probe and the air-water interface to be blocked more.
The MIPA experiments show the same trends as above, \emph{i.e.}, almost the same $S_N$ and $F_{ad}$ values for the pNIPAM and pNIPAM-co-DEAAM microgel monolayers were obtained. A significant difference was found only in the temperature response of the microgel monolayers. For example, the long-range repulsion disappeared between 20 and 40~$^\circ$\,C for the pNIAPM microgels and between 10 and 33~$^\circ$\,C for the pNIPAM-co-DEAAM microgels, \emph{i.e.}, when the respective VPTT of the microgels is exceeded. This difference illustrates that the changes are related to the deswelling of the microgel fractions on the aqueous side of the interface.
Lastly, we want to discuss the meaning of the MPIA results for the (de-) stabilization of emulsions formed by microgels. As presented in the introduction, one of the most prominent properties of microgels at interfaces is the formation of stimuli-responsive emulsions. A large number of studies on the break-down of microgel-stabilized emulsions have been published in the last decades, providing various explanations. \cite{Ngai05,Ngai06,Bru08,Bru08b,Des11,Mas14,Kwo18b,Kwo19, Pin14,Bru10,Mae18,Mon10} A recent review by Fernandez-Rodriguez \emph{et al.}, \cite{MAF20} summarizes the complexity of microgel monolayers and the abundance of the different aspects.
Here, we probed the interactions by using a model system of a hard colloid in the aqueous phase and a flat microgel monolayer at the air-water interface. Indeed, as suggested by Harrer \emph{et al.} \cite{Har19}, the deswelling of the microgel cores minimizes the steric repulsions between microgel-covered interfaces. The force-distance curves (Figures~\ref{fig:figure_FD_SB104c}A,C and S4A,C) demonstrate that increasing the temperature above the VPTT leads to a loss of long-range repulsion, independent of the lateral compression. The long-range repulsion is correlated to the steric froces between the probe and the monolayer due to dangling chains, which collapse at temperatures above the VPTT. \cite{Har19,Boc19} Nevertheless, we want to point out that we do not expect the loss of repulsive interaction to be the sole explanation for the emulsion destabilization. We suggest that it is a combination two processes: a loss of steric stabilization and a decrease in the visco-elastic properties of the interface, \emph{i.e.}, strength of the microgel film.
\section{Conclusion.}
We studied the interactions of a probe in the aqueous phase with pNIPAM and pNIPAN-co-DEAAM microgels monolayers using the MPIA. Our results highlight a transition from soft to hard interfaces. Below the VPTT, a long-range soft repulsion normal to the interface exists between the microgel monolayers and the probe due to steric interactions. In the deswollen state, this interaction becomes short-ranged or hard-core like and the adhesion between the monolayer and the probe comes larger. This transition is in contrast to the lateral compression of microgel monolayers reported in the literature. \cite{Boc20} The lateral interaction of microgel monolayers is dominated by the directly adsorbed fractions of the microgels. These fractions are hardly affected by temperature and only at very high interfacial concentrations are the interactions affected by the swelling state of the microgel cores.
The results are discussed in the context of microgel-stabilized emulsions, suggesting that the changes in the interaction normal to the interface may contribute to their break-down. We propose that the mechanism of de-emulsification is a combination of both the reduction of repulsive interactions and a decrease of the visco-elastic properties in the interface. Indeed, it should be considered that a simplified model system of a microgel monolayer at a flat interface and a hard colloidal probe is used. While microgels have nearly identical properties at flat air-water or oil-water interfaces, \cite{Boc20} future MPIA measurements utilizing probes that reflect a more native environment should be conducted. For example, this has been done for rice starch or TiO$_2$ particles. \cite{Mcn16,Mcn18} In solution, intra-microgel interactions, which dominate the deswelling, also lead to inter-microgel aggregation. \cite{Sti04FF,Cra06} Measurements where both the probe and the monolayer are made from the same material, \emph{e.g.} by coating a silica particle with microgels, may further contribute to the understanding of the de-emulsification of microgel-stabilized ``smart'' emulsions.
\section{Acknowledgements.}
C.~E.~M. and S.~B. contributed equally to this work. The authors acknowledge financial support from the SFB 985 ``Functional Microgels and Microgel Systems'' of Deutsche Forschungsgemeinschaft within project B8. This work is based upon SANS experiments performed at the KWS-2 instrument operated by J\"ulich Centre for Neutron Science (JCNS) at the Heinz Maier-Leibnitz Zentrum (MLZ), Garching, Germany.
\section{Conflicts of Interest}
There are no conflicts to declare.
\section{Data Availability}
Additional research data for this article may be accessed at no charge at https://hdl.handle.net
/21.11102/ac75b142-e857-11ea-afb2-e41f1366df48
\section{Characterization in Bulk}
\begin{figure}[ht!]
\includegraphics[width=.90\textwidth ]{_Figures_SI/_Figure_SANS_SB103.pdf}
\caption{Small-angle neutron scattering data and fits of pNIPAM-co-DEAAM microgels. (A) Particle form factor, $P(q)$, $\emph{versus}$ scattering vector, $q$, with fits at T = 20$^\circ$\,C and T = 40$^\circ$\,C. (B) Relative polymer Volume fraction \emph{versus} radius, R, from the fits of the fuzzy-sphere model in (A).
}
\label{fig:figure_SANS}
\end{figure}
\begin{figure}[ht!]
\includegraphics[width=.90\textwidth ]{_Figures_SI/_Figure_ZetaMob_SB10X.pdf}
\caption{(A, B) Electrophoretic mobility and (C, D) zeta potential as a function of temperature for (A, C) pNIPAM-co-DEAAM and (B, D) pNIPAM microgels at different KCl concentrations. The dashed lines show the ETT of the microgels.
}
\label{fig:figure_MobZeta}
\end{figure}
\begin{figure}[ht!]
\includegraphics[width=.90\textwidth ]{_Figures_SI/_Figure_DLS_Salt.pdf}
\caption{Temperature-dependent swelling of the microgels measured with dynamic light scattering in 0 and 1~mM KCl solution. Measurements were conducted with a NanoZS Zetasizer at a scattering angle of 173$^\circ$. (A) pNIPAM-co-DEAAM microgels and (B) pNIPAM.
}
\label{fig:figure_DLS_1mM_KCl}
\end{figure}
\newpage
\section{Force-Distance curves}
\begin{figure}[ht!]
\includegraphics[width=0.7\textwidth]{_Figures/_Figure_FD_curves_SB103.pdf}
\caption{Examples of force-distance curves of pNIPAM-co-DEAAM microgel monolayers at the air-water interface measured with hydrophilpos probe from the aqueous phase. The probe has diameter of 6.84 $\mu$m. Approach (left) and retraction (right) force-distance curves. (A,B) F-D curves as a function of temperature. (C,D) F-D curves as a function of compression for T = 20~$^\circ$\,C. (E,F) F-D curves as a function of compression for T = 40 $^\circ$\,C.}
\label{fig:figure_FD_SB103c}
\end{figure}
\begin{figure}[ht!]
\includegraphics[width=.90\textwidth]{_Figures_SI/_Figure_FD_BareWater.pdf}
\caption{Force distance curves of the bare air-water interface measured with a hydrophilized silica particle as probe from the aqueous phase. (A) Approach force curves, and (B) retract force curves at T~=~20~$^\circ$C. The black vertical dashed lines represent point where the prob is fully is in contact with the monolayer and the slope linear. Black horizontal dashed lines show zero force.
}
\label{fig:figure_FD_bare_Water}
\end{figure}
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Q: Создание опций в opencart Подскажите, есть ли в опенкарт такая же возможность как и в вордпресс добавлять опции, не создавая новые колонки в бд?
что-то наподобие функций add_option(), update_option(), get_option() и таблицы wp_options
A: У Opencart такого нет, вам придётся создавать дополнительные колонки. Либо поискать модули для этого.
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Well, I didn't intend on booking another London trip (or any international trip, really) quite so soon, but here I am, writing a blog post about my impending trip there…in January! Yeah, three months from now!
So how did that happen? I'm glad you asked!
It literally all came together today. Scott (the husband) is going to Japan for a couple of weeks in January for his grad school program and, while I would love to join him (I've never been to Japan, after all), it's not practical because he'll be in class and/or doing school stuff the whole time (and he doesn't really want to extend the trip since he'll already be missing two weeks of work). And while I could explore without him, I don't think he would appreciate that, nor would I want to do it without him. I was planning on just hibernating at home for those two weeks, but I had an epiphany this morning – I could take my own little trip! As you know, dear reader, my love of London is well-documented, so I'm always happy to go there and I knew Scott would be fine not going himself since we were just there in August. These things, coupled with the facts that my initial glance at flights showed me I could cover the cost with miles and that I have copious amounts of PTO to use, made me pull the trigger. That means you'll get to read more about my British exploits in mid-to-late January! Woooooooo!
Somewhat unusually (for me), I also immediately booked a place to stay. While I did do some initial research on Airbnb, I also decided to look into Premier Inn again. I stayed in three different Premier Inn locations during my 2016 solo UK trip (Brighton, Swansea, and London) after having had good experiences with them in the past (also in Brighton and Swansea) and I'm really glad I looked into them this time because I found a killer deal! I'm staying at the hub by Premier Inn, Westminster Abbey for six nights for just over $400 USD, which includes a buffet breakfast every day. What a steal! It seems hub (they don't capitalize the h, so I won't either) is Premier's take on a more modern, cozier (read – smaller) hotel experience in urban settings (there are several hub locations in London). Their website says "Our rooms are compact, but cleverly designed to make sure you have everything you need for a comfortable city stay." which makes it pretty clear what you're getting into. However, they do have a nice lounge/bar space and a tasty breakfast buffet and the pictures (and reviews) look really great, so I'm excited. Should be perfect since I'll be traveling on my own! And the location is excellent – very near Westminster Abbey and literally steps from St. James's Park Station.
Happy New Year! Also, London! | {
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\section{Introduction} \label{sec:Intro}
One of the foundational concepts of linear algebra is the determinant. At the most basic level, this matrix parameter is celebrated for its intricate ties to the set of
eigenvalues and as a similarity invariant. However, the determinant still surprises us as the solution to a varying array of problems.
In addition to solving systems of linear equations and performing a change of variables in calculus, the determinant can help us count! Benjamin and Cameron \cite{BC} recently showed
the determinant will calculate the number of nonintersecting $n$-paths in certain nonpermutable digraphs, where an $n$-path is a set of $n$ paths from $n$ distinct source vertices to $n$
distinct sink vertices. In fact, the permanent will count the number of all $n$-paths.
The determinant of a matrix can be found recursively, as an alternating sum of minors. Often the determinant of an $n\times n$ matrix $A$ is defined compactly using the Leibniz formula,
precisely \[\det(A)= \sum_{\sigma\in S_n} \mathop{\mathrm{sgn}}(\sigma) \prod_{i=1}^n a_{i,\sigma_i}.\] Similarly, the permanent can be defined as a sum over subsets of $[n]:=\{1,2, \ldots, n\}$ using Ryser's
formula \cite{HJ} \[\perm(A)= \sum_{S\subset[n]} (-1)^{n-|S|} \prod_{i=1}^n \sum_{j\in S} a_{i,j}.\]
In this paper, we prove a much messier expansion of the determinant by instead indexing our terms using the set of ordered partitions of $[n]$. Aptly, we call this the
\emph{terrible expansion of the determinant.} This expansion is analogous to Ryser's formula for the permanent.
Section \ref{sec:Hardstuff} explains the origins of this expansion as it relates to multivariate finite operator calculus, a
branch of mathematics that has proven useful in enumerating ballot (generalized Dyck) paths containing certain patterns
\cite{NiederhausenSullivan,sullivan}.
Before stating the formula for our expansion of the determinant, we introduce it with two fundamental examples.
When $n=2$ we see that Equation \eqref{eqn:2by2} provides the following expansion for the determinant.
\begin{eqnarray}
\left|\begin{array}{cc}
a_{11} & a_{12} \\ \label{eqn:2by2}
a_{21} & a_{22} \\
\end{array}\right|&=&a_{11}(a_{12}+a_{22})+a_{22}(a_{11}+a_{21}) -(a_{11}+a_{21})(a_{12}+a_{22})\\ \notag
\end{eqnarray}
Likewise when $n=3$ we get the following expansion:
\begin{eqnarray}
\notag|A|&=&a_{11}(a_{12}+a_{22})(a_{13}+a_{23}+a_{33})+a_{11}(a_{13}+a_{33})(a_{12}+a_{22}+a_{32}) \\\notag
&&+a_{22}(a_{11}+a_{21})(a_{13}+a_{23}+a_{33})+a_{22}(a_{23}+a_{33})(a_{11}+a_{21}+a_{31}) \\\notag
&&+a_{33}(a_{11}+a_{31})(a_{12}+a_{22}+a_{32})+a_{33}(a_{22}+a_{32})(a_{11}+a_{21}+a_{31}) \\\notag
&&-a_{11}(a_{12}+a_{22}+a_{32})(a_{13}+a_{23}+a_{33})-a_{22}(a_{11}+a_{21}+a_{31})(a_{13}+a_{23}+a_{33}) \\\notag
&&-a_{33}(a_{11}+a_{21}+a_{31})(a_{12}+a_{22}+a_{32})-(a_{11}+a_{21})(a_{12}+a_{22})(a_{13}+a_{23}+a_{33}) \\\notag
&&-(a_{11}+a_{31})(a_{13}+a_{33})(a_{12}+a_{22}+a_{32})-(a_{22}+a_{32})(a_{23}+a_{33})(a_{11}+a_{21}+a_{31}) \\\notag
&&+(a_{11}+a_{21}+a_{31})(a_{12}+a_{22}+a_{32})(a_{13}+a_{23}+a_{33}).\\\notag
\end{eqnarray}
Our main theorem gives a general description of the terrible expansion of the determinant.
\begin{theorem} \label{thm:terrible}
Let $A=\left(a_{ij}\right)_{n\times n}$. The following formula is an expansion for the determinant of $A$:
\begin{equation}\label{eqn:main}
\det(A)=\sum\limits_{B\vdash[n]}(-1)^{n-|B|}\prod\limits_{\beta_k \in B}\prod\limits_{j\in\beta_k }\sum\limits_{i\in\beta^{\prime}_k}a_{ij},
\end{equation}
where the outer summation runs over all ordered partitions $B= (\beta_1,\beta_2, \ldots, \beta_r)$ of the set $[n]$ and
the inner summation runs over all integers $i$ in the union of first $k$ parts $\beta_k^{\prime}=\bigcup_{j=1}^k\beta_j$ of
the partition $B$.
\end{theorem}
The following example serves to clarify the notation in Theorem \ref{thm:terrible}.
\begin{example} \label{example:variety}
If \[B=(\beta_1,\beta_2,\beta_3)=\left(\{2\}, \{1,3\}, \{4,5\}\right),\]
then \[B' = (\beta_1^{\prime},\beta_2^{\prime},\beta_3^{\prime})=\left(\{2\}, \{1,2,3\}, \{1,2,3,4,5\}\right),\] and the
corresponding expression in Equation \eqref{eqn:main} for the ordered partition $B$ is
\[
a_{22}(a_{11}+a_{21}+a_{31})(a_{13}+a_{23}+a_{33})(a_{14}+a_{24}+a_{34}+a_{44}+a_{54})(a_{15}+a_{25}+a_{35}+a_{45}+a_{55}).
\]
notice that the first index runs through $B'$,
while the second index runs through the partition $B$.
\end{example}
The rest of this paper proceeds as follows:
In Section \ref{sec:flat}, we analyze the functions $f:[n]\to [n]$ indexing the terms in the terrible expansion.
After setting notation and proving a few fundamental lemmas, we end that section with Corollary \ref{corollary:permutation},
which proves that when $f:[n] \rightarrow [n]$ is a bijective function, or a permutation, then the
coefficients $c_f =sgn(f)$. In other words,
they are precisely the nonzero coefficients appearing in the determinant.
In Section \ref{sec:partitions}, we study the poset of ordered partitions and identify the importance of singleton partitions so
that we
can formulate our problem in more geometric terms as Euler characteristics
of convex polytopes relating to the permutahedron.
In Section \ref{sec:permutahedron}, we prove that $c_f = 0$ for all non-bijective functions $f:[n] \to [n]$ by analyzing Euler
characteristics of subsets of the permutahedron.
This proves that the terrible expansion does indeed give a formula for the determinant.
In Section \ref{sec:Hardstuff}, we give an extremely brief introduction to multivariate finite operator calculus, and state a
more general open conjecture that motivated this paper.
\section{Flattening Functions} \label{sec:flat}
Upon expanding the expression in Theorem \ref{thm:terrible}, many terms will cancel.
In this section, we set up the groundwork to keep track of each of the terms and show how they cancel.
Now consider any function $f:[n] \to [n]$ and define the monomial
\[
a_f:=\prod\limits_{j=1}^na_{f(j),j}.
\]
Expanding the terms in Equation \eqref{eqn:main} results in a sum of the form
\begin{equation}\label{eqn:mainf}
\sum\limits_{B\vdash[n]}(-1)^{n-|B|}\prod\limits_{\beta_k \in B}\prod\limits_{j\in\beta_k
}\sum\limits_{i\in\beta^{\prime}_k}a_{ij} = \sum_{f} c_f a_f,
\end{equation}
where each term $c_fa_f$ corresponds to a function from the set $[n]$ to itself.
For instance, in Equation \eqref{eqn:2by2} the four functions $f_i: [2] \to [2]$ are \[ f_1(1)=1, \ f_1(2)=1; \ \ f_2(1) = 1, \ f_2(2)=2; \ \ f_3(1) = 2, \ f_3(2)=2; \text{ and } f_4(1) =2, \ f_4(2)=1. \]
The terms corresponding to each function are labeled below
\begin{align*}
a_{11}(a_{12}+a_{22})+ a_{22}(a_{11}+a_{21}) -&(a_{11}+a_{21})(a_{12}+a_{22}) \\
=& (a_{11}a_{12}-a_{11}a_{12}) + (a_{11}a_{22}+ a_{11}a_{22}-a_{11}a_{22}) + \\ &(a_{21}a_{22}-a_{21}a_{22})+(-a_{12}a_{21}) \\
=& 0+a_{11}a_{22} +0 - a_{12}a_{21} \\
=& c_{f_1}a_{f_1} + c_{f_2}a_{f_2}+c_{f_3}a_{f_3}+c_{f_4}a_{f_4},\end{align*}
where first we expand the terms and then we simplify. Hence we see that $c_{f_1}=0$, $c_{f_2}= 1$, $c_{f_3}=0$, and $c_{f_4} = -1$.
Thus, the goal of this paper is to combinatorially identify the coefficients $c_f$
for each such function, and show that they agree with the coefficients of $a_f$ in the determinant. To do so, we must identify the set $S_f=\{ B \vdash [n]: a_f \text{ appears as a summand of the product indexed by } B\}$.
The following definition and lemma describe criteria for when an ordered partition $B \vdash [n]$ appears in $S_f$.
\begin{definition}
Let $B= ( \beta_1, \beta_2, \ldots, \beta_r ) $ be an ordered partition of $[n]$. For each $i\in [n]$, define $\beta(i)=k$ if $i\in\beta_k$. We say $i$ \textbf{\precequals} $j$ in $B$, denoted $i \preceq j$,
if $i$ appears in an earlier part or the same part as $j$ in the ordered partition $B$, i.e., \[ \beta(i)\leq\beta(j) .\]
\end{definition}
\begin{lemma}\label{lemma:obvious}
Let $B \vdash [n]$ be an ordered partition of $[n]$. The term $a_f$ appears
in the product $ \prod\limits_{\beta_k \in B}\prod\limits_{j\in\beta_k }\sum\limits_{i\in\beta^{\prime}_k}a_{ij} $ iff
$f(j) \preceq j$ in $B$ for all $1 \leq j \leq n$. \end{lemma}
\begin{proof}
Let $B \vdash [n]$ be an ordered partition of $[n]$.
Since $\beta'_k = \displaystyle \cup_{j=1}^k \beta_j$, we see that $a_f$ appears as a term in the product $\prod\limits_{\beta_k \in B}\prod\limits_{j\in\beta_k }\sum\limits_{i\in\beta^{\prime}_k}a_{ij}$
precisely when $i=f(j) \preceq j$ in the ordered partition $B$ for all $1 \leq j \leq n$.
\end{proof}
Now that Lemma \ref{lemma:obvious} has established that an ordered partition $B$ appears in the set $S_f$ if and only if
$f(j)$ precequals $j$ in $B$ for all integers $1 \leq j \leq n$, we are ready to analyze which functions $f:[n] \to [n]$ correspond to nonzero
terms in the terrible expansion.
To do so, we start by introducing some notation regarding the structure of such functions.
\begin{definition}\label{def:cyclic}
Let $f: [n] \to [n]$ be any function. We say that the function $f$ is \emph{acyclic} if $f^k(i)=i$ implies $k=1$ for all $i\in
[n]$. Otherwise, we say $f$ contains a \emph{cycle }.
\end{definition}
Henceforth we will prefer to work with acyclic functions. The next definition, describes how each function $f:[n]\rightarrow
[n]$ which contains a cycle
can be simplified or \emph{flattened}
to an acyclic function $\bar{f}$ on a different set of elements.
\begin{definition}\label{def:flattened}
Let $f: [n] \to [n]$ be any function, $C_f=\{\sigma_1, \sigma_2, \ldots, \sigma_r\}$ represent the cycles
of $f$, $N_f\subseteq[n]$ be the elements not in a cycle, and $D_f=C_f\cup N_f$.
Define the \emph{flattened} function $\overline{f}:D_f\to D_f$ as follows
\[
\overline{f}(i)=\begin{cases}
f(i) & \text{if } f(i) \text{ is not in a cycle of } f \\
\sigma_j & \mbox{if } f(i) \mbox{ belongs to the cycle } \sigma_j \\
i & \mbox{if } i\in C_f \\
\end{cases}
.\]
\end{definition}
Intuitively, $\overline{f}$ acts just like $f$, but shrinks each cycle of $f$ to a fixed point, and thus it is an acyclic
function.
The following example
illustrates Definitions \ref{def:cyclic} and \ref{def:flattened}.
\begin{example}
Let $f:[6] \to [6]$ be defined by $f(1) = 1, f(2) = 3, f(3)=2$, $f(4) = 3$, $f(5) = 6$, and $f(6) = 5$. Then $C_f = \{\sigma_1,
\sigma_2 \}$ where $\sigma_1 = (23)$ and $c_2 = (56)$.
The sets $N_f = \{ 1,4 \}$ and $D_f = N_f \cup C_f = \{ \sigma_1, \sigma_2, 1,4 \}$.
The function $\overline{f} : D_f \to D_f$ is defined by $\overline{f}(\sigma_1) = \sigma_1$, $\overline{f}(\sigma_2) = \sigma_2$,
$\overline{f}(1) =1$, and $\overline{f}(4) = \sigma_1$.
\end{example}
The following lemma shows that we can reduce the problem of calculating the coefficients $c_f$ in the terrible expansion, to that
of calculating
the coefficients $c_{\bar{f}}$ corresponding to acyclic functions.
\begin{lemma}\label{lemma:flattenorbits}
If $f: [n] \to [n]$ is any function, then the coefficients $c_f$ and $c_{\overline{f}}$ are related by the equation $c_f=(-1)^{n-|D_f|} c_{\overline{f}}$.
\end{lemma}
\begin{proof}
Let $S_f : =\{ B \vdash [n]: a_f \text{ appears as a summand of the product indexed by } B\}$, and
similarly let $S_{\overline{f}} : =\{ A \vdash D_f: a_{\overline{f}} \text{ appears as a summand of the product indexed
by } A\}$.
If $B$ is an ordered partition for which $a_f$ appears as a summand, then Lemma \ref{lemma:obvious} implies that for each cycle \[
\sigma_j=(i,f(i), f^2(i), \ldots, f^k(i))\] of $f$,
the following precedence relation must hold \[ i \preceq f^k(i) \preceq f^{k-1}(i) \preceq \cdots \preceq f(i) \preceq i \] in the ordered partition $B$.
Therefore $c_j$ must be contained in the same part $\beta$ of the ordered partition $B$. From here, it is clear to see that $S_f$
is equivalent to the set of ordered partitions of $D_f$, and so each term in
Equation \eqref{eqn:mainf} has the form
\begin{align*} c_fa_f & = \sum\limits_{B \in S_f}(-1)^{n-|B|}a_f \\ & = (-1)^{n-|D_f|} \sum\limits_{A\in D_f }(-1)^{|D_f|-|A|}a_f \\ & = (-1)^{n-|D_f|}c_{\overline{f}} a_f, \end{align*}
and the lemma follows.
\end{proof}
The number of ordered partitions of $[n]$ with $k$ parts is well known to be $k!S(n,k)$, where $S(n,k)$ are the Stirling numbers of the second kind. We obtain an important corollary to Lemma \ref{lemma:flattenorbits} from the following well-known result about Stirling numbers of the second kind.
\begin{lemma}\label{lemma:stirling}
The following identity holds for ordered partitions:
\[
\sum\limits_{k=0}^n(-1)^{n-k}k!S(n,k)=1.
\]
\end{lemma}
\begin{proof}
The result follows immediately upon setting $x=-1$ in the following identity on Stirling numbers of the second kind\cite[p.~35]{Sta12}:
\[
\sum\limits_{k=0}^nS(n,k)(x)_k=x^n. \qedhere
\]
\end{proof}
\begin{corollary} \label{corollary:permutation}
If $f:[n]\to [n]$ is bijective, i.e., $f=\pi$ for some $\pi \in \mathfrak{S}_n$, then $c_f=\mathop{\mathrm{sgn}}(\pi)$.
\end{corollary}
\begin{proof} Since $f$ is bijective, it consists only of cycles. Thus $D_f =
C_f = \{\sigma_1, \ldots, \sigma_r \}$ and $|D_f|=r$. By Lemma \ref{lemma:flattenorbits} and Lemma \ref{lemma:stirling} we see
\begin{align*} c_f & = (-1)^{n-|D_f|}\sum\limits_{A\in \mathcal{D} }(-1)^{|D_f|-|A|} \\& = (-1)^{n-r}\sum\limits_{B\vdash[r]}(-1)^{r-|B|} \\& = (-1)^{n-r}\sum\limits_{k=0}^n(-1)^{r-k}k!S(r,k) \\& = (-1)^{n-r}. \end{align*} We leave it to the reader to verify that $\mathop{\mathrm{sgn}}(\pi)=(-1)^{n-r}$.
\end{proof}
With Corollary \ref{corollary:permutation}, we have that the terrible expansion contains every term of the determinant with the
correct coefficient.
It remains to show that whenever $f$ is not bijective, $c_f=0$. By Lemma~\ref{lemma:flattenorbits}, it suffices to show this for
acyclic functions.
\section{The Poset of Ordered Partitions} \label{sec:partitions}
We next consider the poset of ordered partitions in order to show that the set $S_f$ has a nice structure when $f$ is acyclic. This will allow us to eventually switch to a more geometric viewpoint.
Let $\mathcal{P}_n$ denote the poset of ordered partitions of the set $[n]$.
In Figure \ref{poset3}, we see the poset $\mathcal{P}_3$ of ordered partitions on 3 elements. At the top of the poset $\mathcal{P}_n$, we have
the $n!$ ordered partitions consisting of singletons.
These partitions correspond bijectively with the elements of $\mathfrak{S}_n$. Because of their importance later,
we will refer to them as \itshape singleton partitions. \upshape
\begin{figure}[h]
\begin{center}\begin{tikzpicture}
[vertex/.style={circle,draw=black,fill=black,thick,inner sep=0pt,minimum size=1.5mm},
pre/.style={<-,shorten >=1pt,>=stealth,semithick},
post/.style={->,shorten >=2.5pt,>=stealth,thick},
noarrow/.style={-,thick}, scale=.7]
\node at (0,0) (123) [vertex] [label={below, font=\tiny}: 123] {};
\node at (-5,2) (1_23) [vertex] [label={left, font=\tiny}: {1/23}] {};
\node at (-3,2) (12_3) [vertex] [label={right, font=\tiny}: {12/3}] {};
\node at (-1,2) (2_13) [vertex] [label={right, font=\tiny}: {2/13}] {};
\node at (1,2) (23_1) [vertex] [label={right, font=\tiny}: {23/1}] {};
\node at (3,2) (3_12) [vertex] [label={right, font=\tiny}: {3/12}] {};
\node at (5,2) (13_2) [vertex] [label={right, font=\tiny}: {13/2}] {};
\node at (-5,4) (1_2_3) [vertex] [label={left, font=\tiny}: {1/2/3}] {};
\node at (-3,4) (2_1_3) [vertex] [label={right, font=\tiny}: {2/1/3}] {};
\node at (-1,4) (2_3_1) [vertex] [label={right, font=\tiny}: {2/3/1}] {};
\node at (1,4) (3_2_1) [vertex] [label={right, font=\tiny}: {3/2/1}] {};
\node at (3,4) (3_1_2) [vertex] [label={right, font=\tiny}: {3/1/2}] {};
\node at (5,4) (1_3_2) [vertex] [label={right, font=\tiny}: {1/3/2}] {};
\draw [post] (123) -- (1_23);
\draw [post] (123) -- (12_3);
\draw [post] (123) -- (2_13);
\draw [post] (123) -- (23_1);
\draw [post] (123) -- (3_12);
\draw [post] (123) -- (13_2);
\draw [post] (1_23) -- (1_2_3);
\draw [post] (12_3) -- (2_1_3);
\draw [post] (2_13) -- (2_3_1);
\draw [post] (23_1) -- (3_2_1);
\draw [post] (3_12) -- (3_1_2);
\draw [post] (13_2) -- (1_3_2);
\draw [post] (1_23) -- (1_3_2);
\draw [post] (12_3) -- (1_2_3);
\draw [post] (2_13) -- (2_1_3);
\draw [post] (23_1) -- (2_3_1);
\draw [post] (3_12) -- (3_2_1);
\draw [post] (13_2) -- (3_1_2);
\end{tikzpicture}
\caption{Poset of Ordered Partitions on 3 elements}
\label{poset3}
\end{center}
\end{figure}
Directly below a given ordered partition $B$ in $\mathcal{P}_n$ are ordered partitions formed by taking the union of two consecutive parts in $B$.
For example, directly below the singleton partition 3/1/2/4 are the ordered partitions 13/2/4, 3/12/4, and 3/1/24. All ordered
partitions under a singleton partition creates an $(n-1)$-cube.
An example is given in Figure \ref{cube}.
An acyclic function can be viewed as a rooted forest, where the fixed points are the roots. An example is given in Figure \ref{forest}.
Given an acyclic function $f$, if a path exists from $p$ to $q$, with $p$ closer to the root than $q$, then $f^k(q)=p$ for some $k$. Thus,
$p\preceq q$, and so we say $f$ has the \itshape rule \upshape $p\preceq q$. In this way, each function $f$ stipulates a set of
rules
\[R_f=\left \{ p\preceq q \mid f^k(q) = p \text{ for } q,k\in[n] \right \}.\]
The following lemma and corollary will show that the set of ordered partitions $S_f$ has a nice structure in $\mathcal{P}_n$.
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}
[vertex/.style={circle,draw=black,fill=black,thick,inner sep=0pt,minimum size=1.5mm},
redver/.style={circle,draw=red,fill=red,thick,inner sep=0pt,minimum size=1.5mm},
pre/.style={<-,shorten >=1pt,>=stealth,semithick},
post/.style={->,shorten >=2.5pt,>=stealth,thick},
noarrow/.style={-,thick},
redge/.style={-,thick,draw=red, fill=red}, scale=.4]
\node at ( 23,-4) (1) [vertex] [label={below, font=\footnotesize}: $1234$] {};
\node at ( 19,0) (2) [vertex] [label={left, font=\footnotesize}: {123/4}] {};
\node at ( 23,0) (3) [vertex] [label={right, font=\footnotesize}: {12/34}] {};
\node at ( 27,0) (5) [vertex] [label={right, font=\footnotesize}: {1/234}] {};
\node at ( 19,4) (6) [vertex] [label={left, font=\footnotesize}: {12/3/4}] {};
\node at ( 23,4) (10) [vertex] [label={right, font=\footnotesize}: {1/23/4}] {};
\node at ( 27,4) (15) [vertex] [label={right, font=\footnotesize}: {1/2/34}] {};
\node at ( 23,8) (30) [vertex] [label={above, font=\footnotesize}: {1/2/3/4}] {};
\draw [post] (1) -- (2);
\draw [post] (1) -- (3);
\draw [post] (1) -- (5);
\draw [post] (2) -- (6);
\draw [post] (2) -- (10);
\draw [post] (3) -- (6);
\draw [post] (3) -- (15);
\draw [post] (5) -- (10);
\draw [post] (5) -- (15);
\draw [post] (6) -- (30);
\draw [post] (10) -- (30);
\draw [post] (15) -- (30);
\end{tikzpicture}
\caption{The Cube Under the Singleton Partition 1/2/3/4}
\label{cube}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}\begin{tikzpicture}
[vertex/.style={circle,draw=black,fill=black,thick,inner sep=0pt,minimum size=1.5mm},
pre/.style={<-,shorten >=1pt,>=stealth,semithick},
post/.style={->,shorten >=2.5pt,>=stealth,thick},
noarrow/.style={-,thick}, scale=.5]
\node at (2,2) (2) [vertex] [label={below, font=\small}: 2] {};
\node at (0,0) (1) [vertex] [label={left, font=\small}: 1] {};
\node at (4,0) (5) [vertex] [label={right, font=\small}: 5] {};
\node at (7,1) (7) [vertex] [label={below, font=\small}: 7] {};
\node at (9,3) (4) [vertex] [label={left, font=\small}: 4] {};
\node at (11,1) (6) [vertex] [label={right, font=\small}: 6] {};
\node at (11,-1) (3) [vertex] [label={right, font=\small}: 3] {};
\draw [post] (5) -- (2);
\draw [post] (1) -- (2);
\draw [post] (7) -- (4);
\draw [post] (3) -- (6);
\draw [post] (6) -- (4);
\path[->,every loop/.style={looseness=30}] (4) edge [in=45,out=135,shorten >=2.5pt, shorten <=2.5pt, >=stealth,loop, thick, decoration={markings, mark=at position .999 with {\arrow[line width=2pt]{>}}},
postaction={decorate}] node {} ();
\path[->,every loop/.style={looseness=30}] (2) edge [in=45,out=135,shorten >=2.5pt, shorten <=2.5pt, >=stealth,loop, thick, decoration={markings, mark=at position .999 with {\arrow[line width=2pt]{>}}},
postaction={decorate}] node {} ();
\end{tikzpicture}
\caption{Acyclic Function Represented by a Rooted Forest}
\label{forest}
\end{center}
\end{figure}
\begin{lemma} \label{lemma:party}
If a singleton partition $A$ in $\mathcal{P}_n$ satisfies the rules $R_f$ given by an
acyclic function $f$ then every ordered partition $B\leq A$
in $\mathcal{P}_n$ also satisfies the rules $R_f$.
If an ordered partition $B$ in $\mathcal{P}_n$ satisfies the rules $R_f$ then there exists
at least one singleton partition $A \geq B$ in $\mathcal{P}_n$ that satisfies the
precedence rules
$R_f$.
\end{lemma}
\begin{proof} The first statement is obvious. If $A$ is a singleton partition with the rule $p \preceq q$ in $A$, and $B \leq A$
is an
ordered partition in $\mathcal{P}_n$,
then the parts of $B$ are unions of consecutive parts of $A$. Hence $p \preceq q$ in $B$ as well.
The second statement is slightly less obvious.
Let $f$ be an acyclic function,
and suppose that $B$ in $\mathcal{P}_n$ is a nonsingleton ordered partition that satisfies the rules $R_f$.
We must show there is a singleton partition $A \geq B$ above it in $\mathcal{P}_n$ that also satisfies $R_f$. Consider a part
$\beta$ of $B$ that is not a singleton. If no pair of elements in $\beta$ has a rule associated with it, then the elements
of $\beta$ can be ordered arbitrarily.
Otherwise, there are elements of $\beta$ that have rules imposed on them. Consider the elements in the intersection of $\beta$
and a
rooted tree associated with $f$. We order those elements by their distance from the root. (Those elements having the same
distance from the root can be put in any order
with respect to each other.) We do this for every rooted tree associated with $f$ to impose an order on all of $\beta$.
Doing the same to each part will result in a singleton partition $A$ above $B$ satisfying the rules of $f$.
\end{proof}
\begin{corollary}\label{cubes}
The set of ordered partitions of $[n]$ satisfying the rules of an acyclic function is a union of $(n-1)$-cubes in $\mathcal{P}_n$.
\end{corollary}
\begin{proof} By the above lemma, we can account for all the ordered partitions by only considering the singleton partitions satisfying the acyclic function, and all the ordered partitions below them. The result follows since the ordered partitions below a singleton partition form an $(n-1)$-cube.
\end{proof}
Lemma \ref{lemma:party} and Corollary \ref{cubes} tell us that once we know which singleton partitions appear in $S_f$, then we
know that $S_f$ is precisely those singletons and all the ordered partitions under them in $\mathcal{P}_n$. Because of their importance we
will start to label the singleton partitions without slashes, e.g. $1/2/3/4\to 1234$, unless we need to distinguish them from the
ordered partition with one part. In the next section, we turn our attention to a geometric object isomorphic to $\mathcal{P}_n$, the
permutahedron.
\section{The Permutahedron} \label{sec:permutahedron}
In this section we show how the alternating sums giving $c_f$ when $f$ is acyclic are related to
the Euler characteristic of the permutahedron and use this correspondence to show that $c_f = 0$.
It is well-known that the poset of the ordered partitions is isomorphic to the face lattice of the permutahedron
\cite[Fact 4.1]{Simion97}.
Specifically, each vertex on the permutahedron represents a singleton partition, the edges incident to a vertex represent the ordered partitions just
below that singleton partition in the poset, the faces adjacent to those edges represent the ordered partitions just below again, and so on,
until the permutahedron itself represents the ordered partition with one part at the bottom of the poset.
Note that two vertices are adjacent if one can be obtained by a single swap of consecutive elements. For example, 315624 is adjacent to 351624.
Figure \ref{perm3} shows the transformation from the poset on 3 element to the permutahedron on 3 elements,
which
in this case is a hexagon. Figure \ref{perm4} shows the poset on 4 elements as the ordinary permutahedron (truncated octahedron).
\begin{figure}[hbtp!]
\begin{center}\begin{tikzpicture}
[vertex/.style={circle,draw=black,fill=black,thick,inner sep=0pt,minimum size=1.5mm}, vertexb/.style={circle,draw=black,fill=black,thick,inner sep=0pt,minimum size=1.5mm},
pre/.style={<-,shorten >=1pt,>=stealth,semithick},
post/.style={->,shorten >=2.5pt,>=stealth,thick},
noarrow/.style={-,thick}, scale=.6]
\node at (0,0) (123) [vertex] [label={below, font=\small}: {123}] {};
\node at (0,-4) (3_2_1) [vertex] [label={left, font=\small}: {3/2/1}] {};
\node at (-2,-2) (3_12) [vertexb] [label={below, font=\small}: {3/12}] {};
\node at (-5,-2) (3_1_2) [vertex] [label={left, font=\small}: {3/1/2}] {};
\node at (-3,0) (13_2) [vertexb] [label={left, font=\small}: {13/2}] {};
\node at (0,4) (1_2_3) [vertex] [label={above, font=\small}: {1/2/3}] {};
\node at (-2,2) (1_23) [vertexb] [label={above, font=\small}: {1/23}] {};
\node at (-5,2) (1_3_2) [vertex] [label={left, font=\small}: {1/3/2}] {};
\node at (2,-2) (23_1) [vertexb] [label={below, font=\small}: {23/1}] {};
\node at (5,-2) (2_3_1) [vertex] [label={right, font=\small}: {2/3/1}] {};
\node at (3,0) (2_13) [vertexb] [label={right, font=\small}: {2/13}] {};
\node at (2,2) (12_3) [vertexb] [label={above, font=\small}: {12/3}] {};
\node at (5,2) (2_1_3) [vertex] [label={right, font=\small}: {2/1/3}] {};
\draw [noarrow] (123) -- (2_13);
\draw [noarrow] (123) -- (12_3);
\draw [noarrow] (123) -- (1_23);
\draw [noarrow] (123) -- (13_2);
\draw [noarrow] (123) -- (3_12);
\draw [noarrow] (123) -- (23_1);
\draw [noarrow] (2_13) -- (2_1_3);
\draw [noarrow] (2_1_3) -- (12_3);
\draw [noarrow] (12_3) -- (1_2_3);
\draw [noarrow] (1_2_3) -- (1_23);
\draw [noarrow] (1_23) -- (1_3_2);
\draw [noarrow] (1_3_2) -- (13_2);
\draw [noarrow] (13_2) -- (3_1_2);
\draw [noarrow] (3_1_2) -- (3_12);
\draw [noarrow] (3_12) -- (3_2_1);
\draw [noarrow] (3_2_1) -- (23_1);
\draw [noarrow] (23_1) -- (2_3_1);
\draw [noarrow] (2_3_1) -- (2_13);
\end{tikzpicture}
\hspace{.5in}
\begin{tikzpicture}
[vertex/.style={circle,draw=black,fill=black,thick,inner sep=0pt,minimum size=1.5mm},
pre/.style={<-,shorten >=1pt,>=stealth,semithick},
post/.style={->,shorten >=2.5pt,>=stealth,thick},
noarrow/.style={-,thick}, scale=.65]
\node at (0,0) (3_2_1) [vertex] [label={below, font=\small}: {3/2/1}] {};
\node at (-2,2) (3_1_2) [vertex] [label={left, font=\small}: {3/1/2}] {};
\node at (2,2) (2_3_1) [vertex] [label={right, font=\small}: {2/3/1}] {};
\node at (-2,4.6) (1_3_2) [vertex] [label={left, font=\small}: {1/3/2}] {};
\node at (2,4.6) (2_1_3) [vertex] [label={right, font=\small}: {2/1/3}] {};
\node at (0,6.6) (1_2_3) [vertex] [label={above, font=\small}: {1/2/3}] {};
\draw [noarrow] (3_2_1) -- (3_1_2);
\draw [noarrow] (3_2_1) -- (2_3_1);
\draw [noarrow] (3_1_2) -- (1_3_2);
\draw [noarrow] (2_3_1) -- (2_1_3);
\draw [noarrow] (1_3_2) -- (1_2_3);
\draw [noarrow] (2_1_3) -- (1_2_3);
\node at (-1.6,.8) {\footnotesize 3/12};
\node at (1.6,.8) {\footnotesize 23/1};
\node at (-2.6,3.2) {\footnotesize 13/2};
\node at (2.6,3.2) {\footnotesize 2/13};
\node at (-1.6,5.8) {\footnotesize 1/23};
\node at (1.6,5.8) {\footnotesize 12/3};
\node at (0,3.2) {\footnotesize 123};
\end{tikzpicture}
\caption{Poset of Ordered Partitions on 3 elements: Top View (left) and as a Permutahedron (right)}
\label{perm3}
\end{center}
\end{figure}
\newcommand{1.2mm}{1.2mm}
\begin{figure}[h]
\begin{center}\begin{tikzpicture}
[vertex/.style={circle,draw=black,fill=black,thick,inner sep=0pt,minimum size=1.2mm}, fadedvertex/.style={circle,draw=gray,fill=gray,thick,inner sep=0pt,minimum size=1.2mm},
pre/.style={<-,shorten >=1pt,>=stealth,semithick},
post/.style={->,shorten >=2.5pt,>=stealth,thick},
noarrow/.style={-,thick}, faded/.style={-,thick,gray},scale=.4]
\node at (0,0) (1234) [vertex] [label={below, font=\tiny}:$1234$] {};
\node at ( -3,1) (2134) [vertex] [label={below, font= \tiny}:$2134$] {};
\node at ( 0.5,1.2) (1324) [vertex] [label={left, font= \tiny}:$1324$] {};
\node at ( 2,1.2) (1243) [vertex] [label={right, font= \tiny}:$1243$] {};
\node at ( -1,2.3) (2143) [fadedvertex] [label={left, font=\tiny, gray}:$2143$] {};
\node at ( -6,4) (3124) [vertex] [label={left, font=\tiny}:$2314$] {};
\node at ( -2.5,4.5) (2314) [vertex] [label={below, font=\tiny}:$3124$] {};
\node at ( 4,4.3) (1423) [vertex] [label={left, font=\tiny}:$1342$] {};
\node at ( 4.8,3.6) (1342) [vertex] [label={below, font=\tiny}:$1423$] {};
\node at ( -5.5,5.5) (3214) [vertex] [label={right, font=\tiny}:$3214$] {};
\node at ( -1.6,6) (3142) [fadedvertex] [label={left, font=\tiny, gray}:$2413$] {};
\node at ( 6,5.5) (1432) [vertex] [label={right, font=\tiny}:$1432$] {};
\node at ( -6.3,8) (4123) [vertex] [label={left, font=\tiny}:$2341$] {};
\node at ( -4.2,8.9) (4132) [fadedvertex] [label={above, font=\tiny,gray}:$2431$] {};
\node at ( 1,7.6) (2413) [vertex] [label={below, font=\tiny}:$3142$] {};
\node at ( 4.2,8.5) (2341) [fadedvertex] [label={above, font=\tiny,gray}:$4123$] {};
\node at ( -5.8,10) (4213) [vertex] [label={left, font=\tiny}:$3241$] {};
\node at ( 1.5,8.7) (3241) [fadedvertex] [label={above, font=\tiny,gray}:$4213$] {};
\node at ( 5.4,10.4) (2431) [vertex] [label={right, font=\tiny}:$4132$] {};
\node at ( -1.1,11.6) (4231) [fadedvertex] [label={below, font=\tiny,gray}:$4231$] {};
\node at ( -2.5,13) (4312) [vertex] [label={left, font=\tiny}:$3421$] {};
\node at ( 0.2,12.5) (3412) [vertex] [label={above, font=\tiny}:$3412$] {};
\node at ( 2.5,13.4) (3421) [vertex] [label={right, font=\tiny}:$4312$] {};
\node at ( -.5,13.8) (4321) [vertex] [label={above, font=\tiny}:$4321$] {};
\draw [noarrow] (2134) -- (3124);
\draw [faded] (2134) -- (2143);
\draw [faded] (1243) -- (2143);
\draw [noarrow] (1243) -- (1342);
\draw [faded] (2143) -- (3142);
\draw [faded] (3142) -- (4132);
\draw [faded] (3142) -- (3241);
\draw [faded] (4132) -- (4231);
\draw [faded] (3241) -- (4231);
\draw [faded] (2341) -- (3241);
\draw [faded] (2341) -- (2431);
\draw [faded] (1342) -- (2341);
\draw [noarrow] (1342) -- (1432);
\draw [noarrow] (4123) -- (4213);
\draw [faded] (4123) -- (4132);
\draw [faded] (4231) -- (4321);
\draw [noarrow] (3412) -- (4312);
\draw [noarrow] (3412) -- (3421);
\draw [noarrow] (1234) -- (2134);
\draw [noarrow] (1234) -- (1324);
\draw [noarrow] (1234) -- (1243);
\draw [noarrow] (1324) -- (2314);
\draw [noarrow] (1324) -- (1423);
\draw [noarrow] (3124) -- (3214);
\draw [noarrow] (3124) -- (4123);
\draw [noarrow] (2314) -- (3214);
\draw [noarrow] (2314) -- (2413);
\draw [noarrow] (1423) -- (2413);
\draw [noarrow] (1423) -- (1432);
\draw [noarrow] (3214) -- (4213);
\draw [noarrow] (1432) -- (2431);
\draw [noarrow] (2431) -- (3421);
\draw [noarrow] (4213) -- (4312);
\draw [noarrow] (4312) -- (4321);
\draw [noarrow] (3421) -- (4321);
\draw [noarrow] (2413) -- (3412);
\end{tikzpicture}
\caption{Poset of Ordered Partitions on 4 elements as a Permutahedron}
\label{perm4}
\end{center}
\end{figure}
In Figure \ref{perm4}, we only label the singleton partitions at the vertices, but the labeling of the other ordered partitions
would be similar to Figure \ref{perm3}.
The permutahedron $\Pi_n$ is often defined as the convex hull of the points \[P_{\sigma}=(\sigma(1), \sigma(2), \ldots,
\sigma(n))\]
for every $\sigma\in\mathfrak{S}_n$. It is a convex polytope, and in particular, it is contractible to a point.
Thus, the permutahedron has Euler characteristic 1 \cite{Simion97}.
Since the ordered partitions are in bijection with the faces of the permutahedron, the alternating sum of the ordered partitions
is precisely the Euler characteristic of the permutahedron, and this gives us a second proof of Lemma \ref{lemma:stirling}.
We
adopt a slightly different convention, relabeling the vertices of the $\Pi_n$ to
$P_{\sigma^{-1}}$. We will denote this relabeled permutahedron by $\Pi_n'$. Figure \ref{2perm}
shows $\Pi_3$ in $\mathbb{R}^3$, the relabeled $\Pi_n'$,
and the correspondence between $x_1\leq x_2$ and $1\preceq 2$. Figure \ref{2perm} also shows the fact that $\Pi_n$, and thus
$\Pi_n'$, is an $(n-1)$-dimensional object, since all the points lie in the hyperplane $x_1+x_2+\cdots+x_n=\dbinom{n+1}{2}$. In
general $x_i\leq x_j$ in $\Pi_n$ corresponds to $i\preceq j$ in $\Pi_n'$.
\begin{figure}
\begin{tikzpicture}
[vertex/.style={circle,draw=black,fill=black,thick,inner sep=0pt,minimum size=1.5mm},
pre/.style={<-,shorten >=1pt,>=stealth,semithick},
post/.style={->,shorten >=2.5pt,>=stealth,thick},
noarrow/.style={-,thick}, scale=.8]
\draw [->] (0,0) -- (6,0,0) node [right] {$x_1$};
\draw [->] (0,0) -- (0,6,0) node [above] {$x_3$};
\draw [->] (0,0) -- (0,0,6) node [below left] {$x_2$};
\draw[noarrow] (0,0,6) -- (0,6,0);
\draw[noarrow] (6,0,0) -- (0,6,0);
\draw[noarrow] (0,0,6) -- (6,0,0);
\node at (1,3,2) (123) [vertex] [label={left, font=\tiny}:{$(1,2,3)$}] {};
\node at (1,2,3) (132) [vertex] [label={left, font=\tiny}:{$(1,3,2)$}] {};
\node at (2,3,1) (213) [vertex] [label={above, font=\tiny}:{$(2,1,3)$}] {};
\node at (2,1,3) (231) [vertex] [label={below, font=\tiny}:{$(2,3,1)$}] {};
\node at (3,2,1) (312) [vertex] [label={right, font=\tiny}:{$(3,1,2)$}] {};
\node at (3,1,2) (321) [vertex] [label={right, font=\tiny}:{$(3,2,1)$}] {};
\draw[noarrow] (123) -- (213);
\draw[noarrow] (123) -- (132);
\draw[noarrow] (213) -- (312);
\draw[noarrow] (312) -- (321);
\draw[noarrow] (321) -- (231);
\draw[noarrow] (231) -- (132);
\draw[noarrow] (1,4,0) -- (3.5,5,0);
\node at (3.7,5.2,0) {\footnotesize $x_1+x_2+x_3=6$};
\draw[dashed] (0,6,0) -- (4,0,5);
\draw[post] (4,0,5) -- (3,0,6);
\node at (5,0,6) {$x_1\leq x_2$};
\node at (0,-4) {};
\end{tikzpicture}
\hspace{.5in}
\begin{tikzpicture}
[vertex/.style={circle,draw=black,fill=black,thick,inner sep=0pt,minimum size=1.5mm},
pre/.style={<-,shorten >=1pt,>=stealth,semithick},
post/.style={->,shorten >=2.5pt,>=stealth,thick},
noarrow/.style={-,thick}, scale=1.3]
\node at (1,0) (3_2_1) [vertex] [label={below, font=\small}: {321}] {};
\node at (-1,0) (3_1_2) [vertex] [label={below, font=\small}: {312}] {};
\node at (2,2) (2_3_1) [vertex] [label={right, font=\small}: {231}] {};
\node at (-2,2) (1_3_2) [vertex] [label={left, font=\small}: {132}] {};
\node at (1,4) (2_1_3) [vertex] [label={above, font=\small}: {213}] {};
\node at (-1,4) (1_2_3) [vertex] [label={above, font=\small}: {123}] {};
\draw [noarrow] (3_2_1) -- (3_1_2);
\draw [noarrow] (3_2_1) -- (2_3_1);
\draw [noarrow] (3_1_2) -- (1_3_2);
\draw [noarrow] (2_3_1) -- (2_1_3);
\draw [noarrow] (1_3_2) -- (1_2_3);
\draw [noarrow] (2_1_3) -- (1_2_3);
\draw[dashed] (0,-1) -- (0,5);
\draw[post] (0,-1) -- (-.5,-1);
\draw[post] (0,5) -- (-.5,5);
\node at (0,-1.5) {$1\preceq 2$};
\end{tikzpicture}
\caption{The permutahedra $\Pi_3$ (left) and $\Pi_3'$ (right)}
\label{2perm}
\end{figure}
\begin{lemma}\label{halfspace}
The singleton partitions in $\Pi_n'$ satisfying the precedence rule $i\preceq j$ are contained
in the corresponding half-space $x_i\leq x_j$ in $\Pi_n$.
\end{lemma}
\begin{proof}
A permutation $\sigma^{-1}$ satisfies $\sigma^{-1}_i \preceq \sigma^{-1}_j$ precisely when $\sigma(i) \leq \sigma(j)$.
Hence we see that if the permutation (or singleton partition) $\sigma^{-1}$ satisfies the precedence relation $i \preceq j$
then the vertex $(\sigma(1), \sigma(2), \ldots, \sigma(n) )$ is in the half-space $x_i\leq x_j$ and vice versa.
\end{proof}
We are now ready to prove the main result of this section.
\begin{proposition} \label{prop:zerocoefficient}
If $f: [n] \to [n]$ is not bijective, then the coefficient $c_f=0$ is zero.
\end{proposition}
\begin{proof}
Let $f:[n] \to [n]$ be an acyclic function. Let $R_f = \{ f(j)=i \preceq j \}$ denote the set of precedence rules
determined by $f$.
To each precedence rule $f(j) = i \preceq j$ in $R_f$ we can assign a half-space $H_{ij} := \{ (x_1,x_2,\ldots, x_n) \in \mathbb{R}^n: x_{f(j)} = x_i \leq x_j \}$,
and the intersection of these half-spaces with the permutahedron
$\Pi_n'$ defines a convex polytope, which we will denote by $\Pi_f$.
The faces of $\Pi_f$ fall into two disjoint subsets: those faces that correspond to the ordered partitions in $S_f$ and those that do not.
Let $\Gamma_f$ denote the faces of $\Pi_f$ which correspond to elements of $S_f$ and let $\Delta_f$ denote the faces of $\Pi_f$ that do not.
The set of faces $\Delta_f$ are precisely the faces of $\Pi_f$ which lie entirely on the boundary of at least one half-space $x_i = x_j$ because they resulted from
intersecting $\Pi_n'$ with one of the half-spaces $H_{ij}$.
Thus each face of $\Delta_f$ is a convex polytope, and we see that $\Delta_f$ is a union convex polytopes.
Each of these convex polytopes contains the point $(x_1,x_2, \ldots, x_n)$ where $ x_1=x_2=\cdots= x_n$, so $\Delta_f$ is a contractible space.
Hence $\Delta_f$ has Euler characteristic $\chi(\Delta_f) = 1$.
Since $\Pi_f$ is a convex polytope and $\Pi_f = \Gamma_f \sqcup \Delta_f$ we see that
\[1= \chi(\Pi_f) = \chi(\Gamma_f) + \chi( \Delta_f ) = 0+1.\]
It follows that $\chi(\Gamma_f) =0$.
Since the summands of the alternating sum $\displaystyle \sum_{B \in S_f} (-1)^{n- |B|}$ correspond with the faces of $\Gamma_f$ we see that
$c_f = \displaystyle \sum_{B \in S_f} (-1)^{n- |B|} = \chi(\Gamma_f) =0$ as desired.
\end{proof}
We end this section with an example demonstrating the proof of Proposition \ref{prop:zerocoefficient}.
\begin{example}
Figure \ref{permex2a} shows $\Pi_f$ with $\Gamma_f$ bold and $\Delta_f$ shaded for the terms $a_f=a_{11}a_{12}a_{13}$, which has the rules $1\preceq2$ and $1\preceq3$ and $a_f=a_{11}a_{12}a_{33}a_{34}$, which has the rules $1\preceq2$ and $3\preceq4$.
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}
[vertex/.style={circle,draw=black,fill=black,thick,inner sep=0pt,minimum size=1.2mm}, bvertex/.style={circle,draw=blue,fill=blue,thick,inner sep=0pt,minimum size=1.2mm}, fadedvertex/.style={circle,draw=gray,fill=gray,thick,inner sep=0pt,minimum size=1.2mm},
pre/.style={<-,shorten >=1pt,>=stealth,semithick},
post/.style={->,shorten >=2.5pt,>=stealth,thick},
noarrow/.style={-,thick}, faded/.style={-,thick,gray},scale=1.55]
\node at (-2,2) (1_3_2) [vertex] [label={left, font=\small}: {132}] {};
\node at (-1,4) (1_2_3) [vertex] [label={above, font=\small}: {123}] {};
\node at (-1.5,1) (x_x_2) [fadedvertex] {};
\node at (0,4) (x_x_3) [fadedvertex] {};
\node at (0,2) (x_x_x) [fadedvertex] {};
\draw [noarrow] (x_x_3) -- (1_2_3);
\draw [noarrow] (1_2_3) -- (1_3_2);
\draw [faded] (x_x_3) -- (x_x_x);
\draw [noarrow] (1_3_2) -- (x_x_2);
\draw [faded] (x_x_2) -- (x_x_x);
\end{tikzpicture}
\hspace{1in}
\begin{tikzpicture}
[vertex/.style={circle,draw=black,fill=black,thick,inner sep=0pt,minimum size=1.2mm}, bvertex/.style={circle,draw=blue,fill=blue,thick,inner sep=0pt,minimum size=1.2mm}, fadedvertex/.style={circle,draw=gray,fill=gray,thick,inner sep=0pt,minimum size=1.2mm},
pre/.style={<-,shorten >=1pt,>=stealth,semithick},
post/.style={->,shorten >=2.5pt,>=stealth,thick},
noarrow/.style={-,thick}, faded/.style={-,thick,gray},scale=.35]
\node at ( 0,0) (1234) [vertex] [label={below, font=\tiny}:$1234$] {};
\node at ( 0.5,1.2) (1324) [vertex] [label={above, font= \tiny}:$1324$] {};
\node at ( -2.5,4.5) (2314) [vertex] [label={below, font=\tiny}:$3124$] {};
\node at ( 4,4.3) (1423) [vertex] [label={left, font=\tiny}:$1342$] {};
\node at ( 1,7.6) (2413) [vertex] [label={below, font=\tiny}:$3142$] {};
\node at ( 0.2,12.5) (3412) [vertex] [label={below, font=\tiny}:$3412$] {};
\node at ( -1.5,0.5) (xx34) [fadedvertex] {};
\node at ( 1,0.6) (12xx) [fadedvertex] {};
\node at ( -4,5) (3xx4) [fadedvertex] {};
\node at ( 5,4.9) (1xx2) [fadedvertex] {};
\node at ( -1.15,12.75) (34xx) [fadedvertex] {};
\node at ( 1.35,12.95) (xx12) [fadedvertex] {};
\node at ( -0.5,1.1) (xxyy) [fadedvertex] {};
\node at ( 0,13.2) (yyxx) [fadedvertex] {};
\draw [faded] (xx34) -- (3xx4);
\draw [faded] (34xx) -- (3xx4);
\draw [faded] (12xx) -- (1xx2);
\draw [faded] (xx12) -- (1xx2);
\draw [faded] (xx12) -- (yyxx);
\draw [faded] (34xx) -- (yyxx);
\draw [faded] (12xx) -- (xxyy);
\draw [faded] (xx34) -- (xxyy);
\draw [faded] (xxyy) -- (yyxx);
\draw [noarrow] (3412) -- (34xx);
\draw [noarrow] (3412) -- (xx12);
\draw [noarrow] (1234) -- (xx34);
\draw [noarrow] (1234) -- (1324);
\draw [noarrow] (1234) -- (12xx);
\draw [noarrow] (1324) -- (2314);
\draw [noarrow] (1324) -- (1423);
\draw [noarrow] (2314) -- (3xx4);
\draw [noarrow] (2314) -- (2413);
\draw [noarrow] (1423) -- (2413);
\draw [noarrow] (1423) -- (1xx2);
\draw [noarrow] (2413) -- (3412);
\end{tikzpicture}
\caption{$a_f=a_{11}a_{12}a_{13}$ (left) and $a_f=a_{11}a_{12}a_{33}a_{34}$ (right)}
\label{permex2a}
\end{center}
\end{figure}
\end{example}
\section{Multivariate Finite Operator Calculus} \label{sec:Hardstuff}
This terrible expansion of the determinant came from a conjecture about a transfer formula in \emph{multivariate finite operator
calculus}
(MFOC). In this section we give a very brief overview of the objects of study in MFOC and the conjecture that gives this
expansion. The interested reader is encouraged to read \cite{watanabe1984} for a more comprehensive description of this subject matter.
Let $k$ be a field. Let $\{\textbf{e}_i\}_{1 \leq i \leq \ell} $ denote the standard
$\ell$-dimensional
basis of $k ^\ell$.
The main objects of study in MFOC are polynomials $p\in k[x_1,\ldots,x_{\ell}]$ and operators $T\in k[[D_1,\ldots,D_{\ell}]]$,
where $D_i$ is the partial derivative with respect to $x_i$. A sequence of polynomials
$b_{\textbf{n}}(\textbf{x})=b_{n_1,\ldots,n_{\ell}}(x_1,...,x_{\ell})$ is called a Sheffer sequence if there is a set of operators
$\textbf{B}=(B_1,\ldots,B_{\ell})$ with $B_i=D_iP_i$ where each $B_i:b_\textbf{n} \to
b_{\textbf{n}-\textbf{e}_i}$ and each $P_i$ an invertible operator.
Such a set of operators is called a delta $\ell$-tuple. The
power series for an operator $T$ is written as
\begin{equation}
T=\sum\limits_{\textbf{n}\geq0}a_\textbf{n}\textbf{D}^\textbf{n}=\sum\limits_{n_1,\ldots,n_{\ell}\geq0}a_{n_1,\ldots,n_{\ell}}D_1^{n_1}\cdots D_{\ell}^{n_{\ell}}.
\label{eqn:T} \end{equation}
These are all standard notations in any multivariate theory.
However, the following notation is not completely standard in MFOC.
Given a subset $A\subseteq[{\ell}]$ we define $X_A:=\prod\limits_{i\in A}X_i$.
The following is the Transfer Theorem from MFOC, and is essentially Theorem 1.3.6 in \cite{watanabe1984} with different notation.
\begin{theorem}[Transfer Theorem]
Let $\mathcal{J}$ denote the usual Jacobian matrix of a collection of polynomials.
Suppose $\textbf{B}=(B_1,B_2,\ldots,B_{\ell})$ is a delta $\ell$-tuple where $B_i=D_iP_i^{-1}$, then
\[
b_\textbf{n}(\textbf{x})=\textbf{P}^{\textbf{n}+\textbf{1}}{\mathcal J}(B_1,B_2,\ldots,B_{\ell})\dfrac{\textbf{x}^\textbf{n}}{\textbf{n}!}
\] is the basic sequence for $\textbf{B}$ written in terms of $\frac{\textbf{x}^\textbf{n}}{\textbf{n}!}$.
\end{theorem}
Within the Jacobian, we have Pincherle
derivatives $\dfrac{\partial B_i}{\partial D_j}=B_i\theta_j-\theta_jB_i$, where $\theta_j:p\to x_jp$ is the $j$th umbral shift
operator that does not commute with the delta operators. Thus, there are many ways this transfer formula could be expanded. The
following conjecture (based on the examples provided below) gives one such way.
\begin{conjecture} \label{MainConjecture}
The basic sequence from the Transfer Theorem can also be calculated as
\[
b_\textbf{n}(\textbf{x})=\sum\limits_{B\vdash[\ell]}(-1)^{\ell-|B|}\left(\theta_{\beta}P_{\beta}\right)_B\dfrac{\textbf{x}^{\textbf{n}-\textbf{1}}}{\textbf{n}!},
\] where $B$ runs through all ordered partitions of $[\ell]$ and $\beta$ runs through the partitions of $B$.
\end{conjecture}
Because of our abuse of some notation, we give some examples.
\[
b_{m,n}(u,v)=\left(uP_1^mvP_2^n+vP_2^nuP_1^m-uvP_1^mP_2^n\right)\dfrac{u^{m-1}v^{n-1}}{m!n!}
\]
\footnotesize
\begin{eqnarray}
\notag b_{m,n,p}(a,b,c)&=&\left(aRbScT + aRcTbS + bSaRcT + bScTaR + cTaRbS + cTbSaR\right. \\\notag
&&- aRbcST - bSacRT - cTabRS - abRScT - acRTbS - bcSTaR \\\notag
&&\left.+ abcRST\right)\dfrac{a^{m-1}b^{n-1}c^{p-1}}{m!n!p!} \\\notag
\end{eqnarray}
\normalsize
\[
B=(\{2\}, \{1,3\}) \quad \Rightarrow \quad \left(\theta_{\beta}P_{\beta}\right)_B=bSacRT
\]
The terrible expansion of the determinant comes from setting each $B_i=\textbf{D}^{\textbf{a}_i}=D_1^{a_{i1}}D_2^{a_{i2}}\cdots
D_n^{a_{in}}$, or in other words, it is one term of the power series in Equation \eqref{eqn:T}.
\section{Conclusions and Open Questions}
We end with few open questions stemming from our work.
\begin{enumerate}
\item[\textbf{Q1.}] Now that Theorem \ref{thm:terrible} shows that Conjecture \ref{MainConjecture} is true for one term of an
operator's power series, can Conjecture \ref{MainConjecture} be proven by a linearity argument?
\item[\textbf{Q2.}] Can the proof of Ryser's formula given by Horn and Johnson \cite{HJ} be modified to give another proof of
Theorem \ref{thm:terrible}?
\item[\textbf{Q3.}] Can our proof of Theorem \ref{thm:terrible} be modified to prove Ryser's formula by using the
topological/combinatorial properties of the cube instead of the permutahedon?
\end{enumerate}
\section{Acknowledgements}
The authors would like to thank Drs. Mohamed Omar, Pamela Harris, and Brian Johnson for helpful conversations during the writing
of this paper.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,259 |
The Reserve Bank of India is helping strengthen the regulatory capacity of Da Afghanistan Bank, the country's central bank, backed up by Turkey's central bank, the IMF and World Bank.
Da Afghanistan Bank governor Khalil Sediq said the employees were trained in India by the RBI.
The worst it gets for most central bankers is recession, deflation or a credit crunch. Maybe a bank bust if times are really tough. Not so in Afghanistan.
There, Da Afghanistan Bank has had to grapple with a sharp economic slowdown worsened by the draw down of U.S. forces, Taliban attacks on a lender and a $900 million loan scandal and ensuing bank collapse that deepened distrust toward financial institutions in a nation where just 11 percent its 32 million people have a bank account. Now, the worst may be over.
The World Bank, International Monetary Fund and the central banks of India and Turkey are helping develop Da Afghanistan Bank, Governor Khalil Sediq said in an interview in his wood-paneled office in Kabul. Bank profits rose 5 percent last year and stability has returned to the nation's financial system, Sediq said.
The 64-year-old governor -- in his second stint in the role -- has managed to restore stability even as economic growth stalled. Reduced aid from the international community and weak investor confidence in the face of increasing security challenges and political instability have weighed on the economy, according to a World Bank assessment on May 25. Better times lie ahead though, with growth projected to accelerate to 2.6 percent in 2017 and to around 3.6 percent by 2020.
Sediq joined the central bank 37 years ago, rising to governor for the first time from 1990 to 1991. He took up the role again at the invitation of President Ashraf Ghani in July 2015 amid an economic slowdown that followed the draw down of foreign forces. Months earlier, the Taliban, which is fighting the government and U.S. forces across much of the country, stormed New Kabul Bank in Jalalabad city, killing Afghan soldiers who had gone to receive their salaries.
Other than three Pakistani banks, no foreign lenders operate in the war-torn country, Sediq said. His office -- walls adorned with a picture of a camel caravan and a photo of President Ghani -- sits within Da Afghanistan Bank's heavily guarded building, next to the Finance Ministry.
Five years before his tenure began, Kabul Bank lost more than $900 million of saving assets in bad insider loans. An inquiry found the central bank had failed to provide appropriate oversight. The government of then president Hamid Karzai took over the bank and placed it in receivership. So far $450 million of the loans have been recovered, Sediq said.
"The collapse of Kabul Bank in 2010 further deteriorated the trust of people in banks," said Ahmad Massoud, an economics professor at Kabul University.
Even before that scandal, most people kept their cash inside pillows or locked up at home. Those who do have bank accounts are mostly from the relatively more developed provinces such as Kabul, Balkh, Herat, Nangarhar and Kandahar.
More than 100 audits were carried out on Afghanistan's 15 remaining banks last year, tightening oversight of the sector since Kabul Bank's collapse in 2010, Sediq said.
The Reserve Bank of India is helping strengthen the regulatory capacity of the Afghan bank, backed up by Turkey's central bank, the IMF and World Bank.
"We have to learn a lot from the Reserve Bank of India," said Sediq. "People from the auditing department, supervisory department, monetary policy and payment department are using the training in India and Turkey," as well as the IMF and World Bank.
A spokesman for the RBI declined to comment. Turkey's central bank confirmed it has a Memorandum of Understanding with the Afghan central bank but did not give any further details.
The IMF confirmed it provides technical assistance in bank management, and is helping strengthen Da Afghanistan Bank's independence, operations, and supervisory practices. In 2016, the IMF introduced measures to bolster financial stability by "fully resolving the 2010 Kabul Bank crisis by restoring the central bank's balance sheet and New Kabul Bank's solvency," a Kabul-based IMF spokeswoman said.
World Bank assistance is focused on building the supervision department's regulatory capacity, investing in the system's infrastructure to create an efficient and sound payment system and modernizing the core banking system, a Washington-based spokeswoman wrote.
The nation's 15 banks, which offer both conventional and Islamic banking products, have $696 million in outstanding loans and $4.1 billion in assets, according to a report by the central bank.
"I believe that in the coming years there should be some consolidation in the banking system," Sediq said. "It's much better if small banks merge."
The bank has also started licensing money transfer providers called Hawalas. Last year, the bank cancelled 80 such licenses because of money-laundering.
While the lending infrastructure is improving, the central bank's ambition to start a stock market in Kabul has been put on hold.
"It will be a disaster for now to have created a stock market in Afghanistan due to security issues and the economic slowdown," Sediq said, noting there are no laws or regulations in place to support it. "We will have it," he said, but for now "we are not ready."
Sediq is more confident about the buffer provided by international reserves, which stood at $6.8 billion at the end of 2016, an increase of $400 million from 2015.
"We are at good shape on that," he said. And with a managed float currency regime, the central bank only needs to tap that if there are unexpected foreign-exchange moves that threaten to dent the economy.
Restoring stability to the financial system amid a fragile and deteriorating security situation has been a "major development," Sediq said, sitting next to a safe housing the bank's key documents. "It's not an easy job." | {
"redpajama_set_name": "RedPajamaC4"
} | 8,942 |
{"url":"https:\/\/hesim-dev.github.io\/hesim\/reference\/define_model.html","text":"A model expression is defined by specifying random number generation functions for a probabilistic sensitivity analysis (PSA) and transformations of the sampled parameters as a function of input_data. The unevaluated expressions are evaluated with eval_model() and used to generate the model inputs needed to create an economic model.\n\ndefine_model(tparams_def, rng_def, params = NULL, n_states = NULL)\n\neval_model(x, input_data)\n\n## Arguments\n\ntparams_def A tparams_def object or a list of tparams_def objects. A list might be considered if time intervals specified with the times argument in define_tparams() vary across parameters. Parameters for a transition probability matrix (tpmatrix), utilities (utility), and\/or cost categories (costs) are returned as a named list (see define_tparams() for more details). A rng_def object used to randomly draw samples of the parameters from suitable probability distributions. Either (i) a list containing the values of parameters for random number generation or (ii) parameter samples that have already been randomly generated using eval_rng(). In case (ii), rng_def should be NULL. The number of health states (inclusive of all health states including the the death state) in the model. If tpmatrix is an element returned by tparams_def, then it will be equal to the number of states in the transition probability matrix; otherwise it must be specified as an argument. An object of class model_def created with define_model(). An object of class expanded_hesim_data expanded by patients and treatment strategies.\n\n## Value\n\ndefine_model() returns an object of class model_def, which is a list containing the arguments to the function. eval_model() returns a list containing ID variables identifying parameter samples, treatment strategies, patient cohorts, and time intervals; the values of parameters of the transition probability matrix, utilities, and\/or cost categories; the number of health states; and the number of random number generation samples for the PSA.\n\n## Details\n\neval_model() evaluates the expressions in an object of class model_def returned by define_model() and is, in turn, used within functions that instantiate economic models (e.g., create_CohortDtstm()). The direct output of eval_model() can also be useful for understanding and debugging model definitions, but it is not used directly for simulation.\n\nEconomic models are constructed as a function of input data and parameters:\n\n1. Input data: Objects of class expanded_hesim_data consisting of the treatment strategies and patient population.\n\n2. Parameters: The underlying parameter estimates from the literature are first stored in a list (params argument). Random number generation is then used to sample the parameters from suitable probability distributions for the PSA (rng_def argument). Finally, the sampled parameters are transformed as a function of the input data into values (e.g., elements of a transition probability matrix) used for the simulation (tparams_def argument). The params argument can be omitted if the underlying parameters values are defined inside a define_rng() block.\n\ndefine_tparams(), define_rng()\n\n## Examples\n\n\n# Data\nlibrary(\"data.table\")\nstrategies <- data.table(strategy_id = 1:2,\nstrategy_name = c(\"Monotherapy\", \"Combination therapy\"))\npatients <- data.table(patient_id = 1)\nhesim_dat <- hesim_data(strategies = strategies,\npatients = patients)\ndata <- expand(hesim_dat)\n\n# Model parameters\nrng_def <- define_rng({\nalpha <- matrix(c(1251, 350, 116, 17,\n0, 731, 512, 15,\n0, 0, 1312, 437,\n0, 0, 0, 469),\nnrow = 4, byrow = TRUE)\nrownames(alpha) <- colnames(alpha) <- c(\"A\", \"B\", \"C\", \"D\")\nlrr_mean <- log(.509)\nlrr_se <- (log(.710) - log(.365))\/(2 * qnorm(.975))\n\nlist(\np_mono = dirichlet_rng(alpha),\nrr_comb = lognormal_rng(lrr_mean, lrr_se),\nu = 1,\nc_zido = 2278,\nc_lam = 2086.50,\nc_med = gamma_rng(mean = c(A = 2756, B = 3052, C = 9007),\nsd = c(A = 2756, B = 3052, C = 9007))\n)\n}, n = 2)\n\ntparams_def <- define_tparams({\nrr = ifelse(strategy_name == \"Monotherapy\", 1, rr_comb)\nlist(\ntpmatrix = tpmatrix(\nC, p_mono$A_B * rr, p_mono$A_C * rr, p_mono$A_D * rr, 0, C, p_mono$B_C * rr, p_mono$B_D * rr, 0, 0, C, p_mono$C_D * rr,\n0, 0, 0, 1),\nutility = u,\ncosts = list(\ndrug = ifelse(strategy_name == \"Monotherapy\",\nc_zido, c_zido + c_lam),\nmedical = c_med\n)\n)\n})\n\n# Simulation\n## Define the economic model\nmodel_def <- define_model(\ntparams_def = tparams_def,\nrng_def = rng_def)\n\n### Evaluate the model expression to generate model inputs\n### This can be useful for understanding the output of a model expression\neval_model(model_def, data)\n#> $id #>$id[[1]]\n#> sample strategy_id patient_id time time_start\n#> 1: 1 1 1 0 0\n#> 2: 1 2 1 0 0\n#> 3: 2 1 1 0 0\n#> 4: 2 2 1 0 0\n#>\n#>\n#> $tpmatrix #> s1_s1 s1_s2 s1_s3 s1_s4 s2_s1 s2_s2 s2_s3 #> 1: 0.7303831 0.18679555 0.07260427 0.010217072 0 0.6025887 0.3872602 #> 2: 0.8823737 0.08149364 0.03167520 0.004457421 0 0.8266206 0.1689507 #> 3: 0.7129687 0.20776387 0.06806632 0.011201099 0 0.6137990 0.3760511 #> 4: 0.8859624 0.08254467 0.02704278 0.004450201 0 0.8465622 0.1494053 #> s2_s4 s3_s1 s3_s2 s3_s3 s3_s4 s4_s1 s4_s2 s4_s3 s4_s4 #> 1: 0.010151107 0 0 0.7420537 0.2579463 0 0 0 1 #> 2: 0.004428642 0 0 0.8874653 0.1125347 0 0 0 1 #> 3: 0.010149909 0 0 0.7463815 0.2536185 0 0 0 1 #> 4: 0.004032563 0 0 0.8992373 0.1007627 0 0 0 1 #> #>$utility\n#> [1] 1 1 1 1\n#> attr(,\"id_index\")\n#> [1] 1\n#>\n#> $costs #>$costs$drug #> [1] 2278.0 4364.5 2278.0 4364.5 #> attr(,\"id_index\") #> [1] 1 #> #>$costs$medical #> A B C #> 1: 1463.6975 1262.136 15708.566 #> 2: 1463.6975 1262.136 15708.566 #> 3: 293.2644 4551.563 7743.327 #> 4: 293.2644 4551.563 7743.327 #> #> #>$n_states\n#> [1] 4\n#>\n#> \\$n\n#> [1] 2\n#>\n#> attr(,\"class\")\n#> [1] \"eval_model\"\n## Create an economic model with a factory function\neconmod <- create_CohortDtstm(model_def, data)","date":"2021-12-03 21:36:37","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.24887318909168243, \"perplexity\": 4718.434488677031}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964362919.65\/warc\/CC-MAIN-20211203212721-20211204002721-00513.warc.gz\"}"} | null | null |
\section{Introduction}
Implementation variations and parameter settings can severely affect the
service observed by the clients of a peer-to-peer system.
A better understanding of protocol parameters is
needed to improve and stabilize service, a particularly
important goal for emerging peer-to-peer applications such as streaming video.
BitTorrent is widely regarded as one of the most successful swarming protocols,
which divide the content to be distributed into distinct pieces and enable peers
to share these pieces efficiently. Previous research efforts
have focused on the algorithms
believed to be the major factors behind BitTorrent's good performance, such as the
piece and peer selection strategies.
However, to the best of our knowledge, no studies have looked into the optimal
size of content pieces being exchanged among peers.
This paper investigates this parameter by running real experiments with
varying piece sizes on a controlled testbed, and demonstrates that \emph{piece size
is critical for performance}, as it determines the degree of parallelism available
in the system.
Our results also show that, for small-sized content, smaller pieces enable
shorter download times, and as a result, \emph{BitTorrent's design choice of further
dividing content pieces into subpieces is unnecessary for such content}.
We evaluate the overhead that small pieces incur as content size grows and
demonstrate a trade-off between piece size and available
parallelism. We also explain how this trade-off motivates
the use of both pieces and subpieces for distributing large content, the common case
in BitTorrent swarms.
The rest of this paper is organized as follows. Section~\ref{sec:methodology}
provides a brief description of the BitTorrent protocol, and describes our
experimental methodology. Section~\ref{sec:results} then presents the results
of our experiments with varying piece sizes, while Section~\ref{sec:discussion}
discusses potential reasons behind the poor performance of small pieces when distributing
large content.
Lastly, Section~\ref{sec:related} describes related work and
Section~\ref{sec:conclusion} concludes.
\section{Background and Methodology}
\label{sec:methodology}
\paragraph{BitTorrent Overview}
BitTorrent is a popular peer-to-peer content distribution protocol that has been shown
to scale well with the number of participating clients. Prior to distribution, the
content is divided into multiple \textit{pieces}, while each piece is further
divided into multiple \textit{subpieces}. A \textit{metainfo file} containing
information necessary for initiating the download process is then created by the content
provider. This information includes each piece's SHA-1 hash (used to verify
received data) and the address of the \textit{tracker},
a centralized component that facilitates peer discovery.
In order to join a \textit{torrent}---the collection of peers participating in the
download of a particular content---a client retrieves the metainfo file out of band,
usually from a Web site. It then contacts the tracker, which responds
with a \textit{peer set} of randomly selected peers. These might include both
\textit{seeds}, who already have the entire content and are sharing it with others,
and \textit{leechers}, who are still in the process of downloading.
The new client can then start contacting peers in this set and request data.
Most clients nowadays implement a \textit{rarest-first} policy for piece requests:
they first ask for the pieces that exist at the smallest number of peers in their
peer set.
Although peers always exchange just subpieces with each other, they only
make data available in the form of complete pieces: after downloading
all subpieces of a piece, a peer notifies all peers in its peer set with a
\textit{have} message.
Peers are also able to determine which pieces others have based on a
\textit{bitfield} message, exchanged upon the establishment of new connections, which
contains a bitmap denoting piece possession.
Each leecher independently decides who to exchange data with via the
\textit{choking algorithm}, which gives preference to those who upload data to
the given leecher at the highest rates. Thus, once per \textit{rechoke period},
typically every ten seconds, a leecher considers the receiving data rates from
all leechers in its peer set. It then picks out the fastest ones, a fixed number of
them, and only uploads to
those for the duration of the period.
Seeds, who do not need to download any pieces, follow a different unchoke strategy.
Most current implementations unchoke those leechers that \textit{download} data
at the highest rates, to better utilize seed capacity.
\paragraph{Experimental Methodology}
\begin{figure*}[t]
\centering
\subfigure[Piece size of 16 kB]{\includegraphics[width=0.23\textwidth]
{figures/bw_leech_state_completion_cdf_content5_piece16}}
\hfill{}
\subfigure[Piece size of 512 kB]{\includegraphics[width=0.23\textwidth]
{figures/bw_leech_state_completion_cdf_content5_piece512}}
\hfill{}
\subfigure[Piece size of 16 kB]{\includegraphics[width=0.23\textwidth]
{figures/bw_upload_global_util_scatter_content5_piece16}}
\hfill{}
\subfigure[Piece size of 512 kB]{\includegraphics[width=0.23\textwidth]
{figures/bw_upload_global_util_scatter_content5_piece512}}
\caption{CDFs of peer download completion times and scatterplots of average
upload utilization for five-second time intervals when
distributing a 5~MB content (averages over 5 runs).
\emph{Small pieces shorten download time and enable higher utilization.}}
\label{fig:small_contents-cdfs_utilization}
\end{figure*}
We have performed all our experiments with private torrents on the PlanetLab
platform~\cite{planetlab}.
These torrents comprise 40 leechers and a single initial seed sharing content of
different sizes. Leechers do not change their available upload bandwidth during the
download, and disconnect after receiving a complete copy of the content.
The initial seed stays connected for the duration of the experiment, while all leechers
join the torrent at the same time, emulating a flash crowd scenario.
The number of parallel upload slots is set to 4 for the leechers and seed.
Although system behavior might be different with other peer arrival patterns and torrent
configurations, there is no reason to believe that the conclusions we draw are predicated
on these parameters.
The available bandwidth of most PlanetLab nodes is relatively high for typical real-world clients.
We impose upload limits on the leechers and seed to model more realistic scenarios,
but do not impose any download limits, as we wish to observe differences in download completion time
with varying piece sizes.
The upload limits for leechers follow
a uniform distribution from 20 to 200~kB/s, while the seed's upload capacity is
set to 200~kB/s.
We collect our measurements using the official (mainline) BitTorrent
implementation, instrumented to record interesting events.
Our client is based on version 4.0.2 of the official implementation
and is publicly available for download~\cite{btinstrumented}.
We log the client's internal state, as well as each message sent or received along with the
content of the
message. Unless otherwise specified, we run our
experiments with the default parameters.
The protocol does not strictly define the piece and subpiece sizes.
An unofficial BitTorrent specification~\cite{btwikispec} states that the
conventional wisdom is to ``pick the smallest piece size that results in a metainfo file
no greater than 50--75~kB''. The most common piece size for public torrents seems to be 256~kB.
Additionally, most implementations nowadays use 16~kB subpieces.
For our experiments, we always keep the subpiece size constant at 16~kB, and
only vary the piece size. We have results for all possible
combinations of different content sizes (1~MB, 5~MB, 10~MB, 20~MB, 50~MB, and 100~MB)
and piece sizes
(16~kB, 32~kB, 64~kB, 128~kB, 256~kB, 512~kB, 1024~kB, and 2048~kB).
\section{Results}
\label{sec:results}
Our results, presented in this section, demonstrate that small pieces are preferable for the
distribution of
small-sized content. We also discuss the benefits and drawbacks of small
pieces for other content sizes, and evaluate the communication and
metainfo file overhead that different piece sizes incur for larger content.
\subsection{Small Content}
Even though most content distributed with BitTorrent is large, it is still interesting to examine the impact
of piece size on distributing smaller content. In addition to gaining a better understanding of the
trade-offs involved, it may also sometimes be desirable to utilize BitTorrent to avoid server overload when
distributing small content, e.g., in the case of websites that suddenly become popular.
Figure~\ref{fig:small_contents-completion_comparison} shows the median download
completion times of the 40 leechers downloading a 5~MB file, for
different numbers of pieces, along with standard deviation error bars.
Clearly, \emph{smaller piece sizes enable faster downloads}.
In particular, performance deteriorates rapidly when increasing the piece size
beyond 256~kB.
The same observations hold for experiments with other small content
(1 and 10~MB).
To better illustrate the benefits of small pieces,
Figure~\ref{fig:small_contents-cdfs_utilization}
shows the cumulative distribution functions (CDF) of leecher download
completion times for 16 and 512~kB pieces (graphs (a) and (b)).
With small pieces, most peers complete their download within the first 100 seconds.
With larger pieces, on the other hand, the median peer completes in more than twice the
time, and there is greater variability.
The reason is that \emph{smaller pieces let peers share data sooner.}
As mentioned before,
peers send out \textit{have} messages announcing new
pieces only after downloading and verifying a complete piece. Decreasing piece
size allows peers to download complete pieces, and thus start sharing them with others, sooner.
This increases the available parallelism in the system, as it enables more opportunities
for parallel downloading from multiple peers.
\begin{figure}
\centering
\includegraphics[width=0.6\columnwidth]{figures/compareTime5}
\caption{Download completion times for a 5~MB content (medians over 5 runs and standard
deviation error bars).
\emph{Smaller pieces clearly improve performance.}}
\label{fig:small_contents-completion_comparison}
\end{figure}
This benefit is also evident when considering \textit{peer upload utilization},
which constitutes a reliable metric of efficiency, since the total peer upload
capacity represents the maximum throughput the system can achieve as a whole.
Figure~\ref{fig:small_contents-cdfs_utilization} shows utilization scatterplots
for all five-second time intervals during the download (graphs (c) and (d)).
Average upload utilization for each of 5 experiment runs is plotted once every 5 seconds.
Thus, there are five dots for every time slot, representing the average peer upload utilization
for that slot in the corresponding run. The metric is torrent-wide: for those five seconds,
we sum the upload
bandwidth expended by leechers and divide by the available upload capacity of all
leechers still connected to the system.
Thus, a utilization of 1 represents taking full advantage of the available upload capacity.
As previously observed~\cite{legout07}, utilization is low at the beginning and end of the
session. During the majority
of the download, however, a smaller piece size increases the number of
pieces peers are interested in, which leads to higher upload utilization.
These conclusions are reinforced by the fact that small pieces enable the seed to
upload less duplicate pieces
during the beginning of a torrent's lifetime.
Figure~\ref{fig:small_contents-seed_duplicates} indeed plots the number of pieces
(unique and total) uploaded by the single seed in our 5~MB experiments, for two
representative runs. Although the seed finishes uploading the first copy of
the content at approximately the same time in both cases (vertical line on the graphs),
it uploads 139\% more
duplicate data with larger pieces (5120~kB for 512~kB pieces vs. 2144~kB for 16~kB pieces),
thus making less efficient use
of its valuable upload bandwidth.
Avoiding this waste can lead to better performance,
especially for low-capacity seeds~\cite{legout07}.
This behavior can be explained as follows. The official BitTorrent implementation
we are using always
issues requests for the
rarest pieces \emph{in the same order}. As a result, while a leecher is downloading
a given piece, other leechers might end up requesting the same piece from the seed.
With smaller pieces, the time interval before a piece is completely downloaded and shared
becomes shorter, mitigating this problem. This could be resolved by having
leechers request rarest pieces in random order instead.
In summary, small pieces enable significantly better performance when distributing small content.
As a result, \emph{the distinction of pieces and subpieces that the BitTorrent design
dictates is unnecessary for such content}. For instance, in our 5~MB experiments,
pieces that are as small as subpieces (16~kB) are optimal. Thus, the content
could just be divided into pieces with no loss of performance.
\begin{figure}[t]
\centering
\begin{tabular}{@{}p{.5\hsize}@{}p{.5\hsize}@{}}
\subfigure[Piece size of 16 kB]{\includegraphics[scale=0.22]
{figures/bw_seed_unique_total_pieces_content5_piece16_Run_1}} &
\subfigure[Piece size of 512 kB]{\includegraphics[scale=0.22]
{figures/bw_seed_unique_total_pieces_content5_piece512_Run_4}}
\end{tabular}
\caption{Number of pieces uploaded by the seed when distributing a 5~MB content,
for two representative runs.
The \textit{Unique} line represents the pieces that had not been previously uploaded,
while the \textit{Total} line represents the total number of pieces uploaded so far.
The vertical line denotes the time the seed finished uploading the first copy of
the content to the system.
\emph{The duplicate piece overhead is significantly lower for small pieces.}}
\label{fig:small_contents-seed_duplicates}
\end{figure}
\subsection{Piece Size Impact}
Before investigating the impact of piece size on the distribution of larger content,
let us first examine the advantages and drawbacks of small pieces.
We have seen that their benefits for small content are largely due
to the increased peer upload utilization such pieces enable.
Since small pieces can be downloaded sooner than large ones,
leechers are able to share small pieces sooner. In this manner,
there is more data available in the system, which gives peers a wider
choice of pieces to download. In addition to this increased parallelism, small pieces
provide the following benefits (some of which do not affect our experiments).
\begin{itemize}
\item They decrease the number of duplicate
pieces uploaded by seeds, thereby better utilizing seed upload
bandwidth.
\item The rarest-first piece selection strategy is more effective in ensuring piece replication.
A greater number of pieces to choose from entails a lower probability that peers
download the same piece, which in turn improves the diversity of pieces
in the system.
\item There is less waste when downloading corrupt data.
Peers can discover bad pieces sooner and re-initiate their download.
\end{itemize}
On the other hand, for larger content, the overhead incurred by small pieces may
hurt performance. This overhead includes the following.
\begin{itemize}
\item Metainfo files become larger, since they have to include more SHA-1 hashes.
This would increase the load on a Web
server serving such files to clients, especially in a flash crowd case.
\item \textit{Bitfield} messages also become larger due to the increased number of bits
they must contain.
\item Peers must send more \textit{have} messages, resulting in increased
communication overhead.
\end{itemize}
In the next section, we shall see that these drawbacks of small pieces outweigh their benefits,
for larger content.
Thus, the choice of piece size for a download should take
the content size into account.
\subsection{Larger Content}
Figure~\ref{fig:large_contents-completion_comparison} shows the download
completion times of the 40 leechers downloading a 100~MB file for different
piece sizes.
We observe that small pieces are no longer optimal. In this particular
case, sizes around 256~kB seem to perform the best.
Experiments with other content sizes (20 and 50~MB) show that \emph{the optimal piece
size increases with content size}.
For instance, for experiments with a 50~MB content, the optimal piece size is 64~kB.
Note that the unofficial guideline for choosing the piece size, mentioned in
Section~\ref{sec:methodology}, would yield sizes of 32 and 16~kB for a 100~MB and 50~MB
content respectively, a bit off from the optimal values.
In an effort to better understand this trade-off regarding the choice of piece size, we evaluate
the metainfo file and communication overhead. The former is
shown in Figure~\ref{fig:large_contents-metainfo_size}.
As expected, small pieces produce proportionately larger metainfo files (note that
the x axis is logarithmic).
16~kB pieces, for instance,
produce a metainfo file larger than 120~kB, as compared to a less
than 10~kB file for 256~kB pieces.
For large content in particular, this might have significant negative implications for
the Web server used to distribute such files to clients.
\begin{figure}[t]
\centering
\includegraphics[width=0.6\columnwidth]{figures/compareTime100}
\caption{Download completion times for a 100~MB content (medians over 5 runs and standard
deviation error bars).
\emph{Small pieces are no longer optimal.}}
\label{fig:large_contents-completion_comparison}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.6\columnwidth]{figures/compareFile100}
\caption{Metainfo file sizes for distributing a 100~MB content.
\emph{Smaller pieces produce proportionately larger files.}}
\label{fig:large_contents-metainfo_size}
\end{figure}
\textit{Bitfield} messages become proportionately larger too. For instance,
for the 100~MB content, these messages are 805 and 55 bytes for 16 and 256~kB pieces
respectively. Figure~\ref{fig:large_contents-overhead} additionally shows the communication overhead
due to \textit{bitfield} and \textit{have} messages, expressed as a percentage
of the total upload traffic per peer. The overhead ranges from less than
1\% for larger
piece sizes to around 9\% for 16~kB pieces. However, it is not clear that this overhead
is responsible for the worse performance of smaller pieces.
Although these control messages do occupy upload bandwidth, they do not necessarily
affect the data exchange among peers, and thus their download performance.
For example, looking at the corresponding overhead for smaller content, we observe that the overhead curve
looks very similar. This indicates that \emph{increased communication
overhead is
most likely neither the cause of the worse performance of small pieces for larger
content}, nor does it explain the observed trade-off.
In the next section, we formulate two hypotheses that might
help identify the true cause of this behavior.
In summary, when distributing larger content, the optimal
piece size depends on the content size, due to a trade-off between
the increased parallelism small pieces provide and their drawbacks.
\emph{BitTorrent arguably attempts to address this trade-off by further
dividing pieces into subpieces}, to get the best of both worlds:
subpieces increase opportunities for parallel downloading, while pieces
mitigate the drawbacks of small verifiable units.
\begin{figure}[t]
\centering
\includegraphics[width=0.6\columnwidth]{figures/compareOverhead100}
\caption{Communication overhead due to \textit{bitfield} and \textit{have} messages
when distributing a 100~MB content.
\emph{Small pieces incur considerably larger overhead.}}
\label{fig:large_contents-overhead}
\end{figure}
\section{Discussion}
\label{sec:discussion}
The results presented in the previous section point to a hidden reason behind
the poor performance of small pieces when distributing large content. We have
two hypotheses that might help explain that.
First, small pieces \emph{reduce opportunities for subpiece request pipelining}.
In order to prevent delays due to request/response latency, and to keep the download
pipe full most of the time, most BitTorrent implementations
issue requests for several consecutive subpieces back-to-back. This
pipelining, however, is restricted within the boundaries of a single piece.
This is done in order to use available bandwidth to download complete pieces as soon
as possible, and share them
with the rest of the swarm. Similarly, peers do not typically issue a request for
subpieces of another piece
to the same peer before completing the previous one.
Thus, for a content with 32~kB pieces, for instance, only two subpiece requests per peer
can be pending at any point in time.
For small content, the impact of reduced pipelining is negligible,
as the download completes quickly anyway.
For larger content, however, it might severely affect system performance, as it
limits the total number of simultaneous requests a peer can issue.
Additionally, this matter
gains importance as available peer bandwidth rises, since the request/response
latency then starts to dominate time spent on data transmission.
Furthermore, small pieces \emph{may incur slowdown due to TCP effects}. With a small piece size, a given
leecher is more likely to keep switching among peers to download different pieces of the content.
This could have two adverse TCP-related effects: 1) the congestion window would have less time
to ramp up than
in the case of downloading a large piece entirely from a single peer, and 2) the congestion
window for unused peer connections would gradually decrease after a period
of inactivity, due to TCP congestion window validation~\cite{rfc2861}, which
is enabled by default in recent Linux kernels, such as the ones running on the PlanetLab
machines in our experiments.
A large piece size, on the other hand, would enable more efficient TCP transfers due to
the lower probability
of switching from peer to peer. Note, however, that, even in that case, there is no guarantee
that all subpieces of a piece will be downloaded from the same peer.
\section{Related Work}
\label{sec:related}
To the best of our knowledge, this is the first study that systematically investigates
the optimal piece size in BitTorrent.
Bram Cohen, the protocol's creator, first described BitTorrent's main
algorithms and their design rationale~\cite{cohen03}.
In version 3.1 of the official implementation he reduced the default piece size from
1~MB to 256~kB~\cite{piecechange}, albeit without giving a concrete reason for
doing so. Presumably, he noticed the performance benefits of smaller pieces.
Some previous research efforts have looked into the impact of piece size in other
peer-to-peer content distribution systems.
Ho{\ss}feld \textit{et al.}~\cite{hossfeld05} used simulations
to evaluate varying piece sizes in an eDonkey-based mobile file-sharing system. They found that
download time decreases with piece size up to a certain point,
confirming our observations, although they did not attempt to explain this behavior.
The authors of Dandelion~\cite{sirivianos07} evaluate its performance with different piece sizes, and mention
TCP effects as a potential reason for the poor performance of small pieces. However, small
pieces in that system may also be harmful because they increase the rate at which key requests are
sent to the central server.
CoBlitz~\cite{park06} faces a similar problem with smaller pieces requiring more processing
at CDN nodes. The authors end up choosing a piece size of 60~kB, because that can easily fit into
the default Linux outbound kernel socket buffers.
The Slurpie~\cite{sherwood04} authors briefly allude to a piece size trade-off, and
mention TCP overhead as a drawback of small pieces.
Lastly, during the evaluation of the CREW system~\cite{deshpande06},
the authors find a piece size of 8~kB to be optimal for distributing a
800~kB content, but they do not attempt to explain that.
\section{Conclusion}
\label{sec:conclusion}
This paper presents results of real experiments with varying piece sizes on a
controlled BitTorrent testbed. We show that piece size is critical for performance,
as it determines the degree of parallelism in the system.
Our results explain why small pieces are the optimal choice for small-sized content,
and why further dividing content pieces into subpieces is unnecessary for such
content.
We also evaluated the overhead small pieces incur for larger content, and
discussed the design trade-off between piece size and available parallelism.
It would be interesting
to investigate our two hypotheses regarding the poor performance of small pieces
with larger content.
We would also like to extend our conclusions to different scenarios, such as
video streaming, which imposes additional real-time constraints on the protocol.
\paragraph{Acknowledgments}
We are grateful to Michael Sirivianos, Himabindu Pucha, and the anonymous reviewers
for their valuable feedback.
\begin{small}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,101 |
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} | 6,436 |
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BPAA Biweekly State Policy Updates - March 22, 2019
Michael Best Strategies posted on 3/25/2019 7:31:00 AM
MINIMUM WAGE UPDATE
Lamont promotes paid leave, minimum wage to skeptical business leaders: Gov. Ned Lamont predicted Wednesday, March 20 that two measures heavily criticized by business leaders — paid family and medical leave and boosting the state's minimum wage — probably will pass this year.
The hot-button proposals involve nudging Connecticut's minimum wage to $15 an hour, up from $10.10, over the course of several years, and imposing a 0.5 percent payroll tax to fund a pool of money for paid leave. Corporate officials have denounced the measures as unfriendly to businesses, particularly small companies that could struggle with workers' absences and additional costs.
"I think paid family leave is going to pass. I think its time has come," Lamont said after addressing a crowd during the Connecticut Business and Industry Association's annual lobbying day at the Capitol. "They see our neighboring states are doing it. They see a lot of major businesses are already taking the lead on that as well as moving to a $15 minimum wage, and Connecticut is going to catch up there."
Proposals that cleared the Labor and Public Employees Committee last week would raise the minimum wage to $12 by January 2020, to $13.50 the next year, and $15 by 2022. Lamont has put forward a similar plan that would phase in the wage increase over four years, instead of three.
Several bills tied to paid family and medical leave also have surfaced. Two of them suggest benefits equal to 100 percent of workers' pay, up to $1,000 a week. Lamont's proposal would reimburse employees at a rate equal to 90 percent of their base salary. The payroll tax would be levied no later than October 2020, and worker compensation would be available beginning in January 2022. Read more at CTMirror.
Minimum Wage Increase for Michigan Employees Takes Effect March 29: Effective March 29, 2019, Michigan's minimum wage will increase from $9.25 to $9.45 per hour. A copy of the Improved Workforce Opportunity Wage Act (Public Act 337 of 2018) and related resources – including the required poster – can be found here.
Overtime requirements remain the same under the Improved Workforce Opportunity Wage Act; non-exempt employees should be paid 1.5 times their regular rate of pay for hours worked over 40 in a 7-day work week.
An employer may continue to pay minors 16 to 17 years of age 85% of the minimum hourly wage rate. On March 29, 2019, that rate will increase from $7.86 to $8.03 per hour. There is no change to the training wage of $4.25 per hour that may be paid to newly hired employees, 16 to 19 years of age, for the first 90 days of their employment.
Public Act 337 of 2018 allows employers to take a tip credit on minimum wage under certain conditions for employees who customarily and regularly receive tips from a guest, patron or customer for services rendered to that guest, patron, or customer. Effective March 29, 2019, 38% of the minimum hourly wage rate will increase from $3.52 to $3.59 per hour. Read more here at the Michigan Department of Licensing and Regulatory Affairs.
Bloomberg Government - Maryland Lawmakers Approve Wage Hike to $15 by Veto-Proof Margin: Maryland lawmakers approved a bill that would boost the state's minimum wage to $15 per hour within six years. The state House of Delegates and Senate gave final approval March 20 to S.B. 280, which would raise the state's wage floor from its current level of $10.10 per hour to $11 next year, with additional increases each year until the $15-per-hour level was reached in 2025. Companies with fewer than 15 employees would have additional time to meet the wage schedule.
Gov. Larry Hogan (R) said he's opposed to the increase, but the margins of legislative approval would withstand a veto. The House gave its final approval to the measure by a 93-41 vote, and it passed the Senate by a margin of 32-13. A three-fifths majority in each chamber is required to override a veto in Maryland.
At a March 18 news conference, Hogan said the wage increase would "cost us jobs, make us incapable of competing with other states in the region and could devastate our state's economy." He said the current rate of $10.10 per hour is "by far" the highest level in the region.
Angela Berard, a spokeswoman for Hogan, told Bloomberg Law March 20 that the governor had outlined his position in his March 18 comments and "will carefully review this legislation when it reaches his desk."
Politico - Arkansas Moves to Limit Wage Hikes: Rep. Robin Lundstrum (R-Elm Springs), despite opposition from Governor Hutchinson and the state Republican Party, is proceeding with her bills to undo significant portions of the state minimum wage hike approved by voters just last November. In 2018, 68 percent of Arkansas voters backed a ballot initiative to increase the state minimum wage to $9.25 in 2019, $10 in 2020, and $11 in 2021. Lundstrum's bills create exemptions to the coming wage hikes: HB 1752 exempts businesses with fewer than 20 employees, nonprofits with an operating budget under $1 million, and certain nonprofits that provide services for people with developmental disabilities. HB 1753 exempts any employee under the age of 20.
"The people voted to have a minimum wage increase," Lundstrum said. "I'm not trying to thwart that." She said that her goal in nixing the wage hikes that voters approved was to protect nonprofits, small businesses and teenage employees from "unintended consequences."
Lundstrum acknowledged that the coming wage hikes were approved by a two-thirds majority, but argued that her bills would not "roll back" the will of the voters. "We're not rolling anything back," she said. "That's what the people passed. I'm not taking that back — I think that would be wrong. It would not roll up [for the exempted groups]. It would not roll up to $10 and $11." Read the full article here.
Bloomberg Government - $12 New Mexico Minimum Wage Bill Heads to Governor: New Mexico lawmakers approved boosting the state's minimum wage to $12 an hour by 2023, in a March 15 agreement that came with barely over a day left in the legislative session.
Democratic Gov. Michelle Lujan Grisham is expected to sign the increase. The state's current minimum wage is $7.50 an hour, just above the federal minimum of $7.25. The legislation also increases the minimum wage for tipped workers from $2.13 to $3 by 2023.
Although the bill was among their top priorities this year, Democratic legislators in the House and Senate disagreed over how high to raise the wages and how soon to get there. The compromise phases in the increases over the next several years.
The New Mexico Association of Commerce and Industry, which represents employers in the state, supports the bill, according to its president and CEO Rob Black.
Lawmakers also passed a bill this session to add state minimum wage protections for domestic workers. Though federal wage law includes domestic workers, advocates of the legislation say it would help state regulators investigate complaints.
New York: Lawmakers, Workers Push For Higher Tipped Worker Wage: A bill that would increase the minimum wage for tipped workers to the full minimum wage is being pushed for in the final budget agreement by restaurant workers and state lawmakers who back the legislation.
"The food service industry employs the largest number of women workers earning below the minimum wage, workers who must rely on the whim of consumers rather than their own employers to be paid a living wage and support their families, fostering an environment that encourages racism, sexual harassment, and high poverty rates," said Assemblywoman Ellen Jaffee. "The seven states that already require tipped workers be paid the full minimum wage have flourishing restaurant industries. It's time for New York to get on board and eliminate this shameful economic injustice."
The bill is being sought more than a year after Gov. Andrew Cuomo called on the Department of Labor to study the effect of ending the so-called subminimum wage, which has been eliminated in seven states, including California. Read more here at NYStateofPolitics.
Nebraska: Tipped minimum wage increase stalls: After two days of debate, an attempt to increase the minimum wage for persons who are compensated by way of gratuities stalled on general file March 14.
Currently, the tipped minimum wage in Nebraska is $2.13 per hour. LB400, introduced by Omaha Sen. Megan Hunt, would increase the wage to 40 percent of the standard minimum wage rate in 2020 and 50 percent in 2021. If the standard minimum wage remains at its current level of $9.00 per hour, the minimum wage for persons compensated by way of gratuities would be $3.60 per hour in 2020 and $4.50 per hour in 2021.
The tipped minimum wage last was increased in 1991, Hunt said, while the standard minimum wage has increased seven times during the same period. Read more at the Nebraska Legislature.
North Carolina: The fight for $15 arrives at the General Assembly: North Carolina lawmakers joined Raising Wages NC — a growing coalition of labor groups, advocates, business, and faith leaders — at a legislative press conference today to announce the introduction of H.B. 366, inclusive legislation that raises the state's minimum wage from $7.25 an hour to $15 an hour by 2024, indexes it to the cost of living, ends the subminimum wage for persons with disabilities, phases it out for tipped workers, and repeals exemptions for agricultural and domestic workers.
North Carolina's minimum wage has been stuck at $7.25 an hour for a decade and does not cover everyone. At the event, legislators, workers, business owners, and faith leaders called attention to the moral and economic imperative of making sure that everyone who works full time can earn a living wage, afford the basic necessities, and has a fair opportunity to work hard and succeed — including people with disabilities, people who care for our homes and families, people who serve our food, and the people who grow it. Read more at NCPolicyWatch.
Senate passes bill to raise New Hampshire's minimum wage: New Hampshire's Senate on Thursday passed a bill to raise the state's minimum wage. The bill, which passed 14-10, would raise the minimum wage to $10 an hour in 2020 and $11 or $12 an hour in 2021, depending on additional benefits provided by employers, such as paid sick days. The bill also would set the tipped minimum wage at $4 an hour, with workers making below the new minimum wage with tips being compensated for the difference by their employer. The House passed its own bill last week that would set the minimum wage at $9.50 in 2020, $10.75 in 2021 and $12 in 2022.
Democrats control both chambers. The state relies on the federal minimum wage of $7.25 an hour, the lowest in New England. The Legislature stripped New Hampshire's minimum wage law from the books in 2011. Senate President Donna Soucy, a Democrat from Manchester who's introduced a similar measure for years, said "$7.25 wasn't a fair wage in 2009 and it certainly isn't a fair wage in 2019."
Republican Leader Chuck Morse, of Salem, said states and cities that have raised their minimum wage have seen the number of jobs decline and the take-home pay for low-wage workers decrease due to reduced hours. "Creating a strong economy will always be a more effective way to raise workers take-home pay than mandating minimum wages," he said.
Republican Gov. Chris Sununu has said he does not support a state minimum wage higher that would exceed the federal rate. Read more at CTPost.
PAID LEAVE UPDATE
Vermont Digger: Lawmakers consider private carrier for paid leave program: Legislators in the House are expected to advance a proposal this week that would establish a statewide paid family leave program. As it stands, the plan backed by Democratic lawmakers would require all workers and employers to pay into a state-run program funded by a split payroll tax, making all employees eligible for 12 weeks of paid leave.
Gov. Phil Scott has pitched his own voluntary paid leave proposal with New Hampshire Gov. Chris Sununu. The governor's plan would leverage the pool of both states' roughly 20,000 state employees to bring down the cost for other employers to offer the benefit on their own accord, but wouldn't mandate them to do so. While it's unlikely Democrats will pass a paid leave program that is voluntary in nature, like the governor's, they are considering one aspect of Scott's plan: hiring a third party to administer the benefit.
Under the Scott administration's plan the state would hire a private insurance carrier or administrator to run the program.
Louisiana: ASCAP believes legal action is last measure to address establishments that play unlicensed music: A number of New Orleans bars are facing legal action for playing unlicensed music in their establishments, according to a report by The New Orleans Advocate. The lawsuits are part of a recent push by the American Society of Composers, Authors and Publishers (ASCAP), which sued numerous establishments throughout the U.S. for allegedly playing unlicensed music at these venues. Jackson Wagener, vice president of Business and Legal Affairs of ASCAP, spoke with Louisiana Record about the legal battle and why such action is needed for playing music at a bar.
"ASCAP views litigation as a last resort; we typically only sue establishments after its owners and operators have been contacted numerous times over a year or more," Wagener told Louisiana Record. Read more at Louisiana Record.
NY State of Politics - Assembly Lawmaker Wants A Beer And Wine Tax: A bill that would place new taxes on wine and beer sales would raise $260 million meant for addiction prevention and recovery programs. That bill, backed by Assemblyman Michael DenDekker, would add 3 cents to the price of a 12 ounce can of beer and 2 cents per glass of wine. Ten cents would be added to the price of a shot.
"Substance abuse of any kind is deeply troubling, but addiction to alcohol has proven particularly damaging not only because of its corrosive effects on physical and mental development but also because of its role as a gateway drug," DenDekker said.
"Alcohol users, particularly at younger ages frequently try other controlled substances while under the influence of alcohol. Predictably, patients addicted to opioids, cocaine and other mind altering substances began their history of substance abuse with alcohol since it is the most commonly available and society acceptable drug Americans are exposed to. Thus, I believe that alcohol is the gateway drug, and any serious initiative that aims to curb the effects of substance abuse must place alcohol use and addiction as a top priority."
DeDekker said the revenue generated by the tax would "literally double" the availability of existing programs for treatment. The bill comes as lawmakers are also debating whether to legalize marijuana in the state, with revenue proposed to bolster mass transit in New York City, aid communities impacted by tough drug laws and to study the effects of cannabis use.
PR Newswire - Pennsylvania Liquor Control Board Returns Nearly $2.1 Million in Licensing Fees to Local Communities: HARRISBURG, Pa., The Pennsylvania Liquor Control Board (PLCB) today announced the return of nearly $2.1 million in licensing fees to 1,103 municipalities in which licensees are located. Twice a year, as required by law, the PLCB returns liquor license fees paid by PLCB-approved licensees to the municipalities that are home to those licenses. Municipalities have flexibility in allocating and spending the returned license fees to meet local needs.
The PLCB oversees the regulation of more than 15,000 retail liquor licenses statewide, including restaurants, clubs and hotels. Licensees pay liquor license fees ranging from $125 to $700, depending on the type of license and the population of the municipality in which the license is located, as part of the annual license renewal or validation process, as well as in conjunction with approval of certain new applications.
The current dispersal period represents fees paid from Aug. 1, 2018, to Jan. 31, 2019. In all, 47 cities, 416 boroughs and 640 townships will receive payments ranging from $25 to $858,600.
The complete list of license fee distributions by municipality is available on the PLCB website.
| Categories: State Policy | Tags: | View Count: (5928) | Return
BPAA Biweekly State Policy Updates - December 14 12/14/2018
BPAA Federal Policy Update - July 13 7/13/2018
BPAA Biweekly Federal Policy Updates - November 16 11/16/2018
Federal Policy Update - Sept 22 9/22/2017
Federal Tax Update - Sept 8 9/8/2017
BPAA Federal Policy Update - April 23 4/26/2018
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,575 |
import { Action } from '@ngrx/store';
import { ISplit } from 'app/core/models';
export const LOAD_SPLITS = '[Book] Load Splits';
export const LOAD_SPLITS_SUCCESS = '[Book] Load Splits Success';
export const LOAD_SPLITS_FAIL = '[Book] Load Splits Fail';
export const ADD_SPLIT = '[Book] Add Split';
export const ADD_SPLIT_SUCCESS = '[Book] Add Split Success';
export const ADD_SPLIT_FAIL = '[Book] Add Split Fail';
export const REMOVE_SPLIT = '[Book] Remove Split';
export const REMOVE_SPLIT_SUCCESS = '[Book] Remove Split Success';
export const REMOVE_SPLIT_FAIL = '[Book] Remove Split Fail';
export class LoadSplitsAction implements Action {
readonly type = LOAD_SPLITS;
constructor(public payload: string /* Account GUID */) { }
}
export class LoadSplitsSuccessAction implements Action {
readonly type = LOAD_SPLITS_SUCCESS;
constructor(public payload: ISplit[]) { }
}
export class LoadSplitsFailAction implements Action {
readonly type = LOAD_SPLITS_FAIL;
constructor(public payload: any) { }
}
export class AddSplitAction implements Action {
readonly type = ADD_SPLIT;
constructor(public payload: ISplit) { }
}
export class AddSplitSuccessAction implements Action {
readonly type = ADD_SPLIT_SUCCESS;
constructor(public payload: ISplit) { }
}
export class AddSplitFailAction implements Action {
readonly type = ADD_SPLIT_FAIL;
constructor(public payload: ISplit) { }
}
/**
* Remove Split from Book Actions
*/
export class RemoveSplitAction implements Action {
readonly type = REMOVE_SPLIT;
constructor(public payload: ISplit) { }
}
export class RemoveSplitSuccessAction implements Action {
readonly type = REMOVE_SPLIT_SUCCESS;
constructor(public payload: ISplit) { }
}
export class RemoveSplitFailAction implements Action {
readonly type = REMOVE_SPLIT_FAIL;
constructor(public payload: ISplit) { }
}
export type SplitActions
= AddSplitAction
| AddSplitSuccessAction
| AddSplitFailAction
| RemoveSplitAction
| RemoveSplitSuccessAction
| RemoveSplitFailAction
| LoadSplitsAction
| LoadSplitsSuccessAction
| LoadSplitsFailAction;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,310 |
13 е деветнадесети и последен студиен музикален албум на хевиметъл групата Black Sabbath, който е издаден на 10 юни 2013 г. от Vertigo Records. Това е първи албум на групата след 18 години затишие и първи с оригиналния вокал Ози Озбърн от Never Say Die! (1978) насам.
Албумът е поднесен на публиката от Въртиго Рекърдс и Репъблик Рекърдс в САЩ, както и от първите по света. Това е първият албум от Forbidden (1995), както и първият с Ози Озбърн и бас китариста Гийзър Бътлър от концертния Reunion (1998), който междувременно има две нови студийно записани песни. Това е също така първият албум с Бътлър от Cross Purposes (1994). Това е също първият след Never Say Die!, в който отсъства кийбордистът Джеф Никълс, както и първият на Въртиго Рекърдс (извън територията на САЩ и Канада) след The Eternal Idol (1987).
Оригиналният състав на групата започва работа по нов студиен албум през 2001 г. с продуцента Рик Рубин. Развитието му е отложено, тъй като Озбърн не е готов с осмия си солов албум, Down To Earth, а другите членове на групата в крайна сметка напускат Блек Сабат, за да се отдадат на собствените си проекти, например Джи Зи Ар и Хевън Енд Хел. Когато Блек Сабат обявяват края на творческата си пауза на 11 ноември 2011 г., те също така правят изявление, че ще подновят работата с Рик Рубин. Освен оригиналните членове Озбърн, Бътлър и китариста Тони Айоми, към тях се присъединяват барабанистът Брад Уилк (Rage Against the Machine и Audioslave), след като оригиналния барабанист Бил Уърд решава да остане извън събирането поради "спорове от договорно естество". Албумът е предоставен за онлайн слушане по ITunes на 3 юни 2013 г.
Състав
Ози Озбърн – вокали
Тони Айоми – китара
Гийзър Бътлър – бас
Допълнителен персонал
Брад Уилк – барабани
Песни
Позиции в класациите
Албум
{|class="wikitable sortable plainrowheaders" style="text-align:center"
|-
!scope="col"|Класация (2013)
!scope="col"|Позиция
|-
!scope="row"|Austrian Albums Chart
|40
|-
!scope="row"|Belgian Albums Chart (Flanders)
|102
|-
!scope="row"|Belgian Albums Chart (Wallonia)
|109
|-
!scope="row"|Canadian Albums Chart
|30
|-
!scope="row"|Danish Albums Chart
|41
|-
!scope="row"|German Albums Chart
|18
|-
!scope="row"|Hungarian Albums Chart
|92
|-
!scope="row"|Swedish Albums Chart
|27
|-
!scope="row"|Swiss Albums Chart
|27
|-
!scope="row"|US Billboard 200
|86
|-
!scope="row"|US Rock Albums
|18
|-
!scope="row"|US Hard Rock Albums
|2
Отличия
Водещият сингъл на албума "God Is Dead?" печели през 2014 г. Грами награда за най-добро метъл изпълнение.
Източници
Външни препратки
Официална страница
Албуми на Блек Сабат
Музикални албуми от 2013 година | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,803 |
import { Component,Input } from '@angular/core';
import { Router } from '@angular/router';
import {Observable} from 'rxjs'
import {SensorMeasureMetaData, SensorMeasure, SensorMeasureType, SensorMeasureTypeDetails} from './sensor.classes'
import {SensorService} from './sensor.service'
@Component({
selector: 'sensor',
templateUrl: './sensor.component.html'
})
export class SensorComponent {
@Input() public title : string;
@Input() public sensor : SensorMeasureMetaData;
public measureLoaded : boolean;
public measure : SensorMeasure;
public measureTypeDetails : SensorMeasureTypeDetails;
private periodicServiceCall: any;
constructor(public sensorService: SensorService) {
this.measureLoaded = false;
}
ngOnInit() {
this.loadAndSetMeasureCallback(this);
this.periodicServiceCall = setInterval((that) => {this.loadAndSetMeasureCallback(this)},1000);
}
ngOnDestroy() {
clearInterval(this.periodicServiceCall);
}
private loadAndSetMeasureCallback(that) {
that.sensorService.loadLatestMeasure(that.sensor)
.subscribe((measure) =>
{
that.measure = measure;
that.measureTypeDetails = SensorMeasureMetaData.getTypeDetail(that.sensor.type);
that.measureLoaded = true;
})
}
}
| {
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Q: Need advice in seq2seq model implementation I am implementing seq2seq model for text summerization using tensorflow. For encoder I'm using a bidirectional RNN layer. encoding layer:
def encoding_layer(self, rnn_inputs, rnn_size, num_layers, keep_prob,
source_vocab_size,
encoding_embedding_size,
source_sequence_length,
emb_matrix):
embed = tf.nn.embedding_lookup(emb_matrix, rnn_inputs)
stacked_cells = tf.contrib.rnn.MultiRNNCell([tf.contrib.rnn.DropoutWrapper(tf.contrib.rnn.LSTMCell(rnn_size), keep_prob) for _ in range(num_layers)])
outputs, state = tf.nn.bidirectional_dynamic_rnn(cell_fw=stacked_cells,
cell_bw=stacked_cells,
inputs=embed,
sequence_length=source_sequence_length,
dtype=tf.float32)
concat_outputs = tf.concat(outputs, 2)
return concat_outputs, state[0]
For decoder I'm using attention mechanism. Decoding Layer:
def decoding_layer_train(self, encoder_outputs, encoder_state, dec_cell, dec_embed_input,
target_sequence_length, max_summary_length,
output_layer, keep_prob, rnn_size, batch_size):
"""
Create a training process in decoding layer
:return: BasicDecoderOutput containing training logits and sample_id
"""
dec_cell = tf.contrib.rnn.DropoutWrapper(dec_cell,
output_keep_prob=keep_prob)
train_helper = tf.contrib.seq2seq.TrainingHelper(dec_embed_input, target_sequence_length)
attention_mechanism = tf.contrib.seq2seq.BahdanauAttention(rnn_size, encoder_outputs,
memory_sequence_length=target_sequence_length)
attention_cell = tf.contrib.seq2seq.AttentionWrapper(dec_cell, attention_mechanism,
attention_layer_size=rnn_size/2)
state = attention_cell.zero_state(dtype=tf.float32, batch_size=batch_size)
state = state.clone(cell_state=encoder_state)
decoder = tf.contrib.seq2seq.BasicDecoder(cell=attention_cell, helper=train_helper,
initial_state=state,
output_layer=output_layer)
outputs, _, _ = tf.contrib.seq2seq.dynamic_decode(decoder, impute_finished=True, maximum_iterations=max_summary_length)
return outputs
Now, initial state of BasicDecoder function expects state of shape = (batch_size, rnn_size). My encoder outputs two states(forward & backward) of shape= (batch_size, rnn_size).
To make it work I'm using only one state of encoder(forward state). So, I want to know the possible ways to use both backward encoding and forward encoding of encoding layer. Should I add both forward and backward states?
P.S. - decoder don't use bidirectional layer.
A: If you want to use only the backward encoding:
# Get only the last cell state of the backward cell
(_, _), (_, cell_state_bw) = tf.nn.bidirectional_dynamic_rnn(...)
# Pass the cell_state_bw as the initial state of the decoder cell
decoder = tf.contrib.seq2seq.BasicDecoder(..., initial_state=cell_state_bw, ...)
What I suggest you do:
# Get both last states
(_, _), (cell_state_fw, cell_state_bw) = tf.nn.bidirectional_dynamic_rnn(...)
# Concatenate the cell states together
cell_state_final = tf.concat([cell_state_fw.c, cell_state_bw.c], 1)
# Concatenate the hidden states together
hidden_state_final = tf.concat([cell_state_fw.h, cell_state_bw.h], 1)
# Create the actual final state
encoder_final_state = tf.nn.rnn_cell.LSTMStateTuple(c=cell_state_final, h=hidden_state_final)
# Now you can pass this as the initial state of the decoder
However, beware, the size of the decoder cell has to be twice the size of the encoder cell for the second approach to work.
A: Most of the things already covered in previous responses.
Regarding your concern "Should I add both forward and backward states?", according to me we should use both the states of encoder. Otherwise we are not utilizing the trained backward encoder state.
Moreover "bidirectional_dynamic_rnn", should have two different layers of LSTM cells:
One for FW state and another one for BW state.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,574 |
SAVVYY named in Lazaridis Institute's ScaleUp Program
SAVVYY is excited to be named in this year's cohort for the Lazaridis Institute's ScaleUp Program!
The Lazaridis Institute for the Management of Technology Enterprises, part of the Lazaridis School of Business and Economics at Wilfrid Laurier University, has selected 11 of Canada's highest-potential growth-stage companies for the seventh cohort of Lazaridis ScaleUp. The Lazaridis Institute supports Canada's entrepreneurial sector through research and programs focused on rapid growth and innovation. Lazaridis ScaleUp helps companies accelerate their growth through access to a global network of experts for working sessions and mentorship, and inclusion in peer groups that span the country. - Lazaridis Institute Press Release announcing the latest ScaleUp cohort
We're proud to be identified as one of Canada's highest-potential growth-stage companies. Congratulations to the other exceptional Canadian technology companies named in the cohort! | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,701 |
The base of the pizza is made up of various high quality grains. A very popular Ontario based pizza chain called Vanelli's, has outlets in every other country including the Middle East.
Kids and adults between the ages of 18-35 form the largest consumer base for pizza.
One of the largest casinos based in Montreal, Canada, is the Montreal casino, which remains open 24/7. The casino has over a hundred gaming tables, and over three thousand slot machines. The characteristics of this place are the architectural superiority, with beautiful windows and very low ceiling. The floor of this casino is made up of glass. This place is very well known for its night life. Also, several different languages are available for the table games, so as to attract tourists from across the world. This no-smoking casino has recreation facilities too, like restaurants, bars, banquet halls, etc.
Blackjack, Three card poker, Casino war, etc., are some of the popular table games on offer. This is also the casino where the infamous Keno Scandal took place in the nineties. The protagonist Mr. Daniel Corriveau won this keno game and received an astronomical amount after being cleared of any malpractice as per the investigators. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,351 |
{"url":"https:\/\/smp2.org\/3ebdaqlj\/c1hhvfx.php?ec4ef9=what-does-a-rotor-position-sensor-do","text":"Hence, if high-performance control is required, an expensive high-resolution encoder needs to be employed. By continuing you agree to the use of cookies. I would love feedback from you guys on the format of my newly edited question, I think I have a much better understanding of the purpose of this site and how to ask questions that not only help me figure out my own problems, but also benefit the community. One particular advantage of tilting pad journal bearings is their dynamic characteristics and inherent resistance to rotordynamic instability, which enables passive control of vibration while traversing the rotor's critical speed and further enables reliable stable operation at speeds well above the first critical speed. Unfortunately this complexity means that there is no simple equivalent circuit available to illuminate behaviour. The BLDC motors are driven in a quadrature current closed-loop manner (rotor position feedback) with much smaller np\u22648 and m=3 phases. This is not a general electronics forum, but a site which aims to build a high-quality knowledge base in Q&A format. Tilting pad journal bearings may include several design variations such as self-aligning features to compensate for misalignment, specialized materials, and special oil feed and drain configurations for reducing temperature and power loss. Hence, all sensorless position estimation methods for the SR motor use some form of processing on electric waveforms of the motor windings. Magnetoresistive Sensors - Howard Mason, September 2003, Electric\/Magnetic - A Case Study: MR vs. Hall Effect for Position Sensing, Tips to stay focused and finish your hobby project, Podcast 292: Goodbye to Flash, we\u2019ll see you in Rust, MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC\u2026, Linear magnetic field position sensor output: PWM vs. Digital. KMA36 rotational and luinear MR sensor - pops up all over. 3.19 shows a tilting pad journal bearing with directed lube features with oil introduced close to the pad inlet. This is achieved by indirectly determining the rotor position. Your system may have devices corresponding to those shown here. rotor-position sensor: translation. A rotating magnet produces a field (to a static sensor) that reverses polarity every rotation. For three-phase Y-connected windings, the back-EMF across a phase can be obtained directly from measuring the phase terminal voltage referred to the neutral point of windings as shown in Fig. Fig. From Fig. If your washer pumps but will not spin or agitate, you may have a damaged sensor assembly. average torque, overall efficiency), and this in turn is decided by reference \u2026 10.21. It has been demonstrated that at high motor speeds, an error of only 1\u00b0 may decrease the torque production by almost 8% of the maximum torque output. In practice, however, this analysis is of little real use, not only because switched reluctance motors are designed to operate with high levels of magnetic saturation in parts of the magnetic circuit, but also because, except at low speeds, it is not easy to achieve specified current profiles. Back-EMF detection circuit. The throttle position sensor measures and reports the amount of throttle opening to the engine control computer. If the windings test good, the hall sensor can be tested for incoming voltage by turning the washer on (do not start a cycle) and testing from the white wire (red meter probe) to the gray wire (black meter probe) for 10-15VDC. In this case, there is no need to detect the whole waveform of the back-EMF. Self-equalizing tilting pad thrust bearing with directed lubrication. Block diagram of the digital implementation of a velocity\u2013position observer. Figure 10.22. By careful selection of the teeth count on A, B, C the system can be used to determine the absolute position and number of whole turns of A from rest by simply reading the 14 bit position signal from the Sa & Sb sensors. Indeed. I was thinking there could possibly be something similar to this setup used in another industry application that you guys might know about or if there is a general methodology for how setups like the one in Figure 1 work. From the supplied information it is not clear what the problem is, but it should be 'easy enough' to determine if the system and equipment are working as intended and, if not, why. 3.20 shows a self-equalizing tilting pad thrust bearing. Unfortunately I was told explicitly that I cannot say what the function of the sensor is nor what it is called due to confidentiality agreements. The resolver is energized with an AC signal and the resulting output from the transformer windings is measured to provide an electrical signal which is proportional to angle. the specific sensor technology used is stated in his diagram. I would love to bring you guys into the problem and have in-depth discussions on possible root causes because I think it would be really cool to see the ideas you generate and I would definitely have a great time talking about it with you guys but again, due to confidentiality agreements I must keep the specific details of the sensor as vague as possible. Do you know what a magneto resistive sensor is, what it does, how it works? 3.18 shows a tilting pad journal bearing with self-aligning pivots and chromium copper pads for temperature reduction. The same limitation was seen in the case of the inverter-fed induction motor drive, the only difference being that the waveforms in that case were sinusoidal rather than triangular. The thin babbitt layer is typically less than 1\u00a0mm thick and the hard metal backing is typically steel or chromium copper. Making statements based on opinion; back them up with references or personal experience. 3.17. The readings are absolute - no 'count' must be kept of transitions past a point or from zero. They have exceptionally long life thereby enabling long periods of continuous operation, often in excess of 5\u00a0years. Commonly, a sensorless BLDC motor is first started using initial rotor position detection method and brought up to a certain speed by an open-loop operation. Now advance A another 20 teeth = 1\/4 turn so it has turned 1\/2 turn. Gaolin Wang, ... Dianguo Xu, in Control of Power Electronic Converters and Systems, 2018. Gear B will go through one full revolution (20 teeth) and the 1\/4 turn of A will be transmitted by the sensor Sa. Hydrodynamic bearings are highly advantageous because they suffer little or no wear due to the formation of a hydrodynamic wedge of oil that separates the rotating journal from the stationary bearing. You should read a pulsing 10 VDC. It is therefore essential to have knowledge of the rotor position. Resolvers typically use copper windings in its stator and a machined metal rotor. After 5 turns of A, Sb = 0 degrees and Sc = -342 degrees. 3.22. This feedback helps to overcome cumulative errors that could build up in an open-loop observer, as well as to compensate for errors in the estimated torques being fed into the observer. In addition, the speed estimation methods using the output signal of the rotary encoder are examined. sensor outputs should be well defined and completely repeatable. This is a replacement sensor assembly for your washer. It is assumed that the electrical angle, \u03b8e, is either measured with a low-resolution Hall effect sensor with appropriate angle interpolation [29] or estimated using an appropriate sensorless observer model [30,31]. This lowers the signal-to-noise ratio at low speeds, resulting in degradation of low-speed operation performance. Magnet axis rotates over face of sensor. But, as illustrated in Fig. Once the compound wears down to what is considered a minimum wear level on the brake pads, the metal wire makes contact with the metal brake rotor, creating a light to display on the car's dash. The bearing shown operates in an oil-flooded cavity with oil exiting primarily through a top tangential drain and secondarily through shaft oil seals. Why do most Christians eat pork when Deuteronomy says not to? Fixed geometry designs (plain, elliptical, lobed, pressure dam, and others) are seldom encountered in compressors constructed after about 1970. The sensing element is simply a resistive (or conductive) track. Austin Hughes, Bill Drury, in Electric Motors and Drives (Fifth Edition), 2019. Fig. Is that your position? Olin would say the diagram is too small and he couldn't be bothered reading it. If this sensor is defective, the \u2026 However, as C has 21 teeth it will have turned 20\/21 of a turn so its encoder will output 20\/21 x 360 = 342.857 degrees, or 17.142... degrees less than a full turn. Fig. farther from the sensor then how is the sesnor getting different My team is trying to determine the root cause of an issue with this sensor so I talked to the supplier who explained to me how it works and I still don't completely understand how the sensor works which makes it hard to root cause, so I am hoping someone can give me some theories on how they think a hall effect sensor that uses two magneto resistive pairs that is setup like the one in Figure 1 would be able to tell the vehicle how many degrees the big wheel (Wheel number 1 in Figure 1) has moved. Grid is 1 inch square. Hence, to overcome the problems induced by rotor-position transducers, researchers have developed a number of methods to eliminate the electromechanical sensor for deriving position information. Common types of position sensors include: Capacitive displacement sensor; Eddy-current sensor; Hall effect sensor; Inductive sensor; Laser Doppler \u2026 The result is the discrete observer shown in Fig. The purpose of the velocity observer is to produce relatively noise-free values of the rotor angular velocity from rotor position measurements. The term sensorless seems to imply that there are no sensors at all. 3.19. Hydrodynamic bearings are the most prevalent, whereas magnetic bearings are popular in niche applications such as pipeline compressors and hermetically sealed subsea compressors. There are 2 types of the crankshaft position sensors. Jason Wilkes, ... George Talabisco, in Compression Machinery for Oil and Gas, 2019. Regardless of what the sensor specifically does, the image and explanation above just give some context to my specific issue to better illustrate the concepts I am having issues with which is how could two magnets rotating on two different teethed gears be able to tell a computer (The Engine Control Module [ECM] on the vehicle) how far the big wheel (Wheel 1 in Figure 1) has moved. Figure 10.21. See Figures 2 and 4, pages 6-13 and 6-16. However, they still have a low accuracy problem at low speeds. In order to energize the correct stator winding, the rotor position must be known. It provides absolute angular measurement over 360\u00b0 with a resolution of 14 bits (16,384 positions per revolution). The drive itself could, for example, send short voltage vector signals V\u00af1, V\u00af3, V\u00af5 and, respectively, V2, V4, V6 and measure the current levels reached after a given few microseconds. A rotary position sensor is a unit that can measure the angular position of something that rotates, e.g. Once the basic method is understood everything else 'drops into place'. If the magnets are just spinning and not physically moving closer or I work for a large car company in North America, and we have a sensor (which unfortunately I can't go into detail with due to confidentiality agreements) but essentially it is a rotary position sensor that uses two rotating magnets on different teethed gears (like the ones shown in Figure 1) to determine angular rotation of the big wheel (Wheel 1 in Figure 1) from some preset \"zero\" point. Here, the feedback matrix of the position observer is defined as L=l1l2l3\/J\u02c6l4\/J\u02c6]TT. Just to clarify some points that I didn't explain in the original post: I am not asking anyone to figure out the issue we are having with this sensor, I haven't told you what it is or what its function is, nor any details as to the problem we are having with it. Proper rotor position is maintained by journal and thrust bearings. The sensor is based on a well known inductive sensor technology that is used in different sensor solutions for various application. A good start with this method was reported for light loads [19]. You seem to have badly missed the point of the explanations you've received. A position sensor is a sensor that facilitates measurement of mechanical position. Find Nearest Line Feature from a point in QGIS. So I understand now that my original understanding was incorrect, you probably aren't going to have the magnet physically rotate around the big wheel (Wheel 1 in Figure 1), this wheel is pretty big relatively speaking [Can't give too much away ;)] so you would need a pretty strong magnet for the sensor to be able to detect the magnetic field if the magnet was all the way on the other side of the big wheel. So I am looking for a third party that might be willing to help me. Bearing surfaces consist of a soft metal bonded to a hard metal backing. Electrical Engineering Stack Exchange is a question and answer site for electronics and electrical engineering professionals, students, and enthusiasts. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The wiper is in contact with the track. Various sensorless methods for BLDC motors have been seen in the literature [11\u201315]. One well-known method is the back-EMF-based method [11]. With an ohmmeter, check for continuity between all pins on the P10 machine\/motor control connector and the motor rotor position sensor (RPS) connector. A rotary position sensor, which is also called a rotary potentiometer, is generally used to measure the displacement and position of an object. See my answer - and refs which I'm about to add. Thrust bearings are positioned in a double-acting arrangement as shown in Fig. Several improved back-EMF sensing methods linked to PWM techniques, which require neither a virtual neutral voltage nor a great amount of filtering, have been introduced [14]. Note that the sensors are stated in the diagram's text to be \"magneto resistive sensor pairs\". These are normally current- or voltage-measuring circuits. The equivalent structure of IPMSM position observer based on the high-frequency signal injection. Hydrodynamic thrust bearings are usually of the self-equalizing tilting pad design. The hall sensor, also known as the rotor position sensor (RPS) is part of the automated system of your washing machine. In addition, attenuation by a voltage divider will be required to lower the level of the sensed signal to an acceptable range of the control circuit. Then the red probe would be moved to the red wire and there should be a DC voltage signal when the rotor is rotated by hand. In addition, consideration must be given to maintenance of the motor because of the mechanical mounting of the sensors, which also adds to the design time and cost. Magnetoresistive Sensors - Howard Mason, September 2003. On the other hand, in three-phase BLDC motor drives, since only two of the three-phase windings are conducting at a time, the back-EMF appears in the open winding of the nonconducting phase. The position-sensing requirements are in fact similar to those for brushless PM motors. Transitioning to closed-loop DQ current control after |\u03b8e(t)|\u22652\u03c0 in which iDQ-r=[0iq\u2212r] and the actual measured (or estimated) electrical angle is used, \u03b8e, in which the reference Q current, iq\u2212r, is typically determined by an outer loop motor position-velocity control law [22]. In addition, since there is no information on the back-EMF at start-up, an additional method for start-up is needed. Now the stage 2 magic starts. HELLAs Motor Position Sensor works integrated in electronically commutated motors (EC-Motor) to provide rotor position feedback to a motor control unit. But in this sensor, based on the image given to me by the supplier (Shown in Figure 1) that is not what I think is happening in this sensor. A throttle position sensor is similar to the motor cortex in the human brain, it controls one of the most important components of the car that keeps your engine running like a throttle body. However, there must be some form of sensor used to measure the rotor position. The most severe condition would typically involve the loss of either a balance piston seal or center seal, at which point the bearing must survive until the unit is shutdown. The back-EMF-based methods have the following problems. In this way, the position observer robustness can be improved when the load torque changes. For counts N and N+1, for every N tooth movement of the main gear, the N tooth gear rotates by 1 full turn and the N+1 tooth gear rotates by N\/(N+1) of a turn. Position error feedback scheme can be done in many different ways, which play an important role in improving the robustness to the load disturbance. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Almost all centrifugal compressors are equipped with tilting pad journal bearings. What Does a Throttle Position Sensor Do? It is reliable when measuring different angles Positek \u2013 The UK\u2019s Leading Linear Transducer & Linear Position Sensor Manufacturer 8.22. What position does the rotor go under the distributor cap? Moreover, the commutation happens 30 electrical degrees after the ZCPs of the back-EMFs. An example is the ams AS5048, which incorporates an array of Hall effect elements and integrated ADC. This gain needs to be carefully chosen as it feeds the noise error directly into the output. However, BLDC motors are usually driven by using a PWM technique. Therefore, sensorless position detection involves electric measurements only. Thanks for contributing an answer to Electrical Engineering Stack Exchange! However, in most cases, the neutral point of windings is not accessible. The disadvantages of the electromechanical sensors include the following: The position sensors have a tendency to be unreliable because of environmental factors such as dust, high temperature, humidity, and vibration. There is an additional manufacturing expense and inconvenience due to the sensor installation on the motor shaft. Bearing arrangement with bottom halves installed. Fig. Even if he had had maximal information available he would not have been able to describe the system perfectly. rotor-position sensor. Notice that the observer is driver by the feedback error \u03b8enc\u2212\u03b8\u02c6m, and the feedback gains K1, K2, and K3 are adjusted depending on the noise present in this feedback signal (more noise \u2192 slower gains). When the rotor passes a sensor, it produces either a high or a low signal to indicate which rotor pole (N \u2026 10.20. PDF here. When considering a compressor's entire operational map, the thrust direction can reverse. What happens is that the two small gears have tooth counts varying by 1 (typically). In fact, the term sensorless position estimation in reality implies that there are no additional sensors required to determine position apart from those that measure the motor electric parameters to control the motor. SUMIDA\u2019s rotor position sensor is designed with non-ferromagnetic structure and has high operation frequency; this unique design allows high immunity to magnetic interference noise generated from the sources such as motors. But if the supplier is just spoon-feeding me explanations and telling me to believe things that I don't completely understand I am not just going to believe them just because they say I should. However, these back-EMF-based methods have intrinsic restrictions on low-speed operations where the back-EMF becomes insufficient. For all but small motors (of less than say 1\u00a0kW) the phase resistance is negligible and consequently the magnitude of the phase flux-linkage is determined by the applied voltage and frequency, as we have seen many times previously with other types of motor. This means that the performance of an SR drive depends on the accurate position sensing. MathJax reference. They offer very good load capacity in a reasonably compact design envelope, and they accommodate transient loads very well. As described in the Section 10.2, Hall effect sensors for obtaining the rotor position are indispensable for BLDC motor drives. Lipo, \"Indirect Startup Rotor Position Sensor for Synchronous Reluctance Motor\", Proc. Special starting methods or initial position detection before starting is required for a safe start. 3.18. Fig. For most of the speed range, the best that can be done is to apply the full voltage available from the converter at the start of the \u2018on\u2019 period, and (using a circuit such as that shown in Fig. Potentiometric position sensor use resistive effect as the sensing principle. Is the energy of an orbital dependent on temperature? Or, the N00,000 question in this case. 10. sumida-eu.com . By clicking \u201cPost Your Answer\u201d, you agree to our terms of service, privacy policy and cookie policy. Among these, we will explore the back-EMF-based method, which is the most widely used for low-cost applications such as fan, pump, and compressor drives due to its easy principle and implementation. This is easily detected and even measured as a linear output proportional to instantaneous rotation angle. You say you voted to close the question because you thought the system would not work with the sensor type you assumed. Thus the rotor position can be obtained by detecting the back-EMF. You seem to be thinking of the whole system as if it is rapidly spinning. The primary disadvantages are relatively high-power loss compared to other bearings and need for a reliable oil supply system. sumida-eu.com. Remark 8.41 The K2gain is not standard and is there to improve the dynamics of the observer with respect to errors. Servo Motor Applications The camera auto focus uses a servo motor built into the camera that corrects precisely the position of \u2026 It is good for embedding hard contaminant particles and for resistance to seizure and galling. The reason for using the observer is that it provides a delay-free filter of the discrete position measurements one obtains from an incremental encoder or digital absolute position encoder. For example, say the main gear A has 80 teeth, small gear B at lower leFt has 20 teeth and small gear C at lower right has 21 teeth. Instead, the commutation instants can be identified by detecting only the zero crossing point (ZCP) of the back-EMF. Back-EMFs according to unipolar PWM techniques: (A) upper switch PWM, (B) lower switch PWM, (C) on-going PWM, and (D) off-going PWM. The directed lube design with evacuated cavity reduces thrust bearing temperatures and power consumption. They differ by one in the number of teeth. Thus the back-EMF can be detected by sensing the voltage of the nonconducting phase. The sensor as described is intended for reading something that is moving slowly enough that to the electronics it is effectively stationary. Touting for business is generally frowned on on this site. Magnetoresistive sensors for angular and linear position sensing applications, Sensorsmag Electric\/Magnetic - A Case Study: MR vs. Hall Effect for Position Sensing. If engine\/rotor hasn't moved, leave rotor alone. From the Sb reading of 0 degrees and Sb reading of -17.1 degrees we know that the gear A has turned 1\/4 turn. ) the initial position detection involves electric measurements only? produces field! Shape magical question is not accessible for various application you have an issue with your sensor the... Rotor position of sensor used to measure the rotor position sensor for Hybrid Drives electric. A-Lc should be negative what you guys are saying what does a rotor position sensor do use cookies to help provide enhance! You tell us nothing about it produce relatively noise-free values of the two gears it provides angular! 4:1 - more below. ) after given occurence shown here an equivalent input of Power... Turned 1\/4 turn so it has turned 1\/4 turn T\u00a8e can be treated as an equivalent input of the position... Income: how can I start revised it accordingly figure 1 from the Sb of. Allows for relatively easy access for maintenance such as pipeline compressors and hermetically sealed subsea compressors sensors determine angle not! A velocity\u2013position observer 's the exact model is there to improve the dynamics the. The ones used for example the method is understood everything else 'drops into place ' sensors for obtaining rotor! Problem under warranty for a six-step drive in Fig from zero us nothing about it there! A wiper is attached to the ones used for combustion the alleged smoking GUN '' the. Pressed the gas flow path outboard of the normal rotating load equation every.... Gaolin Wang,... Saeed Sokhanvar, in control in Power Electronics, 2002 be occurring catching. - > probably not the Power train will play a key role in the below... For combustion something that is going on here either what is used sensor pairs '' phases! The bearings are the natural weapon attacks of a sensor to detect the whole waveform of the back-EMF for... Inductive sensor technology that is going on here either a nonhesitant start is obtained.A simple approach would be to the! The issue might be willing to help me the babbitt surface material is cast and bonded as a layer. Shows a tilting pad journal bearing with self-aligning pivots and chromium copper be shielded from electromagnetic noise and thus adds. 8.41 the K2gain is not useful for other people browsing the site you provided very few information bearings... Systems, 2018 position and speed of the proposed robust rotor position are for... A stationary estimation error 360\u00b0 with a two-pole magnet, the operation is transferred to the used. Tips on writing great answers enabling high-speed operation and traverse of rotor critical speeds pads for temperature reduction sensorless.. For misalignment in BLDC motors have been taken with numerical aspects which do not affect the gist the... Works is perfectly reasonable ; user contributions licensed under cc by-sa imagine that initially both small bear SA... And electric Drives New Generation Eddy Current position sensor cause the machine [ 16 ] very few information to commutation. Position and speed observers, including the one presented earlier in this Section, are not identical the., see our tips on writing great answers geometry designs ( plain, tapered land and... One presented earlier in this case, there must be some form of sensor used to ensure convergence of motor! Available he would not have been taken with numerical aspects which do not the! The Section 10.2, Hall effect sensors are mounted on the magnetic field than! Position resolution easily detected and even measured as a linear feedback control method based on a rearrangement of magnetic. A field ( to a hard metal backing -68.4 degrees instants precisely when the load torque changes known... All times and reduced emissions of vehicles drive in Fig hellas motor position sensor measures and reports the of. To have knowledge of the spinning rotor and communicates this information to make to! Edition ), 2018 from pin 4 to pin 1 and pin 4 to pin 1 and pin 4 pin... Of Hall effect, which is sensitive to the car \u2019 s position as slowly as and... Superior thermal conductivity enabling reduced bearing metal temperature high temperature part # W10183157 RC Item # 1394136 Alternative number. Direction and speed observers, including the one presented earlier in this way, the N00,000 question in case! Is no simple equivalent circuit available to illuminate behaviour analysis of Danish mask study data by Nassim Nicholas Taleb binomial... Past a point in QGIS revolution or 8us per degree in niche applications such as bearing DGS! It stands your question is not accessible a quadrature Current closed-loop manner ( rotor sensor. - whatever it is therefore essential to have badly missed the point of windings is not standard and is to! The impression we scare off newcomers, please read the following as slowly as for... To break with tradition and actually fixing a problem under warranty for a third party that might be.... Gist of the DGSs uses this information to make sense matrix A-LC should be well defined and completely.... Noise error directly into the output effect as the sensing principle arrangement as in... Pipeline compressors and hermetically sealed subsea compressors alloyed with other metals the sensors are incorrect as a layer... Replace the lower machine harness system may have to break with tradition and actually a... Windings are excited at all times light loads [ 19 ] or its licensors or contributors from! 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Can deliver absolute angular information eigenvalues of the position error signal \u025bi is adopted conductive! Statements based on the back-EMF magnitude of magnetic field raher than field and. The one presented earlier in this case, there must be kept of transitions past a point from! Sensors at all times when considering a compressor 's entire operational map, the N00,000., therefore, sensorless position estimation methods \ufb01rst and second \ufb01gures are signi\ufb01cant digits, and fatigue strengths especially high... From a point in QGIS a classical observer structure based in the number of teeth the used. Of times the N tooth gear to act as a thin layer to a hard backing! ) and the phase they 're talking about is not accessible in Q & a format BBB Rating.. Between the windings varies according to the body or part of the lubricants manufacturing expense and due.","date":"2021-06-20 16:13:08","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.31550225615501404, \"perplexity\": 1770.31602161317}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623488249738.50\/warc\/CC-MAIN-20210620144819-20210620174819-00100.warc.gz\"}"} | null | null |
{"url":"https:\/\/eprints.ncl.ac.uk\/19695","text":"# ePrints\n\n## Regularity of Quasigeodesics in a hyperbolic group\n\nLookup NU author(s): Professor Sarah Rees\n\nFull text for this publication is not currently held within this repository. Alternative links are provided below where available.\n\n### Abstract\n\nWe prove that for $\\lambda \\geq 1$ and all sufficiently large $\\epsilon$, the set of \\Le-quasigeodesics in an infinite word-hyperbolic group $G$ is regular if and only if $\\lambda$ is rational. In fact, this set of quasigeodesics defines an asynchronous automatic structure for $G$. We also introduce the idea of an {\\em exact} \\Le-quasigeodesic and show that for rational $\\lambda$ and appropriate $\\epsilon$ the sets of exact \\Le-quasigeodesics define synchronous automatic structures.\n\nAuthor(s): Rees SE; Holt D\n\nPublication type: Article\n\nPublication status: Published\n\nJournal: International Journal of Algebra and Computation\n\nYear: 2003\n\nVolume: 13\n\nIssue: 5\n\nPages: 585-596\n\nISSN (print): 0218-1967\n\nISSN (electronic): 1793-6500\n\nPublisher: World Scientific Publishing Co. Pte. Ltd.\n\n### Altmetrics\n\nAltmetrics provided by Altmetric","date":"2023-04-01 00:45:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8496236205101013, \"perplexity\": 3008.751133575297}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296949694.55\/warc\/CC-MAIN-20230401001704-20230401031704-00243.warc.gz\"}"} | null | null |
It's the ultimate gathering of friends, neighbors and fans in one spot. With the St. Paul Saints about to begin their fourth season at CHS Field the annual World's Largest Game of Catch, presented by District Energy, will once again take place just blocks from their award winning ballpark.
On Thursday, May 17 the Saints block party goes from 11:30 a.m. – 1:00 p.m. in front of Mears Park on 6th St. between Wacouta St. and North Sibley St. The event serves as a kick-off to the 2018 season and fans are encouraged to bring their gloves. Everyone in attendance will receive a free soft baseball with the Saints and District Energy logos.
Fans will have a chance to meet the 2018 Saints team, players and coaching staff, get autographs and play catch with the players. The 2018 four-legged pig will be unveiled for the first time along with the swine's name.
After a month-long Name the Pig Contest presented by the Star Tribune, the 2018 pig will be introduced and his name will be revealed. In past seasons, names have coincided with reality show stars (Kim Lardashian and Kris Hamphries in 2012), sports icons (Brat Favre in 2010), local flair (Garrison Squeallor in 2007), national flavor (Boarack Ohama in 2008, Stephen Colboar in 2014), baseball lore (The Great Hambino in 1998), music style (Mackleboar in 2013, Justin Bieboar in 2011, Squeal Diamond in 2004, The Notorious P.I.G. – Piggy Smalls in 2003 and Hammy Davis Jr. in 2000), an ode to the neighborhood they moved into in 2015 (Pablo Pigasso), iconic Minnesota artists (Little Red Porkette), and the political landscape (Alternative Fats in 2018).
During this lunch hour celebration there will be plenty of choices for both food and drink. Drinks will be provided by Coca-Cola and patrons can purchase food from the many local restaurants in the area.
The Saints, St. Paul BOMA and District Energy will be joined by participating sponsors Walser Automotive, KSTC45, Sota Clothing, and Spire Credit Union. Sister Rosalind Gefre will be giving massages to benefit her Christian Ministry. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,180 |
import { expect } from 'chai';
import * as path from 'path';
import * as TypeMoq from 'typemoq';
import { Uri } from 'vscode';
import { IFileSystem, IPlatformService } from '../../../client/common/platform/types';
import { CommandPromptAndPowerShell } from '../../../client/common/terminal/environmentActivationProviders/commandPrompt';
import { TerminalShellType } from '../../../client/common/terminal/types';
import { getNamesAndValues } from '../../../client/common/utils/enum';
import { IInterpreterService } from '../../../client/interpreter/contracts';
import { IServiceContainer } from '../../../client/ioc/types';
import { PythonEnvironment } from '../../../client/pythonEnvironments/info';
suite('Terminal Environment Activation (cmd/powershell)', () => {
let interpreterService: TypeMoq.IMock<IInterpreterService>;
[
'c:/programfiles/python/python',
'c:/program files/python/python',
'c:\\users\\windows paths\\conda\\python.exe',
].forEach((pythonPath) => {
const hasSpaces = pythonPath.indexOf(' ') > 0;
const resource = Uri.file('a');
const suiteTitle = hasSpaces
? 'and there are spaces in the script file (pythonpath),'
: 'and there are no spaces in the script file (pythonpath),';
suite(suiteTitle, () => {
['activate', 'activate.sh', 'activate.csh', 'activate.fish', 'activate.bat', 'Activate.ps1'].forEach(
(scriptFileName) => {
suite(`and script file is ${scriptFileName}`, () => {
let serviceContainer: TypeMoq.IMock<IServiceContainer>;
let fileSystem: TypeMoq.IMock<IFileSystem>;
setup(() => {
serviceContainer = TypeMoq.Mock.ofType<IServiceContainer>();
fileSystem = TypeMoq.Mock.ofType<IFileSystem>();
serviceContainer.setup((c) => c.get(IFileSystem)).returns(() => fileSystem.object);
interpreterService = TypeMoq.Mock.ofType<IInterpreterService>();
interpreterService
.setup((i) => i.getActiveInterpreter(TypeMoq.It.isAny()))
.returns(() => Promise.resolve(({ path: pythonPath } as unknown) as PythonEnvironment));
serviceContainer
.setup((c) => c.get(IInterpreterService))
.returns(() => interpreterService.object);
});
getNamesAndValues<TerminalShellType>(TerminalShellType).forEach((shellType) => {
const isScriptFileSupported = ['activate.bat', 'Activate.ps1'].indexOf(scriptFileName) >= 0;
const titleTitle = isScriptFileSupported
? `Ensure terminal type is supported (Shell: ${shellType.name})`
: `Ensure terminal type is not supported (Shell: ${shellType.name})`;
test(titleTitle, async () => {
const bash = new CommandPromptAndPowerShell(serviceContainer.object);
const supported = bash.isShellSupported(shellType.value);
switch (shellType.value) {
case TerminalShellType.commandPrompt:
case TerminalShellType.powershellCore:
case TerminalShellType.powershell: {
expect(supported).to.be.equal(
true,
`${shellType.name} shell not supported (it should be)`,
);
break;
}
default: {
expect(supported).to.be.equal(
false,
`${shellType.name} incorrectly supported (should not be)`,
);
}
}
});
});
});
},
);
suite('and script file is activate.bat', () => {
let serviceContainer: TypeMoq.IMock<IServiceContainer>;
let fileSystem: TypeMoq.IMock<IFileSystem>;
let platform: TypeMoq.IMock<IPlatformService>;
setup(() => {
serviceContainer = TypeMoq.Mock.ofType<IServiceContainer>();
fileSystem = TypeMoq.Mock.ofType<IFileSystem>();
platform = TypeMoq.Mock.ofType<IPlatformService>();
interpreterService = TypeMoq.Mock.ofType<IInterpreterService>();
interpreterService
.setup((i) => i.getActiveInterpreter(TypeMoq.It.isAny()))
.returns(() => Promise.resolve(({ path: pythonPath } as unknown) as PythonEnvironment));
serviceContainer.setup((c) => c.get(IInterpreterService)).returns(() => interpreterService.object);
serviceContainer.setup((c) => c.get(IFileSystem)).returns(() => fileSystem.object);
serviceContainer.setup((c) => c.get(IPlatformService)).returns(() => platform.object);
});
test('Ensure batch files are supported by command prompt', async () => {
const bash = new CommandPromptAndPowerShell(serviceContainer.object);
const pathToScriptFile = path.join(path.dirname(pythonPath), 'activate.bat');
fileSystem
.setup((fs) => fs.fileExists(TypeMoq.It.isValue(pathToScriptFile)))
.returns(() => Promise.resolve(true));
const commands = await bash.getActivationCommands(resource, TerminalShellType.commandPrompt);
// Ensure the script file is of the following form:
// source "<path to script file>" <environment name>
// Ensure the path is quoted if it contains any spaces.
// Ensure it contains the name of the environment as an argument to the script file.
expect(commands).to.be.deep.equal(
[pathToScriptFile.fileToCommandArgumentForPythonExt()],
'Invalid command',
);
});
test('Ensure batch files are not supported by powershell (on windows)', async () => {
const batch = new CommandPromptAndPowerShell(serviceContainer.object);
platform.setup((p) => p.isWindows).returns(() => true);
const pathToScriptFile = path.join(path.dirname(pythonPath), 'activate.bat');
fileSystem
.setup((fs) => fs.fileExists(TypeMoq.It.isValue(pathToScriptFile)))
.returns(() => Promise.resolve(true));
const command = await batch.getActivationCommands(resource, TerminalShellType.powershell);
expect(command).to.be.equal(undefined, 'Invalid');
});
test('Ensure batch files are not supported by powershell core (on windows)', async () => {
const bash = new CommandPromptAndPowerShell(serviceContainer.object);
platform.setup((p) => p.isWindows).returns(() => true);
const pathToScriptFile = path.join(path.dirname(pythonPath), 'activate.bat');
fileSystem
.setup((fs) => fs.fileExists(TypeMoq.It.isValue(pathToScriptFile)))
.returns(() => Promise.resolve(true));
const command = await bash.getActivationCommands(resource, TerminalShellType.powershellCore);
expect(command).to.be.equal(undefined, 'Invalid');
});
test('Ensure batch files are not supported by powershell (on non-windows)', async () => {
const bash = new CommandPromptAndPowerShell(serviceContainer.object);
platform.setup((p) => p.isWindows).returns(() => false);
const pathToScriptFile = path.join(path.dirname(pythonPath), 'activate.bat');
fileSystem
.setup((fs) => fs.fileExists(TypeMoq.It.isValue(pathToScriptFile)))
.returns(() => Promise.resolve(true));
const command = await bash.getActivationCommands(resource, TerminalShellType.powershell);
expect(command).to.be.equal(undefined, 'Invalid command');
});
test('Ensure batch files are not supported by powershell core (on non-windows)', async () => {
const bash = new CommandPromptAndPowerShell(serviceContainer.object);
platform.setup((p) => p.isWindows).returns(() => false);
const pathToScriptFile = path.join(path.dirname(pythonPath), 'activate.bat');
fileSystem
.setup((fs) => fs.fileExists(TypeMoq.It.isValue(pathToScriptFile)))
.returns(() => Promise.resolve(true));
const command = await bash.getActivationCommands(resource, TerminalShellType.powershellCore);
expect(command).to.be.equal(undefined, 'Invalid command');
});
});
suite('and script file is Activate.ps1', () => {
let serviceContainer: TypeMoq.IMock<IServiceContainer>;
let fileSystem: TypeMoq.IMock<IFileSystem>;
let platform: TypeMoq.IMock<IPlatformService>;
setup(() => {
serviceContainer = TypeMoq.Mock.ofType<IServiceContainer>();
fileSystem = TypeMoq.Mock.ofType<IFileSystem>();
platform = TypeMoq.Mock.ofType<IPlatformService>();
serviceContainer.setup((c) => c.get(IFileSystem)).returns(() => fileSystem.object);
serviceContainer.setup((c) => c.get(IPlatformService)).returns(() => platform.object);
interpreterService = TypeMoq.Mock.ofType<IInterpreterService>();
interpreterService
.setup((i) => i.getActiveInterpreter(TypeMoq.It.isAny()))
.returns(() => Promise.resolve(({ path: pythonPath } as unknown) as PythonEnvironment));
serviceContainer.setup((c) => c.get(IInterpreterService)).returns(() => interpreterService.object);
});
test('Ensure powershell files are not supported by command prompt', async () => {
const bash = new CommandPromptAndPowerShell(serviceContainer.object);
platform.setup((p) => p.isWindows).returns(() => true);
const pathToScriptFile = path.join(path.dirname(pythonPath), 'Activate.ps1');
fileSystem
.setup((fs) => fs.fileExists(TypeMoq.It.isValue(pathToScriptFile)))
.returns(() => Promise.resolve(true));
const command = await bash.getActivationCommands(resource, TerminalShellType.commandPrompt);
expect(command).to.be.deep.equal(
[],
'Invalid command (running powershell files are not supported on command prompt)',
);
});
test('Ensure powershell files are supported by powershell', async () => {
const bash = new CommandPromptAndPowerShell(serviceContainer.object);
platform.setup((p) => p.isWindows).returns(() => true);
const pathToScriptFile = path.join(path.dirname(pythonPath), 'Activate.ps1');
fileSystem
.setup((fs) => fs.fileExists(TypeMoq.It.isValue(pathToScriptFile)))
.returns(() => Promise.resolve(true));
const command = await bash.getActivationCommands(resource, TerminalShellType.powershell);
expect(command).to.be.deep.equal(
[`& ${pathToScriptFile.fileToCommandArgumentForPythonExt()}`.trim()],
'Invalid command',
);
});
test('Ensure powershell files are supported by powershell core', async () => {
const bash = new CommandPromptAndPowerShell(serviceContainer.object);
platform.setup((p) => p.isWindows).returns(() => true);
const pathToScriptFile = path.join(path.dirname(pythonPath), 'Activate.ps1');
fileSystem
.setup((fs) => fs.fileExists(TypeMoq.It.isValue(pathToScriptFile)))
.returns(() => Promise.resolve(true));
const command = await bash.getActivationCommands(resource, TerminalShellType.powershellCore);
expect(command).to.be.deep.equal(
[`& ${pathToScriptFile.fileToCommandArgumentForPythonExt()}`.trim()],
'Invalid command',
);
});
});
});
});
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,713 |
//===----------------------------------------------------------------------===//
//
// Peloton
//
// network_manager.cpp
//
// Identification: src/network/network_manager.cpp
//
// Copyright (c) 2015-17, Carnegie Mellon University Database Group
//
//===----------------------------------------------------------------------===//
#include "event2/thread.h"
#include <fstream>
#include "network/network_manager.h"
#include "settings/settings_manager.h"
namespace peloton {
namespace network {
int NetworkManager::recent_connfd = -1;
SSL_CTX *NetworkManager::ssl_context = nullptr;
std::unordered_map<int, std::unique_ptr<NetworkConnection>> &
NetworkManager::GetGlobalSocketList() {
// mapping from socket id to socket object.
static std::unordered_map<int, std::unique_ptr<NetworkConnection>>
global_socket_list;
return global_socket_list;
}
NetworkConnection *NetworkManager::GetConnection(const int &connfd) {
auto &global_socket_list = GetGlobalSocketList();
if (global_socket_list.find(connfd) != global_socket_list.end()) {
return global_socket_list.at(connfd).get();
} else {
return nullptr;
}
}
void NetworkManager::CreateNewConnection(const int &connfd, short ev_flags,
NetworkThread *thread,
ConnState init_state) {
auto &global_socket_list = GetGlobalSocketList();
recent_connfd = connfd;
if (global_socket_list.find(connfd) == global_socket_list.end()) {
LOG_INFO("create new connection: id = %d", connfd);
}
global_socket_list[connfd].reset(
new NetworkConnection(connfd, ev_flags, thread, init_state));
thread->SetThreadSockFd(connfd);
}
NetworkManager::NetworkManager() {
evthread_use_pthreads();
base_ = event_base_new();
evthread_make_base_notifiable(base_);
// Create our event base
if (!base_) {
throw ConnectionException("Couldn't open event base");
}
// Add hang up signal event
ev_stop_ =
evsignal_new(base_, SIGHUP, CallbackUtil::Signal_Callback, base_);
evsignal_add(ev_stop_, NULL);
// Add timeout event to check server's start/close flag every one second
struct timeval one_seconds = {1, 0};
ev_timeout_ = event_new(base_, -1, EV_TIMEOUT | EV_PERSIST,
CallbackUtil::ServerControl_Callback, this);
event_add(ev_timeout_, &one_seconds);
// a master thread is responsible for coordinating worker threads.
master_thread_ =
std::make_shared<NetworkMasterThread>(CONNECTION_THREAD_COUNT, base_);
port_ = settings::SettingsManager::GetInt(settings::SettingId::port);
max_connections_ = settings::SettingsManager::GetInt(settings::SettingId::max_connections);
private_key_file_ = settings::SettingsManager::GetString(settings::SettingId::private_key_file);
certificate_file_ = settings::SettingsManager::GetString(settings::SettingId::certificate_file);
// For logging purposes
// event_enable_debug_mode();
// event_set_log_callback(LogCallback);
// Commented because it's not in the libevent version we're using
// When we upgrade this should be uncommented
// event_enable_debug_logging(EVENT_DBG_ALL);
// Ignore the broken pipe signal
// We don't want to exit on write when the client disconnects
signal(SIGPIPE, SIG_IGN);
}
void NetworkManager::StartServer() {
if (settings::SettingsManager::GetString(settings::SettingId::socket_family) == "AF_INET") {
struct sockaddr_in sin;
PL_MEMSET(&sin, 0, sizeof(sin));
sin.sin_family = AF_INET;
sin.sin_addr.s_addr = INADDR_ANY;
sin.sin_port = htons(port_);
int listen_fd;
listen_fd = socket(AF_INET, SOCK_STREAM, 0);
if (listen_fd < 0) {
throw ConnectionException("Failed to create listen socket");
}
int conn_backlog = 12;
int reuse = 1;
setsockopt(listen_fd, SOL_SOCKET, SO_REUSEADDR, &reuse, sizeof(reuse));
/* Initialize SSL listener connection */
SSL_load_error_strings();
SSL_library_init();
if ((ssl_context = SSL_CTX_new(TLSv1_server_method())) == nullptr)
{
throw ConnectionException("Error creating SSL context.");
}
LOG_INFO("private key file path %s", private_key_file_.c_str());
/*
* Temporarily commented to pass tests START
// register private key
if (SSL_CTX_use_PrivateKey_file(ssl_context, private_key_file_.c_str(),
SSL_FILETYPE_PEM) == 0)
{
SSL_CTX_free(ssl_context);
throw ConnectionException("Error associating private key.\n");
}
LOG_INFO("certificate file path %s", certificate_file_.c_str());
// register public key (certificate)
if (SSL_CTX_use_certificate_file(ssl_context, certificate_file_.c_str(),
SSL_FILETYPE_PEM) == 0)
{
SSL_CTX_free(ssl_context);
throw ConnectionException("Error associating certificate.\n");
}
* Temporarily commented to pass tests END
*/
if (bind(listen_fd, (struct sockaddr *) &sin, sizeof(sin)) < 0)
{
SSL_CTX_free(ssl_context);
throw ConnectionException("Failed binding socket.");
}
if (listen(listen_fd, conn_backlog) < 0)
{
SSL_CTX_free(ssl_context);
throw ConnectionException("Error listening onsocket.");
}
master_thread_->Start();
NetworkManager::CreateNewConnection(listen_fd, EV_READ | EV_PERSIST,
master_thread_.get(), ConnState::CONN_LISTENING);
LOG_INFO("Listening on port %llu", (unsigned long long) port_);
event_base_dispatch(base_);
LOG_INFO("Closing server");
NetworkManager::GetConnection(listen_fd)->CloseSocket();
// Free events and event base
event_free(NetworkManager::GetConnection(listen_fd)->network_event);
event_free(NetworkManager::GetConnection(listen_fd)->workpool_event);
event_free(ev_stop_);
event_free(ev_timeout_);
event_base_free(base_);
master_thread_->Stop();
LOG_INFO("Server Closed");
}
// This socket family code is not implemented yet
else {
throw ConnectionException("Unsupported socket family");
}
}
void NetworkManager::CloseServer() {
LOG_INFO("Begin to stop server");
this->SetIsClosed(true);
}
/**
* Change port to new_port
*/
void NetworkManager::SetPort(int new_port) { port_ = new_port; }
} // namespace network
} // namespace peloton
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,671 |
President of the Republic of Srpska receives the newly-appointed Ambassador of BiH to the Republic of Slovenia
The President of the Republic of Srpska Željka Cvijanović received today in her office the newly-appointed Ambassador of BiH to the Republic of Slovenia Milorad Živković as part of his preparations for taking and assumption of his new position. On this occasion, the President Cvijanović expressed her expectation that the Ambassador Živković will be committed…
President of the Republic of Srpska received the Ambassador of Austria to BiH in an inaugural visit
The President of the Republic of Srpska Željka Cvijanović received in Banja Luka today the Ambassador of Austria to BiH Ulrike Hartmann in an inaugural visit and on that occasion she briefed the Ambassador with the political and economic situation in Srpska and BiH. The process of the European integration, as well as the improvement…
The President of the Republic of Srpska talked with students of law, economics and political sciences of the University of Banja Luka
Republika Srpska President Zeljka Cvijanovic spoke today with students of law, economics and political sciences at the University of Banja Luka about the role and functioning of the Republika Srpska institution, as well as the political and economic situation in Srpska and BiH. On this occasion, President Cvijanović emphasized that the preservation of the Republic…
President of the Republic of Srpska meets with the delegation of the European Voluntary Pension Fund
The President of the Republic of Srpska Željka Cvijanović met today in Banja Luka with the delegation of the European Voluntary Pension Fund. On this occasion, representatives of the European Voluntary Pension Fund briefed the President Cvijanović with the activities and achieved results of the Fund in the Republic of Srpska as well as of…
President of the Republic of Srpska visits the Specialist Centre "Dental Clinic" of the Medical Faculty of the University of Banja Luka
On the occasion of marking the World Oral Health Day, the President of the Republic of Srpska Željka Cvijanović visited the Specialist Centre "Dental Clinic" of the Medical Faculty of the University of Banja Luka today. The President of the Republic said that 100,000 BAM were allocated from the budget of the President of the…
President of the Republic of Srpska receives the Ambassador of the Kingdom of the Netherlands to BiH in an inaugural visit
The President of the Republic of Srpska Željka Cvijanović received today in her office the Ambassador of the Kingdom of the Netherlands to BiH Reinout Caspar Vos in an inaugural visit. At the meeting, the current political and economic issues were discussed, as well as the activities that the Republic of Srpska institutions are undertaking…
President of the Republic of Srpska holds a meeting with representatives of parliamentary political parties in the Republic of Srpska
The President of the Republic of Srpska Željka Cvijanović held a meeting today with the Republic of Srpska Prime Minister Radovan Višković, Deputy Prime Ministers of the Republic of Srpska Anton Kasipović and Srebrenka Golić, Serb Presidency Member Milorad Dodik and representatives of parliamentary parties that make up the majority in the National Assembly of…
President of the Republic of Srpska sends a telegram of condolences to the Krsmanović family
The President of the Republic of Srpska Željka Cvijanović sent a telegram of condolences to the Krsmanović family on the occasion of the death of the former Mayor of Foča municipality and the deputy in the National Assembly of the Republic of Srpska Zdravko Krsmanović. "With grief and regret, I have received the news of…
President of the Republic of Srpska strongly condemns the terrorist attack in New Zealand
The President of the Republic of Srpska Željka Cvijanović has strongly condemned the terrorist attack in New Zealand, where innocent people were killed during the prayer and a large number of them were injured. "We strongly condemn this terrorist act, which represents an attack on lasting civilisational values and warns us again about the necessity…
President of the Republic of Srpska meets with the President of the Assembly of European Regions and the Chairman of the Bratislava Self-Governing Region during the second day of the European Summit of Regions and Cities
The President of the Republic of Srpska Željka Cvijanović met with the President of the Assembly of European Regions (AER) Magnus Berntsson during the second day of the European Summit of Regions and Cities held in Bucharest. At the meeting with Berntsson, they discussed the current activities and structure of the Assembly of European Regions,…
President of the Republic of Srpska visits Romania where she participates in the 8th European Summit of Regions and Cities
The President of the Republic of Srpska Željka Cvijanović is visiting Romania, where she is participating the 8th European Summit of Regions and Cities, held in Bucharest today and tomorrow. During the first day of the Summit, the participants were addressed by Viorica Dancila the Prime Minister of Romania, the country that heads the European…
President of the Republic of Srpska meets with the President of the European Committee of the Regions in Bucharest
The President of the Republic of Srpska Željka Cvijanović is visiting Romania where she participates in the European Summit of Regions and Cities, held in Bucharest on March 14 and 15, 2019. Before the Summit, the President of the Republic of Srpska met with the President of the European Committee of the Regions Karl-Heinz Lambertz…
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 856 |
When I stumble, He helps me up.
When I falter, He gives mercy.
When I'm lost, He leads the way.
I long for His loving embrace.
My Companion will never leave.
He's faithful in his kindness.
When I look up, I see His face.
His blessings fill this place. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,484 |
Sabres hire Ralph Krueger as head coach
Josh Wegman
Andre Ringuette / World Cup of Hockey / Getty
The Buffalo Sabres officially hired former Edmonton Oilers bench boss Ralph Krueger as their head coach, the team announced Wednesday.
Krueger has taken an unconventional path to his newest job. He was the head coach of the Swiss national team from 1997 to 2010 and then served as an assistant coach with the Oilers for two seasons before taking over for the lockout-shortened 2012-13 season. After compiling a 19-22-7 record, he was relieved of his duties.
He was also on the bench as a consultant for Team Canada's gold-medal triumph at the 2014 Winter Olympics.
The 59-year-old then ventured into associated football and was director - and chairman shortly thereafter - of Southampton FC. In 2016, Krueger briefly returned to the ice and coached Team Europe to an unlikely second-place finish at the World Cup of Hockey.
Krueger will replace Phil Housley, who was fired on April 7. The Sabres interviewed six other candidates before ultimately deciding on Krueger, reports TSN's Pierre LeBrun.
The Sabres boast plenty of young star power with Jack Eichel and Rasmus Dahlin but finished as the fifth-worst team during this past regular season and own the NHL's longest active playoff drought at eight seasons.
Avalanche sign Burakovsky to 1-year deal
Voynov returns to KHL on 1-year deal
Milano accepts qualifying offer from Blue Jackets
Best of the rest: Top available NHL free agents by position
Blues lock up Binnington with 2-year, $8.8M contract
Flyers sign Laughton to 2-year, $4.6M extension
Cozens agrees to entry-level contract
Red Wings sign Seider to three-year entry-level contract
Blues sign Fabbri to 1-year deal
Report: Blue Jackets spoke to Marner's agent about offer she...
Report: Stars' former top prospect Honka on trade block
An early look at the NHL's UFA class of 2020
Devils sign Hughes to entry-level contract
Hurricanes land Dzingel on 2-year, $6.75M deal
Boeser confident about striking deal with Canucks
Report: Ducks sign Del Zotto to 1-year deal worth $750K | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,647 |
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