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The Minister for Finance delivered the Budget 2019 proposals today 9th October 2018. The following is a synopsis of the main taxation changes.
Personal Tax:
a) Income Tax Credits & Bands:
Income tax Standard rate band increases by €750:
Single persons: Standard rate band increases from €34,550 to €35,300.
Married persons: Standard rate band increases from €43,550 to €44,300.
Home Carer Tax Credit:
Increase from €1,200 to €1,500.
Earned Income Credit:
The Earned Income Tax Credit of €1,150 increases to €1,350.
b) USC:
Reduction in USC rate of 0.25% on earnings between €19,874 and €70,044.
Earnings between €70,045 and €100,000 remain liable at 8% rate.
11% rate for self employed earning over €100,000 remains in place.
c) Mortgage Interest Relief/ Landlords:
Mortgage interest relief for Landlords will increase from 85% in 2018 to 100% as from 1st January 2019 on loans used to purchase, improve or repair a residential property.
d) BIK on Electric Cars:
The 0% Benefit- in- kind rate for electric vehicles is being extended for a period of 3 years with a cap of €50,000 on the Original Market value of the vehicle.
e) Key Employee Engagement Programme (KEEP).
Increase the ceiling on maximum annual market value of shares that may be awarded to equal the amount of the salary (up from 50%).
Replace the three –year limit with a lifetime limit.
An increase in the quantum of share options that can be granted under the scheme from €250,000 to €300,000.
Corporation Tax/Business Tax
- Film Relief which allows for corporation tax credits of 32% is extended to 2024 and; a new, short-term, tapered regional uplift commencing at 5% is also being introduced, subject to State aid approval, for productions being made in areas designated under the State aid regional guidelines.
- Accelerated Capital Allowances for Gas propelled Vehicles and Equipment: To encourage the use of natural gas and bio-gas.
- Accelerated Capital Allowances for Employer –Provided Fitness and Child care facilities Introduced in Finance Act 2017, the scheme is being amended and commences 1 January 2019. The scheme allows for the write off of capital costs over 8 years.
- Three Year Start Up Relief (Section 486C) which provides corporation tax relief for profit making start-up companies which create and maintain jobs is being extended a further three years, until the end of 2021.
- Exit Tax: A new measure will tax unrealised capital gains where companies migrate or transfer assets offshore such that they leave the scope of Irish tax. The rate for the new tax will be set at 12.5%.
- Controlled Foreign Company (CFC) Rules The Finance Bill will also provide for the introduction of a Controlled Foreign Company (CFC) regime as required by the Anti-Tax Avoidance Directive as laid down by the EU.
- Employer’s PRSI From 1 January 2019 the weekly income threshold for the higher rate of employer’s PRSI will increase from €376 to €386.
- Increase in employer contribution to National Training Fund levy From 1 January 2019 there will be a 0.1% increase (from 0.8% to 0.9%) and from 1 January 2020 there will be a further 0.1% increase (from 0.9% to 1.0%)
- Stock relief for farmers extended for 3 years until end 2021.The extension applies to three separate measures:
- The 25% General Stock Relief on Income Tax.
- The 50% Stock Relief on Income Tax for Registered Farm Partnerships.
- 100% Stock Relief on Income Tax for Certain Young trained Farmers (YTF)
VAT:
- VAT rate on e- books and electronically supplied newspapers is being reduced from 23% to 9% with effect from 01/01/2019.
- VAT rate on tourism activities to increase from 9% to 13.50% from 1st January 2019. Services and goods currently at 9% rate will increase to 13.50% with the exception of newspapers and sporting facilities.
Capital Acquisition Tax:
The tax free threshold on gifts and inheritances passing from parents to children is being increased from €310,000 to €320,000.
Stamp Duty:
Young Trained Farmers Stamp duty Relief (section 81AA SDCA 1999) is being extended for a further three years to 31/12/2021.
Excise Duties:
- Tobacco: 20 cigarettes increased by 50 cents (including VAT) with a pro-rata increase on the other tobacco products.
- Betting Duty: Current rate of 1% is being increased for all bookmakers and 15% rate is being increased to 25% on commission earned by betting intermediaries.
Vehicle Registration Tax ( VRT).
- A 1% VRT surcharge is being brought in for diesel engine passenger vehicles registered from 1st January 2019.
- The VRT relief available for conventional hybrids and plug –in electric hybrids is being extended for a period of one year, until end 2019.
Download a copy of our highlights here. | https://dmnaccountants.ie/latest-news/190-budget-2019 |
EMPOWER AND VENTUM DYNAMICS
Ventum has entered a partnership/collaboration with Empower to enable customers to obtain information about the sources of all materials used in our turbines and to hold ourselves accountable for material use and transparency.
Empower leverages blockchain technology to track all stages of a material’s life cycle in real time, from the initial cleaning process to the finished product.
Each stage is validated and stored in the immutable blockchain-based ledger of the material certificate.
By utilizing the Empower platform, Ventum is able to acquire complete, immutable tracking data following materials from diverse waste sources all the way to the completed product.
After the tracking of materials for one product is completed, Empower can issue a Product Passport for each individual turbine.
The Product Passport enables customers to view the complete story of their turbine by scanning a unique QR code associated to the product.
The Product Passport for the Ventum Turbine prototype will be available later in October.
We are thrilled to embark on this journey with Empower, a forward-thinking company with big aspirations. | https://www.ventumdynamics.com/empower-ing-ventum/ |
Digitalization enables the circular economy transition — it’s time to lean into it
By Peter Moore on August 27, 2021
This article originally appeared on Circulate News.
Steel is not infinitely recycled. For instance, much of the high-grade steels recovered from vehicles are downcycled into a lower-price material and mostly reused in lower-grade applications, such as buildings and infrastructures. Indeed, not aware of the quality of the steel they collect, recyclers mix steel grades, producing a lower-quality recycled material.
Steel recyclers are not alone having little information on materials or products they recover. Plastic recyclers are not always aware of the full chemical makeup of materials they process, including the presence of toxic substances. Repair technicians don’t always have access to disassembly guidelines for electrical equipment. Remanufacturers may not know the number of operating hours of an electric engine to decide if the engine can be restored “as good as new” or whether it should be recycled.
To maintain the value of a product in the economy for as long as possible, information on the product design, composition and condition is critical. With this information, an end-of-life product can be converted back into a valuable resource. In the transition to a circular economy, the equation is no longer “waste = resource” but “waste + information = resource” or, as stated by Idriss J. Aberkane, “waste + knowledge = asset.”
Leveraging digital technologies
Today, various digital technologies enable information to travel with a product. These technologies allow us to identify a product and capture, store, share and analyze data throughout its lifecycle.
To identify and track a product, two types of technologies exist: attached or embedded anchors. These anchors can be physical (such as fluorescent markers or watermarks), digital (such as Radio Frequency Identification or printed electronics), or biological (such as chemical tracers or DNA markers). For example, P&G is experimenting with digital watermarks, imperceptible code the size of postage stamps covering the surface of consumer goods packaging, that can be detected and decoded by a standard high-resolution camera on a waste sorting line. Another example is office furniture company Ahrend, which uses a QR code to identify each piece of furniture in its product as a service range.
In the transition to a circular economy, the equation is no longer ‘waste = resource’ but ‘waste + information = resource.’
Once a product is identified, data regarding its design, condition or location can be retrieved and information can be updated using various technologies. Data capture technologies include sensors or computer vision. For example, robots from ZenRobotics can sort post-consumer mixed material streams through computer vision techniques. Data transmission technologies include Wi-Fi, cellular network, Bluetooth and Low Power Wide Area Network to name a few. The construction equipment manufacturer Komatsu uses satellite communication to collect condition and location data of its machines. Data storage and sharing technologies range from cloud, digital platform, distributed ledger or big data. Thyssenkrupp collects and stores in the cloud operating data of 130,000 of its elevators worldwide to monitor their condition.
Finally, thanks to data analysis technologies, such as artificial intelligence (AI), the capture and exploitation of these often large amounts of data are made possible. For instance, technology company Optoro offers a solution using AI to help retailers and brands manage, process, and sell returns and excess inventory through the highest value channel. Tomra, a manufacturer of advanced collection and sorting machines, uses artificial intelligence to analyze images and data from cameras, near-infrared spectroscopy, X-rays and lasers, sorting waste according to their highest value and best use.
Many firms already leverage digital technologies to manage their products’ information. Still, this information is barely shared along value chains, and as a result, most stakeholders don’t have access to key product data. For example, to extend tire lifecycle, manufacturers equip them with sensors measuring pressure and temperature, but they don’t share information with collectors and processors that could help improve recycling or increase the uptake of recycled rubber in new tires. Thus, due to a lack of opportunity for material valorization, Europe exports more than 50 percent of end-of-life and second-hand tires. Information technology is a key enabler for the transition towards a circular economy, but data sharing across the value chain is also crucial.
Standardize and share data along the value chain
To exchange data along the value chain, stakeholders need to agree on a common language. For example, several businesses from the fashion industry, including H&M, Target and I:CO, have agreed to use a common protocol, Circularity ID, to share information on garment lifecycle. In France, GS1, an organization that develops standards, and Citeo, an Extended Producer Responsibility organization for packaging, worked with brands (French) to support information exchange related to packaging, such as consumer sorting instructions. Driven by the Luxembourg Ministry of the Economy and supported by major international industry companies, the “Circularity Dataset Standardization Initiative” aims at establishing an official standard for communicating data on the circular economy properties of products. In Germany, R-Cycle, a cross-industry consortium, is working on an open and globally applicable tracing standard to ensure the seamless documentation of recyclable packaging along the value chain, based on GS1 standards.
Product passports are a convenient solution to establish a common protocol and share information on the origin, durability, composition, reuse, repair, and dismantling possibilities, and end-of-life handling of a product. For example, shipping company Maersk Line has developed, in partnership with its suppliers, a “Cradle to Cradle Passport,” which lists and describes the materials used to build their vessels, their location and how they can be correctly disassembled and recycled or disposed of. Maersk Line estimates that the passport will increase end-of-life vessels value by 10 percent. The European Commission released a new legislative proposal stating that all batteries (portable, automotive, electric vehicle and industrial), with some exceptions, should have a battery passport. This would enable second life operators to take informed business decisions and allow recyclers to better plan their operations and improve their recycling efficiencies, because it would improve battery sorting, the health and safety conditions of operations, and even has the potential to increase the purity of the recyclable fraction.
If most passports are designed at product level, there are some examples of passports at material level. For example, global steel company SSAB offers a traceability tool called SmartSteel 1.0 that gives steel a digital identity. Customers can identify steel products by scanning its identifier, examine material properties and download certificates.
Once data has been standardized, stakeholders often choose to exchange it using digital platforms. For instance, producers of electrical and electronic equipment use an online platform, the Information for Recyclers Platform (I4R), to share information regarding their equipment with recyclers, and to comply with European legislation. Several large automotive companies have established the International Material Data System, a global data repository that contains information on materials used in the industry that facilitate the recycling of end-of-life vehicles and their materials.
Many digital platforms are also used to support the trade of secondary raw materials. These marketplaces allow secondary material suppliers and buyers to find each other on a web-based platform. They are supposed to create more market liquidity and to provide more supply and demand security for recyclers and their customers. Many marketplaces specialize in specific materials, such as plastics (Scrapo), textiles (Nona Source), construction materials (Backacia [French]), metals (Metalshub) or organic wastes (Organix [French]), while some others cover multiple materials (Recykal). At one point, the World Business Council for Sustainable Development identified more than 100 marketplaces, most operating at a municipal or regional level, but most are failing to attract a large number of suppliers or customers. Many made the faulty assumption that “if we build it, they will come,” but attracting a critical mass of both buyers and sellers can require significant investments.
Stakeholders could also be reluctant to share data on a platform that, most often, is managed by a private company. Distributed ledger, the parent technology of cryptocurrencies and blockchain, offers an alternative technology that could address this concern. A distributed ledger is a type of database that is shared, replicated and synchronized among the members of a decentralized network. For example, in France, organic waste producers, carriers and farmers share information on the organic value chain using a blockchain, ensuring the transparency of the organic waste value chain and, at the end, delivering traceability of the fertilizer produced.
Many companies from the chemical value chain are experimenting with blockchain technology to improve the traceability of chemicals in general, and plastics in particular, through their lifecycle. Mitsui Chemicals and IBM Japan are working on a resource circulation platform using blockchain technology that should ensure traceability throughout the resource life cycle, from raw materials such as monomers and polymers through to the manufacturing, sales, use and recycling of products. Borealis, Covestro and Domo Chemicals have launched a blockchain traceability project, together with Circularise and Porsche. Solvay has begun testing blockchain technology to trace its products throughout the entire value chain in partnership with Chemchain.
Blockchain can also be used to share data with consumers. Using blockchain technology provided by Provenance, Royal Auping has created a Product Passport that describes all the materials that go into their mattresses. The passport can be viewed by scanning an NFC (Near Field Communication) chip on the mattress label with a smartphone, allowing shoppers to view it in-store or at home.
Digital technology is an opportunity for the transition towards a circular economy, but it is not a blanket solution and it comes at a cost for the environment.
Ensure benefits are not offset by environmental costs
Digital technology is an opportunity for the transition towards a circular economy, but it is not a blanket solution and it comes at a cost for the environment. Indeed, digital technologies are responsible for resource depletion. Between 1995 and 2015, the material footprint of digital equipment has quadrupled. The extraction of the raw materials used in this equipment, such as precious metals or rare-earth elements, causes severe environmental damage, including land degradation, water scarcity and biodiversity loss. Most equipment generates a lot of waste. For example, the production of a 3 kg laptop generates 1,200 kg of waste. The short renewal cycle of these types of equipment also causes large amounts of waste. In addition, according to some estimates, digital technologies account for 3.7 percent of global greenhouse gas emissions. In France, digital technologies could be responsible for 7 percent of total greenhouse gas emissions in 2040 (French). And it is worth noting that, over the past 50 years, the development of digital technologies has coincided with the rise of CO2 emissions.
The belief that environmental costs of digital technology can be offset by the gains they could render is still to be confirmed. Therefore, before investing in new digital technologies, industries will have to validate that the negative impacts throughout the technologies’ lifecycles do not offset the expected gains.
Start by building a roadmap
The transition toward a circular economy is unavoidable. Digitalization of material and product flows is a major enabler of this transition. Therefore, businesses, industries and governments alike should start by building digital roadmaps to answer key questions such as: What are the information gaps that prevent the execution of circular strategies? Which stakeholders along the value chain could share these data? Which are the enabling digital solutions to close the information gap? Will environmental costs of these digital technologies be offset by the gains they would render possible? Which open data format should be adopted?
We will not maintain the value of materials and products in the economy for as long as possible until we know what they are. | |
The Arab spring was a series of protests that swept across the middle east and north African region (MENA), by which different governments where either taken down or went through major reforms. One of the core fundamental elements of many for the Arab spring, was social media. The impact social media had was immense as it allowed different groups to connect and plan their protests without much monitoring or awareness from the government officials at the time (Khondker, 2011). Social media provided a platform for many young Arabs to debate and discuss the different responses provided by presidents and governments in response to their protests (Bruns, Highfield and Burgess, 2013). The comebacks every government took used different political rhetoric by which to calm down the public or to address the requests made by protestors. In some instances, some of the rhetoric were focused on buying time for the current regime or government, with hopes of the activity slowing down.
At the heart of every protest, was a different social media platform that was used to plan, debate and coordinate every major standoff gathering. Platforms such as Facebook, were used to spread around videos of police brutality (case study in Tunis) and the different responses provided by governments. Twitter was used for a quick sharing of videos, pictures and spreading information using hashtags (Bruns, Highfield and Burgess, 2013). Users were fuelled by the injustice that was shared in the form of videos spread around through key activists. Most social media users are young Arabs who felt oppressed and had no other means of communicating their perspectives. Youth Engagement with politics peaked in the MENA region throughout the Arab spring due to how these events unfolded. Following the Arab spring, every middle eastern country went down a different pathway either new presidents, civil war, major political reforms and new governments (Howard et al., 2015). Some of these changes were short lived and followed by more protests such as in Egypt 2013. The purpose of this research project is to explore the extent by which political youth engagement continued following the Arab spring. The middle east has had a low engagement with parliamentary and presidential elections primarily due to corruption. Understanding the long-term impact social media had played on Arab youth, and whether it did increase their engagement with their national politics, would allow researchers to recognise how social media can be used as an integral tool for implementing efficient democracies in the Middle east.
Research Question
The primary question this research will aim to answer is the following:
· To what extent did Arab youth continue to use social media for political activism after the Arab spring?
The research question will be followed up by more clarification questions to try and further understand the topic at hand. The follow up questions are the following:
· Has the success/failure of the Arab spring played a role in further engaging or discouraging youth from political participation and activism?
· If there was a decrease in political engagement and activism through social media, why was that the case? (and vice versa)
· If Arab youth continued to engage through social media and political activism even increased, what was the effect and reason?
· How do local governments in the MENA region use social media and has this affected the engagement of Arab youth with regional and local politics?
Research Aim & Approach
The aim of this research is to understand how social media continued to play a role in political activism in the middle east following the Arab spring, using a post-positivism approach with a collection of qualitative methods. This approach will allow a holistic appraisal of the youth political engagement (which will be defined by the research) in different middle eastern countries. The key qualitative method will be an ethnographic study by which the purpose is for the researcher to immerse themselves within the social media environment in two different MENA countries affected by the Arab spring. This would be done by researching the usage of social media before the Arab spring through different platforms and comparing it to the activity after the Arab spring outcomes. To support this monitoring approach of the ethnographic study, 4 focus groups will be conducted to further dig into the mind of Arab youth and understand any development in their usage of social media for political debate, engagement, and activism. One justification for this approach is that a quantitative approach trying to capture many people, would not provide any useful answers to the research questions rather some statistical measures. While this could be integrated within the project, its important to narrow down the tools used to obtain a useful answer and a post-positivism approach that’s based on a all-inclusive ethnographic study, would offer more quality insight into the extent social media was being mobilized for politics by youth. To narrow down the research, two countries were chosen for this ethnographic study. First was Egypt, since the most social media fuelled protests in the Arab spring came from Egypt (Minatullah Sohail and Chebib, 2011). The second country is Tunisia, which was the birthplace of the Arab spring as we know it (El-Khawas, 2012). The age groups will be split into two, 18-23 and 24-29. The older age group would provide more valuable perspectives, as they would have been the same age as the first group during the Arab spring in their respective countries.
Methodology
Research Design
Theoretical framework
Ethical Considerations
References
Bruns, A., Highfield, T. and Burgess, J. (2013) ‘The Arab Spring and Social Media Audiences: English and Arabic Twitter Users and Their Networks’, American Behavioral Scientist. doi: 10.1177/0002764213479374.
El-Khawas, M. A. (2012) ‘Tunisia’s Jasmine Revolution: Causes and Impact’, Mediterranean Quarterly. doi: 10.1215/10474552-1895357.
Howard, P. N. et al. (2015) Opening Closed Regimes: What Was the Role of Social Media During the Arab Spring?, SSRN. doi: 10.2139/ssrn.2595096.
Khondker, H. H. (2011) ‘Role of the New Media in the Arab Spring’, Globalizations. doi: 10.1080/14747731.2011.621287.
Minatullah Sohail, R. and Chebib, N. (2011) ‘The Reasons Social Media Contributed to 2011 Egyptian Revolution’, International Journal of Business Research and Management. | https://www.whizwriters.com/political-science-92/ |
After identifying the “poorest of the poor” through extensive surveying, the government began providing them an opportunity to purchase up to 35 kilograms of rice and wheat at a highly subsidized cost of three rupees per kilogram of rice and two rupees per kilogram of wheat.
Poor families were identified by their respective state rural development facilities through use of surveys. The scheme has been expanded twice, once in June 2003 and then in August 2004, adding an additional 5,000,000 Below Poverty Line(BPL) families each time and bringing the total number of families covered up to 20,000,000.
Purpose of Antyodaya Anna Yojana:
· Giving food to the poor.
· Family of widows or sick person/ handicap/ persons 60 years of age or older who have no assured means of subsistence or social assistance.
· Providing wheat and rice to the poor as per family members.
· Under this scheme, grains of 3 rupees and 2 rupees per kg will be given.
· Providing 35 kg of rice/ wheat per month to the maximum of every family.
· These plans are for BPL only.
· To take advantage of this scheme, grains from Bio-Metric will be given.
Once a family has been recognized as eligible for the AAY, they are to be given a unique “Antyodaya Ration Card.” This card, also called the PDS yellow card, acts as a form of identification, proving that the bearer is authorized to receive the level of rations the card describes. The color of the card is green. | https://basicmahiti.com/govt-schemes/antyodaya-anna-yojanaaay/ |
FIELD OF THE INVENTION
BACKGROUND OF THE INVENTION
DETAILED DESCRIPTION
This invention is generally related to snow removal, and more particularly to a snow removal tool that requires less energy to move snow and facilitates loading a desired depth of snow.
It is well known that manual removal of snow from driveways, walkways, roofs and other places is labor intensive. Perhaps the most widely used tool for manual snow removal is the snow shovel. Use of a snow shovel requires a significant effort because the snow is lifted off of the ground and thrown from the shovel. As a result, shoveling snow can result in lower back strain. A number of more ergonomic variants on the basic snow shovel design are known. However, the basic technique of lifting and throwing the snow has not changed significantly. Various hand plows for pushing snow along the ground are also known. Further, various large scoops for dragging a load of snow along the ground are known. However, these devices do not easily permit one to load snow at less than the full depth on a given surface. Further, the relatively great amount of surface area in contact with the ground tends to increase friction.
FIGS. 1 through 4
100
102
104
108
illustrate a manual snow moving tool. The tool includes a curved scoop and tubular frame . The curvature of the scoop may be a section of a cylinder, elliptical cylinder or other shape. Further, the scoop may have a more complex shape including changes in concavity, e.g., as seen in one cycle of a wave of relatively longer period than amplitude. The scoop is constructed from one or more sections of sheet material such as metal, polycarbonate, fiberglass or other material. For example, 18 gauge sheet metal could be used for the entire scoop. Alternatively, the sheet metal may be attached to a lighter material, with the sheet metal functioning as the leading edge of the scoop to facilitate cutting into the snow. The frame includes tubular members attached to three sides of the scoop, i.e., the three sides other than the leading edge. The tubular members on the sides of the scoop may extend upward and curve back to form handles. The height of the handles can be made adjustable by using two sections of tubing of different diameter such that one section can be inserted into the other sections and fixed in position with a pin or bolt. A rope or cable connects the handles, and a lanyard may be attached to either a handle or the rope.
100
112
The invention has ergonomic advantages over prior art snow removal tools. For example, the curvature of the scoop provides improved control over the depth of cut into the snow relative to flat scoops. In order to adjust the depth of cut the user adjusts the height of the leading edge of the scoop by applying force against the handles to cause the tool to pivot where the scoop contacts the ground, or snow-covered ground. Further, because the pivot point is close to the midway point between the leading and trailing edges the scoop can be pivoted with relatively little effort, even when loaded with snow. The ability to select depth of cut and more easily pivot the load is particularly advantageous when moving deep or wet snow. Small wheels may be attached to the frame in order to facilitate movement across surfaces other than snow and ice.
104
200
200
The leading edge of the scoop which cuts into the snow as it is loaded into the scoop may include a removable edge . The removable edge may be attached to the scoop by means of bolts, screws, clips or other fasteners. Alternatively the scoop and edge may include interlocking features that permit the removable edge to be installed. Note that the edge may extend below the scoop in order to decrease abrasion of the scoop during use, e.g., due to contact with nail or screw heads. In this way the primary wear surfaces are replaceable, thereby extending the useful life of the tool. Various interchangeable edges might also be provided for different surfaces. For example, a plastic edge with a sloped lower surface that maintains separation between the leading edge and the surface from which snow is being removed may be favored for decks because the edge is less likely to catch on nail heads that might be encountered on a deck. Alternatively, a metals edge having little or no separation from the surface being cleaned may be preferable for driveways and walkways.
300
Side panels may be used to enhance load carrying ability. The side panels may be either permanently of temporarily affixed. For example, the side panels may attach to the frame with clips. The sides clip-on sides increase capacity and allow function like a wheelbarrow. The use of side panels generally increases the load carrying capacity of the device by providing lateral support for the load. When used in combination with the wheels described above the side panels help to provide wheel-barrow-like functionality on various surfaces.
100
The scoop is preferably deformable in response to application of unequal force via the handles, e.g., force in different directions. Because the scoop is constructed of sheet material, the scoop can typically be twisted in response to force applied via the handles. In order to avoid compromising such desirable characteristics with the frame, the tubular member attached to the trailing edge of the scoop may be attached to the other tubular frame members with hardware that permits the trailing edge member to freely twist relative to the other frame members. Alternatively, the trailing edge frame member may have a sufficiently low resistance to torsional force that it can be twisted with rigid attachments to the other frame members, and sufficient memory to return to a resting position when the torsional force is removed. Twisting the scoop helps to accommodate an unbalanced load of snow and also to more efficiently remove snow from an uneven surface.
108
402
The safety rope or cable provides enhanced safety when removing snow from an inclined surface such as a roof. For removing snow from a roof the tool may be used, starting at the ridge, to scoop and push snow toward the bottom edge. The curved scoop and balanced pivot point also facilitate dumping the snow over the edge of the roof. Being tied off on the roof, the user simply stops before the roof edge and allows one hand or the other to take the cable and give it a slight tug if necessary to unload the snow quickly. The safety cable helps to prevent loss of the tool over a roofs edge, and also facilitates hands-free return of the tool to the roof ridge. An optional tow bar may also be included.
400
Non-marring surfaces may be attached to the frame. The non-marring surfaces may be constructed from high density plastic. The material should preferable resist cracking at the relatively low temperatures to which a snow removal tool might be subjected. When secured to the lower side of the frame, the non-marring surfaces help reduce the likelihood of scratching the surface from which snow is being removed. Further, the non-marring surfaces may reduce friction between the tool and the surface from which snow is being removed, thereby facilitating movement of the tool. It should also be noted that the non-marring surfaces protect fasteners which may be used to secure the scoop to the frame. For example, countersink holes may be formed in the plastic surface so that the head of the bolt, screw or rivet secures the plastic surface, frame and scoop together.
While the invention is described through the above exemplary embodiments, it will be understood by those of ordinary skill in the art that modifications and variations may be made without departing from the inventive concepts. Further, while the preferred embodiments are described in connection with various illustrative structures, one skilled in the art will recognize that the system may be embodied in connection with other structures. Accordingly, the invention should not be viewed as limited except by the scope and spirit of the appended claims.
BRIEF DESCRIPTION OF THE FIGURES
FIG. 1
is an isometric view of an improved snow removal tool characterized by a curved scoop.
FIG. 2
illustrates a replaceable cutting edge of the snow removal tool.
FIG. 3
is a side view of the snow removal tool.
FIG. 4
is a plan view of the snow removal tool. | |
Contact Information * Notices * Volunteer/Donate * Membership FormThe Florence Warren Latham Museum of Eastford History, 65 John Perry Road will be opened the 2nd and 4th Sunday of each month from 2:00 p.m. until 4:00 p.m., May - October.
Preserving Eastford's Old Cemetery
Click on link to view article.
Eastford State Representatives and Senators from 1849 to 1989
Click on link to view list.
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EASTFORD’S SHARED HISTORY ALBUM
The Eastford Historical Society invites you to join its collaborative Google Photo Album named “Eastford’s Shared History Album”. By using the link below you may join this album, view and comment on photos, and upload photos of your own.
https://goo.gl/photos/pCYo5c8JL5tCvjU6A
This project supports our mission to preserve and share Eastford’s unique history.
If you choose to collaborate on this album by contributing photos, you are agreeing that anyone with the link to it may download and make unrestricted use of your images.
We encourage everyone to share interesting photos that help to tell the story of Eastford’s past. If you ‘add photos’ please tell us all that you know about them in the information section which can be opened by clicking on the ‘i’ symbol. By choosing the ‘pencil’ symbol you can access a tool to edit your photo. There is a ‘comment’ section below each photo where you can add to the story of photos contributed by others.
Anyone can view this album and download photos, but you will need a Google Account (free) in order to join and collaborate by uploading your photos and commenting on photos posted by others.
We welcome your participation as a contributor or simply as a viewer, and hope you enjoy Eastford’s Shared History Album. Please feel free to “share” this invitation and link with others who might find this album interesting.
If you would like us to add your photos to this album; have photos or documents that you would allow us to copy/scan for archival purposes; or have photos, documents, or objects you would consider donating – please contact us.
Thank you for participation in this effort to preserve and share Eastford’s history!
The Eastford Historical Society, Inc. [email protected]
Linda Torgeson (860) 933-0468
Tom Latham (860) 974-1610
Memories of Eastford Past
Click VIEW ALL to Explore Eastford's Fascinating History. Enjoy the Trip! | https://www.eastfordct.org/domain/1137 |
Shrek’s meat cutlets. Sorry, kids… Well… It’s not Shrek’s meat, but these cutlets will make you full for sure! I couldn’t more than one. Cheers!
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INGREDIENTS
600 g (21.2 oz) minced chicken
1 egg
1 tbsp. potato starch (if needed)
Vegetable mix
200 g (7.1 oz) broccoli
100 g (3.5 oz) pumpkin
100 g (3.5 oz) cashew (or different nuts)
4 garlic cloves
1 tsp. marjoram
1 tsp. lovage
1 tsp. black ground pepper
0.5 tsp. salt
A pinch nutmeg
A small piece chili pepper
DIRECTIONS
1. Blend all vegetable mix ingredients together.
2. Mix vegetable mix with minced chicken and an egg. Add a little bit of potato starch if the mixture is too watery.
3. Put a piece of cooking paper to a baking try and pour some oil on the paper. Form flat meat pieces and put them on the cooking paper.
4. Put the cutlets to the preheated oven at 200 °C (390 °F) with heat coming from the bottom and top program (or similar) for about 40-45 minutes or until the cutlets are done.
5. Serve it with whatever you like.
Cheers!
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Daržovių vištienos kotletai
Mūsų knygos Tapkite mūsų rėmėju
INGREDIENTAI
600 g (21,2 uncijos) maltos vištienos
1 kiaušinis
1 v. š. bulvių krakmolo (jei reikia)
Daržovių mišiniui
200 g (7,1 uncijos) brokolių
100 g (3,5 uncijos) moliūgo
100 g (3,5 uncijos) anakardžių (arba kitokių riešutų)
4 česnako skiltelės
1 a. š. mairūno
1 a. š. gelsvės
1 a. š. juodųjų maltų pipirų
0,5 a. š. druskos
Žiupsnelis muskato riešuto
Nedidelis gabaliukas aštraus pipiro
PARUOŠIMAS
1. Sumaišykite visus daržovių mišinio ingredientus.
2. Sumaišykite daržovių mišinį su malta vištiena ir kiaušiniu. Įdėkite truputį bulvių krakmolo, jei mišinys yra per skystas.
3. Įdėkite kepimo popieriaus į kepimo skardą ir užpilkite šiek tiek aliejaus ant popieriaus. Formuokite plokščius mėsos kotletus ir sudėkite juos ant kepimo popieriaus.
4. Įdėkite kotletus į iš anksto iki 200 °C (390 °F) temperatūros įkaitintą orkaitę su kaitinimu iš apačios ir viršaus programa (arba panašia) maždaug 40-45 minutėms, arba, kol kotletai iškeps.
Skanaus! | https://gauchas.online/vegetable-chicken-cutlets/ |
David Blaine: Street Magic1997 - 45 minutes Movie Renowned magician David Blaine performs illusions and tricks up close in street performances that awe and amaze his in-person, on-the-ground audience.
Cast
Production countries US
Popular movies
Popular series
Want a weekly summary of the best additions on Netflix?
Sign up for our weekly newsletter containing the best newly added movies and series on Netflix in South Africa. We won't share your email with anyone. | https://za.flixboss.com/movie/david-blaine-street-magic-80126565 |
These images are unavailable to subscribers, but may be purchased.
Images of folk dancers and other performers in period costume at Howard Davis Park.
Negatives have been scanned at 1,200 dpi and can be downloaded as high quality jpg files. The quality of the scanned image is dependent on the condition of the original.
|Reference|
L/A/75/A3/1/915
|Date|
May 1st 1976 - May 1st 1976
|Names|
|Keywords|
Howard Davis Park | shows | events | Dancing | singing | performances
|Dimensions|
12 35mm negative
|Language|
English
|Level of description|
file
|Creator|
|Access Restriction|
Closed for preservation. Please view digital image.
|Closed until|
2100
|JEP Photographic Job Number||1976/915|
Context: | https://catalogue.jerseyheritage.org/collection/Details/archive/110314108 |
Tonight I would like to introduce fantastic plant - Habenaria medusa! I know, I should keep my emotions low, but I can’t - WOW! That’s how I will describe new candidate.
Habenaria medusa Kraenzl., Bot. Jahrb. Syst. 16: 203 (1892).
Habenaria medusa is a beautiful species native to Southeast Asia (Java, Sumatra, Sulawesi Borneo, and is occasionally also found in Laos and Vietnam). This is a plant that goes dormant in the fall and winter.
There are 27 AOS awards including 4 CCE.
First award - CHM, in August 2006, latest - CCE, in October 2019.
Largest number of flowers and buds in OrchidPro has Habenaria medusa 'Marissa Gittelman' CCE/AOS 91 pts: 195 flowers, 150 buds on 11 inflorescences.
Information from exhibitor:
‘Here is a Habenaria medusa, for cultural award consideration. Growing this plant about 10 years.
Plant - 77cm diameter foliage, 86 cm diameter around flowers
Grown in 37cm tall plastic pot 104cm tall.
16 inflorescenses - at time of photo estimated 224 buds, and 448 flowers, for 672 total (buds still tucked into bracts were not counted)
Verified after all buds had opened - exactly 690 flowers, Max of 48 flowers on two largest inflorescenses.
The longest first flower to tip of infl. is 45cm. the shortest is 34cm (The length of the growths from bottom to top of inflorescence is the only way to accurately measure these - there is no distinct line to distinguish the basal growth from the inflorescence).
# infl. - flower count: 2 - 48; 1 - 47; 1 - 46; 2 - 45; 1 -44; 1 - 43; 2 - 42; 3 - 41; 2 - 40; 1 - 27.
No. Flowers - 448;
No. Buds - 242;
Flower Measurements:
NS H - 8.0 cm; NS V - N/M;
Dorsal Sep. W - 0.8 cm; Dorsal Sep. L - 0.9 cm;
Petal W - N/M; Petals L - N/M;
Lat/Sepal W - N/M; Lat/Sepal L - N/M;
Lip W - N/M; Lip L - N/M.
No. of Inflor. - 16;
Inflor. Length - 45 cm;
Plant Height - 104 cm;
Plant Width - 77 cm.
Inflorescence - Measured from first flower to tip of infl.’
Judges' Comments
Joanna Eckstrom
Some plants simply take one's breath away - this is clearly one of them. Plant appears to be very well-grown, is well-groomed with immaculate foliage, and very evenly flowered. I would not hesitate to nominate for cultural award and would likely score mid-CCE.
Thank you for inviting everyone's input.
Best regards,
Joanna K Eckstrom
Al Messina
This is a plant that warms the heart of everyone: Perfectly grown and groomed. The grower should get a special award for keeping it alive for ten years--a major accomplishment.
As students, we learn that all plants/flowers are scored against a theoretical model of perfection. This plant gets as close as possible ,especially when one considers the longevity and difficulty maintaining the required conditions for growth/regrowth over a long period of time. All conditions for a high CCE appear to have been achieved, most beautifully the arrangement and flower count better than any prior award.
As often as I have asked the presenter of this series to refrain from editorializing, I think we all can agree that this is a special circumstance and forgive his obvious excitement. In my opinion, this plant would be a candidate for the Butterworth.
Thanks for allowing me to participate.
Tom Mirenda
Breathtaking plant! Perfect culture, deserving of a CCE. Perfect even growth, perfect foliage…
little evidence of trimming, and shown at the perfect stage of anthesis
Awesome flower count, all flowers appear to be in excellent shape down to the tips of the spectacular fringes.
(I have seen a larger plant than this in Bloom at the Eric Young Orchid Foundation attaching photo)
….but as spectacular as it was the presentation was not as perfect or uniform as this one.
There were faded flowers….and there were brown tips on its foliage.
The grower deserves extra kudos for the impeccable and almost imperceptible staking.
I am particularly struck by the rhythmic leaflets on the inflorescence which I’d never noticed as a feature of the plant before.
I think a CCE of no less than 95 points is warranted.
It is a tragedy that this masterpiece cannot be actually judged this year.
Deb Bodei
Well, this one was easy. I am handing this in now so you don't have extra to consolidate at the end of the week with the virtual judging demo to plan for.
Habenaria medusa
Observations
Not much for us to research: ). This plant is as pristine as you can get and extremely well-flowered. Even though video and photo indicate it is not bloomed exactly even around the entire circumference, it presents itself as if it is by nature of the blooms and inflorescences. This plant is truly stunning from all angles from what we can observe.
Recommendations
I would nominate this plant for a cultural award and enjoy being part of a team that judged s high-scoring CCE if the photos are representative. The only minor deduction in points I can see is in the bloom condition, specifically the lip. The fimbriation of the lip margin is not entirely uniform and has a somewhat muddy coloring basally. However, these minor bloom conditions do not detract much from the overall breathtaking appearance.
Thank you for including my commentary,
Deb
Ed Weber
THIS plant is what judging is all about! It is successes like this that make the AOS awards system so important... recognizing the outstanding results of the grower's hard work.
This specimen is superior in all ways to all of the CCE entries in OrchidPro. In addition, it is extremely clean and well groomed and the staking is both effective and unobtrusive. I actually didn't notice the staking until my third view of the video clip and I had to actually look for it.
I would have absolutely NO PROBLEM nominating this for a CCE and I would expect to score in the mid to high 90's.
Congratulations to the grower and thank you for letting us see this magnificent display!
Ed Weber
Mark Werther
Habenaria medusa
One of the most spectacular presentations that I have ever seen in 60 years of observation and growing. The display of flowers and condition of plant is close to perfect. Far outshows previous high AM's and 2019 CCE. My general reaction and description is "breathtaking".
There might be a few trimmed leaves and I am leaving room for more inflorescences next flowering with a 96 point evaluation. Congratulations to the grower for superlative cultural mastery.
Mark Werther
Trevor Yee
Even before viewing the comparative AOS cultural awards - my immediate reaction was that this specimen was extremely well grown, flowered and presented. It certainly had the genuine WOW factor!
All the inflorescences are of even height, each at least 2/3 open and presented in a nice circular arrangement.
The highest cultural AOS award was CCE 94 points for 'Cat's Whiskers'. Our candidate is far superior for quality, presentation and floriferousness.
I have no hesitation in recommending a CCE of at least 95 ~ 97 points.
To compare, the AOC criteria for Awards of Excellence of Plant Culture are:
ACC - Award of Cultural Commendation 75 to 79 pts
ACM - Award of Cultural Merit 80 to 84 pts
ACE - Award of Cultural Excellence 85pt +
The elements for consideration are:
SIZE: The plants should be of a significant superior size to that reasonably expected of a normal robust and mature example of its kind.
CONDITION: The plant should have clean, appropriately mature foliage, and be well presented and free from any significant mechanical damage or blemish that detracts from the overall effect. The quality of the foliage should be in proportion to that reasonably expected on a plant of the kind and size.
FLORIFEROUSNESS: The quantity of flowers should be unusually large in number and be in proportion to the size of the plant, having regard to any growths or pseudobulbs capable of blooming at the time it is submitted for the award.
John Sullivan
What a fantastic specimen! The previous CCE of 94 points was awarded to the grower of a plant which is dramatically exceeded by this plant using any number of comparisons. The candidate has 3 times as many flowers of a similar size. If we calculate the number of flowers plus buds per inflorescence on the previously awarded one we get 21.3. The candidate sports a whopping 48! In addition, the plant appears to be superbly grown with immaculate foliage.
The only question in my mind is how far to raise the bar. I am not sure how much more of an increase in bloom is possible for this species. If I had the privilege of judging this plant in person, I think my score would be 98 points.
Thanks for sharing this exercise with us and kudos to the grower for an amazing accomplishment!
Elena Skoropad
WOW!!!!
What a wonderful plant!
This is the easiest plant to judge so far!
It Is more superior and floriferous that all previous CCEs and all previously awarded habenaria medisa. The flower count is unbelievable, the presentation is great; plant condition is immaculate. I would love to nominate this plant for culture award and I would be scoring in CCE 97 points. I wouldn’t mind to go even higher than that.
Kudos to the grower! Only one person can grow habenaria like that on East Cost 👍
Excellent job!
Thank you,
Elena
Carrie Buchman
Habenaria medusa
This is a beautifully grown specimen plant deserving of a cultural award. With ~43 flowers & buds per inflorescence, its floriferousness exceeds that of previously CCE awarded plants. Foliage is clean. The plant is well presented with good spacing between the flowers, creating an ascetically appealing display. I would not nominate it for a flower quality award as the flower size is smaller/on par with than many previous awards, and the basal lip orange, at least in the photos provided, is not as intense as the photos in OP. Kudos to the grower.
Kind Regards, | https://www.virtualorchids.org/archive/week-28-habenaria-medusa |
The next time you jet off on a cross-country vacation flight or a short-hop work trip, say a little thank you to NASA.
The agency, now celebrating its 60th anniversary, is best known for taking humans into space, but it's long been at the forefront of aviation as well. Every flight you take today is made possible through technologies NASA has developed, from the shape of your airplane's wing to the air traffic control systems that guide your flight through the air. You may not be flying to the moon, but you are riding on NASA's decades of aviation research.
The goal of NASA's Aeronautics Research Mission Directorate: Make air travel better for everyone, not just military fighter pilots or astronauts in training.
"[Our] ambition is to help the US aerospace industry be at the leading edge of the global market," says Peter Coen, project manager for NASA's Commercial Supersonic Technology Project. "We're putting the technology out there that essentially the aerospace business can use to make successful products."
To get there, NASA conducts a huge range of research projects for the next generation of aircraft. Some of the models that have been in its wind tunnels don't look like anything carrying passengers today -- say, a blended-wing aircraft that combines the wing and fuselage into a single bulbous structure that looks like something you'd see sneaking a flight out of Area 51. But the agency isn't confining itself to daring experimental designs or Jetsons-like dreams of personal airplanes for crowded cities. It's also working on perfecting existing aircraft to make them safer, cleaner and much, much faster.
Before NASA and after
NASA's history with flight goes back more than a century, starting with the National Advisory Committee for Aeronautics, founded in 1915. At what's now NASA's Langley Research Center, the NACA experimented with wing shapes, built early wind tunnels and helped develop the X-1, the first airplane to break the sound barrier.
When NASA was founded on Oct. 1, 1958, absorbing the NACA's aviation research, the jet age and the era of mass air travel were just taking off. One of its first flight projects was the rocket-powered X-15 research aircraft, part of a long series of experimental X-planes that continues to this day. The X-15 set altitude and speed records, and data from the program was invaluable in helping astronauts handle spaceflight.
Photos: Looking back at NASA's X-15 aircraftSee all photos
Later developments have trickled into every aspect of aviation today. Fuselages made of composite materials make airliners like the Boeing 787 and the Airbus A350 stronger and lighter, winglets save fuel by helping airplanes fly more efficiently, and synthetic vision systems let pilots fly more safely by giving them increased information about the environment around them.
NASA isn't stopping there. Here are three programs it's bringing to the skies.
Supersonic flight
In the nearly three decades that it carried passengers, the Concorde, the world's only supersonic airliner to regularly carry them, delivered the fastest and most exclusive ride in the sky. But its engines were deafening on takeoff, it spewed emissions at high altitudes (about 56,000 feet), and it was massively expensive to operate. Regulations that limited or banned sonic booms over land also meant that it had few places to fly.
The Concorde was permanently grounded in 2003, but NASA is now working to help travelers like you break the sound barrier again. By softening the effects of a sonic boom, which can annoy people, disturb wildlife and damage structures, it hopes to persuade governments that faster-than-sound flight over land isn't a bad thing.
"We want to get to the point where we essentially have supersonic airliners that could fly any route while being efficient and otherwise an environmental good neighbor," Coen says.
In April, NASA announced a partnership with Lockheed Martin to build a test airplane with "low boom" technology, scheduled for a first flight in 2021. Both say the Low-Boom Flight Demonstrator (also known as the LBFD or the X-59 QueSST) will produce a sonic boom that sounds more like a car door slamming than the thunderous noise Concorde produced.
Key to dampening the boom is the LBFD's shape. A long, pointed nose, sharply swept wings and raked canards (small wings positioned forward of the main wings) ensure that the individual shock waves the airplane produces at speeds faster than Mach 1 never converge and cause a traditional sonic boom.
"We've found the best way to reduce [a sonic boom] is to control the strength and position of each wave so that they're relatively similar in strength, and relatively evenly distributed along the length of the airplane" Coen says. "If you do that, most of them don't converge."
Until the LBFD flies, NASA will conduct tests using one of its F/A-18 aircraft to see if a quieter boom is possible. During the tests, which begin in November, the F/A-18 will dive from almost 50,000 feet over the Gulf of Mexico off Galveston, Texas, and go briefly supersonic before leveling off at about 30,000 feet. The sonic boom produced from the dive should sound more like the LBFD's predicted boom (NASA calls it a "sonic thump").
After initial test flights to assess the LBFD's general performance as an airplane, Lockheed and NASA will start flying it supersonically in 2022 over populated areas from California's Edwards Air Force Base. These flights will have a twofold mission: Make sure the LBFD has a low boom and gauge public response to it. Coen says the hope is that residents won't mind the noise overhead.
"We really want to get as broad a response as we can," he says. "We want to understand what the effects of repeated exposures are as well as individual exposures."
But if even the flights are successful, there's still one big hurdle to ordinary people sipping supersonic champagne again. Since 1973, the FAA has prohibited commercial supersonic flight over the United States. Overturning that ban, and similar ones in other countries, is critical before supersonic flight can be economically viable.
"The aviation business is so global these days that any product built in any country is going to be certified in other countries in order to make it successful," Coen says. "So [to go supersonic] you have to keep quiet."
Quieter aircraft
Noise concerns don't afflict just supersonic aircraft. Sit under the final approach to the runway at any airport, and the whine of descending flights is unmistakable, sometimes painfully so.
Though jet engines rightly get most of the blame for aircraft noise, Mehdi Khorrami, an aerospace scientist at Langley, says an aircraft's landing gear can be an even bigger factor. As an airplane comes in to land, air flowing over its extended landing gear causes turbulence, which translates to noise.
"The engine [sound] is still significant, but the airframe is really a prominent portion of the aircraft noise," Khorrami says. "NASA has this long-term goal ... to confine that objectionable part of the aircraft noise to within airport boundaries."
One idea NASA's developed is to put noise-absorbing foam into the well in the fuselage where the gear retracts during flight. It's also developed fairings -- plates that cover landing gear when extended -- with tiny holes that allow air to pass through and reduce noise.
Then there are the surfaces on the wing that slow an aircraft during descent. When flaps are lowered on most existing aircraft, noise comes from air flowing in the gap between the flap and the wing's back end. But with NASA's experimental Adaptive Compliant Trailing Edge flap, there's a seamless transition between it and the wing.
"These concepts will really reduce significantly the noise that is produced," Khorrami says. "So when an an airplane lands, it would be no louder than the surrounding background noise. That's a tall order."
X-Planes are the supersonic daredevils of aviation legendSee all photos
When developing noise-cutting technology, Khorrami starts by conducting computer simulations. That's followed by deploying the technology on models in wind tunnel tests and then on a real aircraft as part of NASA's Acoustic Research Measurement flights based at Edwards. As with the LBFD, microphones on the ground measure the noise level of the test Gulfstream III.
So far, Khorrami says, the landing gear and flap changes have reduced noise on the test aircraft by 70 percent. But his team also has to make sure its tech won't hurt an aircraft's flight performance and will allow air to cool its brakes when it's on the ground.
"It has to be the right balance between significantly reducing the noise and not impacting [a plane's] operation," he says.
Electric aircraft
Even as it works on a quieter aircraft design, NASA isn't ignoring engines. An electric-powered aircraft in development would be cleaner, quieter and more efficient.
Though the notion of an electric airplane may worry you -- maximizing your battery's range is a more pressing issue 35,000 feet in the air than on the freeway -- it's hardly a pie-in-the-sky idea. Besides NASA, others are pursuing the goal of an electric plane, including European low-cost airline Easyjet, which plans to fly one in 2021.
For its part, NASA is developing an all-electric test aircraft called the X-57 Maxwell using a modified Tecnam P2006T aircraft. Though much of the basic design is unchanged, NASA is replacing its gas-powered propellers with electric engines. Using an existing aircraft has a couple of advantages: NASA doesn't have to design an entirely new plane, and it can compare how the electric-powered X-57 performs with how the airplane flew on its original engines.
NASA's wildest rides: Extreme vehicles for Earth and beyondSee all photos
The main goals of the project are to see a 500 percent increase in flight efficiency over a standard P2600T and to set standards for electric propulsion. Other benefits would include a reduction in carbon emissions and engine noise and lower operating and maintenance costs. In other words, they're pretty much the same reasons you'd buy an electric car.
Though the project is aimed at private planes and air taxis for now (current battery technology has a range of around 50 miles), the technology could one day result in a turboelectric passenger airliner. Tom Rigney, project manager for X-57, says lower operating costs for such an aircraft may translate to lower airfares for passengers.
"Electric aircraft are more efficient, quieter and friendlier to the environment than standard fuel-powered aircraft," Rigney said in an email. "Electric propulsion technologies may also allow smaller aircraft to take off and land vertically on roof tops or from parking lots, making them more accessible for applications such as air taxies and commuter aircraft."
The tests will exist in four stages, starting from ground tests of the electric engines and ending with a fully modified X-57 taking to the air. That plane will have a long, experimental wing to give it more lift, two electric motors on the wingtips for propulsion and 12 smaller electric motors to give it more speed for takeoff. NASA also is training pilots in simulators to fly an electric plane and building an 860-pound lithium-ion battery to power the motors.
Sean Clarke, principal investigator for X-57, says some of the battery technology comes from designs NASA developed for the International Space Station.
"This helps ensure that failures in the battery system are contained and won't impose an increased safety risk to the aircraft," Clarke said in an email. "The X-57 Maxwell aircraft uses electric motors that are customized for the power and speed needed for this aircraft, but does not require development of new technologies."
Taking It to Extremes: Mix insane situations -- erupting volcanoes, nuclear meltdowns, 30-foot waves -- with everyday tech. Here's what happens.
Fight the Power: Take a look at who's transforming the way we think about energy. | https://www.cnet.com/news/how-nasa-makes-your-airplane-flights-better-than-ever/?ftag=CAD1acfa04&bhid=27022314629675296327185729131313 |
There are lots of ways to prepare and cook mushrooms. Sauteed mushrooms served up with rich garlic butter are hard to beat, but this is just one of a number of ways that you can prepare these nutrient-rich vegetables. With this in mind, can you steam mushrooms?
To steam mushrooms, pop mushrooms into a strainer or colander over a pot of simmering water for around 10 to 15 minutes until just cooked through.
How to Steam Mushrooms
Of all the methods of preparing mushrooms, steaming them is perhaps the healthiest. Because you don’t add any additional ingredients, you can enjoy the full nutrients of the mushrooms as intended without adding further calories.
Below are the steps you need to follow to steam mushrooms successfully:
- Add Water to a Pan and Heat
Half-fill an appropriately sized pan with cold water and bring it to a boil on the stovetop.
- Prepare Mushrooms
Wash your mushrooms thoroughly and slice them to your preference if you desire. Then leave to one side until the water is boiled.
- Reduce to a Simmer and Add Strainer
When the water is boiling, reduce the heat and maintain the water at a simmer. Then, add a strainer to the top of the pan, ensuring the bottom remains above the waterline.
- Add your Mushrooms
Place the mushrooms in the strainer and then cover with a sheet of aluminium foil. Leave them to steam for between ten and fifteen minutes, depending on the size of the mushrooms.
- Serve and Enjoy
Add seasoning and any other herbs or spices to your mushrooms before serving.
Yes, it is possible to steam mushrooms using the microwave. Pop 120g of sliced mushrooms into a suitable container and microwave for 8 minutes, uncovered until cooked through.
Yes, mushrooms can be steamed in a steamer. They will take between 10 and 15 minutes to steam depending on how much you have cut them up.
How to Steam Mushrooms Successfully
Here are our top 3 must-read tips for when steaming mushrooms:
- Add Some Flavour – While steaming mushrooms is healthy, they will taste a little bland if you don’t add any flavour. Consider adding herbs like rosemary or thyme to the water as it’s boiling, or you could even opt to steam your mushrooms over stock instead of water.
- Use a Rice Cooker – If you have a rice cooker in your home, use it to steam your mushrooms. This saves a lot of time and hassle and will steam them perfectly.
- Serve Over Garlic Butter – Once you’ve steamed your mushrooms, serve them over a knob of garlic butter for an incredible side dish. The butter will melt as you place the mushrooms in the dish, so give them a good stir, so it covers all of the mushrooms equally.
Why You Should Steam Mushrooms?
Given the various ways that you can cook mushrooms, here is why you should steam them:
Healthy
Steaming mushrooms is perhaps the healthiest way to prepare them. They retain all of their nutrients, and you don’t need to add any additional calories or ingredients if you don’t want to.
Easy to Prepare
There’s nothing complicated about steaming mushrooms, as you can see from the steps listed above. Once your water is ready, you need to cover them well, and they should be ready in around ten to fifteen minutes.
Why You Shouldn’t Steam Mushrooms?
In spite of the benefits above, steaming mushrooms isn’t necessarily the best way to prepare them, and here’s why:
They Taste Bland
Steaming or boiling mushrooms doesn’t add anything to their taste. Unless you season them with herbs or steam them over stock, your mushrooms will be relatively tasteless once they’ve been steamed.
They Will Retain Lots of Water
Steaming mushrooms makes them moist and juicy, which is ideal for some people. Others, however, prefer mushrooms to be firm and crispy. Sautéing or frying mushrooms are better options if you don’t want them to taste watery.
Is It Safe to Steam Mushrooms?
It’s safe to eat mushrooms raw if you choose. As such, you don’t need to worry too much about undercooking them, at least from a health perspective.
One of the most important things to check is that your mushrooms have been cleaned thoroughly before you prepare them, as dirt presents the most significant danger to your health.
Do Mushrooms Steam Well?
Steaming mushrooms is an excellent way to enjoy them healthily. They will retain lots of their nutrients and will be juicy and moist, which is a desirable texture for many people.
The main complaint people have when steaming mushrooms is that they can taste a little bland, but you can rectify this by adding seasoning, herbs, and even butter once they’ve been prepared, which offers an improvement in their taste.
Steaming Mushrooms FAQs
Below are a few other questions that get asked when discussing whether you can steam mushrooms or not:
Although all mushrooms can be steamed, we would recommend sticking to cheaper varieties such as button mushrooms. Steaming mushrooms doesn’t show them off in their best light. If you’ve got pricier mushrooms such as chanterelles then you are better off pan-frying in plenty of butter.
Yes, steamed mushrooms are healthy because there is no oil or butter at use during the cooking process. It’s just mushrooms and water.
Yes, enoki mushrooms can be steamed. They actually cook really well this way. Steam them over water for about 10 minutes before dressing with soy sauce, sesame oil and some grated garlic.
Ross is a freelance writer with a passion for delicious food and drinks, having worked in the hospitality industry in the past. He began working with Let’s Foodie when it first launched to provide information on how to reheat, freeze and microwave different foods. | https://letsfoodie.com/how-to-steam-mushrooms/ |
This book provides a practical introduction to fluid simulation for graphics. animation, The Visual Computer: International Journal of Computer Graphics, . J. A. Bærentzen, R. Bridson, Multiphase flow of immiscible fluids on unstructured . Fluid simulation for computer graphics is a special part of Computational Fluid using pbrt and Robert Bridson’s grid-based implicit surface shape plugin for pbrt. where ̄X is a weighted average of nearby particle locations and ̄r is a. grid based and a particle based fluid simulator for computer graph- ics applications. Furthermore, Bridson’s text does not cover particle based methods, like SPH Here r is a the distance between two particles, h is our smoothing width.
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Already read this title? Covers the latest algorithms and techniques, including: Roberts-SC phocomelia syndrome Mathematics. A practical introduction, the second edition of Fluid Simulation for Computer Graphics shows you how to animate fully three-dimensional incompressible flow. Showing of 19 references. It covers all the aspects of fluid simulation, from the mathematics and algorithms to implementation, while making revisions and updates to reflect changes in the field since the first edition.
A positive-definite cut-cell method for strong two-way coupling between fluids and deformable bodies Omar ZarifiChristopher Batty Symposium on Computer Animation What are VitalSource eBooks? Computer Graphics and Geometric Modelling: The title will be removed fo your cart because it is not available in this region.
Fluid Simulation for Computer Graphics – CRC Press Book
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Melting and flowing Mark CarlsonPeter J.
Fluid Simulation for Computer Graphics
All instructor fliid are now available on our Instructor Hub. Citation Statistics Citations 0 20 40 ’10 ’12 ’14 ’16 ‘ BargteilJames F. Simulation Computer graphics Fluid animation.
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Bridson Robert. Fluid Simulation for Computer Graphics [PDF] – Все для студента
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The book changes the order of topics as they appeared in the first edition to make more sense when reading the first time through. The country you have selected will result in the following: Learn More about VitalSource Bookshelf. | https://coldplayer.info/bridson-r-fluid-simulation-for-computer-graphics-89/ |
Moses and the Poultry Bandits
Moses was an old Pyrenees. As a pup he freely roamed the fields around his farm, warding off coon, coyote and hawks from his beloved sheep and poultry. As he grew older so did his neighborhood. A housing development sprang up, covering much of the area where he used to roam. He was okay with this, but the new neighbors were not okay with his visits.
Moses had become “socially unacceptable.”
I first heard of him when I had supper with a group of people from church. His owner asked me if I would like another Great Pyrenees. I told him “No, not at this time.” To which he responded, “Well, I can’t keep him, not even another day. I’m going to have to put him down. He’s taking things from the neighbors’ yards. Just the other day I caught him with a pie he had lifted from someone’s windowsill.” Suddenly I thought of my friend Pete, who lived down by the river and had poultry. I did know someone who needed a LGD for poultry (LGD=livestock guardian dog). I called Pete.
Just weeks ago, my friend Pete had been bemoaning the fact that he’d lost over 50 birds, to predators, that year. He lived near the Illinois River, where a lot of wildlife roamed,
Moses desperately needed a new home!
Pete was happy to take Moses. I brought Moses to my place and Pete picked him up from there. Pete knew little about livestock guardian dogs, but Moses knew a lot about poultry and predators. Several weeks went by and one afternoon Pete dropped in on me. He was beaming. I asked, “How’s Moses working out for you?” Pete replied in his country twang, “Well, I wish I had me one of these dogs a long time ago. In two weeks he killed 25 raccoon and three bobcat! The last bobcat weighed 40 lb. When Moses came out of the brush, after that fight, he had blood all over him, and it wasn’t his own!” Pete had discovered the joys of a LGD for poultry!
Having a LGD for Poultry Usually Takes Training and Commitment.
Most livestock guardian dogs won’t actually bond with the birds. They come to see them as property to protect. Also, since poultry is generally more fragile than other kinds of livestock, it should not be a surprise that a puppy accidentally kills a few birds. In our course we will detail how to train a LGD puppy so that it will guard poultry, or almost anything one would like.
Video of a Pyrenees Who Really Loves Chickens
I’ve noticed that most LGDs treat raccoon, skunk and small predators differently than coyote and larger predators. A lot of times they’ll kill a raccoon rather than chase it away. With other canines and large predators they usually seem content to chase them away. | https://www.homesteadingedu.com/moses-takes-on-poultry-bandits-lgd-for-poultry/ |
Happy birthday Stan, and a good year too.
Posted on: Jan 18, 2019 at 3:33 AM
Happy Birthday John David!
Posted on: Jan 18, 2018 at 3:33 AM
Posted on: Jan 25, 2017 at 10:13 AM
Happy birthday Stan. I hope this day finds you in good health and enjoying it.
Posted on: Sep 24, 2016 at 8:14 PM
It's been awhile since I've made an entry on my profile, but thought that I should share an experience I had Thursday evening. Linda & I went to Parkview's homecoming football game and while we were waiting for the game to begin I heard someone behind us say, "Coach Woolford," when I turned around, sure enough, Mr. Shepp Woolford was sitting behind us. I told him who I was and visited with him some. He was there with his son Kelly who later asked me if I would pose for a picture with his father, which I did. I asked him if he would send the picture to my e-mail so that it could be posted on our website. While there, the word must have gotten around, for several other folk came over to visit with Shepp. I think that he really enjoyed that plus a nice evening being at a football game with his son.
Happy birthday John! Hope your day brings lots of good stuff!
Thank you to Pamela Mason Britton, Jane Miles Moore, and Harold Fox for acknowledging my birthday. It means a lot to know classmates remember you. Our class was (Is) a great one. | http://www.hhsclass1962.com/class_profile.cfm?member_id=4738052 |
A very easy, ready in 20 minutes Egg Curry recipe for weekday dinners. This is a recipe I learnt from my house help and is now my go to Egg Curry recipe for the days I am in a hurry. Steam some rice on the side, slice some onions and you can have a perfect healthy dinner in under 30 minutes.
Egg curry is one of the comfort foods for both me and the husband. Actually for me eggs are, I need an egg almost on a daily basis specially in my breakfast. I have been known to make a sunny side up with even idlies on the side because I didn’t eat any egg the previous day. Probably one of the reasons you might find so many egg recipes on the blog.
Egg curry for our house is one of those I don’t know what to eat or OMG I forgot make dinner days. (Read this post on more details about managing a weekly menu and how to handle such days). A parantha, some steamed rice or boiled millets on the side and it is delicious, healthy meal for a weekday. And what is even better is that it is quick and easy to make.
I have many versions of egg curry in my treasured hand written everyday recipe book. And this one is a recent addition to it. I hired a new cook a while ago and when I asked her to make egg curry she asked me if she can make it her way this time. All for tasting new stuff and flavours, I was happy to let her. 20 minutes she came to me with a bowl of this to taste. One bite and I loved it and instead of having it for dinner we had it with dosa for breakfast that day. Since then it is a regular at home now. We eat it once a week and if left to the child and the man will probably eat more than that.
Last week I asked her for the recipe and she looked at me in that strange what is wrong with you look that I know too well by now. So I stood with her and measured out whatever she was putting for you guys and here is the brilliant recipe for Ballamma’s Egg Curry. The primary addition being roasted channa dal ground with a few spices. It thickness the curry and add volume & a distinct flavour to it.
Easy Egg Curry - 20 minutes Everyday Egg Curry
Ingredients
- 6 medium eggs boiled and peeled
- 1 large onion grated
- 2 meidum tomatoes pureed
- 2 tsp oil
- 1 tsp red chilli powder
- 1/2 tsp turmeric powder
- 2 tsp ginger-garlic paste
- few curry leaves
- Salt to taste
For the masala
- 1/4 cup roasted channa dal
- 2 tsp coriander seeds
- 1 tsp methi seeds
- 2 dry red chilli
Instructions
- Grind all ingredients of the masala together to a fine powder. Keep aside
- In a kadhai, heat oil. Once hot add curry leaves and grated onions to it. Sauté till the onions are slightly browned. This will take 4-5 minutes.
- Add tomatoes and Sauté further for 2-3 minutes. Add the ground masala, around 1 - 1.5 cup of water and bring to boil. Let it simmer for 3-4 minutes
- Add boiled and peeled eggs. Run the knife on the surface of the eggs to create little cuts so that the masala seeps in. Boil further for 2 minutes.
- Switch off and serve warm with rotis, rice or dosa. | https://www.sinamontales.com/recipe-easy-every-day-egg-curry/ |
External Environment Factors of Organisation
The external environment contains an organization’s external factors that shake its businesses indirectly. The organization has no or slight control over these factors. The external environment factors of an organization reside outside the industry, leading to opportunities or threats.
Table of Contents
External Environment Factors of Organisation:
By definition, the external environment factors of an organization are what might affect it externally. For the convenience of analysis, we can divide the external environment factors of the organization into two groups:
- General environment (or remote environment), and
- Industry environment (or the immediate operating environment).
Read more: Internal Environment Factors
General Environment:
In the external environment factors of an organization, the general environment is the first factor. The general atmosphere is the nonspecific forces and dimensions in an organization’s surroundings that might touch its doings. It comprises political, economic, social-cultural, technological, legal, demographical, and international forces. You can learn more explanation of the general environment from PESTLED Analysis.
Industry Environment:
In the external environment factors of an organization, the industry environment is the second factor. The industry environment is the competitive environment of the business organization. Its elements are suppliers, consumers, competitors, new entrants, substitute products, etc.
The strategy-makers must comprehend the encounters and difficulties of both the isolated environmental factors and the instant operating ecological factors. They requisite to increase in value that the unknown environmental factors are widely uncontainable due to their faintly located external landscape. However, they can undertake appropriate strategies to deal with these factors, usually collectively with other industry actors, to reduce the intensity of their impact on the businesses. In contrast, the operating environmental factors are those in the instant competitive circumstances of a business. These are often regarded as ‘specific environment’ or ‘task environment.’ These factors fundamentally establish the competitive environment of a business. When strategists consider both the remote and operating environments, they can become more proactive in strategic planning and more dynamic in grabbing opportunities. In the following deliberations, you will find a wide-ranging explanation of the global environment.
How to Response External Environment Factors of Organisation?
External environment factors of organizations influence businesses in many ways. The influences may be positive or negative. Several forces in the environment may retard the growth of a commercial association. In the same way, specific effects may give an unexpected improvement to the growth of an organization. Thus, it is crucial for strategy managers to recognize the importance of environmental stimuli on the actions of their businesses. Based on such consideration, they can invent ways to respond to environmental forces.
It is essential for the strategy managers to understand the importance of environmental influences on the operations of their business.
A business organization can employ more than a few ways to respond to its situation. We discuss here some standard measures of external environment factors of an organization.
Lobbying:
Firms can appoint solid lobbyists to negotiate with regulators to modify any law or chorus them to endorse new legislation that may unfavorably affect business actions. FBCCI (Federation of Bangladesh Chambers and Commerce and Industry) is a very powerful lobbyist. FBCCI representatives often undertake efforts to influence government agencies or ministries, or committees.
Influencing Customers:
Organizations can influence their customers in different ways. Managers may devise new uses of a product. They may create a new set of products. They may create a new set of customers for products or direct their efforts toward taking customers away from competitors.
Influencing Suppliers:
One of the ways to directly change the environment is to establish a long-lasting relationship with suppliers. Organizations can be done by signing long-standing agreements with static rates. This would serve as a hedge against inflation. Organizations can also protect themselves from the supply-related crisis by establishing backward linkage (that is, producing their materials). For example, a mineral water firm may start creating bottles by itself, or a soft-drink or fruit-processing firm might become its supplier of cans.
Boundary Spanning:
A company may involve itself in boundary-spanning for getting knowledge about what other companies are doing. A person is called a boundary spanner who collects information from outside the organization while he or she is working in the field. Salespeople, purchasing agents, relationship managers are most suitable as boundary spanners.
Environmental Scanning:
Companies can impact the external environment based on info collected through regular analysis and observation. This is known as environmental scanning. When surveyed, ecological scanning can vigorously screen the environment. Using environmental scanning and boundary spanner, organizations can organize and gather relevant info to support managers in making decisions.
Strategic Response:
A company may change its plan to deal with environmental alterations. The strategy-alteration may take any of the forms: a slight change in the strategy, adopting an entirely new approach, or maintain the status quo. Which one would better face the burdens of its competitive environment depends on circumstances that triumph.
Organizational Combinations:
To enter into upholding prominence or new markets in the present market or several other strategic issues, a company may resort to merger, acquisition, takeover, or alliance. Two or more companies may be combined (merge) to form a new company. A firm may buy another firm to acquire its asset. In the case of an acquisition (or takeover), the acquired firm may continue to operate as an acquiring firm’s subsidiary. The acquired (take-over) firm may cease to be existent and become part of the obtaining organization. When two or more organizations assume a new-fangled venture, it converts into a strategic alliance (if the venture is cooperative in nature) or joint venture.
These are the basic idea of the external environment factors of an organization. It will be tried to explain or details external environment factors of organization in the future. | https://higherstudy.org/external-environment-factors-of-organisation/ |
While this code works, it is horribly inefficient (O(n^2)). Your challenge is to make an algorithm with less complexity.
The winning criteria is the solution with the least complexity, but ties will be broken by shortest length in characters.
Package all your code together in a function that meets the following requirements for "correctness:"
Merges elements of both arrays together- Any element in either input array must be in the outputted array.
The outputted array should have no duplicates.
Don't use the standard library's array functions for detecting uniqueness or merging sets/arrays (although other things from the standard library is okay). Let me make the distinction that array concatenation is fine, but functions that already do all of the above are not.
But I am not sure if it is still O(N) based on this JsPerf: http://jsperf.com/unique-array-merging-concat-vs-looping as the concat version is marginally faster with smaller arrays but slower with larger arrays (Chrome 31 OSX).
I'm not great at computing complexity but I believe this is O(N). Would love if someone could clarify.
Edit: Here is a version that takes any number of arrays and merges them.
Should be O(N) assuming a good hash function.
Wasi's 8 character set implementation is better, if you don't think it violates the rules.
Essentially, these are the steps I'm going through in the line above.
For each element el in the concatenated array, put the value el in hash[el] if no value of hash[el] currently exists. The memoization here (hash[el] ||= el) is what ensures the uniqueness of elements.
This should run in O(n) time. Please let me know if I've made any inaccurate statements or if I can improve the above answer either for efficiency or readability.
A quick peek at the source of Array#|, though, seems to confirm that using a hash as an intermediary seems to be the acceptable solution to performing a unique merge between 2 arrays.
This still can cast some doubts, if you want to distinguish 123 and '123', then this would not work..
23 bytes, if we only count the code block inside subroutine. Could be 21, if overwriting global values is allowed (it would remove my from the code). It returns elements in random order, because order doesn't matter. As for complexity, on average it's O(N) (depends on number of hash collisions, but they are rather rare - in worst case it can be O(N2) (but this shouldn't happen, because Perl can detect pathological hashes, and changes the hash function seed when it detects such behavior)).
This should be O(n) as I copy a into c and then check each b against all of c. The last step is to eliminate the garbage that c will contain since it is uninitialized.
The following is testable using Google's Traceur REPL.
If the above snippet uses a = [1,2,3] and b = [1,2,3,4,5,6] then s=[1,2,3,4,5,6].
PHP around 28 characters [ leaving out the example array variables and result variable ].
Not the answer you're looking for? Browse other questions tagged fastest-algorithm or ask your own question. | https://codegolf.stackexchange.com/questions/17127/array-merge-without-duplicates/17152 |
Interstate 81 lane closures in Shenandoah County overnight Aug. 20-21
The Virginia Department of Transportation has scheduled an overnight left lane closure for northbound Interstate 81 in Shenandoah County between mile marker 273 and 275. This closure is expected to take place from 8 p.m. Thursday, August 20, until 7 a.m. Friday, August 21.
During that time, crews will be doing soil and rock testing just north of the overpass for Exit 273 (Mount Jackson/Basye). All work is weather permitting.
Traffic alerts and traveler information can be obtained by dialing 511. Traffic alerts and traveler information also are available at 511Virginia.org. For other assistance call the VDOT Customer Service Center, available 24 hours-a-day, seven days a week. Citizens can dial 1-800-FOR- ROAD (1-800-367-7623) from anywhere in the state to report road hazards, ask transportation questions, or get information related to Virginia’s roads.
The Staunton District Twitter feed is at @VaDOTStaunton. VDOT can be followed on Facebook, Flickr,Twitter and YouTube. RSS feeds are also available for statewide information. The VDOT Web page is located at www.VirginiaDOT.org. | https://augustafreepress.com/interstate-81-lane-closures-in-shenandoah-county-overnight-aug-20-21/ |
I recently built a tower/obelisk type structure by supporting my character's self on a single stack of blocks, building this scaffolding stack underneath myself as I went up and breaking it down from the top after I was done. This seemed like an incredibly risky and failure-prone strategy, and I would not want to attempt it again for concerns of falling. What are some strategies to build a tower or spire structure of arbitrary height from primarily within the safety of inside the tower itself? Preferably, this should not require a full separate scaffolding structure, but this should allow me to occasionally go outside on scaffolding of some sort and decorate the outer surface. Any suggestions or best practices around this? Thanks in advance.
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2If you hold shift, then there's barely any risk when towering or walking around high spaces. Aug 11 '15 at 17:55
Dig a trench around the base of the tower two or three blocks deep and two or three blocks wide on each side, and fill it with water. If you fall off (or jump off when you're done) you can safely land in the water.
If you carry a bucket of water with you, you can use it do float down and then swim back up to the top if you need to run off temporarily. The bucket is reusable and the 'ladder' it makes disappears instantly!
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Can you please clarify how the bucket of water helps (beyond creating the initial moat of water)?– MingAug 11 '15 at 20:10
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2@Ming - if you place the water on the ledge above you, it will overflow down to the ground, and you ride the waterfall down. When you're back at the top later, use the now-empty bucket to pick up the sourceblock (the block of water you originally placed), and the waterfall "shuts off". Aug 11 '15 at 20:20
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1When you "pour out a bucket", you're actually creating a "water generator" cube. Water will flow out from that cube to wherever it can reach, down hills, over cliffs, and it will create currents appropriately. When you place a water generator against a wall above the ground, it will create a column of water that you can swim upwards through. And if you dig a hole out directly below it, the "water-elevator" won't flood the surrounding area.– AyelisAug 11 '15 at 23:55
Build in iterative stages, a floor at a time, with a sort of spiral staircase structure inside. Every new ceiling becomes a new floor, and quickly getting to the top would be a fairly simple task of running in circles. While you're building, every floor or two, build a balcony that extends all the way around the tower. From each balcony, you should be able to decorate the outer walls fairly easily. Torches that could be hidden on the balconies would both prevent mobs from spawning as well as light up the tower at night. | https://gaming.stackexchange.com/questions/231523/minecraft-building-a-tower-of-arbitrary-height-from-inside |
Consumer Price Index: How Much Did Inflation Impact Grocery Prices in 2022?
In 2022, eating a well-balanced diet had a large impact on a well-balanced checkbook as the price of groceries has climbed consistently over the past 12 months.
Learn: Inflation Cools Slightly in November, Costs for Food, Gas and Shelter Remain High
Discover: 3 Easy Tips To Turn Your Credit Woes Into Wows
With the latest data in the Consumer Price Index, released by the U.S. Bureau of Labor Statistics on Tuesday, Dec. 13, groceries went up by 0.5% as of November. While that’s a stark downturn from the record-high price jumps seen earlier in the year, inflation is still a problem for many trying to keep on a budget at the grocery store.
When all 12 months are considered, the “food at home” category in the CPI showed a total 12% spike in cost. In simple math, that means a $4 carton of eggs in 2021 would now cost $4.48 — or, more dramatically, a weekly $200 grocery bill in 2021 would be $224 in 2022. That’s a cost increase of $96 per month.
There are also some particular food groups that continue to bear the brunt of the cost increases. Per the CPI, four of the six grocery food indexes were up month over month. Fresh fruits and vegetables went up 1.4% in November (this, after falling by 0.9% the month prior). Cereals and bakery products are also up 1.1% and dairy and related goods went up 1% for November. Nonalcoholic beverages such as bottled water and soda also jumped by 0.7% last month.
There are some external factors for some of these price hikes, for example, anything using wheat (like cereals and bakery products) is more expensive now than in 2021 (by 16.4% year over year) due to the ongoing Russia-Ukraine war. As the USDA explains, Eastern Europe is one of the biggest exporters of wheat. Per a release by the organization, “Over the last 18 months, wheat prices have risen nearly 110 percent.”
Likewise, Russia is one of the most prominent producers of fertilizer, notes the USDA, which has also impacted farmers who cultivate fresh fruits and vegetables, hence why that food group is also seeing price jumps.
Yet there are a few types of grocery items that have gone down in price in November, as noted by the CPI. The category of meats, poultry, fish and eggs decreased by 0.2% and the beef index dropped by 0.8%; pork went down 0.3%.
Part of the reason, says Marketplace.org, is because supply has now met up with demand and people are “buying cheaper cuts of beef.” Yet, ongoing droughts, avian flu and the cost of feed for animals have impacted the meat market and, as such, that category is still 6.8% more expensive year over year.
Take Our Poll: How Do You Typically Split the Restaurant Bill?
The discussion around food prices will continue to be well into 2023 since the cost of groceries continues to outpace overall inflation. | https://www.gobankingrates.com/money/economy/how-much-inflation-impact-grocery-prices-2022/ |
---
abstract: 'Most of the existing algorithms for approximate Bayesian computation (ABC) assume that it is feasible to simulate pseudo-data from the model at each iteration. However, the computational cost of these simulations can be prohibitive for high dimensional data. An important example is the Potts model, which is commonly used in image analysis. Images encountered in real world applications can have millions of pixels, therefore scalability is a major concern. We apply ABC with a synthetic likelihood to the hidden Potts model with additive Gaussian noise. Using a pre-processing step, we fit a binding function to model the relationship between the model parameters and the synthetic likelihood parameters. Our numerical experiments demonstrate that the precomputed binding function dramatically improves the scalability of ABC, reducing the average runtime required for model fitting from 71 hours to only 7 minutes. We also illustrate the method by estimating the smoothing parameter for remotely sensed satellite imagery. Without precomputation, Bayesian inference is impractical for datasets of that scale.'
author:
- 'Matthew T. Moores[^1] [^2]'
- 'Christopher C. Drovandi'
- Kerrie Mengersen
- 'Christian P. Robert[^3] [^4] [^5]'
bibliography:
- 'abc.bib'
title: 'Pre-processing for approximate Bayesian computation in image analysis'
---
Introduction
============
For many important problems, the computational cost of approximate Bayesian computation (ABC) is dominated by simulation of pseudo-data. This is particularly the case for the Potts model, for which an ABC algorithm was developed by @Grelaud2009. For real world applications in image analysis, the dimension of the state vector can correspond to millions of pixels. The distribution of these states is highly correlated, requiring algorithms such as @Swendsen1987 to simulate efficiently from the generative model. In his comparison of sequential Monte Carlo (SMC-ABC) with a particle Markov chain Monte Carlo (PMCMC) algorithm, @Everitt2012 found in both cases that the computational requirements were dominated by simulation of this state vector.
Adaptive ABC algorithms have been developed to reduce the number of iterations required through more efficient exploration of the posterior. Various particle-based methods have been proposed by @Sisson2007 [@Beaumont2009; @Toni2009; @Drovandi2011; @Jasra2012; @Filippi2013] and @Sedki2013. The SMC-ABC algorithm of @DelMoral2012 uses multiple replicates of the summary statistics for each particle, which accounts for heteroskedasticity in the relationship between the statistics and the parameter values. This algorithm also has the advantage that computation of the importance weights is linear in the number of particles. Both the ABC tolerance $\epsilon$ and the Metropolis-Hastings (MH) proposal bandwidth $\sigma^2_\mathrm{MH}$ are selected adaptively, which reduces the need to tune SMC-ABC for specific applications.
Regression adjustment for ABC was introduced by @Beaumont2002, who post-processed the ABC output by fitting a local linear regression. By modelling the relationship between the simulated parameter values and the corresponding summary statistics, they improved the estimate of the posterior density even with large values of the ABC tolerance. @Blum2010a took a similar approach, except that they used a nonlinear, heteroskedastic regression. They then performed a second ABC run, using the estimate of the posterior to draw parameter values.
The idea proposed in this paper has a strong connection with indirect inference [e.g. @Gourieroux1993]. We assume that the underlying (intractable) distribution of the summary statistic of interest can be approximated well by an alternative parametric model with a tractable distribution. This so-called auxiliary model contains a different set of parameters. The indirect inference nature of our method requires learning the relationship between the parameters of the auxiliary model and the true model, often referred to as the mapping or binding function. @Wood2010 assume a normal parametric model with mean and covariance parameters. Our application involves a single summary statistic and we adopt the approach of @Wood2010 and use a normal distribution with a mean and a variance parameter. @Wood2010 learn the auxiliary parameters for each value of the true parameter proposed during a MCMC run by simulating independently a large collection (say of size $M$) of summary statistics from the true parameter and estimating the corresponding auxiliary parameters based on the sample moments of the generated summary statistics. We note that the resulting target distribution of the method depends on the choice of $M$ [@Drovandi2014]. Our approach differs in that we estimate the binding function prior to running our Bayesian algorithm via producing model simulations over a pre-defined grid (referred to here as the precomputation step). We attempt to recover the true mapping between the mean and variance auxiliary parameters with the true parameters (i.e. the limit as $M \rightarrow \infty$) by using non-parametric regression techniques. An additional advantage is that the output of the precomputation can be reused for multiple datasets that share the same parameter space.
Despite superficial similarities with the “accurate ABC" method of @Ratmann2014, where an auxiliary model is also constructed over a sufficient statistic, our approach cannot be seen as a special case of theirs as accurate ABC requires repeated observations of the summary statistics and consistency of the auxiliary model for its calibration step. We do not claim such proximity with the true posterior distribution and doubt it can be achieved for the models we consider below. Even though we are in the favourable case when the summary statistic is sufficient, most assumptions in @Ratmann2014 do not apply to our setting. The construction of the binding function also highly differs in both motivations and complexity. We nonetheless agree that those different approaches of @Wood2010 [@Drovandi2011a; @Cabras2014; @Ratmann2014], as well as ours, all relate in spirit to the original indirect inference method of @Gourieroux1993. The reader is referred to @Drovandi2014 for more details about the parametric auxiliary model approach adopted in this paper.
The rest of the paper is organised as follows. The ABC rejection sampler and SMC-ABC are reviewed in Sect. \[s:abc\]. Precomputation of the binding function is described in Sect. \[s:method\]. In Sect. \[s:potts\], we illustrate how this method can be applied to the hidden Potts model with additive Gaussian noise. Sect. \[s:results\] contains results from a simulation study as well as real imaging data from the Landsat 7 satellite. The article concludes with a discussion.
Approximate Bayesian Computation {#s:abc}
================================
The ABC rejection sampler introduced by @Pritchard1999 draws values of the parameter from its prior distribution $\pi(\theta)$, then simulates pseudo-data $\mathbf{x}$ from the model. One or more summary statistics $s(\mathbf{x})$ are calculated from the pseudo-data and compared to the values of those statistics in the observed data, $s(\mathbf{y})$. If the difference between the statistics is within the ABC tolerance threshold, then the proposed parameter is accepted.
The simulation of pseudo-data from the generative model is typically the most computationally intensive step in this process. The ABC rejection sampler works best when prior information about the distribution of the parameter is available. The acceptance rate under a sparse or uninformative prior can be extremely low, requiring many pseudo-datasets to be generated for each parameter value that is accepted. The ABC tolerance is a tunable parameter, since a large tolerance means a higher acceptance rate but also increases the error in the estimate of the posterior distribution. If the summary statistic is sufficient for the parameter, then the distribution of the samples approaches the true posterior as $\epsilon$ approaches zero. However, the number of samples that are rejected also increases, to the point that almost none are accepted. Adaptive ABC methods have been developed to address this inefficiency.
Sequential Monte Carlo {#s:smc_abc}
----------------------
The SMC-ABC algorithm of @DelMoral2012 evolves a set of $N$ parameter values, known as particles, through a sequence of $T$ target distributions: $$\pi_t(\theta, \mathbf{x}_{1 \dots M} | \mathbf{y}) \propto \frac{\sum_{m=1}^M \mathbb{I}\left(\Delta(\mathbf{x}_m) < \epsilon_t\right)}{M} \left( \prod_{m=1}^M p(\mathbf{x}_m | \theta) \right) \pi(\theta)$$ where $\epsilon_t$ is the ABC tolerance threshold such that $\epsilon_0 \ge \epsilon_1 \ge \dots \ge \epsilon_T$, $\mathbf{x}_{1 \dots M}$ is a set of $M$ replicates of the pseudo-data that are generated for each particle, $p(\mathbf{y}|\theta)$ is the likelihood, $\mathbb{I}(\cdot)$ is the indicator function, and $\Delta(\mathbf{x})$ is the distance between the summary statistics of the pseudo-data and the observed data, $\| s(\mathbf{x}) - s(\mathbf{y}) \|$, for an appropriate norm. The algorithm involves four major stages: initialisation, adaptation, resampling, and mutation.
#### Initialisation
The algorithm is initialised at $t=0$ by drawing a population of particles $\theta_{i,t}$ from the prior, where $i \in 1 \dots N$. Each particle is associated with $M$ replicates of the summary statistics calculated from pseudo-data. The generation of multiple sets of pseudo-data $\mathbf{x}_{i,m,t}$ for each particle increases the computational cost relative to other ABC methods, but it better handles the situation where there is sizeable variability in the value of the summary statistic for a given parameter. This is the case for the hidden Potts model, as we explain in Sect. \[s:potts\].
#### Adaptation
At each iteration, the particles are assigned importance weights based on the following formula: $$\label{eq:weight}
w_{i,t} \propto w_{i,t-1} \frac{\sum_{m=1}^{M} \mathbb{I}\left( \Delta(\mathbf{x}_{i, m, t-1}) < \epsilon_t\right)}{\sum_{m=1}^{M} \mathbb{I}\left( \Delta(\mathbf{x}_{i, m, t-1}) < \epsilon_{t-1}\right)}$$ These weights are normalised so that $\sum_{i=1}^N w_{i,t} = 1$. These weights gradually degenerate over successive iterations, which is measured by the effective sample size [$ESS$; @Liu2001 pp. 34–36]: $$\label{eq:ess}
ESS_t = \left(\sum_{i=1}^N w_{i,t}^2\right)^{-1}$$ The ABC tolerance $\epsilon_t$ is updated adaptively according to the desired rate $\alpha \in (0,1)$ of the reduction in the $ESS$: $$\label{eq:tolerance}
ESS_t = \alpha ESS_{t-1}$$ This equation must be solved iteratively, e.g. by interval bisection, since $ESS_t$ depends on the weights $w_{\cdot,t}$, which in turn depend on $\epsilon_t$ according to (\[eq:weight\]).
#### Resampling
If $ESS_t$ falls below a threshold value $N_\mathrm{min}$ then the particles are all resampled. The new population of $N$ particles can either be drawn from a multinomial distribution with weights $\lambda_i = w_{i,t}$ or more complicated schemes can be used. @DelMoral2012 employed the systematic resampling scheme of @Kitagawa1996. Once the particles have been resampled, all of the importance weights are set to $N^{-1}$ and thus the $ESS$ is equal to $N$.
#### Mutation
Finally, the particles with nonzero weight are updated using a random walk proposal $q_t(\theta'|\theta_{t-1})$. The bandwidth $\sigma^2_\mathrm{MH}$ can be chosen adaptively using an importance sampling approximation of the variance of $\theta$ under $\pi_{t-1}(\theta|\mathbf{y})$, as in @Beaumont2009. The pseudo-data is also updated using $q(\mathbf{x}'|\theta')$ and jointly accepted with probability $\min(1,\rho_i)$ according to the MH acceptance ratio: $$\label{eq:mh_rho}
\rho_i = \frac{\sum_{m=1}^{M} \mathbb{I}\left( \Delta(\mathbf{x}_{i,m}') < \epsilon_t\right)}{\sum_{m=1}^{M} \mathbb{I}\left( \Delta(\mathbf{x}_{i, m, t-1}) < \epsilon_{t}\right)} \frac{q_t(\theta_{i,t-1}|\theta_i')}{q_t(\theta_i'|\theta_{i,t-1})} \frac{\pi(\theta_i')}{\pi(\theta_{i,t-1})}$$
Precomputation of the Summary Statistic {#s:method}
=======================================
Model fitting with ABC can be decomposed into two separate stages: learning about the summary statistic, given the parameter $\left(s(\mathbf{x})|\theta\right)$; and choosing parameter values, given the summary statistic $\left(\theta|s(\mathbf{y})\right)$. The first stage is achieved by simulating pseudo-data from the generative model in a so-called precomputation step (discussed below), while the second is achieved via a Bayesian algorithm that utilises the output of the precomputation. In the case of latent models there are additional complications that will be discussed further in Sect. \[s:noise\]. To ease the exposition of this section, we assume the data $\mathbf{y}$ are observed without error.
The precomputation step involves simulating pseudo-data for fixed values of the parameter. Firstly we approximate the intractable distribution of the summary statistic, $f(s(\mathbf{y})|\theta)$, with an alternative parametric model that has a tractable likelihood function, $f_A(s(\mathbf{y})|\phi(\theta))$, which contains a different set of parameters $\phi$. There is a strong connection with indirect inference (e.g. @Gourieroux1993) as our method requires learning the mapping between $\theta$ and $\phi$, $\phi(\theta)$, often referred to as the binding function.
@Wood2010 adopts this approach and uses the auxiliary likelihood within a MCMC algorithm. For each proposed value of $\theta$, a set of $M$ independent pseudo-datasets are generated from the true model, $\mathbf{x}_{1:M} = (\mathbf{x}_1,\ldots,\mathbf{x}_M)$, and the corresponding summary statistics are constructed, $s_{1:M} = (s(\mathbf{x}_1),\ldots,s(\mathbf{x}_M))$, which can be viewed as an independent and identically distributed sample from $s(\mathbf{x})|\theta$. The auxiliary model is then fit (using maximum likelihood or the method moments for example) to this sample in order to estimate the parameter $\phi$. We denote this estimate of the mapping $\phi(\theta)$ as $\phi(\theta,s_{1:M})$. @Drovandi2014 note that the target distribution of this method depends on the value of $M$, and if the auxiliary model chosen is suitable, it is desirable to take $M$ as large as possible.
The approach that we use to learn the mapping is different to that in @Wood2010. Here we define a (not necessarily regular) grid over the parameter space $\Theta$. For each $\theta$ within the grid a set of summary statistics, $s_{1:M}$, are generated from the true model and the corresponding auxiliary parameter is estimated, $\phi(\theta,s_{1:M})$. The next step is to use non-parametric regression techniques in order to smooth out the effect due to a finite choice of $M$ and to obtain a direct approximation of the mapping, which we denote $\hat{\phi}(\theta)$. Here each component of $\phi$ is regressed on the model parameter $\theta$. These non-parametric regression models can be used to predict the mapping for $\theta$ values not present in the grid. We refer to this procedure for estimating the mapping as the precomputation step. Our approach, in addition to mitigating the effect of $M$, has the advantage of being very useful when fitting the same model to multiple datasets because the output of the precomputation can be reused, thus amortising its computational cost. For alternative approaches to combining nonparametric regression models with ABC, see the regression adjustment method of @Blum2010a and the meta-modelling or emulation approach of [@Wilkinson2014].
Our application involving the Potts model only has a single summary statistic and we follow @Wood2010 and assume a normal distribution for $f_A$ with mean $\mu$ and variance $\sigma^2$. Therefore our precomputation step requires the generation of $M$ pseudo-datasets for each value of $\beta$ across a grid, recording the sample mean and standard deviation of the summary statistic (i.e. estimates of the auxiliary parameters $\mu$ and $\sigma$) and then applying two separate non-parametric regressions to estimate the mappings $\mu(\beta)$ and $\sigma(\beta)$. It is important to note that for reasonable size images it is not computationally feasible to generate perfect samples from the Potts model. Instead we use an MCMC algorithm to generate a set of $M$ correlated pseudo-datasets and use these to calculate $s_{1:M}$ (see Sect. \[s:pre\_Potts\] for specific details of this for the Potts model).
There are several approximations induced by our method. The first is associated with replacing $f$ with $f_A$. Despite the summary statistic of the Potts model being discrete, our choice of the normal distribution is appropriate due to the size of the images being analysed. The second source of approximation arises from the estimated mapping, $\hat{\phi}(\theta)$, due to the non-parametric regressions. However, the quality of the constructed mapping can be assessed visually and/or by standard data analytic techniques. For convenience in this paper we also introduce a third and seemingly unnecessary approximation effect. The natural implementation, used in @Wood2010 and @Drovandi2014 for example, uses the auxiliary likelihood estimate $f_A(s(\mathbf{y})|\hat{\phi}(\theta))$ directly in a Bayesian algorithm and avoids any comparison of observed and simulated summary statistics (that is, specification of an ABC tolerance $\epsilon$ is not required). In this paper we use a more traditional ABC approach and draw pseudo summary statistics from $f_A(s(\cdot)|\hat{\phi}(\theta))$ for comparison with the observed data (and thus require $\epsilon$). We adopt this approach here for two reasons. The first is for ease of implementation; having already developed an SMC-ABC algorithm, all that is required is to replace the (expensive) simulation of pseudo-data from the model with the (cheap) summary statistic draw from the auxiliary model. The second motivation for our implementation approach is that a more direct comparison of the computational cost can be made between the SMC-ABC method with and without the precomputation step.
Hidden Potts Model {#s:potts}
==================
We illustrate our method using a hidden Potts model. The Potts model is a Markov random field with discrete states $z \in 1\dots k$. It is defined in terms of its conditional probabilities: $$\label{eq:Potts}
p(z_i|z_{i\sim \ell}, \beta) = \frac{\exp\left\{\beta\sum_{i \sim \ell}\delta(z_i,z_\ell)\right\}}{\sum_{j=1}^k \exp\left\{\beta\sum_{i \sim \ell}\delta(j,z_\ell)\right\}}$$ where $i \in 1 \dots n$ are the nodes in the image lattice, also known as pixels, $\beta$ is a scale parameter known as the inverse temperature, $i \sim \ell$ are the neighbouring pixels of $i$, and $\delta(u,v)$ is the Kronecker delta function. In this paper we use the first-order neighbourhood, so $i \sim \ell$ are the four pixels immediately adjacent to an internal node of the lattice. Pixels on the image boundary have less than four neighbours.
The inverse temperature parameter governs the strength of spatial association. A value of zero corresponds with spatial independence, while values greater than zero increase the probability of adjacent neighbours having the same state. The full conditional distribution of the inverse temperature is given by $$\label{eq:beta}
p(\beta|\vec{z}) = \mathcal{C}(\beta)^{-1} \pi(\beta) \exp\left\{ \beta \mathrm{S}(\mathbf{z}) \right\}$$ where $\mathcal{C}(\beta)$ is an intractable normalising constant. It involves a sum over all $k^n$ possible combinations of the labels $\mathbf{z} \in \mathcal{Z}$: $$\label{eq:norm}
\mathcal{C}(\beta) = \sum_{\mathbf{z} \in \mathcal{Z}} \exp\left\{\beta \mathrm{S}(\mathbf{z})\right\}$$
A sufficient statistic is available for this model since it belongs to the exponential family, as noted by @Grelaud2009. If $\mathcal{E}$ is the set of all unique neighbour pairs, or edges in the image lattice, then the sufficient statistic is $$\label{eq:stat}
\mathrm{S}(\mathbf{z}) = \sum_{i \sim \ell \in \mathcal{E}} \delta(z_i,z_\ell)$$ Thus, this statistic represents the total number of like neighbor pairs in the image. As $\beta$ approaches infinity, all of the pixels in the image are almost surely assigned the same label, thus the expectation of $\mathrm{S}(\mathbf{z})$ approaches the total number of edges $|\mathcal{E}|$ asymptotically, while the variance approaches zero. When $\beta=0$, the probability of any pair of neighbours being assigned the same label follows an independent Bernoulli distribution with $p = k^{-1}$, thus $\mathrm{S}(\mathbf{z})$ follows a Binomial distribution with expectation $|\mathcal{E}| /k$ and variance $|\mathcal{E} \; | k^{-1} (1 - k^{-1})$. The distribution of $\mathrm{S}(\mathbf{z})$ changes smoothly between these two extremes, as illustrated by Fig. \[f:exact\_beta\], but its computation is intractable for nontrivial images. The expectations and standard deviations in Fig. \[f:exact\_beta\] were calculated for $k=3$ unique labels and $n=20$ pixels, much less than required for any practical application.
The maximum variance of $\mathrm{S}(\mathbf{z}) | \beta$, which corresponds to the steepest gradient in the expectation, occurs near the critical temperature. This is the point at which the Potts model transitions from a disordered to an ordered state. The phase transition behaviour has analogies in physical systems, such as the Curie temperature in ferromagnetic materials. When $\beta > \beta_\mathrm{crit}$, the values of the labels begin to exhibit long-range dependence and coalesce into clusters of similar values. @Potts1952 showed that this critical point can be calculated exactly for a two-dimensional regular lattice by $\beta_\mathrm{crit} = \log\{1 + \sqrt{k}\}$, so $\beta_\mathrm{crit} \approx 1.005$ for $k=3$ and $\beta_\mathrm{crit} \approx 1.238$ for $k=6$. The nonlinearity and heteroskedasticity evident in Fig. \[f:exact\_beta\] will need to be accounted for in our choice of binding function $\hat\phi(\beta)$.
Precomputation of $S(z)$ {#s:pre_Potts}
------------------------
Since it is impossible to sample from $\beta|\mathbf{z}$ directly, we use ABC methods. This requires simulating pseudo-data from the Gibbs distribution $\pi(z_i|z_{i\sim \ell}, \beta)$ defined by (\[eq:Potts\]). It is difficult to simulate from this distribution because neighbouring pixels are highly correlated with each other, particularly for $\beta > \beta_\mathrm{crit}$, thus pixels can remain in the same state for many iterations. We use the algorithm of @Swendsen1987, which updates clusters of pixels simultaneously. The effect of the approximation error can be seen in Fig. \[f:exact\_beta\] for a trivial image, where computation of the exact likelihood is feasible. Even though Swendsen-Wang is much less computationally intensive than perfect sampling, simulating pseudo-data remains expensive, which is why we approximate $f_A(\mathrm{S}(\mathbf{z}) | \hat\phi(\beta))$ offline using a pre-processing step.
To estimate the binding function for the Potts model, we use 1000 values of the inverse temperature, drawn from a truncated normal distribution $\beta \sim \mathcal{N}\left(\beta_\mathrm{crit}, (\beta_\mathrm{crit} / 2)^2\right) \mathbb{I}(\beta > 0)$. This concentrates computational effort in the vicinity of the critical region, as shown in Fig. \[f:simSz\]. The expectation and standard deviation have been approximated for a regular lattice with $n=125\times 125$ pixels and $k=3$, which corresponds to the simulation study in section \[s:simulation\]. For comparison, we show linear interpolation between 61 values of $\beta$ on a regularly spaced grid, so that the approximation error due to a finite sample size can be seen. The simpler binding function provides a very good fit for the distribution of the expected values, but there is a larger approximation error in the estimate of the standard deviation. This is particularly evident at the critical point, where the variance of the sufficient statistic is much larger than estimated. We found that a binding function based on only 61 values did not provide sufficient accuracy for our purposes.
We perform 1000 iterations of Swendsen-Wang for each value of $\beta$, discarding the first 500 as burn-in. The remaining $M=500$ iterations are used to compute the expectation and variance of the sufficient statistic. It should be noted that this operation is embarrassingly parallel, since the computation for each value of $\beta$ is completely independent. The results of this pre-processing step are stored in a matrix, which can then be used to fit the same model to multiple datasets. During model fitting, these pre-computed values are interpolated to obtain $\hat\mu(\beta')$ and $\hat\sigma(\beta')$, then the conditional distribution of the sufficient statistic is approximated by a Gaussian with these parameters.
Additive Gaussian noise {#s:noise}
-----------------------
![Change in the value of the sufficient statistic according to the current distribution of $\pi_t(\beta|\mathbf{z})$. The horizontal line shows the true value of $\mathrm{S}(\mathbf{z})$.[]{data-label="f:smc-abc_S_z"}](image2_Sz.eps){width="\linewidth"}
In the hidden Potts model, the observed data $\mathbf{y}$ are independently distributed conditional on the latent labels $\mathbf{z}$. Under the assumption of additive Gaussian noise, the observation process is characterised by $$\label{eq:obs}
y_i | z_i\!=\!j \quad\stackrel{iid}{\sim}\quad \mathcal{N}\left(\mu_j,\sigma_j^2\right)$$ Since each unique label value corresponds to the mean and variance of a Normal distribution, this model can be viewed as a spatially-correlated generalisation of the mixture of Gaussians.
When the parameters of these mixture components are unknown, they must be estimated as part of the model fitting procedure. The joint distribution of all random quantities is given by: $$p(\mathbf{y},\mathbf{z},\boldsymbol{\mu},\boldsymbol{\sigma^2},\beta) = p(\mathbf{y}|\mathbf{z},\mu,\sigma^2) \pi(\mu) \pi(\sigma^2) p(\mathbf{z}|\beta) \pi(\beta)$$ where the vectors of noise parameters are $\boldsymbol{\mu} = (\mu_1, \dots, \mu_k)$ and $\boldsymbol{\sigma^2} = (\sigma_1^2, \dots, \sigma_k^2)$.
The conditional distribution of the latent labels is dependent on both the current likelihood of the data as well as the distribution of the particles: $$\label{eq:post_z}
p(z_i | y_i, \boldsymbol\mu, \boldsymbol{\sigma^2}, \beta) = \frac{p(y_i; \mu_{z_i}, \sigma^2_{z_i})}{\sum_{j=1}^k p(y_i; \mu_j, \sigma^2_j)} \; p(z_i|z_{i\sim \ell}, \beta)$$ where $p(y; \mu, \sigma^2)$ is the Gaussian likelihood and $p(z_i | z_{i\sim \ell}, \beta)$ is the Markov random field defined by (\[eq:Potts\]).
This data augmentation approach can be problematic for SMC-ABC because it means that the summary statistic is a moving target, as shown in Fig. \[f:smc-abc\_S\_z\]. The summary statistic in (\[eq:stat\]) is computed from the latent labels, therefore it only depends indirectly on the observed data through the noise parameters $\boldsymbol\mu, \boldsymbol{\sigma^2}$. Accurate posterior inference for $\beta$ is only possible to the extent that the pixel labels and the noise parameters have also been estimated correctly. It can take many SMC iterations for all of the components of the model to converge. This is the third stage of ABC estimation for latent models that was mentioned in section \[s:method\].
One approach to overcoming this circular dependency would be to include the noise parameters $\mu_j,\sigma_j^2$ in the state vector for each SMC particle. Updating these parameters would require generating latent labels from (\[eq:post\_z\]), which would severely limit the scalability of the algorithm. Simulating from the distribution of $\pi(\mathbf{z} | \mathbf{y},\boldsymbol\mu, \boldsymbol{\sigma^2},\beta)$ is even more difficult than drawing pseudo-data from $\pi(\mathbf{z}|\beta)$ [@Hurn1997; @Higdon1998]. It is simply infeasible to do this for each particle individually. Apart from the multiplication of computational cost by serveral orders of magnitude, there is also the issue of memory utilisation when updating particles in parallel. A single copy of the state vector can require several megabytes of memory, depending on the size of the image. Maintaining a copy of $\mathbf{z}$ for each thread would be impractical for massively parallel implementation of the algorithm.
We propose a pragmatic alternative that preserves the scalability of our algorithm, while enabling estimation of all of the components of the hidden Potts model. We run a single MCMC chain in conjunction with our SMC-ABC algorithm, although it would also be possible to maintain a small number of parallel chains. At the end of each SMC iteration, we draw a random sample from the current distribution of $\pi_t(\beta|\mathbf{z})$ according to the importance weights of the particles. These parameter values are used to update the pixel labels according to (\[eq:post\_z\]) by performing multiple iterations of chequerboard sampling [@Winkler2003 chap. 8], one for each value of the parameter. After this sequence of updates, the new pixel labels reflect the distribution of $\beta$ at the current SMC iteration. It is important to note that it requires several iterations to make substantial changes to the distribution of $\mathbf{z}$, thus it is the aggregate effect of all of these updates that transitions from $\mathbf{z}|\pi_{t-1}(\beta)$ to $\mathbf{z}|\pi_t(\beta)$. The new state vector is used to calculate $\mathrm{S}(\mathbf{z})$, as well as the sufficient statistics of the noise parameters $\bar{y}_j | \mathbf{z}$ and $s_j^2 | \mathbf{z}$. We used 1000 MCMC iterations per iteration of SMC to produce the results in section \[s:results\].
One downside of this method is that the MCMC chain can become stuck in a low-probability region of the state space. This is the cause of the outliers that are evident in Fig. \[f:sim\_results\]. Once the Markov chain crosses the phase transition boundary of the Potts model from a disordered to an ordered state, the correlations between neighbouring pixels make the probability of transitioning back extremely low, irrespective of the values of $\beta$ that are used. To mitigate this problem, we initialize $\mathbf{z}, \boldsymbol\mu$ and $\boldsymbol{\sigma^2}$ at $\beta=0$ during the initialisation phase of our algorithm. This increases the probability that the starting value of $\mathrm{S}(\mathbf{z})$ will be below the critical region.
Illustration {#s:results}
============
This section contains experimental results with synthetic data as well as real satellite imagery. In Sect. \[s:simulation\] we evaluate the performance and accuracy of our method for 20 images that have been simulated using the Swendsen-Wang algorithm for known values of $\beta$. We are thus able to compare the posterior distribution with the true parameter value. In Sect. \[s:satellite\] we demonstrate our method on an image obtained from the Landsat 7 satellite. This demonstrates that our pre-computation step enables inference with ABC for datasets of realistic size. We begin by calibrating our method against the SMC-ABC algorithm of @DelMoral2012 as well as the approximate exchange algorithm [@Murray2006; @Cucala2009; @Friel2011].
An source package containing these algorithms is provided in Online Resource 1. Its computational engine is implemented in using RcppArmadillo [@Eddelbuettel2014] with OpenMP for parallelism. Reference implementations of SMC-ABC are available in the supplementary material accompanying @DelMoral2012 and @Everitt2012. Source code for the approximate exchange algorithm has been provided by @Friel2011 and @Everitt2012.
The elapsed times were recorded on 2.66GHz Intel Xeon processors. We used 8 parallel cores for fitting the model to each image and the precomputation was performed on a dual-CPU computer with 16 parallel cores. Memory usage varied depending on the number of pixels and the degree of parallelism. Approximate memory requirements for each computation are reported below.
![Distribution of posterior sampling error for $\beta$, comparing SMC-ABC using $M=50$ replicates of the sufficient statistics with $M=200$.[]{data-label="f:beta_err"}](residuals.eps){width="\linewidth"}
We fit the model to each image using $N=10,000$ SMC particles with $\alpha=0.97$, using residual resampling [@Douc2005]. We drew $M=200$ summary statistics per particle from our precomputed binding function, but found that it was infeasible to simulate that much pseudo-data during model fitting since it took an average of 2 hours 45 minutes per SMC iteration. Instead, we used $M=50$ for SMC-ABC with pseudo-data, which resulted in runtimes that were more reasonable. This made a difference to the accuracy of the posterior, as shown in Fig. \[f:beta\_err\].
![Distribution of effective sample size (ESS) for model fitting with the exchange algorithm in comparison to SMC-ABC with pseudo-data or precomputed $\hat\phi(\beta)$.[]{data-label="f:ess"}](ess.eps){width="\linewidth"}
The exchange algorithm of @Murray2006 is an exact method for intractable likelihoods, but only when it is feasible to use perfect sampling to simulate from $\pi(\mathbf{z} | \beta)$. Even then, many studies such as @McGrory2009 have shown that this algorithm is very computationally intensive and takes longer as the value of $\beta$ increases. For this reason, @Cucala2009 and @Friel2011 replaced the perfect sampling step with 500 iterations of Gibbs sampling, hence creating an approximate exchange algorithm. The effect of this approximation on the accuracy of posterior inference has been studied by @Everitt2012. Since this is a MCMC method, the samples of $\pi(\beta|\mathbf{y})$ are correlated, reducing the effective sample size in comparison to the number of iterations. We found that 100,000 iterations were required to produce an ESS that was comparable with the SMC-ABC methods, as shown in Fig. \[f:ess\].
Simulation Study {#s:simulation}
----------------
Since the inverse temperature cannot be directly observed, we have used a simulation study to evaluate the accuracy of our method where the true value of $\beta$ is known. Following a similar methodology to that introduced by @Cook2006, we first simulated 20 values of $\beta$ from the prior, then generated 20 images from the model that corresponded to those parameter values. Each image had $125 \times 125$ pixels with $k=3$ unique labels. We used a uniform prior on the interval $[0, 1.005]$ for $\beta$ and natural conjugate priors $\pi(\mu_j) \sim \mathcal{N}(0, 100^2)$ and $\pi(\sigma_j^2) \sim \mathcal{IG}(1, 0.01)$ for the additive Gaussian noise.
Precomputation of the binding function took 1 hour 23 minutes for 987 values of $\beta$, using 1000 iterations of Swendsen-Wang for each. The resulting estimates of $\hat\mu(\beta)$ and $\hat\sigma(\beta)$ are illustrated in Fig. \[f:simSz\]. Total CPU time for all 16 parallel threads was 21 hours 50 minutes, indicating over 98% utilisation of the available capacity. Memory usage was less than 1.3GB. Since the same mapping function was reused for all 20 images, this runtime could be amortised across the entire corpus. Thus, the cost of precomputation was less than 5 minutes per image.
![Distribution of CPU times (left) and elapsed (wall clock) times for model fitting with the exchange algorithm in comparison to SMC-ABC with pseudo-data or precomputed $\hat\phi(\beta)$.[]{data-label="f:runtime"}](time.eps){width="\linewidth"}
Across the 20 simulated images, our algorithm took an average of 7 minutes per image for between 49 and 107 SMC iterations. Using pseudo-data to compute the sufficient statistic took an average of 71.4 hours for 45 to 111 SMC iterations. The approximate exchange algorithm took an average of 14.4 hours for 100,000 iterations. Fig. \[f:runtime\] illustrates that this two orders of magnitude difference in the distribution is consistent for both elapsed (wall clock) time and CPU time. This shows that the gain in performance is due to computational efficiency, not because of any increase in parallelism.
Fig. \[f:sim\_results\] shows that both SMC-ABC and the approximate exchange algorithm produced erroneous estimates of $\beta$ for some of the images, with a large difference between the posterior distribution of the particles and the true value of the inverse temperature. This is most likely due to problems with the Markov chain undergoing phase transition, as explained in section \[s:noise\]. We were able to compensate for this problem by increasing the number of replicates of the summary statistic to $M=200$. Thanks to precomputation of the binding function, this had little impact on the runtime of our algorithm.
Satellite Remote Sensing {#s:satellite}
------------------------
We have also illustrated our method on real data, using a satellite image of Brisbane, Australia. The pixel values correspond to the Normalized Difference Vegetation Index (NDVI), which was calculated accordng to: $$\label{eq:ndvi}
\mathrm{NDVI} = \frac{NIR - VIS}{NIR + VIS}$$ In Landsat 7 images, the visible red ($VIS$) band corresponds to light wavelengths of $(0.63 \dots 0.69){\mu}m$ and the near-infrared ($NIR$) band corresponds to $(0.76 \dots 0.90){\mu}m$ [@NASA2011].
The image was cropped to a region of interest that was approximately 40km east to west and 20km north to south, containing a total of 978,380 pixels. This region includes the city centre as well as suburbs and national parks to the south and west. By labelling the pixels, we aim to quantify the levels of vegetation present in the area and identify contiguous clusters of forest and parkland.
Precomputation of the mapping function took 13 hours 23 minutes for 987 values of $\beta$. Total CPU time for all 16 parallel threads was 75 hours 40 minutes, indicating over 98% utilisation of the available capacity. Memory usage was approximately 6.5GB.
We used weakly informative priors $\pi(\beta) \sim \mathcal{U}[0, 3]$, $\pi(\mu_j) \sim \mathcal{N}(\bar{\vec{y}}, 5)$, and $\pi(\sigma_j^2) \sim \mathcal{IG}(1, 0.01)$ for the hidden Potts model with $k=6$ unique labels. Model fitting took 5 hours 36 minutes using our algorithm. CPU time for 8 parallel threads was 39 hours, indicating 88% utilisation. The 95% posterior credible interval for $\beta$ was $[1.243; 1.282]$.
Running the original SMC-ABC algorithm of @DelMoral2012 on this dataset is clearly infeasible, due to the cost of simulating pseudo-data. It takes 89 hours to perform a single SMC iteration on our hardware. We also found that the exchange algorithm of @Murray2006 was unable to scale to data of this dimension, even when using 500 iterations of Gibbs sampling for the auxiliary variable as recommended by @Cucala2009. It took 97 hours for 10,000 MCMC iterations. Discarding the first 5,000 as burn-in left an effective sample size of only 390 due to auto-correlation of the Markov chain.
Discussion {#s:conclusion}
==========
We have demonstrated that the scalability of ABC (and SMC-ABC in particular) can be dramatically improved using a precomputed binding function for indirect inference. We observed two orders of magnitude improvement in runtime in our simulation study, from an average of 71 hours down to only 7 minutes. This enables Bayesian inference for datasets of realistic scale, for which it was previously infeasible. An important example is imaging datasets with a million pixels or more, such as satellite remote sensing. We showed that our algorithm was able to estimate the smoothing parameter of a satellite image, while neither the approximate exchange algorithm nor the SMC-ABC algorithm of @DelMoral2012 were scalable enough to be practical.
The computation of the binding function is embarrasingly parallel and therefore can make full use of modern computer architectures. Once the binding function has been computed, it can be reused to fit the same model to multiple datasets. This is an advantage in many applications such as satellite imaging, where there are a large number of images with approximately the same dimensions. In a longitudinal setting it would also be possible to update the binding function sequentially as each image is processed, to increase its resolution in the region of highest posterior density.
The other major issue that we have addressed is additive Gaussian noise, which is commonly encountered in real world imaging data. @Everitt2012 and @Stoehr2014 have previously looked at ABC for latent models, where the state vector is not directly observed. However, their methods are only applicable where the observed data and the latent model are both discrete, sharing the same state space. The lack of identifiability induced by continuous observations creates a major problem for ABC, since the summary statistics become a moving target. We have introduced a pragmatic approach to reduce the tendency to become stuck in low-probability regions of the parameter space, while preserving the scalability of our method.
Our use of a precomputed binding function appears to be similar to that recently and independently developed in @Cabras2014, who select a multivariate normal auxiliary model for the summary statistic and also use a precomputation step similar to above. It is important to note, however, that @Cabras2014 assume that the covariance matrix of the auxiliary model is independent of $\theta$. Such an assumption is severely violated in our Potts model application (see Fig. \[f:exact\_var\_n\]) and unlikely to hold generally across models with intractable likelihoods. Furthermore, @Cabras2014 assume throughout their paper that the summary statistic must be the same dimension as the parameter. We note here that this assumption is not required to use the preprocessing idea detailed in this paper. Finally, @Cabras2014 use a regular grid over the parameter space. We suggest that for nonlinear, heteroskedastic auxiliary models it may be more appropriate to select a non-regular grid in order to obtain a good estimate of the mapping by including more points around regions where the gradient of the relationship is large.
The degree of complexity required for the binding function is dependent on the dimensionality of the parameter space, the number of summary statistics, and the properties of the relationship between them. These factors will also influence how much pseudo-data (in terms of the grid size and the number of replicate datasets) must be simulated in order to achieve a sufficiently good approximation of the likelihood. The nonlinear, heteroskedastic regression that was applied by @Blum2010a or @Wilkinson2014 would be a good choice in many cases, although simpler techniques for estimating the binding function could also be used. We assume that the summary statistics can be modelled as a continuous and smoothly-varying function of the parameters. The output of the precomputation step can be used to verify empirically that this assumption holds for the specific model under consideration.
@Cabras2014 have shown that ABC with indirect inference can also be applied to multivariate likelihoods, but only where the assumption of homoscedasticity is met. More research is needed to extend this method to Gibbs random fields such as the exponential random graph model (ERGM), which exhibits correlation between its summary statistics dependent on the value of $\theta$.
The current methods for indirect inference would suffer from the curse of dimensionality if applied to models with a huge number of parameters and/or summary statistics, such as those encountered in genetics. Our method degrades very quickly as the number of parameters grows, because of a multi-grid requirement that grows as a power of the dimension of the parameter. We rely on asymptotic arguments as $M \rightarrow \infty$, therefore obtaining a sufficient number of simulations will become much more difficult as the dimension of the parameter space increases. The method that we have described is not universally applicable, nevertheless there exist a wide variety of models to which it could be successfully applied.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank the organisers and attendees of the MCMSki conference for their interest and feedback. In particular, we are grateful to D. P. Simpson, A. Mira, and the anonymous reviewers for their thoughtful comments and suggestions on an earlier version of this manuscript. M. T. Moores acknowledges the financial support of Queensland University of Technology and the Australian federal government Department of Education, Science and Training. C.P. Robert’s research is supported by the Agence Nationale de la Recherche (ANR 2011 BS01 010 01 projet Calibration) and an Institut Universitaire de France senior grant 2010-2016. K. L. Mengersen’s research is funded by a Discovery Project grant from the Australian Research Council. Landsat imagery courtesy of NASA Goddard Space Flight Center and U.S. Geological Survey. Computational resources and services used in this work were provided by the HPC and Research Support Group, Queensland University of Technology, Brisbane, Australia.
[^1]: Mathematical Sciences School, Queensland University of Technology, GPO Box 2434, Brisbane QLD 4001, Australia
[^2]: Current address: Department of Statistics, University of Warwick, Coventry, UK
[^3]: CEREMADE, Université Paris Dauphine, 75775 Paris cedex 16, France
[^4]: CREST, INSEE, France
[^5]: Department of Statistics, University of Warwick, Coventry, UK
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The Large Hadron Collider (LHC) is the largest and most powerful collider in the world. It boosts the particles in a loop 27 kilometres in circumference at an energy of 6.5 TeV (teraelectronvolts), generating collisions at an energy of 13 TeV. The resultant mammoth energy density created is used to break the strong and weak nuclear forces to study the standard model of elementary physics and the smallest building block of the universe. Are there other ways to accelerate particles to high energies over short distances?
Background
Any traditional particle accelerator uses Radio Frequency (RF) cavities to accelerate particles to larger speeds. This is accomplished by alternating the electrical polarity of positively and negatively charged zones within the RF cavity, with the combination of attraction and repulsion accelerating the particles within the cavity. In layman term, Electric field switches from positive to negative at a given frequency, creating radio waves that accelerate particles in bunches and electromagnets are used to steer the particles in the collider.
However, the future of particle accelerators could be smaller particle accelerators achieving similar energies. Advanced Wakefield Experiment (AWAKE) is one such proof of concept experiment at CERN which hopes to achieve this. AWAKE aims to accelerate a low energy electron bunch from 15–20 MeV to several GeV over a short distance (10m). The technical coordinator and project lead Dr Edda Gschwendtner introduces the AWAKE experiment in this video.
In the plasma wakefield experiment, we use plasma instead of cavities. When we inject high energy proton beam into a 10m long plasma cell, strong wakefields are created similar to waves being created after a boat passes by. Hence, When we inject electron beam at lower energies they can bank on these wakefields to get accelerated similar to how nearby surfers to passing boat can bank on waves generated after a boat passes by and get accelerated.
In 2017, AWAKE proved the existence of wakefield after a proton beam was passed through a plasma cell. On 26th May 2018, AWAKE accelerated electrons by passing an electron beam into these wakefields which were previously generated by the proton beam. The electrons were accelerated by a factor of around 100 over a length of 10 metres: injected at an energy of around 19 MeV, they reached an energy of almost 2 GeV. This acceleration gradient achieved is 40 times better than the results achieved at LHC. In this interview, Edda talks about the experiment, results as well as the future of AWAKE.
AWAKE explores the use of plasma to accelerate particles to high energies over short distances. Currently, AWAKE is a proof of concept experiment at CERN. In its experimental year 2017–18, AWAKE discovered a lot of data which needs to be analyzed.
This led to the foundation of the idea for building a library pyAwake.
Google Summer of Code and High Energy Physics (CERN-HSF)
It’s an initiative by Google to link interesting open-sourced problem statement to capable students. It’s one of the most prestigious competitions to be part of and every project receives funding and global recognition.
Particle physics is an exciting field where large collaborations of scientists collect and analyze petabytes data from high-energy physics experiments, such as the Large Hadron Collider, hosted at the CERN laboratory in Geneva, Switzerland. CERN has participated in the GSoC since 2011 and since 2017 the program has expanded to involve the high-energy physics community under the umbrella of the HEP Software Foundation.
On March 25, 2019, the complex problem statement of AWAKE was released as part of the program by mentors Dr Spencer Gessner and Dr Marlene Turner through CERN High Energy Physics Software Foundation. CERN-HSF has 38 projects with 30 organizations. The objectives of the problem statement were as follows :
- To read about 12TB of data created during 2017–18 experimental run and create a faster and simpler AWAKE database.
- To supply API for searching and visualizing multiple datasets in the timestamp range.
- Using NumPy, SciPy, and Matplotlib for data analysis and visualization.
- Porting existing analysis done by AWAKE Analysis tools.
- Encapsulate in library structure and provide example notebooks.
From March 25, 2019 to April 9th, 2019, I was constantly in touch with my mentors at CERN to build a proposal worthy of submission in the program. The proposal had to take in account how to build a database engine and then build generic analysis subroutines on top of that database engine so that it’s convenient for any novice in computer science to analyze on data recorded. On April 2nd, I submitted my proposal on “Building a python based library for AWAKE analysis” (proposal link). Out of 7,555 proposals submitted in GSoC 2019 across 103 countries, 45 proposals were submitted on the problem statement of AWAKE out of which my proposal was finally accepted. In total, CERN-HSF is hosting about 33 interns out of ~1100 interns selected all over the world.
Building a Database
In the first attempt, we worked on writing multi-threaded sub-routines in Python to index the entire database. For each HDF file (Hierarchical Data Format), there will be one CSV file with one-to-one mapping whilst saving only relevant dataset metadata information. This allowed each file to compress from 50–60 MB to 250 KB. In total around 0.5 million files, CSVs were created reducing the 12 TB footprint to 150GB.
Unfortunately, it took about 3 hours to index about 25,000 files with 8 cores. So, we decided to go with Apache Spark technology provided by IT-DB-SAS and jupyter notebooks provide by CERN SWAN service. SWAN (Service for Web based ANalysis) is a platform to perform interactive data analysis in the cloud with the posssibility to offload computations to large Spark clusters, Spark is an open-sourced distributed general-purpose cluster-computing framework. SPARK can process data from AWAKE HDF data repository storaged on EOS (which is a disk-based, low-latency storage service), partition data across serveral cluster nodes, where it can be computed by running python sub-routine and then stored in Resilient distributed dataset (RDD) data format which allows SPARK to hide most of the computational complexity from the users.
With the help of Prasanth Kothuri and Piotr Mrowczynski, I was able to deploy my sub-routine on SPARK (Full code). In a nutshell, the sub-routine will travel in each HDF file using depth-first search technique and extract relevant metadata and attributes stored for each group/dataset and store it in a CSV. Since this was a data reduction task and not a data processing job, SPARK team was kind enough to provide optimal spark configuration (spark.executor.memory and spark.kubernetes.memoryOverheadFactor etc) for the execution of this task and troubleshooting tools like SparkMonitor jupyter plugin and grafana dashboard to live monitor the progress of the job and optimize.
With ~256 nodes, I was able to run the AWAKE data reduction of 12TB data in 3 hours as opposed to 30 days running on single python thread, which is clearly a feat for engineering teams working at CERN. In addition data analysis platform like SWAN allows data analysts and data sciences to focus on the analysis and leave the infrastructure to the capable hands of CERN IT.
If you have access to CERN’s infrastructure you can access the files by logging in to the server using ssh and try the following command and you should see the following in output.
cat /eos/experiment/awake/CSVFiles/2018/06/12/1528812262935000000_130_1130.csv
BUILDING A DATABASE ENGINE AND VISUALIZING TOOLS
After the database for 500,000 files was up and running, it was crucial to build a database engine which would allow scientists to query data and visualize with ease and in a generic manner. Since the users might be a novice in computer science, we decided to build a master search function to cater to every need. The flow of control has been explained in the below diagram and the source code for Searching engine is here.
The master search function has the following input parameter: dataset keywords, comment keywords, starting timestamp, ending timestamp and dataset cuts. The user then gets to choose the correct dataset and the date range for loading multiple datasets in the form of dicts. This array of dicts can be passed to visualizing modules for visualization in the form of movie or collection of images in pyplot. This can be achieved with the following code snippet.
from pyawake.Searching import searching from pyawake.Visualizing import visualizing dataset_cuts = “/AwakeEventData/TT41.BCTF.412340/Acquisition/totalIntensityPreferred < 100.0,/AwakeEventData/TT41.BCTF.412340/Acquisition/totalIntensityPreferred > 1.0” dataset_image = searching.search(“bovwa 04 raw”, “”, “2018–05–12 00:00:00”, “2018–05–12 23:59:59”, dataset_cuts) visualizing.displayMovie(dataset_image)
The displayMovie function uses analyses.frame_analysis python file from AWAKE_ANALYSIS_TOOLS package developed by my mentor Spencer. It applies a necessary filter like scipy.signal.medfilt and displays in a movie fashion. The below example is the result of visualizing streak dataset in May 2018 and this the link for movie.
Streak Camera is an instrument used for measuring the variation in the pulse of light’s intensity with time. In this particular image, there is a beam propagating with batches in high and low electron density alternatively. This information is used to derive whether the experiment is successful by determining whether injected electrons are accelerated to higher energies.
For doing specific analysis, we can use load_column() command instead of search() and it’ll return a 2D array with the values of all the columns. This is extremely useful for doing covariance analysis in the future. For visualizing multiple images at once we can use method visualize_multiple_image(). The source code for visualizing modules can be found here and for AWAKE_ANALYSIS_TOOLS can be found here.
Both these packages are effectively deployed and their example notebooks will be uploaded on SWAN. The testing module and example notebook are also added to my GitHub repository as part of GSoC submission
COVARIANCE ANALYSIS & FUTURE SCOPE
Since the package was completed and deployed, My mentor Spencer was kind enough to work with me in doing a covariance analysis on Streak dataset with the help of application of the package pyAwake.
All images discovered by AWAKE are nothing but a collection of analog signals. So we can perform a Fast Fourier Transform (FFT) on each image to determine the frequency of each signal. The first image shows how FFT tries to match the frequency of the signal and Second shows FFT for 1 month (17,000 files) of data discovered in May 2018. This kind of analysis is very easy to execute after the introduction of pyAwake (code) as we can now compare multiple datasets across thousands of other similar datasets.
The peaks in the above image indicate that these datasets were discovered in similar conditions. We can now scrutinize more by drawing covariance analysis for each dataset. Each dataset is comprised of 512*672 values. Hence, if we apply FFT for each column (i.e 672 columns), we will get the frequency of the signal for each column of the dataset. If we compare each column with other columns and notice similar values, we can conclude that the frequency is bleeding to other columns which can further help in proving that the electrons are getting accelerated
EXPERIENCE, ACKNOWLEDGMENTS AND IMPORTANT LINKS
Google Summer of Code is much more than handsome stipend and fame. It’s a excellent opportunity to connect with some of the most talented people in this world working on great projects. I’d like to formally thank Google for coming up with GSoC program. I would also like to thank CERN SWAN service, CERN IT Hadoop and Spark Service, especially Mr. Prasanth Kothuri who worked with me in parallelizing the data reduction with Spark, providing Spark Kubernetes cluster to run the reduction stage at scale, publishing the software and analysis notebooks in the gallery for wider AWAKE audience and generally mentoring me in new technologies such as SWAN (Jupyter notebooks) and SPARK.
I would like to thank my mentors Dr Marlene Turner who constantly helped me understand the physics in our weekly journal class and Dr Spencer Gessner who selected me, mentored me and introduced me to the AWAKE team at CERN. It has been an amazing journey ! | http://db-blog.web.cern.ch/blog/prasanth-kothuri/2019-08-large-scale-data-reduction-awake-experiment-data-apache-spark-and |
ASU students camp out for Oregon game
TEMPE — It was an eventful night for some ASU students who camped outside Wells Fargo Arena Wednesday.
A few dozen Sun Devils slept on air mattresses and inside blankets. Those who were first in line made sure they received a wristband that would get them into the front two rows of the student section for Thursday night’s game against the No. 2 Oregon Ducks.
“There are 180 priority passes, and after that it’s first come first serve,” said junior Nick Granillo, who was first in line. “We’re trying to get a sold out crowd. It’s going to be a sea of black. We’re going to get crazy and loud in there.”
Those students who stayed the night got a special delivery from ASU football head coach Todd Graham.
“He actually bought us some Hungry Howie’s Pizza,” Granillo said. “Cheese pizza, pepperoni pizza with the good crust.”
Some students didn’t get any sleep at all. Tyler Bell planned to catch up on sleep during class.
“I’m going to make it to Italian, but I’m not going to my honors class. There’s no way,” Bell said.
While students were camped outside, some sneaky Oregon Ducks crossed over enemy lines and painted the “A” on “A” Mountain a bright, neon green. It didn’t take long for Donna Evans and other ASU students to paint the “A” back to black.
“I immediately went into Sun Devil mode and said we have to get paint,” Evans said. “We were running around, and we were like ‘Hey, we need to get this paint. Can people drop a few dollars?’ We got like 80 bucks in less than five minutes. It was awesome.”
It only took Evans and some fellow Devils about 30 minutes to paint the “A” black for tonight’s blackout.
Evans just made it back to Wells Fargo Arena in time to see ASU Head basketball coach Herb Sendek handing out some treats for breakfast.
“We’re rewarding our great students with doughnuts,” said Sendek. “Many of them were here all night and then waiting again today for wristbands for tonight’s big game.”
Kickoff for the Devils showdown with the Ducks is set for 6:00 p.m.
We want to hear from you.
Have a story idea or tip? Pass it along to the KTAR News team here. | https://ktar.com/story/65119/asu-students-camp-out-for-oregon-game/ |
Generalities on meteors
A meteoroid is any solid particle moving in the Solar System with a size between 30 microns and 1m. Beyond these limits, we talk about either nano-dust particles (<30 microns) or asteroids (>1m). The visible phenomenon due to the entrance of a meteoroid into the Earth's atmosphere is called a meteor (or more commonly a shooting star). The vast majority of meteroids are completely burnt at altitudes between 90 and 110 km where they form permanent layers of metallic species such as Iron, Nickel, Sodium. For the largest meteoroids, a part of the object might survive the passing of the atmosphere (that also strongly depends on its trajectory and initial speed). If a small remnant of the meteoroid reaches the ground, it is called a meteorite.
Meteors may occur at specific times in the year, the so-called meteor showers (such as e.g. the Perseids or the Geminids), or more or less randomly at any time, the so-called sporadic meteors. Meteor showers happen when the Earth passes through a dust of particles released along the trajectory of a comet, every time the latter approaches its perihelion (closest point to the Sun along the orbit) .
When a tiny meteoroid (< 10-3g) enters the atmosphere, it does not produce enough light to be dectected from the ground (even with sensitive cameras), but it collides with the molecules/atoms of the upper atmosphere (roughly 80-120 km) and remove one or more electrons from them (ionization). This creates an electron trail along the meteor path on which a radio signal can be reflected and detected by a radio receiver under some geometrical conditions (see below). A meteor detected by this way is called a radio meteor or a meteor echo.
Forward Scattering radio observations
The main principle of forward scattering is illustrated in the figure below. A radio receiver is tuned on the frequency emitted by a transmitter. Direct transmission is impossible because of the curvature of the Earth. When a meteor enters the atmosphere, its trail (illustrated by the arrow in yellow) may reflect temporiraly the radio signal from the transmitter to the receiver. The signal recorded at the receiving station is called a meteor echo. Its duration depends on the lifetime of the trail. The electrons in the trail tend to scatter quickly in the ambient neutral atmosphere. The duration of most meteor echoes is of a fraction of a second but in some rarer cases, it can reach a few seconds up to a few tens of seconds.
An important characteristic of meteor observation by forward scattering is that the reflection is specular. It means that the meteor trail acts like a long mirror on which reflection occurs only at one location (specular point). An optical analogy is shown below where the transmitter is a light bulb, the receiver is our own eyes and the meteor electron trail acts as a mirror. The consequence is that data computed from the observations at one receiving station only apply to this particular part of the trail.
The position of this specular reflection point only depends on the geometry, namely the positions of the transmitter T and receiver R, and the meteor path L. It can be shown that the position of the specular reflection point p is the point along L that is tangential to an ellipsoid which has R and T as focii. At the point p, the radio wave travels the smallest distance between T, L and R (see figure a) below from Wislez & Verbeeck, 2006). This has important consequence : for one fixed configuration T-R, not all meteors can be detected at the receiving station. Indeed, p must be located within the so-called meteor zone (in grey in the figure below) which is approximately located between 80 and 110 km altitude. The upper limit is linked to the fact that the density of the atmosphere at higher altitude quickly drops and therefore not enough ionization can be created by the meteoroid to reflect enough power. The lower limit is mainly due to the fact that most meteors are fully vaporized before reaching such low altitudes. In the case b) below, the reflection point is located too low along L and is never created. In the case c), it is located at too high altitude. In both cases, the meteor echo cannot be detected at R.
Profiles of the meteor echoes
The features of the receiving signal essentially depend on the electron density in the meteor trail. Two cases are mainly discussed: the underdense trails, which correspond to very low electron densities, and the overdense trails, which correspond to very high electron densities.
Underdense meteors
Underdense meteors correspond to very small meteoroids (typically a fraction of a mm in diameter). They are the most frequent ones observed by the BRAMS network since these tiny particles are by far the most numerous. Underdense meteor echoes are very short-libed (a few tenths of second) and their typical profile is illustrated in the figure below, on which the horizontal x-axis represents time and the vertical y-axis represents the power recorded for this meteor echo. It consists in a very rapid rise to a maximum, which corresponds to the time when the specular reflection point is created. It is then immediately followed by an exponential decay which corresponds to the phase where electrons tend to scatter in the ambient neutral atmosphere and the trail eventually disappears. The signal returns then to the background level noise of the receiving station. The duration of a meteor echo is measured as the exponential decay time of the signal.
Overdense meteors
Overdense meteors are rarer and they correspond to bigger meteoroids. These echoes can last between a few seconds up to a few tens of seconds. A typical signal is shown in the figure below.
Benefits of Radio observations
Radio observations can be carried out 24/7, independently of the weather conditions and during daylight. This has led to discover diurnal meteor showers such as e.g. the zeta Perseids in June. Only during strong thunderstorms with a lot of lightnings near the receiving station, the data can become useless due to too many broad-band strong interference.
Faint meteors cannot be detected by visual observations because they do not emit enough light. Radio observations are therefore sensitive to smaller meteoroids and to a larger mass range than visual observations. The two methods of observations are complementary.
Additional source of information
Videos about radio observations of meteors and the BRAMS network prepared for the RMZ
- General information on meteors
- Sporadic meteors vs meteor showers
- Radio observations of meteors
- Specular reflection
- The BRAMS network
External links
Radio Observations on the International Meteor Organization (IMO) website. | https://brams.aeronomy.be/theory |
Scientists have confirmed another detection of gravitational waves, after a violent collision between two black holes roughly a billion light-years from Earth.
It’s been just two years since scientists made the first ever discovery of the remarkable ripples in space-time, proving a century-old prediction in Einstein’s general theory of relativity.
Now, the latest announcement marks the fifth known event caused by a black hole merger, and the sixth detection of gravitational waves overall, suggesting discoveries of this kind are becoming somewhat ‘routine.’
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The event, dubbed GW170608, was detected on June 8. It is the lightest black hole merger spotted yet. The binary, made up of objects 7 and 12 times the mass of the sun, merged to form a single black hole equal to 18 solar masses. This is illustrated above, compared with the others
According to the researchers from the LIGO-VIRGO collaboration, the gravitational waves were created by two relatively light black holes.
The binary, made up of objects 7 and 12 times the mass of the sun, merged to form a single black hole equal to 18 solar masses.
And, the collision cast out ripples known as gravitational waves, with the energy equivalent of roughly 1 solar mass.
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In the new paper, published to arXiv, the authors say the detection ‘extends the range of known stellar-mass binary black hole systems at the low-mass boundary, and hints at connections with other known astrophysical systems containing black holes.’
While the event, dubbed GW170608, was detected on June 8, 2017, the researchers delayed its announcement to dedicate time to two other discoveries.
On August 14, a LIGO-Virgo three-detector observation confirmed gravitational waves from another binary black hole – and, just three days later on August 17, the detectors spotted the first-ever known binary neutron star merger in light, leading to another set of gravitational waves.
Scientists have confirmed another detection of gravitational waves, after a violent collision between two black holes roughly a billion light-years from Earth. The latest announcement marks the fifth known event caused by a black hole merger. An artist's impression is pictured
GRAVITATIONAL WAVES
Scientists view the the universe as being made up of a 'fabric of space-time'. This corresponds to Einstein's General Theory of Relativity, published in 1916.
Objects in the universe bend this fabric, and more massive objects bend it more.
And, gravitational waves are considered ripples in this fabric.
They can be produced, for instance, when black holes orbit each other or by the merging of galaxies.
Gravitational waves are also thought to have been produced during the Big Bang.
In March 2014, a team operating the Bicep2 telescope, based near the South Pole, believed they had found gravitational waves, but their results were proven to be inaccurate.
Then in September 2015, the LIGO Scientific Collaboration made the first ever detection of gravitational waves, proving Einstein’s century-old theory.
Since then, gravitational waves have been confirmed six times – with five from black hole mergers, and one through the collision of two neutron stars.
With gravitational wave detections suddenly piling in, these elusive events that scientists have spent decades hunting are now something of the norm.
‘I think we feel now that with the black hole binaries – unless we come across something that is qualitatively different then it really has started to become cataloguing if you like,’ Professor Ken Strain, a collaboration member from Glasgow University, told the BBC.
The graphic above shows the masses for black holes detected through electromagnetic observations (purple); the black holes measured by gravitational-wave observations (blue); neutron stars measured with electromagnetic observations (yellow); and the masses of the neutron stars that merged, which were detected in gravitational waves (orange)
The LIGO and Virgo detectors completed their second observation run (O2) in August, and will remain offline until next year as scientists work to upgrade the systems to improve their sensitivity.
The next observing run, known as O3, is so far targeted for fall 2018, though there will be occasional test runs in the interim that could give rise to additional detections.
In the months leading up to O3, the scientists will also scour the data from O2, searching for signs of other events.
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Artificial Intelligence Essays, Research Papers & Term Papers (60) essays
Artificial Intelligence essays:
Artificial Inteligence
... institutions. RESEARCH PROJECT TABLE OF CONTENTS Introduction 1 Purpose 1 Source of Information 1 Authorization 1 Overview 2 The First Steps 3 Computer-Synthesized Senses 4 Visual Recognition 4 Current Research 5 Computer-Aided Voice Recognition 6 Current Applications 7 Optical Character Recognition ...
Can Computers Understand? Includes John Searle's arguments
... program. 8) The program is not a product of the computer. 9) A computer does not produce "thoughts" in its brain. 10) A computer cannot understand. John Searle addresses the point of the ability of Artificial Intelligence (AI) to understand, in Mind Brains, and Programs. His main ...
Will robots rule the world one day?
... of artificial intelligence? But it is the fact that AI research was moving fast on its way since 1950s. What is artificial intelligence (AI)? Artificial intelligence is the study of intelligent behaviour in machines. In the 1980s, AI research ...
The impact of AI on Warfare
... neural networks. The development of weapons which use highly complex systems has drastically reduced the number of human casualties in wartime. The bloodshed is minimised because of the accuracy of the computer systems used. This has been an advantage that has brought a lot of ...
A Philosophical Essay Arguing Against the Theories Presented in "The Turing Test" of Artificial Intelligence.
... of one. In order for a computer to "think", it would have to have a relationship with its environment determined by brain processes which define a mental state. Every thought, feeling, and emotion a human is representative of ...
Story of Artificial Intelligence
... of creating intelligent machines. However, it wasn't until the development of the electronic computer in 1941 that technology was available to create machine intelligence. The term ' Artificial Intelligence ' was first ...
Mind and Machine, an essay on A.I
... of neural networks depends on overcoming many technological challenges, such as data cross-talk (caused by radio frequency generation of rapid data transfer) and limited data bandwidth. Real-world applications of these 'intelligent' neural network expert systems include, according to the Artificial ...
Solving N-Queens problem using Genetic Algorithms
... 8", else go to "3" 8. finish Figure 1: The procedures of genetic algorithms 2 Implementation Decisions 2.1 Genetic Algorithm First of all, we need to choose a genetic algorithm from GAlib package. There are four kinds of ...
Data Input/ Output Methods, Storage Devices and Computer Speed
... Optical Data Readers is the appropriate method to use for printed questionnaires. According to the required text for this class Fundamentals of Information Processing:The two categories of optical data readers are for optical mark recognition (OMR) and optical character recognition ... | https://www.writework.com/subject/artificial-intelligence/page/3/pageNo/10 |
Q:
Inheritance in Java with generic elements
I apologize if these empty values/variables are not generic elements, I am really not sure about terminology in Java.
We are given this code:
public class AbstrClass <T, S> {
private T value1;
protected S value2;
protected AbstrClass (T value1, S value2) {
this.value1 = value1;
this.value2 = value2;
}
public T getValue1() {
return value1;
}
public S getValue2() {
return value2;
}
}
public class Impl extends AbstrClass<String, Double> {
protected Integer value2;
public Impl (String value1, int value2) {
super("Ferdinand", (double) value2);
this.value2 = value2 + value2;
}
public void incrementValue2 (Integer value) {
super.value2 += value;
}
public static void main(String[] args) {
Impl impl = new Impl("Horst", 21);
System.out.println(impl.getValue1());
System.out.println(impl.getValue2());
impl.incrementValue2(42);
System.out.println(impl.value2);
System.out.println(impl.getValue2());
}
}
And then we are asked what the result was if we run the code / the main method and why.
I am not sure if my reasoning is correct:
Ferdinand -> b/c it executes a method it inherits from the base AbstrClass, and since there is already a value, it prints out that one.
21.0 -> b/c it exectues a method it inherits from AbstrClass, but there is no value in it, so it gives out the given value of 21, but as a double, since it is defined in Impl as a double value.
42 -> this one I really don't get.
63.0 -> again this exectues from AbstrClass, (see question 2), and we add the 42 from the increment method, which results in 63.
Is my reasoning correct, and why does the third System.out.println result in 42?
A:
In the constructor of Impl is written:
this.value2 = value2 + value2;
Meaning for given input of 21, value2 will have the value of 42.
impl.incrementValue2(42); only affects the variable of the same name in the super class.
System.out.println(impl.value2); will print the value of the variable value2 of Impl - which was set to 42 in the constructor.
Your overall reasoning is good while not 100% on point
it calls getValue1 on the base class which prints value1. value1 was set in the constructor of the parent class to whatever was passed in as first argument. The constructor of Impl ignores its first parameter and always passes "Ferdinand" so the super constructor
it calls getValue2 on the base class which prints value2 of the base class. The fact that Impl has a variable of the same name does not matter. Your reasoning for the .0 is correct.
see the base part of the answer
Maybe specify a bit more precisely why the value of the value2 in the super class changed`
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The tournament was officially inaugurated by Dr. Fakhruddin Ahmed, honourable Chief Adviser, Government of the Peoples' Republic of Bangladesh, on May 8, 2008 at the Grand Ballroom of the Pan Pacific Sonargaon Hotel. Dr. Fakhruddin Ahmed, in his speech, welcomed the foreign delegates to the beautiful Bangladesh.
Aktel Debate Championship '08
North South University and Cider International School clinched the championship titles at the school and university level contests at the final round of the debate competition titled “Aktel Debate Championship-2008” held in Chittagong on 17 May, Saturday.
Winning the Water problem by a Bangladeshi scientist
Dr. Hussam's interest in finding a solution to arsenic poisoning in fact began long before the competition was announced.
As a child he grew up in rural Bangladesh, drinking from one of the government's tube wells that he later suspected, after the first reports of contamination aired, might be contaminated. | https://archive.thedailystar.net/campus/2008/05/04/index.htm |
Financial managers are responsible for cash management. This responsibility demands that the financial manager maintains adequate balances of cash and marketable securities. And also see to the efficient operation of cash management at the lowest cost possible.
What are cash and marketable securities?
Cash is the currency on hand and in the demand deposit (checking account) balances maintained at the bank.
Marketable securities are short term securities ranging from overnight loans to maturities of less than a year. They are securities that are quickly converted to cash.
The financial manager has a job to establish optimal cash balances that minimizes idle cash balances that earn nothing and also maintain sufficient cash balances that meet anticipated and unanticipated cash needs.
Before the financial manager could establish optimal cash balances successfully, they have to examine the following considerations and costs:
- Select a planning timeframe within which to estimate cash inflows and cash outflows expected to occur expected to occur in each sub period within the time frame. This would require the construction of a cash budget with a forecast on the collection of accounts receivable,
2. The cash budget should be constructed to predict times when shortfall could be expected. This is important because every shortfall incurs costs. These costs could be transactions costs associated with raising cash. Or they could be costs associated with deteriorating credit rating.
3. There are also costs associated with having excessive cash balances. Too much idle cash is a indication of missed investment opportunities. Ideally, every fund should be invested, if only temporarily, at a positive rate of interest.
4. The impact of uncertainty on cash management strategy should never be overlooked. To minimize the impact, an improved forecasting and cash flows that give the financial manager control and increased ability to borrow are necessary.
How does the financial manager determine the cash need (an optimal cash balance)?
There are three approaches:
- The Cash budget,
- Minimizing-cost model and,
- Precautionary balances.
- The Cash Budget:
A cash budget is one of the most important budgets prepared in an organization. It is a summary of expected cash receipts and expected cash payments during the budget period.
The consideration of liquidity and cash management as key factors in an organization’s successful operation makes cash budget very important.
Cash budget is the financial manager’s fundamental tool for coordinating and forecasting cash flows. The cash budget shows the effect of budgeted activities, such as cash sales, buying, accounts receivable collections on the cash flow of a business. There are no accruals in the cash budget.
Once estimated cash receipts and expenditures are known, the possibility of having an unexpected shortfall is easily increased.
2. Minimizing- cost model:
The cash budget does not show how the optimal cash balance is arrived at. The job of a good financial manager is to minimize a business’s cost of maintaining cash balances.
For this to be achieved, the financial manager must recognize and minimize the three basic cost categories of short costs, excessive cash costs and procurement costs.
Short costs rise as a business holds less and less cash. On the other hand, long cost (costs incurred for holding idle cash and missing investment opportunities) increases as the idle balances increase.
Another cost that must be considered is the procurement costs associated with operating the cash management system. These are salaries and wages for staff of the cash management unit including overhead of the unit. These costs are fixed.
Now, how are total cash management costs calculated?
Recall that total cash management costs (TC) are made up of;
Procurement costs – P,
Total Short Costs – TSC,
Total Long Costs – TLC
Therefore TC = P + TSC + TLC.
3. Precautionary Cash Balances:
The financial manager has to also determine how much more money to hold in addition to the optimal cash balance. The financial manager has to go back to the history of the business to know what it did when it encountered an unexpected cash need above the optimal balance.
The manager would also have to determine the cost of not having precautionary balances to meet unexpected cash needs.
There are benefits for having precautionary balances. Profit opportunities may suddenly present themselves and a business with precautionary balances would avoid costs associated with the inability to raise cash to meet sudden cash needs.
But there is also a cost for having precautionary balances. One may be forfeited interest that could have been earned if those cash balances were invested.
Therefore, there is need to estimate an optimal level of precautionary cash balances.
Determining the optimal level requires a costs-benefits approach to find where total net profits (benefits minus costs) are maximized.
The benefits (B) could be assumed to be the following:
- The average return on the expected profit opportunity – R,
- Precautionary Cash Balance level – PCB and,
- The fraction of the year over which the opportunity occurred – N.
Therefore, B = R x N x PCB.
The cost (C) of the precautionary balances for the year could be assumed to be:
- The forfeited yield on securities for the period held and,
- Precautionary Cash Balance level – PCB.
Therefore, C = Y x PCB. | https://ayietim.com/2021/12/13/cash-management-what-is-optimal-cash-balance/ |
There are 14 different movement commands. These are used so often that most have very short abbreviations which you ought to use instead of the full-length version - NW is much quicker than NORTHWEST, for example. The main directions are the 8 compass points, N, NE, E, SE, S, SW, W, NW. It is assumed that you "just know" which way is N wherever you are in the game, so there's no use of relative directions like left or right.
As well as these 8 directional commands, there is U and D for UP and DOWN. These are sometimes more convenient than compass points, although they normally match one of them. For example, if you're going up a hillside it's quicker to type U a few times than to read the room description in full to find out exactly which direction actually IS up!
A similar pair of convenience commands are IN and O (for OUT). If you're next to some opening, IN will normally take you through it (unless it leads outside). Although IN won't always work, you can use O almost anywhere - it moves you one room towards the room most centrally located in the game, which is nominally the point where the road fords the river. If you're ever lost, keep repeating O until you find somewhere you recognise.
Similar to O is the SWAMP command, which moves you one room towards the swamp from anywhere on the surface. You may have to do a few Os before SWAMP will work if you're deep underground. See the question on how to get points if you want to know why this command is so handy!
The last movement command is JUMP (or LEAP). This command is not used very often, and implies that there is a drop of some kind, over or down which you wish to move. Don't use JUMP casually, it can easily get you killed. As well as these 14 movement commands, there are other, magical ways to move about, but they're too high-powered to be discussing here. | http://www.mud2.com/CMS/index.php/help/library/59 |
Buried : Contay British Cemetery, Contay, France, Grave III. B. 10.
Memorial : Wychbold St Mary De Wyche Church
Additional information on the memorial: Pte
George Prescott enlisted in the Canadian Overseas Expeditionary Force on 4th January 1915 at Edmonton, Alberta, Canada. On his attestation form he gave his date of birth as 29th June 1890 in Bromsgrove, Worcestershire, his next of kin as his mother, Jessie Prescott, Henbrook, Droitwich Worcestershire and his occupation as a Wood Working Mechanic. He has previously had 5 months military service with the 101st Fusiliers. Henry was 5 feet 5 inches tall with grey eyes, fair hair and a fair complexion.
National Archives of Canada Reference: RG 150, Accession 1992-93/166, Box 7962 – 31
The Canada War Graves Registers (Circumstances of Casualty) records the following information on Private George Prescott, 432089:
“Died of Wounds.”
At No. 9 Casualty Clearing Station.
Buried in Contay British Cemetery, 2 miles West of Warloy-Baillon, 7¼ miles West of Albert, France. | http://www.rememberthefallen.co.uk/casualty/prescott-george-2/ |
Whether you’re a photographer, an Instagram-loving couple, or a third wheel with a smartphone, you might stand to gain from these recommendations for lovebirds.
Photographer couple Sara Byrne and Phil Chester have put together a “quick fire” guide showing standout poses that bring partners’ chemistry to paper.
Examples vary from poses for couples partial to PDA—such as ‘Making A Baby’—to ones for those who are less touchy-feeling, like the ‘Hipster’. Watch the tutorial below for some creative inspiration. | https://designtaxi.com/news/399137/Watch-17-Creative-Photography-Poses-For-Couples-In-Under-Two-Minutes/ |
At Mill Lane Primary School, it is our intent to provide a music curriculum which is enjoyable and inspires a passion for music. It encourages creativity, enthusiasm, self-expression and a love of music which lasts a lifetime. Listening and responding to a range of musical styles, becoming composers and developing as singers and performers, enables pupils to become confident, reflective musicians. At Mill Lane, we have children from many different countries and cultures. Music is a unique form of communication that can affect people on both emotional and intellectual levels, playing an important part in the personal development of an individual and transcending cultures, abilities and generations. We ensure that children face no barriers in the music curriculum. Music is a universal language which is easily accessible to SEND and EAL children.
Implementation
At Mill Lane Primary school, the music curriculum is progressive throughout the whole school, using the EYFS framework and National Curriculum to guide us on the content and focus of each objective. The music curriculum ensures pupils sing, listen, play, perform and evaluate. We employ a specialist music teacher who teaches the majority of the music curriculum. He has great enthusiasm and expert subject knowledge and musical skills which inspire the children to aspire to be musicians. Our progression of skills document ensures that the curriculum is covered, building upon previously taught knowledge and skills so that key concepts are embedded in the children’s long-term memory so they can apply them fluently.
The Early Years Foundation Stage follows the the EYFS guidance. The most relevant areas of learning for music are Communication and Language, Physical Development and Expressive Arts and Design. At Mill Lane, the children in EYFS explore different percussion instruments, sing, listen and move to music.
In Key Stage 1, we build on the skills and knowledge obtained in EYFS. Children are taught to use their voices expressively and creatively by singing songs and speaking chants and rhymes. Through lessons which particularly focus on samba percussion instruments, composition and graphic and music notation, they are taught to play tuned and untuned instruments and to experiment with, create, select and combine sounds using the interrelated dimensions of music. They listen with concentration and understanding to a range of high-quality live and recorded music, including local composers and musicians.
In Key Stage 2, we build further on the skills and knowledge obtained in EYFS and Key Stage 1. The skills and knowledge in Key Stage 2 are particularly developed through lessons which focus on samba percussion instruments, boomwhackers, music theory and GarageBand. In addition, in their classes, years 3 and 4 are taught to play the recorder. Children are taught to sing and play with increasing confidence and control. They develop an understanding of musical composition, organising and manipulating ideas within musical structures and reproducing sounds from aural memory. They are taught to play and perform in solo and ensemble contexts, using their voices and playing musical instruments with increasing accuracy, fluency and expression. They improvise and compose music using the interrelated dimensions of music and using staff and other musical notations. The children listen with attention to detail and recall sounds with increasing aural memory. They appreciate and understand a wide range of high-quality live and recorded music drawn from different traditions and from great composers and musicians, including local composers and musicians. Their understanding of the history of music is developed.
All children in Key Stage 2 are given the opportunity to join the school’s choir, Glee Club. They perform at an annual choir festival, go carol singing in a local shopping centre, perform at the Mayor’s carol service and lead singing in whole school/key stage performances throughout the year.
The children in our nurture class are taught using their own personalised curriculum. They access a hands-on curriculum that follows the thread of the whole school planning. Their work is differentiated following assessments using PIVATs and personal assessments.
Impact
The benefits of music are having a positive attitude to learning and improved well-being. At Mill Lane, we strive so that children will achieve age related expectations in music at the end of their cohort year and they will retain knowledge that is pertinent to music. More importantly, well-being and confidence will be improved. Our music curriculum is of a high quality and teachers have high expectations. It is planned progressively, building on and consolidating prior learning. Children use relevant musical vocabulary and apply skills and knowledge to their work. Their confidence and resilience improve. Music has a high profile in the school, having a prominent position in various whole school/key stage activities. We measure the impact of our curriculum through formative assessment during lessons, termly summative assessments evaluating their skills against those identified on the progressions of skills document, photographs and videos of the children’s practical learning, pupil voice and learning walks.
In summary, children at Mill Lane leave Key Stage 2 having forged their own musical journey. They use relevant musical vocabulary, they can apply skills and knowledge to their work and are confident and resilient performers. | https://www.milllane.org.uk/music/ |
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The Master of Arts with a major in Management and Leadership requires a minimum of 36 graduate credit hours. Coursework is available part-time in the evenings or online. Refer to the Graduate Admission Requirements page for more information regarding admission criteria.
BUS-501, Survey of Business, will be required if the applicant’s GPA is below a 3.0 and/or if the applicant’s undergraduate degree was outside the area of business.
Students must complete all M.A. with a major in Management and Leadership Required Courses (27 credits) AND select either a Concentration (12 credits) or the General Program (9 credits) to reach a total of 36 credits for the degree.
NOTE: Students who complete a concentration in Accounting, Business Data Analytics, Cyber Security, Health Administration, or Software Development and later choose to earn a second Master’s degree in their completed concentration area may only use 6 hours of credit toward fulfillment of requirements for the second degree. Students who complete the MBA degree may not later choose to earn a second Master’s degree in Management & Leadership and vice versa.
(27 credits)
After completion of all required courses, students must then take:
(9 Credits)
Choose any three electives. | https://catalog.maryville.edu/preview_program.php?catoid=20&poid=3561 |
OSHA Requirements and Training Resources for Long Beach CA
Who regulates OSHA training in Long Beach CA? The State of California, under an agreement with Federal OSHA, operates an approved State Plan OSHA Program, commonly known as Cal-OSHA. Initial approval of the California State Plan was published on May 1, 1973, and certification for completing all developmental steps was received on August 19, 1977.
CA State OSHA Training Regulations and Standards
Health and safety standards in the State of California are significantly different than those of Federal OSHA and other state-s that operate state OSHA programs. The Occupational Safety and Health Standards Board is a standard setting agency within the Cal-OSHA program, which includes development and maintenance of standards for certain areas not covered by federal standards or enforcement. These latter standards apply to elevators, aerial passenger tramways, amusement rides, pressure vessels, and mine safety training.
Cal-OSHA health and safety standards (also known as safety orders under Title 8) are available on the California DOL website.
Jurisdiction
The California State Plan applies to all public and private sector places of employment in Long Beach CA and the rest of the state, with the exception of Federal employees, the United States Postal Service (USPS), private sector employers on Native American lands, maritime activities on the navigable waterways of the United States, private contractors working on land designated as exclusive Federal jurisdiction, and employers that require Federal security clearances.
California OSHA Information
The Department of Industrial Relations administers the California Occupational Safety and Health Program, commonly referred to as Cal-OSHA. The Division of Occupational Safety and Health (DOSH) oversees enforcement and consultation. There are several Cal-OSHA offices located throughout the state.
Cal/OSHA District Offices can be located by accessing the Cal-OSHA Enforcement Unit contact page on the Cal/OSHA website. The Cal-OSHA District Offices serving Long Beach CA and the rest of the LA region are:
Cal-OSHA District Office – Los Angeles
320 West 4th Street, Ste. 850
Los Angeles, CA 90013
(213) 576-7451
Cal-OSHA District Office – Monrovia
750 Royal Oaks Drive, Ste. 104
Monrovia, CA 91016
(626) 256-7913
Cal-OSHA District Office – San Bernardino
464 W. 4th St., Ste. 332,
San Bernardino, CA 92401
(909) 383-4321
Cal-OSHA District Office – Santa Ana
2000 E. McFadden Ave., Ste 122
Santa Ana, CA 92705
(714) 558-4451
Cal-OSHA District Office – Torrance
680 Knox Street, Ste. 100
Torrance, CA 90502
(310) 516-3734
Cal-OSHA District Office – Van Nuys
6150 Van Nuys Boulevard, Ste. 405
Van Nuys, CA 91401
(818) 901-5403
Cal-OSHA District Office – West Covina
1906 West Garvey Ave So, Ste. 200
West Covina, CA 91790
(626) 472-0046
Cal-OSHA Field Office – Ventura
1000 Hill Road, Ste. 110
Ventura, CA 93003
(805) 654-4581
Federal OSHA’s National Emergency Hotline telephone number is 1-800-321-OSHA.
OSHA Training Resources
State of California OSHA Posters
OSHA Training Conducted in English or Spanish at Your Site in Long Beach CA (see available courses below)
On-Site OSHA Training Classes IN Long Beach CA
Request a proposal for osha training in Long Beach CA
Long Beach CA - OSHA Training
Disclaimer: This information was obtained from the Federal OSHA website. OSHA Training Services Inc. makes every effort to ensure that the information on this page is accurate and up to date, but changes in state law and procedures affecting the information on this page are beyond our control. Contact Cal-OSHA program staff directly to verify important information about OSHA training requirements in Long Beach CA. | https://www.oshatraining.com/on-site-training/osha-training-long-beach-ca/ |
Q:
What if a normal digital clock used to measure time in twin paradox?
Last night I was watching a youtube video about twin paradox. It still continues to baffle me :).
In that video, the speaker conceptualizes a clock that measures time by reflections of light. If we consider speed of light as reference to our time measurement, I guess it is natural to expect that our time measurements will be warped if we are traveling at speeds comparable to that of light.
I was wondering, what if if we measure time with something else as reference? Say, a digital clock, assuming it works under such extreme speed/acceleration... So, my questions:
Would our time measurements then be still warped?
I think, even if the measured times indeed gets warped, is it not just the perceived time for the traveling twin? Logically, the twins have lived for the same amount of time (as measured from a digital clock on earth) and should have grown/aged similar. Why would the twin who stayed back on earth be aged more?
I might be totally wrong, but I would appreciate if somebody helps me understand where I went wrong.
Thanks for your time.
A:
The type of clock you use makes no difference. It's common to use a clock that measures light using a light beam because it's conceptually simple, but whether you use a digital clock, one with a spring or one with a pendulum you'll get the same result.
I don't want to go into the twin paradox here because it's been discussed to death over the years, not least on this site. I'll just say the effect is quite real. It's not down to some sort of mathematical trick but instead it's a fundamental property of spacetime: specifically the invariance of the proper time.
If there are aspects of the twin paradox (NB it's not a paradox) that you don't understand post a more specific question here and I'm sure there'll be lots of answers. Have a search of the site first though to make sure your question hasn't already been answered.
A:
Yes,the time measurement still shows dilation.
since if a mirror clock and conventional clock in the spacecraft agree with each other on the ground but not when in flight,the disagreement between then could be used to find the speed of the spacecraft independently of any outside frame of reference-which contradicts the fact that all motion is relative
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That the Third Mainland Bridge in Lagos will be closed from July 2020 to January 2021 to enable the Federal Ministry of Works and Housing to carry out repair works on the bridge is no longer news. What is news, instead, is that motorists on that route have the longest six months of their driving experience staring them in the face.
The Lagos never-ending traffic congestion story is never complete without a mention of the snarling, nerves-straining gridlock that defines daily driving experience on that ever-busy 11.8-kilometer bridge that has not, according to federal authorities, received any form of maintenance in eight years.
That experience is even under normal circumstances. With repair work going on coupled with the partial closure of the bridge, it means motorists have to prepare their minds for more tortuous driving and longer travel time. It calls for endurance and tactical scheming.
But the government says there are alternative routes. These alternative routes, according to Olukayode Popoola, the Federal Controller of Works in Lagos, are Carter Bridge through Iddo and Oyingbo to join Adekunle ramp inward Oworoshoki and another from Ijora-Olopa through Western Avenue to Ikorodu Road.
Lagos state government also assures motorists of less stressful travel experience. Fredrick Oladeinde, the state’s Commissioner for Transportation, disclosed at a press briefing in Lagos Tuesday that “a traffic advisory for alternative routes has been carefully planned to ease travelling for commuters plying the route.”
The diversion of traffic to the identified alternative routes means that even motorists who ordinarily have no business going to the island will be affected. Ikorodu Road, for instance, will be under serious pressure as it is going to be the receiver-route for vehicles coming from Eko Bridge, Ijora-Olopa and Western Avenue.
Most, if not all, of these alternative routes, are in deplorable condition. Though Aramide Adeyoye, Special Adviser to Governor Babajide Sanwo-Olu on Works and Infrastructure, says the state government has commenced remediation works on the identified alternative routes to make it motorable for the public, it remains to be seen how far that can go with the daily heavy downpour in the city.
“We expect to see Lagos crawl within this period. The alternative routes which are already congested are not in good motorable condition. More vehicles coming to join those routes will certainly create a driving experience that is better imagined than expressed,” said Frank Onwubuya who lives in Isolo but works on the Island.
Oladeinde assured that officials from Lagos State Transport Management Authority(LASTMA), Federal Road Safety Corps (FRSC) as well as other relevant government agencies would be on the ground to enforce the traffic plan and ensure easy movement during the closure.
But Onwubuya is skeptical, saying, “we have seen this kind of situation before and the experience cannot be forgotten in a hurry. My advice is that people have to plan their movements very well before hitting the road. If you had been spending two or three hours to get to your office or to keep an appointment on the island, make allowance for the expected slow down; add one or two hours to be on the safe side.”
For those who must make use of the Third Mainland Bridge to access the island, Popoola says traffic would be diverted for only the Lagos Island Bound section of the bridge to take motorists from Oworoshoki inward Lagos from 12 midnight to 1 am in the morning.
In the afternoon, from 1pm to 12 midnight daily, he added that the section of the road will be opened to traffic from Lagos Island inward Oworoshoki. “This arrangement will be alternated for the second phase of the project on the Oworoshoki bound lane after the first phase must have been concluded,” he assured. | https://iitcng.com/2020/07/08/what-planned-3rd-mainland-bridge-closure-means-to-lagos-motorists/ |
Introducing our new Student Ambassadors
We’ve recently welcomed our new cohort of students for the 2020-21 academic year, including 15 new Student Ambassadors. Take a look below to hear what our current student ambassadors had to say about their Loughborough University London experience so far and their advice for prospective students:
Andrew
From: the UK
Studying: MSc Risk, Governance & International Management
“Loughborough University is a long standing and prestigious institution with significant global recognition and was a large reason as to why I wanted to attend this university. Visiting the London campus and seeing the facilities and study space available truly put things into practice for me and moving back to London, the city I was born and raised in, made the idea of studying at Loughborough University London a great proposition for me. I love the scope for debate underpinned by quality teaching within my course and at the University. Studying does not feel as rigid and we are regularly encouraged to share our ideas and discuss amongst our classes. Having a range of students from different backgrounds within a variety of opinions truly makes such discussion interesting to see things from different perspectives.”
Ashwin
From: India
Studying: MSc Design Innovation Management
“The course has helped me acquire some practical and innovative business acumen required for excelling in a global company. The strategies and techniques taught in class would surely help me in a business or firm which delivers innovative, functional, and aesthetic solutions for problems and difficulties of people around the world.
The diversity in the students attending the course helps you learn different views and aspects of a topic. Professors and Lecturers from the course are very enthusiastic about what they teach and try to make the classes as interactive and engaging as possible. The block method of teaching also lets you fully immerse in a subject and understand it better.”
Ellie
From: the UK
Studying: MSc Design Innovation Management
“When looking at the different opportunities to further my study, I was drawn to Design Innovation Management because of the Collaborative Project which provides the chance to work with businesses. This would allow me to gain real life project experience as well as develop my network when looking for graduate opportunities. I think the part of the course I enjoy the most is being able to work with people from different backgrounds to my own – there are students from different cultures and educational fields which has given rise to interesting conversations and has opened my mind to other ways of thinking and working.”
Errol
From: The Phillipines
Studying: MSc Security, Peace-building and Diplomacy
“I chose to study at Loughborough University London because it offers a perfect course related to my advocacy of promoting sustainable peace. Aside from that, the roster of world class and top calibre professors is very impressive. The campus location is also amazing. For an international student like me, it is very thrilling to live and study in the UK, more so in London.
The diversity in class is what I enjoy most. Since our classrooms are considered melting pots for cultures around the world – since the students are from different parts of the globe. Not only that, it creates an atmosphere of learning in the global perspective whilst at the same time cultures international understanding.”
Fibi
From: the UK
Studying: MSc Digital Creative Media
“What I enjoy the most about my course is learning new things from different courses every day. For instance, I am now learning coding though I do not have a programming background. I saw it as a real challenge at first, but then I started to adapt to it and work on my assignment bit by bit. I love this feeling of looking back at how far I have come and how much I have learned from the beginning. Besides accumulating knowledge from the courses, meeting and communicating with classmates, lecturers, alumni who have related experiences gives me different perspectives on digital media industry. I do hope that I will reach my future career goals with the help of my master’s degree as a whole.”
Johannes
From: Germany
Studying: MSc Sport Analytics and Technologies
“I love the diversity of students and academics at the University. I have met some amazing fellow students who are passionate about their chosen subject. The academics come from a diverse background with a real passion for the subject they’re teaching. I also love that I can take a stroll in the huge Olympic park or walk along the wonderful canal down the road from the University to relax or have lunch. Some modules had several fantastic guest lecturers talk to us. Some of the guest lecturers have been Loughborough alumni that were working within sports, we also got a talk from the head of research and innovation of a big football club and many more outstanding individuals that have enhanced my learning experience.”
Jokotola
From: Nigeria
Studying: MSc International Management
“What I enjoy most about my course is how the teachers are very invested in seeing you succeed. I also appreciate the new topics I learn about from the business management world. A module I’m taking at the moment is called managing sustainability which is one of my favourites so far. There is so much business can do to make both their customers and employees happy. It’s a great course! Loughborough University London has created such a warm and safe environment for all students especially for international students. I’m glad I’m here!”
Komal
From: India
Studying: MSc Sport Business and Innovation
“I’ve always wanted to study at Loughborough, for its world-class sports programmes and faculty and the diversity of the students that come to the University.
My advice to any students looking to study at Loughborough University London is be ready to learn and embrace the culture, knowledge, and enthusiasm of the Loughborough family. The experience is irreplaceable and goes too quickly so just enjoy it!”
Lauren
From: the UK
Studying: MSc Sport Business and Innovation
“I am really enjoying getting the opportunity to learn from industry experts, who often deliver guest lectures. This provides a great insight and also a fantastic opportunity to network. It is also great studying as part of such a diverse cohort. Within discussions we have such a broad range of different lived experiences that the conversation is always rich, and the differing opinions and approaches challenges us to learn beyond our comfort zones.”
Maria
From: Romania
Studying: MSc Digital Marketing
“I chose to study at Loughborough University London because I was impressed by the high academic standard of the University from my undergraduate studies. I was also attracted to the master’s programme offered on the London campus which I believe will give me an edge in the competitive job market, as well as the diversity of topics covered and its practical focus on the business world. The programme gives me exposure to a variety of issues and challenges that digital marketers face, and we are encouraged to find solutions using both theory and our own reflective thinking.”
Mario
From: Panama
Studying: MSc Sport Business and Innovation
“Studying sports means also passion, emotion, culture those aspects during classes and activities create an immersive experience in theoretical and practical methodology. I really like the connection of Loughborough with stakeholders/industry. In the first month I easily connected with more than 10 start-ups during classes and Future Space activities. The learning content is not just classes, exist a world of activities organized by Future Space (you need to know about them), Loughborough Enterprise Network also has entrepreneurship workshops, seminars. If you need to improve your academic English and writing skills the University also provides workshops for these. It just amazing all the knowledge that you can access through Loughborough University London….. and of course classmates it’s fantastic how much you can learn by making friends.”
Oluwatosin
From: the UK
Studying: MA Media and Creative Industries
“The structure of the degree programmes at Loughborough University London in particular motivates me as I find it to be not only inspiring but also very suited to my aspirations for studying at postgraduate level. Loughborough University is ranked in the top 10 in University league tables and and having studied at the Loughborough campus I figured I’d get the same quality teaching here in London site as well.”
Robert
From: the UK
Studying: MSc Sports Business and Leadership
“The thing I enjoy most about my course is the opportunity to meet like-minded, yet diverse people, and make the most of the discussions and conversations that arise as a result. Furthermore, hearing from such a wide range of sports leaders, each with their own set of stories and careers path, acts as inspiration for us as students as we begin to think about our future beyond studies.”
Shreyans
From: India
Studying: MSc Sport Business and Innovation
“The location of the campus in Here East creates a sense of professionalism which is needed for a master’s course. All the professors are really supportive and friendly which gives you a comfort zone to explore your thoughts and ideas about the sports industry. The most interesting aspect is the association of each concept with the practical world. We learn things by applying concepts in the real world through case studies and group work. Our modules allow us to create a holistic understanding of the sports industry, and this has given me a clarity on what I might want to pick as a prospective career choice in the future in the sports industry. A master’s degree is a good opportunity to fine tune and build on your existence knowledge while you network and explore all the possible career paths.”
Vishal
From: the UK
Studying: MSc Digital Marketing
“London is a great place to be, especially for university students. Speaking from an international student perspective, adapting to a new country is never easy. But trust me, overcoming the anxiety of new climates, cultures and traditions is a life experience. As I sit in Unite Stratford One and write this message, I have a special request for those deciding to join Loughborough University London – just go out, make new friends and explore what London has to offer! Trust me, you will not regret it. In fact, you will feel more confident, strong and on your journey to living a better life. Last but not least, as a university student do not forget – keep a BUDGET. If not, say bye-bye to your bank account!”
To find out more about our Student Ambassadors, please visit our website. | https://blog.lboro.ac.uk/london/students/introducing-our-new-student-ambassadors |
Here are the party registration data for requested, sent, and accepted mail ballots:
In comparison to the requested ballots, here are the percentages of mail absentee ballots that have been returned and accepted by the voter's party registration:
And then the comparison of where we stand, as of 10-15-18 (23 days out from Election Day), with the 2014 final numbers for both requested absentee by mail (67 percent of 2014's totals) and returned & accepted absentee by mail ballots (13 percent of 2014's totals):
The trend lines of comparing the daily totals in 2014 to 2018 (by the same day out from Election Day) continue to show considerable strength in this year's absentee mail ballots, especially noteworthy that this year's election has no major state-wide race (beyond a state supreme court race):
Once we get to the three week mark, I'll add the remaining days until Election Day.
Then, a comparison by voter party registration between 2014's day's requested totals (the dotted lines) and 2018's daily requested totals (the solid lines):
And the percentage comparison of 2018 to 2014's same day totals:
The following chart shows the 2014 total numbers of requested and returned/accepted absentee mail ballots and where things stand as of 2018:
The generational analysis of NC's absentee mail ballots continues to show a skew towards older voters, which is traditional; yet, there is one age group that is intriguing in its performance in this vote method:
Those voters under the age of 21, known as Generation Z, are only 5 percent of the state's voter registration pool of 7 million (as of 10-13-18), but are 8 percent of the requested absentee mail ballots so far.
With North Carolina's biggest contests this mid-term at the congressional level, three of the four districts that are showing the greatest interests are considered the most competitive in the state: the 2nd Congressional District, the 9th Congressional District, and the 13th Congressional District.
In looking at this nearly 70,000 voters who have requested absentee mail ballots, I took their registration data and their voting history to look at how these voters may have voted in 2014, if they were registered after the last mid-term election:
The plurality of voters requesting absentee mail ballots continues to be those voters who were registered in 2014 but didn't vote in 2014.
Next, I looked at the gender breakdowns, and we continue to see a pretty significant advantage to female voters, and within that group, registered Democrats continue their show of strength:
And then looking at gender differences within the voter's region (which could be within a central city within an urban county, within the outlying suburbs within that central city's urban county, the surrounding suburban county, and rural counties):
Urban voters (those who live in a central city within an urban county) continue to be a plurality of all voters, but suburban voters (both those who live in an urban county but outside the central city and those who live in the surrounding suburban counties) are a majority of both female and male voters.
Finally, those requested ballots that have been returned and accepted:
And by party registration, again comparing 2014 (dotted line) to 2018 (solid line):
And by actual numbers: | http://www.oldnorthstatepolitics.com/2018/10/nc-absentee-mail-ballots-as-of-10-15-18.html |
Performed:
About the event
Macbeth is one of Shakespeare’s most familiar and darkest tales. Our production plays out in a not-so-distant, dystopian world, where society is teetering upon the brink of collapse. The Macbeth's tormented struggle for power is fueled by ambition, paranoia, and encouraged by the supernatural, but ultimately becomes a self-fulfilling prophecy rather than a product of outside manipulation. This story is not just about Macbeth, but rather the collaboration and partnership of Macbeth and Lady Macbeth to taste the bittersweet fruits of their sown treachery. From the first moment, the action drives at a frantic pace, with dramatic battles, personal intrigue, and lots of sampling from the playlists of each character's experience. The "weird sisters" bring mystery and prophesy through an intoxicating musical appeal that is sexy, alluring, but extremely dangerous.
Media & Resources
Sorry the images for this performance are not available yet. More Information. | http://theatre.fullcoll.edu/event_season_detail.php?calendarid=1539 |
The NHL Expansion train is once again picking up speed, as League General Manager’s were informed that they could expect a formal decision on the expansion process prior to the coming 2016 NHL Draft, scheduled to be held June 24-25th at the First Niagara Center in Buffalo.
The expansion process, and the potential of adding new teams to the current 30 team League, began on July 6th of 2015, when the NHL made an expansion application available to numerous interested parties and individuals, eventually receiving 16 requests for application.
Of these 16 requests, the most notable of which are undoubtedly those based out of Las Vegas as well as Quebec City. Although previous reports have been denied, some deep within the hockey community have shared the opinion that an NHL team in Sin City could be a done deal. A return to Quebec City, however, is reportedly not as close, leading to the perception that the NHL could expand into a 31 team League for a given amount of time.
How Would an Expansion Draft Work?
With expansion comes the need to fill the roster of an entirely new club, something which ultimately takes place through the process of an expansion draft.To clarify, if the NHL were to add one team, each NHL team would in fact surrender one player, while if two teams were added, each club would in turn surrender two players, leaving the potential for a number of highly skilled players to find new homes with expansion franchises.
Although it has yet to be determined, the NHL would outline a select number or distribution of players which each franchise can in fact protect, allowing them to securely remain with their current franchise.
TSN’s Darren Dreger, who has been covering the current General Manager’s Meetings in Boca Raton, Florida, identified two likely ways in which the draft itself would take place.
In one format, all NHL teams would be allowed to protect three defensemen, seven forwards and one goaltender. Outside of these protected players, the expansion team(s) would be able to select just one player from each NHL club.
In a second format, Dreger forwarded the potential for teams to protect eight players, with no specific allotment of positions, as well as one goaltender. This option would allow teams who are heavy in terms of defensive talent to protect more than three defenseman, or on the other hand, up to eight forwards.
However, prospects, and players who have played less than two seasons, either NHL or AHL come the conclusion of the 2016-17 season would be exempt from selection and would not need to be protected. Therefore, a skater such as Auston Matthews, who will be a rookie next season, will not need to be protected.
However, a player such as Connor McDavid would need to be protected, as come the conclusion of the 2016-17 season, he will have played two seasons of professional hockey.
This process is designed on the expectation than all or any expansion teams would begin play in the 2017-18 NHL season.
The Oilers Protect…
The entire idea of expansion is certainly exciting, however it would force each and every NHL team to heavily consider those who they would protect, as well as those who they would make available to potential selection in an Expansion Draft.
For the Edmonton Oilers, choosing these protected players would be anything but a simple task, especially so considering how weak their lineup has been throughout recent history.
[Related: Who Would the Leafs Protect in an Expansion Draft?]
But, lets attempt to project those who the Oilers would protect in the case of an expansion draft, based on the model of securing three defenders, seven forward and one goaltender.
(Based on the Oilers roster as of the 2015-16 season)
Seven Forwards:
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Connor McDavid
Is there any point in making an argument here? McDavid looks as though he’ll be the best player in the NHL for a long time. Heck, he might already be.
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Leon Draisaitl
Likely the Oilers’ number-two center for the foreseeable future, Draisaitl brings size to the centre of the ice for Edmonton, a characteristic extremely important in the big bad Western Conference. The third overall pick in 2014, the big German has enjoyed a breakout season in 2014-15, yet has been inconsistent. Regardless, he is a vital part of the Oilers future.
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Taylor Hall
One of the best left-wingers in the NHL on a friendly contract with an AAV of $6 Million per year, Hall is another player who is going nowhere anytime soon. The Oilers, who are slightly thin on the wings, would be crazy not to protect Hall, a former first overall pick, in an expansion draft.
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Jordan Eberle
Although Eberle has been the focus of trade rumors in recent seasons, there is absolutely no chance the Oilers would risk letting the skilled winger be taken in an expansion draft. Edmonton’s top right winger has been an offensive staple in the Oilers lineup since entering the NHL in 2010-11, and more recently has found tremendous chemistry with newcomer McDavid.
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Ryan Nugent-Hopkins
Also the target of trade rumors, Nugent-Hopkins could be playing elsewhere by the time the possible expansion draft in 2017 comes into play, however, if he does in fact remain with the team, he is simply too valuable to let loose, unless perhaps the Oilers find themselves in a sudden cap crisis.
-
Benoit Pouliot
Plagued by injuries throughout his career, and again in 2015-16, Benoit Pouliot isn’t the most valuable player, especially on a team laden with such valuable pieces as the Oilers, but he does provide Edmonton with strong forward depth.
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Nail Yakupov
Like Nugent-Hopkins and Eberle, Yakupov too has been the subject of trade talk, and is the most likely of the three to be traded prior to an expansion draft. However, despite his continually diminishing value, the Oilers certainly would not let Yakupov go without a return. Sure, his contract does expire following the 2017-18 season when he becomes an RFA, but chances are he won’t draw an offer sheet, meaning the Oilers could re-sign him at a much lower cost, based on his current career projection.
Three Defensemen:
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Oscar Klefbom
When it comes to the Oilers’ top defenseman, Oscar Klefbom will surely be protected. Drafted by Edmonton 19th overall in 2011, Klefbom was recently signed to a seven-year contract extension worth nearly $30 Million. An gifted offensive player who plays a sound defensive game, there is a reason why the Oilers decided to keep him in town long-term.
-
Darnell Nurse
Selected by the Oilers 7th overall in 2013, Darnell Nurse is a physical defenseman who plays a defense-first style of game, something which Edmonton desperately lacks. Although he does have an offensive upside to his game, he has yet to showcase it at the NHL level, yet when he does, Nurse could become a pillar on the Oilers’ back-end.
-
Brandon Davidson
An incredible surprise for the Oilers in 2015-16 before an injury ended his season, Davidson, like Nurse, is also a sound defender who can chip in offensively. At 6’2″, 210 pounds, Davidson brings decent size to the blue-line, making him a solid piece for the Oilers moving forward. Given his upside and young age, chances are Edmonton wouldn’t risk losing him in an expansion draft.
One Goaltender:
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Cam Talbot
I would love to have Laurent Brossoit’s name here, but as of now, it is simply too early to tell whether or not he will develop into the Oilers’ future number-one netminder. In Talbot, the Oilers have a goaltender with NHL experience who has been nothing short of excellent in his time in Edmonton, despite being shelled on a nightly basis. Recently signed to a three-year extension, Talbot provides the Oilers with the best opportunity to win, yet an expansion team in search of a quality young netminder could come calling for Brossoit in the future.
Who Goes?
To put it frankly, the Oilers could have a much less difficult time determining who to protect in an expansion draft than previously thought.
In terms of forwards, all of the Oilers’ elite young talents would be secured, while a few depth players’ roles on the roster would also be ensured. In fact, of those not mentioned above, there is no players who the Oilers would truly miss if they were selected, with a few exceptions. The selection Matt Hendricks or Mark Letestu by an expansion team would gut the Oilers in terms of forward depth as well as leadership, while the loss of a player such as Patrick Maroon, although not monumental, would strip Edmonton of much needed size.
On defense, the Oilers lack a great deal of depth and skill to begin, making it much more difficult for the loss of a defenseman to significantly impact the team. The most notable defender left off of this list however is undoubtedly Andrej Sekera. Signed by the Oilers to a six-year contract in the Summer of 2015 with an annual cap hit of $5.5 Million, the Oilers could risk letting Sekera go if his play falters, especially so if Edmonton can significantly improve their defense-core in the near future. Otherwise, the losses of Mark Fayne, Eric Gryba or Griffin Reinhart would be less than significant.
In goal, the Oilers lack any sort of goaltending prospect, other than Brossoit of course, and it is too soon to tell whether he will develop into a true number-one NHL goaltender. As a result, Talbot is the only true option for the Oilers to protect. He has played extremely well for the Oilers in his first season and has experience to boot. Given his contract extension, Talbot will call Edmonton home through the 2018-19 season, while Brossoit will have to play his way into the starters role, despite recently signing an extension to keep him with the Oilers through 2017-18. | https://thehockeywriters.com/who-would-edmonton-protect-in-an-expansion-draft/ |
Buffalo Grove High School
We are fortunate to have two methods of communicating information to our student body: televised announcements and scrolling announcements. Both of these methods are effective; however, it is important to realize each method has limitations. As you submit announcements, please realize the following guidelines are in effect due to limitations of time and/or space.
Thank you for your cooperation and understanding. The process of televising daily announcements follows a very tight time line. Your understanding is appreciated. | https://bghs.d214.org/staff/student-announcement-guidelines/ |
Harvard seeks a visionary and collaborative professional to provide leadership in reimagining how the Harvard College Library and the Office for Scholarly Communication and its staff will provide expertise and tools for engaging with the full record of scholarship in new and innovative ways and to advance Harvard Library’s strategy of collective collection development within Harvard and with strategic external partners.
In addition, the Assistant University Librarian for Content Strategies will oversee the Office for Scholarly Communication to ensure that the record of Harvard’s scholarship is available to the world, to promoting Harvard values and influence in scholarly communications, and to influence the migration of scholarly communications toward open, community-supported structures.
By working in collaboration and consultation with collections staff across Harvard libraries, develop strategies that enable Harvard to operationalize collective change in collection development within Harvard’s strong tradition of local management of collections.
By identifying and managing Harvard College Library’s contributions to the collective collection in humanities, social sciences, sciences, and global studies in over seventy languages for the Faculty of Arts and Sciences, demonstrate collaboration that promotes closer integration of collections across Harvard, as well as the ReCAP and Ivy Plus libraries for the benefit of the Harvard and greater scholarly communities.
In collaboration with Library staff focused on digital scholarship, harness the Library’s wealth of resources for the advantage of our faculty and students; work with information vendors for the broadest use rights; advocate for the digitization of the Library’s legacy collections; and promote the use of the collections as data.
Provide management and consultation in the growth of local and collective collecting of distinctive materials in traditional and emerging formats, such as web archiving, digital ephemera, and post-custodial archiving of online information, to ensure the availability of the fullest record of human expression for future scholarship.
Support the ongoing professional development of subject-specialist librarians in Harvard College Library as they further diversify their engagement with faculty and students to meet emerging scholarly needs.
Oversee the Office for Scholarly Communication, to ensure that the record of Harvard’s scholarship is available to the world and to influence the migration of scholarly communications toward open, community-supported structures.
The Assistant University Librarian will oversee the allocation and management of over $20 million annually in funds for collections and related services. This individual will collaborate with collections librarians across Harvard, establish and nurture excellent relationships with faculty and students to align policies and practices with academic priorities, and work with Research and Education librarians on how the Library’s information resources may best support teaching and enriching the student research experience. Beyond Harvard, The Assistant University Librarian will collaborate with peers at MIT, in ReCAP, and across the Ivy Plus confederation to promote effective growth and management of the collective collection in all formats and to investigate further collective work, such as the development of distinctive collections and joint digitization efforts. The Assistant University Librarian will represent Harvard nationally and internationally as needed.
Although this position will remain open until filled, applications will be reviewed beginning October 1st, 2018.
Across the Harvard Library, our work is enriched by our diverse campus community. Our unique and wide-ranging abilities, experiences, and perspectives are integral to achieving Harvard University’s mission of excellence in research, teaching, and learning for our patrons, our collections, and our workplace. We believe that an inclusive environment that cultivates and promotes understanding, respect, and collaboration across our diverse workforce enables our success. The Assistant University Librarian for Content Strategies will promote and embody this commitment.
We encourage individuals with diverse backgrounds, experiences and abilities to apply to be a part of our community of over 700 staff members. Our work with faculty, students and researchers to explore answers to intellectual questions, enduring and new, and to seek solutions to the world’s most consequential problems, requires that we not only reflect, but also champion our diverse society.
Harvard Library inspires collaboration, reflection, experimentation, and discovery. Serving as an information hub, the Library connects users to related disciplines and to University-wide teaching resources. The Library engages users through curated discovery, digital collections, reimagined physical space, and specialized research support, and delivers exceptional experiences to its user communities. Today, Harvard Library’s holdings range from traditional print collections to rapidly expanding access to digital resources. Harvard Library provides the University’s faculty, students, and researchers—now and in the future—with comprehensive access to these materials. | https://acrlny.org/assistant-university-librarian-for-content-strategies-harvard-library-cambridge-ma/ |
FULL-TIME ASSISTANT SUPPORT WORKER POSITION (KILLEIGH, CO. OFFALY)
THE COMPANY:
Bradbury House Ireland provide residential care for adults with Autism
Spectrum Disorder and Intellectual Disabilities. At Bradbury House
Ireland, we provide the highest quality care for service users in
accordance with the regulations and standards applicable, which is
always person centred. We provide a high standard living environment
that is safe, clean and comfortable.
KEY RESPONSIBILITIES:
* To promote a caring environment for service users through high
standards of professional practice which are conducive to the
physical, emotional, social and intellectual needs of the service
users.
* To ensure that each service user receives care appropriate to
their individual needs, service user plan and is always person
centred.
* Capacity to assess and develop individualised support plans.
* To undertake life-skills, programmes and activities are detailed
in individual service user plans and timetables.
* To promote relationships which enable each service user to
participate in the life of the local community to the best of their
ability.
* To ensure the provision of healthcare arrangements which may
include the recording and administration of prescribed medications.
* Encourage and support service users to live as fully and
independently as possible within the local community, providing
information, emotional, organisational and practical support and
training as appropriate.
* Flexible to work relevant shifts including overnights and weekends
as appropriate
* Contribute to the protection of service users from abuse, follow
Bradbury House Ireland’s policies and procedures and comply with
external guidance i.e. HSE, HIQA, etc
ESSENTIAL CRITERIA:
* Minimum level 5 related to Social Care
* Excellent working knowledge of HIQA regulations and standards and
New Directions for Day Services.
* Excellent verbal and written communication skills IT skills,
Report Writing Skills, Systematic approach to make efficient use of
time, Manage Workload, Good Planning Skills, Positive 'can do'
attitude, Enthusiastic, reliable and flexible, Commitment to real
community integration, Ability to initiate strong links within
community.
* Support individuals to acquire valued social roles, Ability to
cope with Challenging Situations, Manage and Supervise Staff,
Self-Motivation and ability to use one's initiative
* Ability to work as part of a team and independently
* Previous experience within the area of social care or supporting
individuals with disabilities.
* Complete Garda Vetting on application
* Full Clean Driving Licence (which permits the holder to drive in
Ireland) is essential for all positions in the Bradbury House Ireland
(Provisional Licences are NOT acceptable / Automatic Licences are NOT
accepted).
* Eligibility to work in Ireland.
DESIRABLE CRITERIA:
* Third Level degree as Social Care Worker
* Previous experience with supporting individuals with Autism
Spectrum Disorder and Intellectual Disability.
* Experience of Supervising a Team.
CONTACT:
Please forward your CV and cover letter by email to **APPLY ON THE
WEBSITE**
CVs must provide full details of all relevant experience &
Qualifications. Any gaps must be clarified.
Interviews will take place periodically in the coming weeks and
positions offered in line with service need
Closing date for applications: **APPLY ON THE WEBSITE**/**APPLY ON THE
WEBSITE**/**APPLY ON THE WEBSITE**. Based in Tullamore area.
We need : English (Good)
Type: Permanent
Payment: | http://dublinwork.com/jobDetail/7415-Irish-Future-Bradbury-House-Ireland:-Assistant-Support-Worker.html |
The procedures to apply RAN include mainly the review of existing data and the assessment of the perception of the population to be targeted in the future. The analysis of documents or statistics, and the application of qualitative rapid assessment procedures are the predominant tools during project identification.
RAN usually takes no more than three weeks and depends on the professional experience, the working conditions, the knowledge on local languages, the capacities of the involved institutions and the experience with participatory and rapid assessment tools.
RAN can be applied as a particular methodology, but it can also be part of a general project identification mission.
RAN is conceptualised for the systematic assessment of nutrition related information during project identification. The presented methodology serves to guide and orient a project finding mission, collecting data under time pressure and within an unknown context. RAN should therefore be adapted to the local context, e.g. when it is applied in conflict post-war regions or in transformation countries.
Figure 1: Three methodological steps of RAN
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Step 1: Data collection at the capital, provincial and district level (government institutions, NGOs etc.)
Methodology:
a) Review of scientific literature, which can provide some poverty oriented cause-effect-relationships for the population in a specific country.
b) Review of information gained in governmental offices (Health department, agricultural sector, statistical offices etc.), which provide useful general data on the demographic, health and socio-economic situation at a regional level.
c) Review of studies conducted by international organisations (World Bank, World Food Program, International Red Cross, Medecins sans Frontière) and national NGOs (Associations working in the area of public health, nutrition, women’s affairs, watershed management or rural development).
The outcomes:
a) First Interim Report.
b) Suggestions and assumptions, based on the review of the literature and secondary data records, for reflection and study in the rural/ urban areas and communities.
Step 2: Collection of information in a predefined urban or rural community, targeting local governments, NGO representatives and the people
Methodology:
a) Several trips to selected communities should be undertaken to talk with the people in the surveyed area.
b) Transects should be made with local government officers and community leaders, through the district or community, in order to observe and compare the secondary data and the verbally commu nicated information with the reality.
c) Group discussion should be carried out with local people, to obtain an idea about food habits, beliefs, customs and perceived needs in the sector of food security.
d) Measuring the body stature of school children could be a tool if no anthropometric data is available in the region. Measuring the height of children with a simple anthropometric tape can provide important information on chronic malnutrition i. e. if children are too small for their age. This reflects the poor conditions or poverty situation in the communities.
The outcomes:
a) Second Interim Report, based on the outcome of the talks with local key persons, semi-structured interviews, and question guidelines applied during group discussions in the field.
Step 3: Data analysis and report for discussions with representatives of relevant national institutions and interested NGOs
Methodology:
a) The analysis of the survey forms provides comparable information between communities or geographical areas.
b) The analysis of the semi-structured interviews provides complementary information about the perception of the people concerning nutrition and poverty.
c) The analysis of the group discussion provides specific topics (problems, ideas, doubts, perceptions, self-assessment of poverty etc.), which could have emerged during the discussion process at community level.
d) Cross checking the different sources of information and persons to be involved during project identification helps clarify the understanding of the nutritional and/or poverty situation of the population.
Outcomes:
a) The final report, including the main findings, hypothesis for the project planning and conceptualisation of areas to be studied in detail, and the general conclusion on the nature of the project (e.g. Integrated Food Security Project, agricultural extension project, community development project, specific nutrition project etc.)
Summary
The following table presents the elements that need to be scheduled while undertaking a RAN during the project identification stage. The table summarises the previous steps 1-3 into 16 smaller steps.
Useful forms for field work Two types of practical forms have been developed to facilitate the application of RAN during project identification:
These forms should be considered as an aid. If necessary, they should be adapted to the specific situation of a survey, and not be considered binding.
1) Analysis Forms (AF)
The Analysis Forms are developed for:
a) Data comparison of national average, province and district to identify limited resources and problems.
b) Partial justification of the pre-selected area for the future project/ program implementation (targeting approach).
Table 1: Summary of steps 1-3
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Note: In general, at the District and Community level, the socioeconomic situation of the population and the supply of resources should be worse than at the national average and province level, to be sure an area with poverty conditions has been selected.
Examples of analysis forms have been prepared for fieldwork:
a) Seasonality in the surveyed area.
b) Data for the agricultural sector.
c) Data for the health sector.
d) Data for the economic sector.
e) Data for the educational sector.
f) Demographic data.
g) Analysis of anthropometric indices (Based on the data obtained from measuring the body stature of the schoolchildren during RAN).
Analysis forms compare the sector related data at the national, provincial and local level, in order to determine the most affected areas. The following table is an example:
Table 2: Analysis form comparing sectoral data at different levels
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2) Survey Forms (SF)
The survey forms are provided to:
a) Guide the practitioners again on the points considered as important for observation, interview and group discussion.
b) Fill in the results of the observation, semi-structured interviews, and group discussions with community members of all communities.
c) Visualise the results obtained in all communities, comparing them and identifying weaknesses and strengths.
d) Include the available resources and existing habits in the analysis of alternatives.
Examples of survey forms have been prepared for fieldwork:
a) Infrastructure in the community.
b) Living conditions of the visited households.
c) Demographic and socio-economic data.
d) Health services in the 5 communities.
e) Survey Form AGRICULTURE – Cultivation.
f) Survey Form AGRICULTURE – Domestic animals.
g) Food consumption habits.
h) Breastfeeding, supplementary feeding and weaning practices.
i) Anthropometric data collected from schoolchildren.
j) Analysis of groups and institutions potentially participating in the project.
Survey forms compare the obtained data between different communities, e.g. table 3.
Table 3: Survey form comparing obtained data between different communities
Additional information: Measurements of body stature of school children
The most reliable way of assessing the nutritional situation of a community is to take the body measurement of the children. Children show body growth failures when they grow up under poor conditions or if they are suffering from food shortage or infectious diseases. These failures in body stature are due to long term nutritional or health problems and can be measured. The relative height for age of a preschool child is internationally recognized as an appropriate indicator for the nutritional problem within a community.
Therefore, anthropometric measurements of the first grade in local schools are a good proxy indicator for nutritional problems within the community. Samples should be taken during project identification according to given field conditions and time constraints.
The standing body height (stature) of the schoolchild is measured in a standing position. The child should stand without shoes on a level floor and should lean with its back against a wall. The child then draws himself or herself to full stature without raising the shoulders, with hands and arms hanging relaxed, with the feet flat on the ground. The legs and heels should be placed against each other. The buttocks, shoulder blades and head should rest against the wall. The estimated line between the eyes and the auditory passages should be level with the floor.
The stature should be measured using a microtoise fixed to the wall (to the nearest 0.1 cm). If a microtoise is not available, a wooden rule or a measuring tape (preferable a tape made out of fibreglass) should be placed against the wall. When measuring stature, the measuring tapes or wooden rules, in which a device is placed on the head when the tape or rule is fixed to a wall should not be used. The scale should commence at 0 cm at the floor. Otherwise, the result can be inaccurate, about 1 cm short.
A wooden or metal right angle should be employed in measuring. This instrument is placed lengthwise against the measuring tape on the wall and is pressed gently against the head so that the stature can be read on the measuring tape in cm.
Measuring tapes are sometimes calibrated in inches and centimetres on the same side. This can cause confusion during measurement.
Table 4: Anthropometric Data Sheet of School children
Presentation of findings
The final report of RAN should present all relevant nutritional and poverty related information of the potential target population as listed below:
a) Description of the situation by sector (Summary of the analysis forms).
b) Description of the situation by region (Summary of the survey forms).
c) Description of the situation from the community point of view (Analysis of the group discussions and key interviews in the field).
d) Estimation of the situation from the point of view of policy makers (National, regional, and local).
e) Results of the anthropometric measurements.
f) Preliminary description of the problem tree.
g) Proposal for the type of project.
h) Proposal for the main project strategies.
i) Analysis of alternatives.
j) Summary.
Comments:
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I’m not normally one for wanton vandalism but some casual graffiti in Galloway has caught my eye. The victim is a road sign warning of cattle. You know the one. A red triangle around the black silhouette of a cow. Except this one has been modified with a small piece of white tape across its midriff, transforming it from generic cow to unmistakable Belted Galloway.
This made me smile, as the real life belties were in the adjoining field, nonchalantly chewing the cud; seemingly oblivious to the yobbo’s efforts to simply make the sign more accurate.
While the petty vandals of Galloway might have a sense of humour, one product of the region that doesn’t is Corserine. This great hulk of a mountain, one of four Corbetts in the national park, is a bit of a goody-two-shoes. Standing at 814m, it’s little more than a grassy hump in a vast landscape. The sensible kid in a class of scallywags. It does its homework on time and wouldn’t dream of defacing a road sign.
It’s probably for this reason that Corserine, like many of the mountains in Galloway, is likely to be quiet. Walkers can’t resist the draw of the proper Scottish mountains just a few hours to the north. You need to invest in Galloway to get the best from it. It requires patience and graft before you can reap its rewards. But for those with the resolve and necessary skillset, time spent in Galloway offers a unique mountain experience.
For all its dour reputation, Corserine does have a playful side if you approach it from the east. A huge and remote corrie, largely devoid of paths and with plenty of grippy outcrops to scramble on, ultimately leads to the bulbous summit. Here you can marvel at the mind-bogglingly huge landscape laid out before you.
Corserine is the high point on a ridge that stretches from north to south, marking the eastern boundary of the Galloway Forest Park. A desolate plantation houses The Merrick and a number of lochs to the west. East is the Forrest Estate then just miles and miles of southern Scotland. It’s a brilliant day out and a step up in terms of wilderness for those used to walking in busier national parks.
A slow start
After a chilly night in the van in the far north of England, I arrive to a crisp November day in the Forrest Estate. A long private road takes me as far as the walker’s car park. From here I’m on my own two feet.
This is no bad thing. The 5km approach gives ample time to loosen up the legs. And the scenery isn’t bad either. Starting on wide forest tracks, I head towards Loch Harrow, a tranquil stretch of water flanked by trees and cradled by the craggy slopes of Craigbrock. Investing my gaze further into the view, I sketch out parts of my planned descent from North Gairy Top in the shadows. But first I’d need to get on to the mountain.
I had in mind to skirt the eastern shores of the loch before skipping across a burn emanating from the frigid waters. Alas, the stream proves to be more wild water rapid than babbling brook, forcing me to retrace my steps to pick up the main forest track again.
Enchanted forest
The path climbs through the trees, gently at first but it becomes more of a workout as I head deeper into the forest. The dark floor under the trees is carpeted with mushrooms. The forest feels utterly enchanted and I wouldn’t have looked twice if there were fairies playfully flitting around the trees.
At an obvious bend in the road, I turn on to an unkempt track, signalling I’m about to enter more serious terrain. I pause by Folk Burn to fill my water bottle and let the sounds of the forest wash over me. It may be a managed forest on a commercial estate but I can’t help but feel I’m the only person for miles around. I sense the forest spirits soothing my nature-depleted soul as I breathe in the clear air.
It’s time to leave the arboreal sanctuary and join the open fell. This is Corserine’s flirtatious side. No more miserable lump in a huge expanse of nothingness. The corrie is rough and pathless. Only the subtle bends in stalks of grass offer reassurance that others have passed this way before.
The waters of Folk Burn cut a deep scar into the otherwise featureless topography as my gaze is drawn to the dizzying heights of the cloudy summit plateau. It’s tempting to retreat immediately to the safety of the forest as the mountain hands me the dice and waits for my next move.
Into the wild
I take my first tentative steps into the wild and trace a route up the steepening slopes. My target is a subtle shoulder just before the worst of the steep ground and it’s a case of trusting my feet to find a natural way through the rugged ground. From here, I plan to alter direction and head north to the more prominent shoulder of Craigrine at 709m. Then it’s a more pleasant push for the summit via the obvious north-eastern spur of the mountain.
Reviewing my progress, I see just how far I’ve come to get to this point. The scale of this place is staggering. A patchwork of greens and browns and blues stretches to the horizon and beyond. I can see the fells of the Lake District and endless, vast skies above. It’s breathtaking.
As I stare to the heavens, a small, red aeroplane buzzes into view. My imagination tells me I am Richard Hannay from John Buchan’s ‘The 39 Steps’ – on the run from an unseen enemy. But the plane pays me no attention, leaving me safe to continue my own adventure.
Appreciative nods
Reaching the summit of Craigrine, an icy wind hits me like a steam train. I furiously dig out more layers and hunker down for the final summit push. On the plateau, I meet a large bank of cloud, which obscures anything of interest. I snatch the odd glance of Carlin’s Cairn, the next summit on the ridge heading north. There’s an occasional glimpse of the huge expanse of hills to the west. I also see my first human of the day. We exchange the briefest of appreciative nods before he seeks refuge from the wind by pressing on towards the Rhinns of Kells.
The cold soon gets the better of me and I continue down the south-east ridge to North Gairy Top. I am engulfed in cloud for a time but the strong winds are on my side and soon clear the fuzz as I lose height.
This part of the walk is fun. A lofty place from which to survey my surroundings and enjoy a spot of lunch. It’s too cold to pause for long. And there’s just the small matter of the tricky descent via North Gairy Top to negotiate.
I inch my way down the steep grassy slope, weaving around small crags and aiming for a fixed point in the depths below. It’s then a case of following the fence north to pick up a break in the trees that will lead me back through the estate to the car. But as is the Galloway style of things, the going is rough and it takes more time and energy than expected to seek the refuge of the enchanted forest.
As the light starts to fade in the late afternoon, I settle into some relaxed walking over the comparatively gentle paths through the forest. I understand that many people simply won’t ‘get’ Galloway. There’s no instant gratification to be had from its lack of spiky ridges and intimidating crags. But put in the effort and you’ll soon realise the sense of ruggedness sometimes offers far greater returns.
Highs and Lowdown
Rating
Start / Finish: Forrest Lodge
Distance: 14.8km
Ascent: 710m
Navigation: Strong navigation skills required in this vast and unforgiving landscape.
Terrain: The going is tough over rugged, pathless terrain. Steep, grassy descent from North Gairy Top. Previous experience of this type of terrain essential.
Facilities: St John’s Town of Dairy
*These routes and descriptions are only ever intended to be a personal record of my adventures, which may inspire your own. Hillwalking involves a degree of risk, so please make sure you are properly equipped and prepared if you choose to follow them. | https://highsandlows.co.uk/corserine |
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PROGRESSIVE STEPS of a carving subject are a visual aid to visualize the development from the opening cuts through steps towards a completed carving. Here are three examples of Progressive Steps in the accompanying photographs of carving a male face in the top illustration and the female face in the second and third illustration. With each Progressive Step illustration the viewer is to study each face in progression of development to imagine what cuts were necessary to accomplish each step. While being a helpful reference tool, the greater function of the Progressive Steps illustration is to help the carver to develop the ability to see a carving develop in one’s imagination.
Studying a Progressive Steps illustration the viewer realizes that one has to read between the lines, so to speak, when following each step and allow imagination to fill in the blanks in between each step. Using the Brasstown Carvers illustration in the top photograph, let us consider each step in the development of the male face. The first example on the left shows the eyebrow notch and nose has been carved as the opening cuts. The second example shows the side of the head at the forehead has been narrowed. The third example shows the hairline has been established, the forehead received its angles of shape and smile line notch has been carved on one side of the nose nostril. The fourth example shows the face has been narrowed down from the cheek bones through the sides of the jaw line, both smile lines notches have been carved to establish the sides of the dental curve with the beginning of a mouth mound. Notice also that the face proportions have been established with three equal division of Hairline to Eyebrows; Eyebrows to Tip of Nose and Tip of Nose to Bottom of Chin. The fifth, sixth and seventh examples show hair styles developed, ears developed, eyes carved, mouth carved into smiles along with facial wrinkles of three individual faces. Example eight shows the beginning of a hat with the head going up into the hat and example nine shows a detailed Santa hat and Santa face with mustache and beard with all the other facial features that proceeded in the earlier examples.
The two Female Face Progressive Steps are to be studied in like manner allowing one’s imagination the fill in the gaps in-between. “If it can be imagine in can be,” is born out in the creative process of being able to see with the mind. The mind will guide the hands that hold the shaping tool in removing the necessary wood into the shape needed for a good foundation for the eventual final step of laying in the detail features.
The next step after studying a Progressive Steps guide with imagination is to practice carving each step with a carving tool being guided by one’s imagination realizing that this practice exercise is a learning process than may be repeated several times to gain the skill of working with imagination and tool guided by hands shaping the wood. During these practice sessions the creativity of imagination will continue to make discoveries and fill in the gaps and read between the lines. Continuing this process one will discover that “the more you carve the better you carve,” and “learning never ends as it follows the Progressive Steps to learn as you carve.” Remember the Wood Bee Carver’s motto: “Would be carvers would be carvers if they would carve wood.”
Below is another example of Progressive Steps for carving a female face created by the late George Keilhofer who was the resident woodcarver in Frankenmuth, MI.
May all your carving steps be “Progressive” towards getting the most out of your carving journey.
You must be logged in to post a comment. | https://woodbeecarver.com/progressive-steps-a-learning-tool/ |
I swear by the book my Aunt Purdy gave me:
The Baker in Me by Daphna Rabinovitch which has many great sweet or savoury recipes with a small section after most recipes on how you can change it up or substitution ideas for certain flavours or for allergy accommodations. At the beginning of the book, it also goes through certain skills and techniques for beginner bakers like me so we don’t mess up… as frequently.
Some other recipes I pulled off google that i’ve tried and loved:
Cajun Sweet Potato Fries – Spicy Mayo Sauce not included but goes very well with it:
https://minimalistbaker.com/cajun-baked-sweet-potato-fries/
More from google:
Zucchini Pasta with Lemon Garlic Shrimp
With Downshiftology and Lisa Bryan
ZUCCHINI PASTA WITH LEMON GARLIC SHRIMP
YIELD: 4 servings TOTAL TIME: 15 minutes PREP TIME: 10 minutes COOK TIME: 5 minutes
INGREDIENTS
- 4 medium zucchini
- 1.5 lb (approx 30) raw shrimp, peeled and deveined
- 2 tbsp olive oil
- 4 garlic cloves, finely chopped
- 2 tbsp butter or ghee
- 1 lemon, juice and zest
- 1/4 cup white wine (or chicken broth)
- 1/4 cup chopped parsley
- pinch of red pepper flakes
- salt and pepper, to taste
DIRECTIONS
- Wash and cut the ends of the zucchini. Using a spiralizer, make the zucchini pasta. Then, set aside.
- Heat the oil in a large pan over medium-high heat. Add the shrimp in one flat layer and sprinkle with salt and pepper. Cook for one minute without stirring, so the bottom side gets a little crispy.
- Add the chopped garlic, then stir the shrimp for another minute or two to cook the other side. Use a large spoon or tongs to remove the shrimp to a plate.
- Add the butter, lemon juice and zest, red pepper flakes and white wine to the pan. Bring to a simmer for 2-3 minutes while stirring.
- Stir in the parsley, then add the zucchini pasta and toss for 30 seconds to warm it up. Add the shrimp back to the pan and stir for another minute. Serve immediately.
December 2, 2019 – I’ve been really into making banana bread as a study snack for finals. It also helps that I have a lot of bananas that were too ripe.
Anyways, here is a B-Bread recipe which is now my go-to after my friend Maddy introduced it to me! (credit to Chrissie Baker)
Link: https://thebusybaker.ca/best-ever-banana-muffins/
BEST EVER BANANA MUFFINS
Baking Notes
Ingredients
- 3 large ripe bananas
- 3/4 cup white sugar
- 1 egg
- 1/3 cup oil vegetable oil is best
- 1 1/2 cups all purpose flour
- 1 tsp baking soda
- 1 tsp baking powder
- 1/2 tsp salt
Instructions
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Preheat your oven to 350 degrees Fahrenheit and line your muffin tin with 12 paper muffin cups.
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In a large mixing bowl, mash the bananas with a fork until no large pieces remain.
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Add the sugar, egg and oil and mix with a wooden spoon until everything is incorporated.
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Measure the flour, baking powder, baking soda and salt into a separate mixing bowl and stir them together with a fork. Once they’re mixed together, dump the dry ingredients into the wet ingredients.
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Using a wooden spoon, carefully mix the wet and the dry ingredients together. The danger here is over-mixing which will result in dense, flat and dry muffins. The best technique for mixing is to scoop underneath the mixture and pull up gently instead of stirring around in a circle (like you’d stir your coffee, for example). Continue mixing this way just until the dry ingredients have been incorporated. It’s fine if there are a few small streaks of flour in the batter; it’s the large pockets of flour you want to make sure are incorporated.
-
Spoon the batter into 12 lined muffin cups. Try your best to ensure you fill the cups evenly. This will ensure each of the muffins is finished baking at the same time (you don’t want some over-baked and some raw in the middle!)
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Bake the muffins in the oven at 350 degrees for about 23 minutes (anywhere from 22-25 minutes, depending on your oven), or until they are golden brown.
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Let them cool in the tins for about 5 minutes and then turn them out of the tins to let them cool completely on a cooling rack. Or, enjoy one warm from the oven slathered in butter.
Recipe Notes
These muffins can be stored in an airtight container for up to 4 days, although they’ve never lasted that long in our house!
You can also cut down on the sugar and add apple sauce for a healthier alternative!
The BEST Red Velvet Cake!!!
Wooow! I made this as a birthday cake, first time making a red velvet cake and one of my first couple attempts at making a cake in general. Turned out DELICIOUS AND MOIST AND DENSE. 10/10
Found the recipe from google below. It suggests using cake flour, which I didn’t have but it gives you recipe to make your own cake flour (flour and corn starch)
Credit to Sally’s Baking Addition
https://sallysbakingaddiction.com/red-velvet-layer-cake-with-cream-cheese-frosting/
How to Make a Homemade Cake Flour Substitute
Step 1: Measure 1 cup all-purpose flour. Remove 2 Tablespoons.
Step 2: Measure 2 Tablespoons cornstarch. Add to the flour. Cornstarch contains less gluten than flour, so it’s a wonderful tenderizing ingredient to help make cake flour.
Step 3: Sift together TWICE. Basically, sift into a mixing bowl once. Then run it through the sifter one more time. Sifting not only mixes the two ingredients together appropriately, it aerates the mixture so the consistency is similar to real cake flour.
Step 4: Measure 1 cup from this mixture. You’ll have about 1 cup anyway, but sometimes sifting can produce more volume since it’s adding air. | http://alysaevans.com/favourite-recipes/ |
1) Vacation. I’m not really very good at taking a vacation. To be more exact, I’m not really very good at BEING on vacation. Once upon a time I actually took many weeks of vacation every year; I wasn’t very good at being on vacation then, either.
It’s not that I don’t enjoy being on vacation, it’s more that it simply takes me too long to realize that I’m actually on vacation. To relax. To be okay with doing exactly… nothing.
I am now day one back from yet another week of failure.
2) Courage. I read the letter from GruntDoc’s blog posted above. The soldier who sacrificed himself so that his comrade might live demonstrated HEROISM, the practical application of courage with a significant aspect of selflessness thrown in. Not too many people have an opportunity to actually do this. What is extraordinary, at least to me, is how frequently this occurs when folks are given the chance. I am awestruck and humbled every time I read one of these stories.
Courage doesn’t always need to be attached to heroism, however. Courage is also the willful refusal to give in to one’s fears. It matters not what others think of any particular fear; fear is intensely personal, as real as any object we might touch. Overcoming that fear, finally screwing up enough courage to move through, then past that fear, is significant and monumental. Always.
I was there last night when “Lovely Daughter” overcame one of her terrible fears. It doesn’t really matter what it was. I was there to witness a little tiny glory, the glory that comes when one finds the courage to win the battle agains fear.
3) The Valley. “… when you start hurting you’re not even CLOSE to the bottom of the valley, and if you don’t panic at the first agonies there’s much, much more of yourself to give.”–Sebastian Junger.
I’m having trouble pushing through the pain. In almost any physical endeavor, but most definitely in Crossfit, it’s necessary to go to a pretty dark place to get the full benefit of the enterprise. I’m having trouble going there recently. I’m not really sure why, to tell the truth. There’s really nothing all that different about today, or yesterday, or last month in comparison with three, four, or six months ago. Nothing different, except for the fact that I’m opening the door to that dark place, but I’m not walking through.
I’ll go there again, of that I am sure. The payback, the benefit, is simply too great to continue to turn back. Perhaps it is the solitude within which I have always traveled when passing through that dark place, or perhaps it is simply the duration of the solitude.
Something is missing, something not exactly like, but something seems like courage.
4) Sonhood. in a few moments I’ll be leaving the warmth and comfort chez bingo, off to attend yet another wake for the parent of a friends. I have reached that stage in life where my friends and acquaintances are losing parents.
That’s not all that’s lost, of course. When we lose a parent, especially when we lose that second parent, we also lose a significant part of who and what we all are. For me it will be losing the role of “Sonhood”. At some time I, you, we will all cease to be someone’s child. The age at which it occurs matters, of course, but in the end when you’ve lost your parents you have become an orphan.
Frankly, my own personal “sonhood” was much easier and much simpler when I was younger. Right up until about the age of 35 it really didn’t take all that much thought or effort. Some rebel, cut the cord, whatever, in their teen years, or even before. I managed to put off whatever rebellion I had in me until around the age of 35. And yet, despite the fact that being the son has actually been a greater challenge as an adult, I revel in the fact that I am still someone’s child.
I guess that’s the point of this, eh? It’s not Father’s Day or Mother’s Day, but it COULD be. Hard or easy, with or without the necessity of thought or planning, in most cases “Sonhood” (or “daughterhood”) is really one of those good things we have, those good things we are.
I’m sad for my friend and his loss, and I’m also a little sad that it sometimes takes something like this to remind me of my “sonhood”, to not take for granted that I am still someone’s child, to remind me to actively engage in being a son.
I’ll see you next week…
This entry was posted on Sunday, February 13th, 2011 at 4:24 pm and is filed under Crossfit, Random Thoughts. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site. | http://drdarrellwhite.com/sunday-musings-21311/ |
The aim of this work was to complement the studies of the mechanisms of control of the degree of circular polarization of emission from resonant tunneling diodes p-type (RTD) by analyzing the optical properties and transport of this type of structure. We focus primarily on the influence of the width of quantum well to these properties and the possible injection of spin-polarized charge carriers, from the two-dimensional hole gas formed in the accumulation layer adjacent to the barriers. First, we discuss the theoretical foundations necessary for understanding the work, followed by the description of samples and experimental methods to perform this study. The results showed that the degree of circular polarization of luminescence of the quantum well is strongly correlated with the transport across the diode and with the separation of the emission energy of each spin component. We note also that reversal signal degree of polarization in the regions of resonant tunneling, which were associated with different Landè g-factors of electrons and holes and the injection through the spin channels. Regarding the issue of contacts emition, we observed the presence of two major contributions, one related to the three-dimensional contact and other related two-dimensional hole gas formed in the accumulation layer. The luminescence of gas did not show significant spin polarization, preventing direct analysis of its influence on the spin properties of quantum well. Finally, we conclude that the variation in width of the quantum well can greatly influence the spin properties of RTDs, and the study of these properties may allow the architecture of new spintronic devices. | https://repositorio.ufscar.br/handle/ufscar/5026 |
Call for applications: Master’s and PhD programs at The New School for Social Research in New York City
Is capitalism sustainable? — Can democracy exist without dissent? — Can critical theory be creative?
Graduate students at The New School for Social Research ask the kind of questions that challenge the status quo across the social sciences and humanities.
Guided by rigorous scholarship and a desire to apply academic discovery to current social problems, our students critically examine interdisciplinary fields to become a force of new knowledge and ideas in the world.
All graduate programs at The New School for Social research can be completed full-time or part-time on our New York City campus. Competitive merit-based scholarships are available in all departments -- in recent years, 85% of master’s students have received merit scholarships at The New School for Social Research.
Social Sciences and Humanities Departments
• Anthropology (MA, PhD)
Combine fundamental concepts of anthropology with critical exploration of the nature and role of ethnography in this leading graduate program.
• Creative Publishing and Critical Journalism (MA)
Prepare to lead the future of serious intellectual publishing in a program that combines print and emerging technologies, rigorous coursework, and design experience.
• Economics (MA, MS, PhD)
Explore evolution of economic thought encompassing a spectrum of heterodox theories and methodologies, including post-Keynesian, Marxian, and neo-Ricardian. Specialized degree option in Global Political Economy and Finance (MA).
• Historical Studies (MA)
Design your own interdisciplinary curriculum as you develop critical, heterodox approaches to historical questions.
• Liberal Studies (MA)
Develop contextual thinking and writing skills across multiple humanities disciplines and engage with leading scholars, journalists, and intellectual writers.
• Philosophy (MA, PhD)Investigate the history of Western thought and modern European philosophy in a program developed by renowned scholars including Hannah Arendt, Hans Jonas, Aron Gurwitsch, and Reiner Schürmann.
• Politics (MA, PhD)
Study the relations and manifestations of power in contexts ranging from the family to the transnational environment, paying close attention to historical and contemporary movements.
• Psychology (MA, PhD)
Investigate modern psychology with an emphasis on social and cultural influences and concerns. Specialized degrees include an MA concentration in Substance Abuse Counseling and PhD programs in Clinical Psychology and in Cognitive, Social, and Developmental Psychology.
• Sociology (MA, PhD)
Examine social inequalities, culture and politics, law and citizenship, and beyond through the lens of The New School’s historical connection to European social science.
Faculty and Research
More than 75 full-time faculty members are engaged in teaching and researching alongside the 800+ graduate students from 70 countries who study at The New School for Social Research. Our faculty is united by a sense of the importance of boldly questioning conventional thinking and expanding the boundaries of social thought.
Faculty continue to publish important books and to engage in dialogue in popular media. Some brief but notable examples include books recently published by Federico Finchelstein (The Dirty War, Oxford University Press), Janet Roitman (Anti-Crisis, Duke University Press), and Dmitri Nikulin (Comedy, Seriously, Palgrave Macmillan), and newspaper columns by Simon Critchley (The New York Times) and Teresa Ghilarducci (The Huffington Post).
About The New School for Social Research
The New School for Social Research was founded in 1919 as a home for progressive thinkers, and housed the University in Exile in 1933, providing an academic haven for scholars persecuted in Nazi Europe. The school became the foundation for a comprehensive university – The New School – and continues the legacy of critical thought, civic engagement, and academic freedom today. | https://pcdn.global/question-convention-generate-progressive-scholarship-new-school-social-research/ |
The South-West Open Youth Orchestra, with support from UWE Bristol Music Technology lecturer and composer, Dr Liz Lane have won the Royal Philharmonic Music Award for Learning and Participation.
The RPS Music Awards, presented in association with BBC Radio 3, are the highest recognition for live classical music in the UK.
Founded in 2015, South West Open Youth Orchestra (SWOYO), run by Open Up Music are the UK's only disabled-led, regional youth orchestra. Liz is Composer in Association for the Orchestra and was influential in their inaugural performances in 2016.
Funding from UWE's Faculty of Environment and Technology Public Engagement and Outreach programme, enabled Liz to rework her World War One inspired composition 'Silver Rose for SWOYO's inaugural performance at a Centre for Performing Arts concert in April last year. This performance was documented in a feature on the BBC's The One Show.
Liz was also honoured to create Silver Song, the first original composition for SWOYO which was performed live in June 2016 for Radio 3 on BBC Music Day.
The Learning and Participation category sponsored by the Associated Board of the Royal Schools of Music celebrates an initiative which inspired wider participation in music-making during 2016 and which created high quality musical experience and understanding.
Dr Liz Lane, says, “It was wonderful to share such a special evening with Clarion player George from the South-West Open Youth Orchestra and the OpenUp Music team. It has been an exciting journey over the last year and a half, and fantastic to collaborate with the young musicians and OpenUp Music, who have worked tirelessly to take SWOYO to this point - next step the National Open Youth Orchestra in 2018! My grateful thanks go to UWE for supporting this work.”
An audience of over 400 guests from the classical music world attended a celebratory presentation and dinner in the City of London. Highlights from the event will be broadcast on BBC Radio 3 at 19:30 on Sunday 14 May. | https://info.uwe.ac.uk/news/uwenews/news.aspx?id=3606 |
A number of commodity analysts continue to believe corn prices will experience on-going volatility, despite the fact that USDA this week predicted this year’s corn crop will be the largest on record at 13.505 billion bushels. The “World Agricultural Supply and Demand Estimates” report this week from the World Agricultural Outlook Board forecast that 92.2 million acres of corn would be planted this spring with 85.1 million acres harvested for grain, 3.7 million more acres than last year. Yield was put at 158.7 bushels per acre, 3.0 bushels below the 1990-2010 trend line reflecting the slow pace of planting progress through early May. Nonetheless, this year’s forecast corn yield is the third largest on record.
USDA forecast ending corn stocks for 2011-12 to be 900 million bushels, 170 million bushels more than the revised upward (55 million bushels) stocks of 730 million bushels for 2010-11. The 900 million bushels represents a very tight 6.6 percent stocks-to-use ratio for 2011-12 and will keep the grain market “on edge,” according to the May 12 issue of the Daily Livestock Report. For 2010-11, the stocks-to-use ration will be 5.4 percent, according the reports this week.
USDA predicted 2011-12 average farm prices for corn to be $5.50-$6.50 per bushel, up significantly from the $5.10-$5.40 average estimated for 2010-11. In 2009-10, the comparable corn price was $3.55 per bushel. “It’s hard to say what yields will be,” Chad Hart, agricultural economist with Iowa State University told AgWeb this week. “In previous years when corn planting has had this sort of delay, we’ve seen a reduction in yield of about 3 to 5 bushels per acre.” “There’s a lot of uncertainty out there,” Hart said. | https://nccwashingtonreport.com/2011/05/13/corn-prices-expected-be-volatile-despite-udsa-new-crop-report/ |
"Meriam Library","39.728066","-121.846302" , named after Ted Meriam, is the library on the Chico State campus. It is the building with the largest square area of any building in Northern California. Their primary function is to assist in fulfilling the mission of teaching and research. Therefore the resources contained therein are mainly for the use of faculty, staff and students of the university. They do have a small selection of fiction, but mainly they have non-fiction, references, journals, maps, and government documents.
Members of the community not connected to the university are allowed to use the collections within the library, and are able to apply for Nonaffiliated User (NAU) card for a fee. This entitles the user to check out materials, but does not give access to Interlibrary Loan, use of computer labs, email account, or remote access to restricted databases. For more information about the NAU card, please see their website.
Study Spots
The second and third floors offer decent places to study. Some of the personal study desks also double as bubble-gum museums. Most desks have power strips available to plug-in laptops and other devices. If you have a Wildcat account, then WiFi is easily tapped into. The south-west end of the 2nd floor is reserved for group studying while most other areas are for individual silent study.
Computer / Internet Access
Computers, including computerized catalogs, are only accessible by those with a Wildcat account. Even though the library is free for the community to utilize, the lack of easily accessible catalogs can be frustrating. Students, of course, have no problems, but they are required to lock the machine out once they are done. The NAU card may provide access. Most machines on the second floor are marked explicitly for research only. Computers in the lab on the first floor and on the third floor can be used in more general ways. Internet is available on all machines, primarily due to a wealth of online research resources. These resources used to be freely accessible to the public without an NAU card.
Resources
- First floor: Circulation and Reserve, main circulation, computer lab, media, limited loan.
- Second floor: Reference Desk, legal documents, and bound periodicals. There are plenty of places to study, including a group study area. There are lockers available for a small fee near the silent study area.
- Third floor: main collection, theses, and oversized books.
- Fourth floor: you'll find an interesting selection of books for those teaching children. There are also a wonderful variety of topographic, and Public Land Survey System (PLSS) maps, and more. An overflow computer lab can be found here along with places to study.
Departments
Several services are housed in Meriam Library. | https://detroit.localwiki.org/chico/Meriam_Library |
Apple has released tvOS 14.4 to the public. tvOS 14.4 is the fourth update to the Apple TV operating system that was released back in mid-September.
The tvOS 14.4 update can be downloaded via the Settings app on the Apple TV by going to “System” -> “Software Update.” If you have automatic software updates turned on tour Apple TV will be upgraded to tvOS 14 automatically.
There were no new tvOS features discovered during tvOS 14.4 beta testing, so it is not clear what’s new in the update.
tvOS updates are usually minor, focusing on bug fixes and performance improvements. Apple provides limited information on what’s new in tvOS updates. | https://gesov.com/apple-releases-tvos-14-4-to-public/ |
Samantha is a freshman this year and is very excited to be apart of Globe for the beginning of their high school career. They've always enjoyed writing and reading, and are happy...
3 Responses to “The Fetshization of Asian Cultures in the West”
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Jaime Punsalan on
January 9th, 2021 11:41 am
Thoughtful and relevant. I am heartened to know that issues like these are examined and thoroughly addressed by the youth of our society. Indeed, being the change they wish to see in the society within which they grow. Little by little, they usher in a world where the inappropriate objectification of women and the appropriation of culture are less and less acceptable. Ms McDonough is a national treasure. Thank you for this article.
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Kareen Punsalan on
January 9th, 2021 3:01 pm
As an older Asian woman, I am inspired to see the younger generation use their voice for change and empowerment. Samantha’s article shines a light onto how thin the line is between admiration and fetishization, how easy it is to cross, and how dangerous it is to normalize. As we (Asians) continue to fight for more representation in everything from leadership to media, we are reminded by their article that it’s not just the amount of representation, but also the type of representation that matters. Their article shows the influence media has on how Asian women are viewed. It is inspiring to have women like Kamala Harris, Tammy Duckworth and Mazie Hirono representing Asian women. While at the same time frustrating to see how much they have to fight for the respect they have earned, as the nation witnessed when VP Elect Harris said, “Mr. Vice President, I’m speaking.”, to claim back her time in the debate. And even more infuriating to see how some media outlets portrayed it as aggressive. Their article is relevant to not only Asian women, but all women. For when media defines women by only one of two stereotypes (be it lotus flower or dragon lady, virgin or whore, mother or career woman) these dangerous tropes are perpetuated for another generation. Articles like Mx. McDonough’s are needed to remind us of the importance of being mindful of not only what media we consume, but how we consume it.
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Crystal on
January 9th, 2021 4:52 pm
Great article! As a Filipina, this spoke to me. There is definitely more need of Asian women representation in media that show us simply as human beings without being hyper sexualized or as sex objects. Keep up the good work, Samantha.
All online comments are reviewed by a member of the editorial staff before being approved. This site is intended to provide information and engage in open and respectful dialogue that is appropriate to the educational environment and fans of all ages. To ensure that exchanges are informative, respectful and lawful, we will NOT post comments that are off topic, spam, personal attacks, illegal, not factual or not appropriate in any other way.
If you want a picture to show with your comment, go get a gravatar. | https://www.chsglobe.com/42206/features/the-fetshization-of-asian-cultures-in-the-west/ |
The consent of the adult victim of human trafficking to their exploitation is one of the fundamental notions that may blur the exploitative condition. Therefore, it is not the core element in the perpetrators’ defense line by chance. Given that the Greek legal framework adequately complies with the EU and international instruments on human trafficking, the Greek case law has gradually developed a harmonized approach. De lege lata, the adult victim’s consent, is considered irrelevant when it has been extracted by any coercive or deceptive means listed in the legal framework. In practice, however, the validity of consent is a matter of legal interpretation (i.e., a process that analyses the real facts, to apply the respective legal provision). This focus on reality recruits unavoidably a prism of social representations. However, the context analysis, such as the business analysis of human trafficking crime in the urban context, may provide valuable input. addresses the need for more interpretative tools and approaches. | https://ojs.panteion.gr/index.php/uc/article/view/282 |
Do you know what it feels like to be adopted, or to create a family through adoption? Do you understand the emotions behind a birth mother's choice?
Real and raw, Adoption Means Love explores the many experiences and emotions of adoptees, adoptive parents, birthparents, foster youth, and foster parents.
With honest and meaningful insight, Adoption Means Love shines a light on this truth: adoption is complicated and it’s beautiful.
Come delve deep into the heart of adoption and travel a journey of loss, love, forgiveness, and triumph. | https://www.everythingadoptionstore.com/ |
We'd all love to feel consistently happy, self-assured, and enthusiastic. Yet often we’re consumed by feelings of failure, regret, and self-doubt. Do our delicate positive emotions have a fighting chance at being in control? Using the book of Tanya as our base, this year's course will delve deep into the nature of our opposing emotions to find out.
Through Torah teachings and contemporary psychology, we'll take a fresh approach to the battle against bad feelings, providing realistic spiritual mechanisms for remaining upbeat no matter what life brings.
Wrestling with feelings of overwhelm, procrastination, or general
not-enough-ness? This session explores the seemingly senseless feelings of inadequacy and charts a course to transform them from adversary to ally. | https://jewishvancouverwa.com/womens-circle |
The Legal Services Corporation (LSC) is the country’s single largest funder of civil legal aid for low-income Americans. Established in 1974, LSC operates as an independent 501(c)3 nonprofit corporation that promotes equal access to justice and provides grants for high-quality civil legal assistance. LSC currently funds 132 independent, nonprofit legal aid organizations with nearly 800 offices throughout the nation.
Basic Function: Under the general direction of the Director of the Office of Program Performance (OPP) or his or her designee, works with grantee programs to provide programmatic oversight and to improve the delivery of quality legal services to the eligible population. Assists programs to improve and expand the development of innovative, comprehensive, and effective best practices and solutions. Extensive travel is required. This position is included in a collective bargaining unit represented by the International Federation of Professional & Technical Engineers, Local 135.
PRINCIPAL DUTIES AND RESPONSIBILITIES:
COMPETENCIES REQUIRED:
General:
Excellent written communication skills with demonstrated ability to produce professional, clear, well-written work product that requires minimal editing. Excellent oral communication and interpersonal skills with demonstrated ability to reach different audiences and manage conflicts. Experience analyzing complicated concepts and offering cogent conclusions. Unimpeachable integrity and adherence to the highest standards of ethics and professionalism. Ability to function in a discreet, diplomatic, and confidential matter. Must possess sound judgment and articulate well-informed and objective recommendations. Ability to understand trends that affect LSC, its grantees, and their clients. Ability to conduct thorough assessments and reviews through effective interviewing, research, and application of relevant regulatory authority. Must be conscientious, disciplined, versatile, and highly motivated. High degree of flexibility, capacity for self-management, and attention to detail. Strong organizational skills. Adaptable and able to manage multiple priorities with shifting and/or competing priorities and tight deadlines. Understanding of and commitment to organizational mission and goals. Ability to influence, motivate, challenge, and inspire confidence among others. Ability to work well with persons from diverse backgrounds, perspectives, and working styles.
Technical/Specialized:
For Program Counsel position, a law degree from an accredited law school and admission to the Bar of the highest court in a jurisdiction. Must be in good legal standing. Must have at least five years of experience in the practice of law for a Program Counsel II position or eight years of experience in the practice of law for a Program Counsel III position. Must have a bachelor’s degree and at least six years of experience related to the administration, management, and delivery of legal services to low-income persons for a Program Analyst II position or a minimum of nine years of experience related to the administration, management, and delivery of legal services to low-income persons for a Program Analyst III.
Capacity to interpret the LSC Act, Appropriations Riders, LSC regulations, policies and rules. Ability to conduct thorough investigations and reviews through effective interviewing and research. Must possess solid research and analytical skills. Management experience within the legal services delivery system and in conducting program quality reviews is preferred. Proficient with Microsoft Word and Excel.
SALARY AND BENEFITS:
Program Counsel II: $103,220
Salary range: $103,220 - $116, 259
Program Counsel III: $116,963
Salary range: $116,963 - $131,738
LSC has a competitive market-based salary structure that establishes a specific salary range for this position. The salary range has eight pay levels separated by seven steps. A new employee will be hired at Step 1 and must wait two years to advance to each of the next three steps (Steps 2 through 4), and one year to advance to each of the next four steps (Steps 5 through 8).
Excellent benefits package.
Position: Program Counsel II or III
Location: 3333 K Street, NW, Washington, D.C. 20007
Department: Office of Program Performance
Employment Status: Regular/Full-Time/Exempt
The Legal Services Corporation (LSC) is the country’s single largest…
APPLICATION PROCEDURE:
To apply, submit a complete application (provide ALL of the information requested and answer ALL of the questions) and attach a résumé and cover letter explaining why you are a good fit for this position. Incomplete applications will not be considered. Apply directly on our website at: www.lsc.gov/about-lsc/careers
Questions can be sent to: [email protected]
Note: If reasonable accommodation in the application process is required, please notify OHR at 202-295-1560 or [email protected].
DIVERSITY STATEMENT:
LSC embraces diversity as a core value. We recognize that our success as a corporation depends upon creating and maintaining a diverse team of talented professionals, and we are committed to a workplace that reflects and supports diverse individual backgrounds and perspectives. Our commitment to diversity, inclusion, and non-discrimination includes race, sex, age, religion, national origin, sexual orientation, gender identity/expression, personal appearance, genetic information, political affiliation, marital status, family responsibilities, disability and status as a veteran, and any other characteristic protected by federal, state, or local laws or regulation. We strive to have a workplace that is comfortable and welcoming for everyone. LSC is an equal opportunity employer.
APPLICATION PROCEDURE: | https://www.idealist.org/en/nonprofit-job/b0f2fffeadc243e78f93e94d1c3241f2-program-counsel-ii-or-iii-legal-services-corporation-washington |
Scottish retail misery intensified ahead of latest lockdown moves
THE year-on-year rate of decline in Scottish retail sales value accelerated again in October, amid a tightening of coronavirus-related restrictions and major economic uncertainty, the latest industry figures reveal.
The data highlight the pressure on the retail sector in Scotland even before yesterday’s news of the impending closure of non-essential shops in 11 local authority areas with a move to tier-four restrictions in these parts of west and central Scotland from 6pm on Friday to slow the spread of coronavirus.
Restrictions in Scotland have been tightened significantly again over recent months amid a resurgence of Covid-19 throughout the UK and elsewhere.
The Scottish Retail Consortium’s latest monthly figures, published today, show the value of retail sales north of the Border in October was down by 8.5 per cent on the same month of last year.
This followed five consecutive months in which the year-on-year decline in Scottish retail sales value had decelerated. The year-on-year fall in sales had slowed from 40.3% in April, at the height of lockdown, to 6% by September.
Flagging the renewed acceleration in the year-on-year decline in Scottish retail sales value in October, SRC director David Lonsdale said: “The recent gradual improvement in retail sales went into reverse in October, slipping to the weakest performance since the ending of the lockdown in June, as greater Covid restrictions and economic uncertainty exerted a vice-like grip. Retail sales in Scotland have still to emerge from the pandemic-induced funk, after eight months, deeply concerning at what is for many stores the start of the all-important festive period.”
The year-on-year rise in food sales value in Scotland accelerated from 3.7% in September to 4.4% in October, with anticipation of lockdown measures and restaurant restrictions cited as factors in the grocery sector’s strength.
However, the year-on-year decline in the value of non-food sales, which tend to represent more discretionary elements of consumer spending, accelerated from 14.2% in September to 19.3% last month.
However, he added: “It isn’t enough to halt an overall decline. With non-food sales down by more than 19% and total year-on-year sales down…retailers are facing a daily fight for survival.”
He added: “Shifting consumer behaviour, driven by Brexit and Covid-19 uncertainty, is having an unprecedented impact on Scotland’s retail industry. We’re just a week away from Black Friday and in the middle of the so-called ‘golden quarter’, when the sector typically witnesses its biggest gains. This winter, to simply survive, retailers will have to double down on innovative strategies, pivot towards profitable online revenue growth and work even harder to tempt some shoppers back to physical retail spaces.”
Commenting on the mix of Scottish retail sales in October, Mr Lonsdale said: “Convenience store and other grocery sales fared well, temporarily buoyed as restrictions on eating out became more prevalent. With many office workers advised to stay home, sales of electronics, household goods and home office products remained high. There were even some signs of early festive shopping, particularly for Christmas decorations and gifting.”
Referring to the Scottish Government’s announcement yesterday that it was moving 11 council areas to tier-four restrictions, the SRC said: “This will see so-called non-essential shops in these council areas shuttered for three weeks, during what is traditionally the key trading period of the year for many.”
It added: “The SRC estimates this will affect over 45% of non-essential shops in Scotland, and those shops will lose out on over £90 million per week in lost revenue whilst they are closed.”
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Fees:
If you are not a citizen of a European Union (EU) or European Economic Area (EEA) country, or Switzerland, you are required to pay application or tuition fees. Formal exchange students will be exempted from tuition fees, as well as the application fee. Read more about fees.
Application Fee: SEK 900
Tuition fee, first semester: SEK 11250
Tuition fee, total: SEK 11250
Please note: The course will be given if resources are available. | http://www.uu.se/en/admissions/freestanding-courses/course/?kKod=5KD031&lasar=13/14&typ=1 |
The LandMarks project was conceived to give diverse indigenous artists the opportunity to work as a community, share experiences and artistic styles, and explore a common spiritual connection to the land. Tamarind Institute brought together two groups of artists from opposite sides of the world to participate in the experience of collaborative printmaking.
The artists were selected based on the quality and subject matter of their work. Native American artists included Chris Pappan (Kaw, Osage, Cheyenne River Sioux); Marie Watt (Seneca); Jewel Shaw (Cree/Metis); and Dyani Reynolds White Hawk (Sicangu Lakota). Australian artists included Djirrirra Wunungmurra (Buku-Larrngay Mulka), Marie Josette Orsto (Tiwi Design), and Alma Sims (Warkulurangu Art Center).
The artists first met in April 2013, when the American artists, together with Tamarind’s Master Printer, Bill Lagattuta, travelled to the Northern Territories, on the northeast coast of Australia, to work at the Buru-Larrngay Mulka Art Center. In addition to using woodblock and etching techniques, the Australian artists shared their traditional methods of bark painting. The artists and printers were taken on an excursion to collect bark and natural pigments (yellow, black and red ochre) used in this process. Lagattuta said, “it was truly the experience of a lifetime to live among the Australian aboriginal people, and to work and make art side-by-side.
During the month of May, 2013, Australian Aboriginal artists from the Northern Territory and Native American artists from various locations in the United States and Canada traveled to Albuquerque to work with printers and students at Tamarind Institute. Many of the artists had not had the opportunity to explore the unique expressive tools and visual language of the lithographic process. For several of the artists visiting from Australia this was their first experience traveling out of the Northern Territories. These artists traveled with companions who facilitated communication, allowing for the success of the project. A key element of the project was not only to give artists a new vocabulary through the printmaking process, but also to open new vistas as the artists absorbed new landscapes and traditions.
(above) Dyani Reynolds White Hawk draws on a stone at Tamarind.
(left) Alma Sims works on her lithographs in the Tamarind studio.
(below) All seven artists and companions enjoyed a festive evening with the Tamarind staff and students, highlighted by a performance of traditional dances by the Australian participants.
LandMarks was partially funded by the National Endowment for the Arts. | http://tamarindo.unm.edu/announcements/view/9-landmarks |
:
David Baldacci
Publisher
:
NA
Language
:
English
ISBN No.
:
Availability
:
Available
No Of Pages
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472
Description:
Military CID investigator John Puller has returned from his latest case to learn that his brother, Robert, once a major in the United States Air Force and an expert in nuclear weaponry and cyber - security, has escaped from the Army's most secure prison. Preliminary investigations show that Robert - convicted of treason - may have had help in his breakout. Now he's on the run and he's the military's number one target.
John Puller has a dilemma. Which comes first - Loyalty to his country, or to his brother? Blood is thicker than water, but Robert has state secrets which certain people will kill for. John does not know for sure the true nature of Robert's crimes, nor if he's even guilty. It quickly becomes clear, however, that his brother's responsibilities were powerful and far - reaching. With the help of US intelligence officer Veronica Knox, both brothers move closer to the truth from their opposing directions. As the case begins to force John Puller into a place he thought he'd never be - on the other side of the law - even his skills as an investigator and his strength as a warrior, might not be enough to save him or his brother.
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A magnetic suspension screw type refrigeration compressor is characterized in that at least a machine body part (1), a driving rotor (2), a driven rotor (3), a magnetic coupler (7), a radial permanent magnet bearing (8) and a bidirectional permanent magnet axial thrust bearing (9) form a whole, and the magnetic suspension screw type refrigeration compressor has the advantages that friction power consumption of a rotary moving part of the compressor is eliminated, and oil-free lubrication operation is achieved; the defects of oil film thermal resistance and pressure drop increase of the filter screen are overcome, comprehensive adaptability and zero leakage to refrigerants are achieved, a refrigeration cycle system is simplified, resources are saved, and the performance and reliability of the compressor are greatly improved. | |
Published On: June 5, 2022 11:37 AM NPT By: Republica | @RepublicaNepal
KATHMANDU, June 5: Electricity generated from the 144 MW connected power plant of Kaligandaki ‘A’ Power House owned by Nepal Electricity Authority (NEA) has been exported.
The NEA started exporting electricity generated from Kaligandaki powerhouse from Saturday midnight.
The NEA started exporting around 39 MW of electricity daily from the Trishuli and Devighat hydropower plants to the Indian Energy Exchange (IEX) from Thursday. NEA will export 177.7 MW of electricity to India. Of which, 140 MW of Kaligandaki ‘A’, 23.2 MW of Trishuli and 14.5 MW of Devighat will be exported to India.
NEA is exporting the surplus energy to India after the power plants started running at full capacity due to the increased flow of water in the rivers with the arrival of monsoon.
NEA Managing Director Kulman Ghising said that the average selling price of electricity exported from Saturday night was Rs 11.2 (INR 7) per unit. "The maximum rate is Rs 19 (INR 12) per unit, the minimum is Rs 2.38 (INR 1.49) and the average rate is Rs 11.2 (INR 7)," Ghising said.
In IEX, 24 hours are divided into 96 blocks of 15/15 minutes and electricity is traded at competitive rates fixed by the market. So the price of each block is different.
Similarly, the authority on Saturday sold electricity at a maximum price of Rs 19.2 (INR 12) and a minimum of Rs 4.08 (INR 2.55) per unit. The average price is Rs 11.48 (INR 7.18).
Nepal has been allowed to sell up to 364 MW of electricity at competitive rates in the Indian Energy Exchange Market. | https://myrepublica.nagariknetwork.com/news/kaligandaki-a-begins-to-export-electricity/ |
Vietnam ranked 6th globally with more than 21,000 students enrolling in American colleges and universities last year.
More than 21,000 Vietnamese students enrolled in colleges and universities in the U.S.
According to the annual Open Doors Report on International Exchange, published by the Institute of International Education, Vietnam ranked sixth globally among the leading sources of foreign students studying at U.S. higher education institutions. The top five last year was China, India, Saudi Arabia, South Korea and Canada.
In Southeast Asia, Vietnam came out on top, sending more than twice the number of students to the U.S. compared to Indonesia, the second largest source of students for the U.S. in the region.
The U.S., followed by Australia and the U.K, is by far the favorite destination for Vietnamese students.
Vietnamese students spent nearly $1 billion for studying at U.S. colleges and universities last year, according to latest data from the U.S. Immigration and Customs Enforcement.
Statistics showed there were only 800 Vietnamese students at U.S. colleges and universities in 1995. Over the past 15 years, Vietnam has remained one of the top sending countries.
U.S. colleges are increasingly seeking international students to bolster enrollment and their budgets. The Open Doors Report pointed out that roughly 75 percent of all international students pay their tuition and fees from personal and family sources as well as assistance from their home country governments or universities.
“The strong increase in Vietnamese student enrollment shows the continued conviction by students and parents that a U.S. degree is a sound investment in their future careers,” according to a statement released by the U.S. Embassy in Vietnam. | https://duhocquoctejns.com/en/vietnamese-students-take-the-largest-number-in-us/ |
During the course of a safety system test being carried out just before a routine maintenance outage, Chernobyl 4 was destroyed as a result of a power transient on 26 April 1986.
The accident at Chernobyl was the product of a lack of safety culture. The reactor design was poor from the point of view of safety and unforgiving for the operators, both of which provoked a dangerous operating state.
The health effects of the Chernobyl accident have been the subject of extensive study by health professionals and unprecedented speculation and exaggeration by parts of the media. This Appendix summarises two authoritative and expert assessments of the situation, and provides links to two others.
Following the 1986 Chernobyl accident, safety concerns over early Soviet reactor designs intensified. As a condition of accession to the European Union, eight Soviet-designed reactors were closed. Sixteen nuclear power reactors of early Soviet design are still operating: one in Armenia and the rest in Russia
Earthquakes and Seismic Protection for Japanese Nuclear Power Plants: Nuclear facilities are designed so that earthquakes and other external events will not jeopardise the safety of the plant. In France for instance, nuclear plants are designed to withstand an earthquake twice as strong as the 1000-year event calculated for each site. It is estimated that, worldwide, 20% of nuclear reactors are operating in areas of significant seismic activity. The International Atomic Energy Agency (IAEA) has a Safety Guide on Seismic Risks for Nuclear Power Plants. Various systems are used in planning, including Probabilistic Seismic Hazard Assessment (PSHA), which is recommended by IAEA and widely accepted.
No harmful health effects were found in 195,345 residents living in the vicinity of the plant who were screened by the end of May 2011. All the 1,080 children tested for thyroid gland exposure showed results within safe limits.
The Fukushima Daiichi reactors are GE boiling water reactors (BWR) of an early (1960s) design supplied by GE, Toshiba and Hitachi, with what is known as a Mark I containment. | https://www.world-nuclear.org/information-library/safety-and-security/safety-of-plants/appendices.aspx |
It appears that the Bolivian negotiators in Washington have had some indication to the effect that a better price than US$0.62½ per pound of fine tin might be paid provided the producers would receive the full benefit of such additional price. This waiver of taxes is the Bolivian Government’s contribution to the negotiations for a better price than US$0.62½. It is suggested by the Assistant Minister of Hacienda, Sr. Palenque, that it would be practically impossible to export a larger quantity than the average of exports to the United States for 1944 and 1945 which, he states, were approximately 21,612 metric tons, and that therefore the offer of a bonus for larger exports during this year than this average is meaningless. The Ministry is suggesting that the basis for a bonus be 16,000 metric tons and that the premium of one-fourth cent for each five percent of exports over and above 16,000 tons up to a total premium of one cent for a twenty percent increase be maintained. The Assistant Minister is also hopeful that in addition to arriving at the basis for a bonus of 16,000 tons that the price before the payment of the bonus will be increased over and above US$0.62½ per pound, in view of the Ministry’s waiver of taxes on such price and such bonuses as may be obtained over and above US$0.62½.
The waiver of taxes in fact only would apply to a price over and above US$0.63½ instead of US$0.62½. It has been the practice of the Government to base its taxes on the c.i.f. price and when the contract price was changed in 1942 to f.o.b. Chilean or Peruvian ports the Government estimated the difference between f.o.b. and c.i.f. at US$0.1½ per pound and added this amount to the f.o.b. price before applying the tax. It will be noted in the attached letters that the Government intends to continue to apply the taxes on a c.i.f. price basis.[Page 390]
The taxes which the Government offers to waive amount to approximately 20% of the value.*
Respectfully yours, | https://history.state.gov/historicaldocuments/frus1946v11/d334 |
Out of home sector - mandatory calorie labelling: consultation
The public is being invited to have their say on plans to add the number of calories to menus in the out of home food sector including cafes, restaurants and takeaways.
10. Enforcement
10.1 As set out in the Scottish Regulators' Strategic Code of Practice, enforcement functions should be delivered in a way that is transparent, accountable, proportionate, consistent, and targeted only where necessary.
10.2 We will work closely with businesses, regulatory bodies and local authorities (LAs) to ensure compliance with any calorie labelling requirements are monitored and enforced in a way that is fair and not overly burdensome. Enforcement approaches may include advice, guidance, inspections, monitoring and enforcement.
10.3 LAs are responsible for enforcing existing food law in the businesses captured in the scope of the mandatory calorie labelling proposals. We acknowledge that LAs are well placed to support with the delivery and enforcement of this policy proposal and welcome views through the consultation on possible approaches to enforcement.
10.4 We recognise the importance of clarity in ensuring the effective implementation of this policy. To support this, we would plan to work with local authorities and industry in developing guidance to local authorities on matters for which they should have regard when discharging their functions.
10.5 We would plan to work closely with the Convention of Scottish Local Authorities, FSS and others to assess resource implications.
Question 16 - Please comment on our proposals for enforcement and implementation outlined in section 10.
Question 17 - How could any requirements be enforced, in a way that is fair and not overly burdensome? | https://www.gov.scot/publications/consultation-mandatory-calorie-labelling-out-home-sector-scotland/pages/11/ |
Also found in: Dictionary, Thesaurus, Legal, Financial, Acronyms, Encyclopedia, Wikipedia.
Related to cardiac index: cardiac output
index(pl. indexes, in´dices) (L.)
1. the numerical ratio of measurement of any part in comparison with a fixed standard.
2. forefinger.
Barthel index an objective, standardized tool for measuring functional status. The individual is scored in a number of areas depending upon independence of performance. Total scores range from 0 (complete dependence) to 100 (complete independence).
bleeding index any of various methods of assessing bleeding in the gingival sulcus before or after treatment.
body mass index (BMI) the weight in kilograms divided by the square of the height in meters, a measure of body fat that gives an indication of nutritional status.
cardiac index cardiac output corrected for body size.
cephalic index 100 times the maximum breadth of the skull divided by its maximum length.
citation index an index listing all publications appearing in a set of source publications (e.g., articles in a defined group of journals) that cite a given publication in their bibliographies.
Colour index a publication of the Society of Dyers and Colourists and the American Association of Textile Chemists and Colorists containing an extensive list of dyes and dye intermediates. Each chemically distinct compound is identified by a specific number, the C.I. number, avoiding the confusion of trivial names used for dyes in the dye industry.
erythrocyte indices the mean corpuscular volume, mean corpuscular hemoglobin, and mean corpuscular hemoglobin concentration. These are all useful for evaluating anemias because they provide information on the size of the erythrocytes and the concentration of hemoglobin. Called also red cell or red blood cell indices.
glycemic index a ranking of foods based on the response of postprandial blood sugar levels as compared with a reference food, usually either white bread or glucose. See table.
left ventricular stroke work index (LVSWI) an index of the amount of work performed by the heart.
leukopenic index a fall of 1000 or more in the total leukocyte count within 1.5 hours after ingestion of a given food; it indicates allergic hypersensitivity to that food.
index Medicus a monthly publication of the national library of medicine in which the world's leading biomedical literature is indexed by author and subject.
opsonic index a measure of opsonic activity determined by the ratio of the number of microorganisms phagocytized by normal leukocytes in the presence of serum from an individual infected by the microorganism, to the number phagocytized in serum from a normal individual.
phagocytic index any arbitrary measure of the ability of neutrophils to ingest native or opsonized particles determined by various assays; it reflects either the average number of particles ingested or the rate at which particles are cleared from the blood or culture medium.
red blood cell indices (red cell indices) erythrocyte indices.
refractive index the refractive power of a medium compared with that of air (assumed to be 1).
short increment sensitivity index (SISI) a hearing test in which randomly spaced, 0.5-second tone bursts are superimposed at 1- to 5-decibel increments in intensity on a carrier tone having the same frequency and an intensity of 20 decibels above the speech recognition threshold.
therapeutic index originally, the ratio of the maximum tolerated dose to the minimum curative dose; now defined as the ratio of the median lethal dose (LD50) to the median effective dose (ED50). It is used in assessing the safety of a drug.
car·di·ac in·dex
the amount of blood ejected by the heart in a unit of time divided by the body surface area; usually expressed in liters per minute per square meter.
cardiac index
a measure of the cardiac output of a patient per square meter of body surface area. Its normal range in a healthy adult is 2.8 to 4.2 L/min/m2.
cardiac indexCardiology The cardiac output of blood–L/min/m2 surface area Lab medicine The ratio of CK-MB to total CK, an indicator of myocardial ischemia, and ↑ risk for acute MI. See Cardiac profile guideline, CK-MB, Troponin I.
car·di·ac in·dex(kahr'dē-ak in'deks)
The amount of blood ejected by the heart in a unit of time divided by the body surface area; usually expressed in liters per minute per square meter. The normal average is 2.8 L.
car·di·ac in·dex(kahr'dē-ak in'deks)
Amount of blood ejected by heart in a unit of time divided by the body surface area.
cardiac
1. pertaining to the heart. See also heart.
2. pertaining to the gastric cardia.
cardiac afterload
the impedance to ventricular emptying presented by aortic pressure.
cardiac area
see precordium.
cardiac biopsy
an uncommon clinical procedure. May be performed via thoracotomy or with a biopsy catheter introduced intravenously.
cardiac catheterization
the insertion of a catheter into a vein or artery and guiding it into the interior of the heart for purposes of measuring cardiac output, determining the oxygen content of blood in the heart chambers, and evaluating the structural components of the heart.
cardiac compensation
in cardiac disease the compensation for the inefficiency of the heart's pump action by enlisting the various reserves of the heart such as hypertrophy, enlargement, increase in rate, so as to maintain circulatory equilibrium and prevent the appearance of signs of congestive heart failure.
cardiac compression
an emergency measure to empty the ventricles of the heart in an effort to circulate the blood, and also to stimulate the heart so that it will resume its pumping action. Involves the application of pressure through the thoracic wall. More commonly used in animals than other forms of cardiac massage.
cardiac conducting cells
specialized cardiac fibers modified to conduct impulses from the A-V node via the septum to the ventricles. Called also Purkinje fibers.
cardiac conducting system
the cardiac tissue responsible for electrical conduction, made up of the sinoatrial node, the atrioventricular node, and the atrioventricular bundle and cardiac conducting fibers.
cardiac depressor nerve
a branch of the vagus nerve composed of afferent nerve fibers which arise around the base of the heart; called also aortic nerve.
cardiac dilatation
the heart volume is increased but the effective mass of cardiac muscle is not. A dilated heart has lost some of its reserve.
cardiac dullness
the area of the chest wall over which a dull sound, indicating the position of the heart, can be elicited by percussion.
cardiac failure
see heart failure.
cardiac fibrillation
see ventricular fibrillation.
cardiac fibrosis
see cardiac cirrhosis.
cardiac flow load
the work required of the heart can be increased by a need for an increased flow rate of blood, e.g. when there is an anastomosis, congenital arteriovenous defect, portosystemic shunt.
cardiac function curves
statistical curves used in modeling the cardiovascular functions, relating e.g. venous return to cardiac output.
cardiac glands
in the cardiac region of the gastric wall; branched, tubular, coiled, mucus-secreting.
cardiac glycosides
the glycosides of Digitalis purpurea (digitoxin, gitalin and gitoxin) and digoxin (from D. lanata). Strophanthin and ouabain are glycosides found in Strophanthus spp. Other cardiac glycosides are present in the skin of toads (Bufo maritimus, B. vulgaris), but are of toxicological rather than therapeutic interest.
cardiac horse sickness
see african horse sickness.
cardiac hypertrophy
enlargement of the heart coincident with an increase in muscle mass; an indication of response to an increase in load which may or may not be associated with disease. It is an expression of cardiac compensation but some of the cardiac reserve has been lost.
cardiac impulse
see cardiac impulse. Called also apex beat.
cardiac index
cardiac output divided by the animal's body surface area in m2. The normal range for dogs is 1.8-3.5 l/m2.
left-sided cardiac enlargement
may involve either the left ventricle or atrium, or both, and can be demonstrated on radiographs and electrocardiography. Seen most commonly in mitral valvular disease in dogs.
cardiac massage
manual massage of the heart or stimulation with an electrical current through an open thoracic wall. The term is sometimes used interchangeably with cardiac compression.
cardiac mucosa
the most cranial of the gastric mucosae; secretes only mucus, except in pigs, in which the area covered by this mucosa is much larger than in the other species and bicarbonate is also secreted.
cardiac murmur
see heart murmur.
cardiac output
the volume of blood pumped per unit of time. May be calculated by oxygen consumption measurement or determined by dilution of indocyanine green or cold saline, using catheters with thermistors placed intravenously (thermodilution method). It can be estimated clinically by measuring heart rate, pulse quality or pressure, and assessment of tissue perfusion, e.g. capillary refill time.
cardiac pacing
employing cardiac pacemakers to control heart rate.
cardiac preload
ventricular end-diastolic volume.
cardiac pressure load
the stress of working against an elevated blood pressure in the arterial circuit; one of the two major groups of causes of heart disease; the other is flow load.
cardiac racing syndrome
a disease of companion birds manifested by a sudden increase in heart rate, up to 1000/min, in the period immediately after being restrained. Death occurs within a few seconds.
cardiac reserve
the reserve mechanisms in the heart to compensate for defects which could make the heart's pumping action ineffective. The reserve mechanisms include hypertrophy, enlargement, increase in heart rate and an increase in stroke volume, a result of the increase in muscle mass and the enlargement of the ventricles.
right-sided cardiac enlargement
may involve either the right ventricle or atrium. Occurs in heartworm disease in dogs.
cardiac rupture
penetration of the myocardium by a reticular foreign body in cows, or rupture of a patch of chronic fibrotic myocarditis in horses, causes cardiac tamponade and sudden death.
cardiac size
may increase as a result of hypertrophy, dilatation or a combination of the two. A common belief with some scientific support is that performance of horses in sprint races is closely related to heart size.
cardiac stroke volume
the amount of blood ejected with each systole.
cardiac thrill
see thrill.
cardiac valve fenestration
the valve surface is incomplete, creating a lattice effect; mostly congenital defects in foals.
cardiac valve hematocysts
congenital, blood-filled cysts on the atrioventricular valves considered to be of no pathogenic significance.
cardiac valve laceration
tearing of the valve tissue or attachment to myocardium may occur spontaneously or as a sequel to endocarditis; adds a significant additional flow load to the heart.
cardiac valve rupture
see cardiac valve laceration (above).
cardiac valves
heart valves formed by evaginations of the cardiac and vascular endothelium supported by connective tissue; includes atrioventricular and semilunar valves on both sides of the heart.
cardiac valvular disease
see valvular disease.
cardiac vascular shunts
includes patent foramen ovale, ventricular septal defect, tetralogy of Fallot, patent ductus arteriosus.
cardiac work
includes effective work—that needed for the onward propulsion of blood through the correct channels against arterial pressure, total work—includes all of the work performed by the heart including some involved in moving blood in the wrong direction. | https://medical-dictionary.thefreedictionary.com/cardiac+index |
Background Cancer sufferers take medicines for coexisting disease and personal medicate with over-the-counter medications (OTCs). took typically 5.5 (range CPI-613 cost 0-13) prescription medications, 2.2 (0-20) OTCs and 1.9 (0-11) supplements. Twenty-one medications had been each used by at least 10% from the topics. Acetaminophen was used by 59.6% from the subjects. One subject matter reported acquiring five acetaminophen-containing medications. The questionnaires awareness was 92.0%, specificity 99.9%. Bottom line Within 3 times to chemotherapy prior, topics took typically 9.6 concomitant medicines, a lot of which alter medication metabolism and or disposition. In scientific trials, multivariate evaluation of most concomitant medicines could increase medically relevant data to recognize medication connections that negate or potentiate the efficiency of cancers treatment regimens. Occasionally, obvious resistance of tumors to chemotherapy could be the total consequence of drug interactions. strong course=”kwd-title” Keywords: Chemotherapy, concomitant medicines, drug interactions, over-the-counter medications Intro Malignancy individuals undergoing treatment with chemotherapy often take additional medications for coexisting medical conditions. In addition to prescribed medicines, many cancer individuals self medicate with over-the counter medications (OTCs), vitamins, dietary supplements, natural herbs and other items (1-3). Level of sensitivity or resistance of a tumor to a chemotherapeutic routine is often assumed to be due to inherent properties of the tumor. However, concomitant medications can alter the efficacy of the chemotherapeutic routine (4) . Use of prescription drugs or alternate medications by malignancy individuals have been reported, but no comprehensive study describes the use of CPI-613 cost prescription drugs, OTCs GPATC3 and health supplements by individuals receiving chemotherapy. One deterrent to such studies is the lack of a validated questionnaire that collects data on the use of all concomitant medications. We have designed and validated a questionnaire for use in prospective studies of drug relationships. The half-life of most medicines is within 72h providing for significant connection between the concomitant drug and the absorption, distribution, rate of metabolism or excretion of the chemotherapy (5). Induction of drug metabolizing enzymes such as cytochrome P450s also happens within 72h (6;7). In addition, in dietary studies, questionnaires requiring a 72h recall have been shown to offer CPI-613 cost valid data (8). As a result, we gathered data on concomitant medication used in 3 times to chemotherapy prior, the perfect timeframe for medication interactions. Herein we details the questionnaires advancement and its own validation within a people of sufferers getting intraperitoneal or intravenous chemotherapy, and report the usage of prescription medications, OTCs, vitamins, herbal remedies and products by these sufferers. Strategies Questionnaire A questionnaire originated that included an exhaustive set of the mostly used prescription drugs, Supplements and OTCs. A summary of prescription drugs was put together in the list, released by RxList.com, CPI-613 cost of the very best 300 Prescriptions for 2004 by Amount folks Prescriptions. The very best 207 medications out of this list had been included. Over the questionnaire, medications were categorized according to the ailment for which they were most commonly prescribed [Heart Medications (Blood Pressure), Pain Relief, Allergy etc.]. Following Phase I of content material validation, a separate section comprising the fifteen anti-nausea medications most commonly used by chemotherapy individuals was included in the questionnaire. Lopressor HCT? (metoprolol with HCTZ) was also added. The questionnaire instructed subjects to indicate whether in the previous 10 days they had taken any of five medicines that are formulated for slow discharge [Fosamax? every week (Alendronate), Actonel? every week (Risedronate), Procrit? (Epoetin Alfa), Neulasta? (Pegfilgrastim), Aranesp? (Darbepoetin Alfa)]. The ultimate questionnaire included a complete of 228 prescription medications. All OTCs was included with the questionnaire ranked by Drugtopics.com, with regards to dollar sales, seeing that the very best 200 OTC/HBC Brands in 2004. For popular makes of OTCs such as for example Tylenol?, Robitussin?, Sudafed?, the questionnaire included every one of the mixture formulations in the producers products. The OTCs had been categorized based on the ailment that they were mostly used. The questionnaire also asked about the usage of deodorants with deodorants and antiperspirants without antiperspirants. All antiperspirants include aluminum which may be utilized through your skin (9). A complete of 211 OTCs had been included on the questionnaire. The questionnaire included 75 vitamin supplements, minerals, products and herbal treatments that were put together from lists from the Top-Selling Organic Dietary Supplements, Common Herbal remedies helpful for Chemotherapy and Cancers, and other resources (10-12). The questionnaire was designed being a self-administered, paper record of eleven web pages. It started with three demographic queries (sex, age, competition) and three queries regarding usage of cigarette, alcohol and unique diets. The medications sections provided, CPI-613 cost to the left of each outlined medication, a package that subjects could examine if they experienced taken that medication within the past 3 days. Each medication was outlined with both its brand names and generic titles. At the end of the prescription drug list, and on the last. | http://www.phytid.org/2019/08/27/background-cancer-sufferers-take-medicines-for-coexisting-disease-and-personal-medicate/ |
This section is from the book "The Gardener V2", by William Thomson. Also available from Amazon: The New Organic Grower: A Master's Manual of Tools and Techniques for the Home and Market Gardener.
This is one of the most useful and showy of stove spring-flowering plants, which produces its orange-yellow bells in great profusion when carefully cultivated. The plant is simple to propagate and cultivate, but is somewhat difficult to ripen into a free-flowering state, unless its requirements are thoroughly understood. Cuttings taken of it during the months of March and April, or small plants of it purchased from the nursery, and grown on; will make nice plants for furnishing next spring, and will with ordinary care develop into fine bushes the following season.
In preparing for inserting the cuttings, the pots should be washed clean, and drained by the ordinary process, and then filled with a mixture of well-decomposed leaf-mould mixed with sharp sand - say in the proportion of one-third of the latter to two-thirds of the former. It will also be advisable to leave space to the depth of an inch on the surface of the pots for a layer of pure sharp sand, which should be pressed down with the ringers and made firm, and into which the cuttings should be inserted. The cuttings should be selected from a clean healthy stock; and of course it is understood that in all cases of propagating, the materials employed should be of the same temperature as that from which the cuttings are taken, and be in good working condition, - that is, to be neither too wet nor too dry. After the cuttings are inserted, the pots should be plunged in a pit or propagating-frame with a bottom-heat of 85°. There is no objection to propagating such plants in a frame with the ordinary bedding plants, such as Verbenas, etc.; but if a moderate bottom-heat is at command in a pit heated by hot-water pipes, the latter should be preferred, as being more conducive to a steady growth than the conditions of the former, as heat fluctuates so rapidly in an ordinary dung-frame when exposed to the external conditions of an ever-varying atmosphere; but, as before remarked, much depends upom circumstances, such as season of propagating, locality, situation, etc. etc.
Assuming, however, that the cuttings have been plunged in a bottom-heat such as that recommended, they will soon form roots if the necessary conditions are observed in the way of shading, syringing, and watering.
After the young plants are sufficiently rooted they ought to be shaken in the plunging material, leaving a vacuum between the pot and the material in which it is plunged, preparatory to placing the cutting-pot upon the surface of the bed, to prevent anything like an abrupt check to growth. The same conditions of atmospheric treatment will still require to be maintained, except that the shading may be dispensed with after the cuttings have furnished themselves with roots, until the plants are ready for potting, which will be as soon as they are fit to be handled, and are large enough to be shifted into 3-inch pots. Before they are shifted, the soil for potting should be got in readiness, by having it mixed and laid up in a heated shed, or at all events in some dry place where it can be taken from and warmed to the necessary temperature a few hours before it is required for potting. The soil should be mainly friable loam, with a dash of leaf-mould and sand mixed with it. Rich soils and overpotting are to be avoided in the cultivation of these plants, especially in the northern parts of the kingdom, where light and sunshine are at a minimum.
The plants would only require all the more ripening if their treatment were such as to induce soft growth, and the probability of flowers would be less also.
When the plants are potted, the soil should be made firmer about their roots than is usually done for soft-wooded plants; and they may be returned to the propagating-bed and placed on the surface of it, or in some other place where the conditions will be somewhat the same, till they root afresh in the new soil, and are found to be growing both at root and top, when the young shoots may be pinched so as to furnish the plant with from six to eight shoots. With regard to pinching, it may be observed that perhaps to this being done too often and too late in the season, and to the plant being kept too long in heat, may be attributed the quantities of flowerless plants that are frequently to be seen. Better have half a dozen firm shoots well ripened and in a flowering condition than double that number in an imperfect state.
As soon as the young shoots break after being pinched, it will be better to shift the plants to a position nearer the glass where they will get plenty of air and light, and where they will make short-jointed stubby growth.
Frorn this date onwards (until the weather becomes warm enough to turn them out into a cold frame) there is no position that would suit them better than that of a low pit with a single hot-water pipe round it, where a regular temperature could be maintained, and where air could be given more or less during the greater part of the day without running the risk of doing injury to other things.
It is not, however, an absolute necessity that this should be the case, as the plants can be very well grown in any Cucumber pit, vinery, or plant-house where there is a little heat to start them in. In this case they should be carefully hardened off by removing them gradually into a lower temperature for some time before they are turned out into their summer quarters.
From 4 to 6 inch pots will be large enough to grow the plants in the first season - certainly for furnishing purposes the 4-inch size will be the best, as the plants in them will be the most likely to flower.
Before the plants are turned out into frames in June, or even where they are grown under the improved facilities elsewhere referred to in the shape of heated pits, their condition should be rigidly inspected, and if they are not in a sufficiently advanced state to undergo the change, they ought to be encouraged, by such means as are at command, to hasten that desirable condition of early well-developed growth before they are exposed to the atmosphere of a cold frame. At the same time, in a warm climate and in a "favourable season," there is no doubt but that these plants could be well managed without a particle of fire-heat after they had a little warmth to start them in; and their growth, although a little shorter than that made in a warmer and closer atmosphere, would be as certain, if not more so, to yield an abundance of flowers, provided it was thoroughly ripened in the autumn. When the plants are housed they ought to be kept near the light and only moderately supplied with water, as they are liable to suffer and lose their leaves if over-watered and kept in a low temperature. When they are required to come into flower, the temperature should be raised gradually to that of a warm greenhouse, and afterwards increased as circumstances may require.
Year-old plants that are trimmed back are treated much the same as plants that are raised from cuttings, except that in the latter case they do not require a high temperature at any time. They are partially shaken out and repotted, and when started in a gentle warmth they grow freely and make fine bushes the second season.
W. Hinds.
Although within the last few years this beautiful and useful greenhouse plant has been several times brought under the notice of your readers by various writers, it is astonishing that we see it so seldom adorning the stages of the amateur's greenhouse. When in full bloom, the appearance of the plant is highly effective. The flowers, which are of a tubular shape, red and yellow in colour, present a telling contrast with the rich glossy foliage, which is a striking feature of the plant. If a proper system of culture has been previously followed, the plants will begin flowering in November, and continue to flower for months. The particular plant in my possession began flowering at the time referred to, and is still in flower at this date (12th March). Younger plants will come on in succession. The propagation and culture of this plant are not necessarily a difficult matter, therefore good growers of the ordinary run of soft-wooded greenhouse plants need have no hesitation on that score in beginning with it. Cuttings of the young wood inserted in the usual light compost root readily at this season, when placed in a warm, moist temperature, and kept close and shaded.
When sufficiently rooted, they should at once be potted off into thumb pots, in a compost of loam, sand, and horse-droppings, or some bone-meal. If leaf-mould is mixed with the soil, it should be only a very little, as it causes a watery growth. Return the plants to the quarters they were struck in till they are established in the pots; and they will require shading from direct sunshine for a few days after being potted; and then two shifts, first into 4-inch, and lastly into 6-inch, pots, will in general be sufficient to grow plants large enough for ordinary purposes. Over-potting the plants should be guarded against, especially those wanted in flower before midwinter.
By timely attention to this matter much disappointment may be avoided. Plants in large pots are apt to continue in free growth till the season is too far advanced to insure the maturity of the young wood, upon which a crop of flowers depend. The plants do well grown in a cold pit or frame from June and onwards during the summer months, when they are attended regularly in regard to the usual items of watering, staking, tying, and pinching to keep them stocky.
The method of cultivation adopted by the "Squire's Gardener," which he detailed in your pages some years ago - viz., that of turning the plants out into earth-pits, is worth the attention of those whose labour power is limited, and who are expected to keep up large numbers of flowering-plants with, at best, inadequate means of doing so. I have adopted this method in the case of Cyclamens with good results. Unless a good position not far from the glass can be afforded them in the greenhouse, there need be no hurry in housing the plants in autumn, as they will do far better in a frame than when huddled along with other plants which already may have too little room. The flowers are produced freely in a comparatively cool greenhouse with a dry atmosphere. They should be carefully and regularly attended to with water in a tepid condition throughout the winter months.
D. Mackie. Ayrshire.
Continue to: | https://chestofbooks.com/gardening-horticulture/Gardener-2/Libonia-Floribunda.html |
US Respiratory & Pulmonary Diseases – VOLUME 3 – ISSUE 1 – FALL 2018
Welcome to this year’s edition of US Respiratory & Pulmonary Diseases introduced by Editorial Board member Robert Balk.
Expert interviews with Don Sin, Juzar Ali, and Ronald Morton touch on the relationship of telomere length of leukocytes and clinical outcomes in individuals with chronic obstructive pulmonary disease (COPD), risk factors for and management of nontuberculous mycobacterial pulmonary infections, and recent updates in the American Thoracic Society and Canadian Thoracic Society clinical guidelines for pediatric home mechanical ventilation, respectively. In editorials, we have a thought-provoking discussion around the dissemination and relevance of medical literature and how this may change in the future (Kaszuba and Alberts), and two important calls to action around the high prevalence of undiagnosed asthma in shelters in conflict zones, and the increasing popularity of waterpipe tobacco smoking (Mohammad). In a review article, Eric Gartman describes the normal changes that occur during sleep, how these are affected in patients with neuromuscular and chest wall disorders, and how to assess patients with neuromuscular disease for the presence of sleep disorders. Finally, Adan Mora presents a case of diffuse pulmonary hemorrhage. Recognition of this rare but fatal condition is important as it may be misdiagnosed as multilobar pneumonia, but its diagnostic workup and treatment is different.
Across the various articles, we hope there’s something of clinical relevance for you to enjoy. We are now accepting submissions for our 2019 edition and welcome enquires through our submission site here.
Foreword – US Respiratory & Pulmonary Diseases. 2018;3(1):13
Welcome to the latest edition of US Respiratory & Pulmonary Diseases. Despite a wealth of advances in the field of respiratory care, respiratory disease remains a leading cause of death and disability worldwide. The articles and expert opinions in this journal reflect the progress and challenges in this active area of clinical research. Our […]
Expert Interviews
Chronic Obstructive Pulmonary Disease—A Disease of Accelerated Aging
Chronic obstructive pulmonary disease (COPD) is a major cause of morbidity and mortality throughout the world. COPD frequently coexists with other age-related co-morbidities,1 and is now known to be a disease of accelerated aging.2 The length of telomeres, which become progressively shorter as cells divide, has been proposed as a biomarker of aging.3 Shorter telomere […]
Nontuberculous Mycobacterial Pulmonary Infections
Pulmonary infections caused by nontuberculous mycobacteria (NTM) are increasing in prevalence, and impose a significant health and economic burden. There are around 150 species of NTM, which are ubiquitous in the environment, and infections are transmitted by environmental spread, particularly from soil and water.1 In an expert interview, Dr Ali discusses risk factors for and […]
Updates in the Guidelines and Treatment of Ventilator-dependent Children
My experience in the care of ventilator-dependent children involves over 20 years’ experience in both the home and long-term care environments. I have managed over 50 ventilator-dependent children in a university-based practice and have weaned at least two children per year completely off mechanical ventilation. In addition, I have helped in the development of […]
Editorials
Relevance and the Dissemination of Medical Literature in the Information Age
With each passing month, the volume of medical data and information seems to grow exponentially. Even with the most finely tuned and calibrated Twitter feeds, relevant journal feeds, emails and bookmarked web pages, the amount of data presented can place physicians in a situation of “data fatigue” or sheer desperation. Medical knowledge and associated databases […]
Underdiagnosis of Asthma in Syrian Shelters in Conflict Zones—Highlighting the Need for a Healthcare Training Program
Underdiagnosis of asthma is an important issue for public health and also at individual level for asthma patients. The 2017 Global Initiative for Asthma used epidemiological surveys to compare the prevalence of asthma symptoms to doctor-diagnosed asthma in low and middle-income countries, revealing that 50% of asthma cases were underdiagnosed, especially in primary care facilities.1 […]
Waterpipe Tobacco Smoking and the Lungs — Short Notice
Waterpipe tobacco smoking (WTS) has been considered a global threat and has been designated as an epidemic by public health officials. WTS is also known as “narghileh” or “narghile”, “shisha” or “hookah”. WTS involves the passage of charcoal-heated air through a perforated aluminum foil and across the tobacco mixture to become smoke that bubbles through […]
Review
Sleep Disorders in Neuromuscular Diseases
Highlights Neuromuscular diseases exhibit a complex array of respiratory physiologic changes that typically progress over time. These physiologic changes can promote significant respiratory abnormalities during sleep, leading to decreased quality of life, significant daytime symptoms, and overall worse survival. There should be a high index of suspicion for the emergence of sleep-related disorders, especially […]
Case Report
The Bleeding Pneumonia — A Review of Diffuse Alveolar Hemorrhage
Highlights Diffuse alveolar hemorrhage (DAH), though rare, can be fatal and requires a high index of suspicion as frank hemoptysis may not be a presenting symptom. Multilobar pneumonia can simulate DAH both clinically and radiographically. Diagnosis relies on a strong clinical suspicion, laboratory studies, imaging, and pathology. Overall incidence of DAH is difficult to […]
Journal articles and more to your inbox
Get the latest clinical insights from touchRESPIRATORYSign me up! | https://www.touchrespiratory.com/journals/us-respiratory-pulmonary-diseases/us-respiratory-pulmonary-diseases-volume-3-issue-1-fall-2018/ |
The purpose of the historic district and the Board of Review, by issuing certificates of approval, is to ensure "that the worthy historical and architectural buildings, sites, monuments, streetscapes, structures, squares and residential neighborhoods are protected and granted proper investigation before any renovation, restoration, preservation, rehabilitation or demolition takes place." This includes basic maintenance of such a nature as new signage, new awnings or repainting in the historic district. It applies only to exteriors, not interiors.
The Downtown Marietta Development Authority (DMDA) continuously seeks ways to encourage further economic development in historic downtown Marietta Square and surrounding district properties. The DMDA is currently offering a 50/50 matching grant (up to $2,000) for business owners located within the DMDA District who would like to enhance the appearance of their storefront by improving their signage and/or awnings. For more information, email Alvin Huff or call him at 770-794-5512.
Requests for applications and placement on the agenda for the Historic Board of Review should be made by calling 770-794-5669.
A scaled drawing of the building facade showing the location of the sign, awning, canopy or painted area. Distance from the bottom of the sign to the ground should be identified. (Side view drawing required for projecting signs.) Only one sign is allowed per building frontage.
A color sample for each color to be used in the sign or painting. Color must be conservative and consistent with other colors in the downtown Marietta historic district. Approved color charts are available from the Mayor's office.
A sample of the materials to be used should be presented to the board at the meeting.
Applicants are asked to turn in their completed applications to the Planning and Zoning office to give staff time before the meeting to make sure that all necessary information is included in the application. | http://mariettapower.com/247/Applications |
NEW DELHI: TDP chief and Andhra Pradesh CM N Chandrababu Naidu has called Friday’s no-confidence motion against the Narendra Modi government in Lok Sabha a “historic opportunity” and asked his party MPs to treat it as a “prestige issue,” since it is because of people of Andhra Pradesh and TDP’s efforts that the PM will be facing his first no-confidence in Parliament, even as he heads a majority government.
“This is a historic opportunity, an inspirational occasion. You should echo the voice of five crore people of Andhra Pradesh in Lok Sabha as this is a prestige issue,” he told his MPs during a teleconference.
Naidu held two teleconferences on Thursday with his MPs from Amravati, advising them on how to conduct themselves in the House, among themselves and with members of other parties.
As part of the opposition benches now, Naidu, a seasoned politician, is not counting on winning the trust vote on the floor of the House in terms of numbers, but is banking on “exposing the Modi government and its betrayal to the people of state” as three TDP MPs -- Jayadev Galla, Ram Mohan Naidu and Kesineni Srinivas – speak in the House.
TDP knows the motion is likely to defeated on the floor of the House, but the party is keen to drive home the point that “it forced PM Modi to face his first no-confidence,” said a party source.
According to TDP sources, the party will put on record the “injustice” meted out to Andhra Pradesh even as it was an ally of BJP, as the state also prepares for assembly elections alongside the 2019 Lok Sabha polls and Naidu seeks a re-election.
TDP also stole a march over its main rival in the state YSRCP, since its members have resigned in protest against the Modi government and hence will not be able to take up the cause of the state in Parliament.
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Description : As the bestselling nursing research text, Understanding Nursing Research: Building an Evidence-Based Practice, 6th Edition offers unique insights into understanding, appraising, and applying published research to evidence-based practice. It is known for its authoritative content, time-tested systematic approach, and unique research example format newly redesigned for the sixth edition to better demonstrate how the steps of the research process relate to evidence-based nursing. "This would be excellent for student nurses, nurses new to research or any nurse with an interest in research." Reviewed by: Helen Reeves, St Giles Walsall Hospice on behalf of Nursing Times, December 2015 Authoritative content is written by two of the true pioneers in nursing research, who offer unique, first-hand insights into the field. Research examples provide practice in working with published studies, with many of the examples including Critical Appraisal and Implications for Practice sections. Clear, step-by-step organization introduces the research process and demonstrates how this systematic framework relates to evidence-based practice. Strong emphasis on evidence-based practice helps you develop skills in studying and appraising published research, so you are prepared for your role in working with research evidence. Critical Appraisal Guidelines boxes provide step-by-step guidance in appraising published research studies. Emphasis on critical appraisal versus critique reflects contemporary usage in nursing research and evidence-based practice. Balanced coverage of qualitative research prepares you to approach research questions and clinical questions with an unbiased view of the researcher's methodology. Presentation of two different appraisal processes covers both the traditional in-depth critical appraisal process that prepares you for graduate-level work in research, and the concise, practice-focused research appraisal process that equips you for quick and accurate evaluation of the applicability of research findings to clinical practice. | https://www.e-bookdownload.net/search/understanding-nursing-research-ebook |
The application closing date is 27 October 2021; you can expect a decision mid December.
Projects with a total project cost in excess of £50,000 should not apply unless all other funding is secured.
Groups awarded funding must complete their project before they can submit another application to the Fund.
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Groups with projects that are located within 10 miles of our sites listed below may apply for grants, subject to The British Steel Community Fund’s eligibility requirements.
- Scunthorpe: DN16 1XA
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- Skinningrove: TS13 4ET
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Please check the map on the previous page to make sure your project is located within the eligible funding area.
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Grants are available for capital projects involving:
- The provision and improvement of public amenities (e.g. village halls, village greens, public playgrounds, sports fields and facilities, nature reserves, community centres, footpaths, cycle-paths and country parks)
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- Commercial organisations
- Individuals
- Membership-based sports clubs and facilities (e.g. bowls and golf clubs), unless membership is open to the general public without undue restriction
- Bus services, minibus services or vehicles
- Any works to public highways
- Projects to deliver visual enhancements (i.e. ‘a view’), as this does not improve, maintain or provide a general public amenity
- CDs, websites or remote interpretation about a site
- Public car parks, unless they are specific to the general public amenity
- Public conveniences
- Allotments (unless the project is for community growing schemes)
- Charity buildings, offices of charities, Citizens’ Advice Bureau and advice centres
- Projects at schools, educational establishments
- Project at hospitals, hospices and day care centres unless the Public Access Requirements, above, are met. Such projects will only be supported where the project benefits the wider community. The project site should have independent access and use of the hospice/hospital etc should not be prioritised over community use during designated public access times. All applicants with projects located at one of these locations should contact GrantScape before applying to ensure projects meet the criteria
- Solar photovoltaic (PV) cells, and wind and water turbines unless: (i) these will be used solely to provide power and/or heat for the community amenity where they will be installed; and (ii) no income will be received from Government Feed-In Tariffs (FITs) or from any similar tariffs introduced in the future
Judging criteria
When considering whether to support your application, we will also be reviewing:
- The level of community support for and involvement in the project
- The local community benefit and enjoyment that will result from the project
- The extent of benefit to the built, social and/or natural environment
Other factors will also be considered, which must be demonstrated in the application:
- The ability of the applicant group to deliver the project, and within a reasonable timeframe
- How the project will be maintained after the project has been completed, i.e. its sustainability (including financial sustainability) and legacy
- The viability of the project, in terms of funds already secured, and value for money
If, after reading the above, you are confident that your project is eligible, please proceed to the next page: | https://www.grantscape.org.uk/fund/british-steel-landfill-community-fund/british-steel-landfill-community-fund-criteria/ |
Thanks for e-mailing me your speech Tuesday to the SMU School of Law. I've been wondering for some time when a member of Congress would raise these issues... and who it might be. I'm really not terribly surprised that it would turn out to be you. You've been one of the few independent voices of reason and prudence in the Senate during our recent bombing campaigns here and there around the world. Your speech puts you at the top of my list as the next Secretary of State, in the administration of whoever wins in 2000. Do you mind if I start a Kay Bailey Fan Club?
* * * * *
America's Engagement in the World at a New Century's Dawn
Legal and Ethical Implications for the Use of Force
Senator Kay Bailey Hutchison Nov. 16, 1999 SMU School of Law
Today I want to talk about America and the World. How do we engage it? Since the Cold War ended, America has been unfocused. Our vast military superiority and our global influence have not been matched by policies that clarify our place and role in the world. Most think of our power as military power. It's much more than that. It's our culture, our economy, our very way of life.
Historians will have a difficult time categorizing the role that America plays in the world today. That is because there is no precedent for the global economic, cultural and military power wielded by America today. I will address each in turn. The American economy is the envy of the world. Each day it is powered to new heights by the creativity and freedom of our people. The millions of daily decisions by individuals result in prosperity the world has never seen. The same is true for our culture -- our music, movies, the English language Internet, English as the common language for the international air traffic control system, our literature, clothing, fitness and sports. It's everything from Burger King to bungee jumping, jet skis to jazzercise.
This is what some call "soft power," the sum total of enticements that flow naturally from genuine freedom. Young people around the globe find it hard to hate America when they love Levis and the Grateful Dead. Our military power too -- our hard power -- knows no competitor. In military terms, we are a colossus astride the world. Our troops are in Japan, Korea, throughout Europe, the Middle East. With their lives we guard countless other nations. We keep tyrants in check -- from Baghdad to Beijing to Pyongyang and Belgrade. No other nation has ever wielded such military power.
What's more: this power is wielded not for conquest, but for peace. America's place in the world is unmatched in all of history. What is that role? It has three elements: First and foremost is the protection of our way of life -- democracy, freedom of speech and religion, the rule of law, and free enterprise. Second, to support and defend allies that share our beliefs and interests. Third, to encourage other nations to free their people and economies.
What does that mean? Leadership on this scale requires discretion. The confidence to know the right course and the will to pursue it. And the confidence to know when not to engage, but encourage others to do so. But true leadership -- and I believe ethical leadership -- is striking out on the right course of action and having others follow by establishing a clear principle for our action. That was the leadership the Senate showed when it recently rejected the Comprehensive Test Ban Treaty. The popular decision would have been to ratify the treaty because -- on the face of it -- who could be against a ban on nuclear testing? The treaty would have left us weaker, not stronger, by threatening the credibility of America's nuclear deterrent, which has helped keep the peace for a half-century.
We rely upon diplomacy to maintain much of this leadership. But when diplomacy fails, global leadership may require the use of military force. But when and how should the United States use military force? There was a time -- it seemed -- when the answer was clear. We should use military force when our vital national interests are clearly threatened. The Cold War forced us to be disciplined. We knew that if we acted, the Soviets might react. And vice versa. Today, because of our superpower status, we are much more free to act. That requires much more discipline. In our political system, that discipline comes from the checks and balances built into it. The Constitution grants the authority to the President as Commander-in-Chief to deploy troops. But we forget that it tilts heavily in favor of the Congress on many related matters.
The Constitution grants Congress a number of specific powers: to declare war; to raise and support armies, and to provide and maintain a navy; the power to make regulations of the land and naval forces; the power to call forth the militia; and the power to provide for organizing, arming, and disciplining the militia. Our framers were quite clear on this issue. They did not break with a monarchy in England only to establish another monarchy in America. In drafting our Constitution, they were chiefly concerned with checking the abuses of Executive power.
One of the framers of our Constitution, James Madison, argued that "the system of checks and balances will not hurry us into war; it is calculated to guard against it. It will not be the power of a single man, or a single body of men, to involve us in such distress."
And Madison remarked "the constitution supposes, what the History of all governments demonstrates, that the executive is the branch of power most interested in war and most prone to it. It has accordingly with studied care, vested the question of war in the legislative." But throughout our history, particularly the last half century, Presidents have sent our troops into conflict without a formal declaration of war. To regain some control in the waning days of the Vietnam War, Congress in 1973 enacted the War Powers Resolution. The act required the President to seek congressional approval to deploy forces not more than 60 days after they were dispatched.
President Nixon vetoed the bill, but Congress overrode his veto. He warned that the War Powers Resolution could "strike from the president a wide range of peace-keeping options...." That may have been the case during the Cold War. Very little has hampered President Clinton from sending peacekeepers all around the world, though. Congress has few tools available to check those excesses. The power of the purse is the most potent weapon, but it's ineffective once troops are on the ground. I would be very reluctant to support a cut-off of funding once troops are on the ground in an overseas mission.
Congress is gradually being excluded from its constitutional role in foreign policy. The consultation process is broken. Congressional involvement in foreign affairs is not and should not be a partisan issue. It is an institutional issue, critical to the future conduct of principled foreign policy. Here, our ethical and legal responsibilities are intertwined. In a representative democracy such as ours, elected officials must stand up and be counted when the most fundamental decisions of war and peace are made. Elected representatives have a bond of trust with their constituents that is never more important than when they are considering sending those constituents into armed conflict. The Constitution gives Congress the obligation to declare war, because the framers of the Constitution understood that our political system demands that accountability. There will always be advocates around the world for the United States to get involved militarily to stop some crisis or another. But who will advocate for the young soldiers that have to get out there and actually do it? If elected American officials do not think first of the Americans going in harm's way, who will? When we ignore that accountability, we threaten our very political system.
It is time for the Congress to reassert its Constitutional role. If the War Powers Resolution of 1973 isn't useful and enforceable, it should be altered, amended, replaced or repealed. I believe it is important for Congress to reclaim the deliberate role intended by the Constitution. I have proposed, for example, limits on the duration and size of a force that could be deployed without Congressional approval. I have proposed that the President be required to identify the specific objectives for a mission prior to its approval by Congress. Too often these operations are open-ended and there are no milestones to measure success or failure and there is no exit strategy.
One of my goals in the Congress has been to reinvigorate the role of Congress in foreign affairs and in the exercise of war powers. The Clinton Administration has difficulty separating the urgencies of regional conflict from the imperatives of superpower leadership. It is a hallmark of this administration for the U.S. to stumble into regional crises and displace friendly local powers who share our goals and could act effectively on their own.
Most recently and disastrously in Kosovo, where we fought to sustain an unsustainable government. We are trying to prevent the realignment of a region where the great powers have tried and failed many times to impose their will on ancient hatreds and atrocities. I do commend the Clinton Administration for not rushing headlong into the recent crisis in East Timor. Instead, the President encouraged the efforts of Australia and New Zealand to take a leading role.
We saw in the Balkans that the United States can bomb a much smaller country into temporary submission. There were ethical questions. We had to weigh the precedent of bombing a sovereign nation that had not attacked the U.S. against reports of ethnic cleansing. Even our method of responding had an ethical component: Is it right to bomb from 30,000 feet -- in a desirable attempt to avoid U.S. casualties -- with the consequence of casualties among the innocent civilians on the ground? This is a growing problem for the United States: What is our ethical responsibility when it comes to ethnic cleansing and other forms of internal terror perpetrated by governments against their own people? There is no easy answer. Generally, if the United States has a unique capability to stop it, I understand the urge to try.
The original mission in Somalia, for example, was simply to move food from the ports -- where it was rotting for lack of distribution -- to the people in the countryside who needed it. That was risky business, but very few countries can marshal the sizable logistic resources needed for such an effort -- thousands of tons of food and the manpower to distribute it. President Bush's intention was that we would leave once that had been accomplished.
But we didn't declare victory and leave. Our troops were ordered to capture the warlord Aideed -- we changed our mission and did not develop an exit strategy. We don't always have a unique capability, though. Not long ago, there was horrific ethnic and tribal cleansing in Liberia. It was as heinous as any we've seen in Bosnia or Kosovo -- in fact, much worse. A group of neighboring countries got together to stop it and intervened between the warring factions. They were successful, and there is now a shaky peace in Liberia. Certainly the U.S. could have done the same thing, but we don't hold a monopoly on peacekeeping.
Years ago, President Nixon laid out principles of how our military forces should be used overseas. Based on his principles, I offer the following outline for a rational superpower foreign policy:
(1) Intervene with troops on the ground only when our own national security is endangered. We could choose to get involved in every regional conflict around the world, but that would prevent us from being able to respond where only we can respond. If we're tied down in Kosovo, who will respond when the North Koreans invade the South and China joins in? This is an ethical question for elected officials. If there is no American security interest threatened by a military action we wish to take, then we have to consider very carefully whether it's right to send young Americans to possibly die in order to defend someone else's national security interest.
(2) Encourage our regional allies to deal with regional problems, with our assistance if needed. This will let us focus our resources where the United States has unique capability; in parts of the world where our interests may be greater or where air power is necessary. Australia is leading in East Timor. We should applaud that and offer the intelligence and logistics support they may require.
(3) Help those who are willing to fight for their own freedom. Too often, we ignore or even oppose local forces who are willing to fight for their own freedom. The Reagan Doctrine was to help freedom fighters help themselves. In the Balkans, President Milosevic was able to suppress the Croats until they ignored our arms embargo and fought to a draw. Congress tried twice to lift the embargo on the Muslims -- but the Clinton administration refused. I am sure the atrocities would have been fewer if there could have been the semblance of a fair fight.
The risks of ignoring these principles are great. Our military forces are being stretched to the breaking point. Just last week, the Army announced that two of its ten combat divisions are unfit for battle -- one fifth of our entire Army. That should alarm anyone who believes that a strong United States is a force for good in the world. Qualified personnel are leaving the services in droves. In 1998, 48 percent of Air Force pilots eligible for continuation opted to leave the service. The Army will fall at least 6,000 below Congressionally authorized troop strength by the end of 1999.
We used up a large part of our weapons inventory in Kosovo -- we were down to fewer than 200 cruise missiles worldwide, for example. That sounds like a lot, but it's about a day's worth for an operation such as Desert Storm. And let's be clear about what that means: Weakness is provocative. It is, in my view, also unethical. Should we have to send our forces into combat, it is simply wrong to do so without the best equipment we can get and all the support they'll need. That's becoming harder to do as we expand our involvement but reduce the size of our military.
By failing to keep our armed forces combat ready, we invite other nations to take advantage. Congress is responding with military pay raises and regular increases in the President's annual defense budget requests. There are other ways we are trying to change course. Congress recently passed a bill I authored that called on the Administration to examine whether we can reduce America's global commitments where the mission has ended or can be accomplished with fewer troops.
As a superpower, the United States must draw distinctions between the essential and the important. Otherwise, we will dissipate our resources and be unable to handle either. To maximize our strengths, we should focus our efforts where they can be best applied. For example, air power -- technology generally -- should clearly be an American responsibility.
But troops on the ground for these operations short of full combat -- such as in Bosnia and Kosovo -- is a much more expensive proposition for us. They require more support and self-protection because America is a bigger target. Some argue that allowing our allies to shoulder the burden of regional conflicts like Kosovo signals a policy of isolationism. That argument is just alarmism and, frankly, rank partisanship. Who could seriously argue that the U.S., with billions of dollars in foreign aid and our commitment to military alliances, is not doing its fair share in the world?
In fact, the truth is the opposite. We must choose more carefully between those areas where direct engagement is called for and those where a supporting role is more appropriate. If we do not, we will not be able to sustain our important global responsibilities. If America's core strength were allowed to dissipate, it would hurt our allies and others yearning for freedom. We cannot and should not do it all. We must carefully guard the strength entrusted to us by the people. It is not an endless supply and it is not easily replaced once exhausted.
Let me close with the wise words of President John Quincy Adams; Secretary of State; President; member of Congress. In 1821, John Quincy Adams defined his vision of America's role in the world. It provides a glimpse at the ethical underpinnings of American engagement by one of our earliest practitioners of foreign policy: "America well knows that by once enlisting under other banners than her own...she would involve herself beyond extrication in all the wars of interest and intrigue.... The fundamental maxims of her policy would insensibly change from liberty to force.... Wherever the standard of freedom and independence has been or shall be unfurled, there will America's heart be. She goes not abroad in search of monsters to destroy. She is the well-wisher to the freedom and independence of all.... The frontlet upon her brows would no longer beam with the ineffable splendor of freedom and independence; but in its stead would soon be substituted an imperial diadem, flashing in false and tarnished luster the murky radiance of dominion and power. She might become the dictator of the world: she would no longer be the ruler of her own spirit."
That warning still applies today, and it frames the ethical questions we must address in a foreign policy that is right for our country: To remain strong, to stay true to our core principles, and to be the beacon of freedom to the world. | http://polyconomics.org/memos/mm-991118.htm |
While no disruptions to electricity grids due to the geomagnetic storm have been reported, people up till New York are expecting to see auroras.
A geomagnetic storm of the moderate scale was supposed to hit Earth on September 27, as per the US space weather tracking agency. It was feared that the space weather phenomenon may affect electricity grids, satellite operations, and some other high latitude systems.
The Geomagnetic Storm Watch for Monday, September 27, was issued by the Space Weather Prediction Center (SWPC) of the National Oceanic and Atmospheric Administration (NOAA). The SWPC predicted a moderate G2-level geomagnetic storm.
It should be noted that a geomagnetic storm is different from a solar storm, resulting from solar wind.
While there were fears of disruption, no issues with power grids due to the geomagnetic storm have been reported till now from anywhere around the world. However, a Twitter user did question if the geomagnetic storm had revealed serious weakness in the power grid of Thurston County, which is located in the US state of Washington.
Geomagnetic storm.
Nevertheless, people living up in New York are expecting to see auroras in the night sky. None have been captured or reported as yet.
Meanwhile, the SWPC has also issued a fresh alert of a Type II Radio Emission that began at 11:50 am IST (0620 UTC). This could be associated with a solar flare event. As per the agency, “Type II emissions occur in association with eruptions on the sun and typically indicate a coronal mass ejection is associated with a flare event.” It estimated the velocity of the emission at 534 km/s
It is caused when the Earth’s magnetosphere is disturbed when there is an interaction between the space environment surrounding the planet and solar wind. The largest storms resulting from such conditions are linked to solar coronal mass ejections (CMEs), as per SWPC.
G2 or moderate level geomagnetic storms do not cause significant damage or disruption. As per SPWC, such storms can impact high-latitude power systems to cause voltage alarms. If the storm continues for a long time, it may result in damage to some transformers.
Satellites and spacecraft can require corrections from ground control, as orbit predictions can be affected. Other systems like high-frequency (HF) radio propagation at higher latitudes can fade. | https://onlypositivethings.com/viral/geomagnetic-storm-update-2021/ |
I can truly say this is the The Ultimate Chicken and Corn Soup. It is velvety and creamy, yet packed with chicken flavor.
The Ultimate Chicken and Corn Soup is the result of trying, recipe after recipe to find the perfect chicken soup. I was after a thick, luscious and rich soup, not a light healthy broth.
Then I recently ate it at a function. The best chicken soup I have ever tasted. I asked for seconds, but I did not have to guts to ask for the recipe. While eating the soup, I made mental notes, flavors I tasted…. textures. I could not wait to get to my kitchen. So here it is, I hope you enjoy it too.
The Ultimate Chicken and Corn Soup
serves 8
Ingredients
2 medium size leeks – finely chopped
a few sprigs fresh thyme
1 bay leaf
45 ml butter
4 chicken breasts with skin ( the skin is important)
2.5 liters chicken stock – I use Woolworths Liquid Stock Concentrate
1 x 410 g KOO Whole Kernel in brine – drained
3-4 cobs of corn – cooked in water until soft 30 ml butter
30 ml butter
5 ml smoked paprika
salt and white pepper
1 packet Cream of White Onion Soup
15 – 30 ml cornflour – depending on how thick you want your soup
250 – 300 ml cream
salt and white pepper
soft herbs to serve
Method
Preheat oven to 200 C. Heat the butter in a big pot, add the thyme and bay leaf and fry the leeks until soft, but not entirely broken up. Remove the skin from the chicken breast and place the skinless breasts in the pot. Add the stock. Poach the chicken on low heat until it is plump and soft. This takes about 20-25 minutes. Remove the chicken breasts and cut into bite-size chunks. Keep aside. Mix the packet of soup and the cornflour with a little cold water and stir it into the stock. Allow to thicken and add the cream. Taste the soup and adjust the seasoning. Add the cut up chicken and canned corn to the soup.
Now for the interesting bits.
Flatten and stretch the chicken skin and place on a baking sheet, lined with baking paper. Season with a little salt. Place another piece of baking paper on top and top that with another baking sheet. You want to press the chicken skin. Place in the oven for about 10-15 minutes. Check regularly. In the meantime, mix the smoked paprika, salt and white pepper with the 30 ml butter and rub it all over the cobs of corn. Place on a baking sheet in the oven and roast until it has char marks. You can also use your blow torch for this. When the corn is done, use a sharp knife and cut the corn off the cobs. Add the corn to the soup, but keep some for garnishing.
Serve the soup with fresh micro herbs, some charred corn and the crispy chicken skin. Please let me know what you think of this soup.
Cook’s Notes:
*You can use fresh corn, but for me, it is all about different textures and the canned corn has a softer texture.
*If you have issues about packets of soup, just increase the amount of cornflour. | https://www.my-easy-cooking.com/2019/08/20/the-ultimate-chicken-and-corn-soup/ |
[**Twists of two or multiple idempotent matrices**]{}
[Yongge Tian]{}
[*Shanghai Business School, Shanghai, China & Central University of Finance and Economics, Beijing, China*]{}
In this article, we revisit some block matrix construction methods and use them to derive various general expansion formulas for calculating the ranks of matrix expressions. As applications, we derive a variety of interesting rank equalities for matrix expressions composed by idempotent matrices, and present their applications in the characterization of some matrix equalities for generalized inverses of partitioned matrices.\
[**Aathematics Subject Classifications (2000):**]{} 15A03; 15A09; 15A27; 47A05
[**Keywords:**]{} rank formula; idempotent matrix; block matrix; generalized inverse
Introduction
============
Throughout this article, let ${\mathbb C}^{m\times n}$ denote the set of all $m\times n$ complex matrices; $A^{\ast}$, $r(A)$, and ${\mathscr R}(A)$ be the conjugate transpose, the rank, and the range (column space) of a matrix $A\in {\mathbb C}^{m\times n}$, respectively; $I_m$ be the identity matrix of order $m$; and $[\, A, \, B\,]$ be a row block matrix consisting of $A$ and $B$. We next introduce the definition and notation of generalized inverses of matrix. The Moore–Penrose inverse of $A \in {\mathbb C}^{m \times n}$, denoted by $A^{\dag}$, is the unique matrix $X \in {\mathbb C}^{n \times m}$ satisfying the four Penrose equations $$\begin{aligned}
{\rm (i)} \ AXA = A, \ \ {\rm (ii)} \ XAX=X, \ \ {\rm (iii)} \ (AX)^{\ast} = AX, \ \ {\rm (iv)} \ (XA)^{\ast} = XA.
\label{11}\end{aligned}$$ A matrix $X$ is called an $\{i,\ldots, j\}$-generalized inverse of $A$, denoted by $A^{(i,\ldots, j)}$, if it satisfies the $i$th,$\ldots,j$th equations. The collection of all $\{i,\ldots, j\}$-generalized inverses of $A$ is denoted by $\{A^{(i,\ldots, j)}\}$. The eight commonly-used generalized inverses of $A$ are $A^{\dag}$, $A^{(1,3,4)}$, $A^{(1,2,4)}$, $A^{(1,2,3)}$, $A^{(1,4)}$, $A^{(1,3)}$, $A^{(1,2)}$, and $A^{(1)}$. Furthermore, let $P_{A} =AA^{\dag}$, $E_{A} = I_m - AA^{\dag}$, and $F_{A} = I_n -A^{\dag}A$ stand for the three orthogonal projectors induced by $A$. Moreover, a matrix $X$ is called a $\{1\}$-inverse of $A$, denoted by $A^{-}$, if it satisfies $AXA = A$; the collection of all $A^{-}$ is denoted by $\{A^{-}\}$. The Drazin inverse of a square matrix $M$, denoted by $X = M^{D}$, is defined to be the unique solution $X$ to the following three matrix equations $M^{t}XM = M^t$, $XMX = X$ and $MX = XM$, where $t$ is the index of $M$, i.e., the smallest nonnegative integer $t$ such that $r(M^t) = r(M^{t+1}).$ When $t =1$, $X$ is also called the group inverse of $M$ and is denoted by $M^{\#}$. See e.g., [@BG; @CM; @RM] for more issues on generalized inverses of matrices.
The rank of matrix is a quite basic concept in linear algebra, which may be defined by different manners and can be calculated directly by transforming the matrix to certain row and/or column echelon forms. One of the most important applications of ranks of matrices is to describe singularity and nonsingularity of matrices, as well as the dimensions of row and column spaces of the matrices. Thus, people would always be of interest in establishing various simple and valuable formulas for calculating the ranks of matrices under various assumptions. One of the best-known fundamental formulas for ranks of matrices is $r(A) = r(PAQ)$ provided $P$ and $Q$ are two nonsingular matrices, and people used it to derive numerous interesting and useful rank equalities for different choice of the matrices $A$, $P$, and $Q$. It is well known in linear algebra that people can establish rank formulas from block matrices and their elementary operations. In this article, we revisit this time-honored trick through some examples and show how to establish various valuable formulas for calculating the ranks of matrices using the block matrix method. As applications, we solve many some matrix equality problems on idempotent matrices and generalized inverses under various assumptions.
How to establish rank formulas for specified block matrices
===========================================================
It has a long history in linear algebra to establish simple and useful equalities for ranks of matrices using various tricky elementary operations of matrices. Especially, there is a major route to derive matrix rank formulas through various specific block matrix constructions. For instance, the following rank formulas in linear algebra $$\begin{aligned}
r(\,I_m - A^2 \,) & = r(\, I_m + A \,) + r(\, I_m - A \,) - m,
\label{hh21}
\\
r(\,A \pm A^2 \,) & = r(A) + r(\, I_m \pm A \,) - m,
\label{hh22}
\\
r(\,A \pm A^3 \,) & = r(A) + r(\, I_m \pm A^2 \,) - m,
\label{hh23}
\\
r[\, A(\, I_m \pm A \,)^2 \,] & = r(A) + r[\, (\, I_m \pm A \,)^2 \,] - m,
\label{hh24}
\\
r(\, I_m - AB \,) + n & = r(\, I_n - BA \,) + m,
\label{hh25}
\\
r(\, A - AXBYA \,) + r(B) & = r(\, B - BYAXB \,) + r(A)
\label{hh27}\end{aligned}$$ for any matrices $A$, $B$, $X$, and $Y$ of the appropriate sizes and the following rank formula $$\begin{aligned}
r[(I_m - P - Q + QP) = m - r(P) - r(Q) + r(PQ)
\label{hh28}\end{aligned}$$ for any two idempotent matrices $P$ and $Q$ of the same size are simple and well known, which can be established by calculating the ranks of the following specified two-by-two block matrices $$\begin{aligned}
& \begin{bmatrix} I_m & I_m + A \\ I_m - A & 0 \end{bmatrix}, \ \ \begin{bmatrix} I_m & I_m \pm A \\ A & 0 \end{bmatrix}, \ \ \begin{bmatrix} I_m & I_m \pm A^2 \\ A & 0 \end{bmatrix}, \ \ \begin{bmatrix} I_m & (I_m \pm A)^2 \\ A & 0 \end{bmatrix},
\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \begin{bmatrix} I_m & A \\ B & I_n \end{bmatrix}, \ \ \ \begin{bmatrix} A & AXB \\ BYA & B \end{bmatrix}, \ \ \ \begin{bmatrix} I_m & Q \\ P & 0 \end{bmatrix},\end{aligned}$$ respectively; see, e.g., [@ASt; @MS:1974; @Sty; @TS4]. These rank formulas can directly be used to characterize algebraic properties of the matrices in the formulas, such as, nullity, singularity, nonsingularity, etc., and of course are basic issues in many textbooks in linear algebra and matrix theory. It seems more natural to consider consecutive subsequences of these block matrices and to extend such kind of rank formulas to general settings under various assumptions. The principal issue dealt with here is to what extent these formulas generalize to cases with multiple matrices. In first half of this section, we derive three general rank formulas using block matrices composed by general solutions of several consistent linear matrix equations. We then present a variety of simple and interesting consequences for idempotent matrices in the formulas.
\[T1\] Let $M \in {\mathbb C}^{m\times m}$ be given$,$ and assume that $X, \, Y \in {\mathbb C}^{m\times m}$ are solutions of the following three matrix equations $$\begin{aligned}
MX = X, \ \ YM = Y, \ \ MY = XM.
\label{z1}\end{aligned}$$ Then the rank of $X - Y$ can be calculated by the expansion formula $$\begin{aligned}
r(\,X - Y\,) = r\!\begin{bmatrix} X \\ Y \end{bmatrix} + r[\,X, \, Y\,] - r(X) - r(Y).
\label{z2}\end{aligned}$$
We first construct a block matrix from $X$ and $Y$ as follows $$\begin{aligned}
N= \begin{bmatrix} - X & 0 & X
\\ 0 & Y & Y \\ X & Y & 0 \end{bmatrix}\!.
\label{z3}\end{aligned}$$ Then it is easy to verify that $$\begin{aligned}
P_1NQ_1 =\begin{bmatrix} I_m & 0 & 0
\\ 0 & I_m & 0 \\ I_m & -I_m & I_m \end{bmatrix}\!N\!\begin{bmatrix} I_m & 0 & I_m
\\ 0 & I_m & -I_m \\ 0 & 0 & I_m \end{bmatrix} = \begin{bmatrix} - X & 0 & 0
\\ 0 & Y & 0 \\ 0 & 0 & X-Y \end{bmatrix}\!,
\label{z4}\end{aligned}$$ and from that $$\begin{aligned}
P_2NQ_2 = \begin{bmatrix} I_m & 0 & M
\\ 0 & I_m & 0 \\ 0 & 0 & I_m \end{bmatrix}\!N\!\begin{bmatrix} I_m & 0 & 0
\\ 0 & I_m & 0 \\ 0 & -M & I_m \end{bmatrix} = \begin{bmatrix} 0 & 0 & X
\\ 0 & 0 & Y \\ X & Y & 0 \end{bmatrix}\!.
\label{z5}\end{aligned}$$ Since $P_1$, $Q_1$, $P_2$, and $Q_2$ are nonsingular matrices, both and imply that $$\begin{aligned}
& r(N) = r(P_1NQ_1) = r\!\begin{bmatrix} - X & 0 & 0
\\ 0 & Y & 0 \\ 0 & 0 & X-Y \end{bmatrix} = r(\,X - Y\,) + r(X) + r(Y),
\label{z6}
\\
& r(N) = r(P_2NQ_2) = r\!\begin{bmatrix} 0 & 0 & X
\\ 0 & 0 & Y \\ X & Y & 0 \end{bmatrix} = r\!\begin{bmatrix} X \\ Y \end{bmatrix} + r[\,X, \, Y\,].
\label{z7}\end{aligned}$$ Combining and leads to .
\[T2\] Let $A, \, B\in {\mathbb C}^{m\times m}$ be given$,$ and assume that $X, \, Y \in {\mathbb C}^{m\times m}$ are solutions of the following equations $$\begin{aligned}
AX = X, \ \ YB = Y, \ \ AY = XB.
\label{z8}\end{aligned}$$ Then the rank of $X - Y$ can be calculated by the expansion formula $$\begin{aligned}
r(\,X - Y\,) = r\!\begin{bmatrix} X \\ Y \end{bmatrix} + r[\,X, \, Y\,] - r(X) - r(Y).
\label{z9}\end{aligned}$$
It is easy to verify from and elementary matrix transformations that the rank of $N$ in is $$\begin{aligned}
r(N) = r\!\begin{bmatrix} 0 & AY & X
\\ 0 & Y & Y \\ X & Y & 0 \end{bmatrix} = r\!\begin{bmatrix} 0 & XB & X
\\ 0 & YB & Y \\ X & Y & 0 \end{bmatrix} =
r\begin{bmatrix} 0 & 0 & X
\\ 0 & 0 & Y \\ X & Y & 0 \end{bmatrix} = r\!\begin{bmatrix} X \\ Y \end{bmatrix} + r[\,X, \, Y\,].
\label{z10}\end{aligned}$$ Combining and leads to .
\[T3\] Let $A, \,B \in {\mathbb C}^{m\times m}$ be given$,$ and assume that $X \in {\mathbb C}^{m\times n}$ and $Y \in {\mathbb C}^{m\times p}$ are solutions of the following equations $$\begin{aligned}
AX = X, \ \ BY = Y, \ \ {{\mathscr R}}(X) \supseteq {{\mathscr R}}(AY), \ \ {{\mathscr R}}(Y) \supseteq {{\mathscr R}}(BX).
\label{z11}\end{aligned}$$ Then the following rank equality holds $$\begin{aligned}
r[\,AY, \, BX\,] = r[\,X, \, Y\,] + r(AY) + r(BX) - r(X) - r(Y).
\label{z12}\end{aligned}$$
Construct a block matrix from $X$, $Y$, $AY$, and $BX$ as follows $$\begin{aligned}
N = \begin{bmatrix} X & 0 & AY & 0
\\ 0 & Y & 0 & BX \\ X & Y & 0 & 0\end{bmatrix}\!.
\label{z13}\end{aligned}$$ Then it is easy to verify under that $$\begin{aligned}
r(N) = r\!\begin{bmatrix} X & 0 & 0 & 0
\\ 0 & Y & 0 & 0 \\ 0 & 0 & -AY & - BX\end{bmatrix} = r[\,AY, \, BX\,] + r(X) + r(Y),
\label{z14}\end{aligned}$$ and that $$\begin{aligned}
r(N) = r\!\begin{bmatrix} 0 & -AY & AY & 0
\\ -BX & 0 & 0 & BX \\ X & Y & 0 & 0\end{bmatrix}
= r\!\begin{bmatrix} 0 & 0 & AY & 0
\\ 0 & 0 & 0 & BX \\ X & Y & 0 & 0\end{bmatrix} = r[\,X, \, Y\,] + r(AY) + r(BX).
\label{z15}\end{aligned}$$ Combining and leads to .
The matrix equations in , , and are quite fundamental in matrix analysis, and have been widely studied in theory and applications,; see [@Mey]. Under the assumptions of these equations, , , and link the solutions of these matrix equations and their operations. In this situation, there is a very strong intrinsic mathematical motivation to establish concrete rank formulas from , , and for various solutions of the matrix equations.
Recall that a square matrix $A$ is said to be idempotent if $A^2 = A$. An idempotent matrix is often called an oblique projector whose null space is oblique to its range, in contrast to orthogonal projector, whose null space is orthogonal to its range. idempotents can be defined in more general algebraic structures, and are important tools in the investigation of the algebraic structures. As is known to all, idempotent matrices have strikingly simple and interesting properties, and one of such properties is that any idempotent matrix $A$ can be decomposed as $A = P{\rm diag}(I_k, 0)P^{-1}$, where $k$ is the rank of $A$, while any Hermitian idempotent matrix $A$ admits the decomposition $A = P{\rm diag}(I_k, 0)P^{\ast}$, where $P^{-1} = P^{\ast}$. Idempotent matrices arise naturally in the theory of generalized inverses of matrices, and are a class of fundamental objects of study in matrix analysis. For instance,
1. the pair of the two ordered products $AA^-$ and $A^-A$ are always idempotent matrices for any generalized inverse $A^-$ of $A$;
2. both $AA^{\dag}$ and $A^{\dag}A$ are Hermitian idempotent matrices for the Moore–Penrose inverse $A^{\dag}$ of $A$;
3. the matrix $X(X^{*}VX)^{\dag}X^{*}V$ is always idempotent. This matrix often occurs in the weighted least-squares estimation problems in linear regression analysis.
Many other types of matrix can be converted into idempotent matrices through some elementary operations. For instance,
1. if $A^2 = -A$, then $(-A)^2 = -A$, i.e., $-A$ is idempotent;
2. if $A^2 = I_m$, then $(\,I_m \pm A \,)/2$ are idempotent; if $A^2 = -I_m$, then $(I_m \pm iA)/2$ are idempotent;
3. the product $B(AB)^{\dag}A$, as well as $BC(ABC)^{\dag}A$, and $C(ABC)^{\dag}AB$ are idempotent.
Any matrix $A$ satisfying a quadratic equation $A^2 + aA + bI_m=0$ can be written as $[\, A - (a/2)I_m \,]^2 = (\, a^2/4 - b\,)I_m$. If $a^2/4 -b \neq 0$, then we can also construct an idempotent matrix from this equality. Through these transformations, various results on idempotent matrices can be extended to other types of quadratic matrices.
There is a substantial literature related to the approaches on idempotent matrices and related topics; see e.g., [@Af; @ACM; @ACS; @AHT; @Ara; @BB; @BBS; @BBSz; @BBo; @BBT; @Bal; @BV; @Bor; @BGK; @BS1; @BS2; @BS3; @Bak; @BT; @BTh; @CX; @CDu; @CD; @Cer; @Cve; @Den1; @Den2; @Den3; @Dok; @DRR; @DD; @BES1; @BES2; @BES3; @BES4; @BES5; @Erd; @FRS; @Fou; @Gab; @Gre; @Gro1; @Gro2; @Gro3; @GT; @HN; @HO1; @HO2; @HP1; @HP2; @HS; @HT; @Ikr1; @Ikr2; @ITP; @Kala; @Kaw; @KR1; @KR2; @KR3; @KRS; @KRa; @KRSa; @Kol; @KS1; @KS2; @Krup; @KRSi; @KSp; @Laf; @LMR; @LWY; @ND; @Nis; @Oml; @OO; @Pas; @Paz1; @Paz2; @Paz3; @PT; @PSI; @Rab0; @Rab1; @Rab2; @RSa; @RR; @RS; @SO; @Shc; @Spi; @Steg; @Ste; @TY; @T2; @T3; @Tian:2011; @Tian:2019b; @TT1; @TT2; @Tos; @Tre; @Vet; @Wan; @WW; @Wu; @XZZ; @YTY; @YL; @ZX], and one of the main contributions in this respect is establishing various analytical and valuable formulas for calculating ranks of various matrix expressions composed by idempotent matrices; see e.g., [@CCLY; @KR2; @TS1; @TS2; @TS3; @ZLLY; @Zuo1; @Zuo2; @ZZY] among others.
Armed with the results in Theorems \[T1\]–\[T3\], we can establish many expansion formulas for calculating ranks of matrix expressions composed by idempotent matrices, and to present their consequences on the relationships among the idempotent matrices which, as we shall see later, are the basis for the theory of ranks of idempotent matrices.
\[T4\] Let $A$ and $B$ be two idempotent matrices of the same size$.$ Then the following rank equalities $$\begin{aligned}
r[\,(AB)^k, \, (BA)^k\,] & = r[\,A, \, B\,] + r[(AB)^k] + r[(BA)^k] - r(A) - r(B),
\label{z16}
\\
r[\,(AB)^kA, \, (BA)^kB\,] & = r[\,A, \, B\,] + r[(AB)^kA] + r[(BA)^kB] - r(A) - r(B),
\label{z17}
\\
r\!\begin{bmatrix} (AB)^k \\ (BA)^k \end{bmatrix} & = r\!\begin{bmatrix} A \\ B \end{bmatrix} + r[(AB)^k] + r[(BA)^k] - r(A) - r(B),
\label{z18}
\\
r\!\begin{bmatrix} (AB)^kA \\ (BA)^kB \end{bmatrix} & = r\!\begin{bmatrix} A \\ B \end{bmatrix} + r[(AB)^kA] + r[(BA)^kB] - r(A) - r(B),
\label{z19}
\\
r[\,(AB)^k - (BA)^k\,] & = r\!\begin{bmatrix} (AB)^k \\ (BA)^k \end{bmatrix} + r[\,(AB)^k, \, (BA)^k\,] - r[(AB)^k] - r[(BA)^k],
\label{z20}
\\
r[\,(AB)^k - (BA)^k\,] & = r\!\begin{bmatrix} A \\ B \end{bmatrix} + r[\,A, \, B\,] + r[(AB)^k] + r[(BA)^k] - 2r(A) - 2r(B),
\label{z21}
\\
r[\,(AB)^kA - (BA)^kB\,] & = r\!\begin{bmatrix} (AB)^kA \\ (BA)^kB \end{bmatrix} + r[\,(AB)^kA, \, (BA)^kB\,] - r[(AB)^kA] - r[(BA)^kB],
\label{z22}
\\
r[\,(AB)^kA - (BA)^kB\,] & = r\!\begin{bmatrix} A \\ B \end{bmatrix} + r[\,A, \, B\,] + r[(AB)^kA] + r[(BA)^kB] - 2r(A) - 2r(B)
\label{z23}\end{aligned}$$ hold for all integers $k \geq 1.$ In particular$,$ the following consequences hold$:$
1. $r[\,(AB)^k, \, (BA)^k\,] = r[(AB)^k] + r[(BA)^k]$ $\Leftrightarrow$ $r[\,A, \, B\,] = r(A) + r(B)$ $\Leftrightarrow$ ${{\mathscr R}}[(AB)^k] \cap {{\mathscr R}}[(BA)^k] = \{0\}$ $\Leftrightarrow$ ${{\mathscr R}}(A) \cap {{\mathscr R}}(B) = \{0\}.$
2. $r[\,(AB)^k, \, (BA)^k\,]= r[\,A, \, B\,]$ $\Leftrightarrow$ ${{\mathscr R}}[(AB)^k] = {{\mathscr R}}(A)$ and ${{\mathscr R}}[(BA)^k] = {{\mathscr R}}(B).$
3. $(AB)^k = (BA)^k$ $\Leftrightarrow$ ${{\mathscr R}}[(AB)^k] = {{\mathscr R}}[(BA)^k]$ and ${{\mathscr R}}[(A^{\ast}B^{\ast})^k] = {{\mathscr R}}[(B^{\ast}A^{\ast}y)^k]$ $\Leftrightarrow$ $r[A, \, B\,] = r(A) + r(B) - r[(AB)^k]$ and $r\!\begin{bmatrix} A \\ B \end{bmatrix} = r(A) + r(B) - r[(BA)^k].$
4. $r[\,(AB)^kA, \, (BA)^kB\,] = r[(AB)^kA] + r[(BA)^kB]$ $\Leftrightarrow$ $r[\,A, \, B\,] = r(A) + r(B)$ $\Leftrightarrow$ ${{\mathscr R}}[(AB)^kA] \cap {{\mathscr R}}[(BA)^kB] = \{0\}$ $\Leftrightarrow$ ${{\mathscr R}}(A) \cap {{\mathscr R}}(B) = \{0\}.$
5. $r[\,(AB)^kA, \, (BA)^kB\,]= r[\,A, \, B\,]$ $\Leftrightarrow$ ${{\mathscr R}}[(AB)^kA] = {{\mathscr R}}(A)$ and ${{\mathscr R}}[(BA)^kB] = {{\mathscr R}}(B).$
6. $(AB)^kA = (BA)^kB$ $\Leftrightarrow$ ${{\mathscr R}}[(AB)^kA] = {{\mathscr R}}[(BA)^kB]$ and ${{\mathscr R}}[(A^{\ast}B^{\ast})^kA^{\ast}] = {{\mathscr R}}[(B^{\ast}A^{\ast})^kB^{\ast}]$ $\Leftrightarrow$ $r[A, \, B\,] = r(A) + r(B) - r[(AB)^kA]$ and $r\!\begin{bmatrix} A \\ B \end{bmatrix} = r(A) + r(B) - r[(BA)^kB].$
Let $X = (AB)^{k-1}A$ and $Y =(BA)^{k-1}B$, as well as $X = (AB)^k$ and $Y =(BA)^k$, as well as respectively. Then they satisfy . In such cases, becomes and , respectively. Eqs. and are established by taking transpose of and , respectively.
Let $M =A$, $X = (AB)^k$, and $Y =(BA)^k$. Then they satisfy , thus becomes .
Let $X = (AB)^kA$ and $Y = (BA)^kB$. Then they satisfy , thus becomes .
Substituting – into and yields and , respectively. Results (a)–(f) follow directly from , , and –.
Some of – were established in the literature; see, e.g., [@TS1; @TS2; @TS3; @ZZY; @ZLLY]. By a similar approach, we can also establish a general rank formula associated with a family of idempotent matrices of the same size.
\[Th25\] Let $A_1, A_2, \ldots, A_k$ be a family of idempotent matrices of the same size$,$ and denote $$\begin{aligned}
A=[A_1, A_2, \ldots, A_k] \ \ and \ \ \widehat{A}_i=[A_1, \ldots, A_{i-1}, \, 0, \, A_{i+1}, \ldots, A_k].
\label{z24}\end{aligned}$$ Then they satisfy the following rank identity $$\begin{aligned}
r[A_1\widehat{A}_1, A_2\widehat{A}_2,\ldots, A_k\widehat{A}_k] = r(A_1\widehat{A}_1) + r(A_2\widehat{A}_2) + \cdots + r(A_k\widehat{A}_k) + r(A) - r(A_1) - r(A_2) - \cdots - r(A_k).
\label{z25}\end{aligned}$$ In particular$,$ the following results hold$:$
1. $r[A_1\widehat{A}_1, A_2\widehat{A}_2,\ldots, A_k\widehat{A}_k] = r(A_1\widehat{A}_1) + r(A_2\widehat{A}_2) + \cdots + r(A_k\widehat{A}_k)$ if and only if $r(A) = r(A_1) + r(A_2) + \cdots + r(A_k).$
2. $r[A_1\widehat{A}_1, A_2\widehat{A}_2,\ldots, A_k\widehat{A}_k] = r(A)$ if and only if ${{\mathscr R}}(A_i\widehat{A}_i) = {{\mathscr R}}(A_i),$ $i = 1, 2, \ldots, k.$
3. If $A_1\widehat{A}_1 = A_2\widehat{A}_2 = \ldots = A_k\widehat{A}_k = 0,$ then $r(A) = r(A_1) + r(A_2) + \cdots + r(A_k).$
4. $r(A) \geq r(A_1) + r(A_2) + \cdots + r(A_k) - r(A_1\widehat{A}_1) - r(A_2\widehat{A}_2) - \cdots - r(A_k\widehat{A}_k)$ holds$.$
From the given matrices, we construct a block matrix as follows $$\begin{aligned}
M = \begin{bmatrix} A_1 & 0 & \cdots & 0 & A_1\widehat{A}_1 & 0 & \cdots & 0
\\
0 & A_2 & \cdots & 0 & 0 & A_2\widehat{A}_2 & \cdots & 0
\\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots
\\
0 & 0 & \cdots & A_k & 0 & 0 & \cdots & A_k\widehat{A}_k
\\
A_1 & A_2 & \cdots & A_k & 0 & 0 & \cdots & 0
\end{bmatrix}.
\label{z26}\end{aligned}$$ We then apply elementary block matrix operations to this $X$ to obtain the following rank equality $$\begin{aligned}
r(X)& = r\!\begin{bmatrix} A_1 & 0 & \cdots & 0 &0 & 0 & \cdots & 0
\\
0 & A_2 & \cdots & 0 & 0 & 0 & \cdots & 0
\\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots
\\
0 & 0 & \cdots & A_k & 0 & 0 & \cdots & 0
\\
0 & 0 & \cdots & 0 & -A_1\widehat{A}_1 & -A_2\widehat{A}_2 & \cdots & -A_k\widehat{A}_k
\end{bmatrix} {\nonumber}\\
& = r(A_1) + r(A_2) + \cdots + r(A_k) + r[A_1\widehat{A}_1, A_2\widehat{A}_2,\ldots, A_k\widehat{A}_k].
\label{z27}\end{aligned}$$ Also by elementary block matrix operations and the idempotency of $A_1, A_2, \ldots, A_k$, we obtain the following rank equality $$\begin{aligned}
r(M) & = r\!\begin{bmatrix} 0 & -A_1A_2 & \cdots & -A_1A_k & A_1\widehat{A}_1 & 0 & \cdots & 0
\\
-A_2A_1 & 0 & \cdots & -A_2A_k & 0 & A_2\widehat{A}_2 & \cdots & 0
\\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots
\\
-A_kA_1 & -A_kA_2 & \cdots & 0 & 0 & 0 & \cdots & A_k\widehat{A}_k
\\
A_1 & A_2 & \cdots & A_k & 0 & 0 & \cdots & 0
\end{bmatrix} {\nonumber}\\
& = r\!\begin{bmatrix} 0 & 0 & \cdots & 0 & A_1\widehat{A}_1 & 0 & \cdots & 0
\\
0 & 0 & \cdots & 0 & 0 & A_2\widehat{A}_2 & \cdots & 0
\\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots
\\
0 & 0 & \cdots & 0 & 0 & 0 & \cdots & A_k\widehat{A}_k
\\
A_1 & A_2 & \cdots & A_k & 0 & 0 & \cdots & 0
\end{bmatrix} {\nonumber}\\
& = r(A_1\widehat{A}_1) + r(A_2\widehat{A}_2) + \cdots + r(A_k\widehat{A}_k) + r(A).
\label{z28}\end{aligned}$$ Combining and leads to . Results (a) and (b) follow directly from .
Eq. shows that all idempotent matrices are linked one another through certain simple but nontrivial rank formulas, so that we can conveniently use them to discuss relationships among all idempotent matrices under various circumstances. It is easy to see that for $k =2$ and for $k =1$ are the same. For $k =3$ in , we obtain the following appealing results on the relationships among any three idempotent matrices of the same size.
\[TW26\] Let $A,$ $B,$ and $C$ be three idempotent matrices of the same size$.$ Then $$\begin{aligned}
r[\,A, \, B, \, C\,] & = r(A) + r(B) + r(C) - r[\,AB, \, AC\,] - r[\, BA, \, BC\,] - r[\, CA, \, CB\,] {\nonumber}\\
& \ \ \ + r[AB, \, AC, \, BA, \, BC, \, CA, \, CB].
\label{z29}\end{aligned}$$ If $AB = BA,$ $AC = CA,$ and $BC = CB,$ then $$\begin{aligned}
r[\,A, \, B, \, C\,] = r(A) + r(B) + r(C) - r[\,AB, \, AC\,] - r[\, BA, \, BC\,] - r[\, CA, \, CB\,] + r[AB, \, AC, \, BC\,].
\label{z30}\end{aligned}$$ In particular$,$ the following results hold$.$
1. $r[\,A, \, B, \, C\,]= r(A) + r(B) + r(C)$ if and only if $r[AB, \, AC, \, BA, \, BC, \, CA, \, CB] = r[\,AB, \, AC] + r[\, BA, \, BC\,] + r[\, CA, \, CB\,].$
2. $r[AB, \, AC, \, BA, \, BC, \, CA, \, CB] = r[\,A, \, B, \, C\,]$ if and only if ${{\mathscr R}}[\,AB, \, AC\,] = {{\mathscr R}}(A),$ ${{\mathscr R}}[\, BA, \, BC\,] = {{\mathscr R}}(B),$ and ${{\mathscr R}}[\, CA, \, CB\,] = {{\mathscr R}}(C).$
3. If $AB = BA =AC = CA = BC = CB =0,$ then $r[\,A, \, B, \, C\,] = r(A) + r(B) + r(C).$
4. $r[\,A, \, B, \, C\,] \geq r(A) + r(B) + r(C) - r[\,AB, \, AC\,] - r[\, AB, \, BC\,] - r[\, AC, \, BC\,]$ holds$.$
\[TW27\] Let $A,$ $B,$ and $C$ be three matrices with the same row number$,$ and denote $P_{A} = AA^{\dag},$ $P_{B} = BB^{\dag},$ and $P_{C} = CC^{\dag}.$ Then $$\begin{aligned}
r[\,A, \, B\,] & = r(A) + r(B) - r(P_{A}P_{B}) - r(P_{B}P_{A}) + r[P_{A}P_{B}, \, P_{B}P_{A} \,],
\label{}
\\
r[\,A, \, B, \, C\,] & = r(A) + r(B) + r(C) - r[\,P_{A}P_{B}, \, P_{A}P_{C}\,] - r[\, P_{B}P_{A}, \, P_{B}P_{C}\,] - r[\, P_{C}P_{A}, \, P_{C}P_{B}\,] {\nonumber}\\
& \ \ \ + r[P_{A}P_{B}, \, P_{A}P_{C}, \, P_{B}P_{A}, \, P_{B}P_{C}, \,
P_{C}P_{A}, \, P_{C}P_{B}].
\label{}\end{aligned}$$ In particular$,$ the following results hold$.$
1. $r[\,A, \, B\,] = r(A) + r(B)$ $\Leftrightarrow$ $r[P_AP_B, \, P_BP_A \,] = r(P_AP_B) + r(P_BP_A)$ $\Leftrightarrow$ ${\mathscr R}(A) \cap {\mathscr R}(B) =\{0\}$ $\Leftrightarrow$ ${\mathscr R}(P_{A}P_{B}) \cap {\mathscr R}(P_{B}P_{A}) =\{0\}.$
2. $r[\,A, \, B\,] = r(A) + r(B) - r(P_{A}P_{B})$ $\Leftrightarrow$ $r[P_AP_B, \, P_BP_A \,] = r(P_AP_B) = r(P_BP_A)$ $\Leftrightarrow$ ${{\mathscr R}}(P_AP_B) = {{\mathscr R}}(P_BP_A)$ $\Leftrightarrow$ $P_AP_B = P_BP_A.$
3. $r[\,A, \, B\,] = r[P_{A}P_{B}, \, P_{B}P_{A} \,]$ $\Leftrightarrow$ $r(A^*B) = r(A) = r(B).$
4. $r[\,A, \, B, \, C\,] = r(A) + r(B) + r(C)$ $\Leftrightarrow$ $r[P_{A}P_{B}, \, P_{A}P_{C}, \, P_{B}P_{A}, \, P_{B}P_{C}, \,
P_{C}P_{A}, \, P_{C}P_{B}] = r[\,P_{A}P_{B}, \, P_{A}P_{C}\,] + r[\, P_{B}P_{A}, \, P_{B}P_{C}\,] + r[\, P_{C}P_{A}, \, P_{C}P_{B}\,].$
5. $r[\,A, \, B, \, C\,] = r(A) + r(B) + r(C) - r(P_{A}P_{B}) - r(P_{A}P_{C}) - r(P_{B}P_{C})$ $\Leftrightarrow$ $r[P_{A}P_{B}, \, P_{A}P_{C}, \, P_{B}P_{A}, \, P_{B}P_{C}, \,
P_{C}P_{A}, \, P_{C}P_{B}] = r[\,P_{A}P_{B}, \, P_{A}P_{C}\,] + r[\, P_{B}P_{A}, \, P_{B}P_{C}\,] + r[\, P_{C}P_{A}, \, P_{C}P_{B}\,] - r(P_{A}P_{B}) - r(P_{A}P_{C}) - r(P_{B}P_{C})$.
One of the well-known problems in the theory of generalized inverses is to determine the relationships among generalized inverses of a block and its submatrices. For instance, $\begin{bmatrix} A^{-} \\ B^{-}
\end{bmatrix}$ is a generalized inverse of $[\,A, \, B\,]$ if and only if $[\,A, \, B\,]\begin{bmatrix} A^{-} \\ B^{-} \end{bmatrix}[\,A, \, B\,] =
[\,A, \, B\,]$ by definition. On the other hand, it is easy to verify that $$\begin{aligned}
[\,A, \, B\,] - [\,A, \, B\,]\begin{bmatrix} A^{-} \\ B^{-}
\end{bmatrix}[\,A, \, B\,] & = [\,A, \, B\,] - [\,(AA^{-} + BB^{-})A, \,(AA^{-} + BB^{-})B\,] {\nonumber}\\
& = - [\,BB^{-}A,\, AA^{-}AB\,]
\label{z31a}\end{aligned}$$ holds for all $A^{-}$ and $B^{-}.$ Applying to yields the following result.
\[TW28\] Let $A \in {\mathbb C}^{m\times n}$ and $B \in {\mathbb C}^{m\times p}.$ Then the two matrices and their generalized inverses $ A^{-}$ and $B^{-}$ satisfy the following rank identity $$\begin{aligned}
r\!\left([\,A, \, B\,] - [\,A, \, B\,]\begin{bmatrix} A^{-} \\ B^{-}
\end{bmatrix}[\,A, \, B\,] \right) = r(AA^{-}B) + r(BB^{-}A) + r[\,A, \, B\,] - r(A) - r(B).
\label{z32}\end{aligned}$$ In particular$,$ the following results hold$.$
1. The maximum and minimum ranks of with respect to $A^{-}$ and $B^{-}$ are given by $$\begin{aligned}
& \max_{A^{-}, B^{-}} r\!\left([\,A, \, B\,] - [\,A, \, B\,]\begin{bmatrix} A^{-} \\ B^{-}
\end{bmatrix}[\,A, \, B\,] \right) = r[A, \, B\,] - |r(A) - r(B)|,
\label{z33}
\\
& \min_{A^{-}, B^{-}} r\!\left([\,A, \, B\,] - [\,A, \, B\,]\begin{bmatrix} A^{-} \\ B^{-}
\end{bmatrix}[\,A, \, B\,] \right) = r(A) + r(B) - r[\,A,\, B\,] = \dim[{{\mathscr R}}(A) \cap {{\mathscr R}}(B)].
\label{z34}\end{aligned}$$
2. $\{[\,A, \, B\,]^{-}\} \cap \left\{\begin{bmatrix} A^{-} \\ B^{-}
\end{bmatrix} \right\} \neq \emptyset \Leftrightarrow r[\,A,\, B\,] = r(A) + r(B) \Leftrightarrow {{\mathscr R}}(A) \cap {{\mathscr R}}(B) = \{0\}.$
3. $\{[\,A, \, B\,]^{-}\} \supseteq \left\{\begin{bmatrix} A^{-} \\ B^{-}
\end{bmatrix} \right\} \Leftrightarrow r[A, \, B\,] =|r(A) - r(B)|$ $\Leftrightarrow$ $A = 0$ or $ B= 0.$
Noting that both $AA^{-}$ and $BB^{-}$ are idempotent and applying to , we obtain $$\begin{aligned}
r[\,AA^{-}BB^{-},\, BB^{-}AA^{-}\,] & = r(AA^{-}BB^{-}) + r(BB^{-}AA^{-}) + r[\,AA^{-},\, BB^{-}\,] -
r(AA^{-}) - r(BB^{-}) {\nonumber}\\
& = r(AA^{-}B) + r(BB^{-}A) + r[\,A, \, B\,] - r(A) - r(B).
\label{z36}\end{aligned}$$ Applying the two known rank formulas $$\begin{aligned}
\max_{A^{-}} r(\, D - CA^{-}B \,) &= \min \left\{ r[ \, C, \, D \, ], \ \
r\!\begin{bmatrix} B \\ D \end{bmatrix}\!, \ \
r\!\begin{bmatrix} A & B \\ C & D
\end{bmatrix} -r(A)
\right\}\!,
\label{z37}
\\
\min_{A^{-}}r(\,D - CA^{-}B \,)&= r(A) + r[\, C, \, D \,] +
r\!\begin{bmatrix} B \\ D \end{bmatrix} + r\begin{bmatrix} A & B \\ C & D \end{bmatrix} -
r\!\begin{bmatrix} A & 0 & B \\ 0 & C & D
\end{bmatrix} -
r\!\begin{bmatrix} A & 0 \\ 0
& B \\ C & D \end{bmatrix}
\label{z38}\end{aligned}$$ in [@T1] to $AA^{-}B$ and $BB^{-}A$ gives $$\begin{aligned}
& \max_{A^{-}} r(AA^{-}B) = \max_{B^{-}} r(BB^{-}A) = \min \{r(A), \ \ r(B)\},
\label{z39}
\\
& \min_{A^{-}} r(AA^{-}B) = \min_{B^{-}} r(BB^{-}A) = r(A) + r(B) - r[\,A, \, B\,].
\label{z40}\end{aligned}$$ Substituting and into yields $$\begin{aligned}
\max_{A^{-}, B^{-}} r[\,AA^{-}BB^{-},\, BB^{-}AA^{-}\,] & = 2\min \{r(A), \ \ r(B)\} + r[\,A, \, B\,] - r(A) - r(B) {\nonumber}\\
& = r[A, \, B\,] - |r(A) - r(B)|,
\label{z41}
\\
\min_{A^{-}, B^{-}} r[\,AA^{-}BB^{-},\, BB^{-}AA^{-}\,] & = 2r(A) + 2r(B) - 2r[\,A, \, B\,] + r[\,A, \, B\,] - r(A) - r(B) {\nonumber}\\
& = r(A) + r(B) - r[\,A, \, B\,].
\label{z42}\end{aligned}$$ Combining with and leads to and , respectively.
More rank formulas can be derived from and their variations. For example, for any $A \in {\mathbb C}^{m\times n}$, $B \in {\mathbb C}^{m\times p}$, and $C \in {\mathbb C}^{m\times q},$ the following rank equality $$\begin{aligned}
& r[\,AA^{-}[\,B, \, C\,], \, BB^{-}[\,A, \, C\,], \, CC^{-}[\,A, \,B\,]\,] {\nonumber}\\
& = r[\,A,\, B, \,C\,] + r[\,AA^{-}B, \, AA^{-}C\,] +r[\,BB^{-}A, \, BB^{-}C\,] + r[\,CC^{-}A, \, CC^{-}B\,] - r(A) - r(B) - r(C)
\label{z43}\end{aligned}$$ holds for all $A^{-}$, $B^{-}$, and $C^{-}$. Thus it is easy to easy to derive from and that $$\begin{aligned}
\min_{A^1, B^{-}, C^{-}} r[\,AA^{-}[\,B, \, C\,], \, BB^{-}[\,A, \, C\,], \, CC^{-}[\,A, \,B\,]\,] = r[\,A,\, B\,] + r[\,A,\,C\,] + r[\,B, \,C\,] - 2r[A,\, B, \,C].
\label{z44}\end{aligned}$$ On the other hand, it was shown in [@Tian:2019] that $$\begin{aligned}
\dim({{\mathscr R}}[\,A,\, B\,]\cap {{\mathscr R}}[\,A,\, C\,] \cap {{\mathscr R}}[\,B, \,C\,]) =
r[\,A,\, B\,] + r[\,A,\,C\,] + r[\,B, \,C\,] - 2r[\,A,\, B, \,C\,].
\label{z45}\end{aligned}$$ Thus we also have the following matrix rank minimization equality $$\begin{aligned}
\min_{A^{-}, B^{-}, C^{-}} r[\,AA^{-}[\,B, \, C\,], \, BB^{-}[\,A, \, C\,], \, CC^{-}[\,A, \, B\,]\,] = \dim({{\mathscr R}}[\,A, \, B\,]\cap {{\mathscr R}}[\,A, \, C\,] \cap {{\mathscr R}}[\,B, \, C\,]).
\label{z46}\end{aligned}$$ Prompted by , we obtain the following equality $$\begin{aligned}
[\,A, \, B, \, C\,] - [\,A, \, B,\, C\,]\begin{bmatrix} A^{-} \\ B^{-} \\ C^{-}
\end{bmatrix}[\,A, \, B, \, C\,] = [\,(\,BB^{-} + CC^{-} \,)A, \, (\,AA^{-} + CC^{-} \,)B, \, (\,AA^{-} + BB^{-} \,)C\,],
\label{z47}\end{aligned}$$ where $AA^{-}$, $BB^{-}$, and $CC^{-}$ are idempotent matrices. In this case, it would be of interest to establish expansion formulas for calculating the rank of the right-hand side of .
Matrix identities composed by two or three idempotent matrices and their applications
=====================================================================================
It has been noticed that two or more given idempotent matrices may satisfy various identities, while these identities can be used to characterize algebraic properties of matrix expressions composed by idempotent matrices. In this section, we first revisit two known identities composed by two idempotent matrices, and then to establish a variety of novel identities composed by two or three idempotent matrices and their applications, including inverses, Moore–Penrose generalized inverses, Drazin generalized inverses, rank, range, and null spaces of the matrix expressions.
\[TK31\] Let $A$ and $B$ be two idempotent matrices of the order $m.$ Then the following two matrix identities $$\begin{aligned}
& (A - B)^2 + (A + B - I_m)^2 = I_m,
\label{v31}
\\
& AB + BA + 4^{-1}I_m = (\, A + B - 2^{-1}I_m \,)^2,
\label{v32}\end{aligned}$$ and the following five rank formulas $$\begin{aligned}
& r[(A - B)^2] = r(A + B) + r(2I_m - A - B) - m,
\label{v33}
\\
& r[(I_m - A - B )^2] = r(I_m + A - B) + r(I_m - A + B) - m,
\label{v34}
\\
& r(AB + BA) = r(I_m - A - B ) + r(A + B) - m,
\label{v35}
\\
& r(I_m - AB - BA) = r[(\sqrt{5} -1)/2I_m + A + B] + r[(\sqrt{5} + 1)/2I_m - A - B] - m,
\label{v36}
\\
& r(2I_m - AB - BA) = r(I_m + A + B ) + r(2I_m - A - B) - m
\label{v37}\end{aligned}$$ hold$.$ In particular$,$ the following facts hold $$\begin{aligned}
& (A - B)^2 =0 \Leftrightarrow (I_m - A - B )^2 = I_m \Leftrightarrow r(A + B) + r(2I_m - A - B) = m,
\label{v38}
\\
& (A - B)^2 = 2^{-1} I_m \Leftrightarrow (I_m - A - B )^2 = 2^{-1}I_m,
\label{v39}
\\
& (A - B)^2 = I_m \Leftrightarrow (I_m - A - B )^2 = 0 \Leftrightarrow r(I_m + A - B) + r(I_m - A + B) - m,
\label{v310}
\\
& AB + BA = - 2I_m \Leftrightarrow (\, A + B - 2^{-1}I_m \,)^2 = - \frac{7}{4}I_m,
\label{v311}
\\
& AB + BA = -I_m \Leftrightarrow (\, A + B - 2^{-1}I_m \,)^2 = - \frac{3}{4}I_m,
\label{v312}\end{aligned}$$ $$\begin{aligned}
& AB + BA = -4^{-1}I_m \Leftrightarrow (\, A + B - 2^{-1}I_m \,)^2 =0,
\label{v313}
\\
& AB + BA = 0 \Leftrightarrow (\, A + B - 2^{-1}I_m \,)^2 = 4^{-1}I_m \Leftrightarrow r(I_m - A - B ) + r(A + B) = m,
\label{v314}
\\
& AB + BA = \frac{3}{4}I_m \Leftrightarrow (\, A + B - 2^{-1}I_m \,)^2 = I_m,
\label{v315}
\\
& AB + BA = I_m \Leftrightarrow (\, A + B - 2^{-1}I_m \,)^2 = \frac{5}{4}I_m \Leftrightarrow r[(\sqrt{5} -1)/2I_m + A + B] + r[(\sqrt{5} + 1)/2I_m - A - B] = m,
\label{v316}
\\
& AB + BA = 2I_m \Leftrightarrow (\, A + B - 2^{-1}I_m \,)^2 = \frac{9}{4}I_m \Leftrightarrow r(I_m + A + B ) + r(2I_m - A - B) = m,
\label{v317}\end{aligned}$$ and $$\begin{aligned}
& r(A - B) =m \Leftrightarrow r(A + B) = r(2I_m - A - B) = m,
\label{vv318}
\\
& r(I_m - A - B ) = m \Leftrightarrow r(I_m + A - B) = r(I_m - A + B) = m,
\label{vv319}
\\
& r(AB + BA) =m \Leftrightarrow r(I_m - A - B ) = r(A + B) = m,
\label{vv320}
\\
& r(I_m - AB - BA) = m \Leftrightarrow r[(\sqrt{5} -1)/2I_m + A + B] = r[(\sqrt{5} + 1)/2I_m - A - B] = m,
\label{vv321}
\\
& r(2I_m - AB - BA) = m \Leftrightarrow r(I_m + A + B ) = r(2I_m - A - B) = m.
\label{vv322}\end{aligned}$$
Eqs. and follow from direct calculations, where was first given in [@Kat]; see also [@ASS; @Bor; @Nis; @Sim]. Applying to and yields –. Eqs. – are direct consequences of –.
We next give a group of matrix identities composed by some linear combinations of two idempotent matrices and their products.
\[TK32\] Let $A$ and $B$ be two idempotent matrices of the order $m,$ and let $k$ be any positive integer$.$ Then the following factorization equalities $$\begin{aligned}
\alpha AB + \beta BA & = (\, \alpha A + \beta B \,)(\, A + B - I_m \,)
= (\, A + B - I_m \,) (\, \beta A + \alpha B \,),
\label{ff31}
\\
\alpha ABA + \beta BAB & = (\, \alpha A + \beta B \,)(\, A + B - I_m \,)^2
= (\, A + B - I_m \,)^2 (\, \beta A + \alpha B \,),
\label{ff32}
\\
\alpha (AB)^k + \beta (BA)^k & = (\, \alpha A + \beta B \,)(\, A + B - I_m \,)^{2k-1}
= (\, A + B - I_m \,)^{2k-1}(\, \beta A + \alpha B \,),
\label{ff33}
\\
\alpha (ABA)^k + \beta (BAB)^k & = (\, \alpha A + \beta B \,)(\, A + B - I_m \,)^{2k}
= (\, A + B - I_m \,)^{2k}(\, \beta A + \alpha B \,)
\label{ff34}\end{aligned}$$ hold for any two scalars $\alpha$ and $\beta.$ In particular$,$ $\alpha AB + \beta BA$ is nonsingular $\Leftrightarrow$ $\alpha (AB)^k + \beta (BA)^k$ is nonsingular $\alpha ABA + \beta BAB$ is nonsingular $\Leftrightarrow$ $\alpha (AB)^k + \beta (BA)^k$ is nonsingular $\Leftrightarrow$ $\alpha (ABA)^k + \beta (BAB)^k$ is nonsingular $\Leftrightarrow$ both $\alpha A + \beta B$ and $A + B - I_m$ are nonsingular; in which cases$,$ the following equalities hold $$\begin{aligned}
(\alpha AB + \beta BA)^{-1} & = (\, A + B - I_m \,)^{-1}(\, \alpha A + \beta B \,)^{-1}
= (\, \beta A + \alpha B \,)^{-1}(\, A + B - I_m \,)^{-1},
\label{dd37}
\\
(\alpha ABA + \beta BAB)^{-1} & = (\, A + B - I_m \,)^{-2}(\, \alpha A + \beta B \,)^{-1}
= (\, \beta A + \alpha B \,)^{-1}(\, A + B - I_m \,)^{-2},
\label{dd38}
\\
[\alpha (AB)^k + \beta (BA)^k]^{-1} & = (\, A + B - I_m \,)^{-2k+1}(\, \alpha A + \beta B \,)^{-1}
= (\, \beta A + \alpha B \,)^{-1}(\, A + B - I_m \,)^{-2k+1},
\label{dd39}
\\
[\alpha (ABA)^k + \beta (BAB)^k]^{-1} & = (\, A + B - I_m \,)^{-2k}(\, \alpha A + \beta B \,)^{-1}
= (\, \beta A + \alpha B \,)^{-1}(\, A + B - I_m \,)^{-2k}.
\label{dd310}\end{aligned}$$
Eqs.– follow from direct expansions and simplifications. Eqs.– follow from –.
It is no doubt that – can be used to approach performances of the matrix expressions on the left-hands under various assumptions, such as, ranks, ranges, nullity, r-potency, nilpotency, nonsingularity, inverses, generalized inverses, norms, etc. We next give some special cases of – and their variations, and present interesting consequences.
\[TK33\] Let $A$ and $B$ be two idempotent matrices of the order $m,$ and let $k$ be a positive integer$.$ Then$,$ the following matrix identities hold $$\begin{aligned}
AB - BA & = (\, A - B \,)(\, A + B - I_m \,) = - (\, A + B - I_m
\,)(\, A - B \,),
\label{w3}
\\
AB + BA & = (\, A + B \,)(\, A + B - I_m \,) = (\, A + B -
I_m\,)(\, A + B\,),
\label{w4}
\\
ABA - BAB & = (\, A - B \,)(\, A + B - I_m \,)^2 = (\, A + B - I_m \,)^2(\, A - B \,),
\label{w14}
\\
ABA + BAB & = (\, A + B \,)(\, A + B - I_m \,)^2 =(\, A + B - I_m \,)^2(\, A + B \,).
\label{w15}\end{aligned}$$ $$\begin{aligned}
(\,AB - BA\,)^k & = (-1)^{k(k-1)/2}(\, A - B \,)^k(\, A + B - I_m \,)^k = (-1)^{k(k-1)/2}(\, I_m - A - B \,)^k(\, A - B \,)^k,
\label{w6}
\\
(\,AB + BA \,)^k & = (\, A + B \,)^k(\, A + B - I_m \,)^k = (\, A + B - I_m \,)^k(\, A + B \,)^k, \ \ \ \ \ \
\label{w7}
\\
(\, ABA - BAB\,)^k & = (\, A - B \,)^k(\, A + B - I_m \,)^{2k} = (\, A + B - I_m\,)^{2k}(\, A - B\,)^k,
\label{w8}
\\
(\,ABA + BAB\,)^k & = (\, A + B \,)^k(\, A + B - I_m \,)^{2k} = (\, A + B - I_m \,)^{2k}(\, A + B\,)^k,
\label{w9}
\\
(AB)^k - (BA)^k & = (\, A - B \,)(\, A + B - I_m \,)^{2k-1} = - (\, A + B - I_m \,)^{2k-1}(\, A - B\,),
\label{w10}
\\
(AB)^k + (BA)^k & = (\, A + B \,)(\, A + B - I_m \,)^{2k-1} = (\, A + B - I_m \,)^{2k-1}(\, A + B \,),
\label{w11}
\\
(ABA)^k - (BAB)^k & = (\, A - B \,)(\, A + B - I_m \,)^{2k} = (\, A + B - I_m\,)^{2k}(\, A - B\,),
\label{w12}
\\
(ABA)^k + (BAB)^k & = (\, A + B \,)(\, A + B - I_m \,)^{2k} =(\, A + B - I_m \,)^{2k}(\, A + B\,).
\label{w13}\end{aligned}$$ In addition$,$ the following matrix identities hold $$\begin{aligned}
& AB - BA + (AB)^2 - (BA)^2 + \cdots + (AB)^k - (BA)^k {\nonumber}\\
& = (A - B)[(\, A + B - I_m \,) + (\, A + B - I_m \,)^3 + \cdots + (\, A + B - I_m \,)^{2k-1}] {\nonumber}\\
& = [(\, A + B - I_m \,) + (\, A + B - I_m \,)^3 + \cdots + (\, A + B - I_m \,)^{2k-1}](B - A),
\\
& AB + BA + (AB)^2 + (BA)^2 + \cdots + (AB)^k + (BA)^k {\nonumber}\\
& = (A + B)[(\, A + B - I_m \,) + (\, A + B - I_m \,)^3 + \cdots + (\, A + B - I_m \,)^{2k-1}] {\nonumber}\\
& = [(\, A + B - I_m \,) + (\, A + B - I_m \,)^3 + \cdots + (\, A + B - I_m \,)^{2k-1}](A+B),
\\
& ABA - BAB + (ABA)^2 - (BAB)^2 + \cdots + (ABA)^k - (BAB)^k {\nonumber}\\
& = (A - B)[(\, A + B - I_m \,)^2 + (\, A + B - I_m \,)^4 + \cdots + (\, A + B - I_m \,)^{2k}] {\nonumber}\\
& = [(\, A + B - I_m \,)^2 + (\, A + B - I_m \,)^4 + \cdots + (\, A + B - I_m \,)^{2k}](A - B),
\\
& ABA + BAB + (ABA)^2 + (BAB)^2 + \cdots + (ABA)^k + (BAB)^k {\nonumber}\\
& = (A + B)[(\, A + B - I_m \,)^2 + (\, A + B - I_m \,)^4 + \cdots + (\, A + B - I_m \,)^{2k}] {\nonumber}\\
& = [(\, A + B - I_m \,)^2 + (\, A + B - I_m \,)^4 + \cdots + (\, A + B - I_m \,)^{2k}](A + B).\end{aligned}$$
\[TK34\] Let $A$ and $B$ be two idempotent matrices of the order $m,$ and let $k$ be a positive integer$.$ Then$,$ the following results hold$.$
1. ${\mathscr R}[\, (AB - BA)^k\,] \subseteq
{\mathscr R}[\,(A - B)^k\,]$ and ${\mathscr R}[\,(AB - BA)^k\,]
\subseteq {\mathscr R}[\,(A + B - I_m)^k\,];$
2. ${\mathscr R}[\, (AB + BA)^k\,] \subseteq
{\mathscr R}[\,(A + B)^k\,]$ and ${\mathscr R}[\,(AB + BA)^k\,]
\subseteq {\mathscr R}[\,(A + B - I_m)^k\,];$
3. ${\mathscr N}[\,(A - B)^k\,] \subseteq {\mathscr
N}[\, (AB - BA)^k\,]$ and ${\mathscr N}[\,(A + B -I_m)^k\,]
\subseteq {\mathscr N}[\, (AB - BA)^k\,];$
4. ${\mathscr N}[\,(A + B)^k\,] \subseteq {\mathscr
N}[\, (AB + BA)^k\,]$ and ${\mathscr N}[\,(A + B -I_m)^k\,]
\subseteq {\mathscr N}[\, (AB + BA)^k\,].$
\[TK35\] Let $A$ and $B$ be two idempotent matrices of the order $m,$ and let $k$ be a positive integer$.$ Then$,$ the following matrix identities hold $$\begin{aligned}
& (\, AB - BA \,)^D = (\, A - B \,)^D(\, A + B - I_m \,)^D = - (\, A + B - I_m \,)^D(\, A - B \,)^D,
\label{w24}
\\
& (\, AB + BA \,)^D = (\, A + B \,)^D(\, A + B - I_m \,)^D = (\, A + B - I_m \,)^D(\, A + B\,)^D,
\label{w25}
\\
& \left[\, (AB)^k - (BA)^k \,\right]^D =
(\, A - B \,)^D \left[\, (\, A + B - I_m \, )^D \,\right]^{2k-1} = - \left[(\, A + B - I_m \,)^D \right]^{2k-1}(\, A - B\,)^D,
\label{w26}
\\
& \left[\, (AB)^k + (BA)^k \, \right]^D = (\, A + B \,)^D\left[(\, A + B - I_m \,)^D\right]^{2k-1} = \left[(\, A + B - I_m \,)^D\right]^{2k-1}(\, A + B\,)^D,
\label{w27}
\\
& \left[ (ABA)^k - (BAB)^k \right]^D = (\, A - B \,)^D \left[(\, A + B - I_m \,)^D \right]^{2k} = \left[(\, A + B - I_m \,)^D \right]^{2k}
(\, A - B\,)^D,
\label{w28}
\\
& \left[ (ABA)^k + (BAB)^k \right]^D = (\, A + B \,)^D \left[(\, A
+ B - I_m \,)^D \right]^{2k} = \left[(\, A + B - I_m\,)^D\right]^{2k} (\, A + B\,)^D.
\label{w29}\end{aligned}$$
It follows from [@Ti-pmd Corollary 5] that $$\begin{aligned}
MN = \pm NM \Rightarrow (MN)^D =N^DM^D.
\label{w30}\end{aligned}$$ Applying this result to , , and – yields the results in this corollary.
\[TK36\] [The Drazin inverses of $A \pm B$ for two idempotents $A$ and $B$ were studied, and some formulas for $(\, A \pm B\,)^D$ were derived, see e.g., [@CD; @Den1; @Den2; @HWW; @ZW]. Substituting the results in these papers into the equalities in Corollary \[TK35\] may yield a variety of equalities for the Drazin inverses of $AB \pm BA$ and $(AB)^k \pm (BA)^k$, etc. ]{}
A group of identities for $(AB)^k$, $(ABA)^k$, and $A - ABA$ are given in the following theorem. Their verifications are straightforward.
\[TK37\] Let $A$ and $B$ be two idempotent matrices of the order $m,$ and let $k$ be a positive integer$.$ Then$,$ the following matrix identities hold $$\begin{aligned}
A - ABA & = A(\, A - B \,)^2 = (\, A - B \,)^2A,
\label{w32}
\\
B - BAB & = B(\, A - B \,)^2 = (\, A - B \,)^2B,
\label{w33}
\\
(\, A - ABA \,)^k & = A(\, A - B \,)^{2k} = (\, A - B \,)^{2k}A,
\label{w34}
\\
(\, B - BAB \,)^k & = B(\, A - B \,)^{2k} = (\, A - B \,)^{2k}B,
\label{w35}
\\
ABA & = A(\, A + B - I_m \,)^2 = (\, A + B -I_m \,)^2A,
\label{w36}
\\
BAB & = B(\, A + B - I_m \,)^2 = (\, A + B - I_m \,)^2B,
\label{w37}
\\
(ABA)^k & = A(\, A + B - I_m \,)^{2k} = (\, A + B -I_m \,)^{2k}A,
\label{w38}
\\
(BAB)^k & = B(\, A + B - I_m \,)^{2k} = (\, A + B - I_m \,)^{2k}B,
\label{w39}
\\
(BA)^2 & = BA(\, A + B - I_m \,)^2 = B(\, A + B -I_m \,)^2A,
\label{w40}
\\
(AB)^2 & = AB(\, A + B - I_m \,)^2 = A(\, A + B - I_m \,)^2B,
\label{w41}
\\
(AB)^k & = A(\, A + B - I_m \,)^{k}B,
\label{w42}
\\
(BA)^k & = B(\, A + B - I_m \,)^{k}A.
\label{w43}\end{aligned}$$
The following result is derived from Theorem \[TK37\] and .
\[TK38\] Let $A$ and $B$ be two idempotent matrices of the order $m,$ and let $k$ be a positive integer$.$ Then$,$ the following matrix identities hold $$\begin{aligned}
(\, A - ABA \,)^D & = A[\,(\, A - B \,)^D\,]^2 =
[\,(\, A - B \,)^D\,]^2A,
\label{w44}
\\
(\, B - BAB \,)^D & = B[\,(\, A - B \,)^D\,]^2 =
[\,(\, A - B \,)^D\,]^2B,
\label{w45}
\\
(ABA)^D & = A[\,(\, A + B - I_m \,)^D\,]^2 =
[\,(\, A + B - I_m \,)^D\,]^2A,
\label{w46}
\\
(BAB)^D & = B[\,(\, A + B - I_m \,)^D\,]^2 =
[\,(\, A + B - I_m \,)^D\,]^2B.
\label{w47}\end{aligned}$$
If $A$ and $B$ happen to be orthogonal projectors of the same size, the results in the previous theorems and corollaries are all valid. In such cases, the matrices $A \pm B$ and $A + B - I_m$ are Hermitian, and the results in the theorems and corollaries can be simplified further. In particular, note that $M^{D} = M^{\dag}$ for a Hermitian matrix. Hence Corollaries \[TK35\] and \[TK38\] reduce to the following results.
\[TK39\] Let $A$ and $B$ be two orthogonal projectors of the order $m,$ and let $k$ be a positive integer$.$ Then$,$ the following matrix identities hold $$\begin{aligned}
(\, AB - BA\,)^{\dag} & = -(\, A - B \,)^{\dag}(\, A + B - I_m \,)^{\dag} =
(\, A + B - I_m \,)^{\dag}(\, A - B \,)^{\dag},
\label{w48}
\\
(\, AB + BA \,)^{\dag} & = (\, A + B \,)^{\dag}(\, A + B - I_m \,)^{\dag} =
(\, A + B - I_m \,)^{\dag}(\, A + B\,)^{\dag},
\label{w49}
\\
\left[\, (AB)^k - (BA)^k \,\right]^{\dag} & = (\, A - B \,)^{\dag}
\left[\, (\, A + B - I_m \,)^{\dag}\,\right]^{2k-1} = - \left[\, (\, A + B - I_m \,)^{\dag} \,\right]^{2k-1}(\, A - B\,)^{\dag},
\label{w51}
\\
\left[\, (AB)^k + (BA)^k \,\right]^{\dag} & = (\, A + B \,)^{\dag}\left[\,
(\, A + B - I_m \,)^{\dag} \,\right]^{2k-1} = \left[\, (\, A + B - I_m \,)^{\dag}\,\right]^{2k-1}(\, A + B \,)^{\dag},
\label{w53}
\\
\left[\, (ABA)^k - (BAB)^k \,\right]^{\dag} & = (\, A - B \,)^{\dag}\left[\,
(\, A + B - I_m \,)^{\dag} \,\right]^{2k} = \left[\,(\, A + B - I_m \,)^{\dag} \,\right]^{2k}(\, A - B\,)^{\dag},
\label{w55}
\\
\left[\, (ABA)^k - (BAB)^k \,\right]^{\dag} & = (\, A + B \,)^{\dag}
\left[\,(\, A + B - I_m \,)^{\dag}\,\right]^{2k} = \left[\,(\, A + B - I_m \,)^{\dag} \,\right]^{2k}
(\, A + B\,)^{\dag},
\label{w57}
\\
(\, A - ABA \,)^{\dag} & = A\left[\,(\, A - B \,)^{\dag}\,\right]^2 =
\left[\,(\, A - B \,)^{\dag}\,\right]^2A,
\label{w58}
\\
(\, B - BAB \,)^{\dag} & = B\left[\,(\, A - B \,)^{\dag}\,\right]^2 =
\left[\,(\, A - B \,)^{\dag}\,\right]^2B,
\label{w59}
\\
(ABA)^{\dag} & = A\left[\,(\, A + B - I_m \,)^{\dag}\,\right]^2 =
\left[\,(\, A + B - I_m \,)^{\dag}\,\right]^2A,
\label{w60}
\\
(BAB)^{\dag} & = B\left[\,(\, A + B - I_m \,)^{\dag}\,\right]^2 =
\left[\,(\, A + B - I_m \,)^{\dag}\,\right]^2B.
\label{w61}\end{aligned}$$
It was shown in [@CT-mp] that $$\begin{aligned}
(AB)^{\dag} & = BA - B[\,(\, I_m - B \,)(\, I_m - A \,)\,]^{\dag}A,
\\
(\, A - B \,)^{\dag} & = (\, A - AB \,)^{\dag} -
(\, B - AB \,)^{\dag},
\\
(\, A - B \,)^{\dag} & = A - B + B(\, A - BA \,)^{\dag} -
(\, B -BA \,)^{\dag}A,
\\
( \, A + B - I_m \, )^{\dag} & = (AB)^{\dag} -
[\, (\, I_m - A \,)(\, I_m - B \, )\, ]^{\dag}\end{aligned}$$ hold for any pair of orthogonal projectors $A$ and $B$. Substituting them into Corollary \[TK39\] will yield many new equalities for the Moore–Penrose inverses of orthogonal projectors. More results and facts on the Moore–Penrose inverses of two orthogonal projectors and their algebraic operations can be found in [@Tian:2019b].
In addition, it is also worth to discuss the following reverse order laws for generalized inverses $$\begin{aligned}
(\alpha AB + \beta BA)^{(i,\ldots,j)} & = (\, A + B - I_m \,)^{(k,\ldots,l)}(\, \alpha A + \beta B \,)^{-1}
\\
& = (\, \beta A + \alpha B \,)^{-1}(\, A + B - I_m \,)^{(i,\ldots,j)},
\\
(\alpha ABA + \beta BAB)^{(i,\ldots,j)} & = [(\, A + B - I_m \,)^{2}]^{(i,\ldots,j)}(\, \alpha A + \beta B \,)^{(i,\ldots,j)}
\\
& = (\, \beta A + \alpha B \,)^{(i,\ldots,j)}[(\, A + B - I_m \,)^{2}]^{(i,\ldots,j)},
\\
[\alpha (AB)^k + \beta (BA)^k]^{(i,\ldots,j)} & = [(\, A + B - I_m \,)^{2k-1}]^{(i,\ldots,j)}(\, \alpha A + \beta B \,)^{(i,\ldots,j)}
\\
& = (\, \beta A + \alpha B \,)^{(i,\ldots,j)}[(\, A + B - I_m \,)^{2k-1}]^{(i,\ldots,j)},
\\
[\alpha (ABA)^k + \beta (BAB)^k]^{(i,\ldots,j)} & = [(\, A + B - I_m \,)^{2k}]^{(i,\ldots,j)}(\, \alpha A + \beta B \,)^{(i,\ldots,j)}
\\
& = (\, \beta A + \alpha B \,)^{(i,\ldots,j)}[(\, A + B - I_m \,)^{2k}]^{(i,\ldots,j)}.\end{aligned}$$
We next explore ranges and null spaces of some matrix expressions composed by two idempotent matrices. It is easy to verify that $$\begin{aligned}
& r[\, A + B, \, I_m + A - B\,] = r[\, A + B, \, I_m - A + B\,] = r[\, A - B, \, I_m + A + B\,] = r[\, A - B, \, I_m - A - B\,] = m\end{aligned}$$ hold for two idempotent matrices $A$ and $B$ of the order $m$. Hence, $$\begin{aligned}
& r[\, A - B, \, I_m - A - B] = r(A + B) + r(I_m - A - B)
\\
& \Leftrightarrow r\!\begin{bmatrix} A \\ B \end{bmatrix} = r(A) + r(B) - r(AB) \ {\rm and} \
r[A, \, B] = r(A) + r(B) - r(BA)
\\
& \Leftrightarrow r\!\begin{bmatrix} A \\ B \end{bmatrix} = r(A) + r(B) - r(BA) \ {\rm and} \
r[A, \, B] = r(A) + r(B) - r(AB).\end{aligned}$$ In Problem 31-4, Issue 31(2003), the Bulletin of the International Linear Algebra Society, the present author proposed such a problem: if $R$ is a ring with unity 1, and $a, \ b \in R$ satisfy $a^2 = a$ and $b^2 = b$, then $$(ab-ba)R = (a - b)R \cap (1 - a - b)R \ \ {\rm and} \ \ R(ab-ba) =
R(a - b) \cap R(1 - a - b).$$ In matrix situation, the problem can be restated as follows: If $A$ and $B$ are a pair of idempotent matrices of order $m$, then $$\begin{aligned}
{\mathscr R}(\,AB - BA\,) = {\mathscr R}(\, A - B \,) \cap
{\mathscr R}(\,I_m - A - B \,).
\label{w62}\end{aligned}$$ It can be seen from this range equality that
1. if $A - B$ is nonsingular, then ${\mathscr R}(\,AB - BA\,) =
{\mathscr R}(\, I_m - A - B \,)$;
2. if $A + B - I_m$ is nonsingular, then ${\mathscr R}(\, AB - BA\,) = {\mathscr R}(\, A - B \,)$;
3. the commutator $AB - BA$ is nonsingular if and only if both $A - B$ and $I_m - A - B$ are nonsingular;
4. $AB = BA$ $\Leftrightarrow$ ${\mathscr R}(\, A - B \,) \cap
{\mathscr R}(\, I_m - A - B \,) = \{ 0 \}$ $\Leftrightarrow$ $r\!\begin{bmatrix} A \\ B \end{bmatrix} = r(A) + r(B) - r(AB) \ {\rm and} \
r[A, \, B] = r(A) + r(B) - r(BA)$ $\Leftrightarrow$ $r\!\begin{bmatrix} A \\ B \end{bmatrix} = r(A) + r(B) - r(BA) \ {\rm and} \
r[A, \, B] = r(A) + r(B) - r(AB).$
The range equality in can be used to find some other analogous range equalities for matrix expressions consisting of a pair of idempotent matrices of the same order.
\[TK310\] Let $A$ and $B$ be two idempotent matrices of the order $m.$ Then$,$ the following range identities hold $$\begin{aligned}
{\mathscr R}(\, AB + BA \,) & = {\mathscr R}(\, A + B \,) \cap
{\mathscr R}(\, A + B - I_m \,),
\label{w63}
\\
{\mathscr R}(\, ABA + BAB \,) & = {\mathscr R}(\, A + B \,) \cap
{\mathscr R}[\, (\, A + B - I_m \,)^2\,],
\label{w64}
\\
{\mathscr R}(\, ABA - BAB \,) & = {\mathscr R}(\, A - B \,) \cap
{\mathscr R}[\, (\, A + B - I_m \,)^2 \,],
\label{w65}
\\
{\mathscr R}[\, (\, AB - BA \,)^2 \,] & =
{\mathscr R}[\,(\, A - B \,)^2\,] \cap {\mathscr R}[\, (\, A + B -
I_m \,)^2 \,].
\label{w66}\end{aligned}$$
It can be derived from Lemma \[TK32\] that $$\begin{aligned}
{\mathscr R}(\, AB + BA \,) & \subseteq {\mathscr R}(\, A + B \,) \cap
{\mathscr R}(\, A + B - I_m \,),
\label{w67}
\\
{\mathscr R}(\, ABA + BAB \,) & \subseteq {\mathscr R}(\, A + B \,) \cap
{\mathscr R}[\, (\, A + B - I_m \,)^2 \,],
\label{w68}
\\
{\mathscr R}(\, ABA - BAB \,) & \subseteq {\mathscr R}(\, A - B \,) \cap
{\mathscr R}[\, (\, A + B - I_m \,)^2 \,],
\label{w69}
\\
{\mathscr R}[\, (\, AB - BA \,)^2 \,] & \subseteq
{\mathscr R}[\,(\, A - B \,)^2\,] \cap {\mathscr R}[\, (\, A + B -
I_m \,)^2 \,].
\label{w70}\end{aligned}$$ If we can show that the dimensions of the subspaces on both sides of – are equal, respectively, then – are valid.
It follows from and that $$\begin{aligned}
r(\, AB + BA \,) = r(\, A + B \,)
+ r(\, A + B - I_m \,) - m.
\label{w74}\end{aligned}$$ On the other hand, using the well-known formula $$\begin{aligned}
\dim [\, {\mathscr R}(M) \cap {\mathscr R}(N) \,] =
r(M) + r(N) - r[\, M, \, N \,],
\label{w75}\end{aligned}$$ we also obtain $$\begin{aligned}
& \dim \left[\, {\mathscr R}(\, A + B \,) \cap
{\mathscr R} (\, A + B - I_m \,) \, \right] {\nonumber}\\
& = r(\, A + B \,) +
r(\, A + B - I_m \,) - r[\,A + B, \, A + B - I_m] {\nonumber}\\
& = r(\, A + B \,) + r(\, A + B - I_m \,) - r[\, A + B, \, I_m ] {\nonumber}\\
& = r(\, A + B \,) + r(\, A + B - I_m \,) - m.
\label{w76}\end{aligned}$$ Eqs. and imply that the dimensions of the subspaces on the both sides of are equal. Thus, holds. Note from that $$ABA - BAB = (\, A - B \,)(\, A + B - I_m \,)^2 =
(\, A - B \,) - (\, A - B \,)^3.$$ Hence, it follows from that $$\begin{aligned}
r(\, ABA - BAB \,) & = r(\, A - B \,) +
r[\, I_m - (\, A - B \,)^2 \,] - m
\\
& = r(\, A - B \,) + r[\,(\, A + B - I_m \,)^2 \,] - m
\\
& = r(\, A - B \,) + r(I_m + A - B) + r(I_m - A + B) - 2m \ \ \ \mbox{(by \eqref{v34})},\end{aligned}$$ and by , $$\begin{aligned}
& \dim \{ \, {\mathscr R}(\, A - B \,) \cap
{\mathscr R}[\, (\, A + B - I_m \,)^2 \,] \, \}
\\
& = r(\, A - B \,) + r[\, (\, A + B - I_m \,)^2 \,]
- r[\, A - B, \ (\, A + B - I_m \,)^2 \, ]
\\
& = r(\, A - B \,) + r[\, (\, A + B - I_m \,)^2 \,] -
r[\, A - B, \ I_m - (\, A - B \,)^2 \, ] \ \ \ \mbox{(by (\ref{v31}))}
\\
& = r(\, A - B \,) + r[\, (\, A + B - I_m \,)^2 \,]
- r[\, A - B, \ I_m \, ]
\\
& = r(\, A - B \,) + r[\, (\, A + B - I_m \,)^2 \,] - m
\\
& = r(\, A - B \,) + r(I_m + A - B) + r(I_m - A + B) - 2m \ \ \ \mbox{(by \eqref{v34})}.\end{aligned}$$ The above two equalities imply that the dimensions of the subspaces on the both sides of are equal. Thus, holds. Similarly, we can show that $$\begin{aligned}
& \dim [\, {\mathscr R}(\, ABA + BAB \,) \,] = r(\, A + B \,) +
r[\,(\, A + B - I_m \,)^2 \,] - m,
\\
& \dim\{ [\, {\mathscr R}(\, A + B \,) \cap {\mathscr R}[\,
(\, A + B - I_m \,)^2 \,] \} =
r(\, A + B \,) + r[\, (\, A + B - I_m \,)^2 \,] - m.\end{aligned}$$ Thus, holds. It can also be shown that $$\begin{aligned}
& \dim{\mathscr R}[\, (\, AB - BA \,)^2 \,] =
r[\,(\, A - B \,)^2\,] + r[\, (\, A + B - I_m \,)^2 \,] - m,
\\
& \dim{\mathscr R}\{ \,[\,(\, A - B \,)^2\,] \cap {\mathscr R}[\, (\, A + B - I_m \,)^2 \,] \, \}
= r[\,(\, A - B \,)^2\,] + r[\, (\, A + B - I_m \,)^2 \,]
- m.\end{aligned}$$ Hence, holds.
We leave the verification of the following result to the reader.
\[TK311\] Let $A$ and $B$ be two idempotent matrices of the order $m.$ Then $$\begin{aligned}
{\mathscr N}(\, AB \pm BA \,) & = {\mathscr N}(\, A \pm B \,) \cap
{\mathscr N}(\, A + B - I_m \,),
\\
{\mathscr N}(\, ABA \pm BAB \,) & = {\mathscr N}(\, A \pm B \,)
\cap {\mathscr N}[\, (\, A + B - I_m \,)^2\,],
\\
{\mathscr N}[\, (\, AB - BA \,)^2 \,] & = {\mathscr N}[\,(\, A - B
\,)^2\,] \cap {\mathscr N}[\, (\, A + B - I_m \,)^2 \,].\end{aligned}$$
Obviously, the formulas and facts in Theorem \[TH313\] can be extended to a family of idempotent matrices and their algebraic operations. It is expected that many more nontrivial algebraic equalities among idempotent matrices and their consequences can be established.
One of the main concerns about a given matrix is to explore the relationships between the matrix and its transpose or conjugate transpose; see e.g., [@Duk; @Gor; @Gow; @Rad; @TZa; @Ver]. It is easy to see that $(A^{*})^2 = A^{*}$ by taking conjugate of both sides of $A^2 = A$. In this case, replacing $B$ with $A^{*}$ in the preceding results, we obtain the following consequences.
\[TL31\] Let $A$ be an idempotent matrix of the order $m.$ Then the following two matrix identities $$\begin{aligned}
& (A - A^{*})^2 + (A + A^{*} - I_m)^2 = I_m,
\\
& AA^{*} + A^{*}A + 4^{-1}I_m = (\, A + A^{*} - 2^{-1}I_m \,)^2,\end{aligned}$$ and the following five rank formulas $$\begin{aligned}
& r(A - A^{*}) = r(A + A^{*}) + r(2I_m - A - A^{*}) - m,
\\
& r(I_m - A - A^{*} ) = 2r(I_m + A - A^{*}) - m,
\\
& r(AA^{*} + A^{*}A) = r[A, \, A^{*}] = r(I_m - A - A^{*}) + r(A + A^{*}) - m,
\\
& r(I_m - AA^{*} - A^{*}A) = r[(\sqrt{5} -1)/2I_m + A + A^{*}] + r[(\sqrt{5} + 1)/2I_m - A - A^{*}] - m,
\\
& r(2I_m - AA^{*} - A^{*}A) = r(I_m + A + A^{*} ) + r(2I_m - A - A^{*}) - m\end{aligned}$$ hold$.$ In particular$,$ $$\begin{aligned}
& A = A^{*} \Leftrightarrow (I_m - A - A^{*} )^2 = I_m \Leftrightarrow r(A + A^{*}) + r(2I_m - A - A^{*}) = m,
\\
& (A - A^{*})^2 = 2^{-1} I_m \Leftrightarrow (I_m - A - A^{*} )^2 = 2^{-1}I_m,
\\
& (A - A^{*})^2 = I_m \Leftrightarrow (I_m - A - A^{*} )^2 = 0 \Leftrightarrow r(I_m + A - A^{*}) + r(I_m - A + A^{*}) - m,
\\
& AA^{*} + A^{*}A = - 2I_m \Leftrightarrow (\, A + A^{*} - 2^{-1}I_m \,)^2 = - \frac{7}{4}I_m,
\\
& AA^{*} + A^{*}A = -I_m \Leftrightarrow (\, A + A^{*} - 2^{-1}I_m \,)^2 = - \frac{3}{4}I_m,
\\
& AA^{*} + A^{*}A = -4^{-1}I_m \Leftrightarrow (\, A + A^{*} - 2^{-1}I_m \,)^2 =0,
\\
& AA^{*} + A^{*}A = 0 \Leftrightarrow (\, A + A^{*} - 2^{-1}I_m \,)^2 = 4^{-1}I_m \Leftrightarrow r(I_m - A - A^{*} ) + r(A + A^{*}) = m,
\\
& AA^{*} + A^{*}A = \frac{3}{4}I_m \Leftrightarrow (\, A + A^{*} - 2^{-1}I_m \,)^2 = I_m,
\\
& AA^{*} + A^{*}A = I_m \Leftrightarrow (\, A + A^{*} - 2^{-1}I_m \,)^2 = \frac{5}{4}I_m \Leftrightarrow r[(\sqrt{5} -1)/2I_m + A + A^{*}] + r[(\sqrt{5} + 1)/2I_m - A - A^{*}] = m,
\\
& AA^{*} + A^{*}A = 2I_m \Leftrightarrow (\, A + A^{*} - 2^{-1}I_m \,)^2 = \frac{9}{4}I_m \Leftrightarrow r(I_m + A + A^{*} ) + r(2I_m - A - A^{*}) = m,\end{aligned}$$ and $$\begin{aligned}
& r(A - A^{*}) = m \Leftrightarrow r(A + A^{*}) = r(2I_m - A - A^{*}) = m,
\\
& r(I_m - A - A^{*}) = m \Leftrightarrow r(I_m + A - A^{*}) = m,
\\
& r(AA^{*} + A^{*}A) = r[A, \, A^{*}] = m \Leftrightarrow r(A + A^{*}) = r(I_m - A - A^{*}) = m,
\\
& r(I_m - AA^{*} - A^{*}A) = m \Leftrightarrow r[(\sqrt{5} -1)/2I_m + A + A^{*}] =
r[(\sqrt{5} + 1)/2I_m - A - A^{*}] = m,
\\
& r(2I_m - AA^{*} - A^{*}A) = m \Leftrightarrow r(I_m + A + A^{*} ) = r(2I_m - A - A^{*}) = m.\end{aligned}$$
\[TL32\] Let $A$ be an idempotent matrix of the order $m,$ and let $k$ be any positive integer$.$ Then the following factorization equalities $$\begin{aligned}
\alpha AA^{*} + \beta A^{*}A & = (\, \alpha A + \beta A^{*} \,)(\, A + A^{*} - I_m \,)
= (\, A + A^{*} - I_m \,) (\, \beta A + \alpha A^{*} \,),
\\
\alpha AA^{*}A + \beta A^{*}AA^{*} & = (\, \alpha A + \beta A^{*} \,)(\, A + A^{*} - I_m \,)^2
= (\, A + A^{*} - I_m \,)^2 (\, \beta A + \alpha A^{*} \,),
\\
\alpha (AA^{*})^k + \beta (A^{*}A)^k & = (\, \alpha A + \beta A^{*} \,)(\, A + A^{*} - I_m \,)^{2k-1}
= (\, A + A^{*} - I_m \,)^{2k-1}(\, \beta A + \alpha A^{*} \,),
\\
\alpha (AA^{*}A)^k + \beta (A^{*}AA^{*})^k & = (\, \alpha A + \beta A^{*} \,)(\, A + A^{*} - I_m \,)^{2k}
= (\, A + A^{*} - I_m \,)^{2k}(\, \beta A + \alpha A^{*} \,)\end{aligned}$$ hold for any two scalars $\alpha$ and $\beta.$ In particular$,$ $\alpha AA^{*} + \beta A^{*}A$ is nonsingular $\Leftrightarrow$ $\alpha (AA^{*})^k + \beta (A^{*}A)^k$ is nonsingular $\alpha AA^{*}A + \beta A^{*}AA^{*}$ is nonsingular $\Leftrightarrow$ $\alpha (AA^{*})^k + \beta (A^{*}A)^k$ is nonsingular $\Leftrightarrow$ $\alpha (AA^{*}A)^k + \beta (A^{*}AA^{*})^k$ is nonsingular $\Leftrightarrow$ both $\alpha A + \beta A^{*}$ and $A + A^{*} - I_m$ are nonsingular; in which cases$,$ the following equalities hold $$\begin{aligned}
(\alpha AA^{*} + \beta A^{*}A)^{-1} & = (\, A + A^{*} - I_m \,)^{-1}(\, \alpha A + \beta A^{*} \,)^{-1}
= (\, \beta A + \alpha A^{*} \,)^{-1}(\, A + A^{*} - I_m \,)^{-1},
\\
(\alpha AA^{*}A + \beta A^{*}AA^{*})^{-1} & = (\, A + A^{*} - I_m \,)^{-2}(\, \alpha A + \beta A^{*} \,)^{-1}
= (\, \beta A + \alpha A^{*} \,)^{-1}(\, A + A^{*} - I_m \,)^{-2},
\\
[\alpha (AA^{*})^k + \beta (A^{*}A)^k]^{-1} & = (\, A + A^{*} - I_m \,)^{-2k+1}(\, \alpha A + \beta A^{*} \,)^{-1}
= (\, \beta A + \alpha A^{*} \,)^{-1}(\, A + A^{*} - I_m \,)^{-2k+1},
\\
[\alpha (AA^{*}A)^k + \beta (A^{*}AA^{*})^k]^{-1} & = (\, A + A^{*} - I_m \,)^{-2k}(\, \alpha A + \beta A^{*} \,)^{-1}
= (\, \beta A + \alpha A^{*} \,)^{-1}(\, A + A^{*} - I_m \,)^{-2k}.\end{aligned}$$
\[TL33\] Let $A$ be an idempotent matrix of the order $m,$ and let $k$ be a positive integer$.$ Then$,$ the following matrix identities hold $$\begin{aligned}
AA^{*} - A^{*}A & = (\, A - A^{*} \,)(\, A + A^{*} - I_m \,) = - (\, A + A^{*} - I_m
\,)(\, A - A^{*} \,),
\\
AA^{*} + A^{*}A & = (\, A + A^{*} \,)(\, A + A^{*} - I_m \,) = (\, A + A^{*} -
I_m\,)(\, A + A^{*}\,),
\\
AA^{*}A - A^{*}AA^{*} & = (\, A - A^{*} \,)(\, A + A^{*} - I_m \,)^2 = (\, A + A^{*} - I_m \,)^2(\, A - A^{*} \,),
\\
AA^{*}A + A^{*}AA^{*} & = (\, A + A^{*} \,)(\, A + A^{*} - I_m \,)^2 =(\, A + A^{*} - I_m \,)^2(\, A + A^{*} \,).
\\
(\,AA^{*} - A^{*}A\,)^k & = (-1)^{k(k-1)/2}(\, A - A^{*} \,)^k(\, A + A^{*} - I_m \,)^k = (-1)^{k(k-1)/2}(\, I_m - A - A^{*} \,)^k(\, A - A^{*} \,)^k,
\\
(\,AA^{*} + A^{*}A \,)^k & = (\, A + A^{*} \,)^k(\, A + A^{*} - I_m \,)^k = (\, A + A^{*} - I_m \,)^k(\, A + A^{*} \,)^k, \ \ \ \ \ \
\\
(\, AA^{*}A - A^{*}AA^{*}\,)^k & = (\, A - A^{*} \,)^k(\, A + A^{*} - I_m \,)^{2k} = (\, A + A^{*} - I_m\,)^{2k}(\, A - A^{*}\,)^k,
\\
(\,AA^{*}A + A^{*}AA^{*}\,)^k & = (\, A + A^{*} \,)^k(\, A + A^{*} - I_m \,)^{2k} = (\, A + A^{*} - I_m \,)^{2k}(\, A + A^{*}\,)^k,
\\
(AA^{*})^k - (A^{*}A)^k & = (\, A - A^{*} \,)(\, A + A^{*} - I_m \,)^{2k-1} = - (\, A + A^{*} - I_m \,)^{2k-1}(\, A - A^{*}\,),
\\
(AA^{*})^k + (A^{*}A)^k & = (\, A + A^{*} \,)(\, A + A^{*} - I_m \,)^{2k-1} = (\, A + A^{*} - I_m \,)^{2k-1}(\, A + A^{*} \,),
\\
(AA^{*}A)^k - (A^{*}AA^{*})^k & = (\, A - A^{*} \,)(\, A + A^{*} - I_m \,)^{2k} = (\, A + A^{*} - I_m\,)^{2k}(\, A - A^{*}\,),
\\
(AA^{*}A)^k + (A^{*}AA^{*})^k & = (\, A + A^{*} \,)(\, A + A^{*} - I_m \,)^{2k} =(\, A + A^{*} - I_m \,)^{2k}(\, A + A^{*}\,).\end{aligned}$$ In addition$,$ the following matrix identities hold $$\begin{aligned}
& AA^{*} - A^{*}A + (AA^{*})^2 - (A^{*}A)^2 + \cdots + (AA^{*})^k - (A^{*}A)^k
\\
& = (A - A^{*})[(\, A + A^{*} - I_m \,) + (\, A + A^{*} - I_m \,)^3 + \cdots + (\, A + A^{*} - I_m \,)^{2k-1}]
\\
& = [(\, A + A^{*} - I_m \,) + (\, A + A^{*} - I_m \,)^3 + \cdots + (\, A + A^{*} - I_m \,)^{2k-1}](A^{*} - A),
\\
& AA^{*} + A^{*}A + (AA^{*})^2 + (A^{*}A)^2 + \cdots + (AA^{*})^k + (A^{*}A)^k
\\
& = (A + A^{*})[(\, A + A^{*} - I_m \,) + (\, A + A^{*} - I_m \,)^3 + \cdots + (\, A + A^{*} - I_m \,)^{2k-1}]
\\
& = [(\, A + A^{*} - I_m \,) + (\, A + A^{*} - I_m \,)^3 + \cdots + (\, A + A^{*} - I_m \,)^{2k-1}](A+A^{*}),
\\
& AA^{*}A - A^{*}AA^{*} + (AA^{*}A)^2 - (A^{*}AA^{*})^2 + \cdots + (AA^{*}A)^k - (A^{*}AA^{*})^k {\nonumber}\\
& = (A - A^{*})[(\, A + A^{*} - I_m \,)^2 + (\, A + A^{*} - I_m \,)^4 + \cdots + (\, A + A^{*} - I_m \,)^{2k}] {\nonumber}\\
& = [(\, A + A^{*} - I_m \,)^2 + (\, A + A^{*} - I_m \,)^4 + \cdots + (\, A + A^{*} - I_m \,)^{2k}](A - A^{*}),
\\
& AA^{*}A + A^{*}AA^{*} + (AA^{*}A)^2 + (A^{*}AA^{*})^2 + \cdots + (AA^{*}A)^k + (A^{*}AA^{*})^k {\nonumber}\\
& = (A + A^{*})[(\, A + A^{*} - I_m \,)^2 + (\, A + A^{*} - I_m \,)^4 + \cdots + (\, A + A^{*} - I_m \,)^{2k}] {\nonumber}\\
& = [(\, A + A^{*} - I_m \,)^2 + (\, A + A^{*} - I_m \,)^4 + \cdots + (\, A + A^{*} - I_m \,)^{2k}](A + A^{*}).\end{aligned}$$
\[TL37\] Let $A$ be an idempotent matrices of the order $m,$ and let $k$ be a positive integer$.$ Then$,$ the following matrix identities hold $$\begin{aligned}
A - AA^{*}A & = A(\, A - A^{*} \,)^2 = (\, A - A^{*} \,)^2A,
\\
A^{*} - A^{*}AA^{*} & = A^{*}(\, A - A^{*} \,)^2 = (\, A - A^{*} \,)^2A^{*},
\\
(\, A - AA^{*}A \,)^k & = A(\, A - A^{*} \,)^{2k} = (\, A - A^{*} \,)^{2k}A,
\\
(\, A^{*} - A^{*}AA^{*} \,)^k & = A^{*}(\, A - A^{*} \,)^{2k} = (\, A - A^{*} \,)^{2k}A^{*},
\\
AA^{*}A & = A(\, A + A^{*} - I_m \,)^2 = (\, A + A^{*} -I_m \,)^2A,
\\
A^{*}AA^{*} & = A^{*}(\, A + A^{*} - I_m \,)^2 = (\, A + A^{*} - I_m \,)^2A^{*},
\\
(AA^{*}A)^k & = A(\, A + A^{*} - I_m \,)^{2k} = (\, A + A^{*} -I_m \,)^{2k}A,
\\
(A^{*}AA^{*})^k & = A^{*}(\, A + A^{*} - I_m \,)^{2k} = (\, A + A^{*} - I_m \,)^{2k}A^{*},
\\
(A^{*}A)^2 & = A^{*}A(\, A + A^{*} - I_m \,)^2 = A^{*}(\, A + A^{*} -I_m \,)^2A,
\\
(AA^{*})^2 & = AA^{*}(\, A + A^{*} - I_m \,)^2 = A(\, A + A^{*} - I_m \,)^2A^{*},
\\
(AA^{*})^k & = A(\, A + A^{*} - I_m \,)^{k}A^{*},
\\
(A^{*}A)^k & = A^{*}(\, A + A^{*} - I_m \,)^{k}A.\end{aligned}$$
\[TL319\] Let $A$ be an idempotent matrix of the order $m,$ and let $k$ be a positive integer$.$ Then$,$ the following results hold$.$
1. ${\mathscr R}(AA^{*} - A^{*}A) \subseteq
{\mathscr R}(A - A^{*})$ and ${\mathscr R}(AA^{*} - A^{*}A)
\subseteq {\mathscr R}(A + A^{*} - I_m).$
2. ${\mathscr R}(AA^{*} + A^{*}A) \subseteq
{\mathscr R}(A + A^{*})$ and ${\mathscr R}(AA^{*} + A^{*}A)
\subseteq {\mathscr R}(A + A^{*} - I_m).$
3. ${\mathscr N}(A - A^{*}) \subseteq {\mathscr
N}(AA^{*} - A^{*}A)$ and ${\mathscr N}(A + A^{*} -I_m)
\subseteq {\mathscr N}(AA^{*} - A^{*}A).$
4. ${\mathscr N}(A + A^{*}) \subseteq {\mathscr
N}(AA^{*} + A^{*}A)$ and ${\mathscr N}(A + A^{*} -I_m)
\subseteq {\mathscr N}(AA^{*} + A^{*}A).$
5. ${\mathscr R}(\,AA^{*} \pm A^{*}A\,) = {\mathscr R}(\, A \pm A^{*} \,) \cap
{\mathscr R}(\, A + A^{*} - I_m \,).$
6. ${\mathscr R}(\, AA^{*}A \pm A^{*}AA^{*} \,) = {\mathscr R}(\, A \pm A^{*} \,) \cap
{\mathscr R}(\, A + A^{*} - I_m \,).$
7. ${\mathscr N}(\, AA^{*} \pm A^{*}A \,) = {\mathscr N}(\, A \pm A^{*} \,) \cap
{\mathscr N}(\, A + A^{*} - I_m \,).$
8. ${\mathscr N}(\, AA^{*}A \pm A^{*}AA^{*} \,) = {\mathscr N}(\, A \pm A^{*} \,)
\cap {\mathscr N}(\, A + A^{*} - I_m \,).$
\[TL312\] Let $A$ be an idempotent matrices of the order $m.$ Then the following two identities $$\begin{aligned}
(\, I_m + \alpha A + \beta A^{*} \,) & = (\, I_m + \alpha A \,)
[\, I_m -(\alpha\beta)(1 + \alpha)^{-1}(1 + \beta)^{-1}AA^{*}\,](\, I_m + \beta A^{*} \,),
\label{pp391}
\\
(\, I_m + \alpha A + \beta A^{*} \,) & = (\, I_m + \beta A^{*}\,)[\, I_m -(\alpha\beta)(1 + \alpha)^{-1}(1 + \beta)^{-1}A^{*}A\,](\, I_m + \alpha A \,)
\label{pp392}\end{aligned}$$ hold for $\alpha \neq -1, 0$ and $\beta \neq -1, 0.$ In this case$,$ both $I_m + \alpha A$ and $I_m + \beta A^{*}$ are nonsingular$.$ In particular$,$ $I_m - \lambda AA^{*}$ is nonsingular if and only if $I_m - \alpha A - \beta A^{*}$ is nonsingular$,$ in which case$,$ $$\begin{aligned}
& (\, I_m - \lambda AA^{*}\,)^{-1} = (\, I_m + \beta A^{*} \,)(\, I_m + \alpha A + \beta A^{*} \,)^{-1}(\, I_m + \alpha A \,),
\label{pp393}
\\
& (\, I_m - \lambda A^{*}A\,)^{-1} = (\, I_m + \alpha A \,)(\, I_m + \alpha A + \beta A^{*} \,)^{-1}(\, I_m + \beta A^{*}\,)
\label{pp394}\end{aligned}$$ hold$,$ where $\lambda = \alpha\beta(1 + \alpha)^{-1}(1 + \beta)^{-1}$.
In the remaining of this section, we present two groups of miscellaneous formulas for two/three idempotent matrices and their operations.
\[TH312\] Let $A$ and $B$ be two idempotent matrices of the order $m.$ Then the following two identities $$\begin{aligned}
(\, I_m + \alpha A + \beta B \,) & = (\, I_m + \alpha A \,)
[\, I_m -(\alpha\beta)(1 + \alpha)^{-1}(1 + \beta)^{-1}AB\,](\, I_m + \beta B \,),
\label{391}
\\
(\, I_m + \alpha A + \beta B \,) & = (\, I_m + \beta B\,)[\, I_m -(\alpha\beta)(1 + \alpha)^{-1}(1 + \beta)^{-1}BA\,](\, I_m + \alpha A \,)
\label{392}\end{aligned}$$ hold for $\alpha \neq -1, 0$ and $\beta \neq -1, 0.$ In this case$,$ both $I_m + \alpha A$ and $I_m + \beta B$ are nonsingular$.$ In particular$,$ $I_m - \lambda AB$ is nonsingular if and only if $I_m - \alpha A - \beta B$ is nonsingular$,$ in which case$,$ $$\begin{aligned}
& (\, I_m - \lambda AB\,)^{-1} = (\, I_m + \beta B \,)(\, I_m + \alpha A + \beta B \,)^{-1}(\, I_m + \alpha A \,),
\label{393}
\\
& (\, I_m - \lambda BA\,)^{-1} = (\, I_m + \alpha A \,)(\, I_m + \alpha A + \beta B \,)^{-1}(\, I_m + \beta B\,)
\label{394}\end{aligned}$$ hold$,$ where $\lambda = \alpha\beta(1 + \alpha)^{-1}(1 + \beta)^{-1}$.
Eqs. and were given in [@Vet]. Eqs. and follow directly from and .
It is also worth to discuss the following two families of reverse order laws for generalized inverses $$\begin{aligned}
& (\, I_m - \lambda AB\,)^{(i,\ldots,j)} = (\, I_m + \beta B \,)(\, I_m + \alpha A + \beta B \,)^{(i,\ldots,j)} (\, I_m + \alpha A \,),
\label{395}
\\
& (\, I_m - \lambda BA\,)^{(i,\ldots,j)} = (\, I_m + \alpha A \,)(\, I_m + \alpha A + \beta B \,)^{(i,\ldots,j)}(\, I_m + \beta B\,)
\label{396}\end{aligned}$$ associated with and .
Concerning relationships among three idempotent matrices of the same size, we have the following fundamental formulas and facts.
\[TH313\] Let $A,$ $B,$ and $C$ be three idempotent matrices of the order $m,$ and denote $M = A + B + C.$ Then the following four identities $$\begin{aligned}
& \alpha(AB + AC) + \beta (BA + BC) + \gamma(CA + CB) = (\alpha A + \beta B + \gamma C \,)(\,M - I_m),
\label{397}
\\
& \alpha(BA + CA) + \beta (AB + CB) + \gamma(AC + BC) = (M - I_m)(\alpha A + \beta B + \gamma C),
\label{398}\end{aligned}$$ and $$\begin{aligned}
&(\alpha + \beta)(AB + BA) + (\alpha + \gamma)(AC + CA) + (\beta + \gamma)(BC + CB) {\nonumber}\\
& = (\alpha A + \beta B + \gamma C)(\,M - I_m) + (M - I_m)(\alpha A + \beta B + \gamma C),
\label{399}
\\
&(\alpha - \beta)(AB - BA) + (\alpha - \gamma)(AC - CA) + (\beta - \gamma)(BC - CB) {\nonumber}\\
& = (\alpha A + \beta B + \gamma C)M - M(\alpha A + \beta B + \gamma C),
\label{3100}
\\
& \alpha(B + C)A(B + C) + \beta (A + C)B(A + C) + \gamma(A + B)C(A + B) {\nonumber}\\
& = (M - I_m)(\alpha A + \beta B + \gamma C)(M - I_m)
\label{3100a}\end{aligned}$$ hold for any three scalars $\alpha,$ $\beta,$ and $\gamma;$ the following identities hold $$\begin{aligned}
& (A + B)^2 + (A + C)^2 + (B + C)^2 = M(I_m + M),
\label{3106}
\\
& (A - B)^2 + (A - C)^2 + (B - C)^2 = M(3I_m - M) = 9/4I_m - (M - 3/2I_m)^2,
\label{3107}
\\
& AB + BA + AC + CA + BC + CB = M(M - I_m) = (M - 2^{-1}I_m)^2 - 4^{-1}I_m,
\label{3108}
\\
& (AB + BA + AC + CA + BC + CB)^k = M^k(M - I_m)^k, \ \ k = 1, 2, \ldots;
\label{3109}\end{aligned}$$ the following rank formulas $$\begin{aligned}
& r[(A + B)^2 + (A + C)^2 + (B + C)^2] = r(M) + r(I_m + M) - m = \dim[{\mathscr R}(M) \cap {\mathscr R}(I_m + M)],
\label{3110}
\\
& r[(A - B)^2 + (A - C)^2 + (B - C)^2] = r(M) + r(3I_m - M) - m = \dim[{\mathscr R}(M) \cap {\mathscr R}(3I_m - M)],
\label{3111}
\\
& r(kI_m - AB - BA - AC - CA - BC - CB) {\nonumber}\\
& = r\!\left[(\sqrt{4k +1} + 1)/2I_m - M \right] +
r\!\left[\,(\sqrt{4k +1} - 1)/2I_m + M \right] - m, \ \ k = 0, 1, \ldots, 6
\label{3112}\end{aligned}$$ hold$;$ and the following range equalities hold $$\begin{aligned}
& {{\mathscr R}}[(A + B)^2 + (A + C)^2 + (B + C)^2] = {{\mathscr R}}(M)\cap {{\mathscr R}}(I_m + M),
\label{3112a}
\\
& {{\mathscr R}}[ (A - B)^2 + (A - C)^2 + (B - C)^2 ] = {{\mathscr R}}(M)\cap {{\mathscr R}}(3I_m - M),
\label{3112b}
\\
& {{\mathscr R}}(AB + BA + AC + CA + BC + CB) = {{\mathscr R}}(M)\cap {{\mathscr R}}(I_m - M),
\label{3112c}\end{aligned}$$ and $$\begin{aligned}
& {{\mathscr N}}[(A + B)^2 + (A + C)^2 + (B + C)^2] = {{\mathscr R}}(N)\cap {{\mathscr N}}(I_m + M),
\label{3112d}
\\
& {{\mathscr N}}[ (A - B)^2 + (A - C)^2 + (B - C)^2 ] = {{\mathscr N}}(M)\cap {{\mathscr N}}(3I_m - M),
\label{3112e}
\\
& {{\mathscr N}}(AB + BA + AC + CA + BC + CB) = {{\mathscr N}}(M)\cap {{\mathscr N}}(I_m - M).
\label{3112f}\end{aligned}$$ In particular$,$
1. the following facts hold $$\begin{aligned}
& \alpha(AB + AC) + \beta (BA + BC) + \gamma(CA + CB) = 0 \Leftrightarrow (\alpha A + \beta B + \gamma C)(M - I_m) = 0,
\\
& \alpha(BA + CA) + \beta (AB + CB) + \gamma(AC + BC) = 0 \Leftrightarrow (M - I_m)(\alpha A + \beta B + \gamma C) = 0,\end{aligned}$$ and $$\begin{aligned}
& (\alpha + \beta)(AB + BA) + (\alpha + \gamma)(AC + CA) + (\beta + \gamma)(BC + CB) =0 {\nonumber}\\
& \Leftrightarrow (\alpha A + \beta B + \gamma C \,)(\,M - I_m) + (M - I_m)(\alpha A + \beta B + \gamma C) =0,
\\
&(\alpha - \beta)(AB - BA) + (\alpha - \gamma)(AC - CA) + (\beta - \gamma)(BC - CB) =0 {\nonumber}\\
& \Leftrightarrow (\alpha A + \beta B + \gamma C \,)M = M(\alpha A + \beta B + \gamma C),
\\
& \alpha(B + C)A(B + C) + \beta (A + C)B(A + C) + \gamma(A + B)C(A + B) =0 {\nonumber}\\
&\Leftrightarrow (M - I_m)(\alpha A + \beta B + \gamma C)(M - I_m) =0;\end{aligned}$$
2. the following facts hold $$\begin{aligned}
& (A + B)^2 + (A + C)^2 + (B + C)^2 = 0 \Leftrightarrow M^2 + M =0,
\\
& (A + B)^2 + (A + C)^2 + (B + C)^2 = I_m \Leftrightarrow M^2 + M = I_m,
\\
& (A - B)^2 + (A - C)^2 + (B - C)^2 = 0 \Leftrightarrow (2M - 3I_m)^2 = 9I_m,
\\
& (A - B)^2 + (A - C)^2 + (B - C)^2 = 9/8I_m \Leftrightarrow (2M - 3I_m)^2 = 9I_m,
\\
& (A - B)^2 + (A - C)^2 + (B - C)^2 = 3I_m \Leftrightarrow (2M - 3I_m)^2 = -3I_m,
\\
& (A - B)^2 + (A - C)^2 + (B - C)^2 = 9/4I_m \Leftrightarrow (2M - 3I_m)^2 = 0,
\\
& AB + BA + AC + CA + BC + CB = kI_m \Leftrightarrow (I_m - 2M)^2 = (4k +1)I_m,
\ \ k = 0, 1, \ldots, 6;\end{aligned}$$
3. $\alpha(AB + AC) + \beta (BA + BC) + \gamma(CA + CB)$ is nonsingular $\Leftrightarrow$ $\alpha(BA + CA) + \beta (AB + CB) + \gamma(AC + BC)$ is nonsingular $\Leftrightarrow$ $\alpha(B + C)A(B + C) + \beta (A + C)B(A + C) + \gamma(A + B)C(A + B)$ $\Leftrightarrow$ both $\alpha A + \beta B + \gamma C$ and $M - I_m$ are nonsingular; in which cases$,$ the following equalities hold $$\begin{aligned}
& [\alpha(AB + AC) + \beta (BA + BC) + \gamma(CA + CB)]^{-1} = (M - I_m)^{-1}(\, \alpha A + \beta B + \gamma C \,)^{-1},
\\
& [\alpha(BA + CA) + \beta (AB + CB) + \gamma(AC + BC)]^{-1} = (\,\alpha A + \beta B + \gamma C \,)^{-1},
(M - I_m)^{-1},\end{aligned}$$ and $$\begin{aligned}
& [\alpha(B + C)A(B + C) + \beta (A + C)B(A + C) + \gamma(A + B)C(A + B)]^{-1}
\\
& = (M - I_m)^{-1}(\alpha A + \beta B + \gamma C)^{-1}(M - I_m)^{-1};\end{aligned}$$
4. the following facts hold $$\begin{aligned}
& r[(A + B)^2 + (A + C)^2 + (B + C)^2] = m \Leftrightarrow r(M) = r(I_m + M) = m,
\\
& r[(A - B)^2 + (A - C)^2 + (B - C)^2] = m \Leftrightarrow r(M) = r(3I_m - M) = m,\end{aligned}$$ and $$\begin{aligned}
& r(kI_m - AB - BA - AC - CA - BC - CB) =m {\nonumber}\\
& \Leftrightarrow r\!\left[(\sqrt{4k +1} + 1)/2I_m - M \right] =
r\!\left[\,(\sqrt{4k +1} - 1)/2I_m + M \right] = m, \ \ k = 0, 1, \ldots, 6.\end{aligned}$$
Eqs., , and follow from expanding both sides of the three equalities. The sum and difference of and result in and , respectively. Eqs.– follow from direct expansions. Applying to – leads to –. Results (a)–(d) follow from –.
In addition, it would be of interest to discuss the following reverse order laws for generalized inverses $$\begin{aligned}
& [\alpha(AB + AC) + \beta (BA + BC) + \gamma(CA + CB)]^{(i,\ldots,j)} = (M - I_m)^{(i,\ldots,j)}(\, \alpha A + \beta B + \gamma C \,)^{(i,\ldots,j)},
\\
& [\alpha(BA + CA) + \beta (AB + CB) + \gamma(AC + BC)]^{(i,\ldots,j)} = (\,\alpha A + \beta B + \gamma C \,)^{(i,\ldots,j)},
(M - I_m)^{(i,\ldots,j)},\end{aligned}$$ and $$\begin{aligned}
& [\alpha(B + C)A(B + C) + \beta (A + C)B(A + C) + \gamma(A + B)C(A + B)]^{(i,\ldots,j)}
\\
& = (M - I_m)^{(i,\ldots,j)}(\alpha A + \beta B + \gamma C)^{(i,\ldots,j)}(M - I_m)^{(i,\ldots,j)}.\end{aligned}$$
Bounds of ranks of some matrix pencils composed by two idempotent matrices
==========================================================================
For a given singular matrix $A$, the two products $AA^{-}$ and $A^{-}A$ are not necessarily unique, but they both are idempotent matrices for all $A^-$. In this case, people are interested in the performance of idempotents associated generalized inverses and their operations; see e.g., [@BR; @CX; @DW; @ND; @T0]. In this section, we approach the ranks of the following four characteristic matrices $$\begin{aligned}
\lambda I_m + AA^{-} \pm BB^{-}, \ \ \lambda I_m + AA^{-}\pm C^{-}C
\label{d11}\end{aligned}$$ associated with the idempotent matrices $A^{-}$, $B^{-}$, and $C^{-}$, where $\lambda$ is a scalar. It is obvious that the ranks of the four matrix expressions in are all functions of $\lambda$, $A^{-}$, $B^{-}$, and $C^{-}$. Thus we are interested in the maximum and minimum ranks, as well as rank distributions of the four matrix expressions. To determine the two ranks, we need to use the following known formulas.
\[TN41\] Let $ A \in {\mathbb C}^{m \times n}, \, B \in {\mathbb C}^{m \times k},$ $C \in {\mathbb C}^{l \times n},$ and $D \in {\mathbb C}^{l \times k}.$ Then $$\begin{aligned}
r[A, \, B] = r(A) + r(E_AB), \ \ \ r\!\begin{bmatrix} A \\ C \end{bmatrix} = r(A) +r(CF_A).
\label{k42}\end{aligned}$$
\[[@T1]\] \[TN43\] The maximum and minimum ranks of the linear matrix-valued function $A - B_1X_1C_1 -B_2X_2C_2$ with respect to the two variable matrices $X_1$ and $X_2$ are given by $$\begin{aligned}
& \max_{X_1, \, X_2}\!r(\, A - B_1X_1C_1 -B_2X_2C_2 \,) = \min\!\left\{ r[\, A, \, B_1, \, B_2 \,], \
r\!\begin{bmatrix} A \\ C_1 \\ C_2 \end{bmatrix}\!, \ r\!\begin{bmatrix} A
& B_1\\ C_2 & 0 \end{bmatrix}\!, \
r\!\begin{bmatrix} A & B_2 \\ C_1 & 0 \end{bmatrix} \right\},
\label{110}
\\
& \min_{X_1, \, X_2}\!r(\, A - B_1X_1C_1 -B_2X_2C_2 \,) = r\!\begin{bmatrix} A \\ C_1 \\ C_2 \end{bmatrix} + r[ \, A, \, B_1, \, B_2 \,] {\nonumber}\\
& \ + \max\left\{\! r\!\begin{bmatrix}A & B_1\\ C_2 & 0\end{bmatrix} - r\!\begin{bmatrix} A
& B_1 & B_2 \\ C_2 & 0 & 0 \end{bmatrix}
-r\begin{bmatrix} A
& B_1 \\ C_1 & 0 \\ C_2 & 0 \end{bmatrix}\!, \ r\!\begin{bmatrix} A
& B_2 \\ C_1 & 0 \end{bmatrix} -r\!\begin{bmatrix} A
& B_1 & B_2 \\ C_1 & 0 & 0 \end{bmatrix} - r\!\begin{bmatrix} A
& B_2 \\ C_1 & 0 \\ C_2 & 0 \end{bmatrix} \!\right\}\!.
\label{111}\end{aligned}$$
We next establish exact expansion formulas for calculating the maximum and minimum ranks of the four matrix pencils with respect to the choice of $\lambda$, $A^{-}$, and $B^{-}$, and use them to derive a variety of simple and interesting properties of the four matrix pencils from the rank formulas.
\[[@BG; @CM; @RM]\] \[TN42\] Let $A \in {\mathbb C}^{m \times n}.$ Then the general expressions of $A^{-},$ $AA^{-},$ and $A^{-}A$ can be written as $$\begin{aligned}
& A^{-} = A^{\dag} + F_{A}U + VE_{A}, \ \ AA^{-} = AA^{\dag} + AVE_{A}, \ \ A^{-}A = A^{\dag}A + F_{A}UA,
\label{112}\end{aligned}$$ where $U,\, V \in {\mathbb C}^{n \times m}$ are arbitrary$.$
\[TN44\] Let $A \in {\mathbb C}^{m \times n}$ and $B \in {\mathbb C}^{m \times p}$ be given$.$
1. If $\lambda \neq 0, -1, -2,$ then the following two formulas hold $$\begin{aligned}
& \max_{A^{-},B^{-}}\!\!r(\, \lambda I_m + AA^{-} + BB^{-}\,) = m,
\label{z110}
\\
& \min_{A^{-},B^{-}}\!\!r(\,\lambda I_m + AA^{-} + BB^{-}\,) = \max\{ \, m + r(A) - r[\, A, \, B \,], \ \
m + r(B) - r[\, A, \, B \,] \,\}.
\label{z111}\end{aligned}$$ In particular$,$ the following results hold$.$
1. There always exist $A^{-}$ and $B^{-}$ such that $\lambda I_m + AA^{-} + BB^{-}$ is nonsingular$.$
2. $\lambda I_m + AA^{-} + BB^{-}$ is nonsingular for all $A^{-}$ and $B^{-}$ $\Leftrightarrow$ The rank of $\lambda I_m + AA^{-} + BB^{-}$ is invariant for all $A^{-}$ and $B^{-}$ $\Leftrightarrow$ $r[\, A, \, B \,] = r(A) = r(B)$ $\Leftrightarrow$ ${{\mathscr R}}(A) = {{\mathscr R}}(B).$
3. There do not exist $A^{-}$ and $B^{-}$ such that $\lambda I_m + AA^{-} + BB^{-} =0.$
2. The following two formulas hold $$\begin{aligned}
& \max_{A^{-},B^{-}}\!\!r(\,AA^{-} + BB^{-}\,) = r[\, A, \, B \,],
\label{z112}
\\
& \min_{A^{-},B^{-}}\!\!r(\,AA^{-} + BB^{-}\,) = \max\{ \, r(A), \ \ r(B) \,\}.
\label{z113}\end{aligned}$$ In particular$,$ the following results hold$.$
1. There exist $A^{-}$ and $B^{-}$ such that $AA^{-} + BB^{-}$ is nonsingular $\Leftrightarrow$ $r[\, A, \, B \,] =m.$
2. $AA^{-} + BB^{-}$ is nonsingular for all $A^{-}$ and $B^{-}$ $\Leftrightarrow$ $r(A) =m$ or $r(B) =m.$
3. There exist $A^{-}$ and $B^{-}$ such that $AA^{-} + BB^{-} = 0$ $\Leftrightarrow$ $AA^{-} + BB^{-} =0$ for all $A^{-}$ and $B^{-}$ $\Leftrightarrow$ $[\, A, \, B \,] = 0.$
4. The rank of $AA^{-} + BB^{-}$ is invariant for all $A^{-}$ and $B^{-}$ $\Leftrightarrow$ ${{\mathscr R}}(A) \supseteq {{\mathscr R}}(B)$ or ${{\mathscr R}}(A) \subseteq {{\mathscr R}}(B).$
3. The following two formulas hold $$\begin{aligned}
& \max_{A^{-},B^{-}}\!\!r(\, -I_m + AA^{-} + BB^{-}\,) = m - |r(A) - r(B)|,
\label{z114}
\\
& \min_{A^{-},B^{-}}\!\!r(\,-I_m + AA^{-} + BB^{-}\,) = m + r(A) + r(B) - 2r[\, A, \, B \,].
\label{z115}\end{aligned}$$ In particular$,$ the following results hold$.$
1. There exist $A^{-}$ and $B^{-}$ such that $- I_m + AA^{-} + BB^{-}$ is nonsingular $\Leftrightarrow$ $r(A) = r(B).$
2. $ - I_m + AA^{-} + BB^{-}$ is nonsingular for all $A^{-}$ and $B^{-}$ $\Leftrightarrow$ ${{\mathscr R}}(A) ={{\mathscr R}}(B).$
3. There exist $A^{-}$ and $B^{-}$ such that $AA^{-} + BB^{-} = I_m$ $\Leftrightarrow$ $r[\, A, \, B \,] = r(A) + r(B) =m.$
4. The rank of $- I_m + AA^{-} + BB^{-}$ is invariant for all $A^{-}$ and $B^{-}$ $\Leftrightarrow$ ${{\mathscr R}}(A) \supseteq {{\mathscr R}}(B)$ or ${{\mathscr R}}(A) \subseteq {{\mathscr R}}(B).$
4. The following two formulas hold $$\begin{aligned}
& \max_{A^{-},B^{-}}\!\!r(\, -2I_m + AA^{-} + BB^{-}\,) = m + r[\, A, \, B \,] - r(A) - r(B),
\label{z116}
\\
& \min_{A^{-},B^{-}}\!\!r(\,-2I_m + AA^{-} + BB^{-}\,) = \max\{ \, m - r(A), \ \ m - r(B) \,\}.
\label{z117}\end{aligned}$$ In particular$,$ the following results hold$.$
1. There exist $A^{-}$ and $B^{-}$ such that $- 2I_m + AA^{-} + BB^{-}$ is nonsingular $\Leftrightarrow$ ${{\mathscr R}}(A) \cap {{\mathscr R}}(B) =\{0\}.$
2. $- 2I_m + AA^{-} + BB^{-}$ is nonsingular for all $A^{-}$ and $B^{-}$ $\Leftrightarrow$ $A =0$ or $B =0.$
3. There exist $A^{-}$ and $B^{-}$ such that $AA^{-} + BB^{-} = 2I_m$ $\Leftrightarrow$ $AA^{-} + BB^{-} =2I_m$ for all $A^{-}$ and $B^{-}$ $\Leftrightarrow$ $r(A) = r(B) =m.$
4. The rank of $- 2I_m AA^{-} + BB^{-}$ is invariant for all $A^{-}$ and $B^{-}$ $\Leftrightarrow$ ${{\mathscr R}}(A) \supseteq {{\mathscr R}}(B)$ or ${{\mathscr R}}(A) \subseteq {{\mathscr R}}(B).$
By , $$\begin{aligned}
&\lambda I_m + AA^{-} + BB^{-} = \lambda I_m + AA^{\dag} + BB^{\dag} + AV_1E_{A} + BV_2E_{B},
\label{z118}\end{aligned}$$ where $V_1\in {\mathbb C}^{n \times m}$ and $V_2\in {\mathbb C}^{p\times m}$ are arbitrary$.$ Applying and to and simplifying gives $$\begin{aligned}
& \max_{V_1, \, V_2} r(\, \lambda I_m + AA^{\dag} + BB^{\dag} + AV_1E_{A} + BV_2E_{B} \,) {\nonumber}\\
& = \min\!\left\{ r[\, \lambda I_m + AA^{\dag} + BB^{\dag}, \, A, \, B \,], \
r\!\begin{bmatrix} \lambda I_m + AA^{\dag} + BB^{\dag} \\ E_{A} \\ E_{B} \end{bmatrix}\!,
\right. {\nonumber}\\
& \ \ \ \ \ \ \ \ \ \ \left. r\!\begin{bmatrix} \lambda I_m + AA^{\dag} + BB^{\dag}
& A\\ E_{B} & 0 \end{bmatrix}\!, \
r\!\begin{bmatrix} \lambda I_m + AA^{\dag} + BB^{\dag} & B \\ E_{A} & 0 \end{bmatrix} \right\} {\nonumber}\\
& = \min\!\left\{ r[\, \lambda I_m, \, A, \, B \,], \ \
r\!\begin{bmatrix} (\lambda +2)I_m \\ E_{A} \\ E_{B} \end{bmatrix}\!, \ \ r\!\begin{bmatrix} (\lambda +1)I_m & A \\ E_{B} & 0 \end{bmatrix}\!, \ \
r\!\begin{bmatrix} (\lambda +1)I_m & B \\ E_{A} & 0 \end{bmatrix} \right\},
\label{z119}\end{aligned}$$ and $$\begin{aligned}
& \min_{V_1, \, V_2}r(\, \lambda I_m + AA^{\dag} + BB^{\dag} + AV_1E_{A} + BV_2E_{B} \,) {\nonumber}\\
& = r\!\begin{bmatrix} \lambda I_m + AA^{\dag} + BB^{\dag} \\ E_{A} \\ E_{B} \end{bmatrix} + r[ \, \lambda I_m + AA^{\dag} + BB^{\dag}, \, A, \, B \,] {\nonumber}\\
& \ \ + \ \max\left\{\! r\!\begin{bmatrix}\lambda I_m + AA^{\dag} + BB^{\dag} & A\\ E_{B} & 0\end{bmatrix} - r\!\begin{bmatrix} \lambda I_m + AA^{\dag} + BB^{\dag}
& A & B \\ E_{B} & 0 & 0 \end{bmatrix}
-r\!\begin{bmatrix} \lambda I_m + AA^{\dag} + BB^{\dag}
& A \\ E_{A} & 0 \\ E_{B} & 0 \end{bmatrix}, \right. {\nonumber}\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left. r\!\begin{bmatrix} \lambda I_m + AA^{\dag} + BB^{\dag}
& B \\ E_{A} & 0 \end{bmatrix} -r\!\begin{bmatrix} \lambda I_m + AA^{\dag} + BB^{\dag}
& A & B \\ E_{A} & 0 & 0 \end{bmatrix} - r\!\begin{bmatrix} \lambda I_m + AA^{\dag} + BB^{\dag}
& B \\ E_{A} & 0 \\ E_{B} & 0 \end{bmatrix} \!\right\} {\nonumber}\\
& = r\!\begin{bmatrix} (\lambda +2)I_m \\ E_{A} \\ E_{B} \end{bmatrix} + r[\, \lambda I_m, \, A, \, B \,] + \ \max\left\{\! r\!\begin{bmatrix} (\lambda +1)I_m & A \\ E_{B} & 0 \end{bmatrix} - r\!\begin{bmatrix} \lambda I_m & A & B \\ E_{B} & 0 & 0 \end{bmatrix}
-r\!\begin{bmatrix} (\lambda +1)I_m
& A \\ E_{A} & 0 \\ E_{B} & 0 \end{bmatrix}, \right. {\nonumber}\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left.
r\!\begin{bmatrix} (\lambda +1)I_m & B \\ E_{A} & 0 \end{bmatrix} -r\!\begin{bmatrix} \lambda I_m
& A & B \\ E_{A} & 0 & 0 \end{bmatrix} - r\!\begin{bmatrix} (\lambda +1)I_m
& B \\ E_{A} & 0 \\ E_{B} & 0 \end{bmatrix} \!\right\}.
\label{z120}\end{aligned}$$ Substituting different values of $\lambda$ into the above formulas and and simplifying yield the rank formulas in –, respectively. The facts in (a)–(d) follow from setting the rank formulas in – equal to $m$ and 0 and applying Lemma \[TN41\], respectively.
\[TN45\] Let $A \in {\mathbb C}^{m \times n}$ and $B \in {\mathbb C}^{m \times p}$ be given$.$
1. If $\lambda \neq 1, 0, -1,$ then the following two formulas hold $$\begin{aligned}
& \max_{A^{-},B^{-}}\!\!r(\,\lambda I_m + AA^{-} - BB^{-}\,) =m,
\label{z121}
\\
& \min_{A^{-},B^{-}}\!\!r(\,\lambda I_m + AA^{-} - BB^{-}\,) = \max\{ \, m + r(A) - r[\, A, \, B \,], \ \ m + r(B) - r[\, A, \, B \,] \}.
\label{z122}\end{aligned}$$ In particular$,$ the following results hold$.$
1. There always exist $A^{-}$ and $B^{-}$ such that $\lambda I_m + AA^{-} - BB^{-}$ is nonsingular$.$
2. $\lambda I_m + AA^{-} - BB^{-}$ is nonsingular for all $A^{-}$ and $B^{-}$ $\Leftrightarrow$ the rank of $\lambda I_m + AA^{-} - BB^{-}$ is invariant for all $A^{-}$ and $B^{-}$ $\Leftrightarrow$ $r[\, A, \, B \,] = r(A) = r(B)$ $\Leftrightarrow$ ${{\mathscr R}}(A) = {{\mathscr R}}(B).$
3. There do not exist $A^{-}$ and $B^{-}$ such that $\lambda I_m + AA^{-} - BB^{-} =0.$
2. The following two formulas hold $$\begin{aligned}
& \max_{A^{-},B^{-}}\!\!r(\,I_m + AA^{-} - BB^{-}\,) =\min \{m, \ \ m + r(A) - r(B) \},
\label{z123}
\\
& \min_{A^{-},B^{-}}\!\!r(\,I_m + AA^{-} - BB^{-}\,) = m + r(A) - r[\, A, \, B \,].
\label{z124}\end{aligned}$$ In particular$,$ the following results hold$.$
1. There exist $A^{-}$ and $B^{-}$ such that $I_m + AA^{-} - BB^{-}$ is nonsingular $\Leftrightarrow$ $r(A) = r(B).$
2. $I_m + AA^{-} - BB^{-}$ is nonsingular for all $A^{-}$ and $B^{-}$ $\Leftrightarrow$ $\Leftrightarrow$ ${{\mathscr R}}(A) \supseteq {{\mathscr R}}(B).$
3. There exist $A^{-}$ and $B^{-}$ such that $BB^{-} - AA^{-} =I_m$ $\Leftrightarrow$ $BB^{-} - AA^{-} =I_m$ holds for all $A^{-}$ and $B^{-}$ $\Leftrightarrow$ $A = 0$ and $r(B) =m.$
4. The rank of $I_m + AA^{-} - BB^{-}$ is invariant for all $A^{-}$ and $B^{-}$ $\Leftrightarrow$ ${{\mathscr R}}(A) \supseteq {{\mathscr R}}(B)$ or ${{\mathscr R}}(A) \subseteq {{\mathscr R}}(B).$
3. The following two formulas hold $$\begin{aligned}
& \max_{A^{-},B^{-}}\!\!r(\,AA^{-} - BB^{-}\,) = \min\{ \, r[\, A, \, B \,], \ \ m + r[\, A, \, B \,] - r(A) - r(B) \},
\label{z125}
\\
& \min_{A^{-},B^{-}}r(\,AA^{-} - BB^{-}\,) = \max\{ \, r[\, A, \, B \,] - r(A), \ \ r[\, A, \, B \,] - r(B) \}.
\label{z126}\end{aligned}$$ In particular$,$ the following results hold$.$
1. There exist $A^{-}$ and $B^{-}$ such that $AA^{-} - BB^{-}$ is nonsingular $\Leftrightarrow$ $r[\, A, \, B \,] = r(A) + r(B) =m.$
2. $AA^{-} - BB^{-}$ is nonsingular for all $A^{-}$ and $B^{-}$ $\Leftrightarrow$ $r(A) =m$ and $B = 0,$ or $A = 0$ and $r(B) =m.$
3. There exist $A^{-}$ and $B^{-}$ such that $AA^{-} = BB^{-}$ $\Leftrightarrow$ ${{\mathscr R}}(A) ={{\mathscr R}}(B).$
4. $AA^{-} = BB^{-}$ holds for all $A^{-}$ and $B^{-}$ $\Leftrightarrow$ $[\, A, \, B \,] = 0$ or $r[\, A, \, B \,] = r(A) + r(B) -m.$
5. The rank of $AA^{-} - BB^{-}$ is invariant for all $A^{-}$ and $B^{-}$ $\Leftrightarrow$ $A=0$ or $B=0$ $r(A)=m$ or $r(B)=m.$
By , $$\begin{aligned}
\lambda I_m + AA^{-} - BB^{-} = \lambda I_m + AA^{\dag} - BB^{\dag} + AV_1E_{A} - BV_2E_{B},
\label{27}\end{aligned}$$ where $V_1\in {\mathbb C}^{n \times m}$ and $V_2\in {\mathbb C}^{p\times m}$ are arbitrary$.$ Applying and to and simplifying gives $$\begin{aligned}
r\!\begin{bmatrix} \lambda I_m \\ E_{A} \\ E_{B} \end{bmatrix}\!, \ \ r\!\begin{bmatrix} (\lambda -1)I_m & A \\ E_{B} & 0 \end{bmatrix}\!, \ \
r\!\begin{bmatrix} (\lambda +1)I_m & B \\ E_{A} & 0 \end{bmatrix} \right\},
$$ and $$\begin{aligned}
& B \\ E_{A} & 0 \\ E_{B} & 0 \end{bmatrix} \!\right\} {\nonumber}\end{aligned}$$ $$\begin{aligned}
& = r\!\begin{bmatrix} \lambda I_m \\ E_{A} \\ E_{B} \end{bmatrix} + r[\, \lambda I_m, \, A, \, B \,] + \ \max\left\{\! r\!\begin{bmatrix} (\lambda -1)I_m & A \\ E_{B} & 0 \end{bmatrix} - r\!\begin{bmatrix} \lambda I_m & A & B \\ E_{B} & 0 & 0 \end{bmatrix}
-r\!\begin{bmatrix} (\lambda -1)I_m & A \\ E_{A} & 0 \\ E_{B} & 0 \end{bmatrix}, \right. {\nonumber}\\
$$ Substituting different values of $\lambda$ into the above formulas and simplifying yield the rank formulas required.
\[TN46\] Let $A \in {\mathbb C}^{m \times n}$ and $C \in {\mathbb C}^{p \times m}$ be given$.$
1. If $\lambda \neq 0, -1, -2,$ then the following two formulas hold $$\begin{aligned}
& \max_{A^{-},C^{-}}\!\!r(\, \lambda I_m + AA^{-} + C^{-}C\,) = m,
\\
& \min_{A^{-},C^{-}}\!\!r(\,\lambda I_m + AA^{-} + C^{-}C\,) = \max\{ \, m - r(CA), \ \ r(A) + r(C) - r(CA) \,\}.
$$ In particular$,$ the following results hold$.$
1. There always exist $A^{-}$ and $C^{-}$ such that $\lambda I_m + AA^{-} + C^{-}C$ is nonsingular$.$
2. $\lambda I_m + AA^{-} + C^{-}C$ is nonsingular for all $A^{-}$ and $C^{-}$ $\Leftrightarrow$ the rank of $\lambda I_m + AA^{-} + C^{-}C$ is invariant for all $A^{-}$ and $C^{-}$ $\Leftrightarrow$ $r[\, A, \, C \,] = r(A) = r(C)$ $\Leftrightarrow$ $CA =0$ and $r(A) + r(C) = m.$
3. There do not exist $A^{-}$ and $C^{-}$ such that $\lambda I_m + AA^{-} + C^{-}C =0.$
2. The following two formulas hold $$\begin{aligned}
& \max_{A^{-},C^{-}}\!\!r(\, AA^{-} + C^{-}C\,) = \min\{ \, m, \ \ r(A) + r(C) \,\},
\\
& \min_{A^{-},C^{-}}\!\!r(\,AA^{-} + C^{-}C\,) = r(A) + r(C) - r(CA).
$$ In particular$,$ the following results hold$.$
1. There exist $A^{-}$ and $C^{-}$ such that $AA^{-} + C^{-}C$ is nonsingular $\Leftrightarrow$ $r(A) + r(C) \geq m.$
2. $AA^{-} + C^{-}C$ is nonsingular for all $A^{-}$ and $C^{-}$ $\Leftrightarrow$ $r(CA) = r(A) + r(C) - m.$
3. There exist $A^{-}$ and $C^{-}$ such that $AA^{-} + C^{-}C =0$ $\Leftrightarrow$ $AA^{-} + C^{-}C =0$ for all $A^{-}$ and $C^{-}$ $\Leftrightarrow$ $A = 0$ and $C= 0.$
4. The rank of $AA^{-} + C^{-}C$ is invariant for all $A^{-}$ and $C^{-}$ $\Leftrightarrow$ $CA =0$ or $r(CA) = r(A) + r(C) - m.$
3. The following two formulas hold $$\begin{aligned}
& \max_{A^{-},C^{-}}\!\!r(\, -I_m + AA^{-} + C^{-}C\,) = \min\{ \, m + r(CA) - r(A) , \ \ m + r(CA) - r(C) \,\},
\\
& \min_{A^{-},C^{-}}\!\!r(\,-I_m + AA^{-} + C^{-}C\,) = \max\{ \, m + r(CA) - r(A) - r(C), \ \ r(CA) \,\}.
$$ In particular$,$ the following results hold$.$
1. There exist $A^{-}$ and $C^{-}$ such that $-I_m + AA^{-} + C^{-}C$ is nonsingular $\Leftrightarrow$ $r(CA) = r(A) = r(C).$
2. $-I_m + AA^{-} + C^{-}C$ is nonsingular for all $A^{-}$ and $C^{-}$ $\Leftrightarrow$ $A = 0$ and $C = 0,$ or $r(A) = r(C) = m.$
3. There exist $A^{-}$ and $C^{-}$ such that $AA^{-} + C^{-}C = I_m$ $\Leftrightarrow$ $CA =0$ and $r(A) + r(C) = m.$
4. $AA^{-} + C^{-}C = I_m$ cannot hold for all $A^{-}$ and $C^{-}.$
5. The rank of $-I_m + AA^{-} + C^{-}C$ is invariant for all $A^{-}$ and $C^{-}$ $\Leftrightarrow$ $A =0$ or $C =0$ or $r(A)= m$ or $r(C) =m.$
4. The following two formulas hold $$\begin{aligned}
& \max_{A^{-},C^{-}}\!\!r(\, -2I_m + AA^{-} + C^{-}C\,) = \min\{ \, m, \ \ 2m - r(A) - r(C) \,\},
\\
& \min_{A^{-},C^{-}}\!\!r(\,-2I_m + AA^{-} + C^{-}C\,) = m - r(CA).\end{aligned}$$ In particular$,$ the following results hold$.$
1. There exist $A^{-}$ and $C^{-}$ such that $-2I_m + AA^{-} + C^{-}C$ is nonsingular $\Leftrightarrow$ $r(A) + r(C) \leq m.$
2. $-2I_m + AA^{-} + C^{-}C$ is nonsingular for all $A^{-}$ and $C^{-}$ $\Leftrightarrow$ $CA = 0.$
3. There exist $A^{-}$ and $C^{-}$ such that $AA^{-} + C^{-}C = 2I_m$ $\Leftrightarrow$ the rank of $-I_m + AA^{-} + C^{-}C$ is invariant for all $A^{-}$ and $C^{-}$ $\Leftrightarrow$ $r(A) = r(C) = m.$
4. The rank of $-2I_m + AA^{-} + C^{-}C$ is invariant for all $A^{-}$ and $C^{-}$ $\Leftrightarrow$ $CA =0$ or $r(CA) = r(A) + r(C) - m.$
By , $$\begin{aligned}
&\lambda I_m + AA^{-} + C^{-}C = \lambda I_m + AA^{\dag} + C^{\dag}C + AV_1E_{A} + F_{C}V_2C,
\label{38}\end{aligned}$$ where $V_1\in {\mathbb C}^{n \times m}$ and $V_2\in {\mathbb C}^{m\times p}$ are arbitrary$.$ Applying and to and simplifying gives $$\begin{aligned}
& \max_{V_1, \, V_2}\!r(\, \lambda I_m + AA^{\dag} + C^{\dag}C + AV_1E_{A} + F_{C}V_2C \,) {\nonumber}\\
& = \min\!\left\{ r[\, \lambda I_m + AA^{\dag} + C^{\dag}C, \, A, \, F_{C} \,], \
r\!\begin{bmatrix} \lambda I_m + AA^{\dag} + C^{\dag}C \\ E_{A} \\ C \end{bmatrix}\!, \right. {\nonumber}\\
& \ \ \ \ \ \ \ \ \ \left. r\!\begin{bmatrix} \lambda I_m + AA^{\dag} + C^{\dag}C
& A\\ C & 0 \end{bmatrix}\!, \
r\!\begin{bmatrix} \lambda I_m + AA^{\dag} + C^{\dag}C & F_{C} \\ E_{A} & 0 \end{bmatrix} \right\} {\nonumber}\\
& = \min\!\left\{ r[\, (\lambda +1)I_m, \, A, \, F_{C} \,], \ \
r\!\begin{bmatrix} (\lambda + 1)I_m
\\ E_{A} \\ C \end{bmatrix}, \ \ r\!\begin{bmatrix} \lambda I_m & A \\ C & 0 \end{bmatrix}, \ \ r\!\begin{bmatrix} (\lambda +2)I_m
& F_{C} \\ E_{A} & 0 \end{bmatrix} \right\}\!,
$$ and $$\begin{aligned}
& \min_{V_1, \, V_2}\!r(\, \lambda I_m + AA^{\dag} + C^{\dag}C + AV_1E_{A} + F_{C}V_2C \,) = r\!\begin{bmatrix} \lambda I_m
+ AA^{\dag} + C^{\dag}C \\ E_{A} \\ C \end{bmatrix} + r[ \, \lambda I_m + AA^{\dag} + C^{\dag}C, \, A, \, F_{C} \,] {\nonumber}\\
& \ \ + \ \max\left\{\! r\!\begin{bmatrix}\lambda I_m + AA^{\dag} + C^{\dag}C & A\\ C & 0\end{bmatrix} - r\!\begin{bmatrix} \lambda I_m + AA^{\dag} + C^{\dag}C
& A & F_{C} \\ C & 0 & 0 \end{bmatrix}
-r\!\begin{bmatrix} \lambda I_m + AA^{\dag} + C^{\dag}C
& A \\ E_{A} & 0 \\ C & 0 \end{bmatrix}\!, \right. {\nonumber}\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left. r\!\begin{bmatrix} \lambda I_m + AA^{\dag} + C^{\dag}C
& F_{C} \\ E_{A} & 0 \end{bmatrix} -r\!\begin{bmatrix} \lambda I_m + AA^{\dag} + C^{\dag}C
& A & F_{C} \\ E_{A} & 0 & 0 \end{bmatrix} - r\!\begin{bmatrix} \lambda I_m + AA^{\dag} + C^{\dag}C
& F_{C} \\ E_{A} & 0 \\ C & 0 \end{bmatrix} \!\right\} {\nonumber}\\
& = r\!\begin{bmatrix} (\lambda + 1)I_m
\\ E_{A} \\ C \end{bmatrix} + r[\, (\lambda +1)I_m, \, A, \, F_{C} \,]
+ \max\left\{\! r\!\begin{bmatrix}\lambda I_m & A\\ C & 0\end{bmatrix} - r\!\begin{bmatrix} \lambda I_m
& A & F_{C} \\ C & 0 & 0 \end{bmatrix}
-r\!\begin{bmatrix} \lambda I_m
& A \\ E_{A} & 0 \\ C & 0 \end{bmatrix}\!, \right. {\nonumber}\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left. r\!\begin{bmatrix} (\lambda +2)I_m
& F_{C} \\ E_{A} & 0 \end{bmatrix} -r\!\begin{bmatrix} (\lambda +1)I_m
& A & F_{C} \\ E_{A} & 0 & 0 \end{bmatrix} - r\!\begin{bmatrix} (\lambda +1)I_m
& F_{C} \\ E_{A} & 0 \\ C & 0 \end{bmatrix} \!\right\}\!.
$$ Substituting different values of $\lambda$ into the above formulas and simplifying yield the rank formulas required.
\[T7\] Let $A \in {\mathbb C}^{m \times n}$ and $C \in {\mathbb C}^{p \times m}$ be given$.$
1. If $\lambda \neq 1, 0, -1,$ then the following two formulas hold $$\begin{aligned}
\\
2. $\lambda I_m + AA^{-} - C^{-}C$ is nonsingular for all $A^{-}$ and $C^{-}$ $\Leftrightarrow$ the rank of $\lambda I_m + AA^{-} - C^{-}C$ is invariant for all $A^{-}$ and $C^{-}$ $\Leftrightarrow$ $CA =0$ and $r(A) + r(C) = m.$
3. There do not exist $A^{-}$ and $C^{-}$ such that $\lambda I_m + AA^{-} - C^{-}C =0.$
2. The following two formulas hold $$\begin{aligned}
& \max_{A^{-},C^{-}}\!\!r(\, AA^{-} - C^{-}C\,) = m - |r(A) + r(C) - m|,
\\
& \min_{A^{-},C^{-}}\!\!r(\,AA^{-} - C^{-}C\,) = r(A) + r(C) - 2r(CA).
$$ In particular$,$ the following results hold$.$
1. There exist $A^{-}$ and $C^{-}$ such that $AA^{-} - C^{-}C$ is nonsingular $\Leftrightarrow$ $r(A) + r(C) =m.$
2. $AA^{-} - C^{-}C$ is nonsingular for all $A^{-}$ and $C^{-}$ $\Leftrightarrow$ $CA = 0$ and $r(A) + r(C) =m.$
3. There exist $A^{-}$ and $C^{-}$ such that $AA^{-} = C^{-}C$ $\Leftrightarrow$ $r(CA) = r(A) = r(C).$
4. $AA^{-} = C^{-}C$ holds for all $A^{-}$ and $C^{-}$ $\Leftrightarrow$ $r(A) = r(C) =m.$
5. The rank of $AA^{-} - C^{-}C$ is invariant for all $A^{-}$ and $C^{-}$ $\Leftrightarrow$ $CA = 0$ or $r(A) + r(C) =m.$
3. The following two formulas hold $$\begin{aligned}
& \max_{A^{-},C^{-}}\!\!r(\, -I_m + AA^{-} - C^{-}C\,) = m - r(A) + r(CA),
\\
& \min_{A^{-},C^{-}}\!\!r(\,-I_m + AA^{-} - C^{-}C\,) = \max\{ \, m - r(A), \ \ r(C) \,\}.
$$ In particular$,$ the following results hold$.$
1. There exist $A^{-}$ and $C^{-}$ such that $-I_m + AA^{-} - C^{-}C$ is nonsingular $\Leftrightarrow$ $r(A) = r(CA).$
2. $-I_m + AA^{-} - C^{-}C$ is nonsingular for all $A^{-}$ and $C^{-}$ $\Leftrightarrow$ $A =0$ or $r(C) =m.$
3. There exist $A^{-}$ and $C^{-}$ such that $AA^{-} - C^{-}C =I_m$ $\Leftrightarrow$ $AA^{-} - C^{-}C = I_m$ holds for all $A^{-}$ and $C^{-}$ $\Leftrightarrow$ $r(A) = m$ and $C =0.$
4. The rank of $-I_m + AA^{-} - C^{-}C$ is invariant for all $A^{-}$ and $C^{-}$ $\Leftrightarrow$ $CA = 0$ or $r(CA) = r(A) + r(C) - m.$
By , $$\begin{aligned}
&\lambda I_m + AA^{-} - C^{-}C = \lambda I_m + AA^{\dag} - C^{\dag}C + AV_1E_{A} - F_{C}V_2C,
\label{47}\end{aligned}$$ where $V_1\in {\mathbb C}^{n \times m}$ and $V_2\in {\mathbb C}^{m\times p}$ are arbitrary$.$ Applying and to and simplifying gives $$\begin{aligned}
r\!\begin{bmatrix} (\lambda +1)I_m \\ E_{A} \\ C \end{bmatrix}\!, \ \ r\!\begin{bmatrix} \lambda I_m
& A \\ C & 0 \end{bmatrix}\!, \
r\!\begin{bmatrix} \lambda I_m & F_{C} \\ E_{A} & 0 \end{bmatrix} \right\}\!,
$$ and $$\begin{aligned}
& \min_{V_1, \, V_2}\!r(\, \lambda I_m + AA^{\dag} - C^{\dag}C + AV_1E_{A} - F_{C}V_2C \,) = r\!\begin{bmatrix} \lambda I_m + AA^{\dag} - C^{\dag}C \\ E_{A} \\ C \end{bmatrix} + r[ \, \lambda I_m + AA^{\dag} - C^{\dag}C, \, A, \, F_{C} \,] {\nonumber}\\
& = r\!\begin{bmatrix} (\lambda +1)I_m \\ E_{A} \\ C \end{bmatrix} + r[ \, (\lambda -1)I_m, \, A, \, F_{C} \,] {\nonumber}\\
& \ \ + \ \max\left\{\! r\!\begin{bmatrix}\lambda I_m & A\\ C & 0\end{bmatrix} - r\!\begin{bmatrix} \lambda I_m
& A & F_{C} \\ C & 0 & 0 \end{bmatrix}
-r\!\begin{bmatrix} \lambda I_m & A \\ E_{A} & 0 \\ C & 0 \end{bmatrix}, \ r\!\begin{bmatrix} \lambda I_m
& F_{C} \\ E_{A} & 0 \end{bmatrix} -r\!\begin{bmatrix} \lambda I_m
& A & F_{C} \\ E_{A} & 0 & 0 \end{bmatrix} - r\!\begin{bmatrix} \lambda I_m
& F_{C} \\ E_{A} & 0 \\ C & 0 \end{bmatrix} \!\right\}\!.
$$ Substituting different values of $\lambda$ into the above formulas and simplifying yield the rank formulas required.
Setting $C= A$ in Theorems \[TK35\] and \[TK36\] leads to the following results.
\[T8\] Let $A \in {\mathbb C}^{m \times m}$ be given$.$
1. If $\lambda \neq 0, -1, -2,$ then the following two formulas hold $$\begin{aligned}
& \max_{A^{-}}r(\, \lambda I_m + AA^{-} + A^{-}A\,) = m,
\\
& \min_{A^{-}}r(\,\lambda I_m + AA^{-} + A^{-}A\,) = \max\{ \, m - r(A^2), \ \ 2r(A) - r(A^2) \,\}.
$$ In particular$,$ the following results hold$.$
1. There always exists $A^{-}$ such that $\lambda I_m + AA^{-} + A^{-}A$ is nonsingular$.$
2. $\lambda I_m + AA^{-} + A^{-}A$ is nonsingular for all $A^{-}$ $\Leftrightarrow$ the rank of $\lambda I_m + AA^{-} + A^{-}A$ is invariant for all $A^{-}$ $\Leftrightarrow$ $A^2 =0$ and $2r(A) = m.$
3. There does not exist $A^{-}$ such that $\lambda I_m + AA^{-} + A^{-}A =0.$
2. The following two formulas hold $$\begin{aligned}
& \max_{A^{-}}r(\, AA^{-} + A^{-}A\,) = \min\{ \, m, \ \ 2r(A) \,\},
\\
& \min_{A^{-}}r(\,AA^{-} + A^{-}A\,) = 2r(A) - r(A^2).
$$ In particular$,$ the following results hold$.$
1. There exists $A^{-}$ such that $AA^{-} + A^{-}A$ is nonsingular $\Leftrightarrow$ $2r(A) \geq m.$
2. $AA^{-} + A^{-}A$ is nonsingular for all $A^{-}$ $\Leftrightarrow$ $r(A^2) = 2r(A) - m.$
3. There exists $A^{-}$ such that $AA^{-} + A^{-}A =0$ $\Leftrightarrow$ $AA^{-} + A^{-}A =0$ for all $A^{-}$ $\Leftrightarrow$ $A = 0.$
4. The rank of $AA^{-} + A^{-}A$ is invariant for all $A^{-}$ $\Leftrightarrow$ $A^2 =0$ or $r(A^2) = 2r(A) - m.$
3. The following two formulas hold $$\begin{aligned}
& \max_{A^{-}}r(\, -I_m + AA^{-} + A^{-}A\,) = \min\{ \, m + r(A^2) - r(A) , \ \ m + r(A^2) - r(A) \,\},
\\
& \min_{A^{-}}r(\,-I_m + AA^{-} + A^{-}A\,) = \max\{ \, m + r(A^2) - 2r(A), \ \ r(A^2) \,\}.
$$ In particular$,$ the following results hold$.$
1. There exists $A^{-}$ such that $-I_m + AA^{-} + A^{-}A$ is nonsingular $\Leftrightarrow$ $r(A^2) = r(A).$
2. $-I_m + AA^{-} + A^{-}A$ is nonsingular for all $A^{-}$ $\Leftrightarrow$ $A = 0$ or $r(A) = m.$
3. There exists $A^{-}$ such that $AA^{-} + A^{-}A = I_m$ $\Leftrightarrow$ $A^2 =0$ and $2r(A) = m.$
4. $AA^{-} + A^{-}A = I_m$ cannot hold for all $A^{-}.$
5. The rank of $-I_m + AA^{-} + A^{-}A$ is invariant for all $A^{-}$ $\Leftrightarrow$ $A =0$ or $r(A) =m.$
4. The following two formulas hold $$\begin{aligned}
& \max_{A^{-}}r(\, -2I_m + AA^{-} + A^{-}A\,) = \min\{ \, m, \ \ 2m - 2r(A) \,\},
\\
& \min_{A^{-}}r(\,-2I_m + AA^{-} + A^{-}A\,) = m - r(A^2).
$$ In particular$,$ the following results hold$.$
1. There exists $A^{-}$ such that $-2I_m + AA^{-} + A^{-}A$ is nonsingular $\Leftrightarrow$ $2r(A) \leq m.$
2. $-2I_m + AA^{-} + A^{-}A$ is nonsingular for all $A^{-}$ $\Leftrightarrow$ $A^2 = 0.$
3. There exists $A^{-}$ such that $AA^{-} + A^{-}A = 2I_m$ $\Leftrightarrow$ the rank of $-I_m + AA^{-} + A^{-}A$ is invariant for all $A^{-}$ $\Leftrightarrow$ $r(A) = m.$
4. The rank of $-2I_m + AA^{-} + A^{-}A$ is invariant for all $A^{-}$ $\Leftrightarrow$ $A^2 =0$ or $r(A^2) = 2r(A) - m.$
\[T9\] Let $A \in {\mathbb C}^{m \times m}$ be given$.$
1. If $\lambda \neq 1, 0, -1,$ then the following two formulas hold $$\begin{aligned}
\\
3. There do not exist $A^{-}$ such that $\lambda I_m + AA^{-} - A^{-}A =0.$
2. The following two formulas hold $$\begin{aligned}
& \max_{A^{-}}r(\, AA^{-} - A^{-}A\,) = m - |2r(A) - m|,
\\
& \min_{A^{-}}r(\,AA^{-} - A^{-}A\,) = 2r(A) - 2r(A^2).
$$ In particular$,$ the following results hold$.$
1. There exists $A^{-}$ such that $AA^{-} - A^{-}A$ is nonsingular $\Leftrightarrow$ $2r(A) =m.$
2. $AA^{-} - A^{-}A$ is nonsingular for all $A^{-}$ $\Leftrightarrow$ $A^2 = 0$ and $2r(A) =m.$
3. There exists $A^{-}$ such that $AA^{-} = A^{-}A$ $\Leftrightarrow$ $r(A^2) = r(A).$
4. $AA^{-} = A^{-}A$ holds for all $A^{-}$ $\Leftrightarrow$ $r(A) =m.$
5. The rank of $AA^{-} - A^{-}A$ is invariant for all $A^{-}$ $\Leftrightarrow$ $A = 0$ or $2r(A) =m.$
3. The following two formulas hold $$\begin{aligned}
& \max_{A^{-}}r(\, -I_m + AA^{-} - A^{-}A\,) = m - r(A) + r(A^2),
\\
& \min_{A^{-}}r(\,-I_m + AA^{-} - A^{-}A\,) = \max\{ \, m - r(A), \ \ r(A) \,\}.
$$ In particular$,$ the following results hold$.$
1. There exists $A^{-}$ such that $-I_m + AA^{-} - A^{-}A$ is nonsingular $\Leftrightarrow$ $r(A^2) = r(A).$
2. $-I_m + AA^{-} - A^{-}A$ is nonsingular for all $A^{-}$ $\Leftrightarrow$ $A =0$ or $r(A) =m.$
3. There exists $A^{-}$ such that $AA^{-} - A^{-}A =I_m$ $\Leftrightarrow$ $AA^{-} - A^{-}A = I_m$ holds for all $A^{-}$ $\Leftrightarrow$ $A =0$ and $r(A) = m.$
4. The rank of $-I_m + AA^{-} - A^{-}A$ is invariant for all $A^{-}$ $\Leftrightarrow$ $A^2 = 0$ or $r(A^2) = 2r(A) - m.$
\[T10\] Let $A \in \mathbb C^{m \times n}, \, B \in \mathbb C^{m \times k},$ $C \in \mathbb C^{l \times n},$ and $D \in \mathbb C^{l \times n}$ be given$,$ and denote $M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$ and $N =\begin{bmatrix} A & 0 \\ 0 & D \end{bmatrix}.$ If $\lambda \neq 1, 0, -1,$ then the following two formulas hold $$\begin{aligned}
& \max_{M^{-},N^{-}}\!\!r\!\left(\, \lambda I_{m+l} + MM^{-} - NN^{-} \right) = m + l,
\label{361}
\\
& \min_{M^{-},N^{-}}\!\!r\!\left(\, \lambda I_{m+l} + MM^{-} - NN^{-} \right) = m + l - r[\, A, \, B \,] - r[\, C, \, D \,] + \max\left\{r(M), \ \ r(A) + r(D) \right\}.
\label{362}\end{aligned}$$ In particular$,$ the following formulas hold $$\begin{aligned}
& \max_{M^{-},N^{-}}\!\!r(\, I_{m+l} + MM^{-} - NN^{-}\,) =\min \{\, m +l, \ \ m + l + r(M) - r(A) - r(D) \},
\label{363}
\\
& \min_{M^{-},N^{-}}\!\!r(\, I_{m+l} + MM^{-} - NN^{-}\,) = m + l + r(M) - r[\, A, \, B \,] - r[\, C, \, D \,],
\label{364}
\\
&\max_{M^{-},N^{-}}\!\!r(\, MM^{-} - NN^{-}\,) = r[\, A, \, B \,] + r[\, C, \, D \,] + \min\{ \, 0, \ \ m + l - r(M) - r(A) - r(D) \},
\label{365}
\\
& \min_{M^{-},N^{-}}\!\!r(\, MM^{-} - NN^{-}\,) = r[\, A, \, B \,] + r[\, C, \, D \,] - \min\{ \, r(M), \ \ r(A) + r(D) \}.
\label{366}\end{aligned}$$
Setting the both sides of – equal to $m + l$ or zero will lead to a group of facts on the singularity, non-singularity, and equality of the corresponding matrix expressions. In addition, it is no doubt that the maximum and minimum ranks of the following matrix pencils $$\lambda_1 I_m + \lambda_2 AA^{-} \pm \lambda_3 BB^{-}, \ \ \lambda_1 I_m + \lambda_2 AA^{-}\pm \lambda_3 C^{-}C$$ with respect to the choices of $A^{-}$, $B^{-}$, and $C^{-}$ can be determined by a similar approach.
Conclusions
===========
We have established some general matrix rank equalities using the block matrix method, and presented many applications of these formulas in dealing with idempotent matrices and their operations. These findings seem exciting and can be selected as constructive issues in textbooks and handbooks on matrices and linear algebra. This kind of work also shows that there exist still many simple but valuable problems on fundamental objects in linear algebra for which we can make deeper exploration and find out various novel and intrinsic conclusions.
Idempotents are really simple and funny in algebras and seem to appear everywhere, so that they are prominent issues for consideration in the field of mathematics. It is expected that numerous analytical formulas can be established for calculating ranks of matrix expressions composed by idempotent matrices using various tricky constructions of block matrices, which we believe will greatly enrich the classic issue in linear algebra from the bottom level, and will also provide applicable tools to deal with various challenging problems in matrix analysis and applications. In addition to this miscellaneous work, it would be interest to consider performances of the sums and differences of several idempotent matrices, such as, $AA^{-} \pm BB^{-} \pm CC^{-}, \ \ [\, A, \, B \,][\, A, \, B \,]^{-} - AA^{-} - BB^{-},
$, etc. In these situations, analytical formulas for calculating the maximum and minimum ranks of $A - B_1X_1C_1 -B_2X_2C_2 - B_3X_3C_3$ will be used. However, it was noticed by the present author in [@Tian:2000] that there may exist no analytical formulas for calculating the minimum ranks of general LMVFs with three or more variable matrices. Thus the rank distributions of these matrix pencils will remain open before solving the minimum rank problems on general LMVFs.
It is well known that idempotents and generalized inverses of elements can be defined in general algebraic structures, which are also important tools in the approaches of the algebraic structures; see e.g., [@AFPR; @AL; @AJL; @AS; @And; @AAK; @Andr1; @Andr2; @ACo; @ACS; @AC; @ACG; @ACMa; @AMMZ; @ASS; @ASh; @BSa; @BBo; @BES1; @BES2; @BES3; @BES4; @BES5; @BSe; @Berg; @Bha1; @Bha2; @BV; @Bik1; @Bik2; @BHKP; @BGK; @BSS; @BS1; @BS2; @BS3; @BH; @BBa; @Bro; @Buc1; @Buc2; @BVa; @Cao; @CX; @CP; @CDu; @CGM; @CMa1; @CMa2; @CG; @CGi; @CM; @CMS1; @CMS2; @CD; @Daw; @Dol; @DRR; @Eck; @Ehr; @ERSS; @Emm1; @Emm2; @FD; @Fil; @Fit; @For; @GW; @HSa; @HO1; @HO2; @HK; @Hen; @HT; @How1; @How2; @HL; @IMO; @ITP; @Lja; @Kal; @Kaw; @KH1; @KH2; @Kol; @KCD; @KR1; @KR2; @KR3; @KR4; @Kir; @KS; @KRSa; @Krup; @KS1; @KS2; @KRSi; @Laf; @LP; @LMR; @Lin; @LW; @MOS; @MV; @Mus; @Neu; @Oml; @PRW; @Park; @PT; @Ped; @Pet; @PTu; @Pru; @Pta; @QFG; @Rab0; @Rab1; @Rab2; @RR; @RSa; @RSS; @Ran; @Rao; @RS; @RSh; @RSi; @Rui; @SZ; @ST1; @ST2; @Sch; @Se; @Sel; @Shc; @SG; @Spi; @Str; @TV; @Vid1; @Vid2; @WW; @WZ; @WDD; @Wei; @Wu; @XZ1; @XZ2; @Yos; @ZW; @Zel; @Zem; @Zuo1; @Zuo2; @Zuo3]. Thus it is believed that the preceding results and facts can be extended with some efforts to idempotents and generalized inverses of elements in other algebraic structures that are somehow close to the matrix case.
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- Creating and Restoring Wetlands: From Theory to Practice by Christopher Craft
At last! A book has been published that brings the ecological theory relevant to restoration of all major wetland types together with the practical, nuts-and-bolts aspects of planning and construction that are key to successful wetland restoration projects. I have taught a course on wetland restoration annually for almost a decade and have found it particularly challenging because there have been no books available that brought together both the theoretical, academic, research-based side with the practical planning, design, construction, and monitoring side. Thus, I would rely on a patchwork of theory-rich book chapters or journal articles written by academics, sections from design or construction manuals, reports written by practitioners or landscape architects, general restoration ecology or ecosystem restoration books, policy-based books, and wetland-type-specific books (see recommended readings below). This patchwork had the benefit of presenting a multitude of perspectives, but inevitably led to repetition of some concepts and lack of depth in others. No longer! Christopher Craft has distilled theoretical and practical knowledge from all major works relating to the creation and restoration of wetlands into a readable, standalone text that is equally valuable for teaching courses on wetland restoration as it is for science-based planning, building, and monitoring wetland restoration projects.
The book is logically organized in a way that proceeds from fundamental wetland science, through current theory and practice, and ends with issues relating to the future of wetland restoration. There are four parts: “Foundations,” “Restoration of Freshwater Wetlands,” “Restoration of Estuarine Wetlands,” and “From Theory to Practice.” Thirteen chapters of 20–35 pages each are distributed across the four parts, meaning that a majority of the book could be assigned and read during a semester-long course or read in entirety in a weekend or two. For those needing to find specific information quickly, each chapter has its own detailed table of contents. References are included at the end of each chapter, making it easy to find literature for each chapter topic. Creating and Restoring Wetlands is particularly well grounded in the scientific literature, including both classic and recent articles and books. The text is generally free of jargon, less formulaic, and more conversational than the writing often found in journal articles, making the book accessible to a wider audience than it might otherwise be.
The first part of the book, “Foundations,” provides the underlying scientific and practical knowledge needed to engage in wetland restoration. This includes: variables controlling wetland ecosystem structure and function, regulations pertaining to wetland definition and restoration, ecological theories applicable to restoration, and the restoration process in general. Although the book is strong on all of these topics, it excels in two areas in particular in this section—ecological theory and landscape placement of restored wetlands. Other works have discussed the many ecological theories that can pertain to restoration projects (in general and specific to wetlands), but Craft takes the clarifying step of distinguishing the theoretical concepts most directly related to restoration ecology (disturbance and succession theories) from the myriad other models that undoubtedly have relevance in some contexts but are less fundamental to understanding the ecological dynamics of restored ecosystems. Wetland restoration projects can be considered catastrophic disturbances of soil, vegetation, and hydrology, comparable to hurricanes, landslides, and floods. As such, much insight can be gleaned from the rich literature on disturbance ecology, dispersal, and colonization (including invasive plants and animals), and community and ecosystem development in naturally occurring ecosystems. Turning to the landscape perspective, it has long been recognized that the position of wetlands in the watershed influences their functions (e.g., water quality improvement and flood peak moderation), and that watershed characteristics (e.g., urbanization, agriculture) in turn affect the structure and function of wetland ecosystems. However, Craft’s chapter on landscape considerations for wetland restoration is a milestone in moving ahead the critical, but often neglected, science and practice of identifying landscape stressors and prioritizing wetland restoration in the... | https://muse.jhu.edu/article/648881/pdf |
Viewpoint: A Quasicrystal for Quantum Simulations
Solids have traditionally been divided into two opposing types: crystals, where the atoms form a long-range periodic pattern, and amorphous materials, where no long-range order of the atoms occurs at all. Somewhere in between are quasicrystals, which lack periodicity yet still display a form of long-range order. To understand this order-disorder frontier, researchers have studied both electrons in quasicrystalline solids, as well as atoms and photons in “synthetic” quasicrystalline lattices. A group led by Ulrich Schneider at the University of Cambridge, UK, has now introduced a new system into the mix—a Bose-Einstein condensate (BEC) of atoms in a 2D quasicrystalline optical lattice . Optical lattices are defect-free and may be engineered more easily than actual crystals, potentially allowing well-controlled simulations of quantum many-body phenomena in a range of new quasicrystal environments.
Mathematically, a periodic structure can have only two-, three-, four-, or sixfold rotational symmetry. But in 1984, Dan Shechtman and collaborators reported their discovery of tenfold rotational symmetry in the electron-diffraction pattern from aluminum-manganese alloys . Although this unusual rotational symmetry couldn’t have come from a crystal, it implied that the atoms in the alloy were ordered in some way. The researchers dubbed the solids quasicrystals, and it was later shown that these strange solids can produce diffraction patterns with other “disallowed” rotational symmetries, including fivefold and eightfold. Shechtman’s pioneering work was recognized with the 2011 Nobel Prize in Chemistry .
Quasicrystals have many other features that aren’t found in conventional solids. They are inherently self-similar, like fractals, meaning that patterns in their structure recur as one zooms in and out. This self-similarity can be seen in diffraction patterns, which exhibit peaks at arbitrarily small momenta. In addition, unlike the periodic (Bloch wave) states that exist incrystals, electronic states in quasicrystalline solids have a peculiar fractal structure. Quasicrystals can also be understood as the projection of a higher-dimensional “parent” crystal onto a lower-dimensional space. Studying these materials therefore provides a window into higher-dimensional phenomena . Thirty-five years after their discovery, quasicrystals continue to fascinate researchers , and they have been realized in many forms, such as photonic structures and twisted bilayer graphene .
An alternative platform for studying quasicrystalline behavior is an atomic gas confined in an optical lattice . Made by interfering two or more laser beams, the typical optical lattice is perfectly periodic. Quasiperiodic lattices can be made by superimposing two lattices whose periodicities are “incommensurate”—meaning their ratio isn’t a rational number. Both 1D and, more recently, 2D quasiperiodic lattices have been produced in this way, but such lattices did not exhibit the forbidden rotational symmetries of a true quasicrystal . A 1997 study succeeded in observing these features with an atom gas in a quasicrystalline optical lattice . But the proximity of the laser frequency to an internal atomic resonance resulted in strong atom-light scattering, which prevented the simulation of quantum many-body behavior.
In their new work , Schneider and collaborators opened a window into this physics. For one, they realized a 2D quasicrystal optical lattice tuned far from any internal atomic resonance, reducing problematic atom-light scattering effects. They also probed this lattice with a BEC, in which all of the atoms are in the same quantum state. They produced the lattice using a planar arrangement of four intersecting 1D optical lattices, with an angle of 45° between each lattice. By tuning the relative strength of each laser, the team could switch between 1D and 2D periodic potentials, and they could create 2D quasicrystals with eightfold rotational symmetry. In the experiment, the optical lattice was flashed “on” for a few microseconds in the same spot as a preformed BEC of potassium atoms. Over the course of this pulse, the researchers observed a series of momentum states emerging in the BEC as the atoms scattered photons from one of the beams of the lattice into another. Organized in a plot, these states formed a self-similar pattern analogous to the diffraction patterns measured by Shechtman et al. 35 years ago . The new pattern could be described as an infinite sequence of progressively larger octagons (Fig. 1).
Schneider’s team also showed that the time evolution of the BEC on the quasicrystal is quite distinct from that on a periodic lattice. In the quasicrystal, the fractal pattern of momentum states emerges as the atoms move to successively lower and more closely spaced momentum states through a sequence of small, photon-induced velocity kicks. The successive population of these smaller momentum states constitutes a quantum walk in the quasicrystal’s momentum space. But in a periodic lattice, these smaller momentum states are inaccessible.
In the present work, Schneider and colleagues interpret their 2D quasicrystal as an incommensurate projection of a 4D cubic lattice onto a 2D plane. The 4D character of this “parent” lattice is reflected in the properties of the BEC quantum walk, which demonstrates the potential of quasicrystalline optical lattices for simulating higher dimensions. Interesting possibilities include the exploration of the quantum Hall effect and other topological properties in 4D, where exotic new phenomena are expected.
Where else might these experiments lead? One avenue might be the study of a weakly interacting BEC in a quasicrystal, a system whose dynamics is expected to present hybrid features of disorder and periodicity. Another direction would be the investigation of strongly correlated behavior—like that described by Hubbard and spin models—in a quasicrystalline environment. With sufficiently deep potentials, one would realize these models with locally variable parameters and explore such behaviors as antiferromagnetism, Mott transitions, and Bose glass phases in the peculiar setting of a quasicrystal. In addition, the new platform might also provide alternative ways of studying the effects of quasiperiodicity on transport, such as anomalous diffusion and many-body localization.
This research is published in Physical Review Letters.
References
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- B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440, 1166 (2006).
- S. J. Ahn et al., “Dirac electrons in a dodecagonal graphene quasicrystal,” Science 361, 782 (2018).
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Directions: In essence, intertextuality is the idea that each text we read will be informed by the previous texts we read. The magic of Neil Gaiman’s Sandman series is that multiple references to mythical beings and other characters in the DC Universe are present. Therefore, those who have read other DC titles and ancient mythology will be informed by these experiences upon reading Sandman. However, we do not need to know about these guests in order to enjoy Gaiman’s story.
Becoming familiar with the guests in the tale, however, does add to the depth of the story being told. Choose a character from Sandman: Preludes and Nocturnes and conduct a little background research on the character (example characters below). In two to three paragraphs, describe your findings. (One paragraph for each myth associated with the object or being you choose. Example below:
Be creative. If, for example, you find out Morpheus is the God of dreams. Then Google search “God of Dreams” and see what other types of characters show up. Hypnos would likely show up, for example. Or Oneiroi. You could write a paragraph on Hypnos, one on Oneiroi, and one on Morpheus as he exists in Ovid’s Metamorphoses.
Because I used Morpheus as an example, he cannot be used for the exercise 🙂
ONCE A CHARACTER HAS BEEN USED, PLEASE DO NOT USE THE SAME CHARACTER (exceptions allowed if you submit within a minute or two of another person, under the premise that you both wrote about the same character around roughly the same time). | https://topcollegeanswers.com/discussion-board-6-intertextuality/ |
Scientists in Oregon have developed a technique that could lead to more accurate dating of historic earthquakes.
The work by researchers at Portland State University and the University of Oregon, published in the Bulletin of the Seismological Society of America, offers a solution to a common problem that arises when using radiocarbon dating to determine the past activity of faults.
“This is what people really want to know: how often do these things happen? And we’re dealing with really messy data to try to give clean numbers on earthquake timing and earthquake frequency,” said Portland State University Professor Ashley Streig, lead author on the paper.
Radiocarbon – or Carbon-14 – dating is one of the primary methods paleoseismologists like Streig use to determine the age of earthquakes.
When earthquakes are large enough, they can rupture the surface of the ground – causing cracks or ridges as the earth rises or falls. Hundreds or thousands of years later, these ruptures are often covered over with new sediment, preserving the record of the original earthquake.
By digging down into the earth at these fault lines, researchers can see a cross-section of sediment layers, including the layers that were shifted, warped or broken during the earthquake. The top of those broken layers would have been the exposed ground when the earthquake happened, and organic material in that layer can be dated with radiocarbon.
Charcoal – organic material burned by long-ago wildfires – is ideal because it is preserved over time. So in theory, the date of charcoal in that top layer is the date of the earthquake. But in reality, inaccuracies can arise.
One is that if that charcoal came from a tree, it might be much older than a pine cone found in the same layer. That’s because trees are time capsules in and of themselves.
Redwoods and Douglas fir trees that are commonly found from Northern California to British Columbia might live 500 years or longer. And every year they add another ring of growth. Each ring would register a different age through radiocarbon testing.
Imagine that one of those trees falls in the woods, starts decaying, and then is burned in a wildfire that happens the same year as an earthquake you want to date. The charcoal you’re trying to date could be from wood burned from the inner part of the tree – say when the tree was 30 years old. Or it could be made from burning wood created when the tree was 300 years old.
“You could sample a piece of charcoal from anywhere within that tree. You don’t know where you are,” Streig said. “That adds uncertainty to the actual recurrence estimates that we come up with for earthquakes.”
While working along the San Andreas Fault in California, Streig and her colleagues discovered a buried stump of logged redwood tree. The stump was buried right next to the spot where two major earthquakes had hit. They were able to determine when the tree was logged and consequently the age of each ring of the tree. With this information, they were able to calibrate the radiocarbon dates of the charcoal they tested and figure out the exact dates of the earthquakes.
“That’s the cool thing of [Streig’s] study is that she used the tree rings to… narrow the range of time in calendar years that that last event occurred,” said Andrew Meigs, a geology professor at Oregon State University.
Using this new method, her team calculated that previous age estimates for the earthquakes were too old by about a century. And the average recurrence rate of earthquakes in the region large enough to rupture the surface of the earth wasn’t every 250 years – it was closer to 150 years – a significant difference if you’re planning a development in the area.
“It’s important for us to understand what the faults are doing over time, how frequently they rupture. Those are all information that we can give to engineers and planners so that we can build a more resilient community,” Streig said.
Using tree rings to study conditions or events in the past – a field called dendrochronology – is not new. Scientists have even used tree rings to date earthquakes in the past – including the last major Cascadia subduction zone earthquake in the Pacific Northwest.
“If you try to go out and use radiocarbon to date the 1700 [Cascadia] earthquake, the radiocarbon is too messy,” Meigs, the OSU professor, said. “The trees of southern Washington, for example, informed [researchers] that the earthquake… had to have occurred between the growing season of the 1699 and 1700. It was the tree rings that narrowed it down to a six-month span.”
Streig’s new method could push the intersection of the two fields further.
“Our hope is that we can take sites where we have a lot of charcoal that’s been dated, and we can try to use the approach that’s presented in this paper to reduce the uncertainty on the age of these earthquakes,” she said.
Streig is now using her new technique try to figure out when the Mount Hood Fault and Gales Creek Fault near Portland were last active.
Copyright 2020 Oregon Public Broadcasting. To see more, visit opb.org
Related Stories: | https://www.nwpb.org/2020/05/27/oregon-scientists-develop-a-more-accurate-way-to-date-past-earthquakes/ |
In a new survey of over 1,000 patients from the Royal College of Physicians, three-quarters (76% - 767 of 1,028 total respondents) felt that their experience of falls prevention services had been positive.
However, among the 24% of respondents reporting a less than positive experience, the majority of issues were related to communication between patients and healthcare professionals, and many of the physical exercise programmes described in the results were not evidence-based, which limits their effectiveness.
Participants also felt that the exercise programmes they were offered did not continue for long enough, were not sufficiently frequent and that the location was not always convenient enough to be really beneficial.
The postal survey, commissioned by the Healthcare Quality Improvement Partnership (HQIP), carried out by the RCP’s Clinical Effectiveness and Evaluation Unit (CEEU) and supported by Age Concern and Help the Aged, showed that the majority of people were positive about their experiences of local falls prevention services.
Key findings:
At least 3 out of 4 people from the total sample group:
- knew why they had been referred to the service (95%)
- felt they had been seen quickly enough to help with their recovery (78%)
- felt they received a thorough health check up (85%)
- felt they had been involved in deciding what actions should be taken after being seen (80%)
- were fully satisfied that good communication took place (85%)
- found it easy or fairly easy to travel to appointments (76%)
- were asked about their experiences of losing their balance or falling (84%).
Overall experience of interaction with Falls Prevention Services:
When asked to describe their overall experience of using their local Falls Prevention Service, of 1,028 respondents, 767 (76%) stated this as ‘Useful for me’. Some 89 patients (9%) marked the ‘Quite useful but could be better’ option, 41 (4%) marked ‘Not useful for me’ and 111 (11%) did not respond.
Some 123 patients (just under 12%) responded when asked to explain why the service was not useful or could have been improved. The key theme identified by this group was poor communication, either between patients and healthcare professionals, or between different healthcare professionals. Examples include delays in referral, no estimate of waiting time once referral made, poor explanation of check up, and uncertainty about whether information about their attendance at the falls prevention service had been provided to their GP.
Of the 650 people in the survey participating in an exercise programme, 602 provided comments about what sort of exercises they were doing. These comments show that many people are continuing to participate in some form of exercise programme, but find self motivation to exercise at home is harder than when attending a class.
Many patients are limited as to what exercise they can do by factors such as pain, tiredness or other health problems, but would like more exercise classes, longer courses, options for home visits, follow up from courses, more local classes and help with transport.
It is of concern that many are participating in exercise programmes that have no evidence base for reducing risk of falls. This means they are not, for example, sufficiently specific, frequent, challenging or progressive. Evidence based programmes include Otago, FaME and Strength and Balance classes.
Recommendations on communication and exercise interventions:
The findings have led to new recommendations for falls prevention services on better communication, both verbal and written, at each stage of the process – assessment, investigation and intervention, including taking more account of individual concerns at each stage of the programme.
Falls prevention services should provide information about the specific types of exercise needed to reduce the risk of falls, to both professionals and the public, promote evidence-based courses and train health and exercise professionals to deliver these programmes. They should also provide education programmes to promote the benefits of healthy active ageing to help prevent the onset of falls and maintain the benefits from attending classes following a fall.
Jackie Riglin MCSP, Falls Prevention Coordinator and Clinical Specialist Physiotherapist, and RCP Clinical Associate for Falls, said:
The findings from this report raise important issues for improving the delivery of a timely, appropriate and effective local Falls Prevention Service. I do hope that both service providers and commissioners will take on board the recommendations around communication and provision of local, long term, evidence based exercise interventions.
Michelle Mitchell, Charity Director at Age Concern and Help the Aged said:
It’s well documented that falling can have a serious impact on an older person’s health and quality of life and falls prevention services play an important role in reducing the number of people who have this type of accident.
While it’s encouraging that most service users in this survey had a positive experience of fall prevention services, it does reveal there is room for improvement around better communication and exercise programmes.
Age Concern and Help the Aged urge local falls prevention services to take adopt the recommendations in this report and obtain feedback from their own service users to help identify areas for improvements.
Dr Jonathan Potter, Clinical Director, RCP Clinical Effectiveness and Evaluation Unit, said:
The Royal College of Physicians is delighted to present this report on the patients’ experience of falls services. While there are some good findings and some less good findings, the most important aspect is that the patients’ view is being heard.
Notes to editors
- The Royal College of Physicians of London provides a huge range of services to our 20,000 Members and Fellows and other medical professionals. These include delivering examinations, training courses, continuous professional development and conferences; undertaking clinical audits; publishing newsletters, guidelines and books through to maintaining the College's historical collections. We also lead medical debate, and lobby and advise government and other decision-makers on behalf of our members.
- HQIP was established in April 2008 to promote quality in healthcare, and in particular to increase the impact that clinical audit has on healthcare quality in England and Wales. It is led by a consortium of the Academy of Medical Royal Colleges, the Royal College of Nursing and National Voices (formerly the Long-term Conditions Alliance). HQIP is a not-for-profit organisation funded by the Department of Health.
- Age UK is the new force combining Age Concern and Help the Aged. We will be known by our new name from Spring 2010. The Age UK family includes Age Scotland, Age Cymru and Age NI. We will help people enjoy a better later life – here and in 70 other countries - by providing life-enhancing services and vital support. We will keep developing products that are specifically designed for people in later life. We will provide advice and information through our helplines, publications and online. We will fund pioneering research into all aspects of getting older. By joining forces, we will ensure more of our funds go where they’re needed. | https://www.rcplondon.ac.uk/news/falls-prevention-services-good-problems-remain |
Note: The term "controlled substance," as used in this policy, means a drug or other substance as defined in applicable federal laws on drug abuse prevention.
Any employee who violates any of these prohibitions will be subject to disciplinary action up to and including termination. Use of alcohol or other drugs before, during or after work may affect you performance on the job. Poor job performance, regardless of the cause, may lead to discipline, and ultimately, termination. Any employee convicted under any criminal drug statute for a violation occurring while on the job, on PVTA or customer premises, or in any PVTA vehicle must notify his/her manager, or PVTA's Human Resources Department no later than five days after such conviction.
PVTA recognizes that employees suffering from alcoholism and controlled substance dependence can often be successfully treated. PVTA encourages any employee with an alcohol or controlled substance dependency problem to voluntarily enter a controlled substance or alcohol rehabilitation program prior to any violation of this policy, and offers assistance through the Employee Assistance Program. However, where a violation of this policy has occurred, an employee's subsequent request to submit to a controlled substance or alcohol rehabilitation program will not prevent PVTA from taking appropriate disciplinary action.
All suppliers doing business with the Pioneer Valley Transit Authority are expected to develop and implement their own policies that are in compliance with this policy. | http://pvta.com/drugPolicy.php |
Working on Ecological Research at UQ
Opportunities to work as part of an ecological research group at UQ come up with some frequency. Such opportunities include positions as short- or long-term research or field assistants, lab managers, postdoctoral fellows and research scientists. All paid positions are posted through the UQ employment page.
Volunteering for Ecological Research projects at UQ
Volunteering can be a great way to gain research experience while studying abroad at UQ, while working full time on an undergraduate degree, or just as a way of getting involved in projects that excite or interest you. Many of the projects highlighted on the Ecology Centre website and many others run by UQ ecologists involve volunteers who periodically help with associated student projects and/or for data collection in the field or in the lab. We post some volunteer positions on this website but many opportunities come up suddenly in the diverse Centre-affiliated research groups. In addition to checking this website for opportunities, you can also contact members of the Centre directly about whether they or their group members need help from a volunteer in the near future.
Student Groups
The iROOS is a student-driven volunteer group of Environmental Science (BEnvSc) undergraduates from The University of Queensland (UQ), though students in other majors such as Bachelor of Science are also welcome. This organization developed out of one of the field-courses offered through the BSc Ecology major - BIOL3016: Outback Ecology, a course that takes student to Idalia National Park in central Queensland for an opportunity to study outback ecology first hand. iROOS developed with the goal of addressing conservation needs of Idalia NP and other rural Queensland parks. Members of this group work closely with UQ staff and the Queensland Parks and Wildlife Services (QPWS). They have, since the development of the group in 2007, been involved in a wide range of conservation, management and research projects based at Idalia National park, including: fresh water rehabilitation and protection projects, surveying the endangered Bridled Nailtail Wallaby, flora and fauna surveys, installing exclusion fencing to aid in the recovery of several endangered species and general park management. Students fund this exciting group through fundraising BBQs and generous gifts from private donors. To find out more about joining or supporting this student organization, please visit their official website.
Domestic & International Opportunities for Graduating Ecologists
Information on opportunites for all levels of graduate ecologists.
Mailing List
We maintain a mailing lists where we post opportunities and relevant marine news. Sign up for the Ecology Centre Mailing List today. | https://ecology.uq.edu.au/opportunities |
I always knew I came into this life ‘born awake’ but then ‘forced to forget’ for the times I was born. The beginning of a new era in consciousness on the planet, at the unique planetary configuration of 1966.
When I worked with the Beings of the Christ Light in a group setting, I was considered ‘young’ of the earthly age in our ‘ungroup’ that met and did this ‘Work’ together… I recently looked at the photos of many of the physical human beings who gathered then - often in numbers between 50-100 on a weekly basis… and I was impressed by the ‘shininess’ of the skin, eyes, and spirits that shone through the Presence of these individuals.. The entire story is mystical. I would say the one constant religion of my incarnation has been one of Synchronicity. I started intellectually ‘studying’ it on a conscious level at the age of 19. Each one of us is a unique prism of god-mythology. Each one of us is sent ‘signs’ … but if I put the synchoncities of how I met ‘The Beings’ in a screenplay, it would remind audiences of ET meets Powder… and all that fall into that genre. Literally mind-blowing and ‘magical’ on all levels.
I met so many people in the 1990s in the City of Angels that were guided by this magical thread of Synchronicity, but the crux - the plateau - of those encounters coalesced with The Beings. I had been studying A Course In Miracles as a ‘Teacher and Student’ for 3 years. I had been involved with Marianne Williamson and Agape (in their teenage ministry - wink)… And even before that, I had been doing Gestalt Therapy, Jungian personal analysis, and ‘inner child’ work based on the John Bradshaw philosophy of ‘healing one’s inner child’. I was living a life of very rapid spiritual, psychological, and emotional body EVOLUTION - but at the same time, like I said - it’s what I signed up for.
The shiny peeps I met through the Work of The Beings - we jokingly referred to as the ‘ungroup’ because indeed, it was — it IS — a Course in Mastery. That means, basically, it’s NOT about consensus reality. I was probably one of the first to ‘Graduate’ from our three-year experience together… because I actually ‘broke up’ with the ‘ungroup’ - and even with the Beings! My own inner “Christed Self “- had come to understand my own ministry, my own Inner Teachings… and I was able to actually Stand for my Self.. in this one ‘Intensive’ weekend they held. The Beings always said that was the goal. And I made it! So this is the opposite of a new religion, or a cult. IT’s LITERALLY about SELF MASTERY. That means to find your own unique pattern of God Self - and learn to Be and Live It!
I believe that is what I came here for. What about you? If you read this and you feel ‘holy bumps’ .. that Sense that you are ‘onto something’ in your own life’s Mystery… then perhaps you are the director and actress in your own “synchronicity movie” — and you may need to consider it’s time to “star in your own life” - and follow the Inner Call to be guided deeper and deeper into your Self. The only ‘teaching’ here, is about how to align one self with his/ her own Truth. However, there is a Presence - that One that was With you ‘Before the Before’ - you WILL feel it - via a sense, a sound, a smell, a vision… we all have a way of sensing… that is again UNIQUELY different .. So what I am asking you - is when you read these words, if you feel or sense this… then all I can say is this Work is for you! There are many that are called. But few are chosen, because few Choose. That’s the paradox. And the riddle. The Answer is in the Quest. You are ready.
I bless you on your journey, Beloved. The world awaits YOUR magical MOVIE! | https://www.innerattainmentmedia.net/jewels-solheim-full-testimony |
WP&UP is a mental health and well-being charitable organization founded by Dan Maby in 2018, whose mission is to promote positive mental health throughout the WordPress ecosystem.
The organization is currently hosting a survey to better understand the needs of the community.
The questions cover the general work environment, general mental health and well-being, and specific work-related mental health and well-being. The survey will close next week and the results will be anonymized, open-sourced, and shared with the community.
Would you like to write for WP Tavern? We are always accepting guest posts from the community and are looking for new contributors. Get in touch with us and let’s discuss your ideas.
Like this: | https://wpsensors.com/take-the-2019-wpup-mental-health-and-well-being-survey-wordpress-tavern/ |
Search for my Tongue
Sujata Bhatt's poem is about what it is like to live in a foreign country, feeling disconnected from your cultural background. The poet feels, at the start of her poem, that she has lost her original language now that she is living abroad.
The poem is also about the experience of colonialism and emigration.
The lost language can be seen as representative of the loss of a cultural heritage, of values and ways of thinking. The fact that Bhatt is Asian may suggest that she is referring to how the English colonised India, imposing laws and language.
The poem is exploratory; it is a form of thinking.By the end of the poem, the poet has changed her mind. In dreams, in thesub-conscious mind, the original language still lives. The imagery of 'your mother tongue would rot' in the first stanza is reversed in the last stanzaas the 'bud' of the mother tongue re-opens.
In the following tables are statements about the poem with supporting quotations...
Quote:
Statement:
'Search for my tongue'
The poet feels she has lost an important part of herself that she needs to recover to feel herself again.
'lost the first one, the mother tongue, and could not really know the other'
The original language is associated with being nurturing, protecting, loving. The second language is seen to be alien. The poet feels it is not possible to fully understand or become completely part of another culture.
'if you lived in a place you had to/speak a foreign tongue'
There is a suggestion that some people would not be allowed to speak their own language, that a foreign language has been imposed.
'You had to spit it out'
'The choice of words makes the original language sound like something disgusting, like phlegm perhaps. This may suggest the way the original language was seen by the foreigners as something inferior to their own language.'
'it ties the other tongue in knots'
The two languages are imagined as being in battle, with the invader being defeated. Clearly this can be seen to refer to colonial powers and independence.
'Every time I think I've forgotten it blossoms out of my mouth'
The first language/culture is shown to be lasting, and in the flower imagery is thought of here as something natural, beautiful and eternal.
Language
The language of the poem is ordinary and conversational 'You ask me','I ask you' - until line 12.
In the following lines, Bhatt uses a metaphor, comparing the mother tongue to some kind of plant, and by implication, the foreign language to some kind of weed or parasite.
This imagery, taken from nature, is picked up in the final stanza where therotten stump of the mother tongue is imagined to grow again into a flower.
Notice also the use of repetition in lines 31-32 & 34.
What is the point of the repetition here?
The language in the middle of the poem is Gujarati. It is given equal status to the English, as it is not translated. We can hear what it sounds like, but unless we speak some Gujarati we cannot know what it means.
In all poetry questions you will be asked to write about the following:
The subject(s) of the poem
The attitude of the poet
The poetic devices the poet uses
In other words, you will be asked about what the poet is writing about, what the poet feels about the subject(s) and how the poem is written.
Subject
The poem can either be simply read as being about one person's experience of moving to a foreign country and speaking two languages.
Or it can be seen as using language to represent the struggle of colonial experience. By trying to control the language colonisers tried to control the thoughts, feelings, values and ideas of the people they colonised. The poem shows how the native culture survives, at a deep level, and is able to overthrow in the end the invading culture.
Attitude
The attitude of the poet is difficult to read in this poem. Clearly there is a sense of wonder about the re-flowering of the native language, and the imagery of rotting suggest some of the pain of losing access to your mother tongue.
The poem however is quietly, subtly rebellious.
It celebrates the overthrowing of a foreign language/power and the resurgence of native language/culture.
Style
Bhatt uses a mixture of language in her poem. There is the conversational of the opening, the extended metaphor of language being like a plant, and there is also the use of Gujarati. This is a word that my computer doesn't even provide a spell-check for.
The language works by concentrating attention on the metaphor and by making us hear but not understand the Gujarati.
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The City of Richmond will mandate COVID-19 vaccines for its employees, Mayor Levar Stoney announced Wednesday, becoming the first locality in Virginia to implement the requirement.
The mandate applies to about 3,600 public employees, including police, fire, public works, social services and parks and recreation workers. It won’t, however, extend to roughly 500 constitutional officers — including the Richmond City Sheriff’s Office and Commonwealth’s Attorney’s Office — or Richmond Public School employees, which are governed by the city’s school board.
There is currently no mandate for state employees to be immunized, making Richmond’s the first requirement of its kind in Virginia. Gov. Ralph Northam is “actively considering” a similar policy, according to Dr. Danny Avula, the state’s vaccine czar. In late July, the Fairfax County Board of Supervisors also passed a motion asking the county executive to “explore” a vaccine requirement for public employees. Currently, though, the county has not implemented its own mandate.
“I think this is the right and proper step at a critical moment here in the summer,” Stoney said at a news briefing on Wednesday. “And I have great confidence that city employees will respond.”
The mayor estimated that roughly 40 to 50 percent of public workers are already vaccinated, similar to the overall immunization rate across the city. Employees are able to apply for medical or religious exemptions, but the city did not clarify the criteria they must meet to qualify for the waivers.
“Should a medical or religious exemption NOT be granted, such employees will be required to be vaccinated or will be subject to discipline in accordance with City of Richmond HR guidelines,” press secretary Jim Nolan wrote in a follow-up email on Wednesday. Workers who are granted an exemption will be required to wear a mask and submit to “regular testing.”
“Employees cannot voluntarily opt out of vaccination and just submit to mask wearing and regular testing as an alternative,” Nolan added. Already vaccinated workers will need to submit documentation to the city by August 18, according to Stoney. Unvaccinated employees are required to receive their first dose by the same date, and be fully vaccinated by October 1.
Richmond, like many Virginia localities, has struggled to boost lagging vaccination rates amid growing concern over the delta variant. Genetic sequencing from the state’s public health laboratory indicates that roughly 70 to 75 percent of new infections are linked to delta, said Avula, who is also director of the Richmond-Henrico Health District. That’s likely an undercount, since many samples are tested outside the public laboratory system.
Federal data suggests that the more transmissible variant now accounts for virtually all new coronavirus cases across the United States. But even as infections and hospitalizations climb, a significant proportion of the population remains unvaccinated. In Richmond, just over 50 percent of adults have been fully immunized. Statewide, it’s 65.3 percent.
“An important thing to note is that the vast majority of these cases, these hospitalizations and these deaths are happening among unvaccinated individuals,” Avula said. “And because of that, we are more and more convinced that our only path out of this is through vaccination.”
Boosting uptake, though, has been a continuing challenge for state and local officials. Virginia has invested more than $20 million in vaccine outreach, but demand has plummeted since the shots became widely available in April.
Throughout much of the last month, the average number of doses plateaued at less than 12,000 administered per day. Three months ago, the daily average was just over 86,000. There has been a slight increase over the last two weeks — fueling hope that the delta variant will drive up demand — but officials have struggled to dispel hesitancy among a significant portion of the population.
As some states continue to experience a surge in cases, vaccine mandates have become increasingly common. Some Richmond businesses, including the restaurants Godfrey’s and Longoven, have implemented their own requirements for staff and customers. New York City recently introduced a mandate for indoor dining and gyms, while California became the first state in the country to require vaccines for its own public employees and health care workers.
At the national level, President Joe Biden also implemented a similar policy for federal workers and contractors. While vaccinated people are capable of catching and transmitting the virus — a growing concern that led the U.S. Centers for Disease Control and Prevention to reimplement universal masking recommendations — immunizations are highly effective at preventing severe symptoms.
The growing number of serious infections among non-immunized individuals has led the CDC to describe the ongoing crisis as a “pandemic of the unvaccinated.”
“What we can all take away is that people who are fully vaccinated don’t need to be concerned about severe outcomes,” Avula said. “It’s very rare for that to happen. But people who are unvaccinated are going to be at much higher risk.”
Our stories may be republished online or in print under Creative Commons license CC BY-NC-ND 4.0. We ask that you edit only for style or to shorten, provide proper attribution and link to our web site. Please see our republishing guidelines for use of photos and graphics. | https://www.virginiamercury.com/2021/08/04/richmond-city-introduces-first-employee-vaccine-mandate-in-virginia/ |
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Nausea and vomiting in adults isn't usually a sign of anything serious and tends to only last 1 or 2 days.
Vomiting is the body’s way of ridding itself of harmful substances from the stomach, or it may be a reaction to something that has irritated the gut.
One of the most common causes of vomiting in adults is gastroenteritis. This is an infection of the gut usually caused by bacteria or a virus, which will normally improve within a few days.
However, vomiting can occasionally be a sign of something more serious, such as appendicitis, so if you are feeling very unwell or are worried about your vomiting, trust your instincts and call your GP.
There is a separate topic on vomiting in children and babies.
Complete this guide to assess your symptoms, and find out what to do if your condition worsens.
Your GP may want to investigate the cause of your vomiting or prescribe treatment.
You should also see your GP if you have diabetes and have been vomiting persistently, particularly if you need to take insulin. This is because prolonged vomiting can affect your blood sugar level.
Occasionally, vomiting can be a sign of a more serious problem.
You should also seek emergency medical help if you think you have swallowed something poisonous.
If you have diarrhoea as well as vomiting, it's likely you have gastroenteritis. This is one of the most common causes of vomiting in adults.
It's often the result of a virus picked up from someone who's ill, such as the norovirus, or food poisoning caused by bacteria found in contaminated food.
Your immune system will usually fight off the infection after a few days and the self-care measures described above can help in the meantime. Read more about treating gastroenteritis.
Pregnant women typically experience repeated episodes of nausea and vomiting, particularly during the early stages of pregnancy. This is often called "morning sickness", although it can occur throughout the day.
In most cases, morning sickness will develop at some point during the first 3 months of pregnancy and will pass by about weeks 16-20. Read more about morning sickness, including things you can do to help reduce your symptoms.
If you have recurrent episodes of vomiting along with intense, throbbing headaches that last for a few hours to days at a time, you may be experiencing migraines.
Conventional painkillers, such as paracetamol and ibuprofen, can sometimes help control the pain and your GP can prescribe anti-sickness medicine to help prevent vomiting. Read more about treating migraines.
If your vomiting is accompanied by dizziness and a feeling of spinning (vertigo), it may be caused by an inner ear infection called labyrinthitis.
Labyrinthitis will usually improve over a few days, and your GP can prescribe medication to reduce your symptoms if necessary.
Nausea and vomiting associated with travelling could be a sign of motion sickness.
These symptoms can sometimes be improved using techniques such as fixing your eyes on the horizon or distracting yourself by listening to music, although medication to prevent and treat motion sickness is also available.
As well as vomiting, appendicitis can cause severe pain in your abdomen (tummy). You should call 999 for an ambulance if you experience pain that suddenly becomes worse and spreads across your abdomen. These are signs that your appendix may have burst.
If you have appendicitis, you will often need surgery to remove your appendix. Read more about treating appendicitis.
Clink on the links above for more information about these conditions and treatments.
In most cases, you won't need any specific treatment and can take care of yourself at home until you feel better.
The most important thing you can do is to keep taking small sips of water frequently so you don't become dehydrated.
A sweet drink such as fruit juice can be useful for replacing lost sugar, although you should avoid sweet drinks if they make you feel sick. Salty snacks, such as crisps, can help replace lost salt.
You may also find ginger helps to relieve your nausea and vomiting. This is available as supplements, or can be found in ginger biscuits and ginger tea. Check with your pharmacist or GP before using ginger supplements. | https://www.nhsinform.scot/illnesses-and-conditions/stomach-liver-and-gastrointestinal-tract/vomiting-in-adults |
Ted Schuur has spent the better part of his career making the connection between climate change and wildfires that are burning an increasing amount of land in Alaska and in sub-Arctic and Arctic forests around the world. So the Northern Arizona University scientist wasn’t all that surprised this summer to find his field stations in the interior of Alaska surrounded by fires on three sides. At the time, the state was well on track to recording its second-worst fire season ever.
The surprise came in mid-summer when Schuur took a few days off from his research to attend a meeting in Colorado. He had hoped the trip would give him a break from Alaska’s noxious smoke. The smoke in Boulder, however, was so thick that the state’s Department of Public Health and Environment was advising parents with young children and people with heart disease and respiratory problems to limit their outdoor activities.
As Schuur soon learned, the pall of smoke in Denver had actually drifted down from a large number of forest fires in the Pacific Northwest and Canada. “I’ve never seen anything quite like this summer,” says Schuur. “It seemed like half the continent was on fire at one time or another.”
Schuur wasn’t exaggerating. In June, as many as 25,000 men and women were fighting thousands of wildfires that were burning out of control in states such as Alaska, Washington, California, and Idaho, and in the Canadian provinces of British Columbia, Alberta, and Saskatchewan. But as severe as the fire situation was this year in the Lower 48 and Canada, it was the really big fires in Alaska and Siberia that captured the attention of meteorologists and fire experts. Because of rapidly rising temperatures, and in some cases, inadequate or antiquated forest management practices, massive wildfires such as these are becoming increasingly common in the Arctic and sub-Arctic regions of the world, scientists say.
‘It seemed like half the continent was on fire at one time or another.’
According to a report recently released by the non-profit organization Climate Central, the number of large wildfires in the Arctic increased nearly tenfold in the first decade of this century compared to the 1950s and 1960s. Only three years in the 1950s and 1960s saw large wildfires in this region. Since 2000, there have been 33 large wildfires in the Arctic, including those in 2004 that burned six percent of Alaska and four percent of the Yukon.
Fire experts agree the situation will likely get much worse as rising temperatures, drier conditions, and increasing thunderstorms and lightning strikes lead to more fires burning even bigger, hotter, faster, and deeper into the ground than at any time in recorded history. According to the U.S. National Climate Assessment, the extent of area burned in Alaskan wildfires is projected to double by 2050 and triple by 2100 if greenhouse gas emissions are not quickly brought under control and runaway Arctic warming continues.
The largest fire on record in North America was the Chinchaga fire that raged in northern Canada for 222 days in 1950 and burned 3. 5 million hectares, or 8.6 million acres. But fire experts like Mike Flannigan of the University of Alberta say that we may see more than one fire burning in excess of 8 million acres in North America in the coming decades. “We’re headed into uncharted territory,” he says. “We’re exceeding thresholds all the time. We better start acting soon.”
The extent of area burned in Alaskan wildfires is projected to double by 2050 and triple by 2100.
The boreal forests of Alaska, the northwestern United States, Canada, Northern Europe, and Russia ring the treeless tundra regions south of the Arctic Circle and easily rival in size the rainforest regions of the world. Within this massive ecosystem — which comprises nearly 30 percent of global forest cover — thousands of rivers and lakes drain into a vast array of bogs, fens, and marshes. An estimated five billion land birds live or nest here.
Some scientists suggest that once the spruce and pines in the boreal forest burn, fire-resistant trees such as aspen will replace them. Flannigan, however, isn’t buying it. So long as there are higher temperatures, decreased precipitation, and more lightning strikes, he says, aspen will burn quite nicely. And once the aspen are gone, the shrubs that will replace them will also burn if they are dry enough.
Fire experts say that as wildfires increase in size and frequency, landscapes will be transformed, air quality will worsen, and wildlife as well as people living in rural settings will increasingly be put in harm’s way. Industry will also suffer, as tar sands operators in northern Alberta discovered this summer when fires interfered with their operations.
Several studies, including one published recently in the Proceedings of the National Academy of Sciences, say that boreal forests and Arctic regions are experiencing more extreme fires because of rising temperatures and changing precipitation patterns. For every degree of warming, there needs to be a 15 percent increase in precipitation to compensate for the moisture loss in the ground. For every degree of warming, however, lightning strikes increase while precipitation drops by about 10 percent. This happened in Alaska in June, when lightning ignited 50 fires a day for five days straight.
For every degree of warming, lightning strikes increase while precipitation drops by 10 percent.
“This is the most on record,” says Sharon Alden, weather program manager of fire-predictive services at the Alaska Interagency Coordination Center. “In 2004 there were only three days with more that 20 new [fire] starts and they were strewn out over the summer.”
With temperatures expected to rise by as much as 10 degrees F by the end of the century, boreal regions could see a 75 to 125 per cent increase in area burned, according to Flannigan. And with deciduous shrubs overwhelming grass and sedge cover on the tundra, Arctic tundra fires — such as the 2007 Anaktuvuk River fire, which accounted for more than half of all of the area that burned on Alaska’s North Slope since 1950 — will be more common, according to a recent report from the National Oceanic and Atmospheric Administration.
There is also the possibility, as Schuur suggests, that more of these fires will burn deep into northern peatlands, where they can smolder through the winter, potentially destroying seedlings and re-igniting once the snow melts in spring.
The prospect for bigger fires burning hotter and faster in the future is especially daunting when one looks at what happened in the 1950 Chinchaga fire. At one point, so much smoke filled the air over North America that flights in Canada and the United States had to be cancelled. In Buffalo, Pittsburgh, Cleveland, and many towns in New York state, it was so dark at mid-day that lights in baseball fields, including those at Yankee Stadium, had to be turned on. Smoke from the fire traveled all the way to Europe.
According to Flannigan and fire expert Cordy Tymstra, who recently wrote a book on the fire, Chinchaga changed the way we fight fires, which now heavily focuses on blazes that threaten people and infrastructure. But they also say that the time has come to rethink much of what we have learned since then because there will be many more fires, and more assets — small towns, oil fields, pipelines, and mines — in the line of fire when another huge blaze comes along. In many cases, traditional fire-suppression techniques will not be successful unless fire managers have maps and 3D models to determine how a fire will behave in a warmer world, and how that fire might affect ecological restoration if it is allowed to burn.
Larger and more frequent fires, logging, and disease are taking a significant toll on Siberian forests.
This is especially true in Siberia, where hundreds of people were killed or made ill by massive wildfires in 2012 and 2015.
There is a lack of reliable data to compare today’s high level of Siberian forest fires with the past, according to Dmitry Schepaschenko, a scientist with the Ecosystems Services and Management Program at the International Institute for Applied Systems Analysis in Austria. Heiko Baltzer, director of the Center for Landscape and Climate Research at the University of Leicester, is using remote sensing to measure and monitor fire and vegetation changes in Siberia’s boreal forest. While it is too early to say definitively that climatic warming is the cause of the vegetation changes — spruce replacing larch, and shrubs replacing spruce —Balzer says it is clear that larger and more frequent fires, logging, and disease are taking a significant toll on Siberian forests.
ALSO FROM YALE e360With Climate Change, Terrifying New Normal for Western Firefighters
Alexey Yaroshenko of Greenpeace Russia, says the situation in his country is even worse since the government began drastically reducing the number of people who were working in the forestry sector in 2006. Where there were once 200,000 people working on forest management and fire suppression in vast regions such as Siberia, there are now just 60,000, he says.
Sobering as future scenarios appear to be, Schuur suggests that an even more alarming development would take place if more fires begin to burn through layers of moss, leaves, and other organic materials that insulate permafrost from surface heat. That could release vast amounts of carbon, he points out, as there is twice as much carbon trapped in permafrost as in the atmosphere.
“As more and more of this carbon is released from the permafrost, temperatures are going to rise even faster,” he says. “Permafrost carbon may not be as visible as changes in sea ice, but you can bet that big changes are happening there and it’s going to have a big impact on the future.”
Correction, October 1, 2015: An earlier version of this story incorrectly stated that Colorado was among the states experiencing a large number of wildfires this June. | https://e360.yale.edu/features/a_new_global_tinderbox_the_worlds_boreal_forests |
For the preparation of an ideal material for the emergency disposal of Cr(VI)-contaminated water, a recyclable nanocomposite of molybdenum disulfide@hybrid acid-doped polyaniline immobilized on porous polyacrylonitrile nanofibers (MoS2@PANI/PAN nanocomposite) was synthesized successively through electrospinning, a hydrothermal process, and in situ polymerization. The advantages of the flowerlike MoS2@PANI/PAN nanocomposite include a fast equilibrium rate (30 min), a high removal capacity (6.57 mmol/g), and good recyclability. Thus, the material’s performance is much better than those of many earlier materials developed for similar applications. MoS2@PANI/PAN nanocomposite still showed high adsorption capacity toward Cr(VI) in the presence of coexisting anions and natural organic matter. The remarkable interaction of this material with Cr(VI) was shown to mainly occur through ion exchange, electrostatic adsorption, and reduction. Furthermore, the MoS2@PANI/PAN nanocomposite could be thoroughly regenerated and successfully reused for more than six cycles without a significant decrease in its removal capacity. Additional favorable properties of MoS2@PANI/PAN nanocomposite are its strong hydrophilic performance and high mechanical strength. Hence, MoS2@PANI/PAN provides a green, economic, and fast method to efficiently remove toxic Cr(VI) from aqueous environments. | https://acs.figshare.com/articles/journal_contribution/Recyclable_Nanocomposite_of_Flowerlike_MoS_sub_2_sub_Hybrid_Acid-Doped_PANI_Immobilized_on_Porous_PAN_Nanofibers_for_the_Efficient_Removal_of_Cr_VI_/5684455/1 |
The reigning Super Bowl champion Kansas City Chiefs head to the Mile High City for a Sunday afternoon showdown with the Denver Broncos in Week 7.
The Chiefs opened as 9.5-point favorites at SportsBetting.com, but with a few factors playing in their favor, it's the Broncos who could have the edge this weekend.
Fans or no fans, Broncos still have home-field advantage
A hostile crowd typically comes to mind when discussing home-field advantage, but even with a limited number of fans in the stands, playing in Denver still comes with a unique challenge for visiting teams due to one natural factor:...
4 Broncos players made PFF's Team of the Week
Following an 18-12 win over the New England Patriots in Week 6, the Denver Broncos had four players make the ProFootballFocus.com Team of the Week -- two on offense and two on defense.
Left tackle Garett Bolles, right guard Graham Glasgow, defensive lineman Shelby Harris and edge defender Bradley Chubb made PFF's TOTW.
Bolles and Glasgow led an offensive line that helped running back Phillip Lindsay rush for 100 yards for the sixth time in his career (Denver is now 6-0 when Lindsay rushes for at least 100 yards in a game).
Harris (91.8) and Chubb (91.1) received...
Broncos DL Shelby Harris 'probably the best' ball-batter Vic Fangio has coached
Denver Broncos defensive lineman Shelby Harris has perfected the art of knocking down passes at the line of scrimmage. Against the New England Patriots last week, one of the passes Harris deflected was intercepted.
“He’s really good at it," Broncos coach Vic Fangio said Monday. "He’s probably the best I’ve been around in that regard. I think the thing with Shelby is that he’s playing a lot better too this year. His overall game is better and when your overall game is better, you’re going to get more opportunities to bat some balls.
"Yesterday’s batted ball that DeShawn got the... | https://broncoswire.usatoday.com/2019/03/22/c-j-anderson-feels-disrespected-about-still-being-a-free-agent/ |
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