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- Support the daily delivery of the service and utilising advanced knowledge and skills with collaboration with medical colleagues, ensure that the clinical requirements of the patients are achieved. - Assess, plan, implement and evaluate care for patients with complex needs and make changes as necessary based on evidence to support the clinical outcome in collaboration with the inter-disciplinary team. - Prepares for and assists with surgical/anaesthetic procedures and interventions using knowledge, skills and experience. - Lead by taking appropriate action in complex critical clinical situations, collaboratively making decisions with the inter-disciplinary team as appropriate - Regularly monitor the quality of care delivered and ensure any recommendations/actions are implemented with further evaluation to achieve agreed standards in conjunction with the Senior Sister and Mtron. - Ensure that all peri-operative documentation-including electronic records, is accurate, legible and complete and adheres to Trust standards. - To provide clinical leadership and develop support mechanisms for sharing and developing good practice Trust wide. If you are a registered band 6 Nurse, send your CV or contact to Email: Telephone: extension 1206 Reference: 36148011 Bank or payment details should not be provided when applying for a job. reed.co.uk is not responsible for any external website content. All applications should be made via the 'Apply now' button.Report this job Replace a job alert Replace a job alert Get Job Alerts straight to your inbox "Office Assistant jobs in London" Your Job Alert has been created and your search saved.	https://www.reed.co.uk/jobs/band-6-scrub-practitioner/36148011?source=searchResults
Buying modern pieces of furniture pieces on the internet or in store is now as popular as it features have you ever been. But just before an individual go store shopping this will be important to research the marketplace to determine the ideal locations to locate often the latest pieces of furniture pieces in addition to effectively update the particular residences interior. 1 Comments Log in to comment or register here.	http://bookmark2020.com/story.php?title=what-you-should-expect-in-the-best-current-furniture-stores
China's tech giant Tencent announced it will invest 500 billion yuan (about 70.11 billion U.S. dollars) in the next 5 years to develop new infrastructure across the country. The funds will be mainly invested in cloud computing, artificial intelligence, blockchain, supercomputer center, quantum computing and data center, said Tang Daosheng, senior executive vice president of Tencent. To spur the innovation of industrial technologies, the firm will also strengthen the development of industrial internet base, innovation center and industrial parks, he said. The Shenzhen-based enterprise will fully leverage resources from leading experts and laboratories and collaborate with top universities at home and abroad to establish research platforms while enhancing industrial research and talent cultivation. Tang called for joint efforts by online and traditional businesses, government departments, research institutions, nonprofit organizations and users to build a community of digital ecology in a bid to put the industrial internet sector into the fast lane. China has pledged fresh efforts to advance the construction of new infrastructure including next-generation information networks and 5G applications, according to an annual government work report.	http://www.china.org.cn/business/2020-06/01/content_76112428.htm
The communications industry is rapidly changing to adjust to emerging technologies and ever increasing customer demand. This customer demand for new applications and increased performance of existing applications is driving communications network and system providers to employ networks and systems having greater speed and capacity (e.g., greater bandwidth). In trying to achieve these goals, a common approach taken by many communications providers is to use packet switching technology, particularly ATM switching technology. Consumers and designers of these systems typically desire high reliability and increased performance at a reasonable price. A common technique for helping to achieve these goals is for these systems to provide multiple paths between a source and a destination. It is typically more cost-effective to provide multiple slower rate links or switching paths, than to provide a single higher rate path. Packets belonging to a packet stream are then distributed (e.g., multiplexed) among multiple paths at a source point. These distributed packets are transported across multiple links and then typically merged back into a single stream of packets at a destination point. One such mechanism for merging these packets into a packet stream is described in “Inverse Multiplexing for ATM (IMA) Specification Version 1.1,” Document No. AF-PHY-0086.001 (Final Ballot—Draft #1), December 1998, hereafter referred to as the “IMA Specification.” An example of such a technique extracted from the IMA Specification is illustrated in FIG. 1A, in which a stream of ATM cells is input into an IMA group device, distributed across three physical links, and merged back into a the original stream of ATM cells. As with most communications devices, there is always a potential for an error on a link or within some other component of the communications system. Once such transient error condition that may occur is an out of IMA frame (“OIF”) anomaly. The IMA Specification provides a state diagram for when a particular link should transition between an IMA working state, an OIF anomaly, or a loss of IMA frame (LIF) defect state. This state diagram reproduced herein in FIG. 1B. The IMA Specification further provides that: (a) on a given link, the IMA receiver shall pass to the ATM layer from the IMA sub-layer any cells accumulated before the occurrence of an OIF anomaly on that link (R-117); (b) the IMA receiver shall pass from the IMA sub-layer to the ATM layer no cells received on a link during an OIF anomaly condition reported on that link (R-118); and (c) the IMA receiver shall replace with Filler cells all ATM layer cells received on a link after an OIF anomaly condition has been detected on that link (R-120). Needed are methods and systems for appropriately handling OIF anomalies.
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Cannavino Library LibGuides Subject Guides Modern Languages & Cultures Subject Guide Reference Search this Guide Search Modern Languages & Cultures Subject Guide: Reference A guide for researching in Spanish, French, Italian, German, Japanese, and more Home Journals Reference French Resources Italian Resources Spanish Resources Resources for Arabic, German, Russian, & More International Newspapers International Films Video Tutorials Plagiarism MLA Citation ConnectNY & Interlibrary Loan Ask-A-Librarian Modern Languages and Culture General Reference Encyclopedia of World Literature in the 20th Century Publication Date: 1998 Covering aspects of literature of the 20th century, this reference title includes more than 2000 entries on authors, 50 national surveys that review the literature of countries around the world and 50 additional entries that discuss topics, such as genres, movements and trends. A Historical Companion to Postcolonial Literatures - Continental Europe and Its Empires by Prem Poddar (Editor); Lars Jensen (Editor); Rajeev Patke (Editor) Publication Date: 2008 Your complete reference to the postcolonial literatures of Continental European Empires. Written by expert scholars in the fields of postcolonial studies, the entries cover major events, ideas, movements and figures in postcolonial histories. They highlight the relevance of colonial histories to the cultural, social, political and literary formations of contemporary postcolonial societies and nations.By outlining the historical contexts of postcolonial literatures, the companion unlocks contemporary debates about race, colonialism & neo-colonialism, politics, economics, culture and language. Literary Movements for Students: Presenting Analysis, Context, and Criticism on Literary Movements by Ira Mark Milne Entries provide in-depth historical background information on each movement as well as modern critical interpretation of each movement's characteristic styles and themes. The Princeton Handbook of World Poetries by Roland Greene (Editor); Stephen Cushman (Editor) Publication Date: 2016 An authoritative and comprehensive guide to poetry throughout the world The Princeton Handbook of World Poetries--drawn from the latest edition of the acclaimed Princeton Encyclopedia of Poetry and Poetics--provides a comprehensive and authoritative survey of the history and practice of poetry in more than 100 major regional, national, and diasporic literatures and language traditions around the globe. Worldmark Encyclopedia of Cultures and Daily Life by Gale Cengage Learning (Editor) Publication Date: 2017 "Covers history, politics, customs, religion, education, human rights issues, rites of passage, and much more for 533 diverse cultural groups in Africa, the Americas, Asia and Oceania, and Europe" Nations of the World by World of Information (Editor) Nations of the World contains profiles of countries with political, economic and business information. The Statesman's Yearbook by Barry Turner (Editor) Call Number: Reference JA51.S7 The Statesman's Yearbook offers detailed information about countries from culture and climate to natural resources and industries. Use these links to find reference books for a specific language French Italian Spanish Resources for Arabic, German, Russian, & More Online Reference Sources Britannica Academic Britannica’s contributors include an extensive network of renowned scholars (including former U.S. presidents and Nobel and Pulitzer Prize winners), advisers, content specialists, and writers—whose job it is to ensure that Britannica is current, accurate, unbiased, comprehensive, relevant, international in scope, and engaging to college-level learners, researchers, and faculty. Credo Reference Provides full-text online access to hundreds of multidisciplinary reference book collections, including art, history, law, medicine, psychology, technology, bilingual dictionaries and encyclopedias. Gale eBooks Encyclopedias, almanacs, and specialized reference sources in the fields of business, education, geography, history, law, literature, medicine, religion, science and the social sciences. Project Gutenberg Online Catalog 42,000+ free ebooks in multiple formats. All books are published prior to 1923. Many foreign language books available. Dictionaries Encyclopedia of Contemporary French Culture by Alex Hughes Call Number: eBook Publisher: Taylor and Francis More than 700 alphabetically organized entries by an international team of contributors provide a fascinating survey of French culture post 1945. Essential cultural context for students of French, Modern History, Comparative European Studies and Cultural Studies. Encyclopedia of Latin American Literature by Verity Smith Call Number: Reference PQ7081.A1 E56 1997 A comprehensive, encyclopedic guide to the authors, works, and topics crucial to the literature of Central and South America and the Caribbean. The encyclopedia also stresses the contribution made by women authors and by contemporary writers. Encyclopedia of Contemporary Spanish Culture by Eamonn Rodgers Call Number: eBook Some 750 alphabetically-arranged entries provide insights into recent cultural and political developments within Spain, including the cultures of Catalonia, Galicia and the Basque country. Coverage spans from the end of the Civil War in 1939 to the present day, with emphasis on the changes following the demise of the Franco dictatorship in 1975. The Oxford Companion to Italian Literature by Peter Hainsworth (Editor); David Robey (Editor) Call Number: eBook Embracing the whole of Italian literature, from the early thirteenth century to the present, this work takes a broad view of what constitutes literature, covering historical writing, travel writing, theatre, and philosophy as well as the novel, poetry, literary dialogues, and critical theory. Biblioteca virtual Miguel de Cervantes Under the auspices of the Foundation, a unique university and private sector (Banco Santander) collaboration, the BVMC has developed through sustained commitment over many years an outstanding corpus of high quality and heterogeneous digital resources of Hispanic culture, supporting global research and academic studies, as well as serving as a rich, cultural resource accessible across the world. ICAA Documents of 20th & 21st Century Latin American and Latino Art Provides access to primary sources and critical documents tracing the development of twentieth-century art in Latin America and among Latino populations in the United States. Encyclopedia of Diderot & d'Alembert This site has been designed to make accessible to teachers, students, and other interested English-language readers translations of articles from the Encyclopédie edited by Denis Diderot and Jean le Rond d'Alembert in the 18th century. Italian Women Writers The Italian Women Writers project (IWW) is a long-term research endeavor to preserve and provide access to an extensive corpus of literature written by Italian women authors. IWW includes authors from the beginning of Italian literature in the late 12th and 13th centuries up to authors born in 1945. The following types of works are included: anthologies, articles and essays, autobiographies, biographies, children's literature, devotional works, dialogues, diaries, dramas, epics, hagiographies, histories and chronicles, interviews and conversations, letters, memoirs, novels, operas, poems, reviews, short stories, and travel literature. Collins Italian Dictionary by HarperCollins UK Staff; HarperCollins Publishers Ltd. Staff Call Number: eBook Publication Date: 2005 Up-to-date coverage of today's language Offers over 40,000 entries and 70,000 translations Easy-to-use format Contains commonly used phrases and idioms Main irregular verb forms given Includes most common abbreviations, acronyms, and geographic names Pronunciations for English and Italian shown in the International Phonetic Alphabet Collins Spanish Dictionary by Collins Publishers Staff Call Number: eBook Publication Date: 2005 The Collins Spanish Dictionary gives you comprehensive coverage of both Spanish and English and the most up-to-date business, political, and Internet terms. Native Spanish and English speakers worked side by side to create a balanced treatment of both languages and to ensure authentic and appropriate translations. Collins French Dictionary and Grammar Call Number: eBook Publication Date: 2014 An up-to-date dictionary and a user-friendly grammar guide in one handy volume. Colour headwords, cultural notes and an easy-to-use grammar section make this the ideal book for intermediate learners. Designed for all intermediate learners of French, whether at school, at home, or for business. 96,000 references and 136,000 translations will help those learning French take their language skills to the next level. Collins German Dictionary by HarperCollins Publishers Ltd. Staff Call Number: eBook Publication Date: 2007 The richest resource for German study Comprehensive and authoritative, including the latest words and phrases from contemporary German and English Key phrases, idioms, and set grammatical structures are highlighted to help you understand more complex entries Culture boxes explain the origins of phrases from literature, film, and popular culture to aid translation and to improve your understanding << Previous: Journals Next: French Resources >> Last Updated: Jan 14, 2022 9:02 AM URL: https://libguides.marist.edu/modl Print Page Login to LibApps Report a problem. Subjects: Modern Languages Contact James A. Cannavino Library 3399 North Road Poughkeepsie, NY 12601 (845) 575-3106 About Cannavino Clarion Archives Disclaimers Staff Only Give us your feedback © Copyright 2007-2020 Marist College. All Rights Reserved.	https://libguides.marist.edu/c.php?g=87317&p=562716
In this episode, I interview Cassandra Falke, professor of English Literature ad UiT, The Arctic University of Norway, about her book The Phenomenology of Love and Reading (Bloomsbury Publishing, 2016). In the text, Falke situates herself within the current revival of the interest in ethics in literary criticism, which coincides with a rise in neuroscientific discoveries about cognition and emotion that similarly have been incorporated into literary studies. Aware of these recent developments, Falke argues that literary study must ground itself philosophically—rather than just scientifically—in order to speak convincingly about literature’s relationship(s) to our ethical lives. To do this, The Phenomenology of Love and Reading recasts French philosopher Jean-Luc Marion’s articulations of a phenomenology of love onto the event of reading. The Phenomenology of Love and Reading accepts Jean-Luc Marion's argument that love matters for who we are more than anything—more than cognition and more than being itself. Falke shows through deft readings of both philosophical and literary texts, as well as ruminations on the experience of reading, how the act of reading can strengthen our capacity to love by giving us practice in love´s habits—attention, empathy, and a willingness to be overwhelmed. Confounding our expectations, literature equips us for the confounding events of love, which, Falke suggests, are not rare and fleeting, but rather constitute the most meaningful and durable part of our everyday life. Britt Edelen is a Ph.D. student in English at Duke University. He focuses on modernism and the relationship(s) between language, philosophy, and literature. You can find him on Twitter or send him an email.	https://newbooksnetwork.com/the-phenomenology-of-love-and-reading?token=npMxf5PNcRcMAz0SxI3mq-KKMr40eILk
Computer Music and GACSS In addition to my works of cut/paste computer music, instrumental compositions, and works for low trumpet, I have also composed a number of pieces for computer music alone or computer music with a solo or ensemble. For a full listing of my musical works, including a link to download them all as MP3s, visit my compositions page. Introduction When I started working with computer music around 1990, the technology was quite different than what we have available today. Computers didn’t come with sound cards, there was no SuperCollider or Max/MSP, and disk space to store created sounds was extremely limited. To engage with the medium, I got my start working in the UIUC Computer Music Project, a lab which provided a home-built digital-to-audio converter (DAC), and a Music-V type language to work with called Music 4C. I authored (coded) various instruments for Music 4C and used them to create new works, including “Uninduced Approximation.” By 1992 a sound card was “affordable” for a typical PC, and I wrote a grant to fund development of a new original software package called GACSS (Genetic Algorithms in Composition and Sound Synthesis). This grant was funded by the UIUC Campus Research Board, and thus began a many year path of creation and collaboration. My collaborator on the grant was Zack Browning, one of my composition teachers at the time. Zack Browning used GACSS in the creation of new works for many years, and produced a CD called Banjaxed. The works on this disc are all for instrument and computer-generated sounds, and the sounds are almost entirely created using GACSS. I highly recommend you get a copy of this disc and check it out. GACSS (Genetic Algorithms in Composition and Sound Synthesis) GACSS is a software package I wrote that allows the composer to graphically visualize sound at the period level and turn this visualization into sound events (waveforms) with variance over time. Periods are designed in terms of length (pitch), curvature, and complexity. A number of breakpoints are specified, and the types of curves between those breakpoints can be set (e.g. linear, exponential, or inverse exponential). To create an event using that period, GACSS then repeats the created period until the duration has been achieved. However, that period is not copied verbatim, but is instead modified on each repetition based on the ‘trans’ value. The trans value defines the window in which temporal transformations of the waveform at the period level can drift from the waveform’s center. One can think of this as a degree of noisyness. The higher the trans, the more the nosie. The principal relies on the fact that as long as a number of successive periods are similar enough, they will be percieved as belonging to a single pitch, even if they change over the course of that event. The more distorted the period becomes from the original, the more the sounds becomes noise. Genetic Algorithms (GAs) were chosen as a method for timbral search. The number of parameters available to the composer can produce an infinite number of possibile sounds. A GA was added to help the composer find timbres of interest within this ‘search space’ of possible sounds. Why GACSS Isn’t a MIDI Synthesizer GACSS can be used to generate both pretty and ugly sounds. But one thing it doesn’t do is produce sounds that mimic traditional instruments. This is by design. Others use technology to recreate those things we already have. I want to use technology to create new sounds and visuals that we can’t create otherwise. I like to find what the machine is good at and exploit that instead of trying to get it do what I’m already good at. lotted ebb mix whit Over the years of 1992-1998 I composed a number of works using GACSS, both for sounds alone (we called it ‘tape’ back then), and for instruments with sound. lotted ebb and mix whit are solo tape pieces and were composed specifically for premieres at the 3:2 New Music Festival in New York City. You can read more about these works on my cut/paste computer music page. If But Or If But Or is another tape piece and was composed for the 3:2 New Music Festival in Wesleyan Connecticut, and was my first work created using GACSS. If But Or was issued on a CD titled waveFORMation, Electronic Music Studios CD EMS 9700, 1997. Not Pitch Not Pitch was written for North Carolina School of the Arts Professor and saxophonist Taimur Sullivan. The work, for baritone saxophone and computer-generated sounds was premiered by Taimur at the Settlement Music School in Philadelphia, PA. In recent years, New Mexico State Professor of Saxophone Rhonda Taylor has toured the work around the United States as part of her 2009-2010 concert season. On March 2, 2012, Rhonda released a new recording of the work on her CD titled Interstice. This is available on iTunes, Amazon, and CD Baby. Epistatic Niche Epistatic Niche uses ideas from natural genetics and the workings of genetic algorithms as a compositional model. The work is for trumpet, piano, percussion, and computer-generated sounds. It was written for the UIUC Contemporary Chamber Players’ 1994 Southeastern United States tour. It’s most recent performance was in 2004 at a Faculty Composer’s Concert at the Krannert Center for the Performing Arts.	https://bengrosser.com/projects/computer-music-gacss/
“Dirty dozen” exceeds 24 in number…. Persistent organic pollutants (POPs) “Dirty dozen” are the first set of Persistent organic pollutants (POPs) banned by the Stockholm convention in 2004. Ever since 14 other chemicals have been added to the list. The committee is now reviewing 5 other chemicals to be added, which increase the list up to a total of 31 POPs. Namely; - Aldrin - Chlordane - DDT - Dieldrin - Endrin - Heptachlor - Hexachlorobenzene (HCB) - Mirex - Toxaphene - Polychlorinated biphenyls (PCB) - Polychlorinated dibenzo-p-dioxins (PCDD) - Polychlorinated dibenzofurans (PCDF) - Chlordecone - Hexabromobiphenyl - Pentachlorobenzene - Lindane - Alpha hexachlorocyclohexane (α-HCH) - Beta hexachlorocyclohexane (β-HCH) - Tetrabromodiphenyl ether and pentabromodiphenyl ether (commercial PentaBDE) - Hexabromodiphenyl ether and heptabromodiphenyl ether (commercial OctaBDE) - Perfluorooctane sulfonate (PFOS), its salts, and PFOSF - Endosulfan - Hexabromocyclododecane (HBCD) - Hexachlorobutadiene (HCBD) - Pentachlorophenol (PCP) - Polychlorinated naphthalenes (PCNs) Chemicals under review to be included in the list; - Decabromodiphenyl ether (DecaBDE) - Short-chain chlorinated paraffins - Hexachlorobutadiene (HCBD) - Dicofol - Perfluorooctanoic acid (PFOA) Why POPs are so significant? Well, these chemicals are “persistent” in the environment. In other words, once released, it will never leave or decay from the environment. Also, they are soluble in organic fluids like oils, fats, and liquid fuels. This character helps them to bio accumulate and become long range travelers. A POP released from Sri Lanka can end up in Antarctic! They tend to bioaccumulate and bio magnify. Which means it travels through food chains and concentrate towards the end. As the figure shows, it enters the food chain through grass and ends up in the infant who feeds on mother’s milk. Just look at the concentration of the chemical.. its low in grass, little high in cow and highest in the infant. There are hundreds of researches that show what these chemicals can do to human, to animals, plants, fungus, algae , bacteria and other organisms in the environment. Ingestion (consumption through mouth), inhalation or absorption through skin may cause developmental defects, cancers, endocrine disruption within reproductive system, central nervous system or immune system. Yet, there are countries still thinking whether to ban or not these chemicals in their land. Some countries have accepted the ban (ratified the Stockholm convention), yet import PCB contaminated transformers, burn POPs contaminated articles in open and use POPs contained or contaminated pesticides, fungicides, etc… It’s pathetic. But; There are things each of us can do to reduce the emission and contamination of POPs. As; - Stop open burning of household waste. There are number of items that can have POP chemicals in them. For example, plastics, PVC, electrical cables, textile, leather, carpets, rubber products, paper packages,.. etc. Thus it is not safe to burn anything in open air. Best thing would be to separate your household waste and submit to recycling centers. Specially electronic items and other non degradable waste. - Reduce application of pesticides/ fungicides/ weedicides / insecticides and other pest controls available over the counter. There are natural substances you can use instead or repellents that can save your cultivation/ business. - Be cautious of what you eat and drink. Even if someone has applied pesticides in your vegetables, you can avoid eating whole amount of those chemicals. Just; - Wash your vegetables thoroughly with flowing water - Avoid eating the peels of vegetables and fruits whenever possible - Allow veggies and fruits to be in open air for sometime before you store them in the fridge. This can help to get rid of volatile residues in them - Try gardening.. at least leafy vegetables can be supplied from your own garden. 4. Even if your meat contains POPs, you can avoid total ingestion by; - Avoiding fatty parts like skins of meat and fish - Avoid eating the gut, gills and heads of fish. Because the adipose tissues in head and other parts involved in food consumption, stores most of the fat soluble pollutants as well as heavy metals. like to read more? here are some links; - The POPs- Stockholm convention - Health effects of POPs - “POPs” by WHO - Sources of POPs - POPs- Air pollution Information System, UK - Sources and Pathways of Persistent Organic Pollutants– IW:LEARN - Sources of by-product POPs and their Elimination by Darryl Luscombe and Pat Costner, Greenpeace International Toxics Campaign, May 2001 - Persistent Organic Pollutant - Persistent Organic Pollutants list - Chemistry of Persistent Organic Pollutants - POPs affects on women – Persistent Organic Pollutants and Early Menopause in U.S. Women, Natalia M. Grindler,Jenifer E. Allsworth,George A. Macones,Kurunthachalam Kannan,Kimberly A. Roehl,Amber R. 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USDA to increase Annual Fee effective 10/01/14 The USDA has announced that there will be an upcoming increase to their Annual Fee. This marks the second time they have increased the Annual Fee since the Annual Fee was first introduced in 2011. Currently the annual fee equals .40% of the mortgage balance. Starting 10/01/14, the Annual Fee will increase to .50% of the mortgage balance. USDA Mortgage Annual Fee The USDA Annual Fee is similar to mortgage insurance. This fee is collected from all borrowers to help the USDA remain financially solvent and to offset potential foreclosure losses. Cost of the USDA Mortgage Annual Fee The annual fee is calculated against the principal balance of the mortgage each year and then this total is charged as a monthly amount. This monthly amount is calculated against the balance annually, so the monthly amount will decrease for payments 13 through 24. The annual fee would then be calculated against the balance after payment 12 and then after payment 24 and then after payment 36 and so on. Gradually this monthly amount decreases with time as the principal balance of the account decreases. Example: a USDA Mortgage with an initial balance of $153,000 would have an initial payment towards the annual fee of $63.28 per month ((153,000 x .05) / 12) . At the end of each calendar year, this annual fee would be recalculated based on the principal balance.	https://www.pennsylvaniausdamortgage.com/news/usda-to-increase-annual-fee-effective-100114
Pressure ulcers create physical restrictions for patients which impose lifestyle changes as well. For example, one’s living arrangements must be altered to accommodate the presence of a pressure ulcer. As a result of these restrictions, social and psychological implications arise. Research has found that physical limitations may severely restrict an individual’s social life, sometimes resulting in social isolation. Some have expressed that their relationships have been impacted. Those with pressure ulcers are often very dependent on others for care, and are affected by the perceived impact these pressure sores impose on others. For example, sufferers develop a fear of becoming a burden upon others, and develop preoccupations regarding the financial impact the pressure ulcer may have on their families. Many have expressed that they have been emotionally or psychologically embarrassed to due odors which stem from the wound, and this has also led to social isolation as a result. In addition, many have felt embarrassed requiring assistance in changing of dressings, and this has impacted relationships and quality of life as well. Psychological implications also arise when individuals develop a preoccupation with the presence and healing of the sore. This, in turn, may impact an individual’s own perception of their body image. When one sustains an injury or wound, particularly one that is visible, it can result in emotional and psychological trauma, including feelings of shame and embarrassment. This is particularly likely in cases of patients suffering from stage IV pressure ulcers, which not only cause pain, but usually also expose muscle and bone. Because pressure ulcers are highly preventable with proper care, monitoring, and treatment, health care providers must remain vigilant in preventing their development. This is not only to prevent accompanying physical pain and discomfort, but to avoid the distressing social and psychological effects that arise as a result of the development of pressure ulcers as well. It is much easier to prevent a pressure ulcer from forming in the first place, as treating an already developed pressure ulcer can be problematic. There are many preventative measures that can be taken, ranging from position changes to proper nutrition. If you believe your loved one is suffering from a pressure ulcer, it is important to ensure that your loved one’s nursing home facility is accurately assessing the wound for infection and providing proper care. The experienced elder abuse attorneys at the Law Offices of Ben Yeroushalmi firmly believe that the formation of preventable pressure ulcers is completely inexcusable, and will fight relentlessly for your loved one’s care and rights. Contact us today at (888) 606-3453 for a free consultation. We are located in both Northern and Southern California and serve cities throughout the state.	https://www.californianursinghomeabuselawyer-blog.com/exploring-the-social-and-psychological-effects-of-pressure-ulcers/
1. Field of the Invention The present invention relates to a diffraction type optical pickup lens and an optical pickup apparatus using the same. In an optical pickup apparatus commonly usable for two or more kinds of optical recording media, in which light beams having respective wavelengths different from each other are employed as irradiation light for the optical recording media depending on the kinds of optical recording media, the diffraction type optical pickup lens can favorably focus these light beams onto their corresponding recording media. 2. Description of the Prior Art In recent years, various kinds of optical recording media have been under development, and optical pickup apparatus which can carry out recording and reproducing while using a plurality of kinds of optical recording media in common have been known. For example, a system which carries out recording and reproducing of DVD (digital versatile disc) and CD-R (recordable optical disc) by using a single optical pickup apparatus has been known. In such two kinds of optical recording media, for example, visible light at about 650 nm is used for DVD in order to improve the recording density, whereas near-infrared light at about 780 nm is required to be used for CD-R since it has no sensitivity for light in the visible region. An optical pickup apparatus which can be used in common for both of them is based on a dual-wavelength beam type which uses two light beams having wavelengths different from each other as irradiation light. In the case where the disc thickness or numerical aperture differs between the above-mentioned two kinds of optical recording media, however, it is necessary for such an optical pickup apparatus to have different focusing actions for the respective wavelengths of light for carrying out reproducing or recording. For responding to such a requirement, a system in which two objective lenses having respective focusing actions different from each other are made exchangeable depending on optical recording media for carrying out reproducing or recording has been known. However, it complicates the structure of the optical pickup apparatus, while opposing demands for making it compact and lowering its cost. As an apparatus satisfying such a requirement, one in which one surface of a convergent lens having aspheric surfaces on both sides is provided with a zonal diffraction grating centered at the optical axis (Japanese Unexamined Patent Publication No. 2000-81566). Since diffracted light on the same order is utilized as effective recording/reproducing light with respect to two wavelengths of light, this diffraction type lens does not function as a diffraction grating having a wavelength selectivity. Therefore, in the diffraction type lens disclosed in this publication, the degree of freedom in setting the focusing position may greatly decrease, so that there may be cases where two luminous fluxes having respective wavelengths different from each other are hard to converge at positions different from each other in practice. On the other hand, a diffraction type lens in which two zone plates having a wavelength selectivity acting on only their corresponding wavelengths of light are formed on respective sides of a flat glass sheet on the light source side of an object lens has already been proposed in commonly-assigned applications Japanese Unexamined Patent Publication Nos. 2001-272516 and 2001-272517). The commonly-assigned technique mentioned above is quite excellent in that two wavelengths of light can easily be focused onto optical recording media having respective NA and thickness values different from each other by using zone plates having a wavelength selectivity. Since the flat sheet itself has no refracting power, however, it is necessary that a diffraction grating acting to converge a parallel luminous flux incident on the flat sheet bear all the focusing functions and aberration-correcting functions. Therefore, even when a certain degree of interval can be taken as a grating pitch in the vicinity of optical axis, the grating pitch may become smaller in the vicinity of marginal areas, whereby it is not so easy to process gratings, such as stepped gratings in particular. Though this may not be a very big problem if the technique for processing fine gratings improves as a matter of course, there is an urgent demand for finding measures against this problem at least at this point of time in view of the time and cost required for processing diffraction gratings.
When you think about your day as a whole, how much time do you spend thinking positive thoughts? What about negative thoughts? And have you ever wondered how these two different lines of thinking are affecting your well-being and your ability to cope with your lung disease? Chronic obstructive pulmonary disease (COPD) is an unbelievably complex disease. Like we’ve discussed in previous posts, it’s a “systemic disease’” meaning it can have manifestations in other areas of the body. So, we can’t even begin to imagine all of the ways it affects our physical and mental health. But despite the complexity of COPD, seemingly small and insignificant things can have a considerable impact on the outcome of the disease. For example, getting on your feet and moving around for just a couple of minutes extra each day could reduce your risk for heart disease and stroke, while helping your body manage hypertension, muscle pain, and stiffness, all of which are common in COPD. What’s more, making small adjustments to your diet like reducing sugar intake and consuming more protein and healthy fats can also have a big impact on your disease outcome. While you may think of “positivity” as something that only impacts your happiness, we’re going to show you in the following sections how it’s inextricably linked to the outcome of your disease and how just a few adjustments to your thought process can make your life a lot easier. As always, if you have any questions for us, feel free to leave them in the comments below or reach out to us via phone or email. Positivity Promotes Productivity While you may think of COPD patients as mostly retired people, there is a surprising amount of them who still work part-time or full-time jobs. According to a study published in the European Respiratory Journal, it’s common for COPD patients to miss work due to exacerbations, recurring lung infections, or simply feeling burnt out from dealing with their disease. Another thing to consider is the types of jobs that people work. Someone who works an office job might be less inclined to miss work than someone who works outside in the sun or in a factory where air pollution can lead to respiratory exacerbations. Equally important to productivity in the workplace is productivity at home. Many COPD patients have obligations at home such as cleaning, paying the bills, and taking care of their children or grandchildren. Completing these obligations can make you feel satisfied and whole so it’s best to stay positive and focus on what you’re trying to accomplish rather than focusing on how challenging it can be with a respiratory disease. But at the same time, everyone has their limits, so you should know what they are and be sure not to cross them. Positivity Results in Better Decision Making Good decision-making is paramount to COPD management. Ultimately, your decisions will mean the difference between managing your respiratory symptoms effectively or letting them run their course. For example, on any given day, you have the opportunity to use your medication as it’s prescribed or deal with the side effects of using it improperly. While it may be tempting to increase your dosage if you feel that it isn’t helping, a better decision would be to connect with your doctor first to see if it will have any unintended consequences. One of the terms you might hear thrown around in the COPD community is something called “shared decision making.” In short, this refers to healthcare professionals working one-on-one with patients to make decisions about the patient’s health. In other words, the doctor isn’t simply telling the patient what to do. The patient is playing an active role in their own health decisions. This benefits the patient because they feel like their personal needs are being met and it benefits the doctor because he/she can learn more about what’s important to COPD patients. Positivity Prevents Anxiety and Depression Anxiety and depression are two of the most common mental health conditions. It’s estimated that around 18.1% of the population or 40 million people in the United States have anxiety. What’s more, these mental health conditions are significantly more common among COPD patients than in the general population. According to this study from the European Respiratory Journal, COPD patients are 85% more likely to experience an anxiety disorder when compared to healthy control subjects, and studies regarding depression have shown similar results. There are several reasons for the high rates of mental illness among COPD patients. One possible reason is that COPD patients spend more time thinking or worrying about their health. From making it to the doctor’s office to taking medication, exercising, and maintaining a strict diet, your illness is something that’s always top of mind. As symptoms escalate, you’ll likely put even more of an emphasis on your health and well-being. Another reason COPD patients have higher rates of mental illness is due to the physical effects of the disease. COPD is known to cause breathlessness, fatigue, and chronic pain, and rapid changes to your diet and sleep routine due to flare-ups and exacerbations can leave you feeling irritable and groggy. Studies have shown that COPD can even affect our hormones and brain chemistry which can manifest itself as behavioral changes. This is why treating your underlying disease is not only important for your physical health, but your mental health as well. Positivity Improves Systemic Health Believe it or not, having a positive attitude can also have beneficial effects on our systemic health. This study found that emotional well-being improves recovery time and survival after physical illness. Other studies have found that positive attitudes in older adults result in a stronger immune system. This is essential for people with COPD because a poor immune system can lead to infections, the primary cause of COPD exacerbation and hospitalization. Several other health-related benefits of positive thinking include lower blood pressure, a lower risk of heart disease, easier weight management, and healthier blood sugar levels. Weight management, in particular, is an issue that many COPD patients face because they expend a lot more energy and burn more calories than they used to, meaning they tend to be underweight. However, the opposite can also be true. Some people tend to “stress eat” when they’re experiencing anxiety or unhappiness, which can lead to unhealthy weight gain. Positivity Leads to More Social Support Social support is extremely important for people with chronic conditions. Between daily conversations with friends and family and interactions with caretakers and healthcare providers, it’s your social support system that is keeping you motivated. If your disease progresses, you may find yourself relying even more on those around you. There’s nothing wrong with this, of course, but you should take care to foster positive relationships early on so that you have that extra support later in life. If you think about it, you probably enjoy spending time around people who are positive and uplifting, so it’s only natural that others would feel the same. What this means is that, despite how difficult the challenges of COPD may become, it’s always beneficial to remain positive and to spread that positive energy to other people. One of the ways many patients like to spread positivity is through online social platforms like COPD360Social. This is a patient engagement platform that’s hosted by the COPD Foundation and allows COPD patients to discuss treatment options and coping mechanisms for their disease. Positivity Helps You Stay in Control “Control” is a very important topic to discuss when you have a chronic disease. Many people who have been diagnosed with a chronic illness suddenly feel like they have lost control of their lives and that their disease now defines who they are and what they are capable of. However, when you look at the big picture, you start to realize that this is not the case at all. Many of the world’s greatest achievers have had some form of chronic illness and the thing that sets them apart the most is that they were able to adapt and overcome the challenges that they were presented with. None of this would be possible without having control. The first step to maintaining control of your life despite your COPD diagnosis is understanding that there is no “right way” to do something. For example, you might be someone who makes use of mobility aids like canes, walkers, manual scooters, or electric scooters. But you shouldn’t feel guilty or self-conscious about using these things even when people around you are not. Instead, you should feel overjoyed knowing that you’re doing what you can in order to stay active and maintain your independence. Another important note about control is that you likely have a lot more of it than you believe. While the lung damage caused by COPD is not reversible, your lungs are not the only things that affect your breathing. Your breathing is also affected by your fitness level, your diet, heart health, blood circulation, and many other things. So, if you want to stay in control, focus on things that you can change like eating right, staying active, drinking plenty of water, keeping up with your oxygen therapy, and visiting the doctor regularly. Once you realize that you have control over all of these things, you’ll be a lot happier and more positive about your condition. The alternative to being “in control” is having a “lack of control.” If you take on the mentality that you’re not in control, you might begin to believe that your disease, healthcare providers, friends, or family members are responsible for your happiness and well-being. And well you should certainly rely on these people for help, it’s ultimately up to you to make a difference in your own life and do what’s necessary to improve your health. How to Increase Positivity in Your Life While negativity has a huge impact on the outcome of your respiratory disease, the good news is that there are many different ways to promote positive thinking in your life. The first and possibly most effective method is something called cognitive behavioral therapy (CBT). This is a form of psycho-social intervention that’s used for a number of different purposes including treating alcohol and drug abuse, eating disorders, marital problems, and mental illness. Recently, it’s come to light how effective CBT can be when it comes to providing coping skills to people with chronic conditions like COPD. CBT is based on the premise that mental health conditions like anxiety and depression are caused by inaccurate thinking patterns and patterns of unproductive behavior. CBT aims to identify these things and take steps to reverse them and replace them with productive and positive thoughts and behaviors. During CBT, you will work one-on-one with a psychologist who will guide you through the process of correcting these things. The ultimate goal is to teach you the skills you need to be able to correct them on your own without the help of a professional. Conclusion COPD is one of the leading causes of morbidity and mortality worldwide. It’s estimated that around 65 million people in the world have been diagnosed with COPD and it affects some 16 million people in the United States alone. While these numbers may be shocking, it helps to know that many of these people have gone on to live long and happy lives by making healthy lifestyle choices like exercising, eating a well-balanced diet, and following their doctor’s instructions when it comes to oxygen therapy and medication. Modern medical research has proven beyond a shadow of a doubt that mental well-being also plays a role in our overall health. People who embrace a positive line of thinking are better equipped to deal with anxiety and depression and in turn, they can mitigate many of the risks associated with COPD such as heart disease, high blood pressure, and stroke. Taking steps to reverse negative thinking can have many beneficial effects in both the short and long term. For many people, cognitive behavioral therapy is the preferred method for accomplishing this. Here at LPT Medical, we want to make it as easy as possible for COPD patients to lead happy and productive lives. That’s why we offer lightweight and reliable portable oxygen concentrators like the Caire FreeStyle Comfort and the Inogen One G5. Unlike oxygen tanks, these devices will allow you to travel wherever and whenever you please. And since they’re so easy to use, you’ll be able to focus more of your attention on living your life rather than worrying about whether or not your oxygen needs are being met. To learn more about portable oxygen concentrators, reach out to us either by phone or email.	https://blog.lptmedical.com/how-positive-thinking-can-improve-outcomes-for-copd-patients
--- abstract: 'For any graph $G$ with $a,b\in V(G)$, a shortest path reconfiguration graph can be formed with respect to $a$ and $b$; we denote such a graph as $S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths from $a$ to $b$ in $G$ while two vertices $U,W$ in $V(S(G,a,b))$ are adjacent if and only if the vertex sets of the paths that represent $U$ and $W$ differ in exactly one vertex. In a recent paper \[Asplund et al., Reconfiguration graphs of shortest paths, Discrete Mathematics 341 (2018), no. 10, 2938–2948\], it was shown that shortest path graphs with girth five or greater are exactly disjoint unions of even cycles and paths. In this paper, we extend this result by classifying all shortest path graphs with no induced $4$-cycles.' author: - \| John Asplund\ [Department of Technology and Mathematics,]{}\ [Dalton State College,]{}\ [Dalton, GA 30720, USA]{}\ [[email protected]]{}\ \ Brett Werner\ [Department of Mathematics,]{}\ [University of Colorado, Boulder,]{}\ [Boulder, CO, 80309, USA]{}\ [[email protected]]{}\ \ bibliography: - 'vdec.bib' title: 'Classification of Reconfiguration Graphs of Shortest Path Graphs With No Induced $4$-cycles\' --- =8.5truein =11truein Introduction ============ In reconfiguration problems, the objective is to determine whether it is possible to transform one feasible solution into a target feasible solution in a step-by-step manner (a reconfiguration), such that each intermediate solution is also feasible. Such transformations can be studied via the reconfiguration graph, in which the vertices represent feasible solutions and there is an edge between two vertices when it is possible to get from one feasible solution to another in a single application of the reconfiguration rule. Many types of reconfiguration problems have been studied with drastically different reconfiguration rules: vertex coloring [@BJLPP; @BC; @CJV; @CJV2; @CJV3], independent sets [@HD; @IDHPSUU; @KMM], matchings [@IDHPSUU], list-colorings [@IKD], matroid bases [@IDHPSUU], and subsets of a (multi)set of numbers [@EW]. This paper focuses on the reconfiguration of shortest paths in a graph. The shortest path graph (SPG) of a graph $G$ with respect to $a,b\in V(G)$ is a graph where each vertex corresponds to a shortest path in $G$ from $a$ to $b$, or an $\mathit{(a,b)}$-geodesic, and two vertices in the SPG are adjacent if and only their corresponding $(a,b)$-geodesics in $G$ differ at exactly one vertex. Kamiński, Medvedev, and Milanič [@KMM] showed that a family of graphs whose size is linear in $k$ has diameter of the reconfiguration graph that is $\Omega(2^k)$. That is, as the size of a graph increases, the diameter of the SPG of $G$ can be exponential. Relatedly, Bonsma [@B] showed that the question of determining if there is a path in a SPG between all pairs of vertices is PSPACE-complete. For these reasons, we suspect characterizing the remaining SPGs will likely be difficult, yet important. Any progress in the direction of characterizing these graphs is worthwhile as mentioned in [@KMM]. Recent studies in the area of reconfigurability have shown an emerging pattern where the most “natural” problems (e.g., finding a spanning tree in $G$) can be done in polynomial-time and its reconfigurability problem (e.g., finding a spanning tree in the SPG of $G$) can also be done in polynomial-time. But since it was shown in [@KMM] that the reconfigurability problem (finding a shortest path between any two vertices in the SPG of $G$) is NP-hard while the “natural” problem is in P (finding a shortest path between any two vertices in $G$), we believe investigation into when it becomes NP-hard is worthwhile. Asplund et al. [@AAEHHNW] showed that cycles are central to characterizing SPGs. We denote a cycle of length $k$ as a $k$-cycle or as $C_k=(v_1,v_2,\ldots,v_k)$. One of the main results of that paper was the classification of all SPGs with girth at least $5$. This paper also established that induced $4$-cycles are extremely prevalent in SPGs and the structure of SPGs containing an induced $4$-cycle can be rather complex. In this paper, we continue investigating the structure of SPGs leading to a classification of all SPGs that do not contain an induced $4$-cycle. This paper is organized as follows. In Section \[notation\], the necessary notation and terminology are introduced, and several results from [@AAEHHNW] that are necessary to prove the results in this paper are given. In Section \[prelims\], there are some preliminary results that will simplify the main results of the paper. The main result of this paper is found in Section \[girth3\], where all SPGs that contain a $3$-cycle, but no induced $4$-cycles are characterized. This result along with the girth $5$ result from [@AAEHHNW] classifies all SPGs with no induced $4$-cycles. The constructions described in Section \[girth3\] will be stronger than is needed to prove the main theorem. In fact, these constructions can be used on a number of SPGs to build larger SPGs. Notation, Terminology, and Previous Results {#notation} =========================================== As discussed in the introduction, the focus of this paper is on a specific class of reconfiguration graphs: shortest path graphs. Let $G$ be a graph with distinct vertices $a$ and $b$. The shortest path graph (SPG) of $G$ with respect to $a$ and $b$, denoted $S(G,a,b)$, is a graph where each vertex corresponds to a shortest path in $G$ from $a$ to $b$, and two vertices $U,W \in V(S(G,a,b))$ are adjacent if and only if their corresponding paths in $G$ differ in exactly one vertex. All graphs considered in this paper are simple. A shortest path from vertex $a$ to vertex $b$ will be called a shortest $(a,b)$-path or an $(a,b)$-geodesic. To reduce notational confusion when discussing vertices in $G$ versus vertices in $S(G,a,b)$, we will adopt the convention of using lower-case letters to denote vertices in $G$ and upper-case letters to denote vertices in $S(G,a,b)$. Furthermore, for convenience, a vertex $U$ in $V(S(G,a,b))$ will be referred to as both a vertex in $S(G,a,b)$ and as an $(a,b)$-geodesic in $G$ when needed. The context will help distinguish to which situation we refer. If ${\mathcal{H}}$ is a SPG where $S(G,a,b)={\mathcal{H}}$ for some graph $G$ and vertices $a,b\in V(G)$, then we say that $G$ is a base graph of ${\mathcal{H}}$. In our classification of SPGs, it will be necessary to identify forbidden induced subgraphs of a graph. Given a graph $G$ and $M\subseteq V(G)$, the subgraph of $G$ induced by $M$ is denoted as $G[M]$. If $G$ and $H$ are graphs, then $G$ is said to be $H$-free if no induced subgraph of $G$ is isomorphic to $H$. A graph with girth $g$, contains a cycle of length $g$, but does not contain a cycle with length smaller than $g$. Two pivotal concepts used throughout this paper are index levels and difference indicies. Let $G$ be a graph and let ${\mathcal{H}}= S(G,a,b)$ where the distance from $a$ to $b$ in $G$ is $n+1$. Then, $(a,b)$-geodesics in $G$ have the form $av_1\ldots v_{n}b$, so we will say that $(a,b)$-geodesics in $G$ have $n$ index levels, and vertex $v_i$ is at index level $i$ in the $(a,b)$-geodesic. Note that if $v=v_i$, for some $i$ with $1 \leq i \leq n$, then $v$ can only appear in $(a,b)$-geodesics at index level $i$. So, the index level of $v$ is defined to be $i$. For the sake of convenience, we will say index level $i$ in a graph $G$ with indicated vertices $a$ and $b$ to be the same as the $i^{\rm th}$ index level of an $(a,b)$-geodesic. Given two $(a,b)$-geodesics $U$ and $W$ in $V(S(G,a,b))$, we define the difference index set of $U$ and $W$, denoted as ${{\rm{diff}}}(U,W)$, as the set of all index levels of the vertices where $U$ and $W$ differ. Note that $U$ and $W$ are adjacent if and only if $\|{{\rm{diff}}}(U,W)\| = 1$. If $UW \in E(S(G,a,b))$, the single index level in ${{\rm{diff}}}(U,W)$ will be called the difference index of $UW$. Much of the work done in this paper revolves around cliques in SPGs. The following theorem provides necessary and sufficient conditions for a SPG to be a clique. \[completeGraph\] [[@AAEHHNW]]{} $S(G,a,b)=K_n$, for some $n\in {\mathbb{N}}$, if and only if each pair of $(a,b)$-geodesics in $G$ differs at the same index. Extending Theorem \[completeGraph\] slightly, we see that for any maximal clique ${\mathcal{K}}$ in a SPG, the difference indices of any two edges in ${\mathcal{K}}$ are the same. Out of convenience, we say the difference index of ${\mathcal{K}}$ is defined to be the common difference index of the edges of ${\mathcal{K}}$. Another important observation is that when a $4$-cycle is present in an SPG, the difference indices on the edges of that $4$-cycle must alternate between the same pair of difference indices. \[4cycle\_ob\] There are exactly two distinct difference indices among all the edges of any induced $4$-cycle in a SPG and those difference indices must alternate as one traverses the edges of the $4$-cycle. The following proposition from [@AAEHHNW] will also be useful. \[manybasegraphs\] [[@AAEHHNW]]{} If ${\mathcal{H}}=S(G,a,b)$ and $d_G(a,b) = n$, then for any $n'\geq n$ there exists a graph $G'$ with vertices $a, b'\in V(G')$ such that $d_{G'}(a, b') = n'$ and ${\mathcal{H}}\cong S(G',a,b')$. One of these properties is that SPGs are $C_5$-free. Another—pivotal for characterizing all SPGs with girth $5$ or more—is that if a SPG contains an induced claw, then it must also contain an induced $4$-cycle. \[noClaw\] [[@AAEHHNW]]{} The claw, $K_{1,3}$, is not a SPG. Furthermore, if a SPG ${\mathcal{H}}$ has an induced claw, then ${\mathcal{H}}$ has an induced $4$-cycle containing two edges of the induced claw. It was also shown in [@AAEHHNW] that if a SPG has an induced $C_k$ for odd $k>5$, then the SPG must contain an induced $4$-cycle. \[oddtoC4\] [[@AAEHHNW]]{} If a SPG ${\mathcal{H}}$ has an induced $C_k$ for odd $k > 5$, then ${\mathcal{H}}$ has an induced $4$-cycle. There were multiple ways in which [@AAEHHNW] showed how new SPGs could be created from two existing SPGs. One of these is the disjoint union. That is, ${\mathcal{H}}_1\cup {\mathcal{H}}_2$ is a SPG if ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ are SPGs. \[disconnect\] [[@AAEHHNW]]{} If ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ are SPGs, then ${\mathcal{H}}_1\cup {\mathcal{H}}_2$ is a SPG. Structural Results {#prelims} ================== To begin characterizing SPGs with $3$-cycles but no induced $4$-cycles, we first analyze some structures that are forced by requiring no induced $4$-cycles and identify some substructures that are forbidden in SPGs. \[3cycle\_adj\] Let ${\mathcal{H}}$ be a SPG containing a $3$-cycle $(U_0,U_1,U_2)$. If $U \in V({\mathcal{H}})\setminus\{U_0,U_1,U_2\}$ with $U \sim U_0$, then either $U$ is adjacent to both $U_1$ and $U_2$ or neither of them. By Theorem \[completeGraph\], we can assume that $U_0$, $U_1$, and $U_2$ pairwise differ in a single index level $i$. If $UU_0$ has difference index $i$ then $U\sim U_1$ and $U\sim U_2$. If $UU_0$ has difference index $j$ where $i\neq j$ then $U\not\sim U_1$ and $U\not\sim U_2$. There are other restrictions when examining the characteristics of SPGs. If $e$ is an edge in a graph $G$, then we denote the graph $G$ with the edge $e$ removed as $G-e$. \[k4-e-free\] SPGs are $(K_4-e)$-free. (See Figure [\[forbiddenGraphs\]]{}.) We say a subgraph $H$ of $G$ is a maximal clique if $H$ is a clique and the induced subgraph of $V(H)\cup\{v\}$ in $G$ is not a clique for all $v\in V(G)\setminus V(H)$. A quick corollary of Proposition \[3cycle\_adj\] which is useful for Lemma \[characterCliques\] is given below. \[atMostOne\] Let ${\mathcal{H}}$ be a SPG and let ${\mathcal{K}}$ be a maximal clique in ${\mathcal{H}}$. Then each vertex in $V({\mathcal{H}})\setminus V({\mathcal{K}})$ is adjacent to at most one vertex in $V({\mathcal{K}})$. The following lemma is pivotal to the constructions in Sections \[girth3\]. In particular, this result asserts there are limitations when it comes to the parts of the SPG which have maximal cliques. Let $E(G_1,G_2)$ be the set of edges joining a vertex in $G_1$ with a vertex in $G_2$. Note that depending on the definition of matching used, conditions $(i)$ and $(ii)$ in Lemma \[characterCliques\] could be merged together if matchings are allowed to be empty sets. We prefer to think of matchings as being non-empty sets since this allows us to highlight the distinct differences between the conditions listed below. \[characterCliques\] Let ${\mathcal{H}}$ be a SPG and let ${\mathcal{K}}_1$ and ${\mathcal{K}}_2$ be two distinct maximal cliques in ${\mathcal{H}}$. Then exactly one of the following must be true: no vertex in ${\mathcal{K}}_1$ is adjacent to a vertex in ${\mathcal{K}}_2$; the set of edges that join vertices in $V({\mathcal{K}}_1)$ to vertices in $V({\mathcal{K}}_2)$ is a matching; or $\|V({\mathcal{K}}_1)\cap V({\mathcal{K}}_2)\|=1$. Suppose that $\|V({\mathcal{K}}_1)\cap V({\mathcal{K}}_2)\|\geq 2$. Then there is an induced $K_4-e$ in ${\mathcal{H}}$, contradicting Corollary \[k4-e-free\]. We see that $\|V({\mathcal{K}}_1)\cap V({\mathcal{K}}_2)\|=1$, is precisely case $(iii)$, so assume $\|V({\mathcal{K}}_1)\cap V({\mathcal{K}}_2)\|=0$. If $\|E({\mathcal{K}}_1,{\mathcal{K}}_2)\|=0$, this falls into case $(i)$. Finally, suppose that $\|E({\mathcal{K}}_1,{\mathcal{K}}_2)\|\geq 1$. By Corollary \[atMostOne\], each vertex in ${\mathcal{K}}_1$ is adjacent to at most one vertex in ${\mathcal{K}}_2$ and vice versa. Thus there is a matching between ${\mathcal{K}}_1$ and ${\mathcal{K}}_2$, completing the proof. We can say a bit more about condition $(iii)$ in Lemma \[characterCliques\]. \[adj\_cliques\] Let ${\mathcal{K}}_1$ and ${\mathcal{K}}_2$ be distinct maximal cliques in a SPG that share a single vertex. Then, ${\mathcal{K}}_1$ and ${\mathcal{K}}_2$ have distinct difference indicies. Let $X$ be the vertex shared by ${\mathcal{K}}_1$ and ${\mathcal{K}}_2$ and let $i$ be the difference index of ${\mathcal{K}}_1$. Then, let $W \in V({\mathcal{K}}_2)$. If ${{\rm{diff}}}(X,W) = \{i\}$, then $W$ is adjacent to all vertices in $V({\mathcal{K}}_1)$, a contradiction. ![Forbidden induced graphs in SPGs[]{data-label="forbiddenGraphs"}](forbiddenGraphs2.pdf) The following proposition shows that $K_{2,3}$ is an induced subgraph that is forbidden in SPGs. \[f3\] SPGs are $K_{2,3}$-free. Let ${\mathcal{H}}$ be a SPG containing an induced subgraph $K_{2,3}$. Label the vertices of the induced $K_{2,3}$ as in Figure \[forbiddenGraphs\]. Let the difference indices of $U_1U_4$ and $U_3U_4$ be $i$ and $j$, respectively, where $i\neq j$ (by Observation \[4cycle\_ob\]). Due to the structure of $4$-cycles in ${\mathcal{H}}$, $U_0U_3$ and $U_0U_1$ have difference indices $i$ and $j$, respectively. Similarly, $U_2U_3$ and $U_1U_2$ must also have difference indices $i$ and $j$, respectively. But if $U_0U_1$ and $U_1U_2$ both have difference index $j$, then $U_0 \sim U_2$ and so ${\mathcal{H}}'$ is not isomorphic to $K_{2,3}$, a contradiction. The following proposition is not necessary for any of the following results; however, it is an interesting result on its own and follows from Corollary \[k4-e-free\] and Proposition \[f3\]. We define the neighborhood of a vertex $v$ in $G$ as ${N}_G(v)$. Let $U$ and $W$ be non-adjacent vertices in a SPG ${\mathcal{H}}$. Then, exactly one of the following holds: - $\|{N}_{\mathcal{H}}(U)\cap{N}_{\mathcal{H}}(W)\|\leq 1$; or - $\|{N}_{\mathcal{H}}(U)\cap{N}_{\mathcal{H}}(W)\|= 2$ and the vertices of ${N}_{\mathcal{H}}(U)\cap{N}_{\mathcal{H}}(W)$, $U$, and $W$ form an induced $C_4$. To get a contradiction, assume there are at least three vertices $Z_1$, $Z_2$, and $Z_3$, contained in ${N}_{\mathcal{H}}(U)\cap {N}_{\mathcal{H}}(W)$. By Lemma \[f3\], at least one pair of vertices among $Z_1$, $Z_2$, and $Z_3$ are adjacent. Without loss of generality, assume $Z_1 \sim Z_2$. But then $U$, $W$, $Z_1$, and $Z_2$ form an induced $(1,2,2)$-graph, a contradiction of Corollary \[k4-e-free\]. Similarly, if $\|{N}_{{\mathcal{H}}}(U)\cap {N}_{\mathcal{H}}(W)\|=2$ and there is an edge between the two vertices in ${N}_{{\mathcal{H}}}(U)\cap {N}_{\mathcal{H}}(W)$, this forms an induced $(1,2,2)$-graph. Again, this is a contradiction of Corollary \[k4-e-free\]. SPGs with 3-cycles but no Induced 4-cycles {#girth3} ========================================== As in [@AAEHHNW], the one-sum of two graphs $G$ and $H$ is defined as the graph formed by joining $G$ and $H$ at a single vertex and preserving the edges. A graph ${\mathcal{H}}$ is a tree of cliques if ${\mathcal{H}}$ is $C_k$-free for $k\geq 4$, claw-free and given any two distinct maximal cliques ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ in ${\mathcal{H}}$, $\|V({\mathcal{H}}_1)\cap V({\mathcal{H}}_2)\|\leq 1$. Figure \[tree\_of\_cliques\] shows an example of a tree of cliques. Note that trees of cliques can be built by starting with a single clique and repeatedly attaching maximal cliques using one-sums. Also note that a tree of cliques can be disconnected. ![A tree of cliques[]{data-label="tree_of_cliques"}](tree_of_cliques.pdf) Trees of Cliques {#Sec:treeOfCliques} ---------------- In this section, it is shown that SPGs that are $C_k$-free for $k\geq 4$ are exactly all trees of cliques. The following lemma shows that a clique can be added to a SPG to form another SPG under certain conditions. Note that this is a stronger result than is necessary to characterize SPGs that are $C_k$-free for any $k\geq 4$. Before stating the lemma, we need a definition. Let ${\mathcal{H}}= S(G,a,b)$ be a SPG and let $U \in V({\mathcal{H}})$. If there is a vertex $v \in V(G)$ such that $U$ is the unique $(a,b)$-geodesic in $G$ passing though $v$, then $U$ is said to be the $v$-geodesic. If it is not necessary to specify the vertex, then we will say that $U$ has the unique vertex-path property. \[Thm:treeClique\] Let ${\mathcal{H}}= S(G,a,b)$ be a SPG and let $U \in V({\mathcal{H}})$ with the unique vertex-path property. Let ${\mathcal{S}}$ be the one-sum of ${\mathcal{H}}$ and a maximal clique ${\mathcal{K}}$ formed by identifying $U$ with any vertex in ${\mathcal{K}}$. Then ${\mathcal{S}}$ is a SPG. Furthermore, the base graph $G'$ of ${\mathcal{S}}$ can be constructed from $G$ so that the following two conditions are satisfied: if $(a,b)$-geodesics in $G$ have at least two index levels, then $(a,b)$-geodesics in $G'$ have the same number of index levels; any vertex in ${\mathcal{H}}$ other than $U$ that has the unique vertex-path property will still have the unique vertex-path property in ${\mathcal{S}}$, and every vertex in ${\mathcal{K}}$ other than the one identified with $U$ will have the unique vertex-path property in ${\mathcal{S}}$. Let $U=av_1\cdots v_i\cdots v_pb$, where $U$ is the $v_i$-geodesic. If $p=1$, by Proposition \[manybasegraphs\], the length of $(a,b)$-geodesics in $G$ can be extended to any length greater than or equal to $1$ without affecting the SPG, so we may assume ${\mathcal{H}}$ and ${\mathcal{K}}$ have the same number of index levels. Thus, we may assume that $p \geq 2$. The one-sum of ${\mathcal{H}}$ and ${\mathcal{K}}= K_n$ is the SPG, $S(G',a,b)$, where $G'$ is constructed as follows for $i< p$. Let $V(G') = V(G) \cup \{w_1,\ldots,w_{n-1}\}$ and $E(G') = E(G) \cup \{v_{i}w_j \,\|\, j = 1, \ldots, n-1\} \cup \{w_jv_{i+2} \,\|\, j = 1, \ldots, n-1\}$. Because $U$ is the unique $(a,b)$-geodesic passing through $v_i$, $G'$ has exactly $n-1$ more $(a,b)$-geodesics than $G$, each of the form $W_j=av_1\cdots v_{i}w_jv_{i+2}\cdots v_pb$, $j = 1,\ldots,n-1$. All of these new $(a,b)$-geodesics are adjacent to one another and to $U$, but are not adjacent to any other $(a,b)$-geodesics in $G$ as desired. The first condition above is clearly satisfied because if $p \geq 2$, $G'$ is constructed from $G$ by adding vertices at a single existing index level of $G$. To see the second condition above is satisfied, note that the only vertices in $V(G)$ that are part of new paths in $G'$ are the vertices $\{v_j \, \| \,j=1,\ldots,p, j \neq i+1\}$. Because $U$, the $v_i$-geodesic, passes through all the vertices in this set, no other $(a,b)$-geodesic in ${\mathcal{H}}$ could be the unique $(a,b)$-geodesic passing through one of these vertices. In addition, recall that the vertices in $V({\mathcal{K}}) \setminus \{U\}$ correspond to $(a,b)$-geodesics of the form $W_j=av_1\cdots v_{i}w_jv_{i+2}\cdots v_pb$, $j = 1,\ldots,n-1$. For each $j = 1,\ldots,n-1$, $W_j$ is the $w_j$-geodesic, and thus the second condition is satisfied. In the case $i = p$, $E(G') = E(G) \cup \{v_{i-2}w_j \,\|\, j = 1, \ldots, n-1\} \cup \{w_jv_{i} \,\|\, j = 1, \ldots, n-1\}$ rather than the set defined above, and the intended SPG is formed by an analogous argument to that above. \[Thm:forestOfCliques\] Let ${\mathcal{H}}$ be a $C_k$-free graph for all $k\geq 4$. Then ${\mathcal{H}}$ is a SPG if and only if ${\mathcal{H}}$ is a tree of cliques. By assumption, ${\mathcal{H}}$ is $C_k$-free for $k \geq 4$. Thus, if ${\mathcal{H}}$ is a SPG, it follows from Theorem \[noClaw\] that ${\mathcal{H}}$ is claw-free. By Lemma \[characterCliques\], ${\mathcal{H}}$ satisfies the remaining conditions to be a tree of cliques. For the other direction of the proof, because the disjoint union of SPGs is a SPG by Theorem \[disconnect\], only a single connected SPG need be considered. Let $G$ be a base graph defined as follows: $V(G) = \{a,v_1, v_2, \ldots, v_n, w, b\}$ and $E(G) = \{av_i, v_iw \| i = 1, \ldots,n\} \cup \{wb\}$. It is easily verifiable that $S(G,a,b) = K_n$. Additionally, each of the $n$ $(a,b)$-geodesics in $G$ is the $v_i$-geodesic for some $i = 1, \ldots, n$, so every vertex in $S(G,a,b)$ satisfies the unique vertex-path property. Using the construction in Lemma \[Thm:treeClique\], it is possible to a create a base graph whose SPG is the one-sum of $S(G,a,b)$ and any clique. In this construction, any vertex in $S(G,a,b)$ that had the unique vertex-path property prior to the construction (other than the vertex being identified in the one-sum) still has the unique vertex-path property in the one-sum. Additionally, each new vertex added to the SPG also has the unique vertex-path property. This can be seen in Figure \[tree\_of\_clique\_ex\]. Thus, by repeatedly applying Lemma \[Thm:treeClique\], it is possible to create any such connected SPG. ![Example of identifying a vertex in an SPG with that of a vertex in the clique we are adding to the SPG[]{data-label="tree_of_clique_ex"}](tree_of_clique_ex.pdf) The following observations will be useful when we generalize this result in the next section. \[cliqueParity\] Let ${\mathcal{H}}= S(G,a,b)$ be a SPG that is a tree of cliques whose base graph $G$ is constructed as in the proof of Theorem [\[Thm:forestOfCliques\]]{}. Then, the following properties hold: - any vertex in ${\mathcal{H}}$ that is in exactly one maximal clique has the unique vertex-path property; - the base graph of ${\mathcal{H}}$ can be constructed so that every $(a,b)$-geodesic in $G$ has exactly two index levels; - if ${\mathcal{K}}_1$ and ${\mathcal{K}}_2$ are two maximal cliques in ${\mathcal{H}}$ that share a single vertex, then the difference index of ${\mathcal{K}}_1$ is $i$ and the difference index of ${\mathcal{K}}_2$ is $i+1$, or vice versa. Tools To Build New SPGs {#Sec:cyclesOfCliques} ----------------------- Building upon the work in Section \[Sec:treeOfCliques\], we develop tools similar to Lemma \[Thm:treeClique\] and Theorem \[Thm:forestOfCliques\] that will characterize all SPGs that possibly contain a $3$-cycle but no induced $4$-cycle. Recall that in [@AAEHHNW], it was shown that the only SPGs with girth five or more are even cycles or paths. We begin by defining three constructions involving connecting two cliques. These constructions will be used to build new SPGs from old SPGs. Although Constructions \[const:A\] and \[const:B\] are not used in the proof of the main theorem in Section \[sec:mainthm\] (Construction \[const:C\] is the only one needed for the proof), they provide intuitive descriptions of how to build the base graph of a given SPG. \[const:C\] Let ${\mathcal{U}}$ and ${\mathcal{W}}$ be vertex disjoint maximal cliques in a SPG ${\mathcal{H}}$, and let $X$ be a vertex that is distinct from the vertices in ${\mathcal{H}}$. Then put an edge between $X$ and all vertices in both ${\mathcal{U}}$ and ${\mathcal{W}}$. Construction \[const:C\] is shown in Figure \[small\_ear\_of\_cliques3\]. ![Construction \[const:C\][]{data-label="small_ear_of_cliques3"}](small_ear_of_cliques3.pdf) \[lem:small\_ears\_of\_cliques\] Let ${\mathcal{H}}= S(G,a,b)$ be a SPG with exactly two index levels, and let ${\mathcal{U}}$ and ${\mathcal{W}}$ be two disjoint maximal cliques in ${\mathcal{H}}$ with difference indicies of opposite parity. Then a SPG ${\mathcal{H}}'$ can be formed from ${\mathcal{H}}$ as described in Construction \[const:C\] (e.g., Figure [\[small\_ear\_of\_cliques3\]]{}). Furthermore, any vertices in ${\mathcal{H}}$ that satisfied the unique vertex-path property prior to the construction still satisfy this property in ${\mathcal{H}}'$. Without loss of generality, let $1$ be the index level of ${\mathcal{U}}$ and $2$ be the index level of ${\mathcal{W}}$. Let $$U_i = au_ivb \;\;\; \text{ and } \;\;\; W_j=av'w_jb$$ be the vertices of ${\mathcal{U}}$ and ${\mathcal{W}}$ for all $i\in\{1,2,\ldots,\|{\mathcal{U}}\|\}$ and $j\in\{1,2,\ldots,\|{\mathcal{W}}\|\}$. Then we build the base graph $G'$ from $G$ by letting $V(G')=V(G)$ and $E(G')=E(G)\cup \{vv'\}$. Under this construction, $S(G',a,b)$ is the graph ${\mathcal{H}}'$ as described in Construction \[const:C\]. Since there are two index levels, it is clear that there are no additional $(a,b)$-geodesics besides $X$. The only edges must join $X$ to all vertices in ${\mathcal{U}}$ and ${\mathcal{W}}$ for the same reason. \[const:A\] Let $U$ and $W$ be non-adjacent vertices in a SPG ${\mathcal{H}}$ that are each in exactly one maximal clique. Let ${\mathcal{K}}_1$ and ${\mathcal{K}}_2$ be two disjoint maximal cliques with $\|V({\mathcal{K}}_i)\| \geq 2$ for $i =1,2$. Let $X$ and $X'$ be distinct vertices in $V({\mathcal{K}}_1)$ and let $Y$ and $Y'$ be distinct vertices in $V({\mathcal{K}}_2)$. Then, build a new graph from ${\mathcal{H}}$, ${\mathcal{K}}_1$, and ${\mathcal{K}}_2$ by identifying $X$ with $U$, $Y$ with $W$, and $X'$ with $Y'$. Construction \[const:A\] is shown in Figure \[small\_ear\_of\_cliques\]. ![Construction \[const:A\][]{data-label="small_ear_of_cliques"}](small_ear_of_cliques_a.pdf) \[const:B\] Let $U$ and $W$ be non-adjacent vertices in a SPG ${\mathcal{H}}$ that are each in exactly one maximal clique. Let ${\mathcal{U}}$ and ${\mathcal{W}}$ be vertex disjoint maximal cliques that contain $U$ and $W$, respectively. Let ${\mathcal{K}}$ be a clique with $\|V({\mathcal{K}})\| \geq 2$. Let $X$ and $X'$ be distinct vertices in $V({\mathcal{K}})$. Then, build a new graph from ${\mathcal{H}}$ and ${\mathcal{K}}$ by identifying $X$ with $U$ and putting an edge between $X'$ and every vertex in ${\mathcal{W}}$. Alternatively, we could build a different graph from ${\mathcal{H}}$ and ${\mathcal{K}}$ by identifying $X$ with $W$ and put an edge between $X'$ and every vertex in ${\mathcal{U}}$. Construction \[const:B\] is shown in Figure \[small\_ear\_of\_cliques2\]. ![Construction \[const:B\][]{data-label="small_ear_of_cliques2"}](small_ear_of_cliques2_a.pdf) \[small\_ears\_of\_cliques\] Let ${\mathcal{H}}= S(G,a,b)$ be a SPG and let $U$ and $W$ be non-adjacent vertices in $V({\mathcal{H}})$. Suppose that $U$ is the $u$-geodesic and $W$ is the $w$-geodesic for vertices $u,w \in V(G)$. If $\|{{\rm{diff}}}(U,W)\| \leq 2$ and $i_1$ and $i_2$ are the indicies of $u$ and $w$, respectively, then the graph ${\mathcal{H}}'$ formed from ${\mathcal{H}}$ by joining $U$ and $W$ to cliques in the following ways is an SPG: If $i_1 \neq i_2$, then ${\mathcal{H}}'$ is the graph defined in Construction \[const:A\] (e.g., Figure [\[small\_ear\_of\_cliques\]]{}). If $i_1 = i_2$, then ${\mathcal{H}}'$ is one of the graphs defined in Construction \[const:B\] (e.g., Figure [\[small\_ear\_of\_cliques2\]]{}). Furthermore, any vertices in $V({\mathcal{H}})\setminus\{U,W\}$ (Construction \[const:A\] only), or either $V({\mathcal{H}})\setminus\{U\}$ or $V({\mathcal{H}})\setminus\{W\}$ (Construction \[const:B\] only), and vertices in the new cliques not including $X,X',Y,Y'$ that satisfied the unique vertex-path property prior to the use of Construction \[const:A\] or \[const:B\] will still satisfy this property in ${\mathcal{H}}'$. Note that since $U$ and $W$ are non-adjacent, $\|{{\rm{diff}}}(U,W)\|=2$. Since the disjoint union of SPGs is a SPG by Lemma \[disconnect\], let ${\mathcal{H}}$ be connected. From Observation \[cliqueParity\], it follows that $U$ and $W$ are in exactly one maximal clique as shown in Figures \[small\_ear\_of\_cliques3\], \[small\_ear\_of\_cliques\], and \[small\_ear\_of\_cliques2\]. Since there are no $4$-cycles in ${\mathcal{H}}$ or ${\mathcal{H}}'$ and $\|{{\rm{diff}}}(U,W)\|=2$, $\|i_1-i_2\|\leq 1$ and if $i_1=i_2$, then the other difference index in ${{\rm{diff}}}(U,W)$ must be $i_1+1$ or $i_1-1$. Notice there are no additional $(a,b)$-geodesics in $G$ passing through $u$ and $w$ as a result. Without loss of generality, assume that $i_1 \leq i_2$ and let $U$ and $W$ be the following: $$U=av_1\cdots v_{i_1-1}v_{i_1}v_{i_1+1}v_{i_1+2}\cdots v_pb \;\;\text{ and }\;\; W=av_1\cdots v_{i_1-1}w_{i_1}w_{i_1+1}v_{i_1+2}\cdots v_pb \text{.}$$ With this notation, note that $u = v_{i_1}$. If $i_1 < i_2 = i_1 +1$, then $w =w_{i_1 +1}$. If $i_1 = i_2$ then $w = w_{i_1}$. Also, note that $v_{i} \neq w_{i}$ for $i \in \{i_1,i_1 +1\}$. First, assume $i_1 < i_2$. To perform Construction \[const:A\], we build the base graph $G_1$ from $G$ as follows. The vertex set $V(G_1)=V(G)\cup \{u_1,\ldots, u_{k_1-2},u_1',\ldots,u_{k_2-2}'\}$, and the edge set $$\begin{aligned} E(G_1)&=E(G)\cup \{v_{i_1}w_{i_1+1}\}\cup\{v_{i_1-1}u_i, u_iw_{i_1+1} \,:\, i\in \{1,\ldots,k_1-2\}\}\\ &\;\;\;\;\;\;\;\;\;\;\;\;\;\cup \{v_{i_1}u_i', u_i'v_{i_1+2}\,:\, i\in\{1,\ldots,k_2-2\}\}\text{,}\end{aligned}$$ where $k_1$ and $k_2$ are the numbers of vertices in the new cliques added to ${\mathcal{H}}$ containing $W$ and $U$, respectively (as shown in Figure \[small\_ear\_of\_cliques\]). Under this construction, $S(G_1,a,b)$ is the graph ${\mathcal{H}}'$ with $U$ and $W$ connected as described in Construction \[const:A\]. To see that all vertices in ${\mathcal{H}}$ (other than $U$ or $W$) that satisfied the unique vertex-path property prior to the construction still satisfy this property after the construction, note that the only vertices in $V(G)$ that are a part of new paths in $G_1$ are in the following set: $\{v_1, \ldots, v_{i_1}, w_{i_1+1}, v_{i_1+2}, \ldots, v_p\}$. Because either $U$ or $W$ passes through each of these vertices, no $(a,b)$-geodesic in $G$ other than $U$ or $W$ can be the unique $(a,b)$-geodesic passing through any of these vertices. Moreover, every new vertex added to ${\mathcal{H}}$, other than $X'$ (or $Y'$), is either the $u_i$-geodesic for some $i \in \{1, \ldots, k_1 - 2\}$ or the $u_i'$-geodesic for some $i \in \{1, \ldots, k_2 - 2\}$. Now, assume $i_1 = i_2$. To perform Construction \[const:B\], we form the base graph $G_2$ with the vertex set $V(G_2)=V(G)\cup \{u_1,\ldots, u_{k_1-2}\}$ and edge set $$E(G_2)=E(G)\cup \{v_{i_1}w_{i_1+1}\}\cup\{v_{i_1}u_i, u_iv_{i_1+2} \,:\, i\in \{1,\ldots,k_1-2\}\}\text{,}$$ where $k_1$ is the number of vertices in the new clique in ${\mathcal{H}}$ containing $U$, as shown in the left graph of Figure \[small\_ear\_of\_cliques2\]. Under this construction, $S(G_2,a,b)$ is the graph ${\mathcal{H}}$ with $U$ and $W$ connected as shown in the first graph of Construction \[const:B\]. To form the second graph in Construction \[const:B\], the edge set $$E(G_2)=E(G)\cup\{w_{i_1},v_{i_1+1}\}\cup\{w_{i_1}u_i,u_iv_{i_1+2}\,:\, i\in \{1,\ldots,k_1-2\}\}$$ is used instead of the edge set described above. If $i_1 = i_2$, it also possible that $W$ has the form, $W=av_1\cdots v_{i_1-2}w_{i_1-1}w_{i_1}v_{i_1+1}\cdots v_p b$. If this is the case, the graph shown in Construction \[const:B\] can be constructed using an argument analogous to the one above. Classification of $C_4$-free SPGs {#sec:mainthm} --------------------------------- Putting this all together, in this section we characterize all SPGs with no induced $4$-cycles. We define a collection of cliques as a graph that is claw-free and given any two distinct maximal cliques ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ in ${\mathcal{H}}$, $\|V({\mathcal{H}}_1)\cap V({\mathcal{H}}_2)\|\leq 1$. That is, a collection of cliques is a tree of cliques that can contain cycles. An example of a collection of cliques is shown in Figure \[cyclesOfCliques\]. Note that a collection of cliques may be disconnected. ![A collection of cliques[]{data-label="cyclesOfCliques"}](cyclesOfCliques.pdf) \[no4cycles\_classification\] Let ${\mathcal{H}}$ be a $C_4$-free graph. Then ${\mathcal{H}}$ is a SPG if and only if ${\mathcal{H}}$ is a collection of cliques that is $C_k$-free for all odd $k \geq 5$. Furthermore, the base graph of ${\mathcal{H}}$ can be constructed with only two index levels. By assumption, ${\mathcal{H}}$ is $C_4$-free. Thus, if ${\mathcal{H}}$ is a SPG, it follows from Theorem \[noClaw\] that ${\mathcal{H}}$ is claw-free. It was shown in [@AAEHHNW] that SPGs are $C_5$-free. From Theorem \[oddtoC4\], it follows that ${\mathcal{H}}$ is $C_k$-free for any odd $k > 5$. By Lemma \[characterCliques\], ${\mathcal{H}}$ satisfies the remaining condition to be a collection of cliques. Now, assume that ${\mathcal{H}}$ is a collection of cliques that is $C_k$-free for all odd $k \geq 5$. By Theorem \[disconnect\], we can assume that ${\mathcal{H}}$ is connected. Any such graph that does not have an induced $C_k$ for $k >3$ is a tree of cliques and thus any such ${\mathcal{H}}$ is an SPG by Theorem \[Thm:forestOfCliques\]. We can now assume that ${\mathcal{H}}$ contains an induced cycle of even length $k >4$. We proceed by induction on the number of vertices in ${\mathcal{H}}$. Let $\|V({\mathcal{H}})\|=n$ and assume any collection of cliques that is $C_4$-free, $C_k$-free for odd $k \geq 5$, and with fewer than $n$ vertices is a SPG, and the base graph of ${\mathcal{H}}$ can be constructed with two index levels. Let $X$ be a vertex in $V({\mathcal{H}})$ shared between two distinct maximal cliques, ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$, that are part of an induced even cycle contained in ${\mathcal{H}}$. Let ${\mathcal{H}}'$ be the graph formed from ${\mathcal{H}}$ by deleting $X$. By the induction hypothesis, ${\mathcal{H}}' = S(G,a,b)$ where $G$ only has two index levels. Because $X$ was the only vertex in $V({\mathcal{H}})$ joining ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$, in ${\mathcal{H}}'$, ${\mathcal{H}}_1-X$ and ${\mathcal{H}}_2-X$ are disjoint. Because ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ are part of an induced cycle with even length in ${\mathcal{H}}$, ${\mathcal{H}}'$ is connected, and from Proposition \[adj\_cliques\], it follows that the difference indicies of ${\mathcal{H}}_1-X$ and ${\mathcal{H}}_2-X$ have opposite parity. By Lemma \[lem:small\_ears\_of\_cliques\], we can form a base graph $G'$ such that $S(G',a,b)={\mathcal{H}}$ and $G'$ has two index levels as desired. By induction, the result follows. Acknowledgments {#acknowledgments .unnumbered} =============== The authors are grateful to AIM and to the organizers of the REUF program at AIM for making this collaboration possible. This project was initiated as part of the REUF program at AIM, NSF grant DMS 1620073. We would also like to thank Beth Novick and Ruth Haas for their helpful suggestions in preparing this paper.
Frederick Beall, Zoo New England General Curator, was honored with a certificate of recognition from the Association of Zoos and Aquariums (AZA) for 30 years of dedication and service to the Wattled Crane program. Beall, who has worked for Zoo New England for more than 20 years, has worked with nearly every crane species throughout his distinguished career. He first began working with wattled cranes, a vulnerable species native to Africa, in the late 1970s. “As a group, cranes are spectacular birds, but they are in deep trouble. They are birds of the marshes and the wetlands and as their habitat is destroyed, these magnificent birds are going to disappear,” said Beall. “I enjoy the challenge of working with these birds – the challenge to better understand their biology and adaptations, and in turn how we can positively impact wild populations to ensure future survival of the species.” Beall began his career in 1966 as an Animal Keeper caring for birds at the Maryland Zoo in Baltimore. By 1978, he served as the Curator for Birds for the Maryland Zoo in Baltimore, a position he held until 1992 when he joined the staff at Zoo New England. Throughout his time in Baltimore, he helped establish the black-footed penguin colony, including husbandry and breeding protocols. Throughout a 20-year-period, more than 500 African black-footed penguins hatched. He established the North American Regional Studbook for African black-footed penguins, which evolved into the African Black-footed Penguin Species Survival Plan. In 1985, Beall established the Regional Studbook (North America) for Wattled Cranes, which expanded in 1987 into the International Studbook (Global) for Wattled Cranes. Since then, he has continuously served as the International Studbook Keeper and the North American Species Survival Plan Coordinator for Wattled Cranes. He has participated in a number of international workshops dedicated to crane species. Throughout his zoo career, Beall has had great success breeding endangered cranes through artificial insemination techniques and environmental manipulation. Under Beall’s leadership, Franklin Park Zoo is the first zoo in North America to have successfully bred endangered Siberian cranes. He has also coordinated with other conservation professionals in the United States and Russia to have red crowned crane eggs, produced here at the Franklin Park Zoo and in other U.S. zoos, transferred to Russia where they were hatched and released into the wild to supplement dwindling wild populations. Beall also serves as the Chair of the AZA Gruiformes Taxon Advisory Group, which coordinates captive management and conservation programs for all crane species.	https://www.zoonewengland.org/zoo-news/2015/september/zoo-new-england-general-curator-recognized-for-years-of-service-by-the-association-of-zoos-and-aquariums/
Mild foaming toothpaste for a targeted cleaning, tooth care and keeping the gums healthy. Sage refines the complexion, and has an anti-inflammatory, antiperspirant, antibacterial and antiviral effect. Sage is therefore ideal for toothpaste, as it counteracts inflammations in the mouth and throat. Peppermint is a very effective medicinal plant at high concentrations. It’s not just refreshing, but is also used as an antibacterial, calming, antiseptic and pain-relieving healing herb. For children under the age of 6: Only use a pea-sized amount of toothpaste. To avoid excessive swallowing, only brush teeth under supervision. If additional fluoride is taken, consult a dentist or doctor.	https://www.logona.de/en/product/complete-protection-daily-care-toothpaste.html
Segregation Law and Legal Definition Segregation is the physical separation of categories of individuals, usually on the basis of gender, race, religion, or class. It can be de jure or de facto—sanctioned by law or custom. Although the word normally implies an involuntary situation, segregation can also reflect voluntary behavior or some mixture of voluntary and involuntary circumstances. Various types of segregation have been common in American history, but the term usually refers to a systemic pattern that has historically affected blacks more than other Americans. The Civil Rights Movement was initiated by Southern blacks in the 1950s and '60s to break the prevailing pattern of racial segregation. This movement spurred the passage of the 1964 Civil Rights Act, which contained strong provisions against discrimination and segregation in voting, education, and the use of public facilities.	https://definitions.uslegal.com/s/segregation/
The Trading Tribe (c) Ed Seykota 2003-2015 - Write for permission to reprint. Ed Seykota's FAQ Home \| Send Mail to FAQ \| Resources \| Ground Rules \| FAQ Index TT_Chartbook FAQ December 01-10, 2015 <-- back \| down \| forward --> Contributors Say Ed Says Dec 10, 2015 Tribe Meeting Results - from Lonely to Full Ed, I'd like to share some results from a recent Tribe Meeting - during which I work on loneliness issues. Almost immediately after the meeting, I notice things changing. Friends start dropping by for a visit - and to connect and catch up on each others' lives. The connections seem unusually caring and open. Surprisingly, I also get a lot of invitations to go out to parties and events. I even have to decline some offers on account ofhaving a full calendar. I find myself meeting new and interesting people, almost effortlesly. Best of all most everyone I meet seems open and kind. Thank you for sharing your process - and results. Dec 10, 2015 Tribe Meeting Results - Calm and Connecting Ed, I wish to share results I see showing up in my life after the Tribe Meeting. Upon arriving home I feel surprise at the level of calm and peace in my home. Weekend afternoons are usually hectic, with lots of excitement for me and my young children, and occasional bickering and whining by all. I feel delight to learn my son, who sometimes likes to hide and act shy, chooses to recite a passage in a large group. This occurs after our Tribe Meeting, and before I return home. Not surprisingly, my main issue at tribe involves "hiding". On weekdays I usually shut myself away in my office to work. Completely out of fashion, I invite a family member to visit and share feelings about a dispute. I view our meeting as a gift. Previously I view it as an interruption in my work day. I notice sharing feelings about the dispute "takes a load off" and help me work more effectively. I find myself with a new, deep desire to receive and share feelings with everyone I encounter. I call on several prospective clients today. The thought is "who else can I call" rather than "ugh who else do I have to call". I sense my own healing just by receiving their feelings. As I receive, the same feelings in me pass through and get set free. Sincerely, Thank you for sharing your process - and results. Dec 9, 2015 How To Start Ed, Thank you for your response. I am near [City], how do I go about taking my feelings about <actually starting to trade> to Tribe as an entry point? Thank you again. Thank you for raising this issue. You might consider reading through FAQ to get an idea how the system works. You might also consider reading The Trading Tribe - see Resources, above. Dec 8, 2015 Standard Deviation - from Trading Ed, I hope you are well. Do you have any sites you recommend for use of standard deviation in trading.? also- do you think becoming a Chartered Market Technician would make a big difference in trading? Thank you for your help. Thank you for raising this issue. You might consider taking your feelings about <actually starting to trade> to Tribe as an entry point. Nothing Happens until you pull. http://successify.net/2013/09/26/pulling-trigger/ Dec 7, 2015 Post-Tribe Jam Ed, I feel a surge of warmth in my face and my eyes get watery as I think about how much kindness you show me over the years. From very first time we meet almost seven years ago, you show very deep caring about me and my feelings. Thank you so much for inviting me into your home again, hosting the Tribe meeting, breakfast and dinner. And thank you for jamming with me, and inviting me to the Wimberley Jam. I feel warm and welcome at the jam, and it really lives up to your description. I plan on sending a full Tribe report before the end of the week. I already notice changes and still feel very much in process. Please hear Whipsaw Song Norwegian Wood Lying Eyes Wah Wah Wah Blues for our recordings. I really enjoy & might have to "borrow" your lick around 1:12 on the Whipsaw Song. Yours Truly, Thank you for attending the Tribe Meeting and for sending me the recordings of our post-Tribe-meeting musical meanderings. Dec 6, 2015 Tribe Report - A Healing Hug Ed, I come to the meeting with a lot of motivation - and support from friends who suggest areas for improvement - and from my sister who encourages me and helps me recall some of our early family dynamics. I take the Hotseat with the issue of wanting to have have great a primary relationship. The PM asks me to show the form of wanting this. I hunch over in my chair and extend my arms out in front of me, reaching for someone, finding nothing but air at the ends of my fingers. I begin sobbing. The PM encourages me to go deeper. I notice a longing to connect with my mother. The PM asks me to recall an incident and this takes me right back into my childhood. When I open my eyes I see my Tribe members, now in roles as my mother, grandmother and grandfather - and all sitting in the room in front of me. My grandfather teaches me that the way to deal with the women in the household includes saying some form of "yes dear" and then doing whatever you want to do instead. He also shows me how to shut down my feelings and to appear to not care. By example, he seems to direct me to treat women with suspicion and disdain. I tell him I appreciate that he wants to inform me and to protect me - and that I'd rather use a different method, namely: establish rapport and share feelings. Then we get to the main event, including me trying to get some affection and caring from my mother. I try various strategies to get affection from her. Nothing seems to work. Seeing my frustration, the PM jumps in and offers to model it for me. I accept his offer and watch as he tries to get affection from her, alternatively demanding affection, pretending not to care and following after her. He winds up angry and expressing frustration. I say, "Wow, you got it. You got how I do it." I consider his showing me how I do it a huge gift, full of insights. With this information, something clicks and I know what to do. I reach out to my mother and ask her to hold on a minute and to let me know she cares about me. She says, dismissively, that of course she does - and then adds we have to run along now and do some errands. I connect with her and look into her eyes and I say I want more; I want her to help me feel, in my heart, that she cares. With that, she approaches me and gives me a hug and tells me that she hears what I want and that she does not know how to do it. She says that she would like to to learn how to do it, working together with me. Then she gives me a hug and tells me she loves me, her son. At that point I begin wailing. I stand there, feeling her arms around me, soaking up the hug I can now feel after a lifetime of longing for it. The hug seems to go on a long time, quenching a long-standing thirst, like rain finally falling on a dry desert after decades of drought. I feel complete and I feel something long dormant starting to grow in me. I also know the hug works both ways and helps to heal her, too. After that hug, I feel quite different - like in a trance - and I still feel that way. Lately, friends report that I seem more accessible and more supportive and less critical of them. I have a sense that, finally, after all these years, I have some idea how a relationship and affection might all fit together. And, incidentally, I get another benefit. I notice much less interest in getting close to people who do not have the ability or desire to help me feel, in my heart, that they care about me. I thank my Tribe and my PM for running this process with me - and my dear friends for helping me see areas for improvement - and my sister for encouraging me and helping me recall our family dynamics. Thank you for sharing your process. Nothing Heals like a hug. https://onehugaday.wordpress.com/2013/10/30/hug-you-so-tight/ Dec 6, 2015 Tribe Report - Five Hotseats Ed, At our recent Tribe Meeting, all five participants arrive ready to take the hot seat, by agreement. I would like to report, briefly, on the five processes while I still have them fresh in mind - and follow up with more details later. The Spanking Hotseat complains about his inability to lose weight. We follow his forms to a critical incident in which his father spanks him - and scolds him for making a mess. Hot seat shuts down and learns to stuff his feeling of anger and resentment. Through the process, Hotseat learns to establish rapport with his father and to ask his father how he feels. His father accepts the invitation, stops beating his son and stats communicates his frustration heart-to-heart. Hotseat notices he no longer has to use food to stuff down his feelings. The Model Hotseat complains about her difficulty in receiving gifts and her frustration that people don't say what they think. We follow her forms to a critical incident in which Hotseat tells her mother that she wants to grow up as a fashion model. Her mother scolds her, telling her she doesn't have enough height, beauty or brains to seek that career. Hotseat cries, curls up in a ball and goes off to her room and falls asleep in her bed. Through the process, Hotseat learns to establish rapport with her mother and to ask her how she feels. Her mother tells her she only wants to protect her daughter. Hotseat explains that she wants her mother's support in exploring interesting life choices and in having adventures. Her mother comes around to support her daughter in developing her own sovereignty. She hugs her daughter and tells her she actually finds her very beautiful and intelligent. Hotseat notices she can establish open and honest, whole-heart rapport and intimacy with all her family members. The Waitress Hotseat complains about people who agree to do something and then let her down. We follow her forms to a critical incident in which her father yells at the dinner table - and throws objects from the table around the room, occasionally hitting someone with the projectile. Hotseat busies herself with trying to clear the dinner table, in order of the most dangerous items first. Through the process, Hotseat learns to establish rapport with her father and to ask him how he feels. Her father tells her he feels frustration about his relationship with his wife and no longer needs to act out violently with his daughter. Hotseat notices she can establish rapport with her business associates and align whole-heart with them, to accomplish objectives. The Closet Hotseat complains about his inability to raise new clients for his money-management business - despite having an excellent track record. We follow his forms to a critical incident in which Hotseat, fearing beatings from both his father and mother, hides in a closet and waits. Through the process, Hotseat learns to establish rapport with his parents and to work with them to keep the house tidy. Hotseat realizes he doesn't have to wait around in fear of his clients - and that he proceed at once and focus on establishing rapport with them. The Barber Shop Hotseat complains about his inability to sustain a loving and supportive relationship with a woman. We follow his forms to a critical incident in which Hotseat, waits, in a barber Shop, for his mother to pick him up and take him home. When she arrives, she looks at him and momentarily engages eye contact. Neither of them show any sign of wanting to go first to recognize the other - perhaps some kind of power struggle. Presently, she exits the room and leaves him behind, to walk home - which he does, defiantly. Through the process, Hotseat comes to learn to reach out first and establish rapport with his mother. She tells him she has many things to do and can't wait around to play games. He tells her he wants her to help him feel she cares about him. She confesses she doesn't know how to do this and that perhaps they can work together to help each other learn how to have a better relationship. Hotseat realizes that he can go the extra mile in establishing rapport with his woman - and to make her feel safe in learning and in building a relationship together. He also realizes in cases in which people do not demonstrate willingness or ability to engage an intimate, rapport-centric relationship, he can, respectfully, release them to go their separate ways. Thank you for documenting the meeting. You might consider checking back in a week or two to report any changes you see showing up in your life. Dec 5, 2015 Childhood Trauma and Addiction Ed, I feel happy hearing from you in this moment of now. I wonder if you might be familiar with some of the work done by Gabor Mate on addiction and the connection between childhood trauma, the chronic "medication" of feelings (especially anger) and disease. I wonder if you ever consider sharing the Trading Tribe Process with a wider audience. I'd love to attend a TTP Workshop in Europe. Thank you for raising this issue. In TTP we hold that during critical childhood events, a child learns, from role models, how to shut down to avoid pain. Later on, this shut-down response pattern (Rock) comes to shape, even dominate, his life. In TTP, we return the client to such critical events by encouraging him to experience the associating feeling and forms. Then, while the client has deep emotional access to his Rocks, he also has an opportunity to fore-give these Rocks back to their donors - and to replace them with the Heart Rock that motivates the establishment of rapport and constructive exchange of feelings. After this process, the client generally finds himself behaving differently in stressful situations - and getting different results - all without having to remember the process or to follow conscious advice. Per your wonder if I plan to take TTP to a wider audience: As I work through my own issues and develop more understanding and compassion for the human condition, and more effectiveness in conducting the process, I may, at some point, engage a project for wider distribution. Dec 5, 2015 Spontaneous AHA Mr. Seykota, Last week, while I was visiting an old friend the funniest thing happened. During one of our conversations, I had a "aha" moment. While its hard to accurately explain in words, I had this great and powerful inner feeling in my chest that seemed to almost follow my spine. It was at this moment that I realized what I honestly wanted to do with my life. Prior to this, I honestly had some conflict, so it felt great to finally sort it out. Also, recently I realized how one of my negative emotional charges is affecting me in other area's of my life. I'm happy that I am finally fully aware of this deeply seated issue and I am excited to work with it. I hope you and the Tribe Community are well, Thank you for sharing your process. Dec 3, 2015 Projecting Disbelief Hi Ed, I am very confused. Why people just don't believe me about trend following? Even my dad...Actually he don't think any kind of trading can make a living in the long term. I showed him my past 2 years trading Chinese futures market result, already total 80% for the 2 years period. But he still hesitates to believe me. This summer, I took my dad to USA for travel. In the mean time, i showed him my card counting in Reno, Nevada. I told him i did a lot of work on testing my card counting strategy on computer simulation and i had proved my strategy real time while i was in US for study. And he did not believe me can gamble for a win until he saw me won real money in Rail City, and Western Village. Even after that, he told me not to be too confident about blackjack. So, I have a question: am i gonna be disbelieved by people for my whole life? Why people don't believe things that they don't normally see people succeed? Best wishes Thank you for sharing your process and for raising this issue. You might consider taking your feelings about <nobody believes me> to Tribe as an entry point. Once you metaform* this feeling from an adversary to an ally, you may notice yourself projecting this drama onto others with less frequency. * Metaform: to transform something by viewing it differently. Dec 2, 2015 Tribe Report - Sister Simultaneity Ed, Hot Seat (HS) arrives at the meeting, hot, ready, and willing to work. HS begins by reading correspondence from HS’s sister. I feel awe, amazement and jealous at the depth of understanding and feeling shared through her correspondence. HS shares feelings of loneliness, rejection and smothering with the Tribe. The HS describes a history of “not being good enough” to parents and companions. A tightness begins in the HS’s throat as these feeling become more intense. The HS is fully willing to feel these feelings and really gets into it. The Tribe is in full support. The Process Manager (PM) asks if the HS can put a smile on the feeling. The HS does not like the feeling. The PM and the tribe begin saying “That’s the best rejection we’ve ever seen.” The HS smiles and laughs. The HS shrugs and says “It’s just a feeling.” During the process, the HS becomes comfortable with the feeling of loneliness and states that loneliness is peaceful. The HS also feels the feeling of smothering and becomes comfortable with this feeling as well. I am impressed at the HS’s ability to get into the feelings so deeply so quickly. At the conclusion of the meeting, a feeling of openness fills the room. After the meeting, I read a text message that I receive from my sister during the meeting. She shares strong feelings with me, is open with me. This is very unusual. The TT process is amazing. Thank you for sharing your process and for documenting the meeting. Dec 2, 2015 Wants to Join a Tribe Hello, I'm interested in joining a Tribe and learning what the Trading Tribe is all about. Can you please direct me on how to get involved and learn about meeting in the Connecticut area? Hope to hear from you soon! Thanks, Thank you for raising this issue. You can check the Tribe Directory for a Tribe near you. You can read The Trading Tribe. You can start your own Tribe.. See Resources, above. Dec 2, 2015 Should and Right Livelihood Ed, I take my snapshot of a creating a “self-sustainably scaling” enterprise to tribe. The Tribe challenges me by sharing that they think this is not really my snapshot or ambition, and that I am (as they observe in prior instances) “playing with concepts in my mind.” They ask me what feelings come up and I share a fear of “letting go.” This becomes my entry point and the tribe encourages me through the Forms Process. As the forms dissipate, I realize that many of my thoughts, including that of creating a “self-sustainably scaling” enterprise – are around what I think I “should” be doing . What I feel clearly (near the zero-point) is that I actually like working closely with people – helping develop the younger consultants that join our firm, and also being involved in delivering project work to our clients. I like building a strong foundation for our enterprise and believe providing a place where people can do the best work of their life - in a balanced way - is more important than how many people work with us, or how many clients we have, or how much we grow our revenues (so long as we can profitably sustain ourselves). Wanting more (or rather, doing what I think I should be doing), seems to take me away from not only enjoying what I have, but also … not even seeing what is right in front of me. Thank you for receiving my process. Thank you for sharing your process. Dec 2, 20156 From Rejection to Affection Ed, During a recent Tribe meeting I experience feelings in my throat that I don't like. My Process Manager helps me to like them and to see their positive intentions. I come to see I have a deep and subconscious pattern in which I train my mate to withhold affection from me and, ultimately, to reject me. I come to see this as a rock I inherit from my parents. Upon further meditation and discussions with friends and family, I see how I have a lifetime pattern of getting myself into this situation - and that I also have a full-time career trying to "fix" the problem by trying to get my mate to want me while simultaneously pushing her away. I now see my desperate and obsessive attempts to get affection as just more setup for the deeper game. I see the only way to win this game - involves letting it all go and not playing anymore. I also see the resources in the Heart Rock in a different way. Previously, I artfully employ "establish rapport" and "share feelings" as tools to carry out, with more efficiency, my push-pull drama. Recently, I see the difference between using these resources as tools and really living by them. As I learn to release this game, I notice some other changes: 1. I do not feel a need to follow my usual method of quickly replacing my mate with someone else with whom to continue the game. 2. I feel a release of tension and an ability to focus on other things, such as my health and on my job and on maintaining my friendships - and on getting to know myself better. 3. I feel more at ease with people in general and more of an overall feeling of contentment. 4. I become aware of other tendencies I use to distance myself from others - such as by feeling superior to them - so I guess I might have some more work to do. 5. I see my parents as doing their best and not trying to harm me. I no longer blame them. I no longer blame ex-wives or girlfriends. I might also guess I have a lot more to learn about getting along with others. So far I have some indication that if I stick with it, it might just work out pretty well. I wish to thank my friends, family and Tribe Members for supporting me in having these learnings. Thank you for sharing your process. The Winning Move Stop playing ! -- from the movie,	https://www.seykota.com/tt/2015/Dec/01-10/default.html
Capablanca y Graupera was born in November 19, 1888 and died on March 8, 1942. He was a Cuban world-class chess player of the mid-twentieth century. He held the title of world chess champion from 1921 to 1927 and was also a highly-regarded chess writer. In 1909, at the age of 20, Capablanca won against US champion Frank Marshall by +8-1=14 which was one of his greatest wins of all times. Then, in a tournament at New York 1911, Capablanca placed 2nd behind Marshall. So Marshall insisted Capablanca should play in a tournament at San Sebastian, Spain in 1911. It was one of the toughest tournaments of that time. All of the world's leading players marked their appearance where Ossip Bernstein and Aaron Nimzowitsch objected to Capablanca’s presence as he hadn't won any other tournaments. But he was allowed to play his first round against Bernstein and had a brilliant win over him which forced Bernstein to acknowledge Capablanca’s talent. Nimzowitsch took offense when Capablanca made a comment that unproven players should hold their tongue in the presence of higher players. Capablanca quickly challenged Nimzowitsch for a series of fast games and won with ridiculous ease. The assembled masters of the game applauded Capablanca. After which Capablanca went on to win the tournament at San Sebastian and this is considered to be his one of the greatest achievements in his chess career. Capablanca plays the Ruy Lopez Opening. Transposing to the Berlin Defense. White takes the exchange option. Many people say the bishop is better than the knight but now Black's pawn formation is compromised. Black doesn't want to play c5 as this will weaken his control on d5. Compelling the bishop to make a decision. Bishop chooses discretion as the better part of valor. Discovered attack on the bishop. White is determined to keep a knight on e4. Again the second knight stands as a ready replacement for his partner should he be needed. Black finally accepts the pawn but now his queen is away from the main battlefield. The queen is offered a second pawn in exchange for another tempo. White's knight's moving towards the enemy king. The queen has regained a central position but is still locked out of the kingside by the knight on h5. White drives the queen away. Now White has achieved a material lead. 30.Qc3 f6 31.Nxf6+ Kg6 32.Nh5! Threatening mate and attacking the rook. You can play through this game here. Black answers the King's Opening with the French Defense. White has allowed Black to win more space and his pieces have a cramped look. Black's pieces on the other hand have much more room to breathe. Increasing the heat on h2. Nimzowitsch liked this idea of supporting the king's direct protector with the bishop from the rear. With the extra space, Black's pieces are taking up good positions but White's pieces are looking forlorn. Choosing not to open the space in front of White's miserable pieces. They must continue to suffer. White gives up the knight for two pawns so that he can release himself from Black's crushing advance and establish his bishop on the a2-g8 diagonal with tempo. The knight leaves his post as h2's protector, White is going for broke now. Black must give up the rook for the knight. Not 19...Kg8? because of 20.Qh5 g6 21.Ng5+ Kg7 22.Qxh7+ Kf6 23.Nh3 Bd6+/= and now White has the upperhand. White wants to open up the long diagonal. Almost even in material but White's king is much more exposed. 27.c5 Nce7 28.Bf3 Bb5 29.Rc2 Nf6 30.a4 Bd3 31.Rcc1 Ne4 32.b5? Now it's over with a forced mate ahead. Better was 32.cxb6 axb6 33.d5 Nxd5-+ Still losing but still fighting. Join in and write your own page! It's easy to do. How? Simply click here to return to Jose Raul Capablanca Anecdotes and Games.	http://www.learn-and-play-online-chess.com/capablanca-san-sebastian.html
WATCH: Mapping Canada’s Carbon Landscapes launch at COP26 WWF-Canada and partners released Canada’s first-ever national carbon analysis and map during a “Panda Pavilion” panel at COP26 in Glasgow on November 10. The study, led by researchers from McMaster University’s Remote Sensing Laboratory, found hundreds of billion of tonnes of carbon stored in ecosystems across Canada, the equivalent to about 30 years of human-caused global greenhouse gas emissions (at 2019 levels). These findings were mapped to show the density of carbon — in different geographic locations — in trees and other plants all the way to two metres below ground. The amount of carbon found in Canada is globally significant, which led to WWF-Canada recommending several actions to help protect carbon stores throughout the country. Click here for more info on Mapping Canada’s Carbon Landscapes The COP26 panel speakers included Mary MacDonald (Senior VP and Chief Conservation Officer, WWF-Canada), James Snider (VP, Science, Knowledge & Innovation, WWF-Canada), Vern Cheechoo (Director of Lands & Resources, Mushkegowuk Council), Alemu Gonsamo (Assistant Professor, McMaster University and Canada Research Chair on Remote Sensing of Terrestrial Ecosystems), and Angela Kane, (CEO, Secwepemcúl’ecwRestoration and Stewardship Society). To learn about the enormous amounts of carbon stored in terrestrial ecosystems throughout the country — and the essential role it can play in fighting climate change — watch a recording of the launch.	https://wwf.ca/stories/mapping-canadas-carbon-landscapes-cop26/
The effects of nickel subsulfide (Ni3S2) and nickel chloride [Ni(II)] on hydroxylation and deglycosylation of pure 2′-deoxyguanosine (dG) and on hydroxylation of guanine (Gu) residues in calf thymus DNA in the absence or presence of hydrogen peroxide (H2O2) and/or ascorbate (Ascb) were studied with the use of high-performance liquid chromatography. Incubation of 0.75 mm dG with 5 mg Ni3S2/ml (particle size <5 µm) at 37°C in aerated 50 mm Tris/HCl buffer, pH 7.4, resulted in slow hydroxylation of dG to 8-hydroxy-2′-deoxyguanosine (8-OH-dG). This effect was greatly enhanced by 20 mm H2O2. Ni(II) alone at concentrations up to 10 mm was inactive but produced 8-OH-dG in the presence of 20 mm H2O2; the latter caused no dG hydroxylation by itself. Both Ni3S2 and Ni(II) increased the formation of 8-OH-dG from dG exposed to H2O2 + Ascb. At pH 7.4 and constant concentrations of H2O2 and Ascb (20 and 8 mm, respectively), Ni(II) over the concentration range 1–10 mm raised the hydroxylation yield by up to five times that without Ni(II). Also, addition of 7.5 mm Ni(II) more than doubled the hydroxylation yield of Gu residues by the 20 mm H2O2 + 8 mm Ascb mixture (pH 7.4) in denatured DNA and doubled it in native DNA. Ni3S2 and Ni(II) alone had no effect on deglycosylation of dG and did not significantly influence the slow rate of Gu production from dG reacting with H2O2 or Ascb at pH 7.4. However, Ni(II), unlike Ni3S2, increased the extent of dG deglycosylation when added to the dG + H2O2 + Ascb system; 10 mm Ni(II) increased deglycosylation by a factor of 2.5 in 24 h. Thus, nickel carcinogens were shown for the first time to cause and/or enhance both hydroxylation and deglycosylation reactions of dG which may contribute to the observed genotoxic and carcinogenic effects of this metal. Footnotes - ↵1 Research sponsored by the National Cancer Institute, Department of Health and Human Services, under contract No. N01-CO-74101 with Bionetics Research, Inc. - ↵2 To whom requests for reprints should be addressed, at Building 538, Room 205, NCI-FCRF, Frederick, MD 21701-1013. - Received March 27, 1989. - Revision received June 26, 1989. - Accepted August 4, 1989. - ©1989 American Association for Cancer Research.	https://cancerres.aacrjournals.org/content/49/21/5964
Dietary Guidelines for Americans I recently read the 2020-2025 Dietary Guidelines for Americans (DGA) executive summary. While I commend their efforts to promote eating more “nutrient-dense” foods, I just can’t get behind the need for dairy, grains, and “little to no added sugar.” The DGA’s purpose is to provide “science-based advice on what to eat and drink to promote health, reduce the risk of chronic disease, and meet nutrient needs.” I think we can do better. Dairy Why are humans the only species that drinks milk after infancy and drinks milk from another species? How did we let this become our reality? Yes, we need calcium, and milk has calcium, but did you know our bodies are more efficient at getting calcium from: - Almonds - Collard greens - Broccoli - Broccoli rabe - Kale, - Bok choy - Figs - Oranges - Sardines - Canned salmon - Okra So why then does the DGA recommend that dairy is one of the “core elements that make up a healthy dietary pattern”? Grains I won’t go into whole versus refined grains in this discussion but suffice to say that anything with the word “refined” in it is simply a non-food and something the human body does not need. We know that whole grains tend to be high in many nutrients, including fiber, B vitamins, magnesium, iron, phosphorus, and selenium. We also know that pears, strawberries, broccoli, Brussels sprouts, and avocado are high in fiber. Beef, chicken, fish, eggs, seeds, and nuts are high in B vitamins. For our minerals like magnesium, iron, phosphorus, and selenium, the following foods are high in these minerals: - Broccoli, cauliflower, swiss chard, and Brussels sprouts - Magnesium, iron, phosphorus, and selenium. - Shellfish - Nuts and seeds - Organ meats - Eggs - Beans - Cocoa - Avocados - Berries - Sardines - Leafy green vegetables Do you see some repeating foods full of what we need like calcium, B vitamins, and minerals? Sugar Humans do not need it, added or not. Period. Okay, I’m going to say a little more on this topic…sugar and the 50+ other names for sugar have no nutritional value whatsoever. The value is that sugar makes us hungry for more food, and that is why the food industry puts sugar or a variety of sugar in almost everything! If you do not believe me, then I challenge you to download the list of 50+ names for sugar at the end of this post and then go through your pantry, refrigerator, and freezer looking for food that does NOT have one of these ingredients in it. Good luck! What’s Next? So, where do we go from here? DGA guidelines, thank you for recommending we consume as many nutrient-dense foods as possible, based on our cultural preferences and budgets. However, I think we can do better to live healthier lives and reverse our global diet crisis. We do not need dairy and if asked can we live without grains, my answer is “yes”. If you or someone you know is suffering from health issues and would like to learn how to reclaim your health and vitality, please consider joining one of our WILDFIT programs or schedule a call to see how we can help transform your life.	https://www.healththenwellness.com/dietary-guidelines-for-americans/
Statement by the Hon. Barry Whiteside, Governor of the Fund for Fiji Mr. Chairman, it is indeed an honor for me to deliver this address on behalf of the delegation of the Republic of Fiji, on the occasion of the International Monetary Fund and the World Bank Annual Meeting. I congratulate you on your appointment to Chair this joint annual discussions. I also warmly congratulate Ms Christine Lagarde for her appointment to the post of Managing Director and Madame Chairman of the Executive Board of the IMF. Over the past 12 months, Mr. Chairman, the global economy has shown signs of slowing and downside risks have heightened, while the expansion remains unbalanced. In most advanced countries, growth has been weak as pressures from persisting fiscal and financial sector imbalances continue to overshadow the economic outlook. Growth in many emerging and developing economies remains strong although signs of overheating have emerged. As a result, global growth prospects have slightly deteriorated to a growth of 4.0 percent for the world economy this year. Although the global economy has bounced back strongly from the depression in 2009, this path to recovery remains uncertain. As such Mr. Chairman, countries need to focus their efforts towards placing the global economy on a strong, even and sustainable growth path. This is by no means an easy task. Fiscal adjustment for some countries that need to restore fiscal sustainability, for instance, will need to be weighed against the risk of not hurting the recovery by implementing consolidation policies too early. As such, policy cooperation amongst countries as was done, following the global financial crisis, is more critical now, given the constraints on monetary policy, continuing banking sector problems and mounting public sector debt. Policy actions will need to be stepped up to help countries secure a trajectory of positive and sustainable growth. Mr. Chairman, growing inter-linkages among economies of the world have made the global economy more complex and as a result made policy making more challenging. The varying weights of countries in terms of their contributions to growth, trade and financial assets and liabilities, all underscore the huge differences and complexity of the world economy. These complexities, while they have provided explanations on policy failures before the crisis, have also, through the greater inter-linkages, been a cause for success in the periods both before and following the crisis. Increased global inter-connectedness Mr. Chairman, through finance and trade had led to the longest sustained period of growth in world history prior to 2009 and also brought about remarkable achievement through greater global cooperation in the face of the recent global financial crisis. The joint efforts by countries to tackle the crisis saw the economic downturn lasting only three calendar quarters, not for a prolonged period. Among current policy challenges is the need to reestablish strong, sustained and balanced growth for the world economy. While global growth both for this year and next year is around 4 percent, this masks the uneven recovery across the globe. Another major challenge facing many advanced economies is the huge public debt, with the debt-toGDP ratios for many countries, increasing by 25 to 30 percentage points. Many of these countries face the need to stabilize their debt levels and return them to pre-crisis levels through substantial fiscal adjustment. The challenge for emerging economies is rapid growth amidst expansionary monetary and fiscal policies. Rising commodity prices in these economies are raising inflationary expectations and placing upward pressure on inflation. Amidst the huge challenges facing the global economy, it is positive to note Mr. Chairman, that the Fund recognizes its need to better understand the interconnectedness between countries and as such provide policy advice to its member countries. In respect of the Fund’s mandate on surveillance, innovations made since the crisis are noteworthy. The introduction of the “early warning exercise”, which has strengthened the monitoring of tail risks faced by the global economy is positive. As well, the mandatory imposition of the Financial Sector Assessment Program (FSAP) on members with systematically important financial systems and the first ever FSAP for the US and China and the FSAP update for Germany are positive achievements. In addition, the increased focus by the Fund on inter-linkages among economies and the spillover effects of policies from one country on to others, should greatly improve the analysis of the Institution. Having said this Mr. Chairman, we are hopeful that the release of the Spillover Report by the Fund will be a timely one. Associated with this is the greater attention now paid to understanding macro-financial linkages as well as the quality of growth and its impact on macroeconomic stability. Mr. Chairman, the enhancements made by the Fund to financial instruments is a welcome development. Streamlining financing programs to focus on core policies needed to reestablish growth and stability is vital. In this light, the introduction of the Flexible Credit Line and the Precautionary Credit Line should provide the insurance-like protection needed by countries in crisis. Enhancing its cooperation with regional financing arrangements is another commendable move by the Fund. It is important that the Fund continues to explore the need to further strengthen the global financial safety net. On Fund resources, whilst we support the higher quota subscriptions by members, which will allow the Fund to draw additional funds from members at short notice for borrowing and financing purposes, we would like to see better balance, than currently is the case, in the utilisation of the funds particularly in program funding. We note that the $330 billion fund allocation to needy countries since the crisis has greatly helped countries from succumbing to financial pressures. Mr. Chairman, we applaud the Fund on the progress on governance reforms. The greater voice now provided to emerging market economies based on their weight in the global economy, is a historic happening. This will ensure the protection of the voices of some of the poorest in countries which are now included because of their significant weight in the global economy. We also recognize and support the Bank’s initiative on creating jobs, renewably energy sector strategy and the development of a corporate scorecard. Mr. Chairman, I also commend the World Bank for the World Development Report 2012 on Gender Equality and Development. The key messages in the Report clearly underscore the importance of gender mainstreaming in economic development and nation building. The Report also makes an important and timely contribution to the global economic development agenda, ahead of the timeline for achieving the Millennium Development Goals. I applaud the work done so far by the Bank to improve gender equality; however, I fully agree that more can be done in order to enhance the development impact of the Bank’s programs and assistance on this front. While low and middle income countries share similar experiences and constraints in terms of gender disparities, a “one-size fits-all” solution will not work. For developing countries like Fiji, addressing gender gaps can make a greater impact in terms of productivity gains and improving standards of living, particularly for those in rural areas. In recognition of this, the Fijian Government has developed a national Women’s Plan of Action 2010 – 2019 to guide resource allocation towards addressing priority areas which range from formal sector employment opportunities to elimination of violence against women and children. The Plan also ensures that Fiji meets its commitments to various international conventions on women, such as the Convention on Elimination on All Forms of Violence. Women in Fiji are also the major and more successful participants of micro finance and Small to Medium Enterprise (SME) schemes funded by Government, civil society and financial institutions. In this regard, Government and key stakeholders are working together to increase awareness and financial support for micro and SME schemes. Therefore, the World Bank’s plans to increase and mainstream gender equality in operations (including financial lending and technical assistance) and policy dialogue will go a long way to guide and complement national programs. Such complementarities will ensure that development outcomes are realized much earlier. In this regard Mr. Chairman, I welcome the Bank’s focus on improving country level gender diagnostics and gender based data as a means for scaling up its lending, research and technical assistance on gender equality. Not only does this approach promote country level ownership and participation, it ensures that development solutions are tailored to country and regional contexts to enhance its impact. To this end, I lend my support to the four focal areas identified by Management to leverage gender interests in the Bank’s work and I look forward to the corporate strategic document by Management that will steer the Bank’s work in promoting gender equality going forward. I also take this opportunity to call on Management to ensure that the plight of low and middle income countries is featured prominently in the plan and design of this strategic direction. Mr. Chairman, the Fiji economy continues to face many challenges. The actual growth for 2010 was a marginal 0.3 percent, lower than the forecast 0.6 percent. The slow pick up in global demand and high oil and food prices remain a major drag on the economy despite the strong growth in tourism and in major export commodities such as sugar, timber, gold, mineral water and fish. Sector performances have been mixed with strong growth in the services sector especially tourism, outweighing the weaker performances in the agriculture and other resource based sectors. The modest level of economic activity has seen continuing weakness in import growth, particularly in mineral fuels. This year’s growth is targeted at 2.7 percent, expected to come from improved performances across most sectors. Foreign reserves continue to grow on the back of better than expected tourism receipts, stable remittance inflows and domestic export earnings. The outlook for the end of year level remains adequate at around 5 months of import cover. Excess capacity in the economy has kept underlying inflation around 3 percent or below. However, headline inflation continues to be driven by high food and oil prices. The outlook for end of year inflation is around 7 percent with an upward bias resulting mainly from recent unexpected increases in domestic food prices. On bilateral relations, Mr. Chairman, we thank the Fund for the continuous consultations and discussions during the annual Article IV Missions, which provide valuable insight for our assessment of the macroeconomic situation, and more importantly advice on policy issues. The authorities continue to examine issues raised by the Fund during these missions as well as make progress on recommendations on the recent Safeguards Assessment report on compliance and independence issues for the Bank. The Reserve Bank of Fiji continues to focus monetary policy towards supporting growth in priority sectors of the economy while ensuring external and financial stability. As such, the accommodative monetary policy stance by the Reserve Bank has seen the cost of borrowing in the market generally decline since the beginning of the year, auguring well for investment intentions in the country and the cost of financing for the Government. The Reserve Bank continues to review its existing prudential policies, formulating new ones to ensure that the financial system remains sound. Moreover, the Reserve Bank in partnership with financial institutions and donors continue to review national strategies to promote greater financial inclusion in Fiji. Countries in the region recently met in Fiji to discuss concepts and tools for the supervision and regulation of non-bank microfinance institutions, which currently do not fall under the ambit of supervision by the Reserve Bank. Mr. Chairman, on other developments in our country, the authorities continue to reaffirm their commitment to the People’s Charter for Change, Peace and Progress and the Roadmap to Democracy – the guiding principle for policy formulation by the Fijian Government and the new Fiji Constitution. Following the endorsement by Pacific leaders of Fiji’s Charter and the Roadmap recently, the Fijian Government has reiterated its commitment towards hosting elections in September 2014, with voter registration targeted to begin in early 2012. This new system of voting will ensure fairness to all and elimination of voter and political party fraud. Under its civil service reform, the Government aims to improve the efficiency, effectiveness and quality of service delivered by the civil service, particularly focusing on creating a leaner and more efficient civil service, which will bring about cost savings and efficiency gains. The Fijian Government continues to implement its strategies for reforming public enterprises. Reforms and restructurings to date have been undertaken in areas including government residences and supplies, utilities, maritime, meteorological, telecommunications and agriculture-livestock. Future plans include the development of plans for outsourcing service delivery and formulating a corporate governance code. Reform initiatives aimed at improving the accessibility to and utilization of land, have progressed well with more land now available for production and social purposes. Under the Land Use Decree 2010, the i-taukei landowners, potential investors, farmers and the state, are able to benefit through certainty of tenure and improved rental return. Furthermore, Mr. Chairman, to promote rural and outer island development, the Fijian Government had set up an “Integrated Rural Development Framework” and facility to provide additional resources for upgrading rural infrastructure. The declaration of tax free regions in these areas supported by tax and non-tax concessions, are expected to boost development in the nation. Mr. Chairman, while the country is beginning to see tangible benefits of reforms, the Fijian Government remains adamant on its course towards addressing systematic corruption in the country. This year, Fiji has gained recognition by the United Nations Convention against Corruption (UNCAC) for its efforts in combating corruption and white collar crime in the country. This was supported by the closure of 30 convicted court cases by the Fiji Independent Commission Against Corruption (FICAC). Mr. Chairman, the Fijian Government is committed to developing and implementing the best political, social and economic policies in order to advance the goals of good governance, prosperity and peace and national unity. The authorities continue to consult widely with the private sector and non-governmental organizations, to identify policies appropriate to the country’s current social and economic situation. In this light Mr. Chairman, we call upon the international community and our development partners, including the Fund and the World Bank, to bear witness to the progress that Fiji has made. Our Government will continue to push for greater engagement and dialogue with the region, as evident in the recent meetings on “Strengthening Partnership Agreement Amongst Pacific Small Island Developing States” and “Economic Partnership Agreement Negotiations”. We reiterate our call to the Fund and World Bank for greater engagement with Fiji, to boost the support it needs to move the country forward quickly. Joint research programs between the Bretton Woods institutions and country officials will greatly enhance understanding of our country situation and challenges and help with capacity building. This will lead to the better design of appropriate policies to achieve our development agenda. At this juncture, I would like to sincerely thank the Fund for the Technical Assistance Fiji continues to receive, and the work done by the IMF Pacific Technical Assistance Centre in Suva. The Reserve Bank and the Finance Ministry have specifically gained from the training provided by the new IMF Macroeconomic Adviser, and regular discussions between the IMF Resident Rep Office and the Reserve Bank have been very useful. I also thank the World Bank for its assistance to Fiji and our island neighbours through its Regional Office in Sydney, Australia. Finally, Mr. Chairman, my best wishes to the Fund and the Bank in their future efforts and we look forward to closely working with both institutions.	https://www.rbf.gov.fj/responses-by-deputy-governor-mr-inia-r-naiyaga-at-the-usp-fiji-update-suva-on-16-september-2011-2/
According to Albert Einstein: The world we've made as a result of the level of thinking we have done thus far creates problems that we cannot solve at the same level at which we created them. To paraphrase Einstein, networking professionals have the ability to create networks that are so complex that when problems arise they can't be solved using the same sort of thinking that was used to create the networks. Add to this the fact that each upgrade, patch, and modification to a network can also be created using complex and sometimes convoluted thinking, and you realize that the result is networks that are hard to understand and troubleshoot. The networks created with this complexity often don't perform as well as expected, don't scale as the need for growth arises (as it almost always does), and don't match a customer's requirements. A solution to this problem is to use a streamlined, systematic methodology in which the network or upgrade is designed in a top-down fashion. Many network design tools and methodologies in use today resemble the "connect-the-dots" game that some of us played as children. These tools let you place internetworking devices on a palette and connect them with local-area network (LAN) or wide-area network (WAN) media. The problem with this methodology is that it skips the steps of analyzing a customer's requirements and selecting devices and media based on those requirements. Good network design must recognize that a customer's requirements embody many business and technical goals including requirements for availability, scalability, affordability, security, and manageability. Many customers also want to specify a required level of network performance, often called a service level. To meet these needs, difficult network design choices and tradeoffs must be made when designing the logical network before any physical devices or media are selected. When a customer expects a quick response to a network design request, a bottom-up (connect-the-dots) network design methodology can be used, if the customer's applications and goals are well known. However, network designers often think they understand a customer's applications and requirements only to discover, after a network is installed, that they did not capture the customer's most important needs. Unexpected scalability and performance problems appear as the number of network users increases. These problems can be avoided if the network designer uses top-down methods that perform requirements analysis before technology selection. Top-down network design is a methodology for designing networks that begins at the upper layers of the OSI reference model before moving to the lower layers. It focuses on applications, sessions, and data transport before the selection of routers, switches, and media that operate at the lower layers. The top-down network design process includes exploring divisional and group structures to find the people for whom the network will provide services and from whom you should get valuable information to make the design succeed. Top-down network design is also iterative. To avoid getting bogged down in details too quickly, it is important to first get an overall view of a customer's requirements. Later, more detail can be gathered on protocol behavior, scalability requirements, technology preferences, and so on. Top-down network design recognizes that the logical model and the physical design may change as more information is gathered. Because top-down methodology is iterative, some topics are covered more than once in this book. For example, this chapter discusses network applications. Network applications are discussed again in Chapter 4, "Characterizing Network Traffic," which covers network traffic caused by application- and protocol-usage patterns. A top-down approach lets a network designer get "the big picture" first and then spiral downward into detailed technical requirements and specifications. Was this article helpful? What you need to know about… Project Management Made Easy! Project management consists of more than just a large building project and can encompass small projects as well. No matter what the size of your project, you need to have some sort of project management. How you manage your project has everything to do with its outcome.	https://www.ccexpert.us/network-design-2/using-a-topdown-network-design-methodology.html
608 N El Camino Real Ave. San Mateo, CA San Mateo and Burlingame Bilingual - English and Chinese We have a great announcement for students and families who are interested in Chinese culture. We will open a Chinese-culture class on Saturdays. We will have great teachers for each class. Our Chinese Immersion Program offer a 100% instructional model from pre-kindergarten through 8th grade. Students spend 2 hours learning in Mandarin on Saturdays. This program is structured on the total language learning approach incorporating content-based instruction, explicit language instruction, and experiential language learning practices. Students learn the simplified Mandarin writing system, or Chinese characters. Expressive and receptive language development is emphasized in all stages of the program. This course helps children learn Chinese via playing games and discussing fun topics. Children will develop their language skills in four aspects: listening, speaking, reading and writing. Chinese lessons for kids are divided into 6 different levels, ranging from beginner to advanced. Preschool Style Chinese Class This class is for 4-year old and up students (3 year old who already know some Chinese). This class is focused on learning Chinese in daily life. In this 50-minute class, children experience a short version of international school life. We start with circle time, then move on to dance and arts (crafts), and end with good-bye circle time. We encourage students to engage in daily Chinese conversation. Your child can learn Chinese like a native speaker. In this class, teachers let students speak Chinese and communicate with other students and teachers. This is the best way to learn Chinese. Several times a year, we will have parties where students can show the parents their Chinese ability. * Please bring your own snack and drink for snack time Chinese Phonics Class This class will help children transform from a non-reader to a child who is capable of reading easy sentences and spelling in Chinese. Starting with the phonetic sounds of the alphabet, we hope the children can read sentences and stories by themselves by the end of the year. Chinese Conversation Class We teach conversational Chinese focused on a monthly theme in this class. Using role play, games, songs, and dances, we encourage children to express themselves in Chinese. When combining math and Chinese immersion, teachers (and parents or tutors) should be mindful of the disparate ways Chinese treats certain concepts, and the confusion this can create for students. For instance, when speaking about fractions in Chinese, the denominator is identified first, followed by the numerator, a pattern that is inverted in English. For large numbers, the groupings are different, with Chinese building on 万 or “ten-thousand,” while English uses multiples of one-thousand. Chinese is also particularly literal when describing math concepts.	https://www.futurevalley.org/chinese-class
The "unprecedented" uncertainty from Brexit is starting to impact small and medium businesses in Ireland. This is according to the latest SME Market Monitor from the Banking and Payments Federation Ireland (BPFI). While the Irish economy has improved over recent years, with a return to more sustainable growth and trading conditions, there exists a number of challenges for SMEs, including the increased possibility of a no-deal Brexit, as well as a slowdown in the global economy, each of which could prove to be extremely difficult for small businesses. The monitor, which was prepared by the EY-DKM Economic Advisory, looked at 15 different indicators including consumer sentiment, manufacturing PMI, domestic demand, disposable income, retail sales, and the rate of employment. It points to a number of negative trends, including a fall in consumer sentiment, declines in retail sales and domestic demand, and a drop in the number of visitors coming to Ireland from Britain. "Given that the average spend by visitors from Britain is €274 per trip, such a trend will be a concern for those SMEs reliant on the tourism industry," the monitor states. Annette Hughes, director, EY-DKM Economic Advisory, said: "Our analysis shows that consumers are struggling to balance the possibility of an undesirable Brexit outcome with the reality of favourable macroeconomic conditions and there is evidence that this is starting to impact the sectors where SMEs tend to operate, such as retail, food and accommodation – sectors which are arguably also some of the most vulnerable to Brexit." While Ms Hughes points to some of the factors that may mitigate the impact of Brexit uncertainty, such as the strength of the Irish economy, a healthy labour market and double-digit growth in new lending to SMEs, she warns that the short-term outlook for SMEs is a growing concern. "Ultimately, the prospect of a no-deal Brexit or indeed an extension of Article 50, in tandem with a slowdown in some of the Eurozone’s largest economies remains a serious downside risk." "As a consequence, and notwithstanding Ireland’s strong macroeconomic performance, a great deal of caution is warranted in regard to the short-term outlook," Ms Hughes added.
“When you cook in a restaurant, it’s like you’re having a party every night. That’s why I fell in love with the whole thing,” says Daniel Costa, chef-owner of Edmonton’s Corso32, where it can take six weeks to score a dinner reservation. But the simple Italian cooking Costa is now known for wasn’t always his passion. “When I was in culinary school, I was making extremely complicated dishes and rebelling against my Italian heritage,” says the 29-year-old, who changed his attitude during a long trip through southern Italy. Eventually, he got it right and the lesson stuck; handmade pastas feature prominently on his menu at Corso32 and in his home kitchen. These fried gnocchi with black kale and pangrattato (a toasted bread-crumb mixture) is a favourite that Costa likes to whip up for the boys after a late night on the town. It might even help stave off a day-after headache. For the gnocchi, mix ricotta, egg, yolk, 1/2 cup pecorino, nutmeg and 1 tsp salt until well combined. Mix in flour. Place dough on a floured work surface and knead for about 45 seconds to ensure ingredients are combined. Cover the dough with a large bowl or damp cloth and let it rest for 30 minutes. Divide the dough into 4 pieces. Roll each piece into 1/2-inch-thick rope on a floured work surface. Cut each cylinder into ½-inch pieces. (If desired, roll each piece on a gnocchi board to give it a dimpled and indented texture.) Bring a large saucepan of salted boiling water to a boil. Gently add gnocchi and cook for one minute after the gnocchi float to the water’s surface. Using a slotted spoon, remove the gnocchi to an ice bath allow them to cool for 5 minutes. Remove the gnocchi from the ice and set aside. Reserve the cooking water. To make the pangrattato, heat 2 tbsp of olive oil in a small non-stick skillet over medium-high. Add 1 garlic clove and fry until garlic begins to turn golden, then discard garlic. Add the bread crumbs and remaining salt to the pan and reduce heat to medium. Cook, stirring, until bread becomes golden, adding up to 1 tbsp more oil as needed. Transfer pangrattato to a plate and set aside. Bring the water the gnocchi were cooked in back up to a boil and add kale, cooking until tender (about 2 minutes). Remove gnocchi to an ice bath with a slotted spoon. Heat remaining olive oil in a large non-stick skillet over medium-high heat. Add remaining garlic clove, fry until golden and discard garlic. Add gnocchi to pan and shake gently to distribute. Fry, stirring occasionally, for 4 minutes or until the gnocchi turn golden, then toss and fry another minute. Remove the kale from the ice bath, blot with paper towels to remove excess moisture and add to pan with the gnocchi, frying for 2 minutes or until kale becomes slightly crisp. Transfer everything to a serving dish and add lemon zest, half the pangrattato and remaining pecorino. Toss to incorporate. Add lemon juice to taste and top with remaining pangrattato.	https://www.southdundasinbox.com/recipes/pasta-italian-styles/chef-daniel-costa-s-fried-ricotta-gnocchi/
The Town of San Anselmo invites you to an Open House on Saturday, December 13th to learn about the local watershed, flooding history, and the Flood Protection & Watershed Program. In partnership with FEMA’s High Water Mark Initiative, the Town will be unveiling permanent signs in downtown indicating the height of historic floods. Join us at 9:30 a.m. for the unveiling ceremony to hear from local and regional leaders and to learn about protecting your family in the event of a flood. Throughout the morning Marin County Flood Control District will be hosting an Open House, which includes hands-on, family friendly activities and updates on flood mitigation projects. Learn more about the Ross Valley Flood Protection & Watershed Program by visiting www.RossValleyWatershed.org. High Water Mark and Open House Details: Saturday, December 13th, 2014 8:00 a.m. to 12:00 p.m. Sign unveiling at 9:30 Ross Valley Fire Department 777 San Anselmo Ave. San Anselmo, CA 94960 Speakers will include: San Anselmo Mayor Tom McInerney Marin County Supervisor Katie Rice Congressman Jared Huffman The Open House is being held in conjunction with the San Anselmo Recreation Department’s Annual Breakfast with Santa. For breakfast and a chance to see the wizened elf, there are two seating choices, 8 a.m. or 10 a.m. If you pre-register, cost is $6/child and $8/adult. At the door, cost is $8/child and $10/adult. Space is limited so make your reservations today. There will also be a toy drive, so please bring an unwrapped toy! For more information or to make reservations, contact the Recreation Department at 258-4640.	https://www.townofsananselmo.org/793/High-Water-Mark-Initiative
Beautiful. Elysian describes a blissful state and comes from the idyllic Greek mythological place called Elysian Fields. METIS - 150 x 180 The greek word 'metis' meant a quality that combined wisdom and cunning. This quality was considered to be highly admirable in the Mycenaean era. EUMENIDES - 150 x 150 Balance, justice, the mind. Also the final play of the Oresteia, called 'The Eumenides', that illustrates how a sequence of events in the trilogy ends up in the development of social order and a proper judicial system in Athenian society. AETHER - 180 x 180 Quintessence, the material that fills the region of the universe above the terrestrial sphere. NOETON - 200 x 160 Plato - "Noeton" means “to think with consciousness” and derives from the word “nous” which means to intellect (mind/reason) or to have intelligence. THEXINOE - 195 x 175 One of the five muses. Thelxis means to charm, to captivate. Enchantment of the mind. MATHEIN - 180 x 170 To learn, acquisition of knowledge. NUMENA - 200 x 180 The 'thinkable'. Numena is a term for the power of either a deity or a spirit that is present in places and objects, an influence perceptible by mind but not by senses. PHRONESIS - 200 x 150 'Practical Wisdom'. The habit of making the right decision and taking the right action in context. The relentless pursuit of excellence for the common good.	http://www.nikoletasekulovic.com/aether-new-york-2108
Creating a will allows an individual to direct what happens to his or her property after death, and who is responsible for managing that process. A trust can be used as an alternative means of passing property after death, or to provide for loved ones, charitable interests, and others during the grantor’s lifetime. But, these important purposes can be disrupted when there is a question as to the validity of the will or trust. These conflicts can be stressful, since they usually involve disputes among family members–often recently bereaved family members. They can also be complicated. If you’re considering contesting a will or trust, or if you are an administrator or trustee facing a contest, it is in your best interest to consult an experienced estate litigation attorney as early as possible. Will Contests As the term suggests, a will contest arises when a beneficiary or other interested party challenges the validity of a will. In Illinois, a party with standing–someone whose financial interests would be affected by the probate of the will–can contest a will on any grounds that would show the will is not the deceased’s valid, current will. The most common grounds include: - Lack of Testamentary Capacity: You’ve probably heard the phrase “being of sound mind.” To create a valid legal document, including a will, the creator must have the mental capacity necessary to make reasoned decisions. If the testator is unable to understand the issues and formulate a plan due to cognitive decline, an impairment such as Alzheimer’s disease or a brain tumor impacting function, or even medications being administered at the time the will was drafted, the probate court may determine that the testator lacked the capacity to execute a valid will. - Fraud or Forgery: Fraud and forgery may take many forms, including someone forging the testator’s signature or the witnesses’ signatures and the testator being tricked into signing the will. Some examples of this type of trickery might include swapping out a will with the terms the testator had agreed to for another document or leading the testator to believe he or she was signing something else. - Undue Influence: Undue influence occurs when another person’s influence over the testator is so significant that the testator isn’t exercising his or her own will. While anyone can exercise undue influence, courts most carefully scrutinize those who enjoy a fiduciary relationship with the testator. That means someone in a position of trust, such as an attorney or guardian. If a fiduciary relationship exists, a presumption of undue influence arises and the burden shifts to the fiduciary to show that there was no undue influence. - Revocation: In some cases, the will submitted to probate may have been executed by the testator, free from influence or deceit and while of sound mind, but has been revoked. The most common means of revoking a will is to execute a new will that explicitly revokes any prior will. But, a will may also be revoked through destruction of the will with the intent to revoke, the creation of a later inconsistent will, or a written revocation executed under the same terms required for a will. How Do Will Contests Work? Any person or entity with standing to contest a will may file a petition to contest the validity of the will. The petition must be filed within six months of the admission of the will to probate. The executor of the estate has a duty to defend the validity of the will. If he or she fails or refuses to do so, the court can appoint a special administrator to defend the will. Will contests can be complicated, particularly when the issue before the court is the testator’s capacity or state of mind at the time the will was executed. Since will contests arise only after the testator’s passing, the most important witness to the process of preparing and executing the will is unavailable. If you are considering challenging a will or are the executor in a probate matter where a will contest has been filed, it is important to talk with an experienced probate litigator as soon as possible. Not all estate planning attorneys have the knowledge and experience necessary to successfully navigate a will contest. Trust Contests Trust contests are similar to will contests in that a trust may be challenged for the same types of reasons. Like will contests, these typically fall into two categories: failure to follow the required legal form, and an allegation that the trust document doesn’t reflect the grantor’s true intentions. One key difference between a will contest and a trust contest is the way those proceedings are initiated. Since a will is submitted to probate and an estate opened through the probate court, there is already a legal proceeding underway. The petition to contest is filed within that case. Trusts aren’t administered through the courts, so a trust contest must be filed as a fresh legal proceeding. Talk to an Experienced Trust and Estate Litigator Will contests and trust contests can be difficult from a legal perspective, draining on a personal level. If these proceedings aren’t managed correctly, they can also deplete estate or trust assets. When conflict arises over the validity of a will or trust document, you’ll want to seek guidance from an attorney experienced in handling this type of litigation. At Roberts & Caruso, we make it easy for family members, beneficiaries, executors and trustees to get the help they need with will and trust contests. We offer free, no-obligation consultations to help you understand the process and make good decisions about your next steps.	https://www.robertsandcaruso.com/practice-areas/contested-estates-contested-wills-and-trusts/
Last week, we traveled to Michigan to gather with my family members to celebrate my dad’s 80th birthday. My parents spend half of the year, including the summer months, at a cottage in northern Michigan. The cabin holds lifetimes of memories, stories told and retold, expanded upon with little additions, much like the cabin has been. My grandfather built the original tiny cottage for his wife as she was dying of cancer at a very young age. My father spent time here as an adolescent, helping to build it with his father and uncles, hauling water up from the lake, running around with other boys vacationing with their families and getting into trouble. Years later, he brought my mother to see the cabin and she fell in love with it. Eventually, every summer, my mother, brother, sister and I stayed at the cabin from early July through Labor Day. My father would join us when he could get away from work. As kids, our days were long and unencumbered. We slept late, wandered through the woods, hunted turtles, swam in the lake, made fudge and played games late into the evenings. Throughout the years, my siblings and I have returned to the cabin with our families, to form new memories. It is rare that any of us are there at the same time, so this past week when most of us were able to gather for at least a few days, inevitably the stories, pictures and home movies came out. As I listened to my family reminisce, I realized that our memories are as varied as we are. Even the experiences we shared as a family are remembered from our unique viewpoints. What the cabin is for me is not the same as what it is for my sister or my nephew. Nevertheless, we all share the common bond of that place. As I walked down to the beach on our last night at the cabin, I smiled to see an old friend who has been there long before my grandfather’s time: A large, white birch tree near the water. This tree was my special place as a child. I would sit on its crooked base and watch the boaters and fishermen on the water. As I touch its beautiful white bark, I consider that it had been there when my father collected water from the lake for his mother to use, and years later when my brother proudly put in his rowboat, earned by working for a man on the other side of the lake. It was there when my father and a much younger me launched our canoe to paddle back into the lagoon, and later, when my sister’s girls played on the beach. More recently, the tree marked our dock as we headed home across the water after my son learned to water ski. This week, it quietly observed my great nephew’s first cast of a fishing line. I don’t know how long that tree will continue to stand on its eroding shore. I hope it lives long enough for my grandchildren to sit in its crook. And so, I share with you my reflections from the week. Make memories where you can. Envelop your family in them and breathe their piney scent whenever you have a moment to reflect. Share the stories, and add a piece here and there. Roast a marshmallow, make a s’more and lick the chocolate off your fingers while telling a ghost story or two. When the next generation comes along, if they’re lucky like I am, they will feel part of a shared special place that their children will also grow to love. Support Local Journalism Sarah Kilgore lives in Edwards with her husband, son and several pets, enjoying all that the Vail Valley has to offer.	https://www.vaildaily.com/opinion/editorial/vail-daily-column-a-week-at-the-cottage/
When choosing toys for your baby or toddler, make sure you inspect them carefully for things that could cause injury. The American Academy of Family Physicians offers these guidelines when choosing safe child toys: - Make sure each toy is sturdy, well-made, and appropriate for your child’s age. - Don’t let your baby play with toys that have sharp edges or points, or small parts. - Make sure parts fit securely and are not loose. - Read labels to check for safety information. Look for toys that are non-toxic, washable and hygienic. - Don’t let your baby play with any toys that are smaller than 1 3/4 inches in diameter or 2 inches long, as these may pose a choking hazard.	http://www.newbornbabyzone.com/health-safety/check-babys-toys-for-safety/
As the debt is only possibly irrecoverable at this stage you may only ‘provide’ for the debt. You will therefore reduce the amounts owing to you at 31 December by £1175. For income tax purposes, the doubtful debt expense of £1,000 is allowable because it is specific and therefore written off in the profit and loss account. Provided that you have paid the VAT over to HMRC, you can reclaim VAT that you paid and which you have not received from the customer, provided that either the Official Receiver has notified you that no monies will be forthcoming or the debt becomes six months old or more. You add the amount of VAT you are reclaiming to the amount of VAT you are reclaiming on your purchases (input tax) and put the total figure in Box 4 of your VAT Return. If you cash account for VAT, you will never have paid the VAT over to HMRC and therefore, you are not due a refund from HMRC. You will merely need to do the accounting adjustment above. If you would like more information of the treatment of bad debts, please contact your local TaxAssist Accountant.	https://www.taxassist.co.uk/resources/questions-and-answers/bad-debt-adjustments
--- author: - 'Anatoly M. Kamchatnov[^1]' - 'Yaroslav V. Kartashov[^2]' title: 'Quasi-one-dimensional flow of polariton condensate past an obstacle' --- Introduction ============ Recent experimental progresses in studying the microcavity polaritons have lead to a huge growth of interest in their collective dynamics (see, e.g., review articles [@keeling-2007; @amo-2010] and references therein). Polaritons possess an extremely small effective mass which allows their condensation at temperatures much greater than that of ultracold atomic vapors. Besides that, parameters of the polariton superfluid can be easily tuned with the use of resonant lasers. However, polaritons have a finite lifetime, and to maintain their steady-state population a continuous pumping is required. Experimentally, it is observed [@kasprzak-2006; @lai-2007] that above a threshold pumping strength an accumulation of low energy polaritons is accompanied by a significant increase of spatial coherence that extends over the entire cloud of polaritons which can then be described by a single order parameter (polariton condensate wave function) obeying an effective Gross-Pitaevskii equation. On the contrary to the atomic condensate situation, the density of the polariton condensate is not an arbitrary parameter anymore. Instead, it is determined by the condition of balance between pumping and dissipation processes. This implies that the non-conservative effects can play a crucial role in the condensate’s nonlinear dynamics. For example, the generation of oblique solitons by the flow of a condensate past a localized obstacle has been observed [@amo-2011] at subsonic speed which is impossible in the case of conservative atomic condensate. The formation of oblique solitons followed by their decay into vortex streets in a non-uniform cloud of polariton condensate has been studied in [@grosso-2011]. Other geometries are also of great interest. In particular, quasi-1D flow of polariton condensate along “quantum wires” was studied experimentally in [@wertz-2010]. In the atomic condensate case such a flow past an obstacle leads to generation of dispersive shock waves propagating upstream and downstream from the obstacle [@hakim-1997; @pavloff-2002; @radouani-2004], as it was observed in the experiment [@ea-2007] and explained theoretically in [@legk]. However, it is easy to see that this theory cannot be applied to the polariton condensate flow which density must be fixed (if pumping and dissipation are balanced) far enough from the obstacle thus preventing formation of jumps in the flow parameters. This means that in the dissipative case the dispersive shock waves generated by the flow must always be attached to the obstacle and relax to the steady-state flow far enough from the obstacle. This qualitative difference between wave patterns in conservative and dissipative cases makes the theory of dissipative flow much more complicated and this Letter is devoted to its development. Theoretical model ================= Several models have been suggested for theoretical description of polariton condensate (see, e.g., [@km-2003]). They were based on various generalizations of the Gross-Pitaevskii (GP) equation. Here we have to take into account the effects of losses and pumping of polaritons on generation of dispersive shock waves by the flow of condensate past an obstacle. To this end, we will use the simple model introduced in [@kb-2008] where nonresonant pumping due to stimulated scattering of polaritons into the condensate and their linear losses are described by the effective “gain” term ${\partial}_t\psi={\gamma}\psi$ where $\psi$ is the polariton condensate wave function. The saturation of gain is modeled by the nonlinear term ${\partial}_t\psi=-{\Gamma}\|\psi\|^2\psi$ which brings the condensate density into equilibrium with the one of external reservoir, $\rho\equiv\|\psi\|^2={\gamma}/{\Gamma}$. The dispersion of the lower branch of polaritons in the effective mass approximation and the nonlinear interaction due to the exciton component of polaritons lead to the following generalized GP equation $$\label{eq1} i\psi_t+\tfrac12\psi_{xx}-\|\psi\|^2\psi=V(x)\psi+i({\gamma}-{\Gamma}\|\psi\|^2)\psi,$$ (written in standard non-dimensional units) where $V(x)$ denotes the potential of the obstacle and $x$ is the direction of the flow of the condensate. The theory developed below can be generalized to other forms of nonlinear gain as, for example, the model considered in [@wc-2007], and the results remain qualitatively similar. However, to be definite, we shall consider here the model (\[eq1\]) for which the description looks especially simple. It is easy to see that eq. (\[eq1\]) without potential ($V(x)\equiv0$) admits a plane wave solution which constant amplitude is determined by the gain and damping coefficients, $$\label{eq2} \psi=ae^{i(u_0x-\mu t)},\quad a=\sqrt{{{\gamma}}/{{\Gamma}}},\quad \mu={{\gamma}}/{{\Gamma}}+\tfrac12u_0^2,$$ where $u_0$ denotes the uniform flow velocity of the condensate and $\rho_0={{\gamma}}/{{\Gamma}}$ is its density. The analysis of modulation stability of such plane waves with respect to harmonic perturbations $\propto \exp[i(Kx-{\Omega}t)]$ shows that the disturbance propagates along the wave (\[eq2\]) with the dispersion law $$\label{eq3} {\Omega}=u_0K-i{\gamma}\pm\sqrt{K^2(a^2+K^2/4)-{\gamma}^2}.$$ The expression under the square root vanishes at the wave number $$\label{eq4} K_c=a\left[2(\sqrt{1+{\Gamma}^2}-1)\right]^{1/2}$$ which separates two different regimes of evolution of a harmonic perturbation; see the plots in fig. \[fig.1\]. In the conservative limit ${\gamma},\,{\Gamma}\to0,\,{\gamma}/{\Gamma}=\mathrm{const}$ we reproduce the standard Bogoliubov dispersion law with sound velocity $c_s=a=\sqrt{\rho_0}$. We shall call this parameter “sound velocity” also for small ${\gamma},\,{\Gamma}\ll1$; it corresponds to the almost linear part ${\Omega}\approx c_sK$ of the dispersion law for $K_c\approx {\Gamma}a \ll K \ll a$ (see fig. \[fig.1\]). Notice that, as long as repulsive nonlinearity is considered, the plane wave solution (\[eq2\]) is modulationally stable, since $\mathrm{Im}(\Omega)<0$ for any $K$-value. ![(Color online) The real and imaginary parts of $\Omega$ as functions of $K$ (see eq. (\[eq3\])) are shown in the case ${\gamma}={\Gamma}=0.05$ and $u_0=0$. []{data-label="fig.1"}](fig1.pdf) It is convenient to transform eq. (\[eq1\]) into a hydrodynamic form by means of the substitution $$\label{eq5} \psi=\sqrt{\rho}\,\exp\left(i\int^x u(x',t)\upd x'\right)e^{-i\mu t}$$ which yields $$\label{eq6} \rho_t+(\rho u)_x=2{\Gamma}\rho(\rho_0 - \rho),$$ $$\label{eq7} u_t+uu_x+\rho_x+\left(\frac{\rho_x^2}{8\rho^2}-\frac{\rho_{xx}}{4\rho}\right)_x=-V_x.$$ The last term in the left-hand side of (\[eq7\]) describes the dispersion effects; the right-hand side of (\[eq6\]) describes the gain and loss effects in the system; the other terms in eqs. (\[eq6\]),(\[eq7\]) have standard hydrodynamic meaning. Hydraulic approximation ======================= In what follows we assume that the size $l$ of the obstacle is much greater than the “healing length” (equal to unity in our non-dimensional variables). In our numerical simulations we will use an obstacle potential in the form $$\label{eq8} V(x)=V_m\exp(-x^2/l^2),\quad l\gg1.$$ All numerical plots below are obtained for $l=5$. We are interested in stationary patterns with $\rho_t\equiv0,\,u_t\equiv0$ generated by the steady flow of the condensate past a localized obstacle ($V(x)\to0$ as $\|x\|\to\infty$), i.e. $\rho$ and $u$ must satisfy the boundary conditions $$\label{eq9} \rho\to \rho_0,\quad u\to u_0\quad \text{as}\quad \|x\|\to\infty.$$ Then eq. (\[eq6\]) can be reduced to $$\label{eq10} (\rho u)_x=2{\Gamma}\rho(\rho_0-\rho)$$ and the stationary eq. (\[eq7\]) can be integrated once to give $$\label{eq10a} \tfrac12{u^2}+\rho+\frac{\rho_x^2}{8\rho^2}-\frac{\rho_{xx}}{4\rho}+V(x)= \tfrac12{u_0^2}+\rho_0$$ For $l\gg 1$ it is natural to assume that the wave pattern has the characteristic length about $l$ and, hence, the dispersive terms in (\[eq10a\]) having higher order derivatives are negligibly small compared with other terms; then we get $$\label{eq11} \tfrac12{u^2}+\rho+V(x)= \tfrac12{u_0^2}+\rho_0.$$ Integration of eq. (\[eq10\]) over space shows that stationary patterns satisfy the condition (see also [@kb-2008]) $$\label{eq12} \int_{-\infty}^{\infty}\rho(\rho_0-\rho)\upd x=0.$$ Excluding $u$ from Eqs. (\[eq11\]) and (\[eq10\]) we get the equation $$\label{eq13} \left(\rho\sqrt{u_0^2+2(\rho_0-\rho-V(x))}\right)_x=2{\Gamma}\rho(\rho_0-\rho)$$ which determines the dependence of $\rho$ on $x$ for a given potential $V(x)$ provided the solution satisfies the condition (\[eq9\]). This equation represents the so-called hydraulic approximation for our system. (It generalizes the well-known Thomas-Fermi approximation to non-zero flow velocity.) At the tails of the wave pattern ($\|x\|\to\infty$) we can neglect the potential $V(x)$ and linearize eq. (\[eq13\]) with respect to small deviations $\rho-\rho_0$ from the steady state. This gives the asymptotic behavior $$\label{eq14} \|\rho-\rho_0\|\propto\exp\left(\frac{2{\Gamma}\rho_0u_0}{\rho_0-u_0^2}x\right)$$ which shows that such tails extending beyond the range of the potential can exist only under the condition $$\label{eq15} \frac{\|\rho_0-u_0^2\|}{{\Gamma}\rho_0u_0}\gg l.$$ This means that the sound velocity $c_s=\sqrt{\rho_0}$ separates two different regimes—subsonic ($u_0<c_s$) and supersonic ($u_0>c_s$)—with drastically different properties. We shall discuss them separately. Subsonic flow ============= The hydraulic approximation describes the profiles of the disturbance well enough practically in the entire subsonic regime $u_0<\sqrt{\rho_0}$ if the velocity is not too close to the sound velocity. In fig. \[fig.2\] we compare the direct numerical solution of eq. (\[eq1\]) with its hydraulic approximation obtained by the numerical solution of eq. (\[eq13\]); quite good agreement is observed (notice that in all simulations we use ${\gamma}={\Gamma}=0.05$ that gives sound velocity $c_s=1$). Naturally, the asymptotic behavior (\[eq14\]) is also confirmed in the upstream flow $x<0$. However, there is no linearized solution decaying to zero at $x\to+\infty$, hence the transition to the asymptotic steady flow in the downstream region $x>0$ has to occur within the range of the potential $V(x)$, and this conclusion is confirmed by direct numerical solution. Indeed, as one observes in fig. \[fig.2\], the stationary disturbance has a smooth shape with a very long monotonically decaying left tail. This tail is attached to the density dip located almost completely within the potential region. The amplitude of the disturbance (i.e. the difference between maximal and minimal density in the condensate) in the subsonic regime monotonically increases with growth of the potential strength. It is convenient to introduce the “number of particles” $$\label{number} N=\int_{-\infty}^{\infty}[\rho^{1/2}(x)-a]^2\upd x$$ disturbed by the flow. This variable monotonically increases with the strength of potential \[fig. \[fig.3\](a)\]. Interestingly, there exists a critical strength of potential $V_m^{cr}$ above which one cannot find stationary profiles. When $V_m\to V_m^{cr}$ the tangent to $N(V_m)$ becomes vertical. For this strength of potential the amplitude of disturbance is maximal. Note that the critical potential strength $V_m^{cr}$ diverges when the incident velocity of the condensate $u_0\to0$ and it monotonically decreases when $u_0$ increases \[fig. \[fig.3\](b)\]. We were able to obtain the critical potential strength only for $0<u_0<0.8c_s$. For higher velocities $[0.8c_s<u_0<c_s]$ the condensate starts developing small oscillations on its right tail. Solutions characterized by different number of oscillations on their right tail may form continuous families that may be obtained even for potentials with $V_m>1$. ![(Color online) Comparison of density profile at $u_0=0.6$, $V_m=0.1$, and ${\gamma}={\Gamma}= 0.05$ obtained numerically from eq. (\[eq1\]) (black curve) and in the framework of the hydraulic approximation (red curve). The dashed line shows plane wave solution (\[eq2\]) in absense of potential. []{data-label="fig.2"}](fig2.pdf) ![(a) Number of particles (\[number\]) versus potential strength for subsonic flow with $u_0=0.6$. The circle corresponds to the parameters used in fig. \[fig.2\]. (b) Critical strength of potential versus $u_0$. In all cases ${\gamma}={\Gamma}=0.05$.[]{data-label="fig.3"}](fig3.pdf) The condition (\[eq15\]) breaks down when the incident velocity $u_0$ is close to the critical value $\sqrt{\rho_0}$. In this case the potential $V(x)$ cannot be neglected in the hydraulic approximation (\[eq13\]. Even more, as $u_0$ approaches the critical value, the upstream density profile steepens and its slope can become so large that the dispersive terms in eq. (\[eq10a\]) can no longer be neglected. As is known, the dispersive effects lead to generation of oscillations in the regions with fast change of the variables. Hence, for $u_0\gtrsim \sqrt{\rho_0}$ we should expect generation of dispersive shock waves and now we proceed to the discussion of the supersonic flow. Supersonic flow =============== If $u_0>\sqrt{\rho_0}$ and the condition (\[eq15\]) is fulfilled, then the linearized solution (\[eq14\]) describes decaying disturbance in the downstream flow at $x\to+\infty$. Thus, contrarily to the subsonic case, the right tail for $x>0$ can extend far beyond the range of the potential. ![(Color online) The density distribution for shock wave at $u_0=1.4$, $V_m=0.8$, and ${\gamma}={\Gamma}=0.05$. Red curves show analytical approximations of tails from eqs. (\[eq14\]) and (\[eq40\]). Dashed line shows plane wave solution in the absence of potential. []{data-label="fig.4"}](fig4.pdf) As mentioned above, the generation of an oscillatory dispersive shock wave is expected in the upstream flow. This is confirmed by numerical solution of (\[eq1\]) with the supersonic boundary conditions; see fig. \[fig.4\]. The dispersive shock waves can be represented as a modulated solution of the undisturbed (${\Gamma}={\gamma}=0,\,V(x)\equiv0$) equation (\[eq1\]) or the system (\[eq6\]), (\[eq7\]) which can be written in the form (see, e.g., [@kamch2000]) $$\label{eq16} \begin{split} \rho(x,t)&=\nu_1+(\nu_2-\nu_1){\mathrm{sn}}^2(\sqrt{\nu_3-\nu_1}\,\theta,m)\\ &=\tfrac14({\lambda}_1-{\lambda}_2-{\lambda}_3+{\lambda}_4)^2+({\lambda}_2-{\lambda}_1)({\lambda}_4-{\lambda}_3)\\ &\times{\mathrm{sn}}^2(\sqrt{({\lambda}_4-{\lambda}_2)({\lambda}_3-{\lambda}_1)}\,\theta,m), \end{split}$$ $$\label{eq17} \begin{split} u(x,t)&=U-{q}/{\rho(x,t)},\\ q&=-\tfrac18({\lambda}_1-{\lambda}_2-{\lambda}_3+{\lambda}_4)({\lambda}_1-{\lambda}_2+{\lambda}_3-{\lambda}_4)\\ &\times({\lambda}_1+{\lambda}_2-{\lambda}_3-{\lambda}_4), \end{split}$$ where $$\label{eq19} \theta=x-Ut,\quad U=\tfrac12({\lambda}_1+{\lambda}_2+{\lambda}_3+{\lambda}_4)$$ $\nu_1,\nu_2,\nu_3$ are related to the Riemann invariants ${\lambda}_i,i=1,2,3,4,$ by the formulae $$\label{eq18} \begin{split} \nu_1&=\tfrac14({\lambda}_1-{\lambda}_2-{\lambda}_3+{\lambda}_4)^2,\\ \nu_2&=\tfrac14({\lambda}_1-{\lambda}_2+{\lambda}_3-{\lambda}_4)^2,\\ \nu_3&=\tfrac14({\lambda}_1+{\lambda}_2-{\lambda}_3-{\lambda}_4)^2, \end{split}$$ and $$\label{eq20} m=\frac{\nu_2-\nu_1}{\nu_3-\nu_1}=\frac{({\lambda}_2-{\lambda}_1)({\lambda}_4-{\lambda}_3)}{({\lambda}_3-{\lambda}_1)({\lambda}_4-{\lambda}_2)}.$$ In the modulated wave the Riemann invariants ${\lambda}_i$ become slow functions of $x$ and $t$ and their evolution is governed by the Whitham equations. They were derived for the case of eq. (\[eq1\]) with ${\Gamma}=0$ in [@kamch2004] and their generalization to the nonzero ${\Gamma}$ is straightforward. Therefore we will write down here the final result without its derivation. In our stationary case ${\lambda}_i$ do not depend on $t$ and the phase velocity $U$ equals to zero, $$\label{eq21} {\lambda}_1+{\lambda}_2+{\lambda}_3+{\lambda}_4=0.$$ Then the Whitham equations can be written in the form $$\label{eq22} \frac{\upd{\lambda}_i}{\upd x}=\frac2{L}\cdot\frac{I_1{\lambda}_i+I_2}{\prod_{m\neq i}({\lambda}_i-{\lambda}_m)},$$ where $$\label{eq23} \begin{split} I_1&={\Gamma}\int_{\nu_1}^{\nu_2}\frac{\nu(\rho_0-\nu)}{\sqrt{\mathcal{R}(\nu)}}\upd\nu,\\ I_2&=\frac{{\Gamma}u_0\rho_0}2\int_{\nu_1}^{\nu_2}\frac{\rho_0-\nu}{\sqrt{\mathcal{R}(\nu)}}\upd\nu, \end{split}$$ $$\label{eq24} \mathcal{R}=(\nu-\nu_1)(\nu-\nu_2)(\nu-\nu_3)$$ and $L$ is the wavelength $$\label{eq25} L=\frac{2{\mathrm{K}}(m)}{\sqrt{({\lambda}_3-{\lambda}_1)({\lambda}_4-{\lambda}_2)}},$$ ${\mathrm{K}}(m)$ being the complete elliptic integral of the first kind. Due to the special structure of Eqs. (\[eq22\]), the symmetric functions of ${\lambda}_i$, $$\label{eq26} \begin{split} s_1&=\sum_i{\lambda}_i,\quad s_2=\sum_{i\neq j}{\lambda}_i{\lambda}_j,\quad s_3=\sum_{i\neq j\neq k}{\lambda}_i{\lambda}_j{\lambda}_k,\\ s_4&={\lambda}_1{\lambda}_2{\lambda}_3{\lambda}_4, \end{split}$$ obey very simple equations $$\label{eq27} \frac{\upd s_1}{\upd x}=0,\quad \frac{\upd s_2}{\upd x}=0,\quad \frac{\upd s_3}{\upd x}=\frac{2I_1}L,\quad \frac{\upd s_4}{\upd x}=-\frac{2I_2}L.$$ The condensate density $\rho$ oscillates in the interval $\nu_1\leq \rho\leq\nu_2$, i.e. $\nu_1$ and $\nu_2$ are the envelopes of the oscillations in the dispersive shock wave. Let us study their asymptotic behavior at $x\to-\infty$. The asymptotic plane wave corresponds to $\nu_2=\nu_1$ (or ${\lambda}_2={\lambda}_1$) so that the parameters of the incident condensate are expressed in terms of the Riemann invariants as $$\label{eq28} \rho_0=\tfrac14({\lambda}_4-{\lambda}_3)^2$$ and $$\label{eq29} u_0=\tfrac12({\lambda}_3+{\lambda}_4-2{\lambda}_1).$$ From (\[eq21\]), (\[eq28\]) and (\[eq29\]) we find the asymptotic values of the Riemann invariants $$\label{eq30} \begin{split} {\lambda}_1&={\lambda}_2=-\frac{u_0}2,\\ {\lambda}_3&=\frac{u_0}2-\sqrt{\rho_0},\quad {\lambda}_4=\frac{u_0}2+\sqrt{\rho_0}. \end{split}$$ The periodic solution corresponds to the ordering ${\lambda}_1\leq{\lambda}_2\leq{\lambda}_3\leq{\lambda}_4$, so that from ${\lambda}_3\geq{\lambda}_2$ we find that $u_0\geq\sqrt{\rho_0}$, i.e. the incident flow must be supersonic for the generation of such a stationary shock wave. The wavelength (\[eq25\]) of small amplitude oscillations ($m=0$) reduces in the limit $x\to-\infty$ to $$\label{eq31} L=\frac{\pi}{\sqrt{u_0^2-\rho_0}}.$$ The asymptotic values of $\nu_1,\nu_2,\nu_3$ at $x\to-\infty$ are $\nu_1(-\infty)=\nu_2(-\infty)=\rho_0$, $\nu_3(-\infty)=u_0^2$. We introduce small deviations from these values, $$\label{eq32} \nu_1=\rho_0+\delta \nu_1,\quad \nu_2=\rho_0+\delta \nu_2,\quad \nu_3=u_0^2+\delta\nu_3,$$ and linearize eqs. (\[eq27\]) with respect to $\delta\nu_i$ with account of the identities $$\label{eq33} \begin{split} &\nu_1+\nu_2+\nu_3=-2s_2,\\ &\nu_1\nu_2+\nu_1\nu_3+\nu_2\nu_3=-s_2^2+4s_4,\quad \nu_1\nu_2\nu_3=s_3^2. \end{split}$$ As a result we arrive at the equations $$\label{eq34} \frac{\upd(\delta\nu_1+\delta\nu_2)}{\upd x}=-\frac{\upd\delta\nu_3}{\upd x}=\frac{2{\Gamma}\rho_0u_0}{u_0^2-\rho_0}(\delta\nu_1+\delta\nu_2).$$ Hence we get $$\label{eq35} \delta\nu_1+\delta\nu_2,\delta\nu_3\propto \exp\left(\frac{2{\Gamma}\rho_0u_0}{u_0^2-\rho_0}x\right).$$ These equations suggest that the sum of deviations $\delta\nu_1+\delta\nu_2$ decays at $x\to-\infty$ faster than the deviations $\delta\nu_1$ and $\delta\nu_2$ separately. If we introduce small deviations of the Riemann invariants from their asymptotic values (\[eq30\]) $$\label{eq36} \begin{split} {\lambda}_1&=-\frac{u_0}2+\delta{\lambda}_1,\quad {\lambda}_2=-\frac{u_0}2+\delta{\lambda}_2\\ {\lambda}_3&=\frac{u_0}2-\sqrt{\rho_0}+\delta{\lambda}_3,\quad {\lambda}_4=\frac{u_0}2+\sqrt{\rho_0}+\delta{\lambda}_4, \end{split}$$ then we find that the asymptotic behavior (\[eq35\]) corresponds to $\|\delta{\lambda}_3\|,\|\delta{\lambda}_4\|\ll\|\delta{\lambda}_1\|,\|\delta{\lambda}_2\|$ so that $$\label{eq37} \begin{split} &\nu_1\cong\tfrac14(2\sqrt{n_0}+\delta{\lambda}_1-\delta{\lambda}_2)^2,\\ &\nu_2\cong\tfrac14(2\sqrt{n_0}-\delta{\lambda}_1+\delta{\lambda}_2)^2,\\ &\nu_3\cong\tfrac14(2u_0-\delta{\lambda}_1-\delta{\lambda}_2)^2, \end{split}$$ and in leading order approximation $\delta\nu_1+\delta\nu_2=0$ which means that (\[eq35\]) corresponds to higher order corrections. Thus, in the main approximation we assume that $\nu_3\cong u_0^2$ and from (\[eq23\]) we obtain the following equations for $\delta{\lambda}_1$ and $\delta{\lambda}_2$: $$\label{eq38} \begin{split} \frac{\upd\delta{\lambda}_1}{\upd x}&=\frac{{\Gamma}\rho_0u_0}{2(u_0^2-\rho_0)}(\delta{\lambda}_1-\delta{\lambda}_2),\\ \frac{\upd\delta{\lambda}_2}{\upd x}&=-\frac{{\Gamma}\rho_0u_0}{2(u_0^2-\rho_0)}(\delta{\lambda}_1-\delta{\lambda}_2). \end{split}$$ From these equations we get $$\label{eq39} \delta{\lambda}_{1,2}\cong \pm C\exp\left(\frac{{\Gamma}\rho_0u_0}{u_0^2-\rho_0}x\right)$$ ($C$ is the integration constant), and hence the envelopes of oscillations in the dispersive shock decay at leading order approximation as $$\label{eq40} \delta\nu_1,\delta\nu_2\propto \exp\left(\frac{{\Gamma}\rho_0u_0}{u_0^2-\rho_0}x\right).$$ This is slower than the dependence in (\[eq35\]) as it should be. ![(a) Number of particles versus potential strength for supersonic flow with $u_0=1.4$. The circle corresponds to solution in fig. \[fig.4\]. (b) Critical strength of potential versus $u_0$. In all cases ${\gamma}={\Gamma}=0.05$. []{data-label="fig.5"}](fig5.pdf) These analytical predictions were confirmed by the direct solution of eq. (\[eq1\]). As one can see from fig. \[fig.4\], a typical shock wave solution may extend far beyond the repulsive potential, both in the positive and negative $x$ directions. Note the excellent agreement at $x\to\pm\infty$ between the numerical density and its analytical fit that uses the expressions (\[eq14\]) and (\[eq40\]). The wavelength of small amplitude oscillations estimated numerically for the parameters of the flow shown in fig. \[fig.4\] is equal to $L_{num}\cong 3.16$ and the analytical formula (\[eq31\]) gives $L_{theor}\cong 3.21$, with an accuracy better than 2%. It should be stressed that the solution in fig. \[fig.4\] is fully stationary (that is, it does not change in time), in contrast to previously reported one-dimensional shock waves in conservative systems. The oscillating left tail of the dispersive shock wave in the supersonic regime becomes more pronounced with increase of potential strength $V_m$ and incident velocity $u_0$. Just as for the subsonic flow the increase of $V_m$ is accompanied by an increase of the amplitude of the shock wave. However, in the supersonic regime the amplitude of shock wave may be comparable with the amplitude of the unperturbed plane wave and the density may decrease almost to zero in the downstream region, especially for strong potentials. The number of particles in the shock wave increases with $V_m$ \[fig. \[fig.5\](a)\] and the character of this dependence also points to the existence of a critical defect strength beyond which shock waves do not exist. A linear stability analysis performed for stationary solutions shows that wave patterns are stable for any strength of potential $V_m$ up to the critical one as long as the incident velocity is small enough, $u_0<0.9c_s$. When $u_0>0.9c_s$, stationary solutions become unstable if the strength of potential exceeds certain limiting value $V_m^{cr}$ (the instability of the dispersive shock waves at $V_m>V_m^{cr}$ is accompanied by the development and emission of small-scale disturbances on the right tail of the shock wave). Still, even shock waves with very long oscillating tails, like the one shown in fig.\[fig.4\], may be stable. The critical value of potential increases monotonically with $u_0$ as it is shown in fig. \[fig.5\](b). Conclusion ========== It is instructive to compare our results with those obtained in the case of the flow of a conservative fluid past an obstacle. As was shown in [@legk], in the conservative case one can distinguish three characteristic ranges of flow velocity—subcritical ($u_0<u_-$), transcritical ($u_-<u_0<u_+$), and supercritical ($u_0>u_+$) where the critical values of the velocity are located at opposite sides of the value of the sound velocity ($u_-<c_s<u_+$). If the flow velocity is subcritical or supercritical, then the disturbance is stationary and has the dimension of the obstacle’s size with definite sign of the difference $\rho-\rho_0$; it has a form of a dip in subcritical region and of a hump in supercritical region. In the present case of non-conservative dynamics the disturbance must have “dips” and “humps” to be compensated in the integral (\[eq12\]). Examples of such disturbances shown in figs. \[fig.2\] and \[fig.4\] demonstrate this property. In the conservative situation the transcritical regime corresponds to non-stationary generation of upstream and/or downstream dispersive shock waves. In the non-conservative situation this regime actually disappears and instead a supersonic flow generates a [stationary dispersive shock wave]{} upstream the obstacle. There are no downstream dispersive shock waves; instead they are replaced by smooth profiles decaying asymptotically to the stationary plane wave. The above properties of the wave patterns generated by the flow of polariton condensate past an obstacle are derived for the model (\[eq1\]). We believe, however, that these properties remain qualitatively the same for other models having stationary plane wave states formed due to balance of pumping and dissipation effects. In particular, this theory can find applications to a so-called “superfluid motion of light” [@pr-1993; @bcz-2001; @lm-2010]. We are grateful to A. Amo, N. Berloff, J. Bloch, A. Bramati, I. Carusotto, C. Ciuti, E. Giacobino, Yu.G. Gladush, N. Pavloff, D. Sanvitto for discussions of superfluidity in the cavity polariton physics. We thank RFBR for partial support. [0]{} . . . . . . . . . . . . . . . . . . . . [^1]: E-mail: [^2]: E-mail:
--- abstract: 'In particular, to minimize the average network delay under any arbitrary spatial distribution of the ground users, the optimal cell partitions of UAVs and terrestrial base stations (BSs) are determined. To this end, using the powerful mathematical tools of optimal transport theory, .' author: - '\' bibliography: - 'references.bib' title: 'Optimal Transport Theory for Cell Association in UAV-Enabled Cellular Networks ' --- Introduction ============ Nevertheless, there are many technical challenges associated with the UAV-based communication systems, which include deployment, path planning, flight time constraints, and cell association. In [@Letter] and [@Irem], the authors studied the efficient deployment of aerial base stations to maximize the coverage performance. The path planning challenge and optimal trajectory of UAVs were addressed in [@Jeong] and [@Qing]. Moreover, UAV communications under flight time considerations was studied in [@HoverTime]. In [@Vishal], the authors analyzed the user-UAV assignment for capacity enhancement of heterogeneous networks. However, this work is limited to the case in which users are uniformly distributed within a geographical area. In fact, none of the previous studies in [@Letter; @HoverTime; @Qing; @orfanus; @mozaffari2; @zhangLetter; @HouraniModeling; @Irem; @Alonso; @Vishal; @Jeong; @zhang; @OTUAV], addressed the delay-optimal cell association problem considering both UAVs and terrestrial base stations, for any arbitrary distribution of users. The main contribution of this paper is to introduce a novel framework for delay-optimal cell association in a cellular network in which both UAVs and terrestrial BSs co-exist. In particular, given the locations of the UAVs and terrestrial BSs as well as any general spatial distribution of users, we find the optimal cell association by exploiting the framework of optimal transport theory [@villani]. we first prove the existence of the optimal solution to the cell association problem, and, then, we characterize the solution space. System Model and Problem formulation ==================================== Consider a geographical area $\mathcal{D}\subset \mathds{R}^2$ in which $K$ terrestrial BSs in set $\mathcal{K}$ are deployed to provide service for ground users that are spatially distributed according to a distribution $f(x,y)$ over the two-dimensional plane. In addition to the terrestrial BSs, $M$ UAVs in set $\mathcal{M}$ are deployed as aerial base stations to enhance the capacity of the network. We consider a downlink scenario in which the BSs and the UAVs use a frequency division multiple access (FDMA) technique to service the ground users. The maximum transmit powers of BS $i$ and UAV $j$ are $P_i$ and $P_j^\textrm{uav}$. Let $W_i$ and $W_j$ be the total bandwidth available for each BS $i$ and UAV $j$. We use $A_i$ and $B_j$ to denote, respectively, the area (cell) partitions in which the ground users are assigned to BS $i$ and UAV $j$. Hence, the geographical area is divided into $M+K$ disjoint partitions each of which is served by one of the BSs or the UAVs. Given this model, our goal is to minimize the average network delay by optimal partitioning of the area. Based on the spatial distribution of the users, we determine the optimal cell associations to minimize the average network delay. Note that, the network delay significantly depends on the cell partitions due to the following reasons. First, the cell partitions determine the service area of each UAV and BS thus impacting the channel gain that each user experiences. Second, the number of users in each partition depends on the cell partitioning. In this case, since the total bandwidth is limited, the amount of bandwidth per user decreases as the number of users in a cell partition increases. . Next, we present the channel models. ![ Network model. ](SystemModel.pdf){width="7.2cm"} \[SystemModel\] UAV-User and BS-User path loss models ------------------------------------- . The path loss between UAV $j$ and a user located at $(x,y)$ is [@HouraniModeling]: $$\label{Pr} \Lambda_j = \left\{\hspace{-0.16cm} \begin{array}{l} K_o^{2}(d_j/d_o)^{2} {\mu _\text{LoS},} \,\,\,\hspace{0.35cm}{\text{LoS link,}}\\ K_o^{2}(d_j/d_o)^{2} {\mu _\text{NLoS},} \,\,\,\hspace{0.21cm}{\text{NLoS link,}} \end{array} \right. \vspace{-0.2cm}$$ where $K_o=\left(\frac{4\pi f_cd_o} {c}\right)^{2}$ , $f_c$ is the carrier frequency, $c$ is the speed of light, and $d_o$ is the free-space reference distance. Also, $\mu_\textrm{LoS}$ and $\mu_ \textrm{NLoS}$ are different attenuation factors considered for LoS and NLoS links. $d_j=\sqrt{(x-x^\textrm{uav} _{j})^2+(y-y^\textrm{uav} _{j})^2+h{^\textrm{uav}_j}^2}$ is the distance between UAV $j$ and an arbitrary ground user located at $(x,y)$. For the UAV-user link, the LoS probability is [@HouraniModeling]: $$\label{PLoS} {\mathds{P}_{\text{LoS},j}} = \alpha {\left( {\frac{180}{\pi}\theta_j - 15} \right)^\gamma}, \,\,\, \theta_j>\frac{\pi}{12},\vspace{-0.15cm}$$ where $\theta_j={\sin ^{- 1}}( \frac {h_j} {d_j})$ is the elevation angle (in radians) between the UAV and the ground user. Also, $\alpha$ and $\gamma$ are constant values reflecting the environment impact. Note that, the NLoS probability is $\mathds{P}_{\text{NLoS},j}=1-\mathds{P}_{\text{LoS},j}$. Considering $d_o=1$m, the average path loss is $K_o{ {{d_j}}^{ 2}}\left[ {{\mathds{P}_{\textrm{LoS},j}}{\mu _\textrm{LoS}} + {\mathds{P}_{\textrm{NLoS},j}}{\mu _\textrm{NLoS}}} \right]$. For the BS-user link, we use the traditional path loss model. Problem formulation ------------------- $$\begin{aligned} &C_j^\textrm{uav}= {\frac{{W_j}}{{N_j^\textrm{uav}}}{{\log }_2}\big( {1 + \frac{\bar P_{r,j}^\textrm{uav}}{\sigma^2}} \big)},\vspace{-0.1cm} \end{aligned}$$ Now, let $\mathcal{L}=\mathcal{K} \cup \mathcal{M}$ be the set of all BSs and UAVs. Also, here, the location of each BS or UAV is denoted by $\boldsymbol{s}_k$, $k\in\mathcal{L}$. We also consider ${D_k} = \left\{ \hspace{-0.15cm}\begin{array}{l} {A_k},\hspace{0.3cm} \textrm{if}\,\,k \in \mathcal{K},\\ {B_k}, \hspace{0.3cm} \textrm{if}\,\, k \in \mathcal{M}, \end{array} \right.$ denoting all the cell partitions, and Then, our optimization problem that seeks to minimize the average network delay over the entire area will be: $$\begin{aligned} \label{Opt1} &\mathop {\min }\limits_{{D_k}} \sum\limits_{k \in \mathcal{L}} {\int_{{D_k}} {Q\left( {\boldsymbol{v},\boldsymbol{s}_k,{D_k}} \right)f(x,y)\textrm{d}x\textrm{d}y} }, \\ \textrm{s.t.}\,\, &\bigcup\limits_{k \in \mathcal{L}} {{D_k}} = \mathcal{D},\,\,\,{D_l} \cap {D_m} = \emptyset ,\,\,\,\forall l \ne m \in \mathcal{L}. \vspace{-0.5cm} \label{Union} \end{aligned}$$ Optimal Transport Theory for Cell Association ============================================= Given the locations of the BSs and the UAVs as well as the distribution of the ground users, we find the optimal cell association for which the average delay of the network is minimized. Now, the optimization problem in (\[Opt1\]) can be rewritten as: $$\begin{aligned} \label{Opt3} &\hspace{-0.9cm}\mathop {\min }\limits_{{D_k}} \begin{small} \sum\limits_{k \in \mathcal{L}} {\int_{{D_k}} {\left[ {{g_k}\left( {\int_{{D_k}} {f(x,y)\textrm{d}x\textrm{d}y} } \right)F(\boldsymbol{v},\boldsymbol{s}_k)} \right]f(x,y)\textrm{d}x\textrm{d}y} }\end{small},\\ \textrm{s.t.}\,\, & \bigcup\limits_{k \in \mathcal{L}} {{D_k}} = \mathcal{D},\,\,\,{D_l} \cap {D_m} = \emptyset ,\,\,\,\forall l \ne m \in \mathcal{L},\end{aligned}$$ where $D_k$ is the cell partition of each BS or UAV $k$. Solving the optimization problem in (\[Opt3\]) is challenging and intractable due to various reasons. First, the optimization variables $D_k$, $ \forall k \in \mathcal{L}$, are sets of continuous partitions which are mutually dependent. Second, $f(x,y)$ can be any generic function of $x$ and $y$ that leads to the complexity of the given two-fold integrations. To overcome these challenges, next, we model this problem by exploiting optimal transport theory [@villani] in order to characterize the solution. Optimal transport theory [@villani] allows analyzing complex problems in which, for two probability measures $f_1$ and $f_2$ on $\Omega \subset \mathds{R}^n$, one must find the optimal transport map $T$ from $f_1$ to $f_2$ that minimizes the following function: $${\mathop {\min }\limits_T \int_\Omega {c\left( {x,T(x)} \right)} f_1(x)\textrm{d}x;\,\,T:\Omega \to \Omega}, \vspace{-0.05cm}$$ where $c(x,T(x))$ denotes the cost of transporting a unit mass from a location $x$ to a location $T(x)$. Let ${a_k} = \int_{{D_k}} {f(x,y)\textrm{d}x\textrm{d}y}$,\ [$E \hspace{-0.2cm}=\hspace{-0.2cm} \left\{ {\boldsymbol{a} = \left( {{a_1},{a_2},...,{a_{K + M}}} \right) \in {\mathds{R}^{K + M}};{a_k} \ge 0, \sum\limits_{k = 1}^{K + M} {{a_k} = 1} } \right\}$]{} . Now, considering $f(x,y)=f(\boldsymbol{v})$ and $c\left( {\boldsymbol{v},{\boldsymbol{s}_k}} \right) = {g_k}({a_k})F\left( {\boldsymbol{v},{\boldsymbol{s}_k}} \right)$, for any given vector $\boldsymbol{a}$, problem (\[Opt3\]) can be considered as a classical semi-discrete optimal transport problem. First, we prove that $c\left( {\boldsymbol{v},\boldsymbol{s}} \right)$ is a semi-continuous function. Considering the fact that $\boldsymbol{s}_k$ is discrete, we have: $\mathop {\lim }\limits_{(\boldsymbol{v},\boldsymbol{s}) \to ({\boldsymbol{v}^},{\boldsymbol{s}_k})} F\left( {\boldsymbol{v},\boldsymbol{s}} \right)\mathop = \mathop {\lim }\limits_{\boldsymbol{v} \to {\boldsymbol{v}^}} F\left( {\boldsymbol{v},{\boldsymbol{s}_k}} \right)$. Note that, given any $\boldsymbol{s}_k$, $k$ belongs to only of $\mathcal{K}$ and $\mathcal{M}$ sets. Given $\boldsymbol{s}_k$, $F(\boldsymbol{v},\boldsymbol{s}_k)$ is a continuous function of $\boldsymbol{v}$. Then, considering the fact that given $a_k$, $g_k(a_k)$ is constant, we have $\mathop {\lim }\limits_{(\boldsymbol{v},\boldsymbol{s}) \to ({\boldsymbol{v}^},{\boldsymbol{s}_k})} g_k(a_k) F\left( {\boldsymbol{v},\boldsymbol{s}} \right) = g_k(a_k)F\left( {{\boldsymbol{v}^},{\boldsymbol{s}_k}} \right)$. Therefore, $c(\boldsymbol{v},\boldsymbol{s})$ is a continuous function and, hence, is also a lower semi-continuous function. Now, we use the following lemma from optimal transport theory: Consider two probability measures $f$ and $\lambda$ on $\mathcal{D} \subset \mathds{R}^n$. Let $f$ be continuous and $\lambda = \sum\limits_{k \in \mathds{N}} {{a_k}{\delta _{{\boldsymbol{s}_k}}}}$ be a discrete probability measure. Then, for any lower semi-continuous cost function, there exists an optimal transport map from $f$ to $\lambda$ for which $\int_\mathcal{D} {c\left( {x,T(x)} \right)} f(x)\textrm{d}x$ is minimized [@Crippa]. Considering Lemma 1, for any $\boldsymbol{a}\in E$, the problem in (\[Opt3\]) admits an optimal solution. Since $E$ is a unit simplex in $\mathds{R}^{M+K}$ which is a non-empty and compact set, the problem admits an optimal solution over the entire $E$. Next, we characterize the solution space of (\[Opt3\]). To acheive the delay-optimal cell partitions in (\[Opt3\]), each user located at $(x,y)$ must be assigned to the following BS (or UAV): Now, consider two partitions $D_l$ and $D_m$, and a point $\boldsymbol{v}_o=(x_o,y_o)\in D_l$. Also, let $B_\epsilon(\boldsymbol{v}_o)$ be a ball with a center $\boldsymbol{v}_o$ and radius $\epsilon >0$. $$\left\{ \begin{array}{l} {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over D} }_l} = D_l\backslash {B_\varepsilon }({\boldsymbol{v}_o}),\\ {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over D} }_m} = D_m \cup {B_\varepsilon }({\boldsymbol{v}_o}),\\ {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over D} }_k} = D_k,\,\,\,\,k \ne l,m. \end{array} \right.$$ Let ${a_\varepsilon } = \int_{{B_\varepsilon }({\boldsymbol{v}_o})} {f(x,y)\textrm{d}x\textrm{d}y} $, and ${{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over a} }_k} = \int_{{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over D} }_k}} {f(x,y)\textrm{d}x\textrm{d}y}$. Considering the optimality of $D_k$, $k\in \mathcal{L}$, we have: $$\begin{aligned} &\hspace{0.1cm}\sum\limits_{k \in \mathcal{K}} {\int_{{D_k}} { {{g_k}\left( {{a_k}} \right)F(\boldsymbol{v},{\boldsymbol{s}_k})} f(x,y)\textrm{d}x\textrm{d}y} } \nonumber \\ & {\mathop \le \limits^{(a)} }\sum\limits_{k \in \mathcal{K}} {\int_{{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over D} }_k}} { {{g_k}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over a} }_k}} \right)F(\boldsymbol{v},{\boldsymbol{s}_k})} f(x,y)\textrm{d}x\textrm{d}y} }. \label{SUM}\\ &\textrm{\textcolor{black} {\normalsize Now, canceling out the common terms in (\ref{SUM}) leads to:}} \nonumber \\ \hspace{0.1cm}&\int_{{D_l}} { {{g_l}\left( {{a_l}} \right)F(\boldsymbol{v},{\boldsymbol{s}_l}) } f(x,y)\textrm{d}x\textrm{d}y} + \int_{{D_m}} { {{g_m}\left( {{a_m}} \right)F(\boldsymbol{v},{\boldsymbol{s}_m})} f(x,y)\textrm{d}x\textrm{d}y}\nonumber\\ & \le \int_{{D_m} \cup {B_\varepsilon }({\boldsymbol{v}_o})} { {{g_m}\left( {{a_m} + {a_\varepsilon }} \right)F(\boldsymbol{v},\boldsymbol{s}_m)} f(x,y)\textrm{d}x\textrm{d}y} \nonumber\\ &+\int_{{D_l}\backslash {B_\varepsilon }({\boldsymbol{v}_o})} { {{g_l}\left( {{a_l} - {a_\varepsilon }} \right)F(\boldsymbol{v},\boldsymbol{s}_l)} f(x,y)\textrm{d}x\textrm{d}y},\nonumber\\ &\int_{{D_l}} { {\left( {{g_l}\left( {{a_l}} \right) - {g_l}\left( {{a_l} - {a_\varepsilon }} \right)} \right)F(\boldsymbol{v},\boldsymbol{s}_l)} f(x,y)\textrm{d}x\textrm{d}y}\nonumber\\ & +\int_{{B_\varepsilon }({\boldsymbol{v}_o})} { {{g_l}\left( {{a_l} - {a_\varepsilon }} \right)F(\boldsymbol{v},\boldsymbol{s}_l)} f(x,y)\textrm{d}x\textrm{d}y}\nonumber\\ &\le \int_{{D_m}} { {\left( {{g_m}\left( {{a_m}}+ {a_\varepsilon } \right) - {g_m}\left( {{a_m}} \right)} \right)F(\boldsymbol{v},\boldsymbol{s}_m)} f(x,y)\textrm{d}x\textrm{d}y}\nonumber\\ & + \int_{{B_\varepsilon }({\boldsymbol{v}_o})} { {{g_m}\left( {{a_m} + {a_\varepsilon }} \right)F(\boldsymbol{v},{\boldsymbol{s}_m})} f(x,y)\textrm{d}x\textrm{d}y},\label{ineq}\vspace{-0.15cm}\end{aligned}$$ \ Theorem 2 provides a precise cell association rule for ground users that are distributed following any general distribution $f(x,y)$. In fact, the inequality given in (\[proof\]) captures the condition under which the user is assigned to a BS or UAV $l$. Under the special case of a uniform distribution of the users, the result in Theorem 2 leads to the classical SNR-based association in which users are assigned to base stations that provide strongest signal. Simulation Results and Analysis =============================== For our simulations, we consider an area of size $4\,\text{km}\times 4 \,\text{km}$ in which 4 UAVs and 2 macrocell base stations are deployed based on a traditional grid-based deployment. The ground users are distributed according to a truncated Gaussian distribution with a standard deviation $\sigma_o$. This type of distribution which is suitable to model a hotspot area. ![ Average network delay per 1Mb data transmission.[]{data-label="Delay"}](Delay_sigma2.eps){width="7cm"} [0.2]{} ![ \[CellPart\]](OPT3.JPG){width="4.6cm"} [0.2]{} ![ \[CellPart\]](SNR3.JPG){width="4.6cm"} Conclusion ========== In this paper, we have proposed a novel framework for delay-optimal cell association in UAV-enabled cellular networks. In particular, to minimize the average network delay based on the users’ distribution, we have exploited optimal transport theory to derive the optimal cell associations for UAVs and terrestrial BSs.
Editor: If you use Key Bridge to leave or enter D.C., or drive through Rosslyn to access Route 50, Interstate 66 or the George Washington Memorial Parkway, you already are aware of the traffic congestion in Rosslyn. What you may not realize is there are four additional large high-rise residential complexes proposed for Rosslyn, which will also include two major hotels and a 40,000-square-foot conference center. And if these new residents and visitors do not own or use their own cars, as the county hopes, they will use Uber and taxis, which will add even more congestion. Why has the county government not done a much-needed traffic study that includes ALL four developers’ proposals to address the impact this increase in traffic and population will have on an already congested area? Although it is important to review the projects’ footprints, the design, the exterior appearance, etc., shouldn’t the county government also determine, independently, the impact of the additional cars, Ubers and taxis?	https://www.insidenova.com/opinion/letters_to_editor/letter-rosslyn-in-on-verge-of-being-overrun-in-traffic/article_bc0344de-b379-11e9-af1a-97dd048e5dc3.html
This wonderful open floor plan offers plenty of space, including a 4-car, side loading garage and an optional Daylight Basement for even more potential space. The massive Great Room of this home features a vaulted ceiling and a fireplace, flanked by doors that lead to the large Outdoor Living/Grilling Porch. The Great Room is also open to the breakfast room and kitchen that features a large eat-at island bar, a prep island bar, a large walk-in pantry and a butler’s pantry. Beyond the butler’s pantry is the Dining Room with the front wall having arched windows to allow for maximum natural light. The very spacious Master Suite boasts a corner fireplace and vaulted ceiling, further adding to the feel of space. The master bath with spoil you with a raised corner whirlpool with a gas fireplace behind it, a large corner glass shower with a seat, and a giant walk-in closet. Just down the hall is the conveniently closet laundry room with plenty of space and even has a pet shower. Off of the foyer and Great Room is the Office/Media/Theater with its front rounded wall floor to ceiling windows, again, for maximum natural lighting. Bedrooms 2, 3, and 4 are on the left side of the home with a Jack and Jill bath between bedrooms 2 and 3 and another bath next to bedroom 4. The Optional Daylight Basement would offer a huge Game Room, complete with a kitchen and a Safe Room off of it, a huge Storage/Workout Room, and access to the Rear Covered Patio. \|Total Living Area:\|\|3713 sq ft\| \|Main Living Area:\|\|3713 sq ft\| \|Unfinished Basement Area:\|\|1405 sq ft\| \|Garage Area:\|\|1233 sq ft\| \|Garage Type:\|\|Attached\| \|Garage Bays:\|\|4\| \|Foundation Types:\|\|Walkout Basement \| \|Exterior Walls:\|\|2x4 \| 2x6 - * $299.00 \|House Width:\|\|78'6\| \|House Depth:\|\|95'5\| \|Number of Stories:\|\|1\| \|Bedrooms:\|\|4\| \|Full Baths:\|\|3\| \|Half Baths:\|\|1\| \|Max Ridge Height:\|\|34'4 from Front Door Floor Level\| \|Primary Roof Pitch:\|\|12:12\| \|Roof Load:\|\|45 psf\| \|Roof Framing:\|\|Stick\| \|Porch:\|\|1279 sq ft\| \|Formal Dining Room:\|\|Yes\| \|FirePlace:\|\|Yes\| \|Main Ceiling Height:\|\|10'0\| View an actual, LIVE cost report. This report is for demo purposes only. Not plan specific. A: The jack n jill bath measures approximately 5’6” x 12’4” And the small bathroom is approximately 5’6” x 8’ A: 6’6” x 6’8” A: We can do this but it would require a modification A: The master closet is 6’6”x19’ and laundry is 7’4”x21’ A: Certainly! A $250 fee applies. A: Yes you can! Please click the "Modifications" tab above to get more information. A: The national average for a house is running right at $125.00 per SF. You can get more detailed information by clicking the Cost-To-Build tab above. Sorry, but we cannot give cost estimates for garage, multifamily or project plans.	https://www.familyhomeplans.com/house-plan-82401
Here's everything you need to know about Super Bowl cities, including which city has hosted the most Super Bowls, how many times the Super Bowl has been in Miami, future Super Bowl locations and a full list of Super Bowl host cities by year. Which city has hosted the most Super Bowls? By hosting the 2020 Super Bowl, Miami will take the lead with 11 Super Bowls hosted. New Orleans is in second place with 10, though it will host the big game in 2024, re-establishing the deadlock. Here are the four cities that have hosted the most Super Bowls: Place City Number (first year, last year) No. 1 Miami 11 (1968, 2020) No. 2 New Orleans 10 (1970, 2013) No. 3 Los Angeles 7 (1967, 1993) No. 4 Tampa 4 (1984, 2009) How many times has the Super Bowl been in Miami? Miami (and nearby Miami Gardens) has hosted the Super Bowl 11 times. It first hosted Super Bowl 2 in 1968. Before this year, its most recent time was Super Bowl 44 in 2010. The Dolphins have never played a Super Bowl in Miami. Here are all the Super Bowls played in Miami. Game Date Result 2 Jan. 14, 1968 Packers beat Raiders, 33-14 3 Jan. 12, 1969 Jets beat Colts, 16-7 5 Jan. 17, 1971 Colts beat Cowboys, 16-13 10 Jan. 18, 1976 Steelers beat Cowboys, 21-17 13 Jan. 21, 1979 Steelers beat Cowboys, 35-31 23 Jan. 22, 1989 49ers beat Bengals, 20-16 29 Jan. 29, 1995 49ers beat Chargers, 49-26 33 Jan. 31, 1999 Broncos beat Falcons, 34-19 41 Feb. 4, 2007 Colts beat Bears, 29-17 44 Feb. 7, 2010 Saints beat Colts, 31-17 Future Super Bowl locations The next four Super Bowl locations have already been chosen. The big game will return to Tampa in 2021, move to the new Los Angeles stadium in 2022, then go to Glendale, Ariz., in 2023 and New Orleans in 2024. Super Bowl Date City (No. of times hosted), stadium 55 Feb. 7, 2021 Tampa (5), Raymond James Stadium 56 Feb. 6, 2022 Los Angeles (8), SoFi Stadium 57 Feb. 5, 2023 Glendale, Arizona (4), State Farm Stadium 58 Feb. 4, 2024 New Orleans (11), Mercedes-Benz Superdome How many times has a host city played in the Super Bowl? On two occasions, a team has played a Super Bowl in its home region. Those teams are 1-1. A team has never played a Super Bowl in its home stadium, however. In 1980, the Los Angeles Rams played Super Bowl 14 at the Rose Bowl in Pasadena, Calif. They lost 31-19 to the Steelers. The Rams played their home games at the Los Angeles Memorial Coliseum, which hosted Los Angeles' first two Super Bowls, but none of its next six. In 1985, the San Francisco 49ers beat the Miami Dolphins 38-16 in Super Bowl 19 in Stanford, Calif. The 49ers played their home games at Candlestick Park, which never hosted a Super Bowl.
Staff PD: We DO NOT meet as a staff this week. You may want to use this time to begin cleaning out your classroom or to work on your report cards. Summer Learning Academy: Summer Learning Academy brochure The schedule is subject to change. Here’s the link to RSVP. Hope to see you there. Fountas and Pinnell Data Entered: Thank you for assessing your students and entering the data on time. I appreciate your hard work!! Summer Clean Up/Clean Out- prep for libraries and modernization: After school dismisses, this summer, Dr. McNamara will walk every classroom in the South Bay Union School District. - If you have old tech, furniture or large items that you no longer want, are broken/damaged or unusable please email me and label items discard that you no longer want (if they are large or tech); - If there are materials and resources in your classroom that are very out of date (no longer board approved, not a current resource or used for instruction in the last two-three years) please email me and label items discard (if you need boxes please let Eunice or Carlos know); - If you have items that need to be shredded, please put them in a box and label them please email me and label items SHRED; - If you would like to store instructional materials in a site storage location, please email me and label the box with your name and the contents (example-math manipulatives). - If you need access to a large trash bin, please let Carlos know when and where. End of Year Checklist for Summer Custodian Deep Clean Prep: Student desks cleaned on top and inside out (marks, stickers, tape, and trash removed). Teacher’s desk cleared off. Student chairs stacked by 5 and placed to the side of the room. Whiteboards cleaned (remove all writing). Closets, cupboards, and windows cleaned (marks, tape, posters, signs, etc.). Walls and windows cleared of paper. All tops of cabinets clear (NO boxes). Unwanted items – stack by door inside classroom – label “DISCARD”. Place all trash in the trash can. Carpets picked up (including staples). Remove all books/toys/blocks from top of bookcases. Personal valuables removed. Clear all tables, also remove all boxes/etc. from under desks and tables so that the carpet/floor can be cleaned. If there are repairs needed in your room, please place a work order in Opra. We will be working with the Warehouse to dispose of our purged items. Please let me know if you have any questions or need any assistance. Shared Leadership: All are welcome to join us at our summer Shared Leadership on Monday, July 16 from 9:30-11:30 with a break for lunch on your own from 11:30-12:30 and then continue from 12:30-2:30. Attendees will be compensated four hours out of contract time. Agenda items include the following: Awards, Duties, Comprehensive School-wide Discipline Plan, School Brand, Modernization, other Summer Book Club: Summer Book Club will meet on Monday, July 16 from 8:30-9:30. We will be discussing Kids First from Day One. A copy was placed in each of your mail boxes. Professional Development/Teacher Prep Week: You return to work on Tuesday, July 17th. We will begin the day with an optional continental breakfast. The day is yours to get prepared for the new school year. We will meet on Wednesday, July 18th at 8:00 at Mendoza for the District Kick Off. Following the Kick Off, you will be free to get lunch, set up your classroom etc. We will meet from 1:00-2:30 on July 18th for Nuts and Bolts. We will meet all day on Thursday, July 19th where lunch will be provided. You will have Friday, July 20th free to finalize planning for the new school year. Apex Fun Run Survey: Please click here to provide feedback about the Apex Fun Run/Leadership Activities/Fundraiser. Water for Field Days: The PTA purchase bottle water for the students to have on Field Day next week. Please contact Teresa to make arrangements. It would be best to help your students label the bottles with their name. Report Cards: The report card window is open. Here are a few reminders:	https://emoryeagles.org/2018/05/28/weekly-update-may-28-2018/
I have built a very complicated macro-free workbook and in one reporting worksheet I have three drop down lists to select the information (ie main categories) that is displayed. Everything worked fine with a drop down list on the Cat2 table to select more information as needed. I then decided to tweak the sheet and added a series of radio buttons (about 20) to select from the displayed information rather than suing a second drop down list to display more details (sub-categories) from the radio button-selected category. Since adding the radio buttons to the worksheet, the arrows on the three Category drop-down list boxes no longer display properly. The arrow shows up when selecting the cell, but disappear when I move the mouse pointer over to the drop down arrow. The drop down indicator reappears after randomly selecting a few radio buttons, and is actually visible in the attached screenshot. Has anyone else come across this issue, and any ideas how to fix it. I am using Excel in Office 365 2016 version.	https://techcommunity.microsoft.com/t5/Excel/Radio-Buttons-interfere-with-Dropdown-lists/td-p/550817
Thai League Co., Ltd. confirmed that there will be an increase in foreign player quotas. In the Thai League 1, starting from the 2022/23 season onwards. Quota of foreign players that can be registered to the system in each season from the original form 3+1+Unlimited (3 non-continent players + 1 Asia quota player + unlimited ASEAN quota players) will increase to 5+1+Unlimited. or can send all 5 foreign players outside the continent The quota of foreign players that can be sent into the field will still be 3+1+3 (3 non-continental players + 1 Asian quota players + 3 ASEAN quota players) as before.	http://snesports.co/thai-league-register-5-foreign-players-since-next-season/
This is a project in my Back-to-School Lisa Frank series! I have so so so many pencils in my supply stash, a few of them are cute, pink, patterned or something, but a lot of them are boring, plain #2 yellow pencils. So, I restyled a few of them and they are now officially WAY cooler. Like, 90’s kid cool. I used a bit of acrylic paint in bright colors + black to make these Lisa Frank pattern inspired pencils. Supplies: Pencils Acrylic Paint #4 Paint Brush #1 Paint Brush Gloss Decoupage Medium Masking or Washi Tape 1. Start by using a bit of tape to tape of the metal part of your pencils so that you don’t paint over it. If you paint over the pointed wood end, you can easily sharpen the pencil. 3. Use a tiny, #1 paint brush to paint on cheetah print and polka dots using black or white acrylic paint. I use pencils a lot, every day. I can’t stand mechanical pencils though, they squeak. Anyone else have that issue?	https://studiokatie.com/diy-lisa-frank-pencils/
What is Radio Astronomy? Black Holes Exoplanets Pulsars Astronomy Explore The VLA Explorer Cosmic Coloring Compositor Milky Way Explorer Interferometry Explained ALMA Explorer NRAO Mission Control Give Home Gallery Baseline #11 – Finding Planets That Have No Star Baseline #11 – Finding Planets That Have No Star Most planets orbit a star, but some planets can escape and "go rogue." But how do astronomers study planets that wander the cold dark of interstellar space? Join our host Summer Ash of the National Radio Astronomy Observatory as she talks about how radio astronomers study rogue planets. Credit: Share this... Facebook Pinterest Twitter Linkedin Categories: Exoplanets Protoplanetary Disks Part of Collections: The Baseline Download Video Have you reviewed our Media Use Policy ? Yes, continue sd 960 x 540 27.28 MB hd 1920 x 1080 93.26 MB sd 640 x 360 13.26 MB hd 1280 x 720 48.25 MB sd 426 x 240 8.91 MB More Exoplanets Images & Videos 3D Model of VLBA Discovery – Exoplanet The Baseline 9: How Do Exoplanets Form? Mind The Gaps! Hydrogen Cyanide in HD 163296 Why Are Gravitational Instabilities Important?	https://public.nrao.edu/gallery/baseline-11-finding-planets-that-have-no-star/
It’s not an uncommon request. When the enquiry comes in, the client thinks the required flowmeter range will be 1 to 10 but, once installed, it’s clear that he should have stated 2 to 20. Then, the question is: what do we do? First: it’s unlikely that the range can be extended beyond the meter maximum rated flow rate. Some flowmeters principles can cope with excessive flow rates, probably at the expense of pressure drop, but most can’t. Second: Consider the magnitude of change and the expectation of accuracy. If it’s a change from 1 to 10 to 1 to 11 with a 5% accuracy requirement when the maximum rate is 12 then that’s an easy one – check with the factory on how the range can be changed within the instrumentation but the range extension will be OK. On the other hand, if it’s a ±0.2% meter then it’s unlikely that any range extension will be within that value without a return to the factory for recalibration. Here’s an example: The client had purchased an LF03 VFF positive displacement meter for the measurement and control of corrosion inhibitor. The viscosity was 55cP and, although the meter is capable of 18 litres per hour, had specified an operational range of 0.2 to 2.3 l/h which we calibrated over. A year later we had a request for a range change – could they up the calibration range to 5.6 l/hr? Of course, yes. Could we provide a statement to this? See below: VFF Flowmeter Extended Range Accuracy – 50cP viscosity The original calibration for meter VFF5112 was up to 2.3 l/hr on 55cP. Litre Meter have analysed the last 17 LF03s calibrated at or around 50cP and can confirm that re-ranging to 5.6 l/hr will have only a minor effect to system accuracy. VFF Analysis: It can be seen from the aggregated performance curves above that extending the flow rate above 2.3 l/hr up to 7 l/hr produces little change in the meter linearity. We would suggest that, in the absence of any higher flow rate information above 2.3 litre per hour, that the meter is unlikely to be outside of ±2% of the 2.3 l/hr pulses per litre value up to 7 litres per hour. Increased confidence and accuracy can be obtained by recalibration. Linearisation: Litre Meter produce a document LM0688 “Technical Description – Linearisation” that explains the linearization process and the flow rate versus pulses per litre table. Also here. Here’s another, more extreme, example: The client bought a series of LF15 VFFs in 2005 with 2500 pound flanges and a flow rate range of 1 to 3 litres per hour on 10cSt fluid. In 2017, on one of the lines, there was a change in viscosity and flow range. The fluid became methanol and the flow rate range expanded to 0.5 to 10 l/hr. The change in viscosity affects positive displacement meter performance at the lower flow rates. In this case, an analysis of the performance curve from the original calibration indicated that it was unlikely to reach 0.5 l/hr on 10 cSt so extending the calibrated range was not possible. Litre Meter constantly develop and improve the VFF. In 2012, the LF05 was introduced. It was followed in 2015 by the LF03. These are smaller rotors and chambers with lower flow capabilities but manufactured in the same size module as the LF15. We were able to offer a replacement module with the lower flow ability, calibrated at the factory, for the customer to swap on site, leaving the 2500 pound flange body in situ. This dramatically sped up the upgrade process and met the new range requirements.	https://www.litremeter.com/extending-the-flowmeters-calibrated-range-an-expert-view/
Tomato plants were transplanted on 7 May 2012 into black (grower standard), white or reflective plastic mulch. Five different varieties of tomato were used: Mt Spring+, Celebrity, Mt Fresh+, Crista and Scarlet Red. When tomato plants began to develop fruit, a 30% shade cloth was randomly placed over 6-10 plants per row. All plants were treated with fungicides four times (chlorothalonil and copper). Shaded plants were sprayed through the shade cloth). Shade cloth covered 50-70% of the plant, bottom of plants were not shaded. At each harvest (5 harvests)) the shade cloth was removed from one side of a row and laid over the other side, once harvest was over shade was placed back in its original position. Two types of yield were taken. Total yield: All ripe tomatoes were picked off a plant and were counted and weighed and Marketable yield: Tomatoes that rated extra large or large with no defects, were counted and weighed. At the end of harvest shade cloth was removed completely and comparisons between plants were taken.	https://extension.umd.edu/learn/tomato-plastic-mulch-and-shade-study
All relevant data are within the paper and its Supporting Information file. Introduction {#sec001} ============ In 2015 the World Health Organisation estimated that about 71 million people are living with the hepatitis C virus \[[@pone.0194396.ref001]\]. Every year, about 700,000 people die of hepatitis C-related pathologies including hepatocellular carcinoma (HCC), cirrhosis and liver failure \[[@pone.0194396.ref002]\]. Recently, the availability of highly-active direct-acting antivirals (DAA) targeting non-structural viral proteins raised the hope of a rapid eradication of HCV. However, the asymptomatic nature of hepatitis C infection is a major problem for management of the disease and targeting the right persons with the right treatments is crucial in order to achieve a sustained virologic response (SVR) at a patient level. Recent recommendations stated that both the genotype and the subtype of the virus are critical for drug selection among those available. Thus, accurate genotyping now plays a key role in the management of HCV patients \[[@pone.0194396.ref003]\]. Commercially available kits for HCV genotyping are scarce. The most commonly used assays are the VERSANT HCV Genotype 2.0 assay (LiPA, Siemens Healthcare, Germany) that is based on a reverse hybridization line probe assay \[[@pone.0194396.ref004]\], and the RealTime HCV Genotype II kit (Abbott, Illinois, USA) that is based on fluorescent-labeled oligonucleotide probes \[[@pone.0194396.ref005]\]. In addition to these a few other kits are available for HCV genotyping \[[@pone.0194396.ref006]\]. The VERSANT HCV Genotype 2.0 assay requires specific equipment (Auto-LiPA Instrument, blot and scan) and presents some typing failures for genotype 1 subtypes \[[@pone.0194396.ref007]\] or undetermined results, especially for samples coming from non-European regions \[[@pone.0194396.ref008]\]. Moreover using 5'UTR and Core regions this assay cannot correctly identify the recombinant form 2k/1b \[[@pone.0194396.ref009]\]. The Abbott assay has been found to misclassify genotype 1 subtypes in 1--5.4% of samples \[[@pone.0194396.ref007],[@pone.0194396.ref010]\]. Consequently, many laboratories still use in-house polymerase chain reaction (PCR) sequencing assays. There is a need for reliable commercially available assays so as to enhance standardization of HCV genotyping. The use of equipment already present in the laboratory for HCV viral load quantification, as for the Abbott assay, is an advantage. Roche Diagnostics has recently developed a new kit for HCV genotyping that allows the genotyping of genotypes 1 to 6 and the determination of subtypes 1a and 1b. It runs on the cobas 4800 platform that is already used for measuring HCV viral loads \[[@pone.0194396.ref011],[@pone.0194396.ref012]\]. This method is based on real-time PCR for the amplification and uses specific fluorescent probes for the detection of the HCV genome. The objective of our study was to evaluate the performances and workflow of the cobas^®^ HCV GT kit from Roche Diagnostics (Mannheim, Germany) in the context of a teaching hospital laboratory, compared with our in-house assay. We therefore evaluated this recently launched method on samples from both routinely managed patients and a selected panel of more complicated cases with sequenced viruses of less frequent genotypes. Materials and methods {#sec002} ===================== 1) Samples {#sec003} ---------- A total of 182 samples from chronically infected HCV patients were retrospectively genotyped using the HCV GT Roche assay. Among them, 101 corresponded to an HCV typing analysis performed for patients followed in the Hepatology care unit of Grenoble Alpes University hospital between February 2014 and April 2016, and for whom more than 650 μL of plasma remained. The viral loads of the samples from this panel were always superior to 4 log10 IU/mL with a mean viral load of 6.00+/-074 log10 IU/mL \[[@pone.0194396.ref013]\]. A further eighty-one samples were added representing two categories of more difficult to type samples. These corresponded either to samples with low viral load (\< 3 log10 IU/mL, n = 42, mean viral load: 1.92+/-0.44 log10 IU/mL), or to samples collected nationwide (n = 39, viral load unknown) received for expertise in our specialized laboratory between February 2014 and April 2016 following the failure of previous typing using the LiPA VERSANT HCV Genotype 2.0 assay. The characteristics of the latter samples are given in [Table 1](#pone.0194396.t001){ref-type="table"}. Samples were stored in the local biobank (DC2008-680) at -80°C. No patient was asked to give an additional blood sample. 10.1371/journal.pone.0194396.t001 ###### Distribution of Hepatitis C virus (HCV) genotypes in studied samples (n = 182). ![](pone.0194396.t001){#pone.0194396.t001g} ------------------------------------------------------------------------------------------------------------ HCV genotype and subtype Number of samples (n = 101) HCV Mean Viral load\ (log10 IU/mL+/-sd) ---------------------------- -------------------------- ----------------------------- ---------------------- Regular samples 1a 35 6.04+/-0.79 1b 30 6.21+/-0.62 2c 2 5.59+/-1.25 3a 11 5.78+/-0.84 4a 7 6.26+/-0.24 4d 12 5.48+/-0.66 4 non-a non-d 3 6.23+/-0.05 6e 1 6.90 Low viral load samples 1a 13 2.03+/-0.46 1b 16 1.92+/-0.39 1e 1 1.3 3 6 1.83+/-0.45 4 6 2.00+/-0.46 LiPA failures 1d 2 / 1g 3 / 1i 1 / 2i 9 6.04+/-0.43 2b 1 / 2c 1 / 2k 1 / RF 2k/1b 1 / 3h 2 / 4f 7 / 4b 2 / 4k 1 6.02 4n 1 / 4o 1 / 4q 1 5.01 4r 2 / 5a 3 / ------------------------------------------------------------------------------------------------------------ 2) Ethics statement {#sec004} ------------------- The collection of biological samples (authorized collection DC2008-680) was approved in December 2008 and reconfirmed in December 2013 by the bioethics committee of the French Ministry of Higher Education and Research. All clinical investigations were conducted according to the principles expressed in the Declaration of Helsinki and the study was approved by the local ethics committee (Comité de Protection des Personnes, Sud-Est V IRB0006705). All patients gave signed informed consent for the use of data and samples for research at the beginning of their hospital stay, as part of the institutional procedures. Authors BN and SL had access to full identification of samples, including full names of patients. Samples were then de-identified prior to any analysis. Other authors did not have any access to identification of samples. 3) Sequencing method {#sec005} -------------------- All samples had been previously genotyped using our in-house sequencing assay considered here as the reference assay. Extraction of RNA from plasma samples was performed using the NucliSENS^®^ EasyMAG system from BioMérieux (Marcy l'Etoile, France). A total volume of 1 mL of plasma was used for the extraction and nucleic acids were eluted in 50 μL. Extracted RNA was stored at -80°C until analysis and re-frozen after RT-PCR and sequencing for eventual re-analysis. First, nucleic acid amplification was performed in the NS3 region of the HCV genome as described by Besse et al \[[@pone.0194396.ref014]\]. Amplification was checked on 1% agarose gels with the GelRed intercalant. When no amplification was obtained, the NS5B region was amplified as described by Sandres-Sauné et al \[[@pone.0194396.ref015]\]. Finally, and only for samples from the difficult to type panels, an amplification of the Core region was performed as described in Pham et al \[[@pone.0194396.ref016]\]. The sequencing reaction was then performed using the Beckman Dye Terminator Cycle Sequencing (DTCS) kit (Beckman Coulter, California, USA), and sequence detection was made by capillary electrophoresis in a Beckman Coulter capillary electrophoresis CEQ 8000 system. Newly obtained sequences were first blasted against the whole nucleotide NCBI database \[[@pone.0194396.ref017]\]. The Max Planck Institute Geno2Pheno \[HCV\] tool was then used to confirm the genotype \[[@pone.0194396.ref018]\]. A final step of phylogenetic analysis was performed using the MAFFT alignment tool \[[@pone.0194396.ref019]\] from the Japanese Computational Biology Research Consortium. The phylogenic tree was built using reference sequences described in Murphy et al \[[@pone.0194396.ref020]\]. Our sequencing assay is regularly assessed by the Quality Control for Molecular Diagnostics (QCMD) scheme (catalogue code: QAV034117). The results from the evaluations received while this study was conducted are highly supportive of our assay. Results form 2016 and 2017 evaluations reported a score of zero (highest grade) for every sample tested. 4) Roche cobas^®^ HCV GT {#sec006} ------------------------ The method evaluated here is the kit (Roche cobas^®^ HCV GT) specifically designed to be used on the Roche cobas^®^ 4800 automated platform. RT-PCR was performed on a Z480 thermocycler from the same automated platform \[[@pone.0194396.ref012]\]. The cobas 4800 system is currently CE marked and FDA approved but the cobas 4800 HCV GT assay is only CE marked and not yet FDA approved. Detection of HCV samples was made by the same thermocycler, and the signal was processed through a proprietary interface \[[@pone.0194396.ref021]\]. The Roche HCV GT assay is designed to differentiate all genotypes from 1 to 6 and also to determine 1a, 1b and 1 non-a non-b subtypes. No typing of subgroups within genotypes 2, 3, 4 or 6 is possible and genotypes 7 are not recognized. The sample volume required for analysis is 400 μL of plasma with 250 μL of dead volume for automated pipetting. Each sample is distributed between three reaction wells for RT-PCR. The PCR is performed using the kit's proprietary designed primers. Amplification of genotypes 2, 3 and 6 relies on the 5'UTR region. For genotypes 1, 4 and 5, the Core region of the genome is amplified and detected. Finally, discrimination between genotypes 1a and 1b is done by amplification of the RNA-dependent-RNA-polymerase (NS5B) gene. Detection of both the HCV genome and its genotype is achieved using fluorescent hydrolysis probes. Internal controls are incorporated in the kit and allow technical validation. A positive control containing extracts from genotypes 1, 4 and 6 allow monitoring of the three RT-PCR reactions. A negative control is also included in the kit. Failure to amplify and detect HCV genotypes is highlighted by the software with three different outcomes: samples considered as 'Invalid' are amplified and detected but invalid internal controls prevent the results from being released. Samples indicated as 'Indeterminate' are amplified and detected but the genotype cannot be accurately determined. Samples indicated as 'Failed' were not correctly extracted or did not perform well in the pre-analytical phase and must be re-tested. Results {#sec007} ======= Samples included in this study were distributed as follows: Genotypes 1 made up 55.5% (n = 101) of the samples. These included subtype 1a (n = 48, 26.4%), subtype 1b (n = 46, 25.3%) and non-a non-b subtype (n = 7, 3.8%). The remaining 81 samples were distributed among the other genotypes: genotype 2 (n = 14, 7.7%), 1 recombinant form 2k/1b, genotype 3 (n = 19, 10.4%), genotype 4 (n = 43, 23.6%) and genotypes 5 and 6 (n = 4, 2.2%). The full distribution of genotypes in the three panels tested is listed in [Table 1](#pone.0194396.t001){ref-type="table"}. Altogether, results could be obtained for 74.7% of the samples in a first-pass and for 77.5% after re-testing failed samples. The cases of samples that failed to give results in the first run and gave results when retested could be explained by either the presence of a clot in the sample or by an insufficient volume of plasma in the analysis tube. Among the 101 samples in the regular genotyping panel, 88 (87.1%) were accurately identified by the Roche kit ([S1 Table](#pone.0194396.s001){ref-type="supplementary-material"}). Results from both techniques are shown in [Table 2](#pone.0194396.t002){ref-type="table"}. When a result was obtained with the Roche kit, there was no mismatch in identification at either the genotype level or at the subtype level. The remaining 13 samples failed to be amplified ("failed" n = 9, 8.9%) or to be correctly classified ("indeterminate" n = 4, 4.0%). Eight of these samples could be re-tested thanks to a sufficient volume of remaining plasma. Among these 8 samples, 5 gave accurate results on second test and 3 had repeated amplification failure. The invalid results were due to a failed internal control that could be due to an inhibition of the PCR reaction in at least one of the 3 PCR steps but why some results were obtained when retested remains unclear \[[@pone.0194396.ref021]\]. Most of the samples which failed were from genotypes 1a (4/35, 11%), 3a (3/11, 27%) or 4d (3/12, 25%). Mean viral load among these failed samples was not significantly different from the rest of the panel (6.5 log10 IU/mL, p = 0.801). Determination of genotype 6 was explored with only one sample of subtype 6e. This sample was not correctly detected by the Roche kit, the system giving an 'indeterminate' result for the sample even though the viral load was elevated (6.9 log10 IU/mL). 10.1371/journal.pone.0194396.t002 ###### Correspondence between results obtained with both techniques for the regular sample panel (n = 101). ![](pone.0194396.t002){#pone.0194396.t002g} ------------------------------ --------------------------------- ------- ---- --- --- --- --- -------- --- ---- ---- Roche HCV GT genotyping results Total 1 1a 1b 2 3 4 5 6 NI ^b^ Sequencing technique results 1a 31 4 35 1b 29 1 30 2c 2 0 2 3a 8 3 11 4a 6 1 7 4d 9 3 12 4k 1 0 1 4n 1 0 1 4r 1 0 1 6e 0 1 1 ------------------------------ --------------------------------- ------- ---- --- --- --- --- -------- --- ---- ---- NI: Not interpretable (Failed, invalid or indeterminate) Among the LiPA failed samples, only 61.5% (n = 24) of the tested samples were correctly genotyped and subtyped by the Roche cobas 4800 kit ([Table 3](#pone.0194396.t003){ref-type="table"}). Four samples could be retested and results were obtained for two of them with a correct genotype classification. Two samples gave different results with both techniques at the subtype level ([S2 Table](#pone.0194396.s002){ref-type="supplementary-material"}). The Roche technique found both to be genotype 1b. However, our in-house sequencing technique classified them as 1d. They were the only two cases for which a difference between the techniques was highlighted. One was checked by resequencing the sample on a different HCV genomic region thanks to the remaining sample volume confirming a 1d subtype. We tested only one recombinant genotype where the sample was determined as 2k/1b by our in-house sequencing technique. The kit identified it correctly as both genotypes 1b and 2. Among the 13 (33.3%) uninterpretable samples, 1 was invalid (1 genotype 4k at 6.02 log10 IU/mL), 7 were considered as indeterminate (4 genotypes 1 non-a non-b, 2 genotypes 3h and 1 genotype 4o) and 5 as failed (4 genotypes 4 non-a non-d and 1 genotype 2c). 10.1371/journal.pone.0194396.t003 ###### Correspondence between results obtained with both techniques for the LiPA failed samples (n = 39). ![](pone.0194396.t003){#pone.0194396.t003g} ------------------------------ --------------------------------- ------- ---- ---- --- --- --- --- --- ---- --- Roche HCV GT genotyping results Total 1 1a 1b 2 3 4 5 6 NI Sequencing technique results 1g 3 3 1d 2 0 2 1i 1 1 2i 9 0 9 2b 0 1 1 2c 1 0 1 2k 1 0 1 2k/1b 1 1 0 1 3h 0 2 2 4f 5 2 7 4b 1 1 2 4k 0 1 1 4n 1 0 1 4o 0 1 1 4q 1 0 1 4r 1 1 2 5a 3 0 3 ------------------------------ --------------------------------- ------- ---- ---- --- --- --- --- --- ---- --- NI: Not interpretable (Failed, invalid or indeterminate) In the whole study, a total of 7 samples were genotype 1 non-a non-b, but none of them was correctly identified by the Roche kit despite correct PCR amplification. Five of those samples gave no result (indeterminate), and the two remaining were the 1d misclassified samples. For viral loads \< 3 log10 IU/mL genotyping assay success was correlated with the viral load (p = 0.0061) ([Fig 1](#pone.0194396.g001){ref-type="fig"}) and failures (20/42) mostly occurred when it was \< 2.3 log10 IU/mL \[200 IU/mL\] ([S3 Table](#pone.0194396.s003){ref-type="supplementary-material"}). Surprisingly, these missing results are essentially indeterminate and invalid thus indicating the correct amplification of the sample ([Table 4](#pone.0194396.t004){ref-type="table"}). 10.1371/journal.pone.0194396.t004 ###### Correspondence between results obtained with both techniques for the low viral load samples (n = 42). ![](pone.0194396.t004){#pone.0194396.t004g} ------------------------------ --------------------------------- ------- --- --- --- --- --- ---- --- ---- ---- Roche HCV GT genotyping results Total 1 1a 1b 2 3 4 5 6 NI Sequencing technique results 1a 2 4 7 13 1b 3 8 5 16 1e 1 1 3a 3 3 6 4a 2 1 3 4d 0 2 2 4g 0 1 1 ------------------------------ --------------------------------- ------- --- --- --- --- --- ---- --- ---- ---- NI: Not interpretable (Failed, invalid or indeterminate) ![Genotyping results with Roche HCV genotyping kit on samples with low HCV viral load.\ (p = 0.006) Successes: filled circles; failures: filled squares; (p = 0.006).](pone.0194396.g001){#pone.0194396.g001} We also evaluated the kit in terms of its ease-of-use and integration into the workflow of our laboratory. The current organization of our in-house procedure leads to the extraction and reverse transcription phases being done on the first day. Then PCR amplification is performed on the second day with control of the amplification by gel electrophoresis. The sequencing reaction is usually done on day three and results are released on the fourth day after verification of chromatograms and analysis. Moreover, experienced staff are required. The Roche kit handles samples in a maximum runtime of three and a half hours. The operator does not need particular skills above those of a laboratory technician and their presence is only required at three different steps (loading of samples and reagents, transfer of extracted samples to the amplification and detection system, and validation of results), for a total of 30 minutes including technical validation. The release of results can be made 4 to 5 hours after the beginning of the run (pretreatment of samples not included), including biological validation. Discussion {#sec008} ========== Like the previous studies published on this assay, our study confirmed the good accuracy of the Roche HCV genotyping assay with no genotype or subtype 1a/1b mismatch when compared to our in-house PCR-sequencing HCV genotyping assay \[[@pone.0194396.ref012],[@pone.0194396.ref022],[@pone.0194396.ref023]\]. When results were available the overall agreement between methods was 98.53%. An exploitable result was available for 74.7% of samples (87.1% for the routine panel, 66.7% forLiPA failures and 52.4% for low viral load samples) in a first-pass analysis. Also no strong conclusion can be drawn considering the relative small number of samples for each genotype, the kit developed by Roche Diagnostics was unable to discriminate between subtypes of genotypes 2 to 6 nor to accurately identify subtypes of genotype 1 that differ from 1a or 1b. Among the 101 genotype 1 samples tested in this study, 94 were accurately identified between 1a or 1b subtypes, two were inaccurately subtyped (1b/1d), and 5 gave indeterminate results. This work also highlighted a quite high rate of uninterpretable results during the first pass (from 13% for regular samples to 52% for viral loads inferior to 1000 IU/mL). Re-testing of failed samples allowed the number of samples without results to be reduced but required a sufficient volume of remaining plasma. Stelzl et al. reported only 3.8% of indeterminate samples \[[@pone.0194396.ref012]\]. Our results on the routine panel exhibited the same number of indeterminate results (4%) but this number increased in low viral load and LiPA failure panels. Moreover we took into account the total number of samples for which no result was obtained. These uninterpretable results corresponded mainly to genotype 3 and 4 non-a samples. The manufacturer's instructions about viral loads set different lower limits of detection for each genotype. The range varies from 2.1 log10 IU/mL \[125 IU/mL\] for genotype 1a to 3 log10 IU/mL for genotype 5 ([Table 5](#pone.0194396.t005){ref-type="table"}). We tested 42 samples below this threshold. Samples with viral load below the manufacturer's limit failed in 47.6% (n = 20) of cases. Interestingly, 17 out of the 42 samples gave interpretable and accurate results. Nevertheless, most HCV genotyping will be performed before introduction of antiviral drugs, when viral loads are usually high. 10.1371/journal.pone.0194396.t005 ###### Characteristics of the commercially available HCV genotyping assays. ![](pone.0194396.t005){#pone.0194396.t005g} ------------------------------------------------------------------------------------------------------------------------------------------------- cobas^®^ HCV GT Siemens Versant HCV GT 2.0 Abbott RealTime HCV GT II -------------------------- ---------------------------- ------------------------------------------------------------- --------------------------- Technology Real time PCR Reverse hybridization technology Real time PCR Target regions 5'UTR, NS5B, Core 5'UTR, Core 5'UTR, NS5B Sensitivity (LoD) Plasma samples:\ 2,106 IU/mL 500 IU/mL GT 1a,2,3,4,6: 125 IU/mL\ GT 1b: 250 IU/mL\ GT 5: 1000 IU/mL\ Serum samples:\ GT 1a,1b,3,4,6: 125 IU/mL\ GT 2: 50 IU/mL\ GT 5: 500 IU/mL Genotype coverage 1--6, 1a/b genotypes 1--6 and subtypes 1a vs. 1b, and subtypes 6 (c-I) 1--6, 1a/b Sample processing volume 400 μL 500--1000 μL 500 μL ------------------------------------------------------------------------------------------------------------------------------------------------- Other failures were mostly due to less common genotypes (1d, 1e, 1g, 1i, 2b, 2i, 2k, 3h, 4b, 4d, 4f, 4g, 4k, 4n, 4o, 4q, 4r) which had already failed when using the LiPA VERSANT HCV Genotype 2.0 assay. We have already described the failure of this last technique to type non-European samples due to the non-specificity of the probes used in 5'UTR and Core regions \[[@pone.0194396.ref008]\]. Despite the addition of amplification in the NS5B region, these samples also exhibited a very high rate of failure with the Roche assay (13/39) most likely due to mismatches with the primers or the probes used. For 5 samples, amplification failed, pointing to primers mismatches. This was mostly the case with genotype 4 samples, probably due to the variability of the Core region used for amplification. Hovwever, the other 8 samples failed to be detected, highlighting the limitations of the probes used. On the other hand, the conjunction of targets on the 5'end of the HCV genome (5\'UTR and Core) and on the 3'end of the genome (NS5B) allows the Roche assay to detect 2k/1b recombinant forms. Taken together our results are in line with the findings of other teams \[[@pone.0194396.ref012], [@pone.0194396.ref022], [@pone.0194396.ref023]\]. Unlike Nieto-Aponte et al, we did not observe any double-infection. Stelzl et al reported an overall agreement of 87.4% on a similar number of plasma samples \[[@pone.0194396.ref012]\]. Fernández-Caballero et al showed more successful results on a comparable number of samples tested, but they did not include rare subtypes \[[@pone.0194396.ref023]\]. Our results show more discrepancies between our in-house sequencing technique and the Roche kit, with up to 26.4% samples unidentified. This is mainly due to our use of the kit at the limits of recommended viral load, as well as the selection of rare genotypes. Unlike PCR-sequencing assays, this new HCV genotyping kit is unable to recognize any new genotype, such as the recently-described genotypes 7a, 7b or 8a \[[@pone.0194396.ref024]--[@pone.0194396.ref026]\]. Embedded controls, which are the same than for viral load assay, will detect the viral RNA but no specific probe is included resulting in an uninterpretable result. Due to its design, the Roche assay also does not differentiate genotype 1 subtypes other than 1a/1b. Hence, subtyping was not as precise as that available with the sequencing method. At present, the international recommendations do not differentiate anti-HCV treatments according to subtype with the exception of genotypes 1. In the near future, despite the arrival of theoretically pan-genotypic therapeutic combinations, some differences may persist such as the lower efficacy of sofosbuvir/velpatasvir/voxilaprevir for 8 weeks on genotype 1a or of the glecaprevir/pibrentasvir combination on previously treated genotype 3a \[[@pone.0194396.ref027]--[@pone.0194396.ref029]\] It has also been shown in previous studies that different subtypes of genotype 4 viruses behave differently in terms of the virologic response patients can achieve. Treatment naïve patients with subtype 4d or 4r showed significantly more failures in virologic response than those with subtype 4a \[[@pone.0194396.ref030],[@pone.0194396.ref031]\]. The latest editions of both the American \[[@pone.0194396.ref032]\] (AASLD---IDSA) and the European recommendations \[[@pone.0194396.ref003]\] (EASL), point out the necessity of looking for resistance associated mutants in the NS5A region of the HCV genome for the treatment of genotype 1a virus using elbasvir/grazoprevir. This will require the addition of another assay. Our study highlighted a few weaknesses of the Roche assay, but the number of samples in each genotype or subtype was too low to draw definitive conclusions as to assess the failure to type one particular subtype and more studies focusing on each of the variants will be needed. Genotype 3a samples were scarce despite being one of the most prevalent genotypes found worldwide, the chosen viral panels being strongly influenced by local epidemiology. Therefore, more data need to be gathered on this genotype to give a full overview of the Roche technique. We did not include any double infection by two different HCV strains in the tested samples and this also should be tested in further studies. With regard to the implementation of the Roche assay in a medical virology laboratory, we found staff training was rapid (1 day) and the instruments needed can also be used for HCV, HBV, and HIV viral load measurements. Unlike our current technique based on customized RT-PCR and Sanger sequencing for which operator time is high, the Roche assay was fast and not time-consuming. As results can be obtained in less than 4 hours, this ease-of-use gives it a distinct advantage over in-house PCR-sequencing assays that require experienced staff, as well as several days before obtaining a result. Of note, if genotyping failed with the Roche assay, the next step was to perform in-house sequencing. Therefore the total working hours became more than 5 days. Nevertheless, considering the costs of implementing this assay, while no complete economic study was performed, it is likely that reduced hands-on time by trained technical staff will lead to reduced costs as seen by Nieto-Aponte et al. \[[@pone.0194396.ref022]\]. In conclusion, the cobas 4800 HCV GT assay from Roche provides a rapid, easy-to-use and accurate solution to first line genotyping of HCV. PCR-sequencing assays performed by an expert center will still be required when the Roche assay fails or when precise data on subtype or NS5A resistance associated mutations are needed for therapeutic decision-making. Supporting information {#sec009} ====================== ###### Raw data for the regular samples. This file includes sequencing results, results obtained with Roche assay on first and second pass and viral load for each sample. (DOCX) ###### Click here for additional data file. ###### Raw data for the LiPa failed samples. This file includes sequencing results, results obtained with Roche assay on first and second pass and viral load for each sample. (DOCX) ###### Click here for additional data file. ###### Raw data for the low viral load samples. This file includes sequencing results, results obtained with Roche assay on first and second pass and viral load for each sample. (DOCX) ###### Click here for additional data file. We thank Alison Foote (Grenoble Alpes University Hospital) for critical editing of the manuscript. We also acknowledge Roche Diagnostics (France) and in particular Bertrand Van Roy from the Roche Molecular Diagnostics team in Grenoble for help and support. [^1]: Competing Interests:None of the authors have any financial or other connection with Roche. Cerba Healthcare provided support in the form of a salary for author JDP, but did not have any additional role in the study design, analysis, decision to publish, or preparation of the manuscript. This does not alter our adherence to PLOS ONE policies on sharing data and materials.
Q: (exam study help) Prove that there are infinitely many integers $m$ such that: $m^3 \equiv n^6 \pmod{19}$ I'm studying for a first year discrete math final exam. I was wondering if anyone could help me with this proof. I started by writing down what I know is true, but I can't seem to bring what I have so far to some meaningful conclusion. Let $n$ be a fixed but arbitrary integer. Prove that there are infinitely many integers m such that: $$m^3\equiv n^6 \pmod {19}$$ $$m^3\equiv n^6 \pmod {19} \Rightarrow 19\|(n^6-m^3)$$ then: $n=19k+r $ for $ k\in \mathbb{R}$ $m=19l+r $ for $ l\in \mathbb{R}$ $n^6=(19k+r)^6 $ $m^3=(19l+r)^3 $ now: $$(n^6-m^3)=(19k+r)^6-(19l+r)^3$$ $$=19(k^6-l^3)+r$$ A: First notice m = $n^2$ is a solution for the equivalence relation. Assume there are only finite many solutions. Let m be the largest. m + 19 is a larger solution. Thus there are infinitely many solutions.
Let α be a permutation of the vertex set V(G) of a connected graph G. Define the total relative displacement of α in G by δα(G) = ∑x,y∈ V(G) \|dG(x,y)-dG(α(x), α(y))\|, where dG(x, y) is the length of the shortest path between x and y in G. Let π(G) be the maximum value of δα(G) among all permutations of V(G). The permutation which realizes π(G) is called a chaotic mapping of G. In this paper, we study the chaotic mappings of complete multipartite graphs. The problem is reduced to a quadratic integer programming problem. We characterize its optimal solution and present an algorithm running in O(n5 log n) time, where n is the total number of vertices in a complete multipartite graph.	https://researchwith.njit.edu/en/publications/quadratic-integer-programming-with-application-to-the-chaotic-map
Aug 21, 2008· A) We have a lot of snails here. My chickens are allowed to free-range in our little backyard during the day and they're very good at finding all the snail hiding spots. I'll frequently find them hiding behind a perennial bashing a snail against concrete or a rock to break it open but they also end up eating the shell. Hazel Slug & Snail is a renewable all natural product made from 100% crushed hazelnut shell. Spread around the base of your vegetables, fruits, & blooms. With the sharp edges of the hazelnut granule & the absorbent powder the slugs & snails start to desiccate while attempting to cross the shell barrier. May 15, 2017· In this work the performance of mussel shell as aggregate in plain concrete has been studied. The mussel shell used came from the cannery industry, which produces more than 1 million tonnes of shell by-product a year worldwide. The mussel shell has been heat-treated at 135 °C for 30 min and then crushed and sieved into sand and gravel. Oct 11, 2016 - Crushed Oyster Shell used as an aggregate in concrete driveway Aug 18, 2020· Concrete production consumes a large amount of fine and coarse aggregates. Therefore, eliminating or reducing the consumption of aggregates in concrete can produce environment-friendly building materials. Considerable research has confirmed that the use of waste materials in concrete addresses the high utilisation of raw materials. Walnut is a common farming product in the north of Iraq. The shell pervious concrete was created by replacing 60% mass of the natural aggregates in control pervious concrete by crushed seashells. The freeze-thaw resistance of the shell pervious concrete is lower than the control pervious concrete and it was not proportional with the mechanical strength. Jul 11, 2018· Crushed Eggshells: Field Test. Since there is still much discussion on this topic, I carried out a simple test. I dried and crushed the shells of uncooked eggs. Then I placed them on a table and formed a little wall of around two inches (five centimeters) wide and put some snails and slugs in … Jul 16, 2018· Mulch from crushed seashells should serve as a slug and snail defense. Does it work? Usually, crushed oyster shells are used to feed laying hens. The shells contain a lot of lime, which is needed by the hens for the production of eggs. In addition, crushed shells are used on sidewalks, where they harden to form a solid surface. Jun 10, 2018· When bought in bulk, seashells are on the more affordable end of the spectrum: comparable to crushed gravel; less than asphalt, concrete, or stone. Clamshells seem to be the least expensive, offered at about $40 per cubic yard, or $50 per ton. We priced crushed oyster shell at … Shell is a great alternative to other types of ground cover and base materials. Our Shells are mined in Florida, supporting the local economy. Our Coastal Double Washed Shell is great for Plant Beds, Decorative landscaping, Parking lots, and Xeriscaping. We deliver Decorative Bulk Shell throughout the … Jan 28, 2018· Someone here recommended cuttle bone for my snails. I'm curious is crushed oyster shell would be comparably beneficial. I did a brief google search and found this Exert: " Oyster Shell for Aquarium Use: Aquatic creatures that require a higher pH and/or love alkaline water, will noticeably respond to the addition of oyster shell. It is perfect for healthy, smooth shell development in snails. Use for exposed concrete surfaces – or polish to achieve an easy to maintain natural stone finish. ... Crushed cockle shells in a riot of waterproof colour. Exclusively manufactured in the UK by Specialist Aggregates using clean recycled shells. Suitable for arts, crafts, event and stage decor. ... Description. Coquina Stone is a sedimentary rock mainly composed of seashells or coral that takes years to produce and is expensive to process. Proline's Seamless Coquina Stone with Sea Shells can provide you a charming aquatic texture with the low cost, durability and the endurance of concrete. Get Your Dirt, Mulch, Rock, Or Shell Today! Visit Us And We'll Guide You Through The Process Of Selecting Your Materials. Pinellas 39660 US Hwy. 19N Tarpon Springs, FL 34689 Directions; Hillsborough/Pasco 14024 S.R. 54 Odessa, FL 33556 Directions; Through a brief call, you can get a quote for your job materials. Pinellas Call us Text us CALL ... study of periwinkle shells as coarse aggregates in concrete works and concluded that 35.4and 42.5% replacement of % crushed granite with periwinkle shells by weight give concrete . Figure 1. Shells of typical (a) snails, (b) periwinkle and (c) cockle. (b) (a) (c) IJSER In other studies for potential use, the shells are mostly used as additive or replacement of part of the cement in concrete. For example, construction material mixed with crushed oyster shell and sand was used for sand compaction piles to improve soft soils underneath a breakwater port in Japan . However, the lime contained in the shells does ... crushed snail shell on concrete This is a discussion on Crushed Oyster Shell within Snail Shells for concrete admixture companies in hyderabad [Live Chat] The Use of Golden Snail (Pomacea sp) as Animal Feed - Tropicultura larized and farmed in concrete tanks, ponds and other controlled golden snail shell meal by 22 . crushed snail shell on concrete; crushed quartzite silicon; crushed rock supply; how can coal be crushed to 1mm; density of crushed granite stones; crushed stone in milwaukee wi; crushed cone; how track ballast is crushed; crushed gravel 20 50mm; lima ohio sevier stone crushed; red crushed sandstone; industrial application of dolomite crushed ... Nov 04, 2013· – The purpose of this paper is to examine the use of a waste marine sea shell product incorporated into a concrete mix as an aggregate replacement. Utilising shells reduces the storage of shell waste, also reducing the need for quarried aggregate and has potential benefits of adding a different material to a design mix concrete mix design for improved performance., – The test … Speaking on radio 94.1FM on Wednesday during a segment on the Giant African Snail, Mohansingh said when crushed, the snails go into a distress response where they emit their eggs. Ewer Solar Snail Figurine Lights Garden Decor, Solar Powered Outdoor Decor Animal Statue LED Decor Figure for Patio Lawn Yard Backyard Decorations, Pond Landscape Light Ornament. 3.3 out of 5 stars 6. $17.99 $ 17. 99 $20.99 $20.99. FREE Shipping. C&F Garden Decor Outdoor Polyresin Snails … Quality Florida Landscaping Shells Shell-A Unique Look in a Long-Lasting Package. Using shell as a landscape accent is a popular choice for Florida homes. Like river gravel, shell is used as durable, environmentally friendly ground cover. Shell can provide drainage and reduce weeds in your garden. Oct 10, 2018· I want to feed some eggshell to my snails for the extra calcium. Can I drop some crushed shells into my freshwater tank? Or is it absolutely necessary that I grind eggshells into a powder? Also, will either method be safe for my goldfish? I am aware that some shells containing calcium and other minerals can raise the ph of my tank. My normal ph ... The shell patch needs to be larger all around than the hole is. The location of this hole isn't terrible but because it is curved, it may not be that easy to find a shell piece that will fit. It need not be snail shell, could use some thin seashell. Make sure the glue does not touch the snail's bodily flesh, it'll burn and sting something fierce. Adekunle stated that the concrete which contained periwinkle shell as partial cement replacement gained the most compressive strength compare to snail and oyster shell recorded at 19 MPa after 28 days curing which was higher than the control sample reading at 17.5 MPa. For the cockle, the concrete of grade 35 MPa had been made [7,18]. 48 concrete cubes, 24 concrete beams, 48 mortar cubes and 48 mortar beams have been prepared for testing purposes. Results revealed that there was a slight decrement in the compressive strength of standard concrete and mortar cubes prepared using raw sand containing 29% sea shells comparing to cubes made with sand removing all exists sea shells 0%. Crushed Snail Shell On Concrete. 1 introduction ijser,15 uses for egg shell,2019 jetir december 2019, volume 6, issue 12,buy crushed sea shells,caribbean coralina stone,characteristics strength and durability of groundnut shell,contemporary hotsell liner plate for coal mill,crushed concrete in pasadena tx,crushed elongated coins machine,crushed snail shell on concrete, Snail Crusher For Sale ires-bibliotecait. crushed snail shell on concrete 46 4178 Ratings The Gulin product line, consisting of more than 30 machines, sets the standard for our industry We plan to help you meet your needs with our equipment, with our distribution and product support system, and the continual introduction and updating of. This review paper emphasis on various sea shells ash such as cockle, clam, oyster, mollusc, periwinkle, snail, and green mussel shell ash as partial cement replacement and its objective is to ... In other studies for potential use, the shells are mostly used as additive or replacement of part of the cement in concrete. For example, construction material mixed with crushed oyster shell and sand was used for sand compaction piles to improve soft soils underneath a breakwater port in Japan . However, the lime contained in the shells does ...	https://www.cafe-kleber.fr/39994_crushed_snail_shell_on_concrete.html
Japan, a unique country where ancestral culture and ultra-modernity blend together, never ceases to fascinate. Although Kyoto is no longer the imperial capital of Japan, it is still its heart. It has preserved its fabulous secrets, such as the old quarter of Gion, the famous Geisha quarter, its numerous Buddhist temples, its Shinto shrines and its magnificent gardens hidden on the surrounding hillsides. Japanese language school The school is located in the famous district of Gion, the oldest and most traditional district of Kyoto, if not of all Japan. The school offers a wide range of facilities: - 9 classrooms equipped with interactive whiteboards, projector and WIFI - Comfortable relaxation areas to provide you with an environment conducive to learning - Game consoles, TV and large DVD library - DVD and CD players available for personal study - Manga Library - Free tea and coffee Courses General information The courses are offered for a duration ranging from 2 to 24 weeks in order to allow European citizens to travel without a visa (it is indeed possible to make 2 stays of 12 weeks) provided that you leave the territory at the end of the first period of 3 months and re-enter again. For stays of at least 6 months, a student visa is required (in this case, you should register at least 6 months in advance, as the time required to obtain one is very long). Stays of 6 months or longer take place at the school in Fukuoka. - Duration of a lesson: 50 minutes - Maximum number of students per class: 8 (exceptionally 9) - Timetable: classes are given in the morning and/or afternoon depending on your level group. - Placement test: on the first day of class - Certificate of participation at the end of the stay - Weekly evaluation test - Personal work to be done at home, corrected and evaluated - The teachers are all qualified in teaching Japanese to foreigners. They are therefore fully aware of the difficulties you may experience and the different learning phases you will go through. Course formulas Japanese Basic Course: 20 lessons per week** These practical courses give you the linguistic and cultural keys that will allow you to express yourself quickly and correctly, and thus be autonomous to fully enjoy your stay in Japan. The courses focus on practical everyday situations that will allow you to quickly become independent in daily life. You attend 2 weekly lessons based on grammar and 2 weekly lessons based on listening, writing and reading. Levels: beginner* to advanced Beginners can start on the 1st Monday of each month all year round and during the summer on the 1st and 3rd Monday of the month. Intensive course: 25 lessons In addition to the 20 Japanese basic course, you also take part in 5 conversation classes per week. Levels: elementary to advanced Combined course : 25 lessons per week: 20 lessons of the basic course + a complementary module of your choice Choice of modules : - Traditional culture: 6 hours of cultural activities per week, on Saturdays. Maximum duration: 4 weeks Levels: beginner to advanced - Pop culture: 6 hours of cultural activities per week, on Saturdays. Maximum duration: 4 weeks Levels: beginner to advanced - Private tuition* : possibility of 5 to 20 hours of private tuition per week in addition to the basic course outside the summer period (beginning of June to end of August). Consult us for details Levels: beginner to advanced - Summer Special "Next Gen" - 16-19 years old** (June, July and August only) : 20 lessons of the basic course per week + 6 hours of cultural activities per week, on Saturdays. The program includes many cultural activities such as: pottery, drawing, manga, tea ceremony, zazen meditation, wearing a kimono, paper making... Duration of the cultural activities module: 4 weeks. Possibility to choose another module after 4 weeks, such as conversation, pop culture or traditional culture. Levels: beginner to advanced - The activities In order to enable students to enjoy a cultural, traditional and modern experience at the same time, the school offers a large number of cultural and linguistic activities, free of charge or at cost price. Free activities: conversation workshop (students are grouped in pairs, 1 foreign student with 1 Japanese student to facilitate conversation), cinema session, city tour, game night (hanafuda, shogi...), festivals, etc. Activities at cost price: cultural modules in addition to the courses, Friday night outing, concerts, events, bowling, karaoke... Sample program for example purposes only: Housing Host family (from 16 years old) Single room, half board This is the ideal formula for any student wishing to immerse himself completely in the Japanese culture and language. You will follow the daily life of a typical family, carefully selected by the school for their kindness and willingness to share their culture. Travel time to the school: maximum 60 min. by public transport. Residences (18+ years old) Single room in shared apartment (up to 8 people per apartment), shared bathroom, living room and kitchen, without meals. Minimum duration: 2 weeks The residences are located in different neighborhoods, close to metro stations. Travel time to school: between 35 and 50 minutes by public transport Rates & Dates 2020 Dates 2020 Possible stays from 2 to 24 weeks, i.e. 2 x 12 weeks. For stays of 6 months or one year, see the school in Fukuoka. Course starting dates (stays of 1 to 12 weeks), except for beginners: every Monday (or Tuesday if Monday is a public holiday) Course start dates for beginners: the 1st Monday of each month all year round and during the summer, the 1st and 3rd Monday of the month. Young learners 16-19 years old : June, July and August Bank holidays and holidays (no classes are given on these dates, an activity is organised instead): 01/01, 13/01, 10/02, 24/02, 20/03, 01/05, 04/05, 05/05, 06/05, 23/07, 24/07, 10/08, 21/09, 22/09, 02/11, 23/11, 31/12 Prices 2020 All taxes and enrolment fees includedSign up for this program \|Lessons and accommodation\| \|Duration\|\|20 lessons per week (family, single room, half board)\|\|20 lessons per week (residence, shared room, no meals provided)\| \|2 weeks\|\|1,750 €\|\|1,430 €\| \|3 weeks\|\|2,335 €\|\|1,855 €\| \|4 weeks\|\|2,930 €\|\|2,295 €\| \|6 weeks\|\|4,125 €\|\|3,165 €\| \|8 weeks\|\|5,370 €\|\|4,095 €\| \|12 weeks\|\|7,810 €\|\|5,895 €\| \|16 weeks\|\|10,280 €\|\|7,730 €\| \|18 weeks\|\|11,405 €\|\|8,535 €\| \|24 weeks\|\|12,850 €\|\|9,020 €\| \|Supplement / week\| \|Module of choice\|\|100 €\| \|Peak season classes (July and August)\|\|30 €\| \|conversation, traditional culture, pop culture and special 14-19\| \|Transfer (per trip)\| \|Kansai or Itami airport\|\|210 €\| \|Kyoto train station\|\|95 €\| Prices include : - The chosen course program - Housing as shown - The teaching material - Orientation on the 1st day of class - Internet access - Registration fees PRices don't include : - The trip to Kyoto - Transfers and local transport - Possible visa fees - The deposit to be paid for the accommodation in residence Practical information Travel Day of arrival : Sunday Day of departure : Sunday **Arrival and departure between 8am and 8pm in residence and before 6pm in a host family Airport of arrival : - Kyoto Airport : Kansai ou Itami Airports Transfer possible on request (subject to a fee, see tab "Dates & rates"). Note: do not book your ticket until we have confirmed your registration with us. Visa info Stays of less than 90 days, renewable once Citizens of the European Union(1) may travel to Japan without a visa if their stay does not exceed 90 days. They must have a passport valid for the duration of their stay and a return plane ticket. The tourist visa can be renewed once after 12 weeks by leaving the territory and re-entering for a further 12 weeks. (1) Other nationalities: consult the website of the Embassy of Japan in your home country. You will also find the list of visa-free countries on the link. A stay of over 24 weeks All stays of more than 24 weeks are considered long-term stays and require a visa. Visa applicants are required to provide a Certificate of Eligibility (COE) (issued by our partner school) in order to apply for a visa.	http://easylanguages.com/languages-abroad/japanese-in-kyoto-16_a546
Unlike the previous productions,this play is not based on any work of literature, biography or historical event, but is a creation of pure imagination. The story is set in a fictive Pear Garden of the Immortal Mountain in the Kingdom of the Chinese Theater. In this place, actor and actress speak with a stage voice and walk to the beat of gongs and drums. The protagonists are post-mortem souls of famous charaters on stage, Emperor Ming of Tang(Tang Ming-huang) and his Royal Consort Yang(Yan Gui-fei). Yan Gui-fei becomes Queen of the Immortals Palace, though she still regrets that there is not one single play that really expresses her feelings. It is only through a session of"character possession" that her resentment of beng left behind and handed over to the mob by Tang Ming-huang can finally be healed. On the other hand, after his death, Tang Ming-huang becomes the God of Chinese Opera, guarding the Xingyun Opera Troupe. One day, the Immortals Palace summons Xingyun Opera Troupe to perform before the Queen. Too frightened at the possibility of meeting Yang Gui-fei, Tang Minghuang refrains from going. As a result, he is not able to protect the actors when they blunder on stage and are driven out of the Palace with their trunks of costumes and accessories all destroyed. Finally, encouraged by the troupe owner, Tang Ming-huang enters the Palace once again to face Yang Gui-fei. When he plays the role of himself, he is able to express his repentance. This play was presented at the Fifth Taiwan International Festival of Arts(TIFA), and sponsored by the National Theater and Concert Hall. It is the third and final piece of A Trilogy of Actors, following Meng Xiaodong, and One Hundred Years on Stage. Director Lee Hsiao-pin considers this play reveals the ongoing dialectic between actors and their characters; therefore, several stage costumes with flowing sleeves appear on stage, while the actors themselves only wear the inner linings of costumes usually worn on rehearsals. The intimate interaction of the two interweaves a landscape indicating the destiny of the opera performers and their troupe. Today the number of：79人 / The total number of visitors：131,041人 / Last Updated：2019/04/19.	https://en.ncfta.gov.tw/cultureevent_138_46.html
This invention relates to a discriminating apparatus and method for detecting the design and color features of a printed pattern such as, for example a note. Document US-A-4 041 456 discloses a method for verifying the denomination of currency in which a bill to be verified is scanned lengthwise by a two track optical sensor. For each bill the resulting analog signals are divided into eight segments or windows each segment producing a binary coded pattern produced by delta modulation. This binary coded pattern is compared to a stored reference pattern and a number is produced representing the dissimilarity between the bill being scanned and the average bill of that denomination with which it is being compared. Thereafter, a processor compares the foregoing numbers with additional quantitative functions which have previously been stored relating the corresponding segments of the bill denomination being scanned to other bill denominations. With the use of limit and weighting functions they are summed over the eight different effective windows and a decision is made as to whether the proper denomination is present. Furthermore document US-A-4183 665 discloses an apparatus for testing the presence of color in a paper. This apparatus includes a sliding tray which is adapted for receiving the paper. Maintained above the tray are a light source and three photocells. The light source casts light upon the paper as the tray is passed thereunder and light is reflected from the paper's surface onto two of the photocells. Appropriate filtering causes each of the two photocells to respond to different wavelengths of light reflected from the paper, these wavelengths corresponding to colors known to be present along the path travelled by the paper as it passes under the photocells. The outputs of the photocells are thus indicative of the presence and relative positions of colored areas upon the surface of the paper. An electric circuit is included to receive and compare the outputs of the photocells with each other and with the output of a reference photocell, thereby determining the authenticity of the paper. Conventionally, to detect the printed pattern of a note, the detecting field is defined by a slit S as shown in Fig. 1. The quantity of light from the detecting visual field is photoelectrically scanned. The note is conveyed past the slit and then the photoelectric conversion signal is sampled to compare the sampling pattern with a predetermined reference pattern. For example, when the printed pattern on the note is as shown in Fig. 1(A), the light from a to b of the detecting visual field S is photoelectrically converted to obtain a waveform shown in Fig. 2(A) and further to obtain the sampling pattern from the waveform. However, when the printed pattern as shown in Fig. 1(B) is scanned over the detecting visual field a to b, the waveform shown in Fig. 2(B) which is the same as Fig. 1(A) is obtained. Therefore, the prior art is deficient in that the patterns (A) and (B) cannot be distinguished from each other although they are obviously different from each other. Accordingly, one object of the present invention is to provide an improved printed matter identifying apparatus and method which scans the printed pattern by dividing the pattern into a plurality of sections in a direction orthogonal to the direction of conveyance and compares the read-out signal from each section with the reference signal for many printed patterns, in order to verify the type of or authenticity of the printed matter. To achieve the above object, a printed matter identifying apparatus comprising conveying means for conveying printed matter through a lighted conveying path in a predetermined direction, said printed matter having a pattern, scanning means for reading at least first and second sections of said printed matter and for generating a signal for each said section representing the portion of said pattern in said respective section, said sections being divided from each other in a direction substantially orthogonal to said predetermined direction of conveyance, operating means connected to said scanning means, for effecting operations between said signals generated by said scanning means and for generating at least one operating signal, and identifying means, connected to said operating means, for identifying said printed matter on the basis of the operating signals, is characterized in that said scanning means includes at least first and second filter means, said first filter means for filtering predetermined wavelengths of light-waves transmitted from said first section and said second filter means for filtering predetermined wavelengths of light-waves transmitted from said second section and at least first and second photoelectric conversion means in optical communication with said first and second filter means respectively, said first photoelectric conversion means for converting said light-waves transmitted through said first filter means into a first electric signal and said second photoelectric conversion means for converting said light-waves transmitted through said second filter means into a second electric signal, in that said scanning means further includes third and fourth filter means, said third filter means for filtering predetermined wavelengths of light- waves transmitted from said first section and said fourth filter means for filtering predetermined wavelengths of light-waves transmitted from said second section, and third and fourth photoelectric conversion means in optical communication with said third and fourth filter means respectively, said third photoelectric conversion means for converting said lightwaves transmitted through said third filter means into a third electric signal and said fourth photoelectric conversion means for converting said lightwaves transmitted through said fourth filter means into a fourth electric signal, in that said operating means includes first means, connected to said first and second photoelectric conversion means, tor combining said first and second electric signals whereby a first operating signal is produced, second means, receiving said first and second electric signals whereby a first pre-operating signal is produced, third means, connected to said third and fourth photoelectric conversion means, for combining said third and fourth electric signals whereby a second pre-operating signal is produced, fourth means, connected to said second and third means, for combining said first and second pre-operating signals whereby a second operating signal is produced. Furthermore, the present invention provides also a printed matter identifying method comprising the steps of: conveying printed matter having a pattern through a lighted conveying path in a predetermined direction, reading at least first and second sections of said printed matter and generating a signal for each said section representing the portion of said pattern in said respective section, said sections being divided from each other in a direction substantially orthogonal to said predetermined direction of conveyance, effecting operations between said signals generated by said scanning means and generating at least one operating signal, and identifying said printed matter on the basis of the operating signals, said printed matter identifying method being characterized by comprising additionally the steps of filtering predetermined wavelengths of light-waves transmitted from said first section by first filter means and filtering predetermined wavelengths of lightwaves transmitted from said second section by second filter means, converting said light-waves transmitted from said first section into a first electric signal by first photoelectric conversion means and converting said light- waves transmitted from said second section into a second electric signal by second photoelectric conversion means, filtering predetermined wavelengths of light-waves transmitted from said first section by third filter means included in said scanning means and filtering predetermined wavelengths of light-waves transmitted from said second section by fourth filter means included in said scanning means, converting said light-waves transmitted through said third filter means into a third electric signal and converting said light- waves transmitted through said fourth filter means into a fourth electric signal, combining said first and second electric signals whereby a first operating signal is produced, producing a first pre-operating signal from said first and second electric signals, combining said third and fourth electric signals whereby a second pre-operating signal is produced and combining said first and second pre-operating signals whereby a second operating signal is produced. According to the present invention as described above, a printed matter identifying device and method can be provided wherein a printed matter is divided in a plurality of sections in a direction orthogonal to a direction to be conveyed, operations between the read-out signals from the respective sections and many printed patterns are identified by comparing the operating signals with the reference signals in order to identify, for example the type of printed matter or the authenticity of the printed matter. Figs. 1(A) and 1(B) are diagrams showing a prior art apparatus for identifying patterns of printed matter; Figs. 2(A) and 2(B) show waveforms read out from the patterns of Fig. 1; Fig. 3 is a perspective view showing one embodiment of an identifying device of the present invention; Figs. 4A and 4B are block diagrams of the device of Fig. 3 for processing signals; Figs. 5(A) through 5(D) illustrate reference patterns; Figs. 6(A) through 6(P) illustrate waveforms for the patterns in Figs. 5(A) through 5(D); and Fig. 7 is a flow chart for the judgment section of the present invention. Other objects and features of the present invention will be apparent from the following description taken in connection with the accompanying drawings, in which: Referring now to the drawings, wherein like reference numerals designate identical or corresponding parts throughout the several views, and more particularly to Fig. 3 thereof, the construction of a note identification device is shown. In the Figure, the device includes a means for dividing the reflected light from the note, in the direction orthogonal to the conveyance direction of the note, into two sections and a receiving means for detecting the reflected light from each of the respective sections of the printed pattern. In Fig. 3, the note 1 is conveyed by a conventional conveying means (not shown), such as the belt driven roller type or any other type well known to those skilled in the art, in the direction A. The central portion of the note 1 is effectively divided into two detecting fields 3 and 3', by separating reflected light-waves from the pattern of the note 1 which are received as having different wavelengths. That is, the light source 2 illuminates the detecting fields 3 and 3' of the note 1. As the note is conveyed, the pattern of the note 1 in each detecting field is scanned. The reflected light- waves from the detecting fields form images on the diffusion plates 5 and 5', respectively, by way of the focusing lenses 4 and 4', respectively. The front of each of the diffusion plates 5 and 5' is provided with the slits 6 and 6'. The slits 6 and 6' limit the size of the patterns which are formed on the diffusion plates 5 and 5'. The rear of each of the diffusion plates 5 and 5' is provided with the light conducting paths 7 and 7' having mirrored inner sides. The light conducting paths 7 and 7' direct the light-waves which pass through the diffusion plates 5 and 5' to the light receivers 10, 11 and 10', 11', such as photodiodes or other such devices well known in the art through color glass filters 8, 9 and 8', 9', respectively. The numerals 8 and 8' denote red color transmitting filters and the numerals 9 and 9' denote blue color transmitting filters. The light receivers 10 and 10' receive only the red component of the reflected light- waves and the light receivers 11 and 11' receive only the blue component of the reflected light- waves from the detecting fields 3 and 3'. The signals 12,12', 13 and 13' from the light receivers 10, 10', 11 and 11', respectively, are amplified by respective amplifiers and are fed to a signal processing section as the signals R, B, R' and B'. The sampling circuits 14, 15, 16 and 17 each comprising a sample and hold circuit connected to the output of the respective amplifiers and an analog to digital converter connected to the output of a respective sample and hold circuit shown in Fig. 4, effect sampling of the photoelectric signals representing the red color components 12 and 12' and blue components 13 and 13' of the respective reflected light-waves from the detecting fields 3 and 3', and produce the respective sampled signals 18, 19, 18' and 19'. If the pattern in Fig. 5(A) is green and the pattern in Fig. 5(B) is blue, the difference signal, representing the difference between the photoelectric signal of the red component of the reflected light-waves from each of the detecting fields 3 and 3', is effective for identifying the patterns in Figs. 5(A) and 5(B). If the pattern in Fig. 5(D) is green and the pattern in Fig. 5(C) is red, the sum signal, representing the sum of the photoelectric signal of the blue component of the reflected light-waves from each of the detecting fields 3 and 3', is effective for identifying the patterns in Figs. 5(C) and 5(D). Therefore, the patterns in Figs. 5(A), 5(B), 5(C) and 5(D) may be identified by the difference signal of the red components and the sum signal of the blue components. Referring again to Fig. 4B, the subtracter 20 calculates the difference between the sampled signals 18 and 18' represented as photoelectric signal of the red component of the reflected light- waves detecting fields 3 and 3', respectively, and produces the difference signal 22. Also, the adder 21 computes the sum of the sampled signals 19 and 19' represented as the photoelectric signal of the blue component of the reflected light-waves from the detecting fields 3 and 3' respectively, and produces the sum signal 23. The subtracter 20 and the adder 21 perform their respective operation in synchronism with a control signal P. Furthermore, the storage section 24, such as a ROM or RAM, stores the red component difference signal and the blue component sum signal obtained from each pattern of the predetermined reference notes (in this example, patterns shown in Figs. 5(A) through 5(D)) and produces the respective reference signals 25 and 26. The comparator 27 compares the difference signal 22 with each of the reference difference signals 25 and the comparator 28 compares the sum signal 23 with each of the reference sum signals 26 to verify which reference pattern and the detected pattern resembles. In the verifying operation, the pattern matching is effected between the sampled signal of the detected pattern and the reference signal to compute the similarity. A similarity value for each of the respective reference patterns for the comparators 27 and 28 is fed to a judgment section 29. The judgment section 29 determines if the sampled signal matches any of the reference signals and produces a signal representing the result of the determination. Thus identification of the note 1 is effected, and if a note does not include a pattern which matches any of the reference patterns, it is processed as a counterfeit note. It should be understood that the judgment section could be incorporated in a microprocessor with at least the comparators 27 and 28 or could be provided as software for a general purpose computer and operates according to the flow chart shown in Fig. 7 which will be explained more fully hereinafter. Referring now to Figs. 5, 6 and 7 the operation of the device will be explained. Figs. 5(A) through 5(D) represent the reference patterns for comparison with the sampled patterns. Figs. 6(A) and 6(E) represent for instance, the red component signals which would be read out from the detecting fields 3 and 3' for the pattern of Fig. 5(A). Fig. 6(1) represents the red component difference signal obtained by subtracting the signal of Fig. 6(E) from the signal of Fig. 6(A). Similarly, the blue component signals (not shown) which should be read out from the detecting fields 3 and 3" and added together to obtain the blue component signal shown in Fig. 6(M). Figs. 6(B) and 6(F) represent the red component signals .for the detecting fields 3 and 3", respectively, of Fig. 5(B). Fig. 6(J) represents the red component difference signal and Fig. 6(N) represents the blue component sum signal for the reference pattern in Fig. 5(B). Figs. 6(C) and 6(G) represent the red component signals for the detecting fields 3 and 3', respectively, of the Fig. 5(C). Fig. 6(K) represents the red component difference signal and Fig. 6(0) represents the blue component sum signal for the reference pattern in Fig. 5(C). Figs. 6(D) and 6(H) represent the red component signals for the detecting fields 3 and 3', respectively of Fig. 5(D). Fig. 6(L) represents the red component difference signal and Fig. 6(P) represents the blue component sum signal for the reference pattern of Fig. 5(D). Therefore, an unknown note is scanned, as shown in Fig. 3, to obtain a sampled red component difference signal 22 and a sampled blue component sum signal 23 which are compared to the reference red component difference signals and the reference blue component sum signals, respectively, stored in the storage section 24 as explained in the description of Fig. 4. Once the comparison of the sampled signals to the reference signals is made, the judgment section determines if the sampled pattern matches any of the reference patterns according to the flow chart of Fig. 7. In the following explanation, the sampled red component difference signal is defined as S1, the sampled blue component sum signal is defined as S2, the reference signals of Figs. 6(1) and 6(M) are defined as R1 and R2 respectively; the signal of Figs. 6(J) and 6(N) are defined as R3 and R4, respectively; the signals of Figs. 6(K) and 6(0) are defined as R5 and R6, respectively; and the signals of Figs. 6(L) and 6(P) are defined as R7 and R8, respectively. If the sampled blue component sum signal S2 is equivalent to signal R2 or signal R4, the sampled red component difference signals is checked. If S1 is equivalent to R1, the sampled pattern is equivalent to the reference pattern of Fig. 5(A). However, if S1 is not equivalent to R1, but is equivalent to R3, the sampled pattern is equivalent to take reference pattern of Fig. 5(B). Further, if S1 is not equivalent to R1 or R3, the sampled pattern (note) is rejected as undefined. If S2 is not equivalent to R2 or R4, S1 is checked against R5 and R7. If S1 is equivalent to R5 or R7, S2 is checked. If S2 is equivalent to R6, then the sampled pattern is equivalent to the reference pattern of Fig. 5(C). However, if S2 is not equivalent to R6, but is equivalent to R8, the sampled pattern is equivalent to the reference pattern of Fig. 5(D). Further, if S2 is not equivalent to R6 or R8, the sampled pattern (note) is rejected as undefined. Therefore, using the above-mentioned method, the sampled patterns can be easily identified and verified. It should be understood that the color separation is not limited to red and blue and the color filter can be changed according to the color of the note. Color separation of more than two colors is also easily accomplished with the present invention. The sampled red component can be added to form a sampled red component sum signal in order to determine the ratio between the blue component sum signal and the red component sum signal, again using a divider. Therefore, the sampled red component difference signal is compared to reference red component difference signals and the sampled blue-red ratio signal is compared to reference blue-red ratio signals. Of course a second adder would be provided to sum the sampled red component signals from the detecting fields and a divider provided to determine the sampled blue-red ratio signal. This increases the reliability of the device for identification. Further, the identifying device according to the present invention is not limited only to notes, but to any printed matter in which the contents of the operations, the variations of colors and the detecting fields are arbitrarily selectable according to the patterns of the printed matter, colors and other such parameters. Obviously, numerous (additional) modifications and variations of the present invention are possible in light of the above teachings. It is therefore to be understood that within the scope of the appended claims, the invention may be practiced otherwise than as specifically described herein.
The resource mix for the U.S. electrical power grid is undergoing rapid change with increased levels of solar photovoltaic (PV) and wind turbine electricity generating capacity. There are potential negative impacts to grid resilience resulting from hurricane damage to wind and solar power stations connected to the power transmission grid. Renewable power sources are exposed to the environment more so than traditional thermal power sources. To our knowledge, damage to power generating stations is not included in studies on hurricane damage to the electrical power grid in the literature. The lack of a hurricane wind damage prediction model for power stations will cause underestimation of predicted hurricane wind damage to the electrical grid with high percentages of total power generation capacity provided by solar photovoltaic and wind turbine power stations.Modeling hurricane wind damage to the transmission grid and power stations can predict damage to electrical grid components including power stations, the resultant loss in power generation capacity, and restoration costs for the grid. This Praxis developed models for hurricane exposure, fragility curve-based damage to electrical transmission grid components and power generating stations, and restoration cost to predict resiliency factors including power generation capacity lost and the restoration cost for electrical transmission grid and power generation system damages. Synthetic grid data were used to model the Energy Reliability Council of Texas (ERCOT) electrical grid. A case study was developed based on Hurricane Harvey. This work is extended to evaluate the changes to resiliency as the percentage of renewable sources is increased from 2017 levels to levels corresponding to the National Renewable Energy Lab (NREL) Futures Study 2050 Texas scenarios for 50% and 80% renewable energy.	https://scholarspace.library.gwu.edu/concern/gw_etds/fq977v030
TECHNICAL FIELD BACKGROUND ART DISCLOSURE Technical Problem TECHNICAL SOLUTION ADVANTAGEOUS EFFECTS DESCRIPTION OF DRAWINGS MODE FOR INVENTION First Embodiment Second Embodiment Third Embodiment Fourth Embodiment x y z The present invention relates to a III-nitride semiconductor light emitting device, and more particularly, to a III-nitride semiconductor light emitting device in which a single layer or plural layers made of SiCN(x≧0, y≧0, x+y>0, z>0) are inserted into or under a nitride active layer. As used herein, the term III-nitride compound semiconductor light emitting device refers to a light emitting device, such as a light emitting diode comprising a compound semiconductor layer made of Al(x)Ga(y)In(1-x-y)N (0≦x≦1, 0≦y≦1, 0≦x+y≦1), and does not exclude the inclusion of either materials made of other group elements, such as SiC, SiN, SiCN, and CN, or a semiconductor layer made of such materials. FIG. 1 FIG. 1 100 200 100 300 200 400 300 500 400 600 500 700 600 800 301 500 400 shows a III-nitride compound semiconductor light emitting device according to the prior art. As shown in , the light emitting device comprises the substrate , the buffer layer epitaxially grown on the substrate , the n-type nitride semiconductor layer epitaxially grown on the buffer layer , the active layer epitaxially grown on the n-type nitride layer , the p-type nitride semiconductor layer epitaxially grown on the active layer , the p-side electrode formed on the p-type nitride semiconductor layer , the p-side bonding pad formed on the p-side electrode , and the n-side electrode formed on the n-type nitride semiconductor layer exposed by mesa-etching of at least the p-type nitride semiconductor layer and the active layer . 100 The substrate can use a GaN-based substrate as a homogeneous substrate, and a sapphire substrate, a silicon carbide substrate or a silicon substrate as a heterogeneous substrate, but can use any other substrates on which nitride semiconductor layers can be grown. 100 The nitride semiconductor layers epitaxially grown on the substrate are usually grown by means of MOCVD (Metal Organic Chemical Vapor Deposition) method. 200 100 The buffer layer serves to reduce differences in lattice constant and the coefficient of thermal expansion between the heterogeneous substrate and the nitride semiconductor. U.S. Pat. No. 5,122,845 discloses a technology in which an AlN buffer layer having a thickness of 100 Å to 500 Å is grown on a sapphire substrate at a temperature ranging from 380° C. to 800° C. U.S. Pat. No. 5,290,393 discloses a technology in which an Al(x)Ga(1-x)N (0≦x<1) buffer layer having a thickness of 10 Å to 5000 Å is grown on a sapphire substrate at a temperature ranging from 200° C. to 900° C. Korean Patent No. 10-0448352 discloses a technology in which a SiC buffer layer is grown at a temperature ranging from 600° C. to 990° C., and an In(x)Ga(1-x)N (0≦x<1) layer is grown on the SiC buffer layer. 300 800 In the n-type nitride semiconductor layer , at least a region (n-type contact layer) in which the n-side electrode is formed is doped with an impurity. The n-type contact layer is preferably made of GaN and is doped with Si. U.S. Pat. No. 5,733,796 discloses a technology in which an n-type contact layer is doped with a desired doping concentration by controlling a mixing ratio of Si and other source materials. 400 400 The active layer is a layer for emitting a photon (light) by recombination of electrons and holes, and is mainly made of In(x)Ga(1-x)N (0≦x<1). The active layer is composed of a single quantum well or multi quantum wells. WO02/021121 discloses a technology in which only some of a plurality of quantum wells and barrier layers are doped. 500 3 The p-type nitride semiconductor layer is doped with an impurity such as Mg, and has a p-type conductivity through an activation process. U.S. Pat. No. 5,247,533 discloses a technology in which a p-type nitride semiconductor layer is activated by means of irradiation of electron beam. U.S. Pat. No. 5,306,662 discloses a technology in which a p-type nitride semiconductor layer is activated through annealing at a temperature of 400° C. or more. Korean Patent No. 10-043346 discloses a technology in which NHand a hydrazine-based source material are used together as a nitrogen precursor for growing a p-type nitride semiconductor layer, so that the p-type nitride semiconductor layer has a p-type conductivity without an activation process. 600 500 The p-side electrode serves to allow the current to be supplied to the entire p-type nitride semiconductor layer . U.S. Pat. No. 5,563,422 discloses a technology of a light-transmitting electrode, which is formed almost on the entire p-type nitride semiconductor layer, in ohmic contact with the p-type nitride semiconductor layer, and made of Ni and Au. U.S. Pat. No. 6,515,306 discloses a technology of a light-transmitting electrode made of ITO(Indium Tin Oxide), which is formed on the n-type superlattice layer formed on the p-type nitride semiconductor layer. 600 600 600 600 Meanwhile, the p-side electrode can be formed to have such a thick thickness that the p-side electrode does not transmit light, i.e., the p-side electrode reflects light toward the substrate. A light emitting device using this p-side electrode is called a flip chip. U.S. Pat. No. 6,194,743 discloses a technology of an electrode structure including an Ag layer of 20 nm or more in thickness, a diffusion barrier layer covering the Ag layer, and a bonding layer made of Au and Al, which covers the diffusion barrier layer. 700 800 P-side bonding pad and n-side electrode are for providing current into the device and for wire-bonding out of the device. U.S. Pat. No. 5,563,422 discloses a technology of an n-side electrode made of Ti and Al. U.S. Pat. No. 5,652,434 discloses a technology of p-side bonding pad directly contacted with p-type nitride semiconductor layer by partially removing the light-transmitting electrode. In a light emitting device, the most integral element is an active layer. Electrons and holes respectively formed in an n-side and a p-side are combined with each other together in a quantum-well layer of the active layer so as to emit light corresponding to the energy band of a quantum-well layer material. Therefore, the efficiency of the light emitting device is greatly influenced by the crystal quality of the active layer, particularly, a quantum-well layer, strain given to the quantum-well layer, the shape of the quantum-well layer and the like. Major parameters of the electrical and optical properties of the light emitting device are also largely dependent upon the above-mentioned factors. 8 2 III-nitride semiconductor light emitting devices usually employ heterogeneous substrates such as a sapphire substrate, a silicon carbide substrate or a silicon substrate. Inherently, lots of crystal defects are thus generated in a grown thin film. It is generally known that dislocations of 10cmor more exist. It is evident that these dislocations reduce the efficiency of the III-nitride semiconductor light emitting devices. However, the reason why the III-nitride semiconductor light emitting devices can show stabilized performance of some degrees even in spite of lots of dislocations is that the active layer has both characteristics of the quantum well and the quantum dot. Accordingly, in order to improve the performance of the light emitting device, especially, brightness, it is necessary to strengthen formation of this quantum dot. It is, however, not easy to strengthen the formation of this quantum dot only with existing thin film growth technology. Further, strain formed in the active layer serves as an important factor having an influence on the properties of the active layer. A variety of differences occur between the photoluminescence (PL) characteristic and the electroluminescence (EL) characteristic of the light emitting device according to the strain. In this connection, the control of strain around the active layer or in the active layer itself becomes an important factor in order to improve the efficiency of the light emitting device. x y z Accordingly, the present invention has been made in view of the above problems occurring in the prior art, and it is an object of the present invention to provide a III-nitride semiconductor light emitting device in which a single layer or plural layers made of SiCN(x≧0, y≧0, x+y>0, z>0) are inserted into or under an active layer to control strain applied to the active layer, and an active layer having an enhanced three-dimensional shape (quasi quantum dots active layer) is formed. x y z To achieve the above object, according to the present invention, there is provided an III-nitride semiconductor light emitting device in which a single layer or plural layers made of SiCN(x≧0, y≧0, x+y>0, z>0) are inserted into or under an active layer. x y z x y z x y z The SiCN(x≧0, y≧0, x+y>0, z>0) material is a direct band gap material. It is known that the energy band gap of the material can vary in a wide range depending upon the composition, and the material can have a cubic structure or hexagonal structure depending upon a growth condition. The present invention has been made in view of these properties of the SiCN(x≧0, y≧0, x+y>0, z>0) material, and it is directed to a technology in which Al(x)Ga(y)In(1-x-y)N (0≦x≦1, 0≦y≦1, 0≦x+y≦1) of the hexagonal structure and SiCN(x≧0, y≧0, x+y>0, z>0) of the hexagonal structure are combined together. x y z x y z According to a first aspect of the present invention, there is provided a III-nitride semiconductor light emitting device including a substrate, and a plurality of nitride semiconductor layers, which are grown on the substrate and have an active layer for generating photons through recombination of electrons and holes, wherein the active layer includes a light emitting layer that generates photons through recombination of electrons and holes, and a barrier layer, which is grown on the light emitting layer and confines the electrons and holes to the light emitting layer, and the light emitting layer includes a SiCN(x≧0, y≧0, x+y>0, z>0) layer, and a nitride semiconductor layer, which is grown on the SiCN(x≧0, y≧0, x+y>0, z>0) layer and includes Ga and N. The first aspect of the present invention is supported by a third embodiment of the present invention. Conventionally, in a III-nitride semiconductor light emitting device, the term “active layer” refers to a region from which light is substantially emitted. The region includes a quantum-well layer made of Al(x)Ga(y)In(1-x-y)N (0≦x≦1, 0≦y≦1, 0≦x+y≦1), which is a light emitting layer that emits photons through recombination of electrons and holes, and a barrier layer made of a single layer or plural layers of Al(x)Ga(y)In(1-x-y)N (0≦x≦1, 0≦y≦1, 0≦x+y≦1), which has an average energy band gap higher than that of the quantum-well layer. The active layer includes at least one quantum-well layer. If the active layer is composed of the multi-quantum-well layers, it is not necessary that all of the quantum-well layers have the same composition or energy band gap, and thickness. The barrier layer is a layer that is adjacent to the quantum-well layer, and refers to a layer that can confine electrons or holes to the quantum-well layer. x y z x y z x y z x y z The present invention includes the SiCN(x≧0, y≧0, x+y>0, z>0) layer within the active layer region, and thus presents a new type of an active layer. That is, the light emitting layer, which was composed of only the Al(x)Ga(y)In(1-x-y)N (0≦x≦1, 0≦y≦1, 0≦x+y≦1) layer in the prior art, is replaced with the stack structure of the SiCN(x≧0, y≧0, x+y>0, z>0) layer and the nitride semiconductor layer, which is grown on the SiCN(x≧0, y≧0, x+y>0, z>0) layer and includes Ga and N. The existence of the SiCN(x≧0, y≧0, x+y>0, z>0) layer affects the growth of the nitride semiconductor layer including Ga and N. Due to this, the light emitting layer has a shape having an enhanced three-dimensional characteristic or a shape having three-dimensional islands (called “Quasi Quantum Dots”) not a simple two-dimensional quantum well structure. In the present invention, the active layer including a new type of the light emitting layer having an three-dimensional characteristic improved unlike the prior art is called a quasi quantum dots active layer. x y z According to a second aspect of the present invention, there is provided a III-nitride semiconductor light emitting device including a substrate, and a plurality of nitride semiconductor layers, which are grown on the substrate and have an active layer having one or more light emitting layers that emits light through recombination of electrons and holes, including a SiCN(x≧0, y≧0, x+y>0, z>0) layer, which is located under a light emitting layer located at the farthest from the substrate among one or more light emitting layers and is not brought into contact with the substrate. The second aspect of the present invention is supported by first and second embodiments of the present invention. In this case, the at least one light emitting layer may have a conventional quantum-well layer or a light emitting layer whose three-dimensional characteristic is strengthened. x y z According to the present invention, a SiCN(x≧0, y≧0, x+y>0, z>0) layer is inserted under or into an active layer, thus strengthening a three-dimensional growth characteristic of the active layer. Accordingly, a III-nitride semiconductor light emitting device having improved optical and electrical characteristics can be provided. Further objects and advantages of the invention can be more fully understood from the following detailed description taken in conjunction with the accompanying drawings in which: FIG. 1 shows the structure of a III-nitride semiconductor light emitting device in the prior art; FIG. 2 shows an atomic force microscope (AFM) image of a CN layer, which is grown about 20 nm in thickness on a GaN layer; FIG. 3 FIG. 2 shows an AFM image when the CN layer of is thickly grown to a thickness more than 100 nm; FIG. 4 x y z shows an AFM image of a SiCN(x>0, y>0, z>0) layer grown on a GaN layer; FIG. 5 is a cross-sectional view of a III-nitride semiconductor light emitting device according to a first embodiment of the present invention; FIG. 6 x y z shows an AFM image of an active layer that is grown after a SiCN(x≧0, y≧0, x+y>0, z>0) layer is inserted under an active layer according to the first embodiment; FIG. 7 x y z shows an AFM image of an active layer grown without the SiCN(x≧0, y≧0, x+y>0, z>0) layer; FIG. 8 is a table showing the electrical and optical properties of the III-nitride semiconductor light emitting device according to the first embodiment of the present invention; FIG. 9 is a cross-sectional view of a III-nitride semiconductor light emitting device according to a second embodiment of the present invention; FIG. 10 is an expanded view of the active layer according to the second embodiment of the present invention; FIG. 11 is a table showing the electrical and optical properties of the III-nitride semiconductor light emitting device according to the second embodiment of the present invention; and FIG. 12 is an expanded view of the active layer of the III-nitride semiconductor light emitting device according to a third embodiment of the present invention. The present invention will now be described in detail in connection with preferred embodiments with reference to the accompanying drawings. x y z x y z A method of growing a SiCN(x≧0, y≧0, x+y>0, z>0) layer on a GaN layer will be first described. A III-nitride semiconductor light emitting device in which the SiCN(x≧0, y≧0, x+y>0, z>0) layer is inserted into an active layer or under the active layer will be then described in connection with embodiments. x y z x y z A GaN layer is grown on a sapphire substrate, and a SiCN(x≧0, y≧0, x+y>0, z>0) layer is grown on the grown GaN layer according to a common method. Characteristics of the grown SiCN(x≧0, y≧0, x+y>0, z>0) layer are analyzed through an AFM. x y z 4 3 x y z 4 4 x y z 4 3 4 2 6 4 4 In order to form the SiCN(x≧0, y≧0, x+y>0, z>0) layer, DTBSi being a kind of a metal organic source was used as a silicon supply source, CBrbeing a kind of a metal organic source was use as a carbon supply source, and NHor DMHy was used as a nitrogen supply source. The reason why these sources were used is that they are easily thermally decomposed at low temperature, and are thus advantageous in forming the SiCN(x≧0, y≧0, x+y>0, z>0) layer at low temperature compared to hydride source such as existing CH, the SiH, and the like. In the present invention, however, it is to be noted that the sources for forming the SiCN(x≧0, y≧0, x+y>0, z>0) layer are not limited to DTBsi, CBr, NHor DMHy, but the SiH, the SiH, etc. can be used as the silicon source, and CCl, CH, etc. can be also used as the carbon source. x y z 3 3 x y z x y z x y z If the SiCN(x≧0, y≧0, x+y>0, z>0) layer is formed on an Al(x)Ga(y)In(1-x-y)N(0≦x≦1, 0≦y≦1, 0≦0≦x+y≦1) layer, NH, a hydrazine-based source or NHand a hydrazine-based source can be used as a nitrogen supplier. Thus, there is an advantage in that it can prevent metal conglomeration in the Al(x)Ga(y)In(1-x-y)N (0≦x≦1, 0≦y≦1, 0≦x+y≦1) layer under the SiCN(x≧0, y≧0, x+y>0, z>0) layer due to the shortage of a nitrogen radical supply while a thin film is grown. In other words, it is possible to grow a desired SiCN(x≧0, y≧0,x+y>0, z>0) layer while not damaging an underlying semiconductor layer while the SiCN(x≧0, y≧0, x+y>0, z>0) layer is grown. FIG. 2 FIG. 2 shows an AFM image of a CN layer, which is grown about 20 nm in thickness on a GaN layer. From , it can be seen that the CN layer is grown on the GaN layer like three-dimensional islands. Since the lattice constant of CN is significantly lower than that of GaN, the CN layer is grown as the island shape due to strain. Island growth means that the grown CN layer has a crystal shape. FIG. 3 FIG. 2 FIG. 3 shows an AFM image when the CN layer of is thickly grown to a thickness more than 100 nm. From , it can be seen that the size of the islands made of CN is increased. That is, the islands made of CN, which are initially grown, serve as seeds, and are thus continuously grown around the seeds. FIG. 4 x y z x y z x y z shows an AFM image of the SiCN(x>0, y>0, z>0) layer grown on the GaN layer. SiCN(x>0, y>0, z>0) has a lattice constant higher than that of CN, and has a similar lattice constant as that of GaN. Thus, SiCN(x>0, y>0, z>0) shows a continuous thin film shape compared to CN, and has a shape that rises high like a summit due to strain. 3 4 x y z In this experiment, NHwas used as the nitrogen source, CBrwas used as the carbon source, and DTBSi was used as the silicon source. After the SiCN(x>0, y>0, z>0) layer was grown around 950° C., a temperature of a reactor dropped to below 100° C. under a hydrogen atmosphere. 2 4 2 4 2 4 x y z x y z x y z x y z x y z In the experiments of FIGS. to , in order to confirm how CN or the SiCN(x>0, y>0, z>0) layer is formed on the GaN layer, it is thickly formed intentionally. It can be seen from FIGS. to that the SiCN(x≧0, y≧0, x+y>0, z>0) layer intended by the present invention can be grown while not externally affecting the underlying GaN layer. If the nitrogen radical is not sufficient or the GaN layer is damaged due to other reasons while the SiCN(x≧0, y≧0, x+y>0, z>0) layer is grown, there occurs a phenomenon that the surface of a thin film is conglomerated or becomes coarse. From FIGS. to , it can be seen that the surface of the GaN layer under the SiCN(x≧0, y≧0, x+y>0, z>0) layer is smooth. Accordingly, it can be seen that the growth of the SiCN(x≧0, y≧0, x+y>0, z>0) layer can be implemented through a combination with the growth of the Al(x)Ga(y)In(1-x-y)N (0≦x≦1, 0≦y≦1, 0≦x+y≦1) layer. x y z Various combinations of the SiCN(x≧0, y≧0, x+y>0, z>0) layer and the active layer and the effects of the combinations will now be described in connection with embodiments. FIG. 5 is a cross-sectional view of a III-nitride semiconductor light emitting device according to a first embodiment of the present invention. 10 20 10 30 20 40 30 50 40 60 50 70 60 80 31 50 40 90 40 30 x y z The light emitting device includes a substrate , a buffer layer epitaxially grown on the substrate , an n-type nitride semiconductor layer epitaxially grown on the buffer layer , an active layer epitaxially grown on the n-type nitride semiconductor layer , a p-type nitride semiconductor layer epitaxially grown on the active layer , a p-side electrode formed on the p-type nitride semiconductor layer , a p-side bonding pad formed on the p-side electrode , and a n-side electrode formed on an n-type nitride semiconductor layer , which is exposed through mesa-etching of at least the p-type nitride semiconductor layer and the active layer . A SiCN(x≧0, y≧0, x+y>0, z>0) layer is formed between the active layer and the n-type nitride semiconductor layer . 30 50 The n-type nitride semiconductor layer and the p-type nitride semiconductor layer are composed of Al(x)Ga(y)In(1-x-y)N (0≦x≦1, 0≦y≦1, 0≦x+y≦1), and can be respectively composed of a single or plural layers. 40 90 40 40 40 40 90 40 x y z x y z x y z x y z In order to have an effect on the growth of the active layer , it is preferred that the location where the SiCN(x≧0, y≧0, x+y>0, z>0) layer is located is immediately under the active layer , or within a range of 0.5 μm from the active layer . In this case, the lowest layer of the active layer can be a barrier layer or a light emitting layer. The closer the SiCN(x≧0, y≧0, x+y>0, z>0) layer toward the active layer , the greater the effect by the SiCN(x≧0, y≧0, x+y>0, z>0) layer . The SiCN(x≧0, y≧0, x+y>0, z>0) layer can be located 0.5 μm apart from the active layer . In this case, however, desired effects cannot be expected. x y z x y z x y z 90 90 90 A thickness of the SiCN(x≧0, y≧0, x+y>0, z>0) layer preferably ranges from 2.5 Å to 500 Å. The thicker the SiCN(x≧0, y≧0, x+y>0, z>0) layer , the greater the influence on variation in morphology of a layer formed thereon. However, if the thickness of the SiCN(x≧0, y≧0, x+y>0, z>0) layer is over 500 Å, the layer grown thereon is excessively grown in three dimensions. In this case, it may be difficult to be applied to the light emitting device. If the thickness is below 2.5 Å, the effect on the layer grown thereon us too small. x y z 3 x y z x y z 30 Regarding the material composition of SiCN(x≧0, y≧0, x+y>0, z>0), in order to protect the underlying Al(x)Ga(y)In(1-x-y)N(0≦x≦1, 0≦y≦1, 0≦x+y≦1) layer , NHor the hydrazine-based source must be injected in a continuous manner. Thus, N (value of z) must be over 0, and Si or C can be excluded, if needed. The composition can be decided according to desired strain. The higher x and the lower C (value of y), the greater the lattice of SiCN. On the contrary, the lower x and the higher C (value of y), the smaller the lattice of SiCN. x y z x y z x y z 90 30 It is preferred that the growth temperature of the SiCN(x≧0, y≧0, x+y>0, z>0) layer ranges from 400° C. to 1100° C. SiCN(x≧0, y≧0, x+y>0, z>0) can be grown with better quality at higher temperature. If SiCN(x≧0, y≧0, x+y>z>0) is grown at a temperature higher than that of Al(x)Ga(y)In(1-x-y)N (0≦x≦1, 0≦y≦1, 0≦x+y≦1), however, the underlying Al(x)Ga(y)In(1-x-y)N (0≦x≦1, 0≦y≦1, 0≦x+y≦1) layer can be damaged. FIG. 6 x y z 90 40 shows an AFM image of an active layer that is grown after the SiCN(x≧0, y≧0, x+y>0, z>0) layer is inserted under the active layer according to the first embodiment. x y z 4 3 x y z 90 90 At this time, the source for growing the SiCN(x≧0, y≧0, x+y>0, z>0) layer was DTBSi of 3 micro mole per minute, CBrof 7 micro mole per minute, and NHof 8 liters per minute. A growth time was 20 seconds, a growth temperature is approximately 1000° C., and an expected growth thickness was 5 to 10 Å. A first light emitting layer was grown on the SiCN(x≧0, y≧0, x+y>0, z>0) layer . FIG. 7 FIG. 7 40 90 x y z shows an AFM image of the active layer that was grown under the same condition without the SiCN(x≧0, y≧0, x+y>0, z>0) layer . From , it can be seen that the active layer looks like a ridgeline, and the ridgelines are connected to each other laterally. As shown in an enlarged image on the underside, the surface has grains and has a bad uniformity irregularly. Further, white points can be shown from the drawing. They indicate portions that rise high, wherein indium metals are conglomerated. It has been known that such indium metal conglomeration phenomenon has a bad influence on the reliability and performance of a device. FIG. 6 40 90 x y z Meanwhile, as shown in , in the active layer into which the SiCN(x≧0, y≧0, x+y>0, z>0) layer is inserted, the active layer does not have grain shape, but has an almost independent island shape. x y z x y z 90 90 It can be seen that the islands are formed relatively in a very regular manner. In the enlarged image on the underside, the islands can be seen well. This is because the three-dimensional growth of the active layer is strengthen due to variation in strain of the active layer by the SiCN(x≧0, y≧0, x+y>0, z>0) layer , as described above. It can be seen that the islands have the size of approximately 0.1 μm to 0.2 μm. It will be, however, evident to those skilled in the art that the size, density and shape of the islands can be changed in various manners depending upon the composition of the SiCN(x≧0, y≧0, x+y>0, z>0) layer . 40 40 FIG. 7 In the event that the active layer is composed of uniform three-dimensional islands, an electron and hole confinement phenomenon can be improved, the uniformity within the active layer can be improved, and optical properties can be improved accordingly. Furthermore, the indium metal conglomeration phenomenon that was shown in can be significantly improved. x y z x y z 90 90 A light emitting device according to the first embodiment of the present invention and a light emitting device without the SiCN(x≧0, y≧0, x+y>0, z>0) layer showed significant difference in a measured wavelength on photoluminescence (PL). In the case of the light emitting device into which the SiCN(x≧0, y≧0, x+y>0, z>0) layer is inserted, it was found that the measured wavelength on PL reduced about 10 nm. It is considered that this is caused by two reasons. FIG. 6 x y z 90 Firstly, quantum dots showed a phenomenon that the wavelength shortens while the energy levels of a conduction band and a valence band more widen than that of the quantum well. As shown in the AFM image of , it can be understood that the wavelength of the active layer shortens while three-dimensional characteristic thereof is enhanced (while having quasi quantum dot characteristic) due to the SiCN(x≧0, y≧0, x+y>0, z>0) layer . 40 90 x y z Secondly, it can be understood that this is caused by variation in piezo characteristic due to variation in strain of the active layer by the SiCN(x≧0, y≧0, x+y>0, z>0) layer . If the piezo characteristic varies, the wavelength can be changed as the bending degree of the energy band is changed. FIG. 8 x y z is a table showing the electrical and optical properties of the III-nitride semiconductor light emitting device according to the first embodiment of the present invention. In view of characteristics of the III-nitride semiconductor light emitting device, if the growth condition of an inserted SiCN(x≧0, y≧0, x+y>0, z>0) layer is optimized and the growth condition of the active layer is optimized accordingly, further improved characteristics can be obtained. 6 8 2 2 The detailed growth condition applied in FIGS. to is as follows. MOCVD method was used using a C face of a sapphire substrate as a main face. Hand/or Nwere used as carrier gases. The pressure of a reactor while growing a III-nitride semiconductor was maintained between 100 Torr to 500 Torr. 3 3 A GaN layer as a buffer layer was first grown on the sapphire substrate at a temperature of 550° C., and was then grown at a temperature of 1050° C. When the GaN layer was grown at the temperature of 550° C., TMG (50 sccm) and NH(15000 sccm) were used as sources. At this time, the GaN layer was grown 300 Å in thickness. On the other hand, when the GaN layer was grown at the temperature of 1050° C., TMG (250 sccm) and NH(18000 sccm) were used as the source. At this time, the GaN layer was grown to a thickness of 2 μm. 3 4 An n-type GaN layer was then grown as an n-type nitride semiconductor layer at a temperature of 1050° C. In this case, TMG (250 sccm) and NH(18000 sccm) were used as sources, and the n-type GaN layer was grown 2 μm in thickness. SiH(8 sccm) was used as an n-type dopant. x y z 4 3 x y z Next, a SiCN(x≧0, y≧0, x+y>0, z>0) layer was grown at a temperature of 1000° C. At this time, DTBSi (3 micro-mole/min) and CBr(7 micro-mole/min) and NH(8 l/min) were used as the sources, and the SiCN(x≧0, y≧0, x+y>0, z>0) layer was grown for 20 seconds. 0.15 0.85 3 0.15 0.85 An InGaN layer was then grown as a light emitting layer at a temperature of 800° C. At this time, TMIn (400 sccm), TMG (30 sccm), and NH(28000 sccm) were used as the sources, and the InGaN layer was grown 25 Å in thickness. 0.01 0.99 3 0.01 0.99 Thereafter, an InGaN layer was grown as a barrier layer at a temperature of 900° C. At this time, TMIn (20 sccm), TMG (30 sccm), and NH(28000 sccm) were used as the sources, and the InGaN layer was grown 100 Å in thickness. The light emitting layer and the barrier layer were then respectively grown three times in an alternate manner under the same growth condition as above. 3 2 Finally, a p-type GaN layer was grown as a p-type nitride semiconductor layer at a temperature of 1000° C. At this time, TMG (100 sccm) and NH(18000 sccm) were used as the sources, and the p-type GaN layer was grown 2000 Å in thickness. CPMg (500 sccm) was used as a p type dopant. FIG. 9 10 20 10 30 20 41 30 50 41 60 50 70 60 80 31 50 40 90 41 x y z is a cross-sectional view of a III-nitride semiconductor light emitting device according to a second embodiment of the present invention. The light emitting device includes a substrate , a buffer layer epitaxially grown on the substrate , an n-type nitride semiconductor layer epitaxially grown on the buffer layer , an active layer , which is epitaxially grown on the n-type nitride semiconductor layer and a plurality of light emitting layers, a p-type nitride semiconductor layer epitaxially grown on the active layer , a p-side electrode formed on the p-type nitride semiconductor layer , a p-side bonding pad formed on the p-side electrode , and a n-side electrode formed on an n-type nitride semiconductor layer , which is exposed through mesa-etching of at least the p-type nitride semiconductor layer and the active layer , wherein a SiCN(x≧0, y≧0, x+y>0, z>0) layer is formed within the active layer . FIG. 10 41 91 41 41 41 x y z b a. is an expanded view of the active layer according to the second embodiment of the present invention. The SiCN(x≧0, y≧0, x+y>0, z>0) layer is inserted into a barrier layer , which forms the active layer together with a light emitting layers x y z x y z x y z x y z 91 41 91 41 41 91 91 a b a A location where the SiCN(x≧0, y≧0, x+y>0, z>0) layer is inserted within the barrier layer can be any place. If the SiCN(x≧0, y≧0, x+y>0, z>0) layer is located at the end or start portion of the barrier layer , however, the light emitting layer and the SiCN(x≧0, y≧0, x+y>0, z>0) layer are brought into contact with each other. In this case, there is a possibility that the performance of the light emitting device may be degraded. This is caused by the memory effect of sources used when the SiCN(x≧0, y≧0, x+y>0, z>0) layer is grown or the diffusion effect within the light emitting device. The silicon source serves as an n-type dopant in the nitride semiconductor layer. In this case, there is a possibility that the performance of a device may be degraded due to doping of the light emitting layer. x y z x y z x y z x y z x y z 91 91 41 41 41 91 91 50 91 a If there is a sufficient purge time after the SiCN(x≧0, y≧0, x+y>0, z>0) layer is grown, this phenomena can be minimized. Further, in the case where the SiCN(x≧0, y≧0, x+y>0, z>0) layer is brought into contact with the light emitting layer of the active layer , this problem can be solved by minimizing the used amount of the silicon source. If the active layer is formed using two or more multi-barrier layers, it is not necessary to insert the SiCN(x≧0, y≧0, x+y>0, z>0) layer into all the barrier layers. In order to maximize the optical characteristic, it is better not to use the SiCN(x≧0, y≧0, x+y>0, z>0) layer in the barrier layer, which is located at the first place or the first and second places from the p-type nitride semiconductor layer . This is because the impurity concentration of the light emitting layer can be increased due to the memory effect of the sources used when the SiCN(x≧0, y≧0, x+y>0, z>0) layer is grown, as described above. x y z x y z 91 91 41 41 41 A thickness of the SiCN(x≧0, y≧0, x+y>0, z>0) layer may be from 2.5 Å to 50 Å. The reason why the thickness is thinly limited is that if the SiCN(x≧0, y≧0, x+y>0, z>0) layer having a thickness of over 50 Å is inserted into the active layer , the quality of the active layer can be abruptly degraded while excessive strain is accumulated within the active layer . FIG. 11 x y z 4 3 91 is a table showing the electrical and optical properties of the III-nitride semiconductor light emitting device according to the second embodiment of the present invention. Sources for growing the SiCN(x≧0, y≧0, x+y>0, z>0) layer included DTBsi of 1 micro mole per minute, CBrof 7 micro mole per minute, and NHof 8 linters per minute. A growth time was 10 seconds, and an expected growth thickness was approximately 2.5 Å to 5 Å. x y z 91 50 The SiCN(x≧0, y≧0, x+y>0, z>0) layer is inserted into a central portion of each of all the barrier layers except for the barrier layer that is the nearest to the p-type nitride semiconductor layer . A growth temperature was 900° C., which was the same as that of the barrier layer. x y z x y z 91 91 As stated above, in order to minimize the memory effect of the sources due to the SiCN(x≧0, y≧0, x+y>0, z>0) layer , there was a purge time of over 1 minute in a high gas flow after the SiCN(x≧0, y≧0, x+y>0, z>0) layer is grown. x y z x y z 91 91 A light emitting device according to the second embodiment of the present invention and a light emitting device without the SiCN(x≧0, y≧0, x+y>0, z>0) layer showed significant difference in a measured wavelength on photoluminescence (PL). In the case of the light emitting device into which the SiCN(x≧0, y≧0, x+y>0, z>0) layer is inserted, it was found that the measured wavelength on PL reduced about 5 nm to 10 nm. It is considered that this is caused by the same principle as that described in the first embodiment. 9 11 2 2 The detailed growth condition applied in FIGS. to is as follows. MOCVD method was used using a C face of a sapphire substrate as a main face. Hand/or Nwere used as carrier gases. The pressure of a reactor while growing a III-nitride semiconductor was maintained between 100 Torr to 500 Torr. 3 3 A GaN layer was first grown as a buffer layer on the sapphire substrate at a temperature of 550° C., and was then grown at a temperature of 1050° C. When the GaN layer was grown at the temperature of 550° C., TMG (50 sccm) and NH(15000 sccm) were used as sources. At this time, the GaN layer was grown 300 Å in thickness. On the other hand, when the GaN layer was grown at the temperature of 1050° C., TMG (250 sccm) and NH(18000 sccm) were used as the source. At this time, the GaN layer was grown to a thickness of 2 μm. 3 4 An n-type GaN layer was then grown as an n-type nitride semiconductor layer at a temperature of 1050° C. In this case, TMG (250 sccm) and NH(18000 sccm) were used as sources, and the n-type GaN layer was grown 2 μm in thickness. SiH(8 sccm) was used as an n-type dopant. 0.15 0.85 3 0.15 0.85 An InGaN layer was then grown as a light emitting layer at a temperature of 800° C. At this time, TMIn (400 sccm), TMG (30 sccm), and NH(28000 sccm) were used as the sources, and the InGaN layer was grown 25 Å in thickness. 0.01 0.99 3 0.01 0.99 Thereafter, an InGaN layer was grown as a barrier layer at a temperature of 900° C. At this time, TMIn (20 sccm), TMG (30 sccm), and NH(28000 sccm) were used as the source, and the InGaN layer was grown 50 Å in thickness. x y z 4 3 x y z A SiCN(x≧0, y≧0, x+y>0, z>0) layer was then grown at a temperature of 1000° C. At this time, DTBSi (1 micro-mole/min), CBr(7 micro-mole/min), and NH(8 l/min) were used as sources, and the SiCN(x≧0, y≧0, x+y>0≦z>0) layer was grown for 10 seconds. 0.01 0.99 3 0.01 0.99 Then, an InGaN layer was grown as a barrier layer at a temperature of 900° C. At this time, TMIn (20 sccm), TMG (30 sccm), and NH(28000 sccm) were used as the source, and the InGaN layer was grown 50 Å in thickness. x y z The light emitting layer, the barrier layer, the SiCN(x≧0, y≧0, x+y>0, z>0) layer, and the barrier layer were then respectively grown twice in an alternate manner under the same growth condition as above. 0.15 0.85 3 0.15 0.85 An InGaN layer was then grown as a light emitting layer at a temperature of 800° C. At this time, TMIn (400 sccm), TMG (30 sccm), and NH(28000 sccm) were used as the source, and the InGaN layer was grown 25 Å in thickness. 0.01 0.99 3 0.01 0.99 Next, an InGaN layer was grown as a barrier layer at a temperature of 900° C. At this time, TMIn (20 sccm), TMG (30 sccm), and NH(28000 sccm) were used as the sources, and the InGaN layer was grown 100 Å in thickness. 3 2 Finally, a p-type GaN layer was grown as a p-type nitride semiconductor layer at a temperature of 1000° C. At this time, TMG (100 sccm) and NH(18000 sccm) were used as sources, and the p-type GaN layer was grown 2000 Å in thickness. CPMg (500 sccm) was used as a p type dopant. FIG. 12 FIG. 10 42 91 41 92 42 42 x y z x y z b b a. is an expanded view of the active layer of the III-nitride semiconductor light emitting device according to a third embodiment of the present invention. The III-nitride semiconductor light emitting device according to the third embodiment is one of the III-nitride semiconductor light emitting device according to the second embodiment. In the second embodiment, the SiCN(x≧0, y≧0, x+y>0, z>0) layer is located within the barrier layer (see ). In this structure, however, a SiCN(x≧0, y≧0, x+y>0, z>0) layer is located at the end of a barrier layer and is thus adjacent to a light emitting layer x y z x y z 92 42 92 42 42 42 42 a a b b In this case, the SiCN(x≧0, y≧0, x+y>0, z>0) layer serves to directly decide the shape of the light emitting layer . Thus, the SiCN(x≧0, y≧0, x+y>0, z>0) layer and the light emitting layer can be conceptually considered as an integral one, and thus considered as one light emitting layer located between the barrier layers and . This constitutes the first aspect of the present invention. x y z 92 As described above, there is a sufficient purge time after the growth of the SiCN(x≧0, y≧0, x+y>0, z>0) layer . Thus, degradation in the performance of the light emitting device can be prevented. At this time, the purge condition may be different depending upon an active layer growth condition. It is, however, preferred that the growth of the thin film is stopped for a given time in a low-pressure and high-speed gas flow and then waits, as possible. 42 a Meanwhile, the light emitting layer according to the third embodiment can be formed to have a three-dimensional active layer having a quasi quantum dot shape, which is more significant than those of the first embodiment or the second embodiment. x y z x y z x y z 92 92 92 42 42 a a It is preferred that a thickness of the SiCN(x≧0, y≧0, x+y>0, z>0) layer has a value of 2.5 Å to 10 Å. This is because if the thickness of the SiCN(x≧0, y≧0, x+y>0, z>0) layer increases since the SiCN(x≧0, y≧0, x+y>0, z>0) layer directly affects the shape of the light emitting layer , the quality of the light emitting layer can be degraded and emission efficiency can be degraded accordingly. 20 50 x 1-y 3 3 In the first and second embodiments, the III-nitride semiconductor light emitting device is formed using the buffer layer using the GaN layer. In the fourth embodiment, however, a non-single crystal SiC seed layer is grown to a thickness of 5 to 200 Å at a temperature of 600 to 990° C., an InGaN (0≦x≦1, 0≦y<1, 0≦x+y≦1) layer is grown 100 to 500 Å in thickness on the SiC seed layer at a temperature of 400 to 900° C., and a p-type nitride semiconductor layer is grown using TMG as a Ga growth source at a temperature of 700 to 1100° C. At this time, DMHy and NHare used as source as a N growth source, the mole ratio of DMHy/TMG ranges from 1 to 300, and the mole ratio of NH/TMG ranges from 100 to 8000. Thus, a III-nitride semiconductor light emitting device having a thickness of 10 Å to 10000 Å is obtained. While the present invention has been described with reference to the particular illustrative embodiments, it is not to be restricted by the embodiments but only by the appended claims. It is to be appreciated that those skilled in the art can change or modify the embodiments without departing from the scope and spirit of the present invention.
Mike Gray [KD7LMO] photo of the lift off of GPSL 2003 balloons. Click picture for slightly higher resolution image. Links to other GPSL recaps are available at the Great Plains Super Launch 2003 Web Page LAUNCH DATE: June 14, 2003 LAUNCH TIME: 07:35 am MST, 13:35 UTC LAUNCH SITE: Deer Trail (directions) Launch Site - Deer Trail ----------------------- Launch Point: 39.6114� lat. -104.0426� long. Grid: X=15.69 Y=29.57 Ascent Rate: 1000 feet per minute Descent Rate: 915 feet per minute Altitude: 5205 feet Predicted Landing Site ----------------------- Landing Point: 39.6754� lat. -103.7823� long. Grid: X=29.5 Y=34.0 Altitude: 4501 feet Flight Time: 131 Minutes Bearing: 72.2� True Range: 14.5 Mi. Actual Landing Site ----------------------- Landing Point: 39.4926� lat. -103.7141� long. Grid: X=33.2 Y=21.4 Bearing: 115.0� True Range: 19.3 Mi. Difference from Predicted to Actual Landing Site -------------------------------------------------- Bearing: 163.9� True Range: 13.1 Mi. If an EOSS payload is highlighted, there is a link to an information page about that payload. (note: the number in parenthesis following the frequency is the time slot in seconds after the GPS minute) \|Location\|\|X Coord\|\|Y Coord\| \|Last Chance\|\|39.5\|\|38.5\| \|Deer Trail\|\|15\|\|30\| \|Arriba\|\|56\|\|8\| \|Akron\|\|60\|\|67\| Grid Calculator centered on Last Chance with a lat long of: EOSS-67 \|Balloon Manufacturer\|\|Kaymont\| \|Balloon Type\|\|latex\| \|Balloon Size\|\|1500 gram\| \|Payload\|\|12 lbs. (5 balloon sats) + EOSS payloads\| \|Free Lift %\|\|25%\| \|Ascent Rate\|\|1200 fpm variable. Probably leak resulted in steadily decreasing ascent rate until it finally went into a slow descent.\| \|Descent Rate\|\|at cutdown ~ 1000 fpm sea level\| \|Parachute\|\|70 in. diameter (HM-2)\| \|Peak Altitude\|\|46,255.9 ft.\| \|Launch Conditions\|\|dead calm\| Arizona Near Space Research (ANSR) - KD7LMO-11 - Text (from www.kd7lmo.net) Montana State (Borealis) - KD7MFJ-11 - Text from Findu.com Edge of Space Sciences (EOSS-67) - W5VSI-11 - K�YUK-11 Experimental Sub-Orbital Society (ES-OS) - KC�JHQ-11 - Text from Findu.Com Treasure Valley Near Space Program (TVNSP) N�KKZ Photos N�LP Photos KC0RPS's Photos No audio recording were made for this flight. by Benj. W�CBH Wow, what a great day. Left at 4:30 am to pick up the driver and got back at around 5:30 pm. We helped unload the helium bottles for the launch teams and we also watched KMark launch his mylar balloon. We left the site before the launch of the five latex balloons. I had trouble with my computer and Nick's tracking program through most of the day. I finally gave up and just decided to use street atlas for the maps. Worked okay then. Had a great talk with the owner of most of the land east of Agate. His brother-in-law owned the land where EOSS 67 landed. I got his phone number and will try to add it to the list of people we know out east of Denver. Might come in handy someday. Marty directed us to go to the aid of the Idaho team on their first balloon which landed southwest of Deer Trail. We went back that way, found the Idaho team and followed them to the proper road to get to their payload. Their computer was working all right and they found their systems without any help from the DFing antennas. We then decided to head back to town and let my new driver meet some of the launch team. We had a great time talking with all of them until KMark's balloon finally decided to land. We then tore off west to help. After many frustrating minutes of trying to find the proper roads, we got the landing grid point computed using Nicks grid calc program and off we went in search of KMark's payload which had become detached from his balloon which was skipping all over the countryside, with N0NDM and crew chasing it. We finally found the signal after Larry K0ANI pulsed the commandable beacon and tracked it to a field full of vicious buffalo. Vicious things. After a sizeable delay, the owner came down and allowed the rest of the tracking crew to go out and retrieve the payload. It took long enough for that to happen that Rick, N0KKZ and N0LP had time to come out and find the landing site and observe our methods of recovering payloads in dangerous areas. Ha! We even found a buffalo pie for Larry, N0NDM, who was still chasing the mylar balloon. We had a great time at lunch in the cafe in Bennett. We met some new people and got to talk to the Idaho group. They live about 12 miles east of where I went to high school and college. I had a great time talking with them about Idaho. Haven't been back since my 30th high school reunion and next year is the 40th. All in all my new DRIVER and I had a great time. We didn't get up on two wheels while making high speed turns, but that is okay. We had a good time, and learned some more about relaxing and listening for weak signals. Now, all I have to do is get my stupid computer to work so that I can read the packets as well as Chris. I did something different this time by mounting my dual band yagi in place of the single band yagi, and it worked well tracking signals on both bands. It also allowed me to communicate through a poorly responding repeater when I was in a bad spot. So, will have to add that to the permanent equipment list. That's all folks. My my, what a day ! I'd like to extend a huge thanks to the EOSS gang and others that helped in the recovery of the "ES-OS" flight string. I thought it would be elementary as I got a position report from 100 feet AGL, but it apparently involved negotiating a yucca field inhabited by man-killing buffalo. The yucca did a number on my bottom package and completely discombooberated the "ChemSat" payload built by Adam's State College students. (Indeed, the trackers initially reported finding only an empty blue box.) Thanks to the EOSS trackers that back-walked the ~1/4 mile drag path to discover the debris field and recover the guts (I hope all) of the ChemSat. Meanwhile, the shredded bag took off and flew another 10+ miles at 0 to 3000 feet AGL. I had a two pound package on top of the balloon, but it just wouldn't tip over for three hours, finally landing at just past 3 pm. The initial thought was that there was an electronic failure of the flight termination systems, but post flight inspection of the flight-termination balloon-top blow-plug revealed two burn marks about the size of a golf-ball. The electronics worked fine, but the blow plug refused to release. All the room temp testing of this system apparently didn't accurately mimic the environment the system encountered. That, or there was a rigging error. I'll dig into that later. Although it was a long day, we all had fun. Finally, extra special thanks to Larry and Dick for - Hauling helium to the launch site and - Helping us with the disposal chores for the 126k cu foot bag. Bruce, NA�BR started the net at or about 06:15 local time (12:15 UTC) on 7.228 MHz. The band was not very cooperative. He moved about between there and 7.240 MHz looking for a clear piece of bandwidth. Propagation was not too good. Distant stations were heard but only sporadically.	https://www.eoss.org/ansrecap/ar_100/recap67.htm
After about five years of being used for its blu-ray capabilities, netflix movies, tv shows, music, video games and the like, my Playstation 3 has finally died. The red blinking light of death is apparently what signifies its death. The fact that my system is years past its warranty, also helps to seal the deal. I can only really speculate at the moment, that the potential problem lies within the software, as I believe there is some sort of problem between the CPU and its power supply. I guess I shouldn’t have left my system running for almost every day I had it, but I’m not overly concerned. At first, I thought that I wouldn’t need to buy another PS3, but I think that’s probably my only real option at the moment. Mostly due to the fact that I own a plethora of blu-ray movies, PS3 games, as well as enjoy the convenience of Netflix on my tv, it’s really my only realistic option. I thought that I’d just buy myself a blu-ray player, but when a PS3 does the same thing blu-ray players do, I might as well get a new one. Thankfully, PS3’s don’t cost as much as they did when I first bought mine. I think the thing that irritates me the most is that I lost all of my save data. So much for finishing Mass Effect 2, and whatever titles I can’t recall. Anyway, I’m in no rush.	https://lnrd.jp/2012/10/05/the-red-blinking-light-of-death/
Fleas are tiny, wingless insects. The adult is an external parasite of people, dogs, cats, and wild animals, including rodents and birds. Responsible in centuries past for the plagues that killed millions of people, the flea is now a far less serious but still troubling health threat and nuisance to humans and animals. Common types Fleas are familiar to most people with a dog or cat. They are the small, flat and normally hidden insects whose quest for blood sets the animal to frantically snapping at its tail or scratching its ears. Cat (and dog) fleas are the most familiar of well over 1,800 species and subspecies. A few other species have been instrumental in spreading serous human diseases, such as bubonic plague. Others pass diseases onto animals and a few burrow into the skin of their hosts. Fleas belong to the order Siphonaptera and are subdivided into three superfamilies. Pulicoidea, with about 25 genera, contains most species of medical and veterinary importance such as the cat, Oriental rat and sticktight fleas. Certopsylloidea is the largest superfamily with 150 genera that are mostly Neotropical in distribution in southern Mexico, Central and South America, and the West Indies. The best known to North Americans is the cat flea (Ctenocephalides felis), which pursues cats and dogs. The very similar dog flea (C. canis) is far less common. These fleas jump onto their hosts and remain there. Few other flea species are familiar, but the Oriental rat flea (Xenopsylla cheopis) which carries the causative bacterium for the plague, retains notoriety for several hundred cases occurring in wild rodents in the U.S. over the last several decades. Biology Adult fleas, which move by running or jumping, feed on the blood of the host. The eggs hatch into worm-like larvae that develop off the host and feed on detritus or the bloody excrement of the adults. The flea structure, form and life cycle show perfect adaptation to life on the host. The Egg – Flea eggs are smooth and white. Most species lay two to six eggs per day and can lay hundreds over a lifetime. Eggs are normally laid in the host’s sleeping and resting areas. When laid on the host, as do cat and dog fleas, the eggs fall out of the host coat, often at the animal’s sleeping quarters. (Hatching occurs in a few days if humidity is above 70% and temperatures are between 65-80 degrees Fahrenheit.) The flea embryo uses a sharp spine on its head, called an egg burster, to cut and tumble out of the egg. The larvae, blind, limbless worms with circlets of hairs around each inter-segmental division, have three molts. Normally intolerant of light, rarely seen but quite mobile, larvae feed on debris on the floor of the nest of their parents. Adult fleas supplement the debris under the sleeping animal with undigested fecal blood of adult fleas The Pupa – The mature larva spins a whitish, ovoid cocoon with debris embedded in it. This resting stage allows a flea to survive for long periods until stimulated to hatch by the appearance of a host. Pupae will not survive if humidity is low (as low as 45% for the rat flea). The Adult – The adult flea, normally 1.5 to 4 mm long, is wingless, flat and brown. The neck is short, as are the antennae, which fit into a protective groove on the head when not in use. The eyes are not well developed. Some species, such as Leptopsylla segnis, the house-mouse flea, lack eyes entirely. Many species have combs, arrays of stiff bristles to keep the flea from being pulled out of the host’s coat, and hind legs adapted for jumping. Two notable exceptions to the active, jumping flea are the sticktight fleas (Echidnaphaga gallinacea of chickens) and the sand flea, chigoe,or jigger (Tunga penetrans) which parasitizes man and animals. The females of these fleas burrow into the skin of the host and remain attached for life. Adult fleas feed by piercing the host’s skin with their mouth parts and penetrating a capillary from which they suck blood, using one or more pumps to convey the blood to the gut. If undisturbed, feeding is complete in 2-10 minutes. Female fleas take up about twice as much blood as males. Health issues Human Health Impact Historically, the tiny rat flea has greatly impacted civilization as carrier of the plague. The tropical or Oriental rat flea is the main carrier of the plague bacterium, Yersinia pestis. Typically feeding on the brown rat, Rattus rattus, and the black or roof rat, R. norvegicus, the rat flea also readily feeds on people. Plague has probably afflicted humans since before the time of Christ. The Philistines are recorded as suffering from a disease with symptoms like those of the bubonic plague. The first pandemic of record was probably in the sixth century, beginning in Egypt. During the late Middle Ages, the Black Death laid waste to Europe in another pandemic. An estimated 25 million people died in the fourteenth century. From 1664 to 1666, 70,000 Londoners dies out of a population of 450,000. Civil disorder broke out. Terrified neighbors even put plague victims to death. What made the disease so frightening was that its cause was unknown. Only in the late 19th and early 20th centuries did a few brave researchers establish that the rat flea had to be present to spread the disease. That discovery came during the third pandemic, which began in China’s Yunnan Province in the 1890s and spread to the West Coast of the U.S. and throughout the world. Deaths in the U.S. occurred in San Francisco, Los Angeles and cities farther east, such as New Orleans. The disease is characterized by rapidly developing high fever, headache, prostration, fatigue and delirium. By the second day, lesions know as bubos (hence bubonic plague) appear in the groin and armpits. Mortality rates are high. Cases of plague still occur in the U.S., primarily in wild rodents. In the western states, 334 cases were reported from 1970-1994. Murine Typhus – Found worldwide, several species of flea, including Xenopsylla cheopis (the main vector), Nosopsylla fasciatus and Leptopsylla segnis carry murine typhus. Flea feces transmit murine typhus when the host human scratches his or her skin to alleviate itching caused by fleabites, allowing the pathogen to enter the body. While thousands of cases occurred each year early in the 20th century, the disease is rare in the U.S., occurring primarily in the South. Symptoms include sudden high fever, headache, nausea, coughing and a spotted rash. Flea-Caused Infections – The female chigoe, jigger or sandflea bores into the skin, usually of the feet, causing extreme irritation. If Tunga penetrans is not removed, it can cause an infection, which may become gangrenous. Chigoes are found in tropical Americas and Africa and are most prevalent in people who walk barefoot. Skin Irritation – Annoying to some, fleabites can cause serious irritation in sensitive individuals. Bird fleas may become a problem when construction disturbs bird nests in buildings. The human flea (Pulex irritans) can also cause irritation but is now much less frequently encountered in the U.S. than cat and dog fleas. Tapeworm – Young children may be at risk if they play in areas where pet excrement is present and the cat or dog has the tapeworm Dipylidium caninum. When tapeworm eggs are defecated by a cat or dog, flea larvae may feed on the excrement and ingest the eggs. The tapeworm eggs hatch in the flea’s larval gut. When the flea completes its development, a cat or dog may ingest the adult flea during grooming or nipping. A child coming into close contact with a pet may ingest a tapeworm-infected adult flea from the pet. Animal Health Impact Cat and Dog Flea – The cat flea, which parasitizes both dogs and cats and can transmit a species of tapeworm to pets, is the main flea of concern to most North Americans. The less common dog flea is similar to the cat flea. Adult cat and dog fleas remain on the host (rather than jumping on the animal only to feed), causing severe irritation and vigorous scratching. This can lead to severe coat loss and frequent visits to a veterinarian. Both flea species may transmit the tapeworm Dipylidium caninum. The tapeworm’s eggs are defecated by the cat or dog and larval fleas consume the feces. Sticktight Flea – An important poultry pest in subtropical America, the sticktight flea remains attached to a chicken, causing ulcers in which flea eggs are laid, and in heavy infestations, anemia. The sticktight flea also will attack cats, dogs, horses and humans. Myxomatosis – This flea-vectored disease of rabbits wiped out nearly the entire rabbit population of the United Kingdom in the 1950s, eliminating rabbit trapping as an income source for many and reducing the food source of raptorial birds and predatory animals. The disease has been deliberately introduced in Australia as a means of controlling huge infestations of rabbits. Flea control Sanitation, insecticides and common sense are keys to efforts to control fleas. Good sanitation measures are important in conjunction with appropriate flea control products used according to label directions on the host or in the host’s habitat. Cat and Dog Fleas Insecticidal pet shampoos and dips, flea collars and total release aerosols, and powders can control cat fleas. Control is best achieved by using several products concurrently so that the flea infestation is attacked both on and off the host. Dipping or shampooing coupled with fogging the home and fitting the treated pets with flea collars is advisable. Normally, no single product is sufficient to control an infestation of cat fleas. For example, treatment of the pet only will not combat fleas present in immature stages in the carpet. The immature stages will emerge as adults and cause reinfestations. Thus a combination of a pet treatment and a house treatment, including the area where the pet sleeps, is usually needed so the cycle can be prevented from recurring. Pet Shampoos and Dips – A veterinarian, professional groomer or the pet owner can apply these products according to label directions. Shampoos are applied and rinsed off after a short time. Dips are ‘leave on’ products. Pet Flea Collars – Flea collars are impregnated plastic strips that allow the slow release of the active ingredient. Collars rely on migration of the active ingredient over the coat of the pet, possibly aided by grooming, to reach the fleas. Total Release Aerosols – Foggers are designed to fill a room with fine particles that settle on exposed surfaces and can penetrate to hidden interior surfaces. Fleas attacking the pet are often also found on the floor, especially in carpet and places where the pet sleeps and rests. Use of a total release aerosol is a convenient means of treating flea infested rooms uniformly and completely. Of course, people and pets should not be in the room during fogging. Be sure to read and follow the directions. In recent years insect growth regulators have been added to foggers and to other flea control products. These materials interrupt the flea’s life cycle before adult fleas emerge to become a nuisance. Because IGRs are slow acting and do not affect the adult fleas, they are normally coupled with a conventional insecticide so that adult fleas can be controlled immediately. Direct Aerosol Sprays – These aerosol products have valves that allow the can to be used in an inverted position so floor areas can be easily sprayed. They are useful for treating limited areas or where foggers might be inappropriate. Powders – Powders applied to the pet are the dry equivalent to dips in that the powder is left on the animal. They also are easier to apply than dips. Systemics – A successful innovation in flea control has been an orally or dermally administered insect growth regulator in pill form for dogs and liquid or gel for cats. Available only from veterinarians, the IGR is administered once a month, preventing eggs from hatching and breaking the life cycle. Since a systemic has no effect on other life stages, including adults, control is not immediate. A conventional adulticide treatment is needed before the IGR can control an infestation. Miscellaneous – Traps with small lights aim at luring fleas to adhesive coated sheets to which they are to become stuck. The impact of these types of traps on flea infestations is unknown. Public Health Control – Control of fleas that may carry plague or other diseases is the responsibility of state and federal public health authorities, which routinely conduct surveys for plague, recording the incidence of plague antibodies in wild and domestic hosts. When surveys have indicated the need, dusting the burrows of rodent hosts with suitable insecticide dusts has controlled potential plague-carrying fleas. The rodents themselves may be controlled by rodenticides. People in western states should avoid contact with wild rodents because of the possibility of contracting plague. Backpackers and campers should be particularly careful. Use of an aerosol insecticide is a wise precaution against rodent fleas. Wood piles and similar cover for the rodents near dwellings should be removed.	https://www.aboutbugsbugsbugs.com/fleas/
PROBLEM TO BE SOLVED: To provide a substrate for mounting an electronic component, in which heat from a peripheral through-hole is not easily dissipated, and also easily and sufficiently conducted to a through-hole. SOLUTION: The substrate for mounting the electronic component includes: the peripheral through-hole 20 disposed at a position near the through-hole 10 to penetrate the substrate, so that when heated and fused solder 50 enters the through-hole 10, the heated and fused solder 50 enters the inner space of the peripheral through-hole, and having a second peripheral wall 20a; and a solid pattern 30 which conducts the heat between a first peripheral wall 10a and the second peripheral wall 20a and is provided in the substrate 3. When the heated and fused solder 50 enters the inner space of the peripheral through-hole 20, the heat is supplied from the second peripheral wall 20a of the peripheral through-hole 20 to the first peripheral wall 10a of the through-hole 10 via the solid pattern 30. COPYRIGHT: (C)2011,JPO&INPIT
Yesterday I mentioned how both the attendees and the presenters at the GF Culinary Summit came from points near and far, and from equally diverse food backgrounds. We were all brought together, though, by one common thread: we’re all gluten-free. But as the weekend moved along, I was continually reminded that for many, being gluten-free is just one component of a broader set of dietary restrictions. This is a topic I’ve addressed before, in a post titled Degrees of Free-dom from back in October 2008. At the GF Culinary Summit in particular, I met folks who were not only gluten-free, but who were also… lactose-free, casein-free, dairy-free, soy-free, corn-free, and citrus-free. There were omnivores and vegetarians and vegans and adherents of the paleo diet. And surely there must have been others who I did not meet. As presenters were demonstrating a given dish, audience questions would inevitably pop up… Can I use Energ-G egg replacer? Can I substitute Smart Balance for butter? Can I use almond milk in place of cow’s milk and cream? Now that Kelli and I are gluten-free bloggers AND cookbook coauthors, I look at those degrees of free-dom through a slightly different lens than when I originally wrote about it in October. I’ve always had a level of understanding with and empathy for people with multiple degrees of free-dom because I’m one of them. Gluten is my dominant food no-no, but there are others, such as grapefruit. But now, the big question for us is: When developing new recipes, who are we developing them for, and how many dietary restrictions do we impose on ourselves in concocting recipes? The answer isn’t quite as straightforward as I might like. On one extreme end of the spectrum, we could take into account the full set of dietary restrictions I listed above, and create recipes and dishes that would be almost universally acceptable. This approach is hampered by two challenges: 1) It is almost unimaginably restrictive in the set of ingredients that would be permitted in our “toolbox,” and 2) It would unfairly (or perhaps more accurately, unnecessarily) constrain people who don’t have such an extreme level of dietary restriction. (And indeed, I don’t know anyone who would be bound by ALL the restrictions I listed… each one of us is likely bound by a small subset of them…) On the opposite end of the spectrum, we could cater to the lowest common denominator – gluten. We could develop recipes that are gluten-free, and in this sense, they would have a different kind of universal applicability since that is the dietary restriction shared by all of us. Of course, this approach has its own pitfalls. Most notably, if you have dietary restrictions beyond simply gluten, you’re left to fend for yourself and modify the recipe in order to make it work with your particular diet. Then there’s what I call the selfish end of the spectrum. If Kelli and I were developing recipes only for ourselves, then this is the logical route to take. In essence, we’d create recipes that adhere to my particular set of dietary restrictions. They’d work for us, and since we’re our own target audience, then it’d be a case of mission accomplished. But we don’t create recipes only for ourselves, so the selfish approach wouldn’t necessarily work because it’s…self-serving. Lastly, there’s what I’d call the customization approach. In short, we’d customize our recipes based on the needs of the moment. We do this already when having friends over for dinner. We’ll be making a gluten-free dinner for my sake. But sometimes we’ll also have a vegetarian at the dinner table, or someone who can’t do refined, processed sugars. And so we tailor the meal to meet the needs of everyone present. We can do this on the blog to a degree, answering questions and comments and offering suggestions for modifying a recipe to suit someone’s particular needs. But a cookbook is a much more static thing, and we can’t predict who might pick up a copy off the shelf at their local bookstore. And therein lies the rub. So what do you think? If we assume that gluten is the most frequent, common denominator, what are the most prevalent secondary dietary restrictions? Lactose? Dairy in general? Something else? And when you build a recipe, do you do it for yourself? Or for a friend or family member who has a certain set of restrictions? These are good questions to answer, I think. As gluten-free foodies, we’re accustomed to asking others to accomodate our needs…whether a family member or friend, a restaurant, whatever. But it helps, at times, to think about the dietary needs and restrictions of others, which may be more restrictive – or simply different – than our own. And as people who do have a dietary restriction, we’re in a unique position to understand how that feels.	http://nogluten-noproblem.com/2009/10/no-soup-for-you-maybe.html
Q: C++ use std::enable_if to create std::tuple specialisations up to 10 arguments I want to create a tuple that has specializations for up to 10 args, similar to how std::pair is a specialization for two args. i.e tuple<int,float,bool> will have the members first(), second(), and third() Here is my attempt so far: #pragma once #include <tuple> #include <type_traits> template<typename... Types> struct tuple : std::tuple<Types...> { using std::tuple<Types...>::tuple; static constexpr size_t size = sizeof...(Types); template<size_t N> using elem_n = std::tuple_element_t<N, std::tuple<Types...>>; template<size_t N> const elem_n<N>& get() const { return std::get<N>(this); } template<size_t N> elem_n<N>& get() { return std::get<N>(this); } template<bool F = false> std::enable_if_t< (size >= 1) \|\| F, const elem_n<0>&> first() const { return get<0>(); } template<bool F = false> std::enable_if_t< (size >= 2) \|\| F, const elem_n<1>&> second() const { return get<1>(); } template<bool F = false> std::enable_if_t< (size >= 3) \|\| F, const elem_n<2>&> third() const { return get<2>(); } template<bool F = false> std::enable_if_t< (size >= 4) \|\| F, const elem_n<3>&> fourth() const { return get<3>(); } template<bool F = false> std::enable_if_t< (size >= 5) \|\| F, const elem_n<4>&> fith() const { return get<4>(); } template<bool F = false> std::enable_if_t< (size >= 6) \|\| F, const elem_n<5>&> sixth() const { return get<5>(); } template<bool F = false> std::enable_if_t< (size >= 7) \|\| F, const elem_n<6>&> seventh() const { return get<6>(); } template<bool F = false> std::enable_if_t< (size >= 8) \|\| F, const elem_n<7>&> eighth() const { return get<7>(); } template<bool F = false> std::enable_if_t< (size >= 9) \|\| F, const elem_n<8>&> ninth() const { return get<8>(); } template<bool F = false> std::enable_if_t< (size >= 10) \|\| F, const elem_n<9>&> tenth() const { return get<9>(); } template<bool F = false> std::enable_if_t< (size >= 1) \|\| F, elem_n<0>&> first() { return get<0>(); } template<bool F = false> std::enable_if_t< (size >= 2) \|\| F, elem_n<1>&> second() { return get<1>(); } template<bool F = false> std::enable_if_t< (size >= 3) \|\| F, elem_n<2>&> third() { return get<2>(); } template<bool F = false> std::enable_if_t< (size >= 4) \|\| F, elem_n<3>&> fourth() { return get<3>(); } template<bool F = false> std::enable_if_t< (size >= 5) \|\| F, elem_n<4>&> fith() { return get<4>(); } template<bool F = false> std::enable_if_t< (size >= 6) \|\| F, elem_n<5>&> sixth() { return get<5>(); } template<bool F = false> std::enable_if_t< (size >= 7) \|\| F, elem_n<6>&> seventh() { return get<6>(); } template<bool F = false> std::enable_if_t< (size >= 8) \|\| F, elem_n<7>&> eighth() { return get<7>(); } template<bool F = false> std::enable_if_t< (size >= 9) \|\| F, elem_n<8>&> ninth() { return get<8>(); } template<bool F = false> std::enable_if_t< (size >= 1) \|\| F, elem_n<9>&> tenth() { return get<9>(); } }; I have also tried it with: template<size_t N> using elem_n = std::conditional_t<(size >= N), std::tuple_element_t<N, std::tuple<Types...>>, void>; But when testing with using my_tripple = tuple<int, std::string, float>; my_tripple a; a.first() = 6; a.second() = "hello"; a.third() = 0.1f; I get the compile errors: /usr/include/c++/9/tuple:1303: error: static assertion failed: tuple index is in range 1303 \| static_assert(__i < tuple_size<tuple<>>::value, \| ~~~~^~~~~~~~~~~~~~~~~~~~~~~~~~~~ And from fourth() to tenth() error: no type named ‘type’ in ‘struct std::tuple_element<3, std::tuple<int, std::__cxx11::basic_string<char, std::char_traits<char>, std::allocator<char> >, float> >’ 26 \| template<bool F = false> std::enable_if_t< (size >= 4) \|\| F, const elem_n<3>&> fourth() const { return get<3>(); } \| ^~~~~~ This is related to the question c++ Use std::enable_if to conditionally add getters to a variadic variant template, but the solution from that doesn't work here. Thanks A: Your approach of using a defaulted template parameter for second(), third() was the right direction, but what you missed is that you needed to make get<>'s template parameter dependent on the default template parameter, so it does not get resolved until template instantiation time, and never gets resolved if it's never used. Unless it's dependent on the template parameter, it gets resolved at declaration time, and it fails for the reasons given. A short example using just second(). third(), fourth(), et. al. would get declared the same way, using size_t n=2, size_t n=3, and so on: #include <tuple> template<size_t n=1> auto second() const { return get<n>(); } }; tuple<int> foo; tuple<int, float> bar; float foobar() { return bar.second(); // compiles, foo.first() would be a compilation error } In foo's case, since get<1> never actually exists unless second() gets explicitly called, there is no compilation error.
Recently, portable equipment which uses a battery, such as a mobile phone, a digital camera and so on, has been widely developed. Such portable equipment commonly uses a constant voltage circuit which supplies a constant power supply voltage. The constant voltage circuit widely employs a voltage step-up/step-down switching regulator to obtain a regulated voltage. A conventional voltage step-up/step-down switching regulator employs two DC-DC converters, i.e., a step-up converter and a step-down DC-DC converter. The conventional voltage step-up/step-down switching regulator performs a step-down operation by activating the step-down DC-DC converter when an input voltage is larger than an output voltage so as to output a predetermined constant voltage. Meanwhile, the conventional voltage step-up/step-down switching regulator performs a step-up operation by activating the step-up DC-DC converter when the input voltage is smaller than the output voltage. FIG. 1 illustrates a conventional voltage step-up/step-down switching regulator 100. FIG. 2 illustrates a timing chart of waveforms of each part of the circuit of FIG. 1. The voltage step-up/step-down switching regulator 100 includes an oscillator 110, an operational amplifier 116, PWM (pulse width modulation) comparators 112 and 114, and resistors R101 and R102. A signal A is an output signal from an error amplifier (not shown) and is an amplified signal of a difference voltage between a reference voltage and a proportional-voltage proportional to the output voltage. The signal A is defined as an error signal A. A signal B is an upper peak voltage V2 and a signal C is a low voltage V1 which is lower than the upper peak voltage V2. The oscillator 110 generates a triangular wave G which swings between the upper peak voltage V2 as an upper limit voltage and the lower peak voltage V1 as a lower limit voltage. The operational amplifier 116 forms an inverting amplifier to have a reference voltage with the upper limit voltage V2 of the triangular wave. If the resistors R101 and R102 have equal resistance value from each other, a signal D becomes an waveform equivalent to an inverted signal of the error signal A with respect to the upper limit voltage V2. The error signal A is input to an input of the PWM comparator 112. The triangular wave G, which is an output of the oscillator 110, is input to another input of the PWM comparator 112. The signal D, which is the output signal of the operational amplifier 116, is input to an input of the PWM comparator 114. Similarly to the PWM comparator 112, the triangular wave G is input to another input of the PWM comparator 114. The voltage step-up/step-down switching regulator 100 further includes NMOS transistors S101 and S102, diodes D101 and D102, a coil L101 and a capacitor C101. When the voltage step-up/step-down switching regulator 100 performs the step-down operation, the NMOS transistor S102 is off and the NMOS transistor S101 only performs on/off operation. If an input voltage Vin becomes larger than an output voltage Vout, an on-time of the NMOS transistor S101 becomes shorter. If input voltage Vin becomes closer to the output voltage Vout, an on-time of the NMOS transistor S101 becomes longer. When the voltage step-up/step-down switching regulator 100 performs the step-up operation, the NMOS transistor S101 is on and the NMOS transistor S102 only performs on/off operation. If the input voltage Vin becomes smaller than the output voltage Vout, an on-time of the NMOS transistor S102 becomes longer. If the input voltage Vin becomes closer to the output voltage Vout, an on-time of the NMOS transistor S102 becomes shorter. The PWM comparator 112 compares the voltage of the error signal A with the voltage of the triangular wave G. If the voltage of the error signal A is larger than the voltage of the triangular wave G, the PWM comparator 112 outputs a signal F with a high level. If the voltage of the error signal A is smaller than the voltage of the triangular wave G, the PWM comparator 112 outputs the signal F with a low level. The PWM comparator 114 compares the output voltage D of the operational amplifier 116 with the voltage of the triangular wave G. If the voltage of the triangular wave G is larger than the output voltage D, the PWM comparator 114 outputs a signal E with a high level. If the voltage of the triangular wave G is smaller than the output voltage D, the PWM comparator 114 outputs a signal E with a high level. When the error signal A is within the voltage range of the triangular wave G, the PWM comparator 112 outputs a signal F with a pulse wave and performs the step-down operation by making the NMOS transistor S101 on/off. The output voltage D of the operational amplifier 116 exceeds the upper limit voltage V2 of the triangular wave G during this time period. As a result, the output signal E of the PWM comparator 114 becomes low level and the NMOS transistor S102 becomes off. When the error signal A exceeds the upper limit voltage V2 of the triangular wave G, the output signal F of the PWM comparator 112 becomes high level and the NMOS transistor S101 becomes on. Meanwhile, when the output voltage D of the operational amplifier 116 is within the voltage range of the triangular wave G, the PWM comparator 114 outputs a signal E with pulse wave and performs the step-up operation by making on/off with the NMOS transistor S102. Thus, the voltage step-up/step-down switching regulator 100 controls the output voltage to obtain a predetermined constant voltage by exchanging the operational mode between step-up and step-down modes in accordance with the input voltage Vin. However, the conventional voltage step-up/step-down switching regulator 100 needs two PWM comparators, i.e., the PWM comparator 112 to control the step-down switching element of the NMOS transistor S101 and the PWM comparator 114 to control the step-up switching element of the NMOS transistor S102. Further, in addition to the error amplifier which is commonly used, the operational amplifier 116 is needed to invert the error signal A with respect to the upper limit voltage V2 and to input the inverted error signal to the PWM comparator 114 for the step-up operation. Furthermore, the conventional voltage step-up/step-down switching regulator 100 has a cost penalty because the PWM comparators and the operational amplifier are generally formed of an analog circuit which needs many circuit elements and requires the circuit elements to have high precision.
Items purchased from VON DUTCH are eligible for a partial to full refund if returned within 15 days of purchase. A small number of heavily discounted items may be marked FINAL SALE, in which case they are not eligible for refund, exchange, or store credit. Items must be returned unworn, unwashed, undamaged, unused, and with all original tags attached. Shoes scuff very easily on any hard surface. Always try on shoes on carpet. Shoe boxes are an important part of the product presentation and must be returned undamaged, along with the shoes. We will send you an email when your return package is delivered back to our warehouse. Please note, returns do take us 2-3 business days to process upon reaching our facility. Once your return is processed and your refund is issued, you will receive a refund confirmation email. Refunds can take several business days to reflect on your credit card account. This varies depending on your financial institution. If you receive an item that is damaged, please contact customer service immediately.	https://vondutch.com/pages/returns-exchanges
RELATED APPLICATIONS GOVERNMENT RIGHTS TECHNICAL FIELD BACKGROUND BRIEF SUMMARY DETAILED DESCRIPTION This application claims the benefit of U.S. Provisional Patent Application Ser. No. 61/542,629 filed Oct. 3, 2011 to the same inventor. This invention was made with government support under contract NNA08BB37C awarded by NASA and under contract HR0011-08-C-0101 awarded by DARPA. The government has certain rights in the invention. The present invention generally relates to small storage tanks for fluids, and more particularly relates to small-scale tanks with high tank factors. Storage of high pressure gases and liquids is a critical requirement for many applications, e.g. rocket and aircraft propulsion components, automotive airbags, pneumatic and hydraulic systems, etc. The science for design and manufacture of suitable tanks for this purpose is well documented, with many examples of commercially available tanks. Typical tanks are made in the form of spheres or cylinders, and may be manufactured from metals or composite (with or without a liner). FIG. 1 FIG. 2 Pictures of representative commercially available tanks for high pressure storage of gases and liquids are shown in and as examples of commercially available tanks. While such tanks are relatively common in large sizes with diameters in excess of six inches, they less common in the extremely small size-class (i.e. diameters of the order of a few inches). The problem is especially difficult in extremely weight sensitive applications (i.e. rocket engines), and in applications where the pressure of the stored fluid is very high (several hundred pounds per square inch). The realization of small high-pressure tanks has proved challenging for several reasons including that, given a limitation of minimum gage thickness for conventional materials, the mass of the walls ends up being much higher than what is required, thereby making the tanks much heavier than they need to be and it is difficult to form conventional materials into suitable cylindrical or spherical shapes at the small scale. An exemplary conventional metal tank is welded together from pieces bent sheet metal. For example, a first sheet is rolled into a cylinder, and two hemispherical ends are then formed in a press. The hemispherical ends are then welded onto the ends of the cylinder. The smallest gage aluminum which can be worked in such a process is 30 mil, and even that is very difficult and expensive. This is the practical gage limitation that prevents conventional methods from making thinner-walled aluminum tanks. Consequently, there are currently no commercially available high tank factor storage tanks in the 1-10 cubic inch size class. FIG. 3 A key figure-of-merit commonly used in this context is the “tank factor” which is defined as: “Failure Pressure” times “Storage Volume” divided by “Tank Weight” (the lower the tank weight for a given failure pressure and volume, the better the tank, and hence, higher the tank factor). depicts the tank factors for commonly available tanks as a function of storage volume, and clearly shows that while one can achieve high tank factors (nearing 30,000 meters for storage volumes in the 100-10,000 cubic inches range), the achievable tank factor decreases with size, there being no tanks with similar performance in the small size-scale (i.e. storage volumes of 1-10 cubic inches). The tanks that do exist in the small size scale (less than 10 cubic inches) are either single-use disposable cylinders, for example, those used to inflate life-jackets, or “sample cylinders” used for capturing and transporting small samples of gas for analysis. These are limited to cylindrical shapes and have tank factors of less than 2500 meters. Accordingly, it is desirable to manufacture a tank with a high tank factor (approximately 8,000 meters) in the 1-10 cubic inches volume range. In addition, it is desirable to devise a method of manufacturing such tanks that is effective and economical. Furthermore, other desirable features and characteristics of the present invention will become apparent from the subsequent detailed description and the appended claims, taken in conjunction with the accompanying drawings and the foregoing technical field and background. An apparatus is provided for storing fluids at high pressures in small volumes. The apparatus comprises one or more pressure vessels that are made up of multiple arrays of internal chambers with a single gas inlet and outlet for each vessel as well as gas feeder and connector lines. A method is described for manufacturing small-volume tanks with high tank factors by aligning and stacking a plurality of patterned layers into a 3D shape, sandwiching the stacked layer between end wall structures, and diffusion bonding the multiple layers into a single monolithic tank with automatic fluid interconnects between internal chambers. The present invention uses a micro layer metal foil etching and diffusion bonding methodology to realize high-pressure tanks in the small size-class. FIG. 4 FIG. 4 An exemplary embodiment of the invention is shown in in cross section form. Herein, the 2″×2″ square piece consists of two separate pressure vessels on the left and right that are made up of multiple honeycomb shaped internal chambers with a single gas inlet and outlet for each vessel as well as gas feeder and connector lines. The alignment pin referred to in is used to align the different layers and ensure a good diffusion bond between the layers for structural integrity of the internal chambers in the final structure. The creation of such smaller chambers within the pressure vessels reduces the structural requirements on the outermost metal walls, thereby allowing for a light weight structure. FIGS. 20A-20D A key element of the present invention is the method used to manufacture the tanks. As discussed in regard to , the process involves: slicing a CAD model of the geometry into multiple layers; generating the necessary “pattern” artwork for each layer; using the pattern to etch each metal layer and create the pre-formed shapes; aligning and stacking of each of the layers into a 3D shape, and sandwiching between end wall structures; diffusion bonding the multiple layers into a single monolithic tank with automatic fluid interconnects between internal chambers; and external machining of the structure to release the final geometry and create access ports. The invention provides A small scale metal tank for high pressure storage of fluids including: a tank factor of at least three thousand meters and a tank volume of at most ten cubic inches. The tank, including: an enclosure including a plurality of outer tank walls; an array of internal chambers within the enclosure; a plurality of fluidic interconnections between each of the internal chambers of the array of internal chambers and each other internal chamber of the array of internal chambers; and a fluidic conduit between an internal chamber of the a array of internal chambers and a point external to the enclosure. The tank, where the outer tank wall of the plurality of outer tank walls includes a flat outer tank wall. The tank, where the enclosure includes a shape that is adapted to and/or conformal to a particular mechanical application. The tank, where the array of internal chambers is formed of diffusion-bonded metal layers having diffusion-bonded seams between adjacent layers. The tank, where each chamber of the array of internal chambers has: opposed first and second end walls: a plurality of side walls extending between the opposed first and second end walls; an internal junction between a side wall of the plurality of side walls and one of the opposed first and second end walls; and a filet at the internal junction, where the filet includes no fusion-bonding seams. The tank, where either the opposed first and second end walls include a portion of an outer tank wall of the plurality of outer tank walls and the portion of the outer tank wall includes an arcuate shape that is internal and/or external. The tank, where a side wall of the plurality of side walls includes a portion of an outer tank wall of the plurality of outer tank walls and the portion of the outer tank wall includes an arcuate shape that is internal and/or external. The tank, where the at least one array of chambers includes two or more arrays of chambers, each forming an independent vessel within the enclosure and each having fluidically interconnected chambers within each of the two or more arrays of chambers and each vessel having a fluidic conduit external to the enclosure. A small scale metal tank for high pressure storage of fluids having: a tank factor of at least three thousand meters; and a tank volume of at most ten cubic inches; where the tank includes: an enclosure including a plurality of outer tank walls; at least one array of internal chambers within the enclosure; an internal junction between a side wall of the plurality of side walls and one of the opposed first and second end walls; and a filet at the internal junction, where the filet includes no the fusion-bonding seams. The tank, where the outer tank wall of the plurality of outer tank walls includes a flat outer tank wall. The tank, where the enclosure includes: a shape adapted to fit adaptively and/or conformally with a particular mechanical device; and a shape that is not spherical. The tank, where the array of internal chambers is formed of diffusion-bonded metal layers having diffusion-bonded seams between adjacent diffusion-bonded layers. The tank, where each chamber of the array of internal chambers has: opposed first and second end walls: a plurality of side walls extending between the first and second end walls; an internal junction between a side wall of the plurality of side walls and one of the first and second end walls; and a filet at the internal junction, where the filet includes no the diffusion-bonding seams. The tank, where one of the first and second end walls includes a portion of an outer tank wall of the plurality of outer tank walls and the portion of the outer tank wall includes an arcuate shape that is internal and/or external. The tank, where one side wall of the plurality of side walls includes a portion of an outer tank wall of the plurality of outer tank walls and the portion of the outer tank wall includes an arcuate shape that is internal and/or external. The tank, where the at least one array of chambers includes two or more arrays of chambers, each forming an independent vessel within the enclosure and each having fluidically interconnected chambers within each of the two or more arrays of chambers and each vessel having a fluidic conduit terminating external to the enclosure. A small scale metal tank for high pressure storage of fluids having: a tank factor of at least three thousand meters and a tank volume of at most ten cubic inches; where the tank includes: an enclosure including a plurality of outer tank walls; at least one array of internal chambers within the enclosure; an internal junction between a side wall of the plurality of side walls and one of the opposed first and second end walls; and a filet at the internal junction, where the filet includes no the fusion-bonding seams; where the enclosure includes: a shape adapted to fit adaptively and/or conformally with a particular mechanical device; a shape that is not spherical; and a shape that does not have a hemispherical tank end; where each chamber of the array of internal chambers includes: a plurality of diffusion-bonded metal layers having diffusion-bonded seams between adjacent the diffusion-bonded layers; opposed first and second end walls each including one diffusion-bonded layer of the plurality of the diffusion-bonded layers; a plurality of side walls each comprised of a stack of the fusion bonded layers and extending between the opposed first and second end walls; an internal junction between a side wall of the plurality of side walls and one of the opposed first and second end walls; and a filet at each the internal junction, where the filet includes no diffusion-bonding seams. The tank, where either the first end wall, the second end wall, and a side wall of the plurality of side walls of the chamber includes a portion of an outer tank wall of the plurality of outer tank walls and the portion of the outer tank wall includes an arcuate surface that is internal and/or external. The tank, further including the tank attached to the particular mechanical device. The following detailed description is merely exemplary in nature and is not intended to limit the invention or the application and uses of the invention. Furthermore, there is no intention to be bound by any expressed or implied theory presented in the preceding technical field, background, brief summary or the following detailed description. FIG. 1 100 100 is a perspective view illustrating a prior art tank in the 100-10,000 cubic inches volume range. Tank is a spherical tank. FIG. 2 200 is a perspective view illustrating a plurality of prior art tanks in the 100-10,000 cubic inches volume range. Two spherical tanks and two cylindrical tanks with hemispherical tank ends are shown. FIG. 3 300 300 is a chart illustrating tank factor vs. storage volume for prior art tanks. The prior art has no tank factors above 3,000 meters for tanks in the 1-10 cubic-inch volume range. The upper volume limit is actually slightly greater than ten cubic inches, as shown. More precisely, the chart shows no tank factors above zero meters in tanks under ten cubic inch volume. Tanks that do exist in the small size scale (less than 10 cubic inches) are either single-use disposable cylinders, for example, those used to inflate life-jackets, or “sample cylinders” used for capturing and transporting small samples of gas for analysis. These are limited to cylindrical shapes and have tank factors of less than 2500 meters. FIG. 4 FIG. 9 400 410 410 406 408 404 402 404 410 400 is a perspective view illustrating an exemplary embodiment of the foil-layer stack structure showing the internal tank structure for a small volume, high tank factor, tank, according to an embodiment of the present invention. The internal tank structure includes first vessel and second vessel made of an interconnected (see ) array of chambers (one of 128 labeled) in a frame . The chambers are illustrated as hexagonal in cross-section, but the invention is not so limited. In various embodiments, various cross-sectional shapes may be used, as will be discussed and illustrated in greater detail below. The internal tank structure is made by bonding foil layers together in a vertical stack . The fingers in the illustration are not part of the invention, but give an approximate size reference. FIG. 4 FIG. 9 408 406 404 914 916 406 408 904 906 An embodiment of the invention is shown in in cross section form. Herein, the 2″×2″ square piece consists of two separate pressure vessels and on the left and right, respectively, that are made up of a honeycomb of hexagonal-shaped internal chambers with a single gas inlet and (see ) for each vessel and as well as gas feeder and connector lines and , respectively. 508 412 414 404 400 FIG. 5 Alignment pins , such as the one shown in , are inserted into alignment holes and to align the various layers and ensure a good diffusion bond between the layers for structural integrity of the internal chambers and in the final light-weight structure . 404 406 408 402 400 The creation of such smaller chambers within the pressure vessels and reduces the structural requirements on the outermost metal frame , thereby allowing for a light-weight structure . FIGS. 20A-20D 1. Slicing a CAD model of the geometry into multiple layers; 2. Generating the necessary “pattern” artwork for each layer; 3. Using the pattern to etch each metal layer and create the pre-formed shapes; 4. Aligning and stacking of each of the layers into a 3D shape, and sandwiching between end wall structures; 5. Diffusion bonding the multiple layers into a single monolithic tank with automatic fluid interconnects between internal chambers; and 6. External machining of the structure to release the final geometry and create access ports. A key element of the present invention is the method used to manufacture the tanks. As discussed in greater detail in regard to , the process involves: FIG. 5 FIG. 6 FIG. 16 FIG. 6 506 504 510 500 502 516 514 514 514 500 508 1004 1616 1510 1620 512 500 504 602 606 506 is a perspective view illustrating another exemplary embodiment of an internal tank structure with an end wall being added to a stack for a small volume, high tank factor, tank , according to an embodiment of the present invention. Edge chambers (one of ten labeled) have arcuate internal surfaces and a flat external surface , as shown. In a preferred embodiment, the flat external surface will be machined away, as illustrated in . In another embodiment, at least a portion of the flat surface may be retained to assist in fitting tank into another mechanical device or application. Alignment pin is used to align the various layers, similar to layers , , , and (See ), and to ensure a good diffusion bond between the layers for structural integrity of the internal chambers and in the final structure . The fingers in the illustration are not part of the invention, but give an approximate size reference. The present invention realizes flat end walls (also and in ) in the frame (uncommon in pressure vessels) without sacrificing tank factor and performance. FIG. 6 600 610 602 606 604 608 600 610 612 614 600 610 is a perspective view illustrating two additional embodiments of walled tanks and with chambers and (one of thirty-six labeled in each), respectively, trimmed via electrical discharge machining (EDM), with respective trimmed external material and for a small volume, high tank factor, tank, according to an embodiment of the present invention. The EDM trimming reduces the weight of the tanks and without sacrifice of required strength. Fluidic couplings and provide both an inlet for charging and discharging the tank and , respectively, through a single tube. FIG. 7 700 702 600 600 704 404 702 is a perspective view illustrating an exemplary embodiment of the internal tank structure with end walls for a small volume, high tank factor, tank under hydrostatic testing, according to an embodiment of the present invention. Hydrostatic testing verifies the ability of the tank to withstand operational pressures. Bulging of the individual chamber end wall portions can be seen. FIG. 8 800 802 600 600 802 404 is a perspective view illustrating another exemplary embodiment of the internal tank structure with end walls for a small volume, high tank factor, tank under hydrostatic testing, according to an embodiment of the present invention. Testing to failure defines the limits of the tanks' design capability. As shown, the end wall has delaminated between some of the internal chambers , but pressure loss has not occurred. FIG. 9 FIG. 4 404 406 408 400 906 404 406 916 400 904 404 408 914 400 914 916 400 914 916 400 914 916 402 902 908 910 912 406 408 914 916 400 918 404 404 918 404 is a top plan view diagrammatic view illustrating an exemplary arrangement of chambers into two exemplary first and second vessels and , according to the exemplary tank embodiment of . Fluid inlet lines feed fluid to the chambers (one of sixty-four labeled) of first vessel from an inlet conduit that extends outside of the tank . Fluid inlet lines feed fluid to the chambers (one of sixty-four labeled) of second vessel from an inlet conduit that extends outside of the tank . In a particular preferred embodiment, fluid inlet conduits and may also be used as outlet conduits in an application that first pressurizes the tank with fluid through the inlet conduits and and then releases pressurized fluid out of the tank through conduits and . Frame includes alignment pin apertures and , as well as first and second mounting apertures and . In a preferred embodiment, each vessel and additionally has its own fluid outlet (not shown, but similar to inlets and ). The design enables realization of a complete tank with automatic interconnects (one of ten diagonals labeled) between internal chambers to allow for fluid connectivity to each of the internal chamber volumes. Interconnects have a lesser depth than the depth of internal chamber . FIG. 10 FIG. 9 FIG. 4 FIG. 11 FIG. 12 1000 1002 1002 1004 1002 1010 1002 1000 400 1006 1008 1000 1012 1012 is a cut-away perspective view illustrating an exemplary annular small volume, high tank factor, tank , according to another embodiment of the present invention. Each arcuate chamber (one of many labeled) is fluidically connected to each other arcuate chamber via fluid conduits (not shown, but see for example). The outer end wall seals the top layer of arcuate chambers in a three-dimensional array of arcuate chambers . Tank has first and second vessels (not visible in this view), as with the embodiment of , and has first and second fluid inlets and for first and second vessels, respectively. The tank is formed in a disk-like flat shape that may adaptively and/or conformally shaped to be easily integrated with other devices by attachment or otherwise. and show applications in small satellite and rocket propulsion systems, respectively. The opening is shaped adaptively to a particular application and so may be conformal to a mechanical device to which it will be attached or may provide access for any pipes, regulators, valves, or other structures that may pass through opening in the particular application. FIG. 11 1100 1000 1100 1102 1104 1102 1104 1000 1108 1106 1000 1110 1100 1012 1008 1102 1104 1000 600 1000 1000 is a cross-sectional perspective view illustrating a first exemplary rocket propulsion system for a small satellite using the exemplary annular tank , according to an embodiment of the present invention. An advantage of the inventive method is the ability to produce an external shape that can be conformal and/or adaptive with an application. The rocket propulsion system includes first and second fuel tanks and . In a particular embodiment, first and second fuel tanks and may each hold a propellant, such as monopropellant hydrazine. Annular tanks may hold a pressurant gas, such as nitrogen, to provide pressure to the hydrazine to move the hydrazine through regulator to one or more thrusters (one of four labeled). The radially exterior outer wall of tank is shaped conformally to a housing for the rocket propulsion system to make efficient use space and its inner opening is shaped adaptively to the space requirements of the regulator . In another exemplary embodiment, first fuel tank may hold a bi-propellant, such as monomethylhydrazine, and second fuel tank may hold an oxidizer, such as nitrogen tetroxide, each separately pressurized using pressurant gases from annular tanks . Those of skill in the art, enlightened by the present disclosure, will appreciate the many variations of rocket engine systems that may be advantageously created using small tanks and with high tank factors, including the use of small tanks to hold propellant, including cold gas propellant. FIG. 12 1200 1214 1215 1216 1217 1214 1215 1216 1217 1202 1204 1206 1210 1208 1202 1202 1208 1212 1200 1214 1215 1216 1217 1214 1215 1216 1217 1204 1202 1200 is a perspective view illustrating a second exemplary rocket propulsion system using exemplary semi-annular tanks , , , and , according to an embodiment of the present invention. Four semi-annular tanks , , , and equatorially surround spherical monopropellant tank and are supported by frame . Pressurant valve supplies pressurant gas over line to pressurant intake valve of monopropellant tank . The pressurant gas entering monopropellant tank through pressurant intake valve forces the monopropellant into thruster and valve assembly to provide thrust for the rocket propulsion system . In various additional embodiments, the mounting of the semi-annular tanks , , , and may be non-equatorial. The radially outer wall of tanks , , , and are shaped adaptively to the frame and the curvature of the inner walls is shaped conformally to spherical monopropellant tank . Rocket propulsion system is exemplary of the broad variation in possible shapes for tanks of the present invention. FIG. 13 FIG. 16 1306 1308 1310 1310 1300 1300 1306 1308 1310 1310 1302 1304 1306 1310 1313 1308 1310 1314 1308 1312 1306 1308 1310 1312 1004 1616 1610 1620 is a perspective cut-away view of a first additional exemplary embodiment of arranging exemplary tank chambers , , , and into vessel , according to an embodiment of the present invention. Vessel is preferably a corner portion of a larger vessel (not shown). Considerable variation in the shapes and wall thicknesses of tank chambers , , , and is within the scope of the present invention. The minimum wall thickness consistent with required tank strength is preferred and is found using a CAD system or structural analysis. In the illustrated embodiment, only wall has a thickness of 0.016 inches, while other walls, such as wall , have a thickness of 0.020 inches. Chamber is a tank interior chamber, chamber is a tank corner chamber, and chambers and are tank edge chambers. Tank corner chamber has an arcuate substitute for its two outer walls, having an arcuate surface both internally and externally. Edge chambers and each have one arcuate wall. The overall strategy is to provide square interior chambers and exterior chambers , , and with arcuate outer walls. The apparatus reflects the method's ability to realize a very wide variety of internal and external shapes and geometrical flexibility in the plane (using CAD to convert the designs into artwork for etching of the metal layers, such as , , , and shown in ). FIG. 14 1406 1408 1410 1412 1414 1416 1400 1400 1406 1408 1412 1418 1410 1416 1420 1424 1414 1422 1402 1404 1406 1408 1410 1412 1414 1416 1420 1418 1422 1424 1400 is a perspective cut-away view of a second alternate exemplary embodiment of arranging exemplary tank chambers , , , , , and into a vessel , according to an embodiment of the present invention. Vessel is preferably a corner portion of a larger vessel (not shown). Internal tank chambers and , illustrated in a cut-away view, are hexagonal in cross section, as shown. Corner tank chamber has four of its six hexagonal sides merged into an arcuate wall , as shown. A first type of tank edge chamber and have two of their outer walls merged into an arcuate outer wall and , as shown. A second type of edge tank chamber has one arcuate outer wall , as shown. The minimum wall thickness consistent with required tank strength is preferred and is found using a CAD system. In the illustrated embodiment, only wall has a thickness of 0.008 inches, while other walls range in thickness up to a thickness of 0.022 inches, such as wall . The overall strategy is to provide hexagonal interior chambers and and hexagonal exterior chambers , , , and with arcuate outer walls , , , and , respectively. An advantage of the inventive method is the ability to produce an external shape that can be conformal with an application. Another advantage of the method used to make vessel is the ability to make external shapes that are not necessarily spherical or cylindrical, thereby allowing for more efficient usage of available space and the ability to make tanks that are conformal to the devices that use the tanks. FIG. 15 1506 1508 1510 1512 1500 1506 1510 1512 1508 1514 1518 1516 1502 1504 1506 1508 1510 1514 1518 1512 1516 is a perspective cut-away view of a third alternate exemplary embodiment of arranging square tank chambers , , , and into a vessel, according to an embodiment of the present invention. Vessel is preferably a corner portion of a larger vessel (not shown). Internal tank chamber has a square cross section. Corner chamber has an arcuate substitute for two of its walls, providing both an arcuate interior surface and an arcuate exterior surface. Edge chambers and each have a an arcuate outer wall and , respectively, as shown. The minimum wall thickness consistent with required tank strength is preferred and is found using a CAD system. In the illustrated embodiment, only wall has a thickness of 0.0075 inches, while other walls range in thickness up to a thickness of 0.020 inches, such as wall . The overall strategy is to provide square interior chambers and also to provide exterior chambers , , and with arcuate outer walls , and , as shown. FIG. 16 FIG. 10 1000 1000 1002 1004 1602 1002 1601 1604 1610 1608 1004 1616 1608 1604 1004 1610 1620 1612 1004 1620 1610 1622 1601 1622 1610 1004 1622 1004 1608 1610 1608 1624 1608 1610 1004 1620 1622 1624 1608 is a composite of perspective, cut-away, and diagrammatic views illustrating exemplary inner details of a first annular tank , according to the embodiment of . Annular tank is shown in a cut-away perspective view and defines radial section AA′. Arcuate chambers, such as chamber (one of many labeled), are stacked radially and axially in a two-vessel configuration (not shown). The top foil layer seals the top layer of arcuate chambers . The radial cross sectional array illustrates top edge chamber , with a floor , a side wall and a top layer . With ten foil layers (one of ten labeled) per side of chamber , plus top layer , floor , and bottom layers, a stack of one hundred foil layers that are bonded together is shown. The top layer , bottom , and floor layers have filets (one of one thousand and eight in cross section labeled) to avoid a destructive concentration of forces at the corners. Filets are formed by etching a sixteen mil foil layer down to floor thickness and a twelve mil layer down to outside wall thickness, for example. Filets are used at all corners where chamber walls , , , and back and front chamber walls (not shown, but same as ) meet. The seams (one labeled) between the side and the floor or top layer or bottom are outside of the filet , so any stress at the chamber corners is engaged by solid material and not by a seam . Side walls are thinner than can be achieved by other production methods, due to minimum gauge limitations. FIG. 17 FIG. 16 1700 1708 1704 1716 1706 1710 1720 1716 1712 1714 1722 1701 1724 1722 1722 1722 1706 1708 1710 1708 1702 1604 is a composite of perspective, cut-away, and diagrammatic views illustrating exemplary inner details of a second annular tank , according to an embodiment of the present invention. Chamber walls of chambers (one of sixty-five labeled) each have twenty foil layers (one of twenty labeled). Top layer , floor layers (one of four labeled) and bottom layer , together with the wall layers forma stack of one hundred and six foil layers. Enlarged portion more clearly illustrates the use of filets (one of two hundred and sixty in cross section BB′ shown as array ) to resist stress concentrations at the corners. Seams are preferably outside the filet . Filets are formed by etching a sixteen mil foil layer down to floor thickness, for example. Filets are used at all corners where chamber walls , , , and back and front internal chamber walls (not shown, but same as ) meet. Actual chambers are shorter along their arcuate length than the chambers of the embodiment of , as shown. FIG. 18 1802 1800 1802 1808 1804 1800 1806 1802 1810 1800 1806 is a diagram illustrating exemplary domed end wall portions for the end wall of a tank, according to an embodiment of the present invention. Domes terminate chambers while flat portions of the end wall rest on inner chamber walls . The domed portions , which may be regarded as double filets, avoid stress concentrations at the seam between the end wall and the chamber walls . FIG. 19 FIG. 3 1900 is a chart illustrating the comparative performance of the present invention and prior art, according to all embodiments of the present invention. The present invention creates tanks in a region bounded by the storage volume range of one-to-ten cubic inches that have tank factors in the neighborhood of eight-thousand meters, depending on the particular embodiment. None of the prior art (see also ), can match this performance. Accordingly, the present invention is novel. FIG. 20A FIG. 20B FIGS. 20A-20D 2000 2008 2009 2007 2004 2005 2006 2009 2004 2005 2007 2010 is a diagrammatic illustration of a first exemplary step in the process of making an exemplary device using stacked (see ) etched foil layers (one of six labeled), according to an embodiment of the present invention. Each metal foil sheet is etched with patterns and , for example, and cut along demarcation lines into smaller sheets . The exemplary patterns and are determined by slicing a 3D CAD model of the device into slices having the same thickness as the metal foil sheet . The device illustrated in is a small thruster , but the technique is broadly applicable to the small tank factor tanks of the present invention as well. FIG. 20B 2001 2010 2008 2009 2009 2008 2009 2010 2004 2005 is a diagrammatic illustration of a second exemplary step in the process of making an exemplary device using stacked etched foil layers , according to an embodiment of the present invention. The layers are stacked in an aligned configuration, with approximately four hundred layers per device . Considerable complexity in the patterns, such as patterns and , is possible with the present method. The illustrated patterns are not intended to be limiting. FIG. 20C 2002 2010 2008 2009 2002 2014 2008 2016 2012 2008 2010 2014 is a diagrammatic illustration of a third exemplary step in the process of making an exemplary device using stacked etched foil layers , according to an embodiment of the present invention. In the third exemplary step , the entire stack , of which stack is a part is subjected to pressure in a mechanical press , as well as heat sufficient to bond the metal foil layers together. The device has taken form within the entire stack . FIG. 20D 2008 2009 2010 2018 2020 2014 2010 2010 2003 is a diagrammatic illustration of a fourth exemplary step in the process of making an exemplary device using stacked etched foil layers , according to an embodiment of the present invention. Internal surfaces of the device may be machined smooth using a finishing tool intruded into the entire stack against the interior surfaces of the device . The exterior of the device may be trimmed by cutting and finished with grinders and polishers. External flanges, features, and couplings, if desired, may be formed in the trimming and finishing portion of step . FIG. 19 1608 1708 1. Presence of internal walls, exemplified as walls and , to provide structural integrity and strength while reducing overall weight and external wall thickness; 600 610 1000 1700 918 404 1002 1704 404 1002 1704 2. Realization of a complete tank , , , or with automatic interconnects between internal chambers , , or to allow for fluid connectivity to each of the internal chamber , , or volumes; 1304 1420 1518 1614 1714 3. Ability to realize wall thicknesses, such as for walls , , , , and , that are much smaller than those allowable by minimum gage limitations; 1300 1400 1500 2009 4. Ability to realize a very wide variety of internal shapes (, , and ) and geometrical flexibility in the plane (using CAD to convert the designs into artwork for etching of the metal layers ); 600 610 1000 5. Ability to realize an external shape , , that can be conformal with an application; 600 610 1000 1700 6. Ability to make external shapes , , , and that are not necessarily spherical or cylindrical, thereby allowing for more efficient usage of available space; 504 602 606 7. Ability to realize flat end walls , , and (uncommon in pressure vessels) without sacrificing tank factor and performance; 1622 1722 1000 1700 8. Placement of end wall fillets and in the small-scale tanks and to remove stress concentrations and improve performance; 1000 1700 1108 1000 9. Provision for annular and and other shapes so as to allow for plumbing channels and other structure through the tank (in the middle hole or elsewhere); and 1802 10. Use of scalloped or domed end walls to further reduce the size and thickness of the external walls for a given level of pressure. The present invention overcomes the limitation of low tank factors in the small size-class by realizing highly-efficient and light-weight tanks for high-pressure storage of liquids and gases in small storage volumes. As shown in , use of the present invention to realize such small-scale tanks allows for tank factors nearing 8,000 meters in storage volumes as low as 1-10 cubic inches. Other key unique features of the present invention include: While at least one exemplary embodiment has been presented in the foregoing detailed description, it should be appreciated that a vast number of variations exist. It should also be appreciated that the exemplary embodiment or exemplary embodiments are only examples, and are not intended to limit the scope, applicability, or configuration of the invention in any way. Rather, the foregoing detailed description and following claims will provide those skilled in the art with a convenient road map for implementing the exemplary and additional embodiments. It should be understood that various changes can be made in the function and arrangement of elements without departing from the scope of the invention. BRIEF DESCRIPTION OF THE DRAWINGS The present invention will hereinafter be described in conjunction with the following drawing figures, wherein like numerals denote like elements, and FIG. 1 is a front perspective view illustrating a prior art tank in the 100-10,000 cubic inches volume range; FIG. 2 is a front perspective view illustrating a plurality of prior art tanks in the 100-10,000 cubic inches volume range; FIG. 3 is a chart illustrating tank factor vs. storage volume for prior art tanks; FIG. 4 is a perspective view illustrating an exemplary embodiment of the foil-layer stack and internal tank structure for a small volume, high tank factor, tank, according to an embodiment of the present invention; FIG. 5 is a perspective view illustrating another exemplary embodiment of the internal tank structure with an end wall being added to a stack for a small volume, high tank factor, tank, according to an embodiment of the present invention; FIG. 6 is a perspective view illustrating two additional embodiments of walled tank structures trimmed via electrical discharge machining (EDM) with respective trimmed external material for a small volume, high tank factor, tank, according to an embodiment of the present invention; FIG. 7 is a perspective view illustrating another exemplary embodiment of the internal tank structure with end walls for a small volume, high tank factor, tank under hydrostatic testing, according to an embodiment of the present invention; FIG. 8 is a perspective view illustrating another exemplary embodiment of the internal tank structure with end walls for a small volume, high tank factor, tank under hydrostatic testing, according to an embodiment of the present invention; FIG. 9 FIG. 4 is a top plan view diagrammatic view illustrating an exemplary arrangement of chambers into two exemplary vessels, according to the exemplary embodiment of ; FIG. 10 is a cut-away perspective view illustrating an exemplary annular small volume, high tank factor, tank, according to another embodiment of the present invention; FIG. 11 is a cross-sectional perspective view illustrating a first exemplary application of the exemplary annular tank, according to an embodiment of the present invention; FIG. 12 is a perspective view illustrating a second exemplary rocket propulsion system using exemplary semi-annular tanks, according to an embodiment of the present invention; FIG. 13 is a perspective cut-away view of a first alternate exemplary embodiment of arranging chambers into chambers, according to an embodiment of the present invention; FIG. 14 is a perspective cut-away view of a second alternate exemplary embodiment of arranging chambers into chambers, according to an embodiment of the present invention; FIG. 15 is a perspective cut-away view of a third alternate exemplary embodiment of arranging chambers into chambers, according to an embodiment of the present invention; FIG. 16 FIG. 10 is a composite of a perspective, cut-away, and diagrammatic views illustrating exemplary inner details of a first annular tank, according to the embodiment of ; FIG. 17 is a composite of a perspective, cut-away, and diagrammatic views illustrating exemplary inner details of a second annular tank, according to an embodiment of the present invention; FIG. 18 is a cross-sectional diagrammatic view illustrating exemplary domed end wall portions for the end walls of a tank, according to an embodiment of the present invention; FIG. 19 is a chart illustrating the comparative performance of the present invention and prior art, according to embodiments of the present invention; FIG. 20A is a diagrammatic illustration of a first exemplary step in the process of making an exemplary device using stacked etched foil layers, according to an embodiment of the present invention; FIG. 20B is a diagrammatic illustration of a second exemplary step in the process of making an exemplary device using stacked etched foil layers, according to an embodiment of the present invention; FIG. 20C is a diagrammatic illustration of a third exemplary step in the process of making an exemplary device using stacked etched foil layers, according to an embodiment of the present invention; and FIG. 20D is a diagrammatic illustration of a fourth exemplary step in the process of making an exemplary device using stacked etched foil layers, according to an embodiment of the present invention.
Create an Account - Increase your productivity, customize your experience, and engage in information you care about. PHOTO #1: From left, Victoria Fire Department Chief Tracy Fox poses for a photo with Firefighter/EMTs John Rudd, Austin Muñoz and Enrique Cavazos. The new firefighter/EMTs were sworn in during a Nov. 5 ceremony at the Community Center annex. They will now... PHOTO: A cardiac monitor and electrocardiogram (EKG) defibrillator are displayed at Victoria Fire Department Station Two, 2708 Miori Lane. This equipment is used by firefighter/EMTs to provide first aid to patients with heart attack symptoms. The City of V... Victoria Fire Department employees eat breakfast while Fire Chief Tracy Fox, back left, and Battalion Chief Tim Hunter, back right, share statistics about the Victoria Fire Department’s treatment of cardiac arrest patients during an EMS Life Saving Awards... PHOTO: DeTar Hospital Navarro provided complimentary donuts to EMS workers May 17 in celebration of National EMS Week. Citizens Medical Center and DeTar Healthcare System will provide meals and snacks for EMS workers throughout the week. Shown from left a... PHOTO: Team leader Brandy Marek with the Victoria Fire Department administers a dose of the COVID-19 vaccine Feb. 5 to a Victoria Meals on Wheels client. The Victoria Fire Department is no longer accepting registrations for its COVID-19 vaccination program... The Victoria Fire Department swore in two new employees, honored five recently promoted fire engineers and recognized an employee who completed her probationary period during a ceremony at the Victoria Community Center annex Friday. Shown from left are ne... PHOTO #1: Members of the Victoria Fire Department and volunteer registered nurse Megan Garcia prepare doses of the COVID-19 vaccine Friday for distribution to Victoria Meals on Wheels clients. The fire department partnered with Meals on Wheels to deliver ... PHOTO: Victoria Fire Department Emergency Medical Services carries equipment, shown here, to provide blood transfusions to critically ill or injured patients. Today marks one year since the Victoria Fire Department began providing whole blood transfusions ... The Victoria Fire Department is partnering with local health care providers and nonprofits to provide a no-cost flu shot clinic for homeless and low-income residents 11 a.m.-1 p.m. Oct. 26-28 at Queen City Park. The clinic will be funded by Community Devel... Victoria Fire Department employees are producing a collection of videos to teach the public about fire safety, and residents will have the opportunity to vote for their favorites in honor of Fire Prevention Week. “Normally at this time of year, we host our... Victoria Fire Department fire medic Luis Malone-Ordaz was named Fire Fighter of the Year by the Victoria Northside Rotary Club. Malone-Ordaz has been with the Victoria Fire Department since May 2015. Before that, he worked for Gemini Ambulance Service in ... PHOTO #1: Members of the Victoria Fire Department pose for a photo at the San Antonio 110 9/11 Memorial Climb on Sept. 11, 2019. This year’s event will be hosted remotely at multiple locations, including Memorial Stadium in Victoria. PHOTO #2: Co... From top left, firefighter/paramedic Leanna Richter, firefighter/EMT Madelyn Bishop, firefighter/EMT Rebekah Thompson, firefighter/EMT Ethan Strzelcyk, firefighter/EMT Angelo Stafford, firefighter/EMT Wyatt Andrews, firefighter/EMT Jalen Rangel and inspec... City of Victoria Fire Department engineers Joel Gomez and Raul Liendo, who will both retire this month, have been with the department long enough to remember when Fire and EMS were separate operations. Gomez and Liendo both started as members of Victoria E... PHOTO #1: Firefighter paramedic Pamela Yanta with the Victoria Fire Department prepares May 21 to help test nursing home residents for COVID-19. In order to comply with Governor Greg Abbott’s May 11 order that all nursing home residents and staff be teste... The City of Victoria’s Fire Department has earned the American Heart Association’s 2020 Mission: Lifeline EMS Gold Plus recognition for its efforts to improve care for patients suffering from an ST-elevation myocardial infarction (STEMI), a serious type o...	https://www.victoriatx.gov/CivicAlerts.aspx?CID=13
The most common problem when being faced with overwinding of a clock is that the lubricant or oil inside the mechanism has worn off due to time and the inside workings have began to rub against each other, causing friction and damages inside. Can you Overwind a grandfather clock? Luckily, the truth is that you can’t over-wind your clock. Over-winding is basically a myth! Let’s take a look at what actually causes a clock to quit running or chiming after winding it up… A clock mainspring is made of spring steel and is about the width of a ruler (but not quite as thick). Can you wind a grandfather clock too tight? WHAT ABOUT THE SAYING THAT “THE CLOCK IS WOUND TOO TIGHT”? You can only wind a spring drive clock so far and then either the spring will break, or the key will break, or one of the gears will be ripped loose. How do you release an overwound grandfather clock? A clock can be overwound, resulting in the weights being stuck in the highest position. When this happens, gently move the minute hand past the quarter hour and listen for the chime. This should release the weight. When winding, be sure you can see the top of the weight and then stop to avoid overwinding. What happens if a clock is wound too tight? Some clocks will run faster if they are wound too tightly in the first 24 hours after you’ve tightened the mainspring. Ideally, if you stop just short of the mainspring’s full potential, your clock will run just fine for a full week without having to get any attention from you. How do I fix an overwound clock? To rid the timepiece of lint & dust, you can spray it with compressed air after the initial cleaning. You can also apply small amounts of clock cleaning fluid, which should be available to find online with a bristle brush if required, however most find that warm water may tackle the job just as well. Is it OK to turn a clock backwards? If your timepiece has a mechanical movement, then it is best not to put the clock in reverse. If your timepiece is a quartz movement, you are probably safe to turn it backward. If you have a very old cuckoo clock, it may not even go in reverse without breaking the mechanism. Should the weights on a grandfather clock drop evenly? Yes, the order is critical for proper function. On 99% of floor clocks (grandfather/grandmother) with three weights, the heavier weight goes on the right-hand chain hook or pulley as you are FACING the clock. How do you fix a grandfather clock movement? The tips below can help you fix the faulty movement of your grandfather clock. - Step 1 – Weight. Weights control the grandfather clock movements. … - Step 2 – Checking the Weight. Examine and see if the weights are aligned properly. … - Step 3 – The Hands. … - Step 4 – The Pendulum. … - Step 5 – The Moon Dial. … - Step 6 – Check the Level. How do you wind a 30 day clock? Try to wind the clock as close to the time when it stopped as possible. Move the long minute hand on the clock face clockwise to set the current time. The hour hand should travel along with the minute hand. Wait for the clock to strike each hour if you have to move the time up several hours. How much does it cost to repair a grandfather clock? How much does clock repair cost? \|Type of Repair\|\|Average Cost\| \|Grandfather clock setup\|\|$60 to $150\| \|House call\|\|$55 to $100 within 25 miles ($1-$2 per additional mile)\| \|Spring replacement\|\|$10 to $30 per spring\| \|Repairing broken teeth\|\|$10 to $30 per tooth\| How do you wind an old mantle clock? The most effective way to wind your clock would be to open the front door, insert the crank, hold the clock steady with your left hand, and turn the crank with your right hand. After winding the clock, set the correct time by moving the minute hand either clockwise or counterclockwise. How often should you wind a grandfather clock? Wind weekly, or as necessary. Almost all grandfather clocks are made to run for seven or eight days without winding, so winding them on the same day each week will ensure that it never stops.	https://thewatchesmen.com/about-time/what-happens-if-you-overwind-a-grandfather-clock.html
Q: If $f(0) = 1$ and $f'(0) = -1$ and $f(x) > 0$ always, what can be said about $f''(x)$ Suppose we have a function $f: [0,\infty) \to R$ so that $f(0) = 1$ and $f'(0) = -1$ and $f(x) > 0$ always, what can be said about $f''(x)$ ? I think we can't say much about $f''$ because there is no restriction but my answer book says $f''(x) < 0$ for all $x$. I think this is not possible at all, as it would mean $f(x) = 0$ for some $x \in (0,1)$. Help me. Also I tried considering $f(x) = \exp(-x)$ A: You can pretty much fix any value you want for $f''(0)$. For instance $g(x)=1-x+\frac a2x^2$ verifies $\begin{cases}g(0)=1\\g'(0)=-1\\g''(0)=a\end{cases}\quad$ for any $a\in\mathbb R$. By continuity the condition $g(x)>0$ is true in a neighbourhood of $0$, but since $g$ is polynomial it grows to infinity at infinity, we have to make it quickly decreasing in order to build a suitable $f$. This is possible by multiplying $g$ by $\exp(-\alpha x^4)$ (I choose $x^4$ in order to have no problem with negative numbers, and also such that the second derivative in not impacted by the value of $\alpha$). Finding a good $\alpha(a)$ so that $f(x)\neq 0$ can be done empirically, but anyway the exponential will always win (just make the coefficient bigger). For instance $f(x)=(1-x+\frac a2x^2)\exp\left(-\max(1000,a^4)x^4\right)$ is working well. And $f^{(i)}(0)=g^{(i)}(0)$ for $i=0,1,2$ along with $f(x)>0$ for all $x\in\mathbb R$. And this construction can be extended to fix any $n$-th derivative in $0$, provided you take care of the growth of the function far away of $0$ by multiplying by a suitable quickly decreasing exponential. A: You're right. They must have omitted some important hypothesis or who knows what. Note also that the answer is not $f''(x)>0$ either. While that could be the case (as in the example you gave), you could have a change of convexity/concavity. For instance, consider $f(x)=\frac1{1+(1+2x)^2}$. One could add some trigonometric functions to the mix and obtain a function which changes the sign of $f''$ indefinitely. You can say, nevertheless, that is not true that $f''(x)<0$ always. ;)
Written in a clear, non-technical manner, Introduction to Video Production focuses on the fundamental principles of video production and the technologies used in production. This book discusses video aesthetics, technologies, and production practice in a clear and concise manner. It also emphasizes the importance of teamwork and planning in the production process. Chapters are clearly organized and heavily illustrated, with key terms identified in boldface. With Introduction to Video Production, readers will learn not only how the technology works, but how to work with the technology and with each other. Table of contents - Cover - Title Page - Copyright Page - Table of Contents - Preface - Chapter 1 Introduction - Chapter 2 Video Production Environments - Chapter 3 Lighting - Chapter 4 The Video Camera - Camera Equipment Quality and Target Markets - Camera Configurations - Parts of a Video Camera - How the Video Camera Makes a Picture: Analog and Digital Standards - Camera Operational Controls - Camera Supports - The Zoom Lens - Aperture Control and Depth of Field - Camera Movement - Operating the Video Camera - Video Connectors - Chapter 5 Audio and Sound Control - Chapter 6 The Video Switcher - Chapter 7 The Role of the Producer - Chapter 8 Directing: Part 1 - Chapter 9 Directing: Part 2 - Chapter 10 Video Recording and Editing - Chapter 11 Video Editing Techniques - Chapter 12 Graphics and Set Design - Chapter 13 Video on the Web - Chapter 14 Anatomy of Production: Two Case Studies - Appendix A Brief History of Audio Recording Media Technologies - Glossary - Bibliography - Index Product information - Title: Introduction to Video Production - Author(s):	https://www.oreilly.com/library/view/introduction-to-video/9781317347149/
Mingjun Teng, Lixiong Zeng, Wenjie Hu, Pengcheng Wang, Zhaogui Yan, Wei He, Yu Zhang, Zhilin Huang, Wenfa Xiao, The impacts of climate changes and human activities on net primary productivity vary across an ecotone zone in Northwest China, Science of The Total Environment, 10.1016/j.scitotenv.2020.136691, 714, (136691), (2020). The Effects of Human Activities on Environment – Explained! Some important impacts of human activities on environment are outlined below: 1. Deforestation: In order to provide timber and farm land to increased population, large number of forest trees are cut and forest area is converting to farm lands. The rate of deforestation is so faster that around 1.5 million hectare of forest cover is lost every Impacts on Mangroves – South Florida Aquatic Environments Human activities commonly affect the distri-bution, quantity, and chemical quality of water resources. The range in human activities that affect the interaction of ground water and surface water is broad. The following discussion does not provide an exhaustive survey of all human effects but emphasizes those that are relatively widespread. To (PDF) Impacts of human induced activities on species Human impacts can however be both positive and negative. Whilst in previous times, human activity has caused different types of problems for the mangrove ecosystem, the council and public nowadays have strived to implement strategies and regulations that need to be followed so that it benefits the environment in as many ways as possible. Human activities that threaten biodiversity - YouTube (PDF) Negative impacts of man-made activities on water 6/26/2020 · Teaching about the human impacts on climate is supported by five key concepts: Teaching this principle is supported by five key concepts: a. The overwhelming consensus of scientific studies on climate indicates that most of the observed increase in global average temperatures since the latter part of the 20th century is very likely due to human activities, primarily from increases in Impact Of Human Activities On Environment Essay Pdf A final way that humans ca have a positive impact on the desert biome is by gaining knowledge about the biome. If a human has more information about how gentle this ecosystem is and how their actions may affect this desert biome and all of its intricate interactions, then they are less likely to … 11 Ways Humans Impact the Environment The impacts of climate change and human activities on Human activities affect the environment by contributing to air pollution, or the emission of harmful substances into the air. While it can be difficult to understand which pollutants are associated with specific effects on the environment or public health, it is generally accepted that air pollution can indeed cause public health problems and Essay on the Impact of Human Activities on Environment 4/8/2020 · Human activities have impacted the environment more than any other species, including deforestation, natural resource depletion, reduced biodiversity, and pollution of the air, land and water. How Do Humans Affect the Environment ( 6 + Negative Ways) Humans impact the physical environment in many ways: overpopulation, pollution, burning fossil fuels, and deforestation. Changes like these have triggered climate change, soil erosion, poor air quality, and undrinkable water. These negative impacts can affect human behavior and can prompt mass migrations or battles over clean water. Help your students understand the impact humans have on the Human activities are impacting the climate system. - CLEAN Impact of human activities on the hydrosphere. The activities of modern society are having a severe impact on the hydrologic cycle.The dynamic steady state is being disturbed by the discharge of toxic chemicals, radioactive substances, and other industrial wastes and by the seepage of mineral fertilizers, herbicides, and pesticides into surface and subsurface aquatic systems. What is the impact of human activities on natural Tundra - Tundra - Effects of human activities and climate change: Earth’s tundra regions are harsh and remote, so fewer humans have settled there than in other environments. However, humans have a long history in the tundra. For example, the first people who went to North America from Asia more than 20,000 years ago traveled through vast tundra settings on both continents. Human impacts on plants and animals - SlideShare Impacts of human induced activities on species composition and diversity in Miombo woodlands of Bagamoyo district, Tanzania The Effects of Human Activity on Coastal Landforms \| Bartleby Such activities, when carried out without planning, may generate significant impacts in the structuring and functioning of global freshwater ecosystems (Steffen et al., 2015;Isbell et al., 2017). Human Impacts on Marine Species \| National Geographic Society Human Impacts on Marine Species Students learn about three examples of human impacts on marine life: migration patterns and shipping, algal blooms and water chemistry, and marine debris. Some of these impacts are due to human activity in the ocean, and some impacts on the ocean are due to human … Breeding biology of Southern Ground Hornbill Bucorvus Positive: 1. Human Activity in the Rainforest Biome has allowed for the discovery and implementation of many different medicines, nearly 120 prescription medications today are derived from plants in the tropical rainforest, nearly ⅔ of these drugs are known to have cancer fighting properties, not only are the drugs used to combat cancer, they are key in fighting malaria, heart disease Human Activities Can Have an Impact on Natural Disasters 6/18/2011 · Human activities affect both water qu ality and quantity. It leads to a change of land use and land cover, which changes the water balance and usually changes the EFFECTS OF HUMAN ACTIVITIES ON THE INTERACTION OF Human Impact - Desert All of these activities take resources and habitats away from plants and animals. - NatureWorks. share this page . Home About Watch Nature Files Teachers Order DVD Contact : Natural and Human Impacts on Wildlife : The Human Touch Humans are now responsible for causing changes in the environment that hurt animals and plant species. The impacts of climate changes and human activities on net Identify five threats to ecosystems from human activities. Understand what factors affect climate change and what the current trajectory and implications of climate change are for the next century. Discuss the role of population, affluence, and technology in driving human impacts. Human impacts on marine ecosystems - GEOMAR - Helmholtz The Effects of Human Activity on Coastal Landforms Human activities add another layer of complexity to the natural processes of coastal lands and materials. These activities may have direct or indirect effects on our changing coastlines. They may effect sources of new sediment to the coast and the movement of sediment within the coastal Impact of Human Activities on the Nitrogen Cycle ADVERTISEMENTS: The following points highlight the five main impacts of agriculture on environment. The impacts are:- 1. Degradation of Land 2. Deforestation 3. Biodiversity 4. Pest Problem 5. Disposal of Industrial & Agricultural Wastes. Impact # 1. Degradation of Land: The degradation of land in one form or the other is matter of serious concern […] The Impact of Human Activities on the Environment Essay 5/15/2018 · Biogeochemical Cycles and Human Activities 1. Biogeochemical Cycles and Human Activities WATENTENA AMOS MAY, 2018 2. Objectives o Identify and describe the flow of nutrients in each biogeochemical cycle. o Explain the impact that humans have on the biogeochemical cycles.	http://ershary.gq/impacts-of-human-activities-155082.html
Christmas with the Rat Pack is a 2002 album compiling Christmas songs by Frank Sinatra, Dean Martin, and Sammy Davis Jr. Track listing. I've Got My Love to Keep Me Warm" – Dean Martin "Mistletoe and Holly" Frank Sinatra "Christmas Time All Over the World" – Sammy Davis Jr. "The First Noel" – Frank Sinatra Frank Sinatra Sings for Only the Lonely. ... A Jolly Christmas From Frank Sinatra. 1957. In the Wee Small Hours. 1955. Similar artists. Dean Martin. Nat King Cole ...	https://tszsy.l0l.in.net/solo-a-star-wars-story-online.html
--- author: - 'C'' esar Domínguez [^1]' - 'Dominique Duval [^2]' date: 'August 25., 2009' title: A parameterization process as a categorical construction --- - Abstract. The parameterization process used in the symbolic computation systems Kenzo and EAT is studied here as a general construction in a categorical framework. This parameterization process starts from a given specification and builds a parameterized specification by transforming some operations into parameterized operations, which depend on one additional variable called the parameter. Given a model of the parameterized specification, each interpretation of the parameter, called an argument, provides a model of the given specification. Moreover, under some relevant terminality assumption, this correspondence between the arguments and the models of the given specification is a bijection. It is proved in this paper that the parameterization process is provided by a free functor and the subsequent parameter passing process by a natural transformation. Various categorical notions are used, mainly adjoint functors, pushouts and lax colimits. Introduction ============ Kenzo [@Kenzo] and its predecessor EAT [@EAT] are software systems developed by F. Sergeraert. They are devoted to Symbolic Computation in Algebraic Topology. In particular, they carry out calculations of homology groups of complex topological spaces, namely iterated loop spaces. By means of EAT and Kenzo, some homology groups that had never been obtained with any other method, neither theoretical nor automatic, have been computed. In view of the obtained results, some years ago, the first author of this paper began the formal study of the programs, in order to reach a good understanding on the internal calculation processes of these software systems. In particular, our study of the data types used in EAT and Kenzo [@LPR03; @DLR07; @DRS06] shows that there are two different layers of data structures in the systems. In the first layer, one finds the usual abstract data types, like the type of integers. In the second layer, one deals with algebraic structures, like the structure of groups, which are implemented thanks to the abstract data types belonging to the first layer. In addition, we realized that in a system such as EAT, we do not simply implement one group, but more generally parameterized families of groups. In [@LPR03] an operation is defined, which is called the imp construction because of its role in the implementation process in the system EAT. Starting from a specification ${\Sigma}$ in which some operations are labelled as “pure” [@DRS06], the imp construction builds a new specification ${\Sigma}_A$ with a distinguished sort $A$ which is added to the domain of each non-pure operation. It follows that each implementation of ${\Sigma}_A$ defines a family of implementations of ${\Sigma}$ depending on the choice of a value in the interpretation of $A$. Besides, working with the imp construction in [@LPR03] we were able to prove that the implementations of EAT algebraic structures are as general as possible, in the sense that they are ingredients of terminal objects in certain categories of models; this result is called the exact parameterization property. Later on, led by this characterization of EAT algebraic structures, in [@LPR03] we reinterpreted our results in terms of object-oriented technologies like hidden algebras [@GM00] or coalgebras [@Ru00]. This paper deals with generalization by parameterization in the sense of Kenzo and EAT, so that our parameters are symbolic constants of a given type, that will be replaced by arguments which are elements in a given set. The notion of parameterization in programming and specification languages bears several meanings, where the parameter may be a type or a specification. For instance, in object-oriented programming, parametric polymorphism is called generic programming, in C++ it is characterized by the use of template parameters to represent abstract data types. On the other hand, in algebraic specifications, a parameterized specification is defined as a morphism of specifications where the parameter is the source and the parameter passing is defined as a pushout [@ADJ80]. The framework for this paper is provided by equational logic, considered from a categorical point of view. An equational theory, or simply a theory, is a category with chosen finite products. A model $M$ of a theory ${\Theta}$ is a functor $M\colon {\Theta}\to{\mathit{Set}}$ which maps the chosen products to cartesian products. A theory ${\Theta}$ can be presented by a specification ${\Sigma}$, this means that ${\Sigma}$ generates ${\Theta}$. In this paper, we are not interested in specifications for themselves, but as presentations of theories. So, specifications are used mainly in the examples, and we feel free to modify a specification whenever needed as long as the presented theory is not changed. The parameterization process studied in this paper is essentially the “imp construction” of [@LPR03]. Starting from a theory ${\Theta}$ it provides a parameterized theory ${\Theta}_A$ by adding a type of parameters $A$ and by transforming each term $f\colon X\to Y$ in ${\Theta}$ into a parameterized term $f'\colon A\times X\to Y$ in ${\Theta}_A$. Then clearly ${\Theta}_A$ generalizes ${\Theta}$: the models of ${\Theta}$ can be identified to the models of ${\Theta}_A$ which interpret the type of parameters $A$ as a singleton. There is another way to relate ${\Theta}$ and ${\Theta}_A$, called the parameter passing process, which runs as follows. By adding to ${\Theta}_A$ a constant $a$ (called the parameter) of type $A$ we get a theory with parameter ${\Theta}_a$, such that for each parameterized term $f'\colon A\times X\to Y$ in ${\Theta}_A$ there is a term $f'(a,-)\colon X\to Y$ in ${\Theta}_a$. Then the parameter passing morphism $j\colon {\Theta}\to {\Theta}_a$ maps each term $f\colon X\to Y$ in ${\Theta}$ to $f'(a,-)\colon X\to Y$ in ${\Theta}_a$. Given a model $M_A$ of ${\Theta}_A$ an argument $\alpha$ is an element of the set $M_A(A)$, it provides a model $M_{A,\alpha}$ of ${\Theta}_a$ which extends $M_A$ and satisfies $M_{A,\alpha}(a)=\alpha$. Thanks to the parameter passing morphism, the model $M_{A,\alpha}$ of ${\Theta}_a$ gives rise to a model $M$ of ${\Theta}$ such that $M(f)=M_A(f')(\alpha,-)$ for each term $f$ in ${\Theta}$. Moreover, under some relevant terminality assumption on $M_A$, this correspondence between the arguments $\alpha\in M_A(A)$ and the models of ${\Theta}$ is a bijection: this is the exact parameterization property. The parameterization process and its associated parameter passing process have been described for each given theory ${\Theta}$, but in fact they have the property of preserving the theory structure, which can be stated precisely in a categorical framework: this is the aim of this paper. The parameterization process is defined as a functor: the construction of the parameterized theory ${\Theta}_A$ from the given theory ${\Theta}$ is a functor left adjoint to the construction of a coKleisli category, and more precisely it is a free functor in the sense of section \[sec:free\]. The parameter passing process is defined as a natural transformation, along the following lines. First, the construction of the theory with parameter ${\Theta}_a$ from the parameterized theory ${\Theta}_A$ is simply a pushout construction, such that the construction of ${\Theta}_a$ from ${\Theta}$ is a functor. Then, each parameter passing morphism $j:{\Theta}\to{\Theta}_A$ is defined from a lax colimit of theories, in such a way that the parameter passing morphisms are (essentially) the components of a natural transformation from the identity functor to this functor. A first version of this approach can be found in [@DDLR05], it relies on diagrammatic logic [@Du03; @Du07]. In this paper, the explicit use of diagrammatic logic is postponed to the appendix. With respect to the previous papers like [@LPR03], we provide a new interpretation of the parameterization process and in addition an interpretation of the parameter passing process. Moreover, we take into account the fact that there is a pure part in the given theory, and we derive the exact parameterization property from a more general result which does not rely on the existence of a terminal model. In section \[sec:defi\] equational theories are defined and several examples are presented. The parameterization process and the parameter passing process are defined categorically in section \[sec:const\]. In section \[sec:free\] free functors are defined as left adjoint functors associated to morphisms of limit sketches, and it is proved that the parameterization functor is free. The diagrammatic point of view on equational logic is presented in appendix \[sec:dia\]. Most of the categorical notions used in this paper can be found in [@MacLane98] or in [@BW99]. We omit the size issues: for instance most colimits should be small. A graph always means a directed multigraph, and in order to distinguish between various kinds of structures with an underlying graph, we speak about the objects and morphisms of a category, the types and terms of a theory (or a specification) and the points and arrows of a limit sketch. Examples and definitions {#sec:defi} ======================== Equational theories and specifications {#subsec:equa} -------------------------------------- In this paper, equational logic is seen from a categorical point of view, as for instance in [@Pitts]. \[defi:equa-thry\] The category ${\mathbf{T}}_{{\mathit{eq}}}$ of equational theories is made of the categories with chosen finite products together with the functors which preserve the chosen finite products. In addition, ${\mathbf{T}}_{{\mathit{eq}}}$ can be seen as a 2-category with the natural transformations as 2-cells. Equational theories are called simply theories. For instance, the theory ${\mathit{Set}}$ is made of the category of sets with the cartesian products as chosen products. \[rema:equa-thry\] The correspondence between equational theories in the universal algebra style (as in [@LEW96]) and equational theories in the categorical style (as defined here) can be found in [@Pitts]. Basically, the sorts and products of sorts become objects, still called types, the operations and terms become morphisms, still called terms (the variables correspond to projections, as in example \[exam:sgp\]) and the equations become equalities: for instance a commutative square $g_1\circ f_1=g_2\circ f_2$ means that there is a term $h$ such that $g_1\circ f_1=h$ and $g_2\circ f_2=h$. However a more subtle point of view on equations is presented in appendix \[sec:dia\]. \[defi:equa-mod\] A (strict) model $M$ of a theory ${\Theta}$ is a morphism of theories $M\colon {\Theta}\to{\mathit{Set}}$ and a morphism $m\colon M\to M'$ of models of ${\Theta}$ is a natural transformation. This forms the category ${\mathit{Mod}}({\Theta})$ of models of ${\Theta}$. For every morphism of equational theories ${\theta}\colon {\Theta}_1\to{\Theta}$, we denote by ${\theta}^{}\colon {\mathit{Mod}}({\Theta})\to{\mathit{Mod}}({\Theta}_1)$ the functor which maps each model $M$ of ${\Theta}$ to the model ${\theta}^{}(M)=M\circ {\theta}$ of ${\Theta}_1$ and each morphism $m\colon M\to M'$ to $m\circ {\theta}$. In addition, for each model $M_1$ of ${\Theta}_1$, the category of models of ${\Theta}$ over $M_1$ is denoted ${\mathit{Mod}}({\Theta})\|_{M_1}$, it is the subcategory of ${\mathit{Mod}}({\Theta})$ made of the models $M$ such that ${\theta}^{}(M)=M_1$ and the morphisms $m$ such that ${\theta}^{}(m)={\mathit{id}}_{M_1}$. Whenever ${\theta}$ is surjective on types, the category ${\mathit{Mod}}({\Theta})\|_{M_1}$ is discrete. A theory ${\Theta}$ can be described by some presentation: a presentation of an equational theory ${\Theta}$ is an equational specification ${\Sigma}$ which generates ${\Theta}$; this is denoted ${\Theta}\dashv{\Sigma}$. Two specifications are called equivalent when they present the same theory. An equational specification can be defined either in the universal algebra style as a signature (made of sorts and operations) together with equational axioms, or equivalently, in a more categorical style, as a finite product sketch, see [@Lellahi89], [@BW99], and also section \[subsec:sketch\] and appendix \[subsec:dia-equ\]. The correspondence between the universal algebra and the categorical points of view runs as in remark \[rema:equa-thry\]. \[defi:equa-spec\] The category ${\mathbf{S}}_{{\mathit{eq}}}$ of equational specifications is the category of finite product sketches. With (generalized) natural transformations as 2-cells, ${\mathbf{S}}_{{\mathit{eq}}}$ can be seen as a 2-category. Equational specifications are called simply specifications. The category ${\mathbf{T}}_{{\mathit{eq}}}$ can be identified to a subcategory of ${\mathbf{S}}_{{\mathit{eq}}}$ (more precisely, to a reflective subcategory of ${\mathbf{S}}_{{\mathit{eq}}}$). When ${\Sigma}$ is a presentation of ${\Theta}$, a model of ${\Theta}$ is determined by its restriction to ${\Sigma}$, which is called a model of ${\Sigma}$, and in fact ${\mathit{Mod}}({\Theta})$ can be identified to the category ${\mathit{Mod}}({\Sigma})$ of models of ${\Sigma}$. $$\begin{array}{\|l\|l\|c\|} \hline & \textrm{subscript} \; {\mathtt{E}}& {\Sigma}_{{\mathtt{E}}} \\ \hline \hline \textrm{type (or sort)} & {\mathtt{Type}}& X \\ \hline \textrm{term (or operation)} & {\mathtt{Term}}& \xymatrix@C=3pc{X\ar[r]^{f} & Y} \\ \hline \textrm{selection of identity} & {\mathtt{Selid}}& \xymatrix@C=3pc{X\ar[r]^{{\mathit{id}}_X} & X} \\ \hline \textrm{composition} & {\mathtt{Comp}}& \xymatrix@C=3pc{X\ar[r]^{f} \ar@/_3ex/[rr]_{g\circ f}^{=} & Y\ar[r]^{g} & Z} \\ \hline \textrm{binary product} & {\mathtt{2\texttt{-}Prod}}& \xymatrix@C=3pc@R=.7pc{ X & \\ & X{\!\times\!}Y \ar[lu]_{p_X} \ar[ld]^{p_Y} \\ Y & \\ } \\ \hline \textrm{pairing (or binary tuple)} & {\mathtt{2\texttt{-}Tuple}}& \xymatrix@C=3pc@R=1pc{ & X & \\ Z \ar[ru]^{f} \ar[rd]_{g} \ar[rr]\|{\,{\langlef,g\rangle}\,} & \ar@{}[u]\|{=}\ar@{}[d]\|{=} & X{\!\times\!}Y \ar[lu]_{p_X} \ar[ld]^{p_Y} \\ & Y & \\ } \\ \hline \end{array}$$ We will repeatedly use the fact that ${\mathbf{T}}_{{\mathit{eq}}}$ and ${\mathbf{S}}_{{\mathit{eq}}}$, as well as other categories of theories and of specifications, have colimits, and that left adjoint functors preserve colimits. In addition every specification is the colimit of a diagram of elementary specifications. The elementary specifications are the specifications respectively made of: a type, a term, an identity term, a composed term, a $n$-ary product and a $n$-ary tuple, for all $n\geq0$, as in figure \[fig:elem\] (where only $n=2$ is represented). Let us consider a theory ${\Theta}$ presented by a specification ${\Sigma}$, then ${\Sigma}$ is the colimit of a diagram $\Delta$ of elementary specifications, and ${\Theta}$ is the colimit of the diagram of theories generated by $\Delta$. Examples {#subsec:exam} -------- \[exam:term\] Let us consider the theory ${\Theta}_{{\mathit{op}},0}$ presented by two types $X,Y$, and the three following theories extending ${\Theta}_{{\mathit{op}},0}$ (the subscript ${\mathit{op}}$ stands for “operation”, since ${\Theta}_{{\mathit{op}}}$ is presented by the elementary specification for terms or operations ${\Sigma}_{{\mathtt{Term}}}$). The unit type is denoted ${1}$ and the projections are not given any name. $$\begin{array}{c\|c\|c\|c\|} \cline{2-2} {\Theta}_{{\mathit{op}},A} \dashv & \xymatrix@R=1pc{A & A{\!\times\!}X\ar[l]\ar[d]\ar[rd]^{f'} & \\ & X & Y \\ } & \multicolumn{2}{c}{} \\ \cline{2-2} \cline{4-4} \multicolumn{2}{c}{} & \qquad {\Theta}_{{\mathit{op}},a} \dashv & \xymatrix@R=1pc{A & A{\!\times\!}X\ar[l]\ar[d]\ar[rd]^{f'} & \\ {1}\ar[u]^{a} &X & Y \\ } \\ \cline{2-2} \cline{4-4} {\Theta}_{{\mathit{op}}} \dashv & \xymatrix{ & X\ar[r]^{f} &Y \\ } & \multicolumn{2}{c}{} \\ \cline{2-2} \end{array}$$ These theories are related by various morphisms (all of them preserving ${\Theta}_{{\mathit{op}},0}$): ${\theta}_{{\mathit{op}},A}\colon {\Theta}_{{\mathit{op}},A}\to{\Theta}_{{\mathit{op}}}$ maps $A$ to ${1}$ and ${\theta}_{{\mathit{op}},a}\colon {\Theta}_{{\mathit{op}},a}\to{\Theta}_{{\mathit{op}}}$ extends ${\theta}_{{\mathit{op}},A}$ by mapping $a$ to ${\mathit{id}}_{{1}}$, while $j_{{\mathit{op}},A}\colon {\Theta}_{{\mathit{op}},A}\to{\Theta}_{{\mathit{op}},a}$ is the inclusion. In addition, here are two other presentations of the theory ${\Theta}_{{\mathit{op}},a}$ (the projections are omitted and $ {1}\times X$ is identified to $X$): $$\begin{array}{\|c\|} \cline{1-1} \xymatrix@R=1pc{A & A{\!\times\!}X\ar[rd]^{f'} & \\ {1}\ar[u]^{a} & X \ar@{}[ru]\|(.3){=} \ar[u]^{a{\!\times\!}{\mathit{id}}_X} \ar[r]_{f''} & Y \\ } \\ \cline{1-1} \end{array} \qquad\qquad \begin{array}{\|c\|} \cline{1-1} \xymatrix@R=1pc{A & A{\!\times\!}X\ar[r]^{f'} \ar@{}[rd]\|{=} &Y \\ {1}\ar[u]^{a} & X \ar[u]^{a{\!\times\!}{\mathit{id}}_X} \ar[r]_{f''} & Y \ar[u]_{{\mathit{id}}_Y} \\ } \\ \cline{1-1} \end{array}$$ It is clear from anyone of these new presentations of ${\Theta}_{{\mathit{op}},a}$ that there is a morphism $j_{{\mathit{op}}}\colon {\Theta}_{{\mathit{op}}}\to{\Theta}_{{\mathit{op}},a}$ which maps $f$ to $f''$. In addition, ${\theta}_{{\mathit{op}},a}\circ j_{{\mathit{op}},A} = {\theta}_{{\mathit{op}},A}$ and there is a natural transformation $t_{{\mathit{op}}} \colon j_{{\mathit{op}}} \circ {\theta}_{{\mathit{op}},A} {\Rightarrow}j_{{\mathit{op}},A}$ defined by $(t_{{\mathit{op}}})_X={\mathit{id}}_X$, $(t_{{\mathit{op}}})_Y={\mathit{id}}_Y$ and $(t_{{\mathit{op}}})_A=a\colon {1}\to A$. $$\xymatrix@R=.8pc{ {\Theta}_{{\mathit{op}},A} \ar[dd]_{{\theta}_{{\mathit{op}},A}} \\ \mbox{ } \\ {\Theta}_{{\mathit{op}}} \\ } \qquad \qquad \xymatrix@R=.8pc{ {\Theta}_{{\mathit{op}},A} \ar[dr]^{j_{{\mathit{op}},A}} \ar[dd]_{{\theta}_{{\mathit{op}},A}} \\ \ar@{}[r]\|(.4){=} & {\Theta}_{{\mathit{op}},a} \ar[dl]^{{\theta}_{{\mathit{op}},a}} \\ } \ar@{}[r]\|(.4){{\begin{turn}{-20}\ensuremath{\Uparrow}\end{turn}}}\|(.25){t_{{\mathit{op}}}} & {\Theta}_{{\mathit{op}},a} \\ {\Theta}_{{\mathit{op}}} \ar[ur]_{j_{{\mathit{op}}}} \\ }$$ Parameterization process (construction of ${\Theta}_{{\mathit{op}},A}$ from ${\Theta}_{{\mathit{op}}}$). The theory ${\Theta}_{{\mathit{op}},A}$ is obtained from ${\Theta}_{{\mathit{op}}}$ by adding a type $A$, called the type of parameters, to the domain of the unique term in ${\Theta}_{{\mathit{op}}}$. Then ${\Theta}_{{\mathit{op}},A}$ can be seen as a generalization of ${\Theta}_{{\mathit{op}}}$, since each model $M$ of ${\Theta}_{{\mathit{op}}}$ can be identified to a model of ${\Theta}_{{\mathit{op}},A}$ where $M(A)$ is a singleton. We will also say that ${\Theta}_{{\mathit{op}},A}$ is the expansion of ${\Theta}_{{\mathit{op}}}$. Parameter passing process (construction of ${\Theta}_{{\mathit{op}},a}$ from ${\Theta}_{{\mathit{op}},A}$ and of a morphism from ${\Theta}_{{\mathit{op}}}$ to ${\Theta}_{{\mathit{op}},a}$). The theory ${\Theta}_{{\mathit{op}},a}$ is obtained from ${\Theta}_{{\mathit{op}},A}$ by adding a constant term $a\colon {1}\to A$, called the parameter. A model $M_a$ of ${\Theta}_{{\mathit{op}},a}$ is made of a model $M_A$ of ${\Theta}_{{\mathit{op}},A}$ together with an element $\alpha=M_a(a)\in M_A(A)$, so that we can denote $M_a=(M_A,\alpha)$. Now, let $M_A$ be some fixed model of ${\Theta}_{{\mathit{op}},A}$, then the models $M_a$ of ${\Theta}_{{\mathit{op}},a}$ over $M_A$ correspond bijectively to the elements of $M_A(A)$ by $M_a \mapsto M_a(a)$, so that we get the parameter adding bijection (the category ${\mathit{Mod}}({\Theta}_{{\mathit{op}},a})\|_{M_A}$ is discrete): $${\mathit{Mod}}({\Theta}_{{\mathit{op}},a})\|_{M_A} {\stackrel{\simeq}{\rightarrow}}M_A(A) {\hspace{5mm}}\mbox{by} {\hspace{5mm}}M_a = (M_A,\alpha) \mapsto M_a(a)=\alpha \;.$$ On the other hand, each model $M_a=(M_A,\alpha)$ of ${\Theta}_{{\mathit{op}},a}$ gives rise to a model ${j_{{\mathit{op}}}}^{}(M_a)$ of ${\Theta}_{{\mathit{op}}}$ such that ${j_{{\mathit{op}}}}^{}(M_a)(X)=M_a(X)=M_A(X)$, ${j_{{\mathit{op}}}}^{}(M_a)(Y)=M_a(Y)=M_A(Y)$ and ${j_{{\mathit{op}}}}^{}(M_a)(f)=M_a(f'')=M_A(f')(\alpha,-) $. Now, let $M_A$ be some fixed model of ${\Theta}_{{\mathit{op}},A}$ and $M_0$ its restriction to ${\Theta}_{{\mathit{op}},0}$, then for each model $M_a=(M_A,\alpha)$ of ${\Theta}_{{\mathit{op}},a}$ over $M_A$ the model ${j_{{\mathit{op}}}}^{}(M_a)$ of ${\Theta}_{{\mathit{op}}}$ is over $M_0$. This yields the parameter passing* function (the categories ${\mathit{Mod}}({\Theta}_{{\mathit{op}},a})\|_{M_A}$ and ${\mathit{Mod}}({\Theta}_{{\mathit{op}}})\|_{M_0}$ are discrete): $${\mathit{Mod}}({\Theta}_{{\mathit{op}},a})\|_{M_A} \to {\mathit{Mod}}({\Theta}_{{\mathit{op}}})\|_{M_0} {\hspace{5mm}}\mbox{by} {\hspace{5mm}}M_a \mapsto {j_{{\mathit{op}}}}^{}(M_a) \;.$$ Exact parameterization.* Let $M_0$ be any fixed model of ${\Theta}_{{\mathit{op}},0}$, it is made of two sets ${\mathbb{X}}=M_0(X)$ and ${\mathbb{Y}}=M_0(Y)$. Let $M_A$ be the model of ${\Theta}_{{\mathit{op}},A}$ over $M_0$ such that $M_A(A)={\mathbb{Y}}^{{\mathbb{X}}}$ and $M_A(f')\colon {\mathbb{Y}}^{{\mathbb{X}}} \times {\mathbb{X}}\to {\mathbb{Y}}$ is the application. It can be noted that $M_A$ is the terminal model of ${\Theta}_{{\mathit{op}},A}$ over $M_0$. Then the parameter passing function is a bijection, and composing it with the parameter adding bijection we get (where ${\ulcorner\!{M(f)}\!\urcorner}\in{\mathbb{Y}}^{{\mathbb{X}}}$ corresponds by currying to $M(f)\colon {\mathbb{X}}\to{\mathbb{Y}}$): $${\mathit{Mod}}({\Theta}_{{\mathit{op}}})\|_{M_0} {\cong}M_A(A) {\hspace{5mm}}\mbox{by} {\hspace{5mm}}M_{A,\alpha} \leftrightarrow \alpha {\hspace{5mm}}\mbox{i.e., by} {\hspace{5mm}}M \leftrightarrow {\ulcorner\!{M(f)}\!\urcorner} \;.$$ \[exam:sgp\] Let ${\Theta}_{{\mathit{sgp}}}$ be the theory for semigroups presented by one type $G$, one term ${\mathit{prd}}\colon G^2 \to G$ and one equation ${\mathit{prd}}(x,{\mathit{prd}}(y,z))={\mathit{prd}}({\mathit{prd}}(x,y),z)$ where $x$, $y$, $z$ are variables of type $G$. As usual with the categorical point of view, in fact the variables are projections; here, $x,y,z\colon G^3\to G$ are the three projections and for instance ${\mathit{prd}}(x,y)$ is ${\mathit{prd}}\circ{\langlex,y\rangle}\colon G^3\to G$, composed of the pair ${\langlex,y\rangle}\colon G^3\to G^2$ and of ${\mathit{prd}}\colon G^2 \to G$ (more details are given in the appendix, example \[exam:dia-sg\]). Parameterization process. In order to get parameterized families of semigroups, we consider the theory ${\Theta}_{{\mathit{sgp}},A}$ presented by two types $A$ and $G$, one term ${\mathit{prd}}'\colon A\times G^2 \to G$ and one equation ${\mathit{prd}}'(p,x,{\mathit{prd}}'(p,y,z))= {\mathit{prd}}'(p,{\mathit{prd}}'(p,x,y),z)$ where $x$, $y$, $z$ are variables of sort $G$ and $p$ is a variable of sort $A$. Parameter passing process. The theory ${\Theta}_{{\mathit{sgp}},a}$ is ${\Theta}_{{\mathit{sgp}},A}$ together with a parameter $a\colon {1}\to A$, hence with ${\mathit{prd}}''={\mathit{prd}}'\circ(a\times{\mathit{id}}_{G^2}) \colon G^2 \to G$ (where ${1}\times G^2$ is identified to $G^2$). Each model $M_A$ of ${\Theta}_{{\mathit{sgp}},A}$ gives rise to a family of models of ${\Theta}_{{\mathit{sgp}},a}$, all of them with the same underlying set $M_A(G)$ but with different interpretations of $a$ in $M_A(A)$. Mapping ${\mathit{prd}}$ to ${\mathit{prd}}''$ defines a morphism from ${\Theta}_{{\mathit{sgp}}}$ to ${\Theta}_{{\mathit{sgp}},a}$. So, each model $M_a$ of ${\Theta}_{{\mathit{sgp}},a}$ gives rise to a model $M$ of ${\Theta}_{{\mathit{sgp}}}$ such that $M(G)=M_a(G)$ and $M({\mathit{prd}})(x,y)=M_a({\mathit{prd}}')(\alpha,x,y) $ for each $x,y\in M_a(G)$, where $\alpha=M_a(a)$ is the argument. \[exam:list\] This example motivates the existence of pure terms in the given theory. Let us consider the theory ${\Theta}_{{\mathit{nat}}}$ “of naturals” presented by a type $N$ and two terms $z\colon {1}\to N$ and $s\colon N\to N$, and let us say that $z$ is pure. Let ${\Theta}_{{\mathit{nat}},0}$ be the subtheory presented by $N$ and $z$, it is called the pure subtheory of ${\Theta}_{{\mathit{nat}}}$. We define the theory ${\Theta}_{{\mathit{nat}},A}$ as made of two types $A$ and $N$ and two terms $z\colon {1}\to N$ and $s'\colon A\times N\to N$. It should be noted that ${\Theta}_{{\mathit{nat}},A}$ contains ${\varepsilon}_{1}\colon A\times {1}\to {1}$ and $z'=z\circ {\varepsilon}_{1}\colon A\times {1}\to N$. Then ${\Theta}_{{\mathit{nat}},A}$ is a theory “of lists of $A$”, with $z$ for the empty list and $s'$ for concatenating an element to a list. In this way, the theory of lists of $A$ is built as a generalization of the theory of naturals; indeed the naturals can be identified to the lists over a singleton. \[exam:dm\] Here is another example where pure terms are required, this is a simplified version of many structures in Kenzo/EAT. Let ${\Theta}_{{\mathit{mon}}}$ be the theory for monoids presented by one type $G$, two terms ${\mathit{prd}}\colon G^2 \to G$ and ${e}\colon \to G$, and the equations ${\mathit{prd}}(x,{\mathit{prd}}(y,z)) = {\mathit{prd}}({\mathit{prd}}(x,y),z)$, ${\mathit{prd}}(x,{e}) = x$, ${\mathit{prd}}({e},x) = x$ where $x$, $y$, $z$ are variables of type $G$. Let ${\Theta}_{{\mathit{dm}}}$ be the theory for differential monoids, presented by ${\Theta}_{{\mathit{mon}}}$ together with one term ${\mathit{dif}}\colon G \to G$ and the equations ${\mathit{dif}}({\mathit{prd}}(x,y)) = {\mathit{prd}}({\mathit{dif}}(x),{\mathit{dif}}(y))$, ${\mathit{dif}}({e}) = {e}$, ${\mathit{dif}}({\mathit{dif}}(x)) = {e}$, and with the terms in ${\Theta}_{{\mathit{mon}}}$ as its pure terms. In order to get parameterized families of differential structures on one monoid, we define the theory ${\Theta}_{{\mathit{dm}},A}$ presented by two types $G$, $A$ and three terms ${\mathit{prd}}\colon G^2 \to G$, ${e}\colon {1}\to G$ and ${\mathit{dif}}' \colon A\times G \to G$ and the equations ${\mathit{prd}}(x,{\mathit{prd}}(y,z)) = {\mathit{prd}}({\mathit{prd}}(x,y),z)$, ${\mathit{prd}}(x,{e}) = x$, ${\mathit{prd}}({e},x) = x$, ${\mathit{dif}}'(p,({\mathit{prd}}(x,y))) = {\mathit{prd}}({\mathit{dif}}'(p,x), {\mathit{dif}}'(p,y))$, ${\mathit{dif}}'(p, {e}) = {e}$, ${\mathit{dif}}'(p,{\mathit{dif}}'(p,x)) = {e}$. Each model $M_A$ of ${\Theta}_{{\mathit{dm}},A}$ gives rise to a family of models of ${\Theta}_{{\mathit{dm}}}$, all of them with the same underlying monoid $(M_A(G),M_A({\mathit{prd}}),M_A({e}))$: there is a model $M_a$ of ${\Theta}_{{\mathit{dm}}}$ over $M_A$ for each element $\alpha$ in $M_A(A)$, with its differential structure defined by $M_a({\mathit{dif}})= M_A({\mathit{dif}}')(\alpha,-)$. \[exam:pi\] In the next sections we will use the theories with the following presentations: $$\begin{array}{c\|c\|c\|c\|} \cline{2-2} \Pi_A \dashv & \xymatrix{A} & \multicolumn{2}{c}{} \\ \cline{2-2} \cline{4-4} \multicolumn{2}{c}{} & \qquad \Pi_a \dashv & \xymatrix@R=1pc{ A \\ {1}\ar[u]^{a} \\ } \\ \cline{2-2} \cline{4-4} \Pi \dashv & \xymatrix{{1}} & \multicolumn{2}{c}{} \\ \cline{2-2} \end{array}$$ These theories are related by several morphisms: $\pi_A\colon \Pi_A\to \Pi$ maps $A$ to ${1}$, both $i\colon \Pi\to \Pi_a$ and $i_A\colon \Pi_A\to \Pi_a$ are the inclusions, and $\pi_a\colon \Pi_a\to \Pi$ extends $\pi_A$ by mapping $a$ to ${\mathit{id}}_{{1}}$, so that $\pi_A$ and $\pi_a$ are epimorphisms. In addition, $\pi_a\circ i_A=\pi_A$ and there is a natural transformation $p\colon i \circ \pi_A {\Rightarrow}i_A$ defined by $p_A=a\colon {1}\to A$. The diagram below on the right is the lax colimit of $\pi_A$, which means that it enjoys the following universal property: for each $\Pi'_a$ with $i'_A\colon \Pi_A\to \Pi'_a$, $i'\colon \Pi\to \Pi'_a$ and $p'\colon i' \circ \pi_A {\Rightarrow}i'_A$, there is a unique $h\colon \Pi_a\to \Pi'_a$ such that $h\circ i_A = i'_A$, $h\circ i = i'$ and $h\circ p=p'$. For instance, given $\Pi$, $\pi_A\colon \Pi_A\to \Pi$, ${\mathit{id}}_{\Pi}\colon \Pi\to \Pi$ and ${\mathit{id}}_{\pi_A}\colon \pi_A {\Rightarrow}\pi_A$, then $\pi_a\colon \Pi_a\to \Pi$ is the unique morphism such that $\pi_a\circ i_A = \pi_A$, $\pi_a\circ i = {\mathit{id}}_{\Pi}$ and $\pi_a\circ p={\mathit{id}}_{\pi_A}$. $$\xymatrix@R=.8pc{ \Pi_A \ar[dd]_{\pi_A} \\ \\ \Pi \\ } \qquad \qquad \xymatrix@R=.8pc{ \Pi_A \ar[dr]^{i_A} \ar[dd]_{\pi_A} \\ \ar@{}[r]\|(.4){=} & \Pi_a \ar[dl]^{\pi_a} \\ } \ar@{}[r]\|(.4){{\begin{turn}{-20}\ensuremath{\Uparrow}\end{turn}}}\|(.25){p} & \Pi_a \\ \Pi \ar[ur]_{i} \\ }$$ Some other kinds of theories {#subsec:thry} ---------------------------- For every theory ${\Theta}$, the coslice category of theories under ${\Theta}$ is denoted ${\Theta}{\!\!\downarrow\!\!}{\mathbf{T}}_{{\mathit{eq}}}$. It can be seen as a 2-category, with the natural transformations which extend the identity on ${\Theta}$ as 2-cells. \[defi:A-th\] A parameterized theory ${\Theta}_A$ is a theory ${\Theta}$ with a distinguished type, called the type of parameters and usually denoted $A$. The 2-category of parameterized theories is the coslice 2-category ${\mathbf{T}}_A=\Pi_A{\!\!\downarrow\!\!}{\mathbf{T}}_{{\mathit{eq}}}$ of theories under $\Pi_A$. A theory with a parameter ${\Theta}_a$ is a parameterized theory with a distinguished constant of type $A$, called the parameter and usually denoted $a\colon {1}\to A$. The 2-category of theories with a parameter is the coslice 2-category ${\mathbf{T}}_a=\Pi_a{\!\!\downarrow\!\!}{\mathbf{T}}_{{\mathit{eq}}}$ of theories under $\Pi_a$. According to the context, ${\Theta}_A$ usually denotes the parameterized theory $\gamma_A \colon \Pi_A\to{\Theta}_A$, and sometimes it denotes the equational theory ${\Theta}_A$ itself. Similarly for ${\Theta}_a$, which usually denotes $\gamma_a\colon \Pi_a\to{\Theta}_a$ and sometimes ${\Theta}_a$ itself. In addition, it can be noted that $\Pi$ is the initial theory (which may also be presented by the empty specification) so that $\Pi{\!\!\downarrow\!\!}{\mathbf{T}}_{{\mathit{eq}}}$ is isomorphic to ${\mathbf{T}}_{{\mathit{eq}}}$. The 2-categories ${\mathbf{S}}_A$ and ${\mathbf{S}}_a$ of parameterized specifications and specifications with a parameter, respectively, are defined in a similar way. On the other hand, the input of the parameterization process is a theory ${\Theta}$ together with a wide subtheory ${\Theta}_0$ (wide means: with the same types), such a structure is called a decorated theory. \[defi:dec-th\] A decorated theory is made of a theory ${\Theta}$ with a wide subtheory ${\Theta}_0$ called the pure subtheory of ${\Theta}$. A morphism of decorated theories is a morphism of theories ${\theta}\colon {\Theta}\to{\Theta}'$ which maps the pure part of ${\Theta}$ to the pure part of ${\Theta}'$. This forms the category ${\mathbf{T}}_{{\mathit{dec}}}$ of decorated theories. So, a decorated theory ${\Theta}$ is endowed with a distinguished family of terms, called the pure terms, such that all the identities and projections are pure and every composition or tuple of pure terms is pure. Pure terms are denoted with “${\rightsquigarrow}$”. When there is no ambiguity we often use the same notation ${\Theta}$ for the theory ${\Theta}$ itself and for the decorated theory made of ${\Theta}$ and ${\Theta}_0$. The decorated specifications are defined in a straightforward way. For instance, we may consider the decorated specification made of a type $N$, a pure term $z\colon {1}{\rightsquigarrow}N$ and a term $s\colon N\to N$ (see example \[exam:list\]). Constructions {#sec:const} ============= The parameterization process is a functor {#subsec:gene} ----------------------------------------- In this section we prove that the parameterization process is functorial, by defining a functor $F_{{\mathit{exp}}}\colon {\mathbf{T}}_{{\mathit{dec}}}\to{\mathbf{T}}_A$ (“${\mathit{exp}}$” for “expansion”) which adds the type of parameters to the domain of every non-pure term. In addition, theorem \[theo:gene\] states that $F_{{\mathit{exp}}}$ is left adjoint to the functor $G_{{\mathit{exp}}}\colon {\mathbf{T}}_A\to{\mathbf{T}}_{{\mathit{dec}}}$, which builds the coKleisli category of the comonad $A\times-$. Moreover, we will see in section \[sec:free\] that $F_{{\mathit{exp}}}$ is a free functor associated to a morphism of limit sketches, and in appendix \[sec:dia\] that this morphism of limit sketches underlies a morphism of diagrammatic logics. $$\begin{array}{\|l\|l\|c\|c\|} \hline & \textrm{index} \; {\mathtt{E}}{\mathtt{.x}}& {\Sigma}_{{\mathtt{E}}{\mathtt{.x}}} & F_{{\mathit{exp}}}{\Sigma}_{{\mathtt{E}}{\mathtt{.x}}} \\ \hline \hline \textrm{type} & {\mathtt{Type}}{\mathtt{.p}}& X & X \\ \hline \textrm{pure term} & {\mathtt{Term}}{\mathtt{.p}}& \xymatrix@C=3pc{X\ar@{~>}[r]^{f} & Y} & \xymatrix@C=3pc{X\ar[r]^{f} & Y} \\ \hline \textrm{term} & {\mathtt{Term}}{\mathtt{.g}}& \xymatrix@C=3pc{X\ar[r]^{f} & Y} & \xymatrix@C=3pc{A{\!\times\!}X\ar[r]^{f'} & Y} \\ \hline \textrm{sel. of identity} & {\mathtt{Selid}}{\mathtt{.p}}& \xymatrix@C=3pc{X\ar@{~>}[r]^{{\mathit{id}}_X} & X} & \xymatrix@C=3pc{X\ar[r]^{{\mathit{id}}_X} & X} \\ \hline \textrm{pure composition} & {\mathtt{Comp}}{\mathtt{.p}}& \xymatrix@C=3pc{X\ar@{~>}[r]^{f} \ar@/_3ex/@{~>}[rr]_{g\circ f}^{=} & Y\ar@{~>}[r]^{g} & Z} & \xymatrix@C=3pc{X\ar[r]^{f} \ar@/_3ex/[rr]_{g\circ f}^{=} & Y\ar[r]^{g} & Z} \\ \hline \textrm{composition} & {\mathtt{Comp}}{\mathtt{.g}}& \xymatrix@C=3pc{X\ar[r]^{f} \ar@/_3ex/[rr]_{g\circ f}^{=} & Y\ar[r]^{g} & Z} & \xymatrix@C=3pc{A{\!\times\!}X \ar[r]^{{\langle{\mathit{pr}}_X,f'\rangle}} \ar@/_3ex/[rr]_{g'\circ{\langle{\mathit{pr}}_X,f'\rangle}}^{=} & A{\!\times\!}Y\ar[r]^{g'} & Z \\} \\ \hline \textrm{binary product} & {\mathtt{2\texttt{-}Prod}}{\mathtt{.p}}& \xymatrix@C=3pc@R=.7pc{ X & \\ & X{\!\times\!}Y \ar@{~>}[lu]_{p_X} \ar@{~>}[ld]^{p_Y} \\ Y & \\ } & \xymatrix@C=3pc@R=.7pc{ X & \\ & X{\!\times\!}Y \ar[lu]_{p_X} \ar[ld]^{p_Y} \\ Y & \\ } \\ \hline \textrm{pure pairing} & {\mathtt{2\texttt{-}Tuple}}{\mathtt{.p}}& \xymatrix@C=3pc@R=1pc{ & X & \\ Z \ar@{~>}[ru]^{f} \ar@{~>}[rd]_{g} \ar@{~>}[rr]\|{{\langlef,g\rangle}} & \ar@{}[u]\|{=}\ar@{}[d]\|{=} & X{\!\times\!}Y \ar@{~>}[lu]_{p_X} \ar@{~>}[ld]^{p_Y} \\ & Y & \\ } & \textrm{pairing} & {\mathtt{2\texttt{-}Tuple}}{\mathtt{.g}}& A{\!\times\!}Z \ar[ru]^{f'} \ar[rd]_{g'} \ar[rr]\|{{\langlef',g'\rangle}} & In order to define the functor $F_{{\mathit{exp}}}$ we use the fact that it should preserve colimits. It has been seen in section \[subsec:equa\] that every specification is the colimit of a diagram of elementary specifications. Similarly, every decorated specification is the colimit of a diagram of elementary decorated specifications, denoted ${\Sigma}_{{\mathtt{E}}{\mathtt{.x}}}$ where $\mathtt{x}=\mathtt{p}$ for “pure” or $\mathtt{x}=\mathtt{g}$ for “general”. Informally, the functor $F_{{\mathit{exp}}}$ explicits the fact that every general feature in a decorated specification gets parameterized, while every pure feature remains unparameterized. Figure \[fig:F-elem\] defines the parameterized specification $F_{{\mathit{exp}}}{\Sigma}_{{\mathtt{E}}{\mathtt{.x}}}$ for each elementary decorated specification ${\Sigma}_{{\mathtt{E}}{\mathtt{.x}}}$ (many projection arrows are omitted, and when needed the projections from $A\times X$ are denoted ${\mathit{pr}}_X\colon A\times X\to A$ and ${\varepsilon}_X\colon A\times X\to X$). The morphisms of parameterized specifications $F_{{\mathit{exp}}}{\sigma}$, for ${\sigma}$ between elementary decorated specifications, are straightforward. For instance, let $c\colon {\Sigma}_{{\mathtt{Term}}{\mathtt{.g}}}\to{\Sigma}_{{\mathtt{Term}}{\mathtt{.p}}}$ be the conversion morphism, which corresponds to the fact that every pure term can be seen as a general term, then $F_{{\mathit{exp}}}c$ maps $f'\colon A\times X\to Y$ in $F_{{\mathit{exp}}}{\Sigma}_{{\mathtt{Term}}{\mathtt{.g}}}$ to $f\circ {\varepsilon}_X\colon A\times X\to Y$ in $F_{{\mathit{exp}}}{\Sigma}_{{\mathtt{Term}}{\mathtt{.p}}}$. Now, given a decorated theory ${\Theta}$ presented by the colimit of a diagram $\Delta$ of elementary decorated specifications, we define $F_{{\mathit{exp}}}{\Theta}$ as the parameterized theory presented by the colimit of the diagram $F_{{\mathit{exp}}}\Delta$ of parameterized specifications. \[defi:gene\] The functor $F_{{\mathit{exp}}}:{\mathbf{T}}_{{\mathit{dec}}}\to{\mathbf{T}}_A$ defined above is called the parameterization functor. Clearly the parameterization functor preserves colimits. In addition, let ${\Theta}_A$ be the parameterized theory $F_{{\mathit{exp}}}{\Theta}$, it follows from the definition of $F_{{\mathit{exp}}}$ that the equational theory ${\Theta}_A$ is a theory under ${\Theta}_0$. Now the functor $G_{{\mathit{exp}}}$ is defined independently from $F_{{\mathit{exp}}}$. Let ${\Theta}_A$ be a parameterized theory. The endofunctor of product with $A$ forms a comonad on ${\Theta}_A$ with the counit ${\varepsilon}$ made of the projections ${\varepsilon}_X\colon A\times X\to X$ and the comultiplication made of the terms $\delta_X\colon A\times X\to A\times A\times X$ induced by the diagonal on $A$. Let ${\Theta}$ be the coKleisli category of this comonad: it has the same types as ${\Theta}_A$ and a term ${[f]}\colon X\to Y$ for each term $f\colon A\times X\to Y$ in ${\Theta}_A$. There is a functor from ${\Theta}_A$ to ${\Theta}$ which is the identity on types and maps every $g\colon X\to Y$ in ${\Theta}_A$ to ${[g\circ {\varepsilon}_X]}\colon X\to Y$ in ${\Theta}$. Then every finite product in ${\Theta}_A$ is mapped to a finite product in ${\Theta}$, which makes ${\Theta}$ a theory. Let ${\Theta}_0$ denote the image of ${\Theta}_A$ in ${\Theta}$, it is a wide subtheory of ${\Theta}$. In this way, any parameterized theory yields a decorated theory. The definition of $G_{{\mathit{exp}}}$ on morphisms is straightforward. The next result can be derived directly, or as a consequence of theorem \[theo:free\]. \[theo:gene\] The parameterization functor $F_{{\mathit{exp}}}$ and the functor $G_{{\mathit{exp}}}$ form an adjunction $F_{{\mathit{exp}}}\dashv G_{{\mathit{exp}}}$: $$\xymatrix@C=4pc{ {\mathbf{T}}_{{\mathit{dec}}} \ar@/^/[r]^{F_{{\mathit{exp}}}} \ar@{}[r]\|{\bot} & {\mathbf{T}}_A \ar@/^/[l]^{G_{{\mathit{exp}}}} }$$ The next result states that ${\Theta}$ can be easily recovered from ${\Theta}_A$, by mapping $A$ to ${1}$. \[prop:general\] Let ${\Theta}$ be a decorated theory with pure subtheory ${\Theta}_0$ and $\gamma_A\colon \Pi_A\to{\Theta}_A$ the parameterized theory $F_{{\mathit{exp}}}{\Theta}$. Let $\gamma\colon \Pi\to{\Theta}$ be the unique morphism from the initial theory $\Pi$ to the theory ${\Theta}$. Then there is a morphism ${\theta}_A\colon {\Theta}_A\to{\Theta}$ under ${\Theta}_0$ such that the following square is a pushout: $$\xymatrix{ \ar@{}[rd]\|{{{\scriptstyle [P.O.]}}} \Pi_A \ar[d]_{\pi_A} \ar[r]^{\gamma_A} & {\Theta}_A \ar[d]^{{\theta}_A} \\ \Pi \ar[r]^{\gamma} &{\Theta}\\ }$$ It can easily be checked that this property is satisfied by each elementary specification. Then the result follows by commuting two colimits: on the one hand the colimit that defines the given theory from its elementary components, and on the other hand the pushout. When there is an epimorphism of theories ${\theta}\colon {\Theta}_1\to{\Theta}_2$, we say that ${\Theta}_1$ is the generalization of ${\Theta}_2$ along ${\theta}$. Indeed, since ${\theta}$ is an epimorphism, the functor ${\theta}^{}\colon {\mathit{Mod}}({\Theta}_2)\to{\mathit{Mod}}({\Theta}_1)$ is a monomorphism, which can be used for identifying ${\mathit{Mod}}({\Theta}_2)$ to a subcategory of ${\mathit{Mod}}({\Theta}_1)$. \[coro:general\] With notations as in proposition \[prop:general\], ${\Theta}_A$ is the generalization of ${\Theta}$ along ${\theta}_A$. Clearly $\pi_A\colon \Pi_A\to\Pi$ is an epimorphism. Since epimorphisms are stable under pushouts, proposition \[prop:general\] proves that ${\theta}_A\colon {\Theta}_A\to{\Theta}$ is also an epimorphism. Let $F_{{\mathit{exp}}}:{\mathbf{T}}_{{\mathit{dec}}}\to{\mathbf{T}}_A$ be the parameterization functor and let $U\colon {\mathbf{T}}_A\to{\mathbf{T}}_{{\mathit{eq}}}$ be the functor which simply forgets that the type $A$ is distinguished, so that $U \circ F_{{\mathit{exp}}}\colon {\mathbf{T}}_{{\mathit{dec}}}\to{\mathbf{T}}_{{\mathit{eq}}}$ maps the decorated theory ${\Theta}$ to the equational theory ${\Theta}_A$. $$\xymatrix@C=4pc{ {\mathbf{T}}_{{\mathit{dec}}} \ar[r]^{F_{{\mathit{exp}}}} & {\mathbf{T}}_A \ar[r]^{U} & {\mathbf{T}}_{{\mathit{eq}}} \\ }$$ Every theory ${\Theta}$ can be seen as a decorated theory where the pure terms are defined inductively as the identities, the projections, and the compositions and tuples of pure terms. Let $I\colon {\mathbf{T}}_{{\mathit{eq}}}\to{\mathbf{T}}_{{\mathit{dec}}}$ denote the corresponding inclusion functor. Then the endofunctor $U \circ F_{{\mathit{exp}}}\circ I\colon {\mathbf{T}}_{{\mathit{eq}}}\to{\mathbf{T}}_{{\mathit{eq}}}$ corresponds to the “imp* construction” of [@LPR03], which transforms each term $f\colon X\to Y$ in ${\Theta}$ into $f'\colon A\times X\to Y$ for a new type $A$. The parameter passing process is a natural transformation {#subsec:passing} --------------------------------------------------------- A theory ${\Theta}_a$ with a parameter is built simply by adding a constant $a$ of type $A$ to a parameterized theory ${\Theta}_A$. Obviously, this can be seen as a pushout. \[defi:adding\] Let $\gamma_A\colon \Pi_A\to{\Theta}_A$ be a parameterized theory. The theory with parameter extending $\gamma_A$ is $\gamma_a\colon \Pi_a\to{\Theta}_a$ given by the pushout of $\gamma_A$ and $i_A$: $$\xymatrix{ \ar@{}[rd]\|{{{\scriptstyle [P.O.]}}} \Pi_A \ar[d]_{i_A} \ar[r]^{\gamma_A} & {\Theta}_A \ar[d]^{j_A} \\ \Pi_a \ar[r]^{\gamma_a} &{\Theta}_a \\ }$$ The pushout of theories in definition \[defi:adding\] gives rise to a pullback of categories of models, hence for each model $M_A$ of ${\Theta}_A$ the function which maps each model $M_a$ of ${\Theta}_a$ over $M_A$ to the element $M_a(a)\in M_A(A)$ defines a bijection: $$\label{eq:adding} {\mathit{Mod}}({\Theta}_a)\|_{M_A} {\stackrel{\simeq}{\rightarrow}}M_A(A) \;.$$ Let us assume that the parameterized theory $\gamma_A\colon \Pi_A\to{\Theta}_A$ is $F_{{\mathit{exp}}}{\Theta}$ for some decorated theory ${\Theta}$ with pure subtheory ${\Theta}_0$. Then the pushout property in definition \[defi:adding\] ensures the existence of a unique ${\theta}_a\colon {\Theta}_a\to {\Theta}$ such that ${\theta}_a \circ \gamma_a = \gamma\circ\pi_a$ (which means that ${\theta}_a$ maps $A$ to ${1}$ and $a$ to ${\mathit{id}}_{{1}}$) and ${\theta}_a \circ j_A = {\theta}_A$. Then ${\Theta}_A$ is a theory under ${\Theta}_0$ and the composition by $j_A$ makes ${\Theta}_a$ a theory under ${\Theta}_0$ with $j_A$ preserving ${\Theta}_0$. $$\xymatrix@R=.8pc{ {\Theta}_A \ar[dr]^{j_A} \ar[dd]_{{\theta}_A} & \\ \ar@{}[r]\|(.4){=} & {\Theta}_a \ar[dl]^{{\theta}_a} \\ {\Theta}& \\ }$$ Lax cocones and lax colimits in 2-categories generalize cocones and colimits in categories. For each decorated theory ${\Theta}$ with pure subcategory ${\Theta}_0$, let ${\Theta}_A=F_{{\mathit{exp}}}{\Theta}$ and ${\theta}_A\colon {\Theta}_A\to{\Theta}$ be as in section \[subsec:gene\], and let ${\Theta}_a$ and $j_A\colon {\Theta}_A\to{\Theta}_a$ be as above. Let $j\colon {\Theta}\to{\Theta}_a$ be the morphism under ${\Theta}_0$ which maps each type $X$ to $X$ and each term $f\colon X\to Y$ to $f'\circ(a\times{\mathit{id}}_X)\colon X\to Y$. Let $t\colon j\circ{\theta}_A {\Rightarrow}j_A$ be the natural transformation under ${\Theta}_0$ such that $t_A=a\colon {1}\to A$. Then the following diagram is a lax cocone with base ${\theta}_A$ in the 2-category ${\Theta}_0{\!\!\downarrow\!\!}{\mathbf{T}}_{{\mathit{eq}}}$, for short it is denoted $({\Theta}_a,j_A,j,t)$, and it is called the lax colimit associated to ${\Theta}$ because of lemma \[lemm:passing\]. $$\xymatrix@R=.8pc{ {\Theta}_A \ar[dr]^{j_A} \ar[dd]_{{\theta}_A} & \\ \ar@{}[r]\|(.4){{\begin{turn}{-20}\ensuremath{\Uparrow}\end{turn}}}\|(.25){t} & {\Theta}_a \\ {\Theta}\ar[ur]_{j} & \\ }$$ \[lemm:passing\] Let ${\Theta}$ be a decorated theory with pure subcategory ${\Theta}_0$. The lax cocone $({\Theta}_a,j_A,j,t)$ with base ${\theta}_A$ defined above is a lax colimit in the 2-category of theories under ${\Theta}_0$. This means that the given lax cocone is initial among the lax cocones with base ${\theta}_A$ in ${\Theta}_0{\!\!\downarrow\!\!}{\Theta}$, in the following sense. For every lax cocone $({\Theta}'_a,j'_A,j',t')$ with base ${\theta}_A$ under ${\Theta}_0$ there is a unique morphism $ h\colon {\Theta}_a\to{\Theta}'_a$ such that $ h\circ j_A=j'_A$, $ h\circ j=j'$ and $ h\circ t = t'$, it is defined from the pushout in definition \[defi:adding\] by $h\circ j_A=j'_A$, so that $h(A)=A$, and $h\circ \gamma_a(a)=t'_A:{1}\to A$. For instance, given ${\Theta}$, ${\theta}_A\colon {\Theta}_A\to {\Theta}$, ${\mathit{id}}_{{\Theta}}\colon {\Theta}\to {\Theta}$ and ${\mathit{id}}_{{\theta}_A}\colon {\theta}_A {\Rightarrow}{\theta}_A$, then ${\theta}_a$ is the unique morphism such that ${\theta}_a\circ j_A = {\theta}_A$, ${\theta}_a\circ j = {\mathit{id}}_{{\Theta}}$ and ${\theta}_a\circ t={\mathit{id}}_{{\theta}_A}$. Let ${\Theta}$ be a decorated theory with pure subtheory ${\Theta}_0$ and let $({\Theta}_a,j_A,j,t)$ be its associated lax colimit, with base ${\theta}_A\colon {\Theta}_A\to {\Theta}$. Let $M_A$ be a model of ${\Theta}_A$ and $M_0$ its restriction to ${\Theta}_0$, and let $\{ (M,m) \mid m\colon {{\theta}_A}^{}M \to M_A \}\|_{M_0}$ (where as before ${{\theta}_A}^{}M=M\circ{\theta}_A$) denote the set of pairs $(M,m)$ with $M$ a model of ${\Theta}$ over $M_0$ and $m$ a morphism of models of ${\Theta}_A$ over $M_0$. A consequence of the lax colimit property is that the function which maps each model $M_a$ of ${\Theta}_a$ over $M_A$ to the pair $(j^{}M_a, t^{}M_a)=(M_a\circ j,M_a\circ t)$ defines a bijection: $$\label{eq:passing} {\mathit{Mod}}({\Theta}_a)\|_{M_A} {\cong}\{ (M,m) \mid m\colon {{\theta}_A}^{}M \to M_A \}\|_{M_0} \;.$$ The bijections \[eq:adding\] and \[eq:passing\] provide the next result, which does not involve ${\Theta}_a$. \[prop:passing\] Let ${\Theta}$ be a decorated theory with pure subtheory ${\Theta}_0$ and let ${\Theta}_A=F_{{\mathit{exp}}}{\Theta}$ and ${\theta}_A\colon {\Theta}_A\to {\Theta}$. Then for each model $M_A$ of ${\Theta}_A$, with $M_0$ denoting the restriction of $M_A$ to ${\Theta}_0$, the function which maps each element $\alpha\in M_A(A)$ to the pair $(M,m)$, where $M$ is the model of ${\Theta}$ such that $M(f)=M_A(f')(\alpha,-)$ and where $m:{{\theta}_A}^{}M \to M_A$ is the morphism of models of ${\Theta}_A$ such that $m_A:M({1})\to M_A(A)$ is the constant function $\star\to\alpha$, defines a bijection: $$\label{eq:adding-passing} M_A(A) {\cong}\{ (M,m) \mid m\colon {{\theta}_A}^{}M \to M_A \}\|_{M_0} \;.$$ As an immediate consequence, we get the exact parameterization* property from [@LPR03]. \[coro:exact\] Let ${\Theta}$ be a decorated theory with pure subcategory ${\Theta}_0$, and let ${\Theta}_A=F_{{\mathit{exp}}}{\Theta}$. Let $M_0$ be a model of ${\Theta}_0$ and $M_A$ a terminal model of ${\Theta}_A$ over $M_0$. Then there is a bijection: $$\label{eq:exact} M_A(A) {\cong}{\mathit{Mod}}({\Theta})\|_{M_0}$$ which maps each $\alpha\in M_A(A)$ to the model $M_{A,\alpha}$ of ${\Theta}$ defined by $M_{A,\alpha}(X)=M_0(X)$ for each type $X$ and $M_{A,\alpha}(f)=M_A(f')(\alpha,-)$ for each term $f$, so that $M_{A,\alpha}(f)=M_A(f)$ for each pure term $f$. The existence of a terminal model of ${\Theta}_A$ over $M_0$ is a consequence of [@Ru00] and [@HR95]. Corollary \[coro:exact\] corresponds to the way algebraic structures are implemented in the systems Kenzo/EAT. In these systems the parameter set is encoded by means of a record of Common Lisp functions, which has a field for each operation in the algebraic structure to be implemented. The pure terms correspond to functions which can be obtained from the fixed data and do not require an explicit storage. Then, each particular instance of the record gives rise to an algebraic structure. Clearly the construction of $\gamma_a$ from $\gamma_A$ is a functor, which is left adjoint to the functor which simply forgets that the constant $a$ is distinguished. So, by composing this adjunction with the adjunction $F_{{\mathit{exp}}}\dashv G_{{\mathit{exp}}}$ from theorem \[theo:gene\] we get an adjunction $F'_{{\mathit{exp}}}\dashv G'_{{\mathit{exp}}}$ where $F'_{{\mathit{exp}}}$ maps each decorated theory ${\Theta}$ to ${\Theta}_a$, as defined above: $$\xymatrix@C=4pc{ {\mathbf{T}}_{{\mathit{dec}}} \ar@/^/[r]^{F'_{{\mathit{exp}}}} \ar@{}[r]\|{\bot} & {\mathbf{T}}_a \ar@/^/[l]^{G'_{{\mathit{exp}}}} }$$ Let $U'\colon {\mathbf{T}}_a\to{\mathbf{T}}_{{\mathit{eq}}}$ be the functor which simply forgets that the type $A$ and the constant $a$ are distinguished. Then the functor $U' \circ F'_{{\mathit{exp}}} \colon {\mathbf{T}}_{{\mathit{dec}}}\to{\mathbf{T}}_{{\mathit{eq}}}$ maps the decorated theory ${\Theta}$ to the equational theory ${\Theta}_a$. $$\xymatrix@C=3pc{ {\mathbf{T}}_{{\mathit{dec}}} \ar[r]^{F'_{{\mathit{exp}}}} & {\mathbf{T}}_a \ar[r]^{U'} & {\mathbf{T}}_{{\mathit{eq}}} \\ }$$ The morphism of theories $j\colon {\Theta}\to{\Theta}_a$ depends on the decorated theory ${\Theta}$, let us denote it $j=J_{{\Theta}}$. Let $H\colon {\mathbf{T}}_{{\mathit{dec}}}\to{\mathbf{T}}_{{\mathit{eq}}}$ be the functor which maps each decorated theory ${\Theta}$ to the equational theory ${\Theta}$. The next result is easy to check. \[theo:passing\] The morphisms of theories $J_{{\Theta}}\colon {\Theta}\to{\Theta}_a$ form the components of a natural transformation $J\colon H {\Rightarrow}U'\circ F'_{{\mathit{exp}}}\colon {\mathbf{T}}_{{\mathit{dec}}}\to{\mathbf{T}}_{{\mathit{eq}}}$. $$\xymatrix@C=3pc{ {\mathbf{T}}_{{\mathit{dec}}} \ar[r]^{F'_{{\mathit{exp}}}} \ar@/_4ex/[rr]_{H}^(.4){\Uparrow}^(.45){J} & {\mathbf{T}}_a \ar[r]^{U'} & {\mathbf{T}}_{{\mathit{eq}}} \\ }$$ \[defi:passing\] The natural transformation $J\colon H {\Rightarrow}U'\circ F'_{{\mathit{exp}}}\colon {\mathbf{T}}_{{\mathit{dec}}}\to{\mathbf{T}}_{{\mathit{eq}}}$ in theorem \[theo:passing\] is called the parameter passing natural transformation. \[exam:adding\] Starting from ${\Theta}_{{\mathit{op}}}$ and ${\Theta}_{{\mathit{op}},0}$ as in example \[exam:term\], the pushouts of theories from proposition \[prop:general\] and definition \[defi:adding\] are respectively: $$\begin{array}{\|c\|c\|c\|c\|c\|c\|c\|} \cline{1-1} \cline{3-3} \cline{5-5} \cline{7-7} \xymatrix{A \\} & \longrightarrow & \xymatrix@R=1pc{A & A{\!\times\!}X \ar[l]\ar[d]\ar[rd]^{f'} & \\ & X & Y \\} & \qquad \qquad & \xymatrix@R=1pc{A \\} & \longrightarrow & \xymatrix@R=1pc{A & A{\!\times\!}X \ar[l]\ar[d]\ar[rd]^{f'} & \\ & X & Y \\} \\ \cline{1-1} \cline{3-3} \cline{5-5} \cline{7-7} {\multicolumn{1}{c}{\downarrow}} & {\multicolumn{1}{c}{}} & \cline{1-1} \cline{3-3} \cline{5-5} \cline{7-7} \xymatrix@R=1pc{ \\ {1}\\ } & \longrightarrow & \xymatrix@R=1pc{ \\ & X\ar[r]^{f} &Y \\ } & & \xymatrix@R=1pc{A \\ {1}\ar[u]^{a} \\ } & \longrightarrow & \xymatrix@R=1pc{ A & A{\!\times\!}X \ar[l]\ar[d]\ar[rd]^{f'} & \\ {1}\ar[u]^{a} & X & Y \\ } \\ \cline{1-1} \cline{3-3} \cline{5-5} \cline{7-7} \end{array}$$ We have seen in example \[exam:term\] two other presentations of the vertex ${\Theta}_{{\mathit{op}},a}$ of the second pushout, with $f''=f'\circ(a\times{\mathit{id}}_X):X\to Y$. For each decorated theory ${\Theta}$, the morphism of equational theories $j_{{\mathit{op}}}=J_{{\Theta}_{{\mathit{op}}}}:{\Theta}\to{\Theta}_a$ maps $f$ to $f''$, as in example \[exam:term\]. A model $M_0$ of ${\Theta}_{{\mathit{op}},0}$ is simply made of two sets ${\mathbb{X}}=M_0(X)$ and ${\mathbb{Y}}=M_0(Y)$. On the one hand, a model of ${\Theta}$ over $M_0$ is characterized by a function $\varphi\colon {\mathbb{X}}\to{\mathbb{Y}}$. On the other hand, the terminal model $M_A$ of ${\Theta}_{{\mathit{op}},A}$ over $M_0$ is such that $M_A(A)={\mathbb{Y}}^{\mathbb{X}}$ and $M_A(f')\colon {\mathbb{Y}}^{{\mathbb{X}}} \times {\mathbb{X}}\to {\mathbb{Y}}$ is the application. The bijection ${\mathit{Mod}}({\Theta})\|_{M_0}{\cong}M_A(A)$ then corresponds to the currying bijection $\varphi\mapsto{\ulcorner\!{\varphi}\!\urcorner}$. \[exam:dm-terminal\] Let ${\Theta}_{{\mathit{dm}}}$ be the theory for differential monoids from example \[exam:dm\], with the pure subtheory ${\Theta}_{{\mathit{dm}},0}={\Theta}_{{\mathit{mon}}}$ of monoids. They generate the parameterized theory ${\Theta}_{{\mathit{dm}},A}$ as in example \[exam:dm\]. Let $M_0$ be some fixed monoid and $M_A$ any model of ${\Theta}_{{\mathit{dm}},A}$ over $M_0$, then each element of $M_A(A)$ corresponds to a differential structure on the monoid $M_0$. If in addition $M_A$ is the terminal model of ${\Theta}_{{\mathit{dm}},A}$ over $M_0$, then this correspondence is bijective. \[exam:state\] When dealing with an imperative language, the states for the memory are endowed with an operation ${\mathit{lookup}}$ for observing the state and an operation ${\mathit{update}}$ for modifying it. There are two points of view on this situation: either the state is hidden, or it is explicit. Let us check that the parameterization process allows to generate the theory with explicit state from the theory with hidden state. First, let us focus on observation: the theory ${\Theta}_{{\mathit{st}}}$ is made of two types $L$ and $Z$ (for locations and integers, respectively) and a term $v\colon L\to Z$ for observing the values of the variables. The pure subtheory ${\Theta}_{{\mathit{st}},0}$ is made of $L$ and $Z$. We choose a model $M_0$ of ${\Theta}_{{\mathit{st}},0}$ made of a countable set of locations (or addresses, or “variables”) ${\mathbb{L}}=M_0(L)$ and of the set of integers ${\mathbb{Z}}=M_0(Z)$. Let ${\mathbb{A}}={\mathbb{Z}}^{\mathbb{L}}$, then as in example \[exam:adding\] the terminal model $M_A$ of ${\Theta}_{{\mathit{st}},A}$ over $M_0$ is such that $M_A(A)={\mathbb{A}}$ and $M_{{\mathit{st}},A}(v')\colon {\mathbb{A}}\times{\mathbb{L}}\to{\mathbb{Z}}$ is the application, denoted ${\mathit{lookup}}$. The terminal model $M_A$ does correspond to an “optimal” implementation of the state. Now, let us look at another model $N_A$ of ${\Theta}_{{\mathit{st}},A}$ over $M_0$, defined as follows: $N_A(A)={\mathbb{A}}\times {\mathbb{L}}\times {\mathbb{Z}}$ and $N_A(v')\colon {\mathbb{A}}\times {\mathbb{L}}\times {\mathbb{Z}}\times {\mathbb{L}}\to{\mathbb{Z}}$ maps $(p,x,n,y)$ to $n$ if $x=y$ and to ${\mathit{lookup}}(p,y)$ otherwise. The terminality property of $M_A$ ensures that there is a unique function ${\mathit{update}}\colon {\mathbb{A}}\times {\mathbb{L}}\times {\mathbb{Z}}\to {\mathbb{A}}$ such that ${\mathit{lookup}}({\mathit{update}}(p,x,n),y)$ is $n$ if $x=y$ and ${\mathit{lookup}}(p,y)$ otherwise. So, the updating operation ${\mathit{update}}$ is defined coinductively from the observation operation ${\mathit{lookup}}$. Free functors {#sec:free} ============= In this section some basic facts about limit sketches and their associated adjunction are mentioned, and it is proved that the parameterization functor $F_{{\mathit{exp}}}$ from section \[subsec:gene\] is a free functor, in the sense that it is the left adjoint associated to a morphism of limits sketches. Limit sketches {#subsec:sketch} -------------- It is quite usual to define a free functor as the left adjoint of a forgetful functor, but there is no unique definition of a forgetful functor. In this section forgetful functors are defined from morphisms of limit sketches, they are not always faithful. There are several definitions of limit sketches (also called projective sketches) in the litterature, see for instance [@CL84] or [@BW99]. These definitions are different but all of them serve the same purpose: each limit sketch generates a category with limits, so that limit sketches generalize equational specifications in allowing some interdependence between the variables. In this paper, limit sketches are used at the meta level, in order to describe each category of theories or specifications as the category of realizations (or models) of a limit sketch. While a category with limits is a graph with identities, composition, limit cones and tuples, satisfying a bunch of axioms, a limit sketch is a graph with potential identities, composition, limit cones and tuples, which are not required to satisfy any axiom. Potential limit cones, or simply potential limits, may also be called specified limits or distinguished cones. \[defi:sketch\] A limit sketch is a graph where some points $X$ have an associated potential identity arrow ${\mathit{id}}_X\colon X\to X$, some pairs of consecutive arrows $f\colon X\to Y$, $g\colon Y\to Z$ have an associated potential composed arrow $g\circ f\colon X\to Z$, some diagrams $\Delta$ have an associated potential limit, which is a cone with base $\Delta$, and when there is a potential limit with base $\Delta$ then some cones with base $\Delta$ have an associated potential tuple, which is a morphism of cones with base $\Delta$ from the given cone to the potential limit cone. A morphism of limit sketches is a morphism of graphs which preserves the potential features. This yields the category of limit sketches. Whenever this definition is restricted to potential limits with a finite discrete base (called potential finite products), we get the category of finite product sketches: this is the category ${\mathbf{S}}_{{\mathit{eq}}}$ of equational specifications, from section \[subsec:equa\]. \[defi:realization\] Given a limit sketch ${\mathbf{E}}$ and a category ${\mathbf{C}}$, a realization (or loose model) of ${\mathbf{E}}$ with values in ${\mathbf{C}}$ is a graph homomorphism which maps the potential features of ${\mathbf{E}}$ to real features of ${\mathbf{C}}$. A morphism of realizations is (an obvious generalization of) a natural transformation. This gives rise to the category ${\mathit{Real}}({\mathbf{E}},{\mathbf{C}})$ of realizations of ${\mathbf{E}}$ with values in ${\mathbf{C}}$. By default, ${\mathbf{C}}$ is the category of sets. By default, ${\mathbf{C}}$ is the category of sets. A category is called locally presentable if it is equivalent to the category of set-valued realizations of a limit sketch ${\mathbf{E}}$; then ${\mathbf{E}}$ is called a limit sketch for this category. Let ${{\overline}{{\mathbf{E}}}}$ denote the category generated by ${\mathbf{E}}$ such that every potential potential feature of ${\mathbf{E}}$ becomes a real feature of ${{\overline}{{\mathbf{E}}}}$. The Yoneda contravariant realization ${\mathcal{Y}}_{{\mathbf{E}}}$ of ${\mathbf{E}}$ is the contravariant realization of ${\mathbf{E}}$ with values in ${\mathit{Real}}({\mathbf{E}})$ such that ${\mathcal{Y}}_{{\mathbf{E}}}(E)={\mathit{Hom}}_{{{\overline}{{\mathbf{E}}}}}(E,-)$ for every point or arrow $E$ in ${\mathbf{E}}$. Then for each theory ${\Theta}$ and each point $E$ in ${\mathbf{E}}$, the set ${\Theta}(E)$ is in bijection with ${\mathit{Hom}}_{{\mathit{Real}}({\mathbf{E}})}({\mathcal{Y}}_{{\mathbf{E}}}(E),{\Theta})$. The Yoneda contravariant realization is injective on objects and faithful. In addition it is dense: although ${\mathit{Real}}({\mathbf{E}})$ may be “much larger” than ${\mathbf{E}}$, every realization of ${\mathbf{E}}$ is the vertex of a colimit with its base in the image of ${\mathcal{Y}}_{{\mathbf{E}}}$. Let ${\mathbf{e}}\colon {\mathbf{E}}_1\to{\mathbf{E}}_2$ be a morphism of limit sketches and $G_{{\mathbf{e}}}\colon {\mathit{Real}}({\mathbf{E}}_2)\to{\mathit{Real}}({\mathbf{E}}_1)$ the precomposition with ${\mathbf{e}}$. A fundamental result due to Ehresmann states that there is an adjunction, that will be called the adjunction associated with ${\mathbf{e}}$: $$\xymatrix@C=4pc{ {\mathit{Real}}({\mathbf{E}}_1) \ar@/^/[r]^{F_{{\mathbf{e}}}} \ar@{}[r]\|{\bot} & {\mathit{Real}}({\mathbf{E}}_2) \ar@/^/[l]^{G_{{\mathbf{e}}}} }$$ Moreover, the functor $F_{{\mathbf{e}}}$ (contravariantly) extends ${\mathbf{e}}$ via the Yoneda contravariant realizations, in the sense that there is an isomorphism: $$F_{{\mathbf{e}}} \circ {\mathcal{Y}}_{{\mathbf{E}}_1} {\cong}{\mathcal{Y}}_{{\mathbf{E}}_2} \circ {\mathbf{e}}\;.$$ Our definition of forgetful and free functors relies on this adjunction. \[defi:free\] A forgetful functor is a functor of the form $G=-\circ{\mathbf{e}}\colon {\mathit{Real}}({\mathbf{E}}_2)\to{\mathit{Real}}({\mathbf{E}}_1)$ for a morphism of limit sketches ${\mathbf{e}}\colon {\mathbf{E}}_1\to{\mathbf{E}}_2$. A free functor is a left adjoint to a forgetful functor (as every adjoint functor, it is unique up to a natural isomorphism). \[rema:free\] It is easy to describe the forgetful functor $G_{{\mathbf{e}}}$, using its definition: for each realization $R_2$ of ${\mathbf{E}}_2$, the realization $R_1=G_{{\mathbf{e}}}(R_2)$ of ${\mathbf{E}}_1$ is such that $R_1(E)=R_2({\mathbf{e}}(E))$ for every point or arrow $E$ in ${\mathbf{E}}_1$. It is also quite easy to describe the left adjoint functor $F_{{\mathbf{e}}}$, using the fact that $F_{{\mathbf{e}}}$ extends ${\mathbf{e}}$: let $R_1$ be a realization of ${\mathbf{E}}_1$ and $R_2=F_{{\mathbf{e}}}(R_1)$, if $R_1={\mathcal{Y}}_{{\mathbf{E}}_1}(E)$ for some point $E$ in ${\mathbf{E}}_1$ then $R_2={\mathcal{Y}}_{{\mathbf{E}}_2}({\mathbf{e}}(E))$, and the general case follows thanks to the density of ${\mathcal{Y}}_{{\mathbf{E}}_1}$ and to the fact that $F_{{\mathbf{e}}}$ preserves colimits (since it is a left adjoint). A limit sketch for equational theories {#subsec:sk-equa} -------------------------------------- The construction of various “sketches of categories” and “sketches of sketches” is a classical exercise about sketches [@CL84; @CL88; @BW99]. Here we build (a significant part of) a limit sketch ${\mathbf{E}}_{{\mathit{eq}}}$ for the category ${\mathbf{T}}_{{\mathit{eq}}}$ of equational theories, i.e., for the category of categories with chosen products. ### $\bullet$ Graphs {#bullet-graphs .unnumbered} Let us start from the following limit sketch ${\mathbf{E}}_{{\mathit{gr}}}$ for the category of graphs, simply made of two points ${\mathtt{Type}}$ and ${\mathtt{Term}}$ (for types and terms) and two arrows ${\mathtt{dom}}$ and ${\mathtt{codom}}$ (for domain and codomain): $$\xymatrix@C=4pc{ \;{\mathtt{Type}}\; & \;{\mathtt{Term}}\; \ar[l]_{{\mathtt{dom}}} \ar@<1ex>[l]^{{\mathtt{codom}}} \\ }$$ The image of ${\mathbf{E}}_{{\mathit{gr}}}$ by its Yoneda contravariant realization is the following diagram of graphs: $$\begin{array}{\|c\|c\|c\|} \cline{1-1}\cline{3-3} \xymatrix{ X \\} & \xymatrix@C=3pc{ \mbox{} \ar[r]^{X\mapsto X} \ar@<-2ex>[r]_{X\mapsto Y} & \mbox{} \\} & \xymatrix{ X \ar[r]^{f} & Y \\} \\ \cline{1-1}\cline{3-3} \end{array}$$ ### $\bullet$ Categories {#bullet-categories .unnumbered} First, let us build a limit sketch ${\mathbf{E}}'_{{\mathit{gr}}}$ by adding to ${\mathbf{E}}_{{\mathit{gr}}}$ a point ${\mathtt{Cons}}$ for consecutive terms, as the vertex of the following potential limit, where the projections ${\mathtt{fst}}$ and ${\mathtt{snd}}$ stand for the first and second component of a pair of consecutive terms and ${\mathtt{middle}}$ stands for its “middle type” (codomain of the first component and domain of the second one): $$\xymatrix@=1.5pc{ & {\mathtt{Cons}}\ar[dl]_{{\mathtt{fst}}} \ar[dr]^{{\mathtt{snd}}} \ar[d]\|{{\mathtt{middle}}} & \\ {\mathtt{Term}}\ar[r]_{{\mathtt{codom}}} & {\mathtt{Type}}& {\mathtt{Term}}\ar[l]^{{\mathtt{dom}}} \\ }$$ hence the equations $ {\mathtt{codom}}\circ{\mathtt{fst}}= {\mathtt{middle}}\,,\; {\mathtt{dom}}\circ{\mathtt{snd}}= {\mathtt{middle}}$ hold, so that ${\mathtt{middle}}$ may be omitted. Adding such a potential limit, with new vertex and projections over a known base, is an equivalence of limit sketches: the realizations of ${\mathbf{E}}'_{{\mathit{gr}}}$ are still the graphs. Now, a limit sketch ${\mathbf{E}}_{{\mathit{cat}}}$ for categories is obtained by adding to ${\mathbf{E}}'_{{\mathit{gr}}}$ two arrows ${\mathtt{selid}}$ for the selection of identities and ${\mathtt{comp}}$ for the composition and several equations: $$\xymatrix@C=4pc{ \;{\mathtt{Type}}\; \ar@<1ex>@/^/[r]^{{\mathtt{selid}}} & \;{\mathtt{Term}}\; \ar[l]_{{\mathtt{dom}}} \ar@<1ex>[l]^{{\mathtt{codom}}} & \;{\mathtt{Cons}}\; \ar[l]_{{\mathtt{fst}}} \ar@<1ex>[l]^{{\mathtt{snd}}} \ar@<-1ex>@/_/[l]_{{\mathtt{comp}}} \\ }$$ $${\mathtt{dom}}\circ{\mathtt{selid}}= {\mathtt{id}}_{{\mathtt{Type}}} \,,\; {\mathtt{codom}}\circ{\mathtt{selid}}= {\mathtt{id}}_{{\mathtt{Type}}} \,,\; {\mathtt{dom}}\circ{\mathtt{comp}}= {\mathtt{dom}}\circ{\mathtt{fst}}\,,\; {\mathtt{codom}}\circ{\mathtt{comp}}= {\mathtt{codom}}\circ{\mathtt{snd}}\,,$$ and with the equations which ensure that the three axioms of categories are satisfied. The image of this part of ${\mathbf{E}}_{{\mathit{cat}}}$ by its Yoneda contravariant realization is the following diagram of categories: $$\begin{array}{\|c\|c\|c\|c\|c\|} \cline{1-1}\cline{3-3} \cline{5-5} \xymatrix{ X \ar@(lu,ru)^(.6){{\mathit{id}}_X} \\} & \xymatrix@C=3pc{ \mbox{} \ar[r]^{X\mapsto X} \ar@<-2ex>[r]_{X\mapsto Y} & \mbox{} \ar@<-3ex>@/_/[l]_{f\mapsto{\mathit{id}}_X} \\ } & \xymatrix{ X \ar@(lu,ru)^(.6){{\mathit{id}}_X} \ar[r]^{f} & Y \ar@(lu,ru)^(.6){{\mathit{id}}_Y} \\} & \xymatrix@C=3pc{ \mbox{} \ar[r]^{f\mapsto f} \ar@<-2ex>[r]_{f\mapsto g} \ar@<3ex>@/^/[r]^{f\mapsto g\circ f} & \mbox{} \\} & \xymatrix{ X \ar@(lu,ru)^(.6){{\mathit{id}}_X} \ar[r]^{f} \ar@/_4ex/[rr]_{g\circ f} & Y \ar@(lu,ru)^(.6){{\mathit{id}}_Y} \ar[r]^{g} & Z \ar@(lu,ru)^(.6){{\mathit{id}}_Z} \\} \\ \cline{1-1}\cline{3-3} \cline{5-5} \end{array}$$ ### $\bullet$ Theories {#bullet-theories .unnumbered} We build a limit sketch ${\mathbf{E}}'_{{\mathit{cat}}}$ by adding to ${\mathbf{E}}_{{\mathit{cat}}}$ for each $n\in{\mathbb{N}}$ the following potential limits, with vertex ${\mathtt{Type}^\mathtt{n}}$ for $n$-tuples of types and ${\mathtt{n\texttt{-}Cone}}$ for $n$-ary discrete cones: $$\xymatrix@=1pc{ & {\mathtt{Type}^\mathtt{n}}\ar[dl]_{{\mathtt{b1}}} \ar[dr]^{{\mathtt{bn}}} & \\ {\mathtt{Type}}& \dots \qquad \dots & {\mathtt{Type}}\\ } \qquad\qquad \xymatrix@=1pc{ & {\mathtt{n\texttt{-}Cone}}\ar[dl]_{{\mathtt{c1}}} \ar[dr]^{{\mathtt{cn}}} \ar[dd]\|(.4){{\mathtt{vertex}}} & \\ {\mathtt{Term}}\ar[rd]_{{\mathtt{dom}}} & \dots \qquad \dots & {\mathtt{Term}}\ar[ld]^{{\mathtt{dom}}} \\ & {\mathtt{Type}}& \\ }$$ The arrow ${\mathtt{vertex}}$ may be omitted when $n>0$. When $n=0$, the potential limits mean that ${\mathtt{Type}^\mathtt{0}}$ is a unit type (also denoted ${\mathtt{Unit}}$) and ${\mathtt{0\texttt{-}Cone}}$ is isomorphic to ${\mathtt{Type}}$. We also add the tuple ${\mathtt{n\texttt{-}base}}={\langle{\mathtt{codom}}\circ{\mathtt{c1}}, \dots, {\mathtt{codom}}\circ{\mathtt{cn}}\rangle}$ which maps each cone to its base: $$\xymatrix@C=4pc{ {\mathtt{Type}^\mathtt{n}}& {\mathtt{n\texttt{-}Cone}}\ar[l]_{{\mathtt{n\texttt{-}base}}} }$$ The realizations of ${\mathbf{E}}'_{{\mathit{cat}}}$ are still the categories. Now, a limit sketch ${\mathbf{E}}_{{\mathit{th}}}$ for equational theories is obtained by adding to ${\mathbf{E}}'_{{\mathit{cat}}}$ the following features, for each $n\in{\mathbb{N}}$. First an arrow ${\mathtt{n\texttt{-}prod}}\colon {\mathtt{Type}^\mathtt{n}}\to{\mathtt{n\texttt{-}Cone}}$ together with the equation ${\mathtt{n\texttt{-}base}}\circ{\mathtt{n\texttt{-}prod}}={\mathtt{id}}_{{\mathtt{Type}^\mathtt{n}}}$, for building the product cone of each family of $n$ types. Then for building tuples, an arrow ${\mathtt{n\texttt{-}tuple}}\colon {\mathtt{n\texttt{-}Cone}}\to{\mathtt{Term}}$ together with the equations ${\mathtt{dom}}\circ{\mathtt{n\texttt{-}tuple}}={\mathtt{vertex}}$ and ${\mathtt{codom}}\circ{\mathtt{n\texttt{-}tuple}}={\mathtt{vertex}}\circ{\mathtt{n\texttt{-}prod}}\circ{\mathtt{n\texttt{-}base}}$ and with several additional equations for ensuring that the universal property of a product is satisfied. So, here is a relevant part of this limit sketch ${\mathbf{E}}_{{\mathit{th}}}$ for theories (equations are omitted, and only one arity $n$ is represented): $$\xymatrix@C=4pc@R=3pc{ \;{\mathtt{Type}}\; \ar@<1ex>@/^/[r]^{{\mathtt{selid}}} & \;{\mathtt{Term}}\; \ar[l]_{{\mathtt{dom}}} \ar@<1ex>[l]^{{\mathtt{codom}}} & \;{\mathtt{Cons}}\; \ar[l]_{{\mathtt{fst}}} \ar@<1ex>[l]^{{\mathtt{snd}}} \ar@<-1ex>@/_/[l]_{{\mathtt{comp}}} \\ \;{\mathtt{Type}^\mathtt{n}}\; \ar@<1ex>[u]^{{\mathtt{b1}}} \ar@{}[u]\|{\dots} \ar@<-1ex>[u]_{{\mathtt{bn}}} \ar@<-1ex>@/_/[r]_{{\mathtt{n\texttt{-}prod}}} & \;{\mathtt{n\texttt{-}Cone}}\; \ar@<1ex>[u]^{{\mathtt{c1}}} \ar@{}[u]\|{\dots} \ar@<-1ex>[u]_{{\mathtt{cn}}} \ar@<-2ex>@/_/[u]_{{\mathtt{n\texttt{-}tuple}}} \ar[l]_{{\mathtt{n\texttt{-}base}}} \ar[ul]\|{{\mathtt{vertex}}} & }$$ Let us focus on the following part of ${\mathbf{E}}_{{\mathit{th}}}$: $$\xymatrix@C=4pc{ \;{\mathtt{Type}^\mathtt{2}}\; \ar@<-1ex>@/_/[r]_{{\mathtt{2\texttt{-}prod}}} & \;{\mathtt{2\texttt{-}Cone}}\; \ar[l]_{{\mathtt{2\texttt{-}base}}} \ar@/_/[r]_{{\mathtt{2\texttt{-}tuple}}} & \;{\mathtt{Term}}\; \\ }$$ and its image by the Yoneda contravariant realization (only presentations are given): $$\begin{array}{\|c\|c\|c\|c\|c\|} \cline{1-1}\cline{3-3}\cline{5-5} \xymatrix@C=3pc@R=1pc{ X & \\ & X{\!\times\!}Y \ar[lu]_{p} \ar[ld]^{q} \\ Y & \\ } & \xymatrix@C=3pc@R=1pc{ \\ \mbox{} \ar@<1ex>[r]^{\subseteq} & \mbox{} \ar@<1ex>@/^/[l]^{Z\mapsto X{\!\times\!}Y} \\ } & \xymatrix@C=3pc@R=1pc{ & X & \\ Z \ar[ru]^{f} \ar[rd]_{g} \ar[rr]\|{\,{\langlef,g\rangle}\,} & \ar@{}[u]\|{=}\ar@{}[d]\|{=} & X{\!\times\!}Y \ar[lu]_{p} \ar[ld]^{q} \\ & Y & \\ } & \xymatrix@C=3pc@R=1pc{ \\ \mbox{} & \mbox{} \ar@/^/[l]^{h\mapsto {\langlef,g\rangle}} \\ } & \xymatrix@C=3pc@R=1pc{ \mbox{} \\ W \ar[r]^{h} & T \\ } \\ \cline{1-1}\cline{3-3}\cline{5-5} \end{array}$$ The parameterization process is a free functor {#subsec:sk-free} ---------------------------------------------- ### $\bullet$ Parameterized theories {#bullet-parameterized-theories .unnumbered} A limit sketch ${\mathbf{E}}_A$ for parameterized theories is obtained by adding to ${\mathbf{E}}_{{\mathit{th}}}$ an arrow ${\mathtt{A}}\colon {\mathtt{Unit}}\to {\mathtt{Type}}$. ### $\bullet$ Decorated theories {#bullet-decorated-theories .unnumbered} A limit sketch ${\mathbf{E}}_{{\mathit{dec}}}$ for decorated theories comes with a morphism ${\mathbf{e}}_{{\mathit{undec}}}\colon {\mathbf{E}}_{{\mathit{dec}}}\to{\mathbf{E}}_{{\mathit{th}}}$ which forgets about the decorations (“$\mathit{undec}$” for “undecoration”). Here are two slightly different choices, the first one is simpler but the second one better reflects the idea of decoration. A limit sketch ${\mathbf{E}}_{{\mathit{dec}}}$ for decorated theories is made of two related copies of ${\mathbf{E}}_{{\mathit{th}}}$: one copy ${\mathbf{E}}{\mathtt{.p}}$ for the pure features and another copy ${\mathbf{E}}{\mathtt{.g}}$ for the general features, together with a monomorphic transition arrow ${\mathtt{t}}_{{\mathtt{E}}} \colon {\mathtt{E}}{\mathtt{.p}}\to {\mathtt{E}}{\mathtt{.g}}$ for each point ${\mathtt{E}}$ in ${\mathbf{E}}_{{\mathit{th}}}$ with ${\mathtt{t}}_{{\mathtt{Type}}}$ an identity and with the transition equations ${\mathtt{t}}_{{\mathtt{E}}'}\circ{\mathtt{e}}{\mathtt{.p}}={\mathtt{e}}{\mathtt{.g}}\circ{\mathtt{t}}_{{\mathtt{E}}}$ for each arrow ${\mathtt{e}}\colon {\mathtt{E}}\to{\mathtt{E}}'$ in ${\mathbf{E}}_{{\mathit{th}}}$. The morphism ${\mathbf{e}}_{{\mathit{undec}}}$ maps both copies ${\mathbf{E}}{\mathtt{.p}}$ and ${\mathbf{E}}{\mathtt{.g}}$ to ${\mathbf{E}}$. Another limit sketch for decorated theories, still denoted ${\mathbf{E}}_{{\mathit{dec}}}$, is the sketch of elements (similar to the more usual category of elements) of a model ${\mathit{\Delta}}$ of ${\mathbf{E}}_{{\mathit{th}}}$ with values in ${\mathbf{T}}_{{\mathit{eq}}}$, then the morphism ${\mathbf{e}}_{{\mathit{undec}}}$ is provided by the construction. This model ${\mathit{\Delta}}$ formalizes the fact that identities and projections are always pure while the composition or pairing of pure terms is pure. Precisely, the theory ${\mathit{\Delta}}({\mathtt{Type}})$ is generated by one type $D$ and the theory ${\mathit{\Delta}}({\mathtt{Term}})$ by two types $p$, $g$ (for “pure” and “general”, respectively) and a monomorphism $p\to g$ (for “every pure term can be seen as a general term”). As for the functors, ${\mathit{\Delta}}({\mathtt{selid}})$ maps $D$ to $p$, ${\mathit{\Delta}}({\mathtt{n\texttt{-}prod}})$ maps ${\langleD,\dots,D\rangle}$ to $p$, while ${\mathit{\Delta}}({\mathtt{comp}})$ maps ${\langlep,p\rangle}$ to $p$ and ${\langlep,g\rangle},{\langleg,p\rangle},{\langleg,g\rangle}$ to $g$ and ${\mathit{\Delta}}({\mathtt{n\texttt{-}tuple}})$ maps ${\langlep,\dots,p\rangle}$ to $p$ and everything else to $g$. The resulting sketch of elements ${\mathbf{E}}_{{\mathit{dec}}}$ is made of one point ${\mathtt{Type}}\mathtt{.D}$ over the point ${\mathtt{Type}}$ of ${\mathbf{E}}_{{\mathit{th}}}$, two points ${\mathtt{Term}}{\mathtt{.p}}$ and ${\mathtt{Term}}{\mathtt{.g}}$ over the point ${\mathtt{Term}}$ of ${\mathbf{E}}_{{\mathit{th}}}$, four points over ${\mathtt{Cons}}$, $2^n$ over ${\mathtt{Type}^\mathtt{n}}$ and ${\mathtt{n\texttt{-}Cone}}$, a monomorphic arrow ${\mathtt{Term}}{\mathtt{.p}}\to {\mathtt{Term}}{\mathtt{.g}}$, and so on. ### $\bullet$ From decorated theories to parameterized theories {#bullet-from-decorated-theories-to-parameterized-theories .unnumbered} Let us consider the functors $F_{{\mathit{exp}}}\colon {\mathbf{T}}_{{\mathit{dec}}}\to{\mathbf{T}}_A$ and $G_{{\mathit{exp}}}\colon {\mathbf{T}}_A\to{\mathbf{T}}_{{\mathit{dec}}}$ from section \[subsec:gene\]. We can now prove that $G_{{\mathit{exp}}}$ is a forgetful functor and $F_{{\mathit{exp}}}$ is its associated free functor, in the sense of definition \[defi:free\]. \[theo:free\] There is a morphism of limit sketches ${\mathbf{e}}_{{\mathit{exp}}}\colon {\mathbf{E}}_{{\mathit{dec}}} \to {\mathbf{E}}_A$ such that the associated adjunction is $F_{{\mathit{exp}}}\dashv G_{{\mathit{exp}}}$ from section \[subsec:gene\]. In section \[subsec:gene\] the functor $F_{{\mathit{exp}}}$ has been defined on ${\mathcal{Y}}({\mathbf{E}}_{{\mathit{dec}}})$ (see figure \[fig:F-elem\]). Since $F_{{\mathit{exp}}}$ extends ${\mathbf{e}}_{{\mathit{exp}}}$ via the Yoneda contravariant realizations, this provides the definition of a unique morphism ${\mathbf{e}}_{{\mathit{exp}}}$ with associated left adjoint $F_{{\mathit{exp}}}$. For instance, the point ${\mathtt{Term}}{\mathtt{.p}}$ is mapped to ${\mathtt{Term}}$ and the point ${\mathtt{Term}}{\mathtt{.g}}$ to the point of ${\mathbf{E}}_A$ characterized by the fact that its image by Yoneda is presented by $X$, $Y$ and $f'\colon A\times X\to Y$. Then it is easy to check that the precomposition with ${\mathbf{e}}_{{\mathit{exp}}}$ is the functor $G_{{\mathit{exp}}}$. ### $\bullet$ A span of limit sketches {#bullet-a-span-of-limit-sketches .unnumbered} Altogether, the following span of limit sketches provides a framework for the process that starts from an equational theory, choose the pure terms, and forms the corresponding parameterized theory: $$\xymatrix@C=4pc { {\mathbf{E}}_{{\mathit{eq}}} & {\mathbf{E}}_{{\mathit{dec}}} \ar[l]_{{\mathbf{e}}_{{\mathit{undec}}}} \ar[r]^{{\mathbf{e}}_{{\mathit{exp}}}} & {\mathbf{E}}_A \\ \\ }$$ A limit sketch for equational specifications {#subsec:sk-spec} -------------------------------------------- In this section we build a limit sketch ${\mathbf{E}}_{{\mathit{sp}}}$ for equational specifications from the limit sketch ${\mathbf{E}}_{{\mathit{th}}}$ for theories, thus providing another point of view on the elementary specifications in section \[subsec:equa\]. This construction can be seen as an illustration of the factorization theorem in [@Du03]. A direct detailed construction of a limit sketch for equational specifications can be found in appendix \[subsec:dia-spec\]. In the part of ${\mathbf{E}}_{{\mathit{th}}}$ shown in section \[subsec:sk-equa\] there are four arrows that are neither in ${\mathbf{E}}_{{\mathit{gr}}}$ nor projections in a potential limit: ${\mathtt{selid}}$, ${\mathtt{comp}}$, ${\mathtt{n\texttt{-}prod}}$, ${\mathtt{n\texttt{-}tuple}}$. These arrows stand for features that are always defined in a theory but only partially defined in a specification. So, ${\mathbf{E}}_{{\mathit{sp}}}$ is obtained by replacing each of these arrows ${\mathtt{e}}\colon {\mathtt{E}}_1\to{\mathtt{E}}_2$ by a span: $$\xymatrix@C=3pc{ {\mathtt{E}}_1 & \;\;{\mathtt{E}}'_1\;\; \ar@{>->}[l]_{{\mathtt{e}}'_1} \ar[r]^{{\mathtt{e}}'} & {\mathtt{E}}_2 \\ }$$ where the arrow “$\rightarrowtail$” stands for a potential monomorphism (which can be expressed as a potential limit). So, here is (a relevant part of) ${\mathbf{E}}_{{\mathit{sp}}}$: $$\xymatrix@C=4pc@R=3pc{ \;{\mathtt{Selid}}\;\; \ar@{>->}[r] \ar@/^4ex/[rr]^{{\mathtt{selid}}} & \;{\mathtt{Type}}\; & \;{\mathtt{Term}}\; \ar[l]_{{\mathtt{dom}}} \ar@<1ex>[l]^{{\mathtt{codom}}} & \;{\mathtt{Cons}}\; \ar[l]_{{\mathtt{fst}}} \ar@<1ex>[l]^{{\mathtt{snd}}} & \;\;{\mathtt{Comp}}\; \ar@{>->}[l] \ar@/_4ex/[ll]_{{\mathtt{comp}}} \\ \;{\mathtt{n\texttt{-}Prod}}\;\; \ar@{>->}[r] \ar@/_4ex/[rr]_{{\mathtt{n\texttt{-}prod}}} & \;{\mathtt{Type}^\mathtt{n}}\; \ar@<1ex>[u]^{{\mathtt{b1}}} \ar@{}[u]\|{\dots} \ar@<-1ex>[u]_{{\mathtt{bn}}} & \;{\mathtt{n\texttt{-}Cone}}\; \ar@<1ex>[u]^{{\mathtt{c1}}} \ar@{}[u]\|{\dots} \ar@<-1ex>[u]_{{\mathtt{cn}}} \ar[l]_{{\mathtt{n\texttt{-}base}}} \ar[ul]\|{{\mathtt{vertex}}} & \;\;{\mathtt{n\texttt{-}Tuple}}\; \ar@{>->}[l] \ar@/^/[ul]_(.4){{\mathtt{n\texttt{-}tuple}}} & \\ }$$ The elementary specifications from section \[subsec:equa\] are the images by the Yoneda contravariant realization of the points in ${\mathbf{E}}_{{\mathit{sp}}}$ which are not vertices of potential limits, namely: ${\mathtt{Type}}$, ${\mathtt{Term}}$, ${\mathtt{Selid}}$, ${\mathtt{Comp}}$, ${\mathtt{n\texttt{-}Prod}}$, ${\mathtt{n\texttt{-}Tuple}}$: our notations are such that ${\Sigma}_{{\mathtt{E}}}={\mathcal{Y}}({\mathtt{E}})$ for each of these points ${\mathtt{E}}$. Conclusion ========== This paper provides a neat categorical formalization for the parameterization process in Kenzo and EAT. Future work includes the generalization of this approach from equational theories to other families of theories, like distributive categories, and to more general kinds of parameters, like data types. [99]{} Michael Barr, Charles Wells. Category Theory for Computing Science. Centre de Recherches Mathématiques (CRM) Publications, 3rd Edition, 1999. Hartmut Ehrig, Hans-Jörg Kreowski, James Thatcher, Eric Wagner, Jesse Wright. Parameterized Data Types in Algebraic Specification Languages. Springer. Lecture Notes in Computer Science 85, p. 157–168 (1980). Laurent Coppey, Christian Lair. Leçons de Théorie des Esquisses. Diagrammes 12 (1984). Laurent Coppey, Christian Lair. Leçons de Théorie des Esquisses. Diagrammes 19 (1988). César Domínguez, Laurenano Lambán, Julio Rubio. Object-Oriented Institutions to Specify Symbolic Computation Systems. Rairo - Theoretical Informatics and Applications 41, p. 191–214 (2007). César Domínguez, Julio Rubio, Francis Sergeraert. Modeling Inheritance as Coercion in the Kenzo System. Journal of Universal Computer Science 12 (12), p. 1701–1730 (2006). César Domínguez, Dominique Duval, Laureano Lambán, Julio Rubio. Towards Diagrammatic Specifications of Symbolic Computation Systems. In: Mathematics, Algorithms, Proofs. T. Coquand, H. Lombardi, M. Roy (Eds.). Dagstuhl Seminar 05021 (2005). <http://drops.dagstuhl.de/portals/index.php?semnr=05021>. Xavier Dousson, Francis Sergeraert, Yvon Siret. The Kenzo Program. Institut Fourier, Grenoble (1999). [http://www-fourier.ujf-grenoble.fr/\~ sergerar/Kenzo](http://www-fourier.ujf-grenoble.fr/~ sergerar/Kenzo). Dominique Duval. Diagrammatic Specifications. Mathematical Structures in Computer Science 13, p. 857–890 (2003). Dominique Duval. Diagrammatic Inference. arXiv:0710.1208v1 (2007). 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Journal of Pure and Applied Algebra 115, p. 49–79 (1997). Andrew M. Pitts. Categorical Logic. Chapter 2 of S. Abramsky and D. M. Gabbay and T. S. E. Maibaum (Eds). Handbook of Logic in Computer Science, Volume 5. Algebraic and Logical Structures. Oxford University Press, 2000. Julio Rubio, Francis Sergeraert, Yvon Siret. EAT: Symbolic Software for Effective Homology Computation. Institut Fourier, Grenoble (1997). <ftp://fourier.ujf-grenoble.fr/pub/EAT>. J.J.M.M. Rutten. Universal Coalgebra: a Theory of Systems. Theoretical Computer Science 249 (1), p. 3–80 (2000). Diagrammatic logics {#sec:dia} =================== In this paper we have introduced the equational logic in a categorical way, considering equational theories as categories with chosen finite products. An equational theory can be presented by an equational specification, which means that this specification generates the theory. In section \[sec:free\] we have outlined the construction first of a limit sketch for the equational theories and then of a limit sketch for the equational specifications. This appendix provides a detailed description of these limit sketches, with slightly more subtle definitions of equational theories and specifications, which are better suited for formalizing equational proofs. In addition, as often in the framework of algebraic specifications (as for instance in [@LEW96] and in [@DDLR05]) we consider first the specifications, then we get the theories by using the inference rules of the equational logic. Finally, the parameterization process is presented from this point of view. The framework of diagrammatic logics [@Du03; @Du07] is well suited for dealing with “usual” logics like the equational logic as well as with more “unusual” ones like the decorated equational logic, and also for dealing with various morphisms of logics, for instance we will see that the parameterizing functor stems from a morphism of logics. This appendix can be seen as an introduction to diagrammatic logics, based on section \[subsec:sketch\] about limit sketches. Equational logic, revisited {#subsec:dia-equ} --------------------------- As in section \[sec:defi\], instead of the algebraic definition of equational specifications given for instance in [@LEW96], we define equational specifications from finite product sketches. In the main text we have defined equational specifications exactly as finite product sketches, so that the equations become equalities of arrows: this is all right for defining models but this makes every proof trivial. In this appendix we give a more subtle definition of equational specifications as finite product sketches with equations; then definition \[defi:equa-spec\] is easily recovered by mapping equations to equalities. In spite of this minor difference, we use the same notations (${\mathbf{S}}_{{\mathit{eq}}}$, ${\mathbf{T}}_{{\mathit{eq}}}$) in this appendix as in the main part of the paper. \[defi:dia-spec\] An equational specification is a limit sketch (definition \[defi:sketch\]) where all the potential limits are potential finite products, together with a set of pairs of parallel terms called the equations and denoted $t_1 \equiv t_2$. A morphism of equational specifications is a morphism of limit sketches which preserves the equations. This yields the category of equational specifications ${\mathbf{S}}_{{\mathit{eq}}}$. Similarly, in this appendix, the equations in an equational theory need not be equalities. Roughly speaking, an equational theory is an equational specification where the equations form an equivalence relation and all the potential features become real up to equations. For this reason the relation $\equiv$ is called a congruence. So, an equational theory is not a category, it is only a bicategory (the congruence defines its 2-cells), but it becomes a category with chosen finite products, as in definition \[defi:equa-thry\], as soon as both members in each equation get identified. Conversely, every category with chosen finite products can be seen as an equational theory where the equations are the equalities. \[defi:dia-thry\] An equational theory is an equational specification where each type has a potential identity, each pair of consecutive terms has a potential composition, each list of types has a potential product, each list of terms with a common domain has a potential tuple, and in addition the equations form a congruence, which means that the relation $\equiv$ is an equivalence relation compatible with composition and that the usual axioms for categories with products are satisfied up to $\equiv$. The category of equational theories ${\mathbf{T}}_{{\mathit{eq}}}$ is the full subcategory of ${\mathbf{S}}_{{\mathit{eq}}}$ with objects the equational theories. It may be noted that the inclusion of ${\mathbf{T}}_{{\mathit{eq}}}$ in ${\mathbf{S}}_{{\mathit{eq}}}$ is faithful. In fact, for products and tuples, only the arities $n=2$ and $n=0$ will be considered: the general case may easily be guessed, or alternatively one can use the fact that all finite products may be recovered from the binary products and a terminal object. A set of inference rules for the equational logic, for generating an equational theory from an equational specification, is presented in figure \[fig:equ\]. When there is no ambiguity, we often omit “equational” and “potential”. $$\begin{array}{\|l\|c\|} \hline \multicolumn{1}{\|c\|}{\textrm{name}} & \textrm{rules} \\ \hline \hline \textrm{composition} & \frac{f:X\to Y \quad g:Y\to Z}{g\circ f:X\to Z} \\ \textrm{identity} & \frac{X}{{\mathit{id}}_X:X\to X} \\ \hline \textrm{equivalence} & \frac{f}{f\equiv f} \qquad \frac{f\equiv g}{g\equiv f} \qquad \frac{f\equiv g \quad g\equiv h}{f\equiv h} \\ \hline \textrm{substitution} & \frac{f:X\to Y \quad g_1\equiv g_2:Y\to Z}{g_1\circ f\equiv g_2\circ f:X\to Z} \\ \textrm{replacement} & \frac{f_1\equiv f_2:X\to Y \quad g:Y\to Z}{g\circ f_1\equiv g\circ f_2:X\to Z} \\ \hline \textrm{associativity} & \frac{f:X\to Y \quad g:Y\to Z \quad h:Z\to W}{(h\circ g)\circ f \equiv h\circ (g\circ f)} \\ \textrm{unit rules} & \frac{f:X\to Y}{f\circ{\mathit{id}}_X\equiv f } \qquad \frac{f:X\to Y}{{\mathit{id}}_Y\circ f \equiv f} \\ \hline \textrm{binary product} & \frac{Y_1 \quad Y_2} {Y_1\times Y_2} \qquad \frac{Y_1 \quad Y_2} { {\mathit{pr}}_1:Y_1\times Y_2\to Y_1} \qquad \frac{Y_1 \quad Y_2} {{\mathit{pr}}_2:Y_1\times Y_2\to Y_2} \\ \textrm{pairing} & \quad \frac{f_1:X\to Y_1 \quad f_2:X\to Y_2} {{\langlef_1,f_2\rangle} :X\to Y_1\times Y_2} \qquad \frac{f_1:X\to Y_1 \quad f_2:X\to Y_2} {{\mathit{pr}}_1\circ {\langlef_1,f_2\rangle} \equiv f_1} \qquad \frac{f_1:X\to Y_1 \quad f_2:X\to Y_2} {{\mathit{pr}}_2\circ {\langlef_1,f_2\rangle} \equiv f_2} \quad \\ \textrm{pairing uniqueness} & \frac{f_1:X\to Y_1\quad f_2:X\to Y_2 \quad f :X\to Y_1\times Y_2 \quad {\mathit{pr}}_1\circ f \equiv f_1 \quad {\mathit{pr}}_2\circ f \equiv f_2 } {{\langlef_1,f_2\rangle} \equiv f} \\ \hline \textrm{terminal type} & \frac{}{\;{1}\;} \\ \textrm{collapsing} & \frac{X}{{\langle\,\rangle}_X:X\to {1}} \\ \textrm{collapsing uniqueness} & \frac{f:X\to {1}} {{\langle\,\rangle}_X \equiv f} \\ \hline \end{array}$$ The models of a specification ${\Sigma}$ with values in a theory ${\Theta}$ are defined as the morphisms of specifications from ${\Sigma}$ to ${\Theta}$. In addition, the morphisms of models of ${\Sigma}$ with values in ${\Theta}$ can be defined in the usual natural way, so that there is a category of models ${\mathit{Mod}}({\Sigma},{\Theta})$ of ${\Sigma}$ with values in ${\Theta}$. The category of set-valued models of ${\Sigma}$ is the category of models of ${\Sigma}$ with values in the category of sets seen as an equational theory, with the cartesian products as potential products and the equalities of functions as equations. Each equational specification in the algebraic sense ${\mathrm{Sp}}$ gives rise to an equational specification ${\Sigma}$: each sort of ${\mathrm{Sp}}$ becomes a type of ${\Sigma}$, each list of sorts $X_1,\dots,X_n$ of ${\mathrm{Sp}}$ becomes a type $X_1\times\dots\times X_n$ of ${\Sigma}$, each operation or term $f\colon X_1\dots X_n\to Y$ of ${\mathrm{Sp}}$ becomes a term $f\colon X_1\times\dots\times X_n\to Y$ of ${\Sigma}$, and each equation $f_1\equiv f_2$ of ${\mathrm{Sp}}$ becomes an equation $f_1\equiv f_2$ of ${\Sigma}$; for this purpose, both terms $f_1$ and $f_2$ in ${\mathrm{Sp}}$ must be considered as terms in all the variables that appear in $f_1$ or in $f_2$, as explained for instance in [@BW99]. Then, the category of models of ${\mathrm{Sp}}$ in the algebraic sense is isomorphic to the category of set-valued models of ${\Sigma}$. \[exam:dia-sg\] The equational specification ${\Sigma}_{{\mathit{sgp}}}$ for semigroups can be represented as a graph with an equation: $$\begin{array}{c\|cc\|} \cline{2-3} {\Sigma}_{{\mathit{sgp}}} = & \xymatrix{ G^2 \ar[r]^{{\mathit{prd}}} & G \\ } & \xymatrix{ \txt{${\mathit{prd}}\circ{\langlex,{\mathit{prd}}\circ{\langley,z\rangle}\rangle}\equiv{\mathit{prd}}\circ{\langle{\mathit{prd}}\circ{\langlex,y\rangle},z\rangle} $} \\ } \\ \cline{2-3} \end{array}$$ However, many details are implicit in this illustration. More precisely, the equational specification ${\Sigma}_{{\mathit{sgp}}}$ can be built as follows. First an equational specification ${\Sigma}_{{\mathit{mgm}}}$ for magmas (a magma is simply a set with a binary operation) is made of two types $G$ and $G^2$, three terms $u,v,{\mathit{prd}}:G^2\to G$ and one potential product $G{\stackrel{u}{\longleftarrow}}G^2{\stackrel{v}{\longrightarrow}}G$. Then, a second equational specification ${\Sigma}_{{\mathit{mgm}}}'$ is obtained by adding to ${\Sigma}_{{\mathit{mgm}}}$ a type $G^3$, terms $x:G^3\to G$, $w:G^3\to G^2$, a potential product $G{\stackrel{x}{\longleftarrow}}G^3{\stackrel{w}{\longrightarrow}}G^2$, and also the terms $f_1={\mathit{prd}}\circ{\langlex,{\mathit{prd}}\circ w\rangle}:G^3\to G$ and $f_2={\mathit{prd}}\circ{\langle{\mathit{prd}}\circ {\langlex,u\circ w\rangle},v\circ w\rangle}:G^3\to G$. We also add $y=u\circ w$ and $z=v\circ w$, and the equations $w\equiv {\langley,z\rangle}$, $f_1\equiv {\mathit{prd}}\circ{\langlex,{\mathit{prd}}\circ{\langley,z\rangle}\rangle}$ and $f_2\equiv {\mathit{prd}}\circ{\langle{\mathit{prd}}\circ{\langlex,y\rangle},z\rangle}$. Then ${\Sigma}_{{\mathit{mgm}}}'$ is equivalent to ${\Sigma}_{{\mathit{mgm}}}$. Finally ${\Sigma}_{{\mathit{sgp}}}$ is made of ${\Sigma}_{{\mathit{mgm}}}'$ with the equation $f_1\equiv f_2$, or equivalently with the equation ${\mathit{prd}}\circ{\langlex,{\mathit{prd}}\circ{\langley,z\rangle}\rangle}\equiv{\mathit{prd}}\circ{\langle{\mathit{prd}}\circ{\langlex,y\rangle},z\rangle}$. In sections \[subsec:dia-spec\] and \[subsec:dia-thry\], these notions are embedded in the definition of a diagrammatic equational logic ${\mathcal{L}}_{{\mathit{eq}}}$ [@Du07]. This means that we build a limit sketch ${\mathbf{E}}_{{\mathit{eq}},S}$ for ${\mathbf{S}}_{{\mathit{eq}}}$, a limit sketch ${\mathbf{E}}_{{\mathit{eq}},T}$ for ${\mathbf{T}}_{{\mathit{eq}}}$, and a morphism of limit sketches ${\mathbf{e}}_{{\mathit{eq}}}:{\mathbf{E}}_{{\mathit{eq}},S}\to{\mathbf{E}}_{{\mathit{eq}},T}$ such that the inclusion functor $G_{{\mathit{eq}}}:{\mathbf{T}}_{{\mathit{eq}}}\to{\mathbf{S}}_{{\mathit{eq}}}$ is the precomposition with ${\mathbf{e}}_{{\mathit{eq}}}$ and its left adjoint $F_{{\mathit{eq}}}:{\mathbf{S}}_{{\mathit{eq}}}\to{\mathbf{T}}_{{\mathit{eq}}}$ (as we saw in section \[subsec:sketch\]) maps each equational specification to its generated theory. Equational specifications {#subsec:dia-spec} ------------------------- In this section we provide a detailed construction of a limit sketch ${\mathbf{E}}_{{\mathit{eq}},S}$ for the category ${\mathbf{S}}_{{\mathit{eq}}}$ of equational specifications; except for equations, we will get essentially the same sketch as ${\mathbf{E}}_{{\mathit{sp}}}$ in section \[subsec:sk-free\]. We begin with the sketch ${\mathbf{E}}_{{\mathit{gr}}}$ for graphs: $$\xymatrix@C=4pc{ \;{\mathtt{Type}}\; & \;{\mathtt{Term}}\; \ar[l]_{{\mathtt{dom}}} \ar@<1ex>[l]^{{\mathtt{codom}}} \\ }$$ Then, we extend ${\mathbf{E}}_{{\mathit{gr}}}$ for each kind of potential features; each limit sketch is followed by its image by its Yoneda contravariant realization. Finally, by glueing together these extensions of ${\mathbf{E}}_{{\mathit{gr}}}$ (by a colimit of limit sketches) we get the limit sketch ${\mathbf{E}}_{{\mathit{eq}},S}$. ### $\bullet$ Composites {#bullet-composites .unnumbered} A sketch ${\mathbf{E}}_{{\mathit{gr\_comp}}}$ for graphs with potential composites is obtained by extending ${\mathbf{E}}_{{\mathit{gr}}}$ as follows, with its potential limit and equalities: $$\xymatrix@C=3pc{ && {\rule[-5pt]{0pt}{0pt}}{\mathtt{Comp}}\ar@{>->}[d]^{{\mathtt{i}}} \ar@<-1ex>[dl]_{{\mathtt{comp}}} \\ {\mathtt{Type}}& {\mathtt{Term}}\ar@<1ex>[l]^{{\mathtt{codom}}} \ar@<-1ex>[l]_{{\mathtt{dom}}} & {\mathtt{Cons}}\ar@<1ex>[l]^{{\mathtt{snd}}} \ar@<-1ex>[l]_{{\mathtt{fst}}} \\ } \qquad \xymatrix@=1.5pc{ & {\mathtt{Cons}}\ar[dl]_{{\mathtt{fst}}} \ar[dr]^{{\mathtt{snd}}} \ar[d]\|{{\mathtt{middle}}} & \\ {\mathtt{Term}}\ar[r]_{{\mathtt{codom}}} & {\mathtt{Type}}& {\mathtt{Term}}\ar[l]^{{\mathtt{dom}}} \\ } \qquad \begin{array}{l} \\ {\mathtt{dom}}\circ{\mathtt{comp}}= {\mathtt{dom}}\circ{\mathtt{fst}}\circ{\mathtt{i}}\\ {\mathtt{codom}}\circ{\mathtt{comp}}= {\mathtt{codom}}\circ{\mathtt{snd}}\circ{\mathtt{i}}\\ \end{array}$$ The point ${\mathtt{Comp}}$ stands for the set of composable terms, the potential mono ${\mathtt{i}}$ for the inclusion, and the arrow ${\mathtt{comp}}$ for the composition of composable terms. The image of ${\mathbf{E}}_{{\mathit{gr\_comp}}}$ by its Yoneda contravariant realization is the following morphism of realizations of ${\mathbf{E}}_{{\mathit{gr\_comp}}}$; as required, the image of the mono ${\mathtt{i}}$ is an epimorphism. $$\begin{array}{\|c\|c\|c\|c\|c\|} \cline{5-5} \multicolumn{4}{c\|}{ } & \xymatrix{ X \ar[r]^{f} \ar@/_4ex/[rr]_{g\circ f} & Y \ar[r]^{g} & Z \\} \\ \cline{5-5} \multicolumn{3}{c}{ } & \multicolumn{1}{c}{ \xymatrix@C=3pc@R=1.5pc{ \mbox{ } & \mbox{ } \\ \mbox{ } \ar[ru]^{f\mapsto g\circ f} & \mbox{ } \\ } } & \multicolumn{1}{c}{ \xymatrix@C=3pc@R=1.5pc{ \mbox{} \\ \mbox{} \ar@{->>}[u]_{{\begin{turn}{90}\ensuremath{\subseteq}\end{turn}}} \\} } \\ \cline{1-1}\cline{3-3}\cline{5-5} \xymatrix{ X \\} & \xymatrix@C=3pc{ \mbox{} \ar@<2ex>[r]^{X\mapsto X} \ar@<-2ex>[r]_{X\mapsto Y}& \mbox{} \\} & \xymatrix{ X \ar[r]^{f} & Y \\} & \xymatrix@C=3pc{ \mbox{} \ar@<2ex>[r]^{f\mapsto f} \ar@<-2ex>[r]_{f\mapsto g}& \mbox{} \\} & \xymatrix{ X \ar[r]^{f} & Y \ar[r]^{g} & Z \\} \\ \cline{1-1}\cline{3-3}\cline{5-5} \end{array}$$ ### $\bullet$ Identities {#bullet-identities .unnumbered} A sketch ${\mathbf{E}}_{{\mathit{gr\_id}}}$ for graphs with potential identities is obtained by extending ${\mathbf{E}}_{{\mathit{gr}}}$ as follows: $$\xymatrix@C=3pc{ {\rule[-5pt]{0pt}{0pt}}{\mathtt{Selid}}\ar@{>->}[d]_{{\mathtt{i0}}} \ar@<1ex>[rd]^{{\mathtt{selid}}} & \\ {\mathtt{Type}}& {\mathtt{Term}}\ar@<1ex>[l]^{{\mathtt{codom}}} \ar@<-1ex>[l]_{{\mathtt{dom}}} \\ } \qquad\qquad \begin{array}{l} {\mathtt{dom}}\circ{\mathtt{selid}}= {\mathtt{id}}_{{\mathtt{Selid}}} \\ {\mathtt{codom}}\circ{\mathtt{selid}}= {\mathtt{id}}_{{\mathtt{Selid}}} \\ \end{array}$$ The point ${\mathtt{Selid}}$ stands for the set of types with a selected identity, the potential mono ${\mathtt{i0}}$ for the inclusion, and the arrow ${\mathtt{selid}}$ for the selection of the identities. $$\begin{array}{\|c\|c\|c\|} \cline{1-1} \xymatrix{ {\rule[-10pt]{0pt}{25pt}}X \ar@(lu,ld)_{{\mathit{id}}_X} \\} & \multicolumn{2}{c}{ } \\ \cline{1-1} \multicolumn{1}{c}{ \xymatrix@C=3pc@R=1.5pc{ \mbox{} \\ \mbox{} \ar@{->>}[u]^{{\begin{turn}{90}\ensuremath{\subseteq}\end{turn}}} \\} } & \multicolumn{1}{c}{ \xymatrix@C=3pc@R=1.5pc{ \mbox{ } & \mbox{ } \\ \mbox{ } & \mbox{ } \ar[lu]_{f\mapsto{\mathit{id}}_X} \\ } } & \multicolumn{1}{c}{ } \\ ### $\bullet$ Binary products {#bullet-binary-products .unnumbered} A sketch ${\mathbf{E}}_{{\mathit{gr\_prod}}}$ for graphs with potential binary products is obtained by extending ${\mathbf{E}}_{{\mathit{gr}}}$ as follows: $$\xymatrix@C=3pc{ & & & {\rule[-5pt]{0pt}{0pt}}{\mathtt{2\texttt{-}Prod}}\ar@{>->}[d]^{{\mathtt{j}}} \ar@<-1ex>[ld]_{{\mathtt{2\texttt{-}prod}}} \\ {\mathtt{Type}}& {\mathtt{Term}}\ar@<1ex>[l]^{\;{\mathtt{codom}}} \ar@<-1ex>[l]_{{\mathtt{dom}}} & {\mathtt{2\texttt{-}Cone}}\ar@<1ex>[l]^{{\mathtt{c2}}} \ar@<-1ex>[l]_{{\mathtt{c1}}} \ar[r]^{{\mathtt{2\texttt{-}base}}} & {\mathtt{Type}^\mathtt{2}}\ar@/^7ex/@<2ex>[lll]^{{\mathtt{b2}}} \ar@/^7ex/[lll]_{{\mathtt{b1}}} \\ } \; \xymatrix@R=1pc@C=.5pc{ & {\mathtt{Type}^\mathtt{2}}\ar[dl]_{{\mathtt{b1}}} \ar[dr]^{{\mathtt{b2}}} & \\ {\mathtt{Type}}&& {\mathtt{Type}}\\ & {\mathtt{2\texttt{-}Cone}}\ar[dl]_{{\mathtt{c1}}} \ar[dr]^{{\mathtt{c2}}} \ar[d]\|(.45){{\mathtt{vertex}}} & \\ {\mathtt{Term}}\ar[r]_{{\mathtt{dom}}} & {\mathtt{Type}}& {\mathtt{Term}}\ar[l]^{{\mathtt{dom}}} \\ } \; \begin{array}{l} \\ {\mathtt{b1}}\circ{\mathtt{2\texttt{-}base}}= {\mathtt{codom}}\circ{\mathtt{c1}}\\ {\mathtt{b2}}\circ{\mathtt{2\texttt{-}base}}= {\mathtt{codom}}\circ{\mathtt{c2}}\\ \end{array}$$ The point ${\mathtt{2\texttt{-}Prod}}$ stands for the set of binary products, the mono ${\mathtt{j}}$ for the inclusion, and the arrow ${\mathtt{2\texttt{-}prod}}$ for the operation which maps a binary product to its underlying binary cone. $$\begin{array}{\|c\|c\|c\|c\|c\|c\|c\|} \cline{7-7} \multicolumn{6}{c\|}{ } & \xymatrix@=1pc{ & Y_1{\!\times\!}Y_2 \ar[ld]_{{\mathit{pr}}_1} \ar[rd]^{{\mathit{pr}}_2} & \\ Y_1 && Y_2 \\} \\ \cline{7-7} \multicolumn{5}{c}{ } & \multicolumn{1}{c}{ \xymatrix@C=3pc@R=1.5pc{ \mbox{ } & \mbox{ } \\ \mbox{ } \ar[ru]_{f_i\mapsto {\mathit{pr}}_i}^{X\mapsto Y_1{\!\times\!}Y_2} & \mbox{ } \\ } } & \multicolumn{1}{c}{ \xymatrix@C=3pc@R=1.5pc{ \mbox{} \\ \mbox{} \ar@{->>}[u]_{{\begin{turn}{90}\ensuremath{\subseteq}\end{turn}}} \\} } \\ \cline{1-1}\cline{3-3}\cline{5-5}\cline{7-7} \xymatrix{ X \\} & \xymatrix@C=2pc{ \mbox{} \ar@<2ex>[r]^{X\mapsto X} \ar@<-2ex>[r]_{X\mapsto Y} & \mbox{} \\} & \xymatrix{ X \ar[r]^{f} & Y \\} & \xymatrix@C=2pc{ \mbox{} \ar@<2ex>[r]^{f\mapsto f_1} \ar@<-2ex>[r]_{f\mapsto f_2} & \mbox{} \\} & \xymatrix@=1pc{ & X \ar[ld]_{f_1} \ar[rd]^{f_2} & \\ Y_1 && Y_2 \\} & \xymatrix@C=2pc{ \mbox{} & \mbox{} \ar[l]_{\supseteq} \\} & \xymatrix@=1pc{ & \mbox{ } & \\ Y_1 && Y_2 \\} \\ \cline{1-1}\cline{3-3}\cline{5-5}\cline{7-7} \end{array}$$ ### $\bullet$ Pairing {#bullet-pairing .unnumbered} Now, a sketch ${\mathbf{E}}_{{\mathit{gr\_pair}}}$ for graphs with potential binary products and with potential pairings (or 2-tuples) is obtained by extending ${\mathbf{E}}_{{\mathit{gr\_prod}}}$ as follows: $$\xymatrix@C=3pc{ & & {\rule[-5pt]{0pt}{0pt}}{\mathtt{Pair}}\ar[r]^{{\mathtt{2\texttt{-}codom}}}\ar@{>->}[d]_{{\mathtt{2\texttt{-}dom}}}\ar[ld]_{{\mathtt{pair}}} & } \qquad \begin{array}{l} {\mathtt{j}}\circ{\mathtt{2\texttt{-}codom}}= {\mathtt{2\texttt{-}base}}\circ{\mathtt{2\texttt{-}dom}}\\ {\mathtt{dom}}\circ{\mathtt{pair}}= {\mathtt{vertex}}\circ{\mathtt{2\texttt{-}dom}}\\ {\mathtt{codom}}\circ{\mathtt{pair}}= {\mathtt{vertex}}\circ{\mathtt{2\texttt{-}prod}}\circ{\mathtt{2\texttt{-}codom}}\\ \end{array}$$ $$\begin{array}{\|c\|c\|c\|c\|c\|c\|c\|} \cline{5-5}\cline{7-7} \multicolumn{4}{c\|}{ } & \xymatrix@=1pc{ & Y_1{\!\times\!}Y_2 \ar[ld]_{{\mathit{pr}}_1} \ar[rd]^{{\mathit{pr}}_2} & \\ Y_1 && Y_2 \\ & X \ar[lu]^{f_1} \ar[ru]_{f_2} \ar[uu]^{g} & \\} & \xymatrix@C=2pc{ \mbox{} & \mbox{} \ar[l]_{\supseteq} \\} & \xymatrix@=1pc{ & Y_1{\!\times\!}Y_2 \ar[ld]_{{\mathit{pr}}_1} \ar[rd]^{{\mathit{pr}}_2} & \\ Y_1 && Y_2 \\} \\ \cline{5-5}\cline{7-7} \multicolumn{3}{c}{ } & \multicolumn{1}{c}{ \xymatrix@C=3pc@R=1.5pc{ \mbox{ } & \mbox{ } \\ \mbox{ } \ar[ru]^{f\mapsto g} & \mbox{ } \\ } } & Y_1 && Y_2 \\ & X \ar[lu]^{f_1} \ar[ru]_{f_2} & \\} & \xymatrix@C=2pc{ \mbox{} & \mbox{} \ar[l]_{\supseteq} \\} & \xymatrix@=1pc{ Y_1 && Y_2 \\ & \mbox{ } & \\ } \\ \cline{1-1}\cline{3-3}\cline{5-5}\cline{7-7} \end{array}$$ ### $\bullet$ Terminal type {#bullet-terminal-type .unnumbered} A terminal (or final) type is a nullary product, and a nullary cone is simply a type (its vertex). With this correspondence in mind, the construction of ${\mathbf{E}}_{{\mathit{gr\_fin}}}$ below is similar to the construction of ${\mathbf{E}}_{{\mathit{gr\_prod}}}$ above, with ${\mathtt{0\texttt{-}Cone}}={\mathtt{Type}}$, ${\mathtt{Type}^\mathtt{0}}={\mathtt{Unit}}$ and ${\mathtt{0\texttt{-}base}}:{\mathtt{Type}}\to{\mathtt{Unit}}$, where ${\mathtt{Unit}}$ is the vertex of a potential limit cone with an empty base, and in addition ${\mathtt{0\texttt{-}Prod}}={\mathtt{Final}}$ and ${\mathtt{0\texttt{-}prod}}={\mathtt{final}}$ $$\xymatrix@C=3pc{ {\rule[-5pt]{0pt}{0pt}}{\mathtt{Final}}\ar@{>->}[d]_{{\mathtt{j0}}} \ar@<1ex>[rd]^{{\mathtt{final}}} & & \\ {\mathtt{Unit}}& {\mathtt{Type}}\ar[l] & {\mathtt{Term}}\ar@<1ex>[l]^{{\mathtt{codom}}} \ar@<-1ex>[l]_{{\mathtt{dom}}} \\ } \qquad\qquad \xymatrix@R=1pc{ & {\mathtt{Unit}}& \\ \mbox{} \ar@{}[rr]\|{\txt{(empty base)}} & & \mbox{} \\ }$$ The point ${\mathtt{Unit}}$ stands for a singleton, the point ${\mathtt{Final}}$ with the arrow ${\mathtt{j0}}$ for a set with at most one element, and the arrow ${\mathtt{final}}$ stands for the selection of the terminal type. $$\begin{array}{\|c\|c\|c\|c\|c\|} \cline{1-1} \xymatrix{ {1}\\} & \multicolumn{4}{c}{ } \\ \mbox{ } & \mbox{ } \ar[lu]_{X\mapsto {1}} \\ } } & \multicolumn{1}{c}{ } \\ \cline{1-1}\cline{3-3}\cline{5-5} \mbox{ } & \xymatrix@C=3pc{ \mbox{} \ar[r]_{\subseteq} & \mbox{} \\} & \cline{1-1}\cline{3-3}\cline{5-5} \end{array}$$ ### $\bullet$ Collapsing {#bullet-collapsing .unnumbered} Now, a sketch ${\mathbf{E}}_{{\mathit{gr\_coll}}}$ for graphs with a potential terminal type and with potential collapsings (or 0-tuples) is obtained by extending ${\mathbf{E}}_{{\mathit{gr\_fin}}}$ as follows: $$\xymatrix@C=3pc{ {\rule[-5pt]{0pt}{0pt}}{\mathtt{0\texttt{-}Prod}}\ar@{>->}[d]_{{\mathtt{j0}}} \ar@<1ex>[rd]^{{\mathtt{0\texttt{-}prod}}} & {\rule[-5pt]{0pt}{0pt}}{\mathtt{Coll}}\ar[l]_{{\mathtt{0\texttt{-}dom}}}\ar@{>->}[d]^{{\mathtt{0\texttt{-}codom}}}\ar[rd]^{{\mathtt{coll}}} & \\ {\mathtt{Unit}}& {\mathtt{Type}}\ar[l]_{{\mathtt{0\texttt{-}base}}} & {\mathtt{Term}}\ar@<1ex>[l]^{{\mathtt{codom}}} \ar@<-1ex>[l]_{{\mathtt{dom}}} \\ } \qquad\qquad \begin{array}{l} {\mathtt{dom}}\circ{\mathtt{coll}}= {\mathtt{0\texttt{-}codom}}\\ {\mathtt{0\texttt{-}prod}}\circ{\mathtt{0\texttt{-}dom}}\\ \end{array}$$ $$\begin{array}{\|c\|c\|c\|c\|c\|} \cline{1-1}\cline{3-3} \xymatrix{ {1}\\} & \xymatrix@C=3pc{ \mbox{} \ar[r]_{\subseteq} & \mbox{} \\} & \xymatrix@=1pc{ X \ar[r]^{g} & {1}\\}& \multicolumn{2}{c}{ } \\ \cline{1-1}\cline{3-3} \mbox{ } & \mbox{ } \ar[lu]_{f\mapsto g} \\ } }\\ ### $\bullet$ Equations {#bullet-equations .unnumbered} A sketch ${\mathbf{E}}_{{\mathit{gr\_eq}}}$ for graphs with equations is obtained by extending ${\mathbf{E}}_{{\mathit{gr}}}$ with two points ${\mathtt{Para}}$ and ${\mathtt{Equa}}$ which stand for the set of pairs of parallel arrows and the set of equations, respectively. The arrows ${\mathtt{left}}$ and ${\mathtt{right}}$ extract the two terms from a pair of parallel terms. The potential limit establishes that ${\mathtt{Para}}$ represents pairs of parallel terms. $$\xymatrix@C=3pc{ {\mathtt{Type}}& {\mathtt{Term}}\ar@<1ex>[l]^{{\mathtt{codom}}} \ar@<-1ex>[l]_{{\mathtt{dom}}} & {\mathtt{Para}}\ar@<1ex>[l]^{{\mathtt{right}}} \ar@<-1ex>[l]_{{\mathtt{left}}} &\ {\mathtt{Equa}}\ar@{>->}[l]^{{\mathtt{equa}}} \\ } \qquad \xymatrix@C=1.5pc{ && {\mathtt{Para}}\ar[dll]_{{\mathtt{left}}} \ar[drr]^{{\mathtt{right}}} && \\ {\mathtt{Term}}\ar@/_4ex/[rrr]_(.3){{\mathtt{codom}}}\ar[r]_{\,{\mathtt{dom}}}& {\mathtt{Type}}&& {\mathtt{Type}}& {\mathtt{Term}}\ar@/^4ex/[lll]^(.3){{\mathtt{dom}}}\ar[l]^{{\mathtt{codom}}} \\ \\ }$$ $$\begin{array}{\|c\|c\|c\|c\|c\|c\|c\|} \mbox{} \ar@<2ex>[r]^{f\mapsto f} \ar@<-2ex>[r]_{f\mapsto g} & \mbox{} \\} & \xymatrix@=1pc{ X \ar@/^/[rr]^{f}\ar@/_/[rr]_{g}& &Y \\} & \xymatrix@C=2pc{ \mbox{} \ar@{->>}[r]_{\subseteq} & \mbox{} \\} & \xymatrix@=1pc{ X\ar@/^/[rr]^{f}\ar@{}[rr]\|{\equiv}\ar@/_/[rr]_{g} & & Y \\} \\ \cline{1-1}\cline{3-3}\cline{5-5}\cline{7-7} \end{array}$$ ### $\bullet$ Equational specifications {#bullet-equational-specifications .unnumbered} Finally, a sketch ${\mathbf{E}}_{{\mathit{eq}},S}$ for equational specifications is obtained as the colimit of the sketches ${\mathbf{E}}_{{\mathit{gr\_comp}}}$, ${\mathbf{E}}_{{\mathit{gr\_id}}}$, ${\mathbf{E}}_{{\mathit{gr\_eq}}}$, ${\mathbf{E}}_{{\mathit{gr\_pair}}}$ and ${\mathbf{E}}_{{\mathit{gr\_coll}}}$ over ${\mathbf{E}}_{{\mathit{gr}}}$. Here is its underlying graph (with ${\mathtt{Type}}$ repeated twice for readablity), in addition it has all the potential limits and all the equalities from the component sketches. $$\xymatrix@C=3pc{ & && {\rule[-5pt]{0pt}{0pt}}{\mathtt{Equa}}\ar@{>->}[d]_{{\mathtt{equa}}} & & & \\ {\rule[-5pt]{0pt}{0pt}}{\mathtt{Final}}\ar@{>->}[d]_{{\mathtt{j0}}} \ar@<1ex>[rd]\|{{\mathtt{final}}} & {\rule[-5pt]{0pt}{0pt}}{\mathtt{Coll}}\ar[l]_{{\mathtt{0\texttt{-}codom}}} \ar@{>->}[d]^{{\mathtt{0\texttt{-}dom}}} \ar@/^7ex/@<1ex>[rrd]^(.4){{\mathtt{coll}}} & {\rule[-5pt]{0pt}{0pt}}{\mathtt{Selid}}\ar@{>->}[d]_{{\mathtt{i0}}} \ar@<1ex>[rd]\|{{\mathtt{selid}}} & {\mathtt{Para}}\ar@<-.5ex>[d]_(.2){{\mathtt{left}}}\ar@<.5ex>[d]^(.2){{\mathtt{right}}} & {\rule[-5pt]{0pt}{0pt}}{\mathtt{Comp}}\ar@{>->}[d]^{{\mathtt{i}}} \ar@<-1ex>[dl]\|{{\mathtt{comp}}} & {\rule[-5pt]{0pt}{0pt}}{\mathtt{Pair}}\ar[r]^{{\mathtt{2\texttt{-}codom}}}\ar@{>->}[d]_{{\mathtt{2\texttt{-}dom}}}\ar@/_7ex/@<-1ex>[lld]_(.4){{\mathtt{pair}}}& {\rule[-5pt]{0pt}{0pt}}{\mathtt{2\texttt{-}Prod}}\ar@{>->}[d]^{{\mathtt{j}}} \ar@<-1ex>[ld]\|{{\mathtt{2\texttt{-}prod}}} \\ {\mathtt{Unit}}& {\mathtt{Type}}\ar[l]_{{\mathtt{0\texttt{-}base}}} \ar@{=}[r] & {\mathtt{Type}}& {\mathtt{Term}}\ar@<1ex>[l]^{{\mathtt{codom}}} \ar@<-1ex>[l]_{{\mathtt{dom}}} & {\mathtt{Cons}}\ar@<1ex>[l]^{{\mathtt{snd}}} \ar@<-1ex>[l]_{{\mathtt{fst}}} & {\mathtt{2\texttt{-}Cone}}\ar@/^5ex/@<2ex>[ll]^(.4){{\mathtt{c2}}} \ar@/^5ex/[ll]_(.4){{\mathtt{c1}}} \ar[r]^{{\mathtt{2\texttt{-}base}}} & {\mathtt{Type}^\mathtt{2}}\ar@/^10ex/@<2ex>[llll]^(.3){{\mathtt{b2}}} \ar@/^10ex/[llll]_(.3){{\mathtt{b1}}} \\ }$$ \[exam:dia-spec-nat\] Let us consider the equational specification ${\Sigma}_{{\mathit{nat}}}$: $$\begin{array}{c\|cc\|} \cline{2-3} {\Sigma}_{{\mathit{nat}}} : & \xymatrix@R=0.6pc{ \mbox{} && N' \ar@/^1pc/[dd]^{p} & \\ \mbox{} && & \\ {1}\ar[rr]^{z} && N \ar@/^1pc/[uu]^{s} & \\ } & \xymatrix@R=0pc{ \mbox{ } \\ \txt{ terminal type : ${1}$} \\ \txt{ equation: $p\circ s \equiv {\mathit{id}}_N$} \\ } \\ \cline{2-3} \end{array}$$ The specification ${\Sigma}_{{\mathit{nat}}}$ has a model “of naturals” $M_{{\mathit{nat}}}$ which maps the type ${1}$ to a singleton $\{\star\}$, the type $N$ to the set ${\mathbb{N}}$ of non-negative integers, the type $N'$ to the set ${\mathbb{N}}^$ of positive integers, the term $z$ to the constant function $\star\mapsto 0$, and the terms $s$ and $p$ to the functions $x\mapsto x+1$ and $x\mapsto x-1$, respectively. So, the model $M_{{\mathit{nat}}}$ of ${\Sigma}_{{\mathit{nat}}}$ is illustrated by a diagram in the equational theory of sets, which has the same form as the diagram for ${\Sigma}_{{\mathit{nat}}}$: $$\begin{array}{cc} M_{{\mathit{nat}}} : & \xymatrix@R=0.6pc{ \mbox{} && {\mathbb{N}}^ \ar@/^1pc/[dd]^{x\mapsto x-1} & \\ \mbox{} &&& \\ \{\star\} \ar[rr]^{\star\mapsto0} && {\mathbb{N}}\ar@/^1pc/[uu]^{x\mapsto x+1} & \\ } \\ \end{array} \qquad\qquad$$ Besides, ${\Sigma}_{{\mathit{nat}}}$ can be seen as a set-valued realization of ${\mathbf{E}}_{{\mathit{eq}},S}$: $$\xymatrix@C=.9pc{ {\Sigma}_{{\mathit{nat}}}: & && {\rule[-5pt]{0pt}{0pt}}\{p\circ s \equiv {\mathit{id}}_N\} \ar@{>->}[d] & & & \\ {\rule[-5pt]{0pt}{0pt}}\{\star\} \ar@{>->}[d] \ar@<1ex>[rd]\|{\star\mapsto{1}} & {\rule[-5pt]{0pt}{0pt}}\emptyset \ar[l] \ar@{>->}[d] \ar@/^7ex/@<1ex>[rrd] & {\rule[-5pt]{0pt}{0pt}}\{N\} \ar@{>->}[d] \ar@<1ex>[rd]\|{N\mapsto{\mathit{id}}_N} & \{{\langlep\circ s, {\mathit{id}}_N\rangle},\dots\} \ar@<-1ex>[d] \ar@<1ex>[d] & {\rule[-5pt]{0pt}{0pt}}\{{\langles,p\rangle}\} \ar@{>->}[d] \ar@<-1ex>[dl]\|{<s,p>\mapsto p\circ s} & {\rule[-5pt]{0pt}{0pt}}\emptyset\ar[r]\ar@{>->}[d]\ar@/_7ex/@<-1ex>[lld] & {\rule[-5pt]{0pt}{0pt}}\emptyset \ar@{>->}[d] \ar[ld] \\ \{\star\} & \{{1},N,N'\} \ar[l] \ar@{=}[r] & \{{1},N,N'\} & \{z,s,p,{\mathit{id}}_N,p\circ s\} \ar@<1ex>[l] \ar@<-1ex>[l] & \{{\langlez,s\rangle},{\langles,p\rangle},\dots\} \ar@<1ex>[l] \ar@<-1ex>[l] & \{{\langles,{\mathit{id}}_N\rangle},\dots\} \ar@/^5ex/@<2ex>[ll] \ar@/^5ex/[ll] \ar[r] & \{{\langleN',N\rangle},\dots\} \ar@/^10ex/@<2ex>[llll] \ar@/^10ex/[llll] \\ \mbox{ } \\ }$$ Equational logic {#subsec:dia-thry} ---------------- In section \[subsec:dia-spec\], a limit sketch ${\mathbf{E}}_{{\mathit{eq}},S}$ for equational specifications has been defined. Now, we describe simultaneously a limit sketch ${\mathbf{E}}_{{\mathit{eq}},T}$ for equational theories and a morphism ${\mathbf{e}}_{{\mathit{eq}}}:{\mathbf{E}}_{{\mathit{eq}},S}\to{\mathbf{E}}_{{\mathit{eq}},T}$, by translating at the sketch level the fact that the equational theories are the equational specifications which satisfy the rules of the equational logic, as described in section \[subsec:dia-equ\]. It happens that this can be done simply by mapping some arrows in ${\mathbf{E}}_{{\mathit{eq}},S}$ to identities, thereby some pairs of points in ${\mathbf{E}}_{{\mathit{eq}},S}$ get identified in ${\mathbf{E}}_{{\mathit{eq}},T}$. Let us call (syntactic) entailment any arrow $t$ in ${\mathbf{E}}_{{\mathit{eq}},S}$ which will become an identity in ${\mathbf{E}}_{{\mathit{eq}},T}$. Let us look more closely at the rules of the equational logic (figure \[fig:equ\]). Each rule may be considered as a fraction in the sense of [@GZ67], i.e., as a span $r={{s}/{t}}$ from the hypothesis $H$ to the conclusion $C$, where the denominator $t$ is an entailment, which is illustrated as follows: $$\xymatrix@C=3pc{ H \ar@<1ex>@{-->}[r] & H' \ar[l]^{t} \ar[r]^{s} & C \\ }$$ The image of a fraction $r={{s}/{t}}$ by the Yoneda contravariant realization is a cospan $\rho={{{\sigma}}\backslash{\tau}}$ in the category ${\mathbf{S}}_{{\mathit{eq}}}$ of equational specifications, where $\tau={\mathcal{Y}}(t)$, which will become an idenitity in ${\mathbf{T}}_{{\mathit{eq}}}$, is also called an entailment; this is illustrated in a similar way: $$\xymatrix@C=3pc{ {\mathcal{H}}\ar[r]_{\tau} & {\mathcal{H}}' \ar@<-1ex>@{-->}[l] & {\mathcal{C}}\ar[l]_{{\sigma}} \\ }$$ The numerators of the rules are used for easily composing the rules, but here only the denominators matter, and in addition it may happen that several rules have the same denominator. Let $F_{{\mathit{eq}}}\dashv G_{{\mathit{eq}}}$ be the adjunction associated to ${\mathbf{e}}_{{\mathit{eq}}}$, then the functor $F_{{\mathit{eq}}}$ is obtained by mapping the denominators of the rules to identities. A similar approach can be found in [@Makkai97]. For instance, the reflexivity rule means that “for each term $f$ there is an equation $f\equiv f$”, the corresponding fraction is: $$\xymatrix@C=3pc{ {\mathtt{Term}}\ar@<1ex>@{-->}[r] & {\mathtt{Refl}}\ar[l]^{t} \ar[r]^{s} & {\mathtt{Equa}}\\ }$$ where ${\mathtt{Refl}}$, $s$ and $t$ are defined by the potential limit: $$\xymatrix{ & {\mathtt{Refl}}\ar[dl]_{t} \ar[d] \ar[dr]^{s} & \\ {\mathtt{Term}}& {\mathtt{Para}}\ar@<1ex>[l]^{{\mathtt{right}}} \ar@<-1ex>[l]_{{\mathtt{left}}} & {\mathtt{Equa}}\ar[l]^{{\mathtt{equa}}} \\ }$$ The specification ${\mathcal{Y}}({\mathtt{Refl}})$ is made of a term $f:X\to Y$ and an equation $f\equiv f$, which is represented as $ \xymatrix@C=2pc{ X \ar[r]^(.3){f}\|{\,\equiv\,} & Y \\ } $. The image of this rule by Yoneda is: $$\begin{array}{\|c\|c\|c\|c\|c\|} \cline{1-1}\cline{3-3}\cline{5-5} {\rule[-10pt]{0pt}{25pt}}\xymatrix{ X \ar[r]^{f} & Y \\ } & \!\!\!\!\!\! \xymatrix@C=3pc{ \mbox{} \ar[r]_{f\mapsto f} & \mbox{} \ar@<-1ex>@{-->}[l] \\} \!\!\!\!\!\! & \xymatrix{ X \ar[r]^(.3){f}\|{\,\equiv\,} & Y \\ } & \!\!\!\!\!\! \xymatrix@C=3pc{ \mbox{} & \mbox{} \ar[l]_{f\mapsto f}^{g\mapsto f} \\} \!\!\!\!\!\! & \xymatrix{ X \ar@/^/[r]^{f} \ar@/_/[r]_{g} \ar@{}[r]\|{\equiv} & Y \\ } \\ \cline{1-1}\cline{3-3}\cline{5-5} \end{array}$$ In figure \[fig:dia\], for several rules of the equational logic we give the corresponding denominator in ${\mathbf{E}}_{{\mathit{eq}},S}$ (on the left) and its image by ${\mathcal{Y}}$ in ${\mathbf{S}}_{{\mathit{eq}}}$ (on the right). This entailment has the form $\xymatrix@C=2pc{{\mathcal{H}}\ar[r]_{\tau} & {\mathcal{H}}' \ar@<-1ex>@{-->}[l] \\ }$ and can be read as: “as soon as there is an occurrence of ${\mathcal{H}}$ in a specification ${\Sigma}$, it may be extended (up to equivalence) as an occurrence of ${\mathcal{H}}'$”. [\|cccccc\|]{} entailment &\ \ $\xymatrix@C=3pc{ {\mathtt{Cons}}\ar@<1ex>@{-->}[r] & {\mathtt{Comp}}\ar[l]^{{\mathtt{i}}} \\ }$ & $\quad$ & & [$\!\!\!\!\!\! \xymatrix@C=3pc{ \mbox{} \ar[r]_{\subseteq} & \mbox{} \ar@<-1ex>@{-->}[l] \\} \!\!\!\!\!\!$]{}& & $\quad $\ \ $\xymatrix@C=3pc{ {\mathtt{Type}}\ar@<1ex>@{-->}[r] & {\mathtt{Selid}}\ar[l]^{{\mathtt{i0}}} \\ }$ & & & [$\!\!\!\!\!\! \xymatrix@C=3pc{ \mbox{} \ar[r]_{\subseteq} & \mbox{} \ar@<-1ex>@{-->}[l] \\} \!\!\!\!\!\!$]{}& &\ \ $\xymatrix@C=3pc{ {\mathtt{Term}}\ar@<1ex>@{-->}[r] & {\mathtt{Refl}}\ar[l]^{t} \\ }$ & & & [$\!\!\!\!\!\! \xymatrix@C=3pc{ \mbox{} \ar[r]_{\subseteq} & \mbox{} \ar@<-1ex>@{-->}[l] \\} \!\!\!\!\!\!$]{}& &\ \ $\xymatrix@C=3pc{ {\mathtt{Type}^\mathtt{2}}\ar@<1ex>@{-->}[r] & {\mathtt{2\texttt{-}Prod}}\ar[l]^{{\mathtt{j}}} \\ }$ & & & [$\!\!\!\!\!\! \xymatrix@C=3pc@R=0pc{ \\ \mbox{} \ar[r]_{\subseteq} & \mbox{} \ar@<-1ex>@{-->}[l] \\} \!\!\!\!\!\!$]{}& &\ \ $\xymatrix@C=3pc{ {\mathtt{2\texttt{-}Cone}}\ar@<1ex>@{-->}[r] & {\mathtt{Pair}}\ar[l]^{{\mathtt{2\texttt{-}dom}}} \\ }$ & & & [$\!\!\!\!\!\! \xymatrix@C=3pc@R=0pc{ \\ \mbox{} \ar[r]_{\subseteq} & \mbox{} \ar@<-1ex>@{-->}[l] \\} \!\!\!\!\!\!$]{}& &\ \ $\xymatrix@C=3pc{ {\mathtt{Unit}}\ar@<1ex>@{-->}[r] & {\mathtt{0\texttt{-}Prod}}\ar[l]^{{\mathtt{j0}}} \\ }$ & & & [$\!\!\!\!\!\! \xymatrix@C=3pc{ \mbox{} \ar[r]_{\subseteq} & \mbox{} \ar@<-1ex>@{-->}[l] \\} \!\!\!\!\!\!$]{}& &\ \ $\xymatrix@C=3pc{ {\mathtt{Type}}\ar@<1ex>@{-->}[r] & {\mathtt{Coll}}\ar[l]^{{\mathtt{0\texttt{-}codom}}} \\ }$ & & & [$\!\!\!\!\!\! \xymatrix@C=3pc{ \mbox{} \ar[r]_{\subseteq} & \mbox{} \ar@<-1ex>@{-->}[l] \\} \!\!\!\!\!\!$]{}& &\ [^1]: Departamento de Matemáticas y Computación, Universidad de La Rioja, Edificio Vives, Luis de Ulloa s/n, E-26004 Logroño, La Rioja, Spain, [email protected]. [^2]: Laboratoire Jean Kuntzmann, Université de Grenoble, 51 rue des mathématiques, BP 53, F-38041 Grenoble Cédex 9, France, [email protected].
The 9 mile race follows the same course as the 2019 race that we called 8.6 miles. After receiving input as to course distance from a few of the 2019 racers with GPS devices, which varied by over a half mile, we decided to call the race a 9 miler. The GPS route presented below shows the route to be about 8.7 miles, which we believe is a little short. All 9 mile runners will receive a swag bag with goodies, a technical running shirt or running cap, and a burrito breakfast. The breakfast will be served from 8:00 AM until about 10:30 AM by the Mountain Village Foundation, a local Pine non-profit. The race starts at the end of Camp Lo Mia deep in the Pine Canyon. The course leaves the camp on a dual track trail for about .15 miles to the junction with the Pine Canyon trail, which allows racers to jostle into single file. It then follows the Pine Canyon trail about 6.5 miles through various forest flora and soil conditions until it reaches the Pineview trail junction. The course then makes a sharp left turn onto the Pineview trail. It continues on the Pineview about 1.3 miles to the Highline trail, then down the Highline about 1 mile to the finish line at the Pine Trailhead. You will start at Camp Lo Mia at 5750 feet elevation and climb at an almost constant ascent to 6190 feet in 1.4 miles (6% grade), then descend 150 feet in the next half mile (-6% grade). You will then ascend the next 1.3 miles to the course peak of 6325 feet. You then do a steep descent that quickly becomes gradual for 1.3 miles to 6190 feet. A steeper descent, passing the Goodenough trail junction aid station (liquid refreshments) will take you to about the 6.6 mile mark and 5525 feet (-6.3% grade). This brings you to the Pineview trail junction and the 2nd aid station (liquid refreshments). Turning onto the Pineview trail, you will ascend to 5800 feet in 1 mile (5% grade), where you will be afforded beautiful views of the valley and surrounding forest. Descending the next .4 miles you will reach the Highline trail junction at a beautiful shaded area next to a creek. You will continue your descent on the Highline trail 1 mile to the Pine Trailhead finish at 5380 feet. The free parking is located off Hwy 87 about a half mile south of the Pine trailhead (.1 mile north of mile marker 266 – look for the event signs). All runners will be bused from the parking area to the starting line in Camp Lo Mia. Please do not try to park at the Pine Trailhead or try to drive into Camp Lo Mia. Only our busses will be allowed in Camp Lo Mia There will be a gear drop at the starting line (use your swag bag, it will have a tag with your bib number attached). Your drop bag will be transported to the Pine Trailhead finish area. Covid Precautions: We will be limiting the number of runners on each bus to 25 (less than 50% capacity). Out of respect for all of our runners we ask that you wear a mask for the short ride from the parking area to the starting line. We will have masks available as you enter the bus and trash receptacles as you exit the bus. - Friday May 6th - 5:00 PM – 7:00 PM Late registration & packet pickup, Ponderosa Market, Pine (Hwy 87 & Hardscrabble Mesa Rd.) - Race Day – Saturday, May 7th - 5:30 AM – 7:00 AM: Parking, late registration & packet pickup, parking area (Hwy 87, just N. of mile marker 266 – watch for event signs) - 6:45 AM: Mandatory prerace brief, near registration at parking area - 7:00 AM: Start loading buses - 7:40 AM: First wave start, Camp Lo Mia - 7:30 AM – 11:30 AM: Bus transportation available between parking area & Pine Trailhead - 8:00 AM – 10:30 AM: Burrito breakfast - 9:30 AM: 9 mile award presentation, Pine Trailhead - Medals for all finishers - Medals for top 3 finishers in each 10 year age group - Plaques for top 3 female and male overall finishers - Plaque for top overall age-graded winner Participation in the Pine Trail Run 9 Miler is open to runners twelve years of age and older and is limited to 150 runners. All participants must cross the finish line in 3.5 hours. - Register by December 31, 2021: $55 - Register by February 28, 2022: $60 - Register by April 23, 2022: $65 - Register onsite May 6th & May 7th: $70 Online registration closes April 23, 2022. In addition to the fees shown there is a $5 processing fee.	https://psfuelreduction.org/pinetrailrun/races/9mile/
Taxi and private hire drivers may have to pass an enhanced criminal record check before they are granted a licence under a new government proposal. The Department for Transport has put forward proposals which aim to improve the safety of passengers of taxis and minicabs under strict licensing guidelines. A consultation was launched in February 2019 on the “robust new rules for licensing authorities outlining how they should use their powers to protect vulnerable passengers from harm”. The BBC reported: “Under the proposals, every council in England would be told to carry out checks on all applicants. Current guidelines allow councils to set their own driver standards, including whether to make the checks. The plans also include introducing national minimum standards and a database to stop applicants applying to councils after being refused elsewhere.” Taxis Minister Nusrat Ghani said: “While the vast majority of drivers are safe and act responsibly, we have seen too many cases where taxi and minicab drivers have used their job to prey on vulnerable people, women and children. These rules would make sure that drivers are fit to carry passengers, keeping people safe while stopping those with bad intentions from getting behind the wheel of a taxi or minicab.” The consultation will run up until April 22nd, 2019, and comes as part of the Government’s response to the Task and Finishing Group’s report on taxi and private hire vehicle licensing. The Government has also pledged to legislate on national minimum standards for drivers, establish a national licensing database and look at restricting drivers operating far away from where they are licensed. The department will also consider whether vehicles should be fitted with CCTV as part of these minimum standards. These encrypted systems mean footage can only be accessed if there is a crime reported. Do you have an interest in vehicle and driving laws? Have a look at our recruitment board and see if there are any legal jobs you would like to apply for.	https://www.lawabsolute.com/recruitment-news/article/enhanced-checks-for-taxi-and-private-drivers-proposed
Perth zoo is set on 17ha (42 acres) of land. History In 1986 the Western Australia Acclimatization society first met. They had two intentions one was introduce European animals to the Australian wild a role which quickly ended and the other was to establish a zoo. They invited Melbourne Zoo’s director, Albert Le Souef to chose the site for the zoo. The current site where the zoo sits was selected by him and his son Ernest was appointed to direct the zoo. Building work began in June 1987. As soon as the site was selected work on extensive gardens began. Two bear caves, mammal and monkey houses and a model castle for guinea pigs were the first habitats completed. Into these moved orangutans, two monkeys, four ostriches, a pair of lions and a tiger. On October 17 1898 the governor of Western Australia, Lieutenant Colonel Sir Gerad Smith opened the zoo. Entry was set at six pence for adults and three pence for children. Le Souef spent 35 years running the zoo. He developed the zoo’s botanical collection along with the animal collection. They grew a number of crops for the animals a tradition which continues today. During 1919 he lectured part time in agriculture and veterinary science at The University of Western Australia. A museum was established at the zoo where students could learn anatomy and farmers could receive free vet lessons. He worked through the financial issues of the depression by taking a voluntary pay cut himself along with cutting wages and staff. Unfortunately the zoo fell into disrepair and he recommended control of the zoo be handed to the State Gardens Board. The zoo hosted a range of entertainment alongside the animals. Fashion parades, tennis tournaments, croquet matches, music, beautiful baby contests and mineral baths have all been hosted at Perth Zoo. In 1909 the Australasian Open (now the Australian Open) was held there. Up until 1961 people could ride elephants and pony rides have occurred at the zoo. The carousel was installed in 1947 and continues operating to this day. Number of animals Perth zoo is home to over 1100 animals from 190 species Main Exhibits Australian Walkabout This habitat features a range of animals including waterbirds, crocodiles, penguins, wombats, kangaroos, dingoes, koalas, echidnas, Tasmanian devils and a range of other Australian animals. The reptile house is also in this area which features reptiles from around the world. A main feature of this area is the numbat habitat. This showcases Perth Zoos work to save this species. They are one of very few zoos to hold this rare species and the only that breeds them. Asian Rainforest Numerous Asian species including Asian elephants, white cheeked gibbons, orang-utans, tigers, sun bears, red pandas, otters and silvery gibbons live in this habitat. They have one of the most successful oranguatan breeding programs in the world. They have breed 27 since 1970. African Savannah At the time of its construction this was Perth zoo’s largest ever construction projects. Currently it houses rhinos, painted dogs, hyenas, lions, zebras, baboons, meerkats, giraffes and tortoises. Amazonia This area of the zoo features animals from South America. Species housed here include coati, squirrel monkey’s, golden lion tamarin’s and conures. Primates This are of the zoo is home to a variety of primate species. Nocturnal House This exhibit reverses the lighting so that creatures of the night are active during the day. Also located around the zoo are exhibits for Galapagos giant tortoise, tree kangaroo and cassowary. On the main lake are islands for silvery gibbon and ruffed lemur. Star Animals Temara and Semeru Temara and Semeru are two oranguatans bred at Perth Zoo. In 2006 and 2011 respectively they were released into the wild’s of Indonesia. They form part of a program to form a new wild colony of Sumatran oranguatans. Attractions Carousel Kids can ride the historic carousel which is the last of its kind still permanently installed in Western Australia. Zoo HQ Learn about what it takes to care for the animals housed at Perth Zoo in this interactive display. You can experience life as a zookeeper, vet or even an animal by trying some of the enrichment puzzles which keepers give the animals.	https://www.theanimalfacts.com/go-2-zoo/perth-zoo/
The Night parrot was considered as extinct species by many biologists and researchers. A living specimen had not been captured since the 1890s. In 2013, the Australian ornithologist John Young found and also took photos of this species. However, there were still some doubters who didn’t believe that the Night Parrot really survived. A new discovery is an unquestionable evidence of existence of this mysterious bird. The first specimen was captured after 18 months of an intensive research by an ornithologist Steve Murphy on 4 April. He decided to take feather samples for DNA analysis. The bird was tagged with a small tracker. A battery inside can last for 21 days so this tracking brought much of new information about ecology and ethology of the Night Parrot. Before this observation researchers didn’t know anything about the secret life of this species. a a For better monitoring about 30 remote cameras have been places all over the area. “Before this research, we didn’t know what they ate, where they got their water from or anything. We’re really starting from ground zero with the night parrot.” said Rob Murphy, executive manager of conservation group Bush Heritage Australia for The Guardian. That information about exact location is highly confidential. “This is the biggest story of conservation in Australia today. For as long as we can, we’ll keep it as secret as we can. It’s just such a critical thing that we do everything that we can to save this species to bring it back from the brink of extinction.“ said Mr Murphy a a Another member of the successful research team is Dr Steve Murphy. According to his statement, public would have an opportunity to see them if more birds will be found. „But without finding those other populations the risks are still really too great“, said the researcher for ABC news.	http://www.parrotsdailynews.com/the-night-parrot-is-not-extinct-after-more-than-a-century-a-living-specimen-was-captured-in-queensland/
This project intends to apply research to significantly upgrade community-based, off-grid, renewable energy access solutions in remote and rural areas to create greater social and economic value and to advance progress on multiple Sustainable Development Goals (SDGs). Research teams from the Center for Energy and Society, Arizona State University, and the Institute for Technology Assessment and Systems Analysis, Karlsruhe Institute of Technology, jointly with partner organizations, will evaluate off-grid energy projects using the “Multi-layer design framework for social value creation.” Based on the results of the evaluations, the research teams will work with the partner organizations to improve energy solutions and community sustainable development (measured via Sustainable Development Goal number 7 and additional SDGs). The core design approach in the multi-layer framework is to enable and empower users of new sustainable energy systems (e.g. households, businesses and local governments) to use their new access to energy to create diverse forms of social value, such as income, health, education, clean water, food security and reduced inequality. Surrounding its social value core, the framework design strategies include: - designing socio-technical arrangements to deliver the energy services required to create the desired forms of social value; - designing local energy management organizations to imagine, design, build, operate, repair, and scale sustainable energy solutions; - designing ownership arrangements to ensure reinvestment of energy proceeds in local economies and societies; - designing regional energy innovation ecosystems; and - designing policy and governance to support this work. The project sees an application of the multi-layer framework in projects of four partner organizations: Sunbridge Solar (Nepal); Solar Solutions (Philippines); ENVenture (Uganda) and Practical Action (Bolivia). These partners are already engaged in achieving SDG 7 and other SDGs. The multi-layer framework will also be improved by using the cross-impact balance method, which is used for the systematization and analysis of complex-socio-technical systems.	https://www.itas.kit.edu/english/projects_poga18_eneeb.php
This is an advance summary of a forthcoming article in the Oxford Research Encyclopedia of Natural Hazard Science. Please check back later for the full article. Worldwide, low-lying coastal land margins are becoming increasingly vulnerable to natural and manmade disasters due to the effects of climate change, population dynamics, saltwater intrusion, loss of coastal ecosystems, and erosion of coastlines. In 2003, it was estimated that 1.2 billion people (23% of the world’s population) lived within 100 km of a shoreline and 100 m in elevation of mean sea level. As populations increase, coastal areas are also susceptible to additional stresses due to land-use and hydrological changes. In addition to human communities, the coastal land margin includes ecologically and economically significant estuaries and wetlands. Coastal wetlands and marshes provide food, shelter, and nursery areas for commercially harvested fish and shellfish. Wetlands also help protect coastal communities by mitigating impacts of storm surge and erosion. A System of Systems (SoS) approach is best for assessing potential future coastal hazards and their impacts. Employing an SoS framework permits new patterns and properties to emerge (i.e., nonlinear and dynamic effects of climate change) that would otherwise be unobserved using simplified models. The SoS framework also allows the sea level rise (SLR) projections, and other subsystems, to be linked to carbon emission scenarios so the full climate change impact is considered for all subsystems. Furthermore, this approach to studying coastal hazards supports the translation of science to application as coastal managers require scientific data regarding the potential impacts of SLR to make informed decisions to manage human and natural communities. Synergetic studies that integrate the dynamic interaction among physical, ecological, and anthropogenic environments are required to better predict the impacts to the coastal system in a more holistic fashion. Individually, observations and modeling are insufficient for making scientifically defensible, detailed, and credible assessments of the dynamic response of the coastal region under future SLR conditions. The capability exists to model the bio-geo-physical system, link that modeling to the historic record, and produce a dynamic coastal response to SLR using a SoS framework. Further, incorporating economic and ecosystem services valuations into the SoS enables stakeholders to better understand and assess future coastal hazards and enhance coastal resiliency.	http://naturalhazardscience.oxfordre.com/browse?btog=chap&pageSize=10&sort=titlesort&subSite=orenhs&t0=ORE_NHS%3AREFNHS052
"Wings," created by Rockport Elementary School children, is on view at the Rockport Arts Association & Museum, 12 Main St. in Rockport. The butterflies represent the 25 Rockport residents lost to COVID-19 since the pandemic began.Rockport Arts Association & Museum/Courtesy photo "Wings," created by Rockport Elementary School children, is on view at the Rockport Arts Association & Museum, 12 Main St. in Rockport. The butterflies represent the 25 Rockport residents lost to COVID-19 since the pandemic began.Rockport Arts Association & Museum/Courtesy photo "Wings," an art installation marking the death of 25 Rockport residents due to COVID-19 and created by Rockport Elementary School children, is on view at the entrance to the Rockport Arts Association & Museum, 12 Main St. in Rockport.Rockport Arts Association & Museum/Courtesy photo "Wings," an art installation marking the death of 25 Rockport residents due to COVID-19 and created by Rockport Elementary School children, is on view at the entrance to the Rockport Arts Association & Museum, 12 Main St. in Rockport.Rockport Arts Association & Museum/Courtesy photo 'Wings' for 25 lost to COVID-19 Schoolchildren's public art project memorializes Rockport losses By Michael Cronin Staff Writer Mar 16, 2021 2 hrs ago 1 of 4 "Wings," created by Rockport Elementary School children, is on view at the Rockport Arts Association & Museum, 12 Main St. in Rockport. The butterflies represent the 25 Rockport residents lost to COVID-19 since the pandemic began.Rockport Arts Association & Museum/Courtesy photo "Wings," created by Rockport Elementary School children, is on view at the Rockport Arts Association & Museum, 12 Main St. in Rockport. The butterflies represent the 25 Rockport residents lost to COVID-19 since the pandemic began.Rockport Arts Association & Museum/Courtesy photo "Wings," an art installation marking the death of 25 Rockport residents due to COVID-19 and created by Rockport Elementary School children, is on view at the entrance to the Rockport Arts Association & Museum, 12 Main St. in Rockport.Rockport Arts Association & Museum/Courtesy photo "Wings," an art installation marking the death of 25 Rockport residents due to COVID-19 and created by Rockport Elementary School children, is on view at the entrance to the Rockport Arts Association & Museum, 12 Main St. in Rockport.Rockport Arts Association & Museum/Courtesy photo ROCKPORT — March marks the one-year anniversary of the COVID-19 lockdowns. Since the pandemic began, the state Health Department has reported 2,223 total confirmed and probable deaths in Essex County due to the disease. Rockport Elementary School kindergartners through fifth-graders hope to honor those lives lost with a new art installation outside the Rockport Art Association & Museum on 12 Main St. "Wings" features 25 hand-decorated paper butterflies hung up over the association's entrance. Each represents one of the reported 25 Rockport residents that died from the virus. The butterflies are accompanied by a printed excerpt from "[i carry your heart with me (i carry it in)]," a poem by e.e. cummings. “Here is the deepest secret nobody knows/(here is the root of the root and the bud of the bud/in the sky of the sky of a tree called life which grows higher than the soul can hope or mind can hide)/and this is the wonder that’s keeping the stars a part/I carry your heart (i carry it in my heart),” it reads. The exhibit was organized by Rockport Art Association President Heidi Caswell Zander and Rockport Elementary School art teacher Sarah Tetrault, who became an association board member last August. "One thing I have a vision for would be to have more involvement with the youth and provide more opportunities for people to see youth art," said Caswell Zander. "So it was great to have Sarah come on to the board last August. We worked together to see how we could do something positive in remembrance for those who've passed." The program was made possible in part because of Rockport Art Association's Creative Community initiative, which aims to build a community of like-minded Cape Ann artists. Before the pandemic hit, more than 80 artists would gather at RAA each week to practice and create their art. "Projects like ("Wings") are little satellites that came out of Creative Community," explained Caswell Zander. "We've done mail-in art through the whole year, where people can send us 4- by 6-inch postcards of their work. In the summer we'll have them all hanged." "Wings" will be on view through "the next couple of weeks," according to Caswell Zander. When we asked for stories from breast cancer survivors and others, the calls started coming and did not stop. We found instances of courage, hope, determination, fear, survival, even loss. Read them in this special section. Sign up now to get our FREE breaking news coverage delivered right to your inbox. First Amendment: Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.
\|Name: _________________________\|\|Period: ___________________\| This quiz consists of 5 multiple choice and 5 short answer questions through Part 2: Chapters 9-11. Multiple Choice Questions 1. How much money did Malcolm make in the stock market that day? (a) Twenty-six thousand dollars. (b) He didn't make any money in the stock market. (c) Fifty thousand dollars. (d) Twenty-four thousand dollars. 2. What did Olivia consider sweet about Malcolm? (a) Neither of these. (b) That he had given her the day to herself. (c) That he didn't want her tiring herself before the reception. (d) Both of these. 3. Why is Malcolm to transfer a million dollars into trust for each of the boys? (a) So they can run their own lives. (b) So they can leave their father's side. (c) So they have some control over their own lives. (d) So they have money. 4. Why did Alicia finally choose to tell Olivia what was happening? (a) She knew Garland would have wanted her to tell Olivia. (b) She didn't want to be in Foxworth Hall anymore. (c) She was attempting to leave. (d) She was pregnant with Malcolm's child. 5. What did Malcolm attempt to get Olivia to do? (a) Malcolm didn't attempt to get Olivia to do anything. (b) Turn over her fortune to him. (c) Come out with him and tour the businesses. (d) Come out with him to redecorate the house. Short Answer Questions 1. What thought cheered Olivia when she had seen the small room where she was to work? 2. What name(s) was etched on the desks? 3. What news did Malcolm bring Olivia in the nursery? 4. What mountains were "looming like sleeping giants?" 5. What seemed to be happening to Garland?	http://www.bookrags.com/lessonplan/garden-of-shadows-dollanganger/quiz2a.html
The invention discloses a data quality management method and a data quality management system. The data quality management method comprises the following steps: managing a quality knowledge library; analyzing the data quality characteristics, and presetting a quality problem domain, a quality dimension domain, a quality rule domain and a quality standard domain; collecting quality information: selecting the quality dimension and the quality rules required by a user from the quality knowledge library, and extracting a data set meeting the user requirements from the original data sets; evaluating the data quality: evaluating the data quality according to the collected quality information, and generating and submitting a data quality report to the user or a quality manager according to the quality problem domain and the quality standard domain in the quality knowledge library; and improving the data quality: correcting the data quality problem detected in the evaluation of the data quality for improvement. The data quality management method and the data quality management system are applied to monitoring, evaluating and continuously improving the data quality in the whole hydrology industry based on the whole data processing procedure of the hydrology industry.
Computers have an enormous number of jobs that they complete even within a matter of seconds, prompting the need for a central processing unit (CPU) that helps manage and coordinate these tasks. In this blog, you’ll learn what a CPU is and how it operates as the central component of a computer to ensure it operates at maximum efficiency. What is a CPU, and what is its purpose? A CPU (central processing unit, or simply, processor) is the main chip in a computer that is responsible for carrying out all of its tasks. Often referred to as the “brain,” the processor tells all of the other components in a computer what to do based on the instructions it is given by the software running on that computer. CPUs exist in lots of devices other than traditional computers like smartphones, TVs, and tablets. Where is the CPU located? In a computer, the CPU is generally found at the center of the system directly connected to the motherboard. It is usually under a cooling fan or heat sink, as the CPU would become damaged from overheating without a proper cooling mechanism. Socketed CPUs can be removed and replaced as needed over time. In many modern applications, the CPU may be integrated directly onto a single integrated circuit with memory interfaces and input/output devices, becoming a system-on-a-chip (SoC). This is particularly common in edge and mobile solutions. What are the main parts of a CPU? There are three main parts of a CPU: the control unit (CU), the arithmetic logic unit, (ALU), and the registers. - Control Unit (CU): This regulates the flow of input and output (I/O). It fetches instructions from the main memory and decodes into specific commands. - Artithmetic Logic Unit (ALU): This is where all of the processing happens, including mathematical calculations and logical operations for decisions making, like comparing data. - Registers: This is an extremely fast memory location. The data and instructions that are currently being processed during the fetch-execute cycle are stored there for quick access by a processor. How does a CPU work? A CPU can execute millions of instructions per second, but it can only carry out one instruction at a time. It first receives some type of input, typically from an input device–such as a monitor display screen, a keyboard, a mouse, or a microphone–from an application/system software program, like your web browser or operating system, or from memory. It is then in charge of four tasks: fetching, decoding, executing, and storing. (More on that in the next section.) Finally, there is an output of some kind, such as printing something to the screen. This process is called the fetch-execute cycle, and it happens millions of times per second. Source: doc.ic.ac.uk. A CPU’s main cycle is called the fetch-execute cycle, and it occurs millions of times per second. What are the main tasks of a CPU? Let’s take a look at a CPU’s four primary tasks: - Fetching includes getting instructions from memory, so the CPU knows how to handle the input and knows the corresponding instructions for that particular input data it received. Specifically, it looks for the address of the corresponding instruction and forwards the request to the RAM (random access memory). The CPU and the RAM constantly work together in a process called “reading from memory.” - Decoding involves translating the instructions into a form the CPU can understand, which is machine language. - Executing means carrying out the given instructions. - Storing is the result of the execution back to memory for later retrieval if and when requested. This is also called writing to memory. Key CPU Terms Clock speed Expressed in gigahertz (GHz), clock speed is a rough indication of how many calculations a processor can make each second. The higher the clock speed, the more calculations the processor can perform. Threads A thread is a virtual component that helps deliver workloads to the CPU. The more threads you have, the faster workloads are delivered and the easier they are organized, leading to increased efficiency. Threads are vital to a computer’s functionality because they determine how many tasks a computer can perform at any given time. The number of threads you have depends upon the number of cores in your CPU. Each core may have two threads depending on the specific processor and if hyperthreading is supported. For example, a dual core processor may have four threads and a processor with four cores may have eight threads. Hyperthreading Many modern CPUs support a technology called hyperthreading. Hyperthreading works by making a single physical core appear as multiple physical cores, allowing the operating system (OS) take advantage of concurrent instruction processing and enhancing the computational power. Cores Think of a human body: if threads are the hands, then cores are the mouth. Cores are separate physical devices within the main CPU chip that act as independent processors, taking data from the threads and performing computational tasks. Software applications can be written so that multiple cores can work concurrently on processing program data, generally referred to as multithreading. How quickly a CPU can process data is affected by the number of available cores. The more cores a CPU has, the greater the computational power it has. As a result, more tasks can be run and completed simultaneously. For example, a dual-core CPU has two cores, meaning that there are, in essence, two CPUs on the same chip and can run two instructions at the same time. An eight-core processor would be able to run 8 instructions at the same time. Most modern server class CPUs have at least 8 cores with some configurations supporting more than 30 cores per processor. Motherboards can contain multiple processors connected together by the UPI, or Intel® Ultra Path Interconnect. Source: extreme tech.com. Cores are separate physical devices within the main CPU chip that act as independent processors, taking data from the threads and performing computational tasks. CPUs and Trenton Systems Without its “brain” operating at maximum efficiency, a computer’s functionality is compromised, posing a risk to critical data and vital parts and components. Equipped with high core counts and advanced cybersecurity technologies, CPUs help computers securely process and analyze data to enhance computer power across a wide range of environments. At Trenton, we design our high-performance computers with next-gen Intel® CPUs to enhance throughput and ensure optimal performance in real-time. For example, our TAC is equipped with dual Intel Xeon D 1700 CPUs. With 2.32x faster processing speed and 5.73x faster AI inferencing, these processors accelerate concurrent workloads and control throughput to improve performance at the tactical edge. We are also a member of the Intel Partner Alliance and a member of the Intel Early Access Program, which allows our customers to have access to the latest Intel technologies before they go on the market. Through the increased efficiency provided by CPUs, we provide customized hardware and software solutions that provide needed insights to make critical decisions ianywhere, anytime.	https://marinecorpgifts.com/what-is-a-cpu-central-processing-unit/
WINDHOEK, FEB 10 – Namibia imports a lot of fuel, which will inevitably cost more in Namibia Dollar terms. Meanwhile, international oil prices remain on the advance, adding further pressure. Global uncertainty continues to pressure commodity prices. The sovereign wealth funds or relatively low public debt levels will weather the storm. Namibia plans to establish a sovereign wealth fund by the end of this year that will be used to serve as a buffer against future economic shocks. By working together, pooling our skills, knowledge, and experience and building on our strengths, we can accomplish great things. The impact on financial markets in turn provides additional channels through which the oil price increase would affect economic variables. However, given cyclical developments in the world economy, it is unclear to what extent the imminent increase in the oil price has been directly responsible for the turbulence in advanced country financial markets and movements in currency markets. There are clear indications that the increase in oil prices has had an adverse effect on economy by affecting the pace of activity and corporate earnings, as well as confidence. The disruption caused by an oil price hike also depends on the state of the business cycle, the response of macroeconomic policies, and the flexibility of the underlying economies. I concurred with Mr. Shiimi asserted that “Sovereign Wealth Fund will be split into short- and long-term funds and be financed with proceeds from the renewable-energy industry, mining royalties, fishing quotas and the sale of state-owned assets”. I therefore, urge all Namibians to join hands. If we don`t speak up against disunity, if we keep quiet and remain under the radar, the enemies of peace and those who want to promote disunity will have their ways. H.E. Dr. Sam Nujoma, Founding President and Father of the Namibian Nation always say” “a people united, striving to achieve common good for all members of the society, will always emerge victorious”! Indeed, it is important to take action to pre-empt the second-round effects of the consequent inflationary pressures. If the monetary authorities accommodate an oil price shock, the resulting increase in inflation tends to get incorporated into inflationary expectations, which become persistent and significantly raise the costs of the subsequent disinflation. In practice, the situation is more complex because oil prices and GDP growth run both ways. High oil prices could dampen economic growth. In addition to increasing international prices, a weaker foreign exchange rate plays an extensive role in higher oil prices, within Namibia. Increase in oil price indirectly and directly affect the prices of a variety of goods and services that are dependent on oil in the production and the delivery of these goods and services. These increases can stifle economic growth by their negative effect on supply and demand. Increasing prices of goods and services reduces the supply and production of other goods and services due to increased production costs. Additionally, demand for these goods and services will also decline due to higher purchasing costs. These elements can lead to a rise in inflation and suppression of economic growth. Because Namibia imports a substantial part of the energy it needs, energy price increases have a larger negative effect on the standard of living of Namibians. If all of the petroleum used in Namibia were produced domestically, higher oil prices would not lower the overall Namibia standard of living in the long run by any more than they would lower GDP. The loss of business and consumer confidence resulting from an oil shock could lead to significant shifts in levels and patterns of investment, savings and spending. Higher oil prices would undoubtedly drive up the prices of other fuels, magnifying the overall macroeconomic impact. Unfortunately, those changes in spending patterns can be quite disruptive for certain key economic sectors and seem to be part of the mechanism by which the earlier oil price shocks had contributed to previous economic recessions. Oil prices remain an important macroeconomic variable. Higher prices can still inflict substantial damage on the economies of oil-importing countries and on the global economy as a whole. Companies are less able to pass through higher energy-input costs in higher prices of goods and services because of strong competition in wholesale and retail markets. As a result, higher oil prices have so far eroded profits more than they have pushed up inflation. Yet the economic threat posed by higher oil prices remains real. Fiscal imbalances would worsen, pressure to raise interest rates would grow and the current revival in business and consumer confidence would be cut short, threatening the durability of the current cyclical economic upturn.	https://namibiadailynews.info/the-impact-of-rising-oil-prices-on-the-namibian-economy/
Tungurahua, Ecuador's 10th-highest peak, is a 5016m high active stratovolcano also known as the "The Black Giant." It has a 600 ft. (183 m) wide crater. Most of the volcano is covered by snow. Don't be fooled by descriptions of this mountain as "easy." People have died on this mountain and you need to be in good physical condition to climb it and enjoy it. It is nowadays dangerous to climb (since 1999) as its increased volcanic activity has unpredictable eruptions as a consequence. You do not want to be caught on the volcano when major explosions of gases, ashes and lava occur. When the volcano increased its activity in 1999, the ice cap melted away and the peak is since then ice free. In spite of this many people still attempt to climb the volcano. ALWAYS check with locals about the mountains condition. Baños (20.000 inhabitants) is 25km from the province capital Ambato. It's easy to reach by bus from all over the country as it's a main tourist spot. The bus from the capital Quito takes 3,5 hours. You can start walking from Banos (this is a very long option): follow the road on the Ambato side of town, then take the first trail to the right of the store and follow this path to town of Pondoa --don't follow the road to the baths. You can hire camionetas (small trucks) in Banos to take you to Parque Nacional Sangay where Tunguraghua is located. The camionetas will take you to the entrance to the park Baños, at 1800m, is still the main entry into the climb and you will enter initially Sangay National Park which encompasses the Volcano, normally you'll pay $10 but if on a single day trip the fees are not charged. When we entered the National Park the office on the mountain was closed and deserted (July 2003). The trail, from the Park Entrance, to the refugio is obvious and well-marked but fairly steep and takes 3-4 hours. Most guides/climbers depart early in the morning from the refugio between 3 to 6 am for a 4 to 6 hours climb to the summit with the last hour through snow and ice parts usually requiring ropes to keep ones grip as the gradient is severe in parts. The glacier is small and represents only the final 40 minutes of the climb. Crampons are recommended. The rest of the climb is scree and demanding, especially if it is muddy. Descending to the Refuge is then only around 90 minutes and a further 2 to 3 hours walk back down to the entrance of the park Best months for ascent to Tungurahua are from December to March for clearest views, but the weather is as unpredictable as Ecuador itself. The Refugio (3800 m) is around 4 hours ascent from the Park entrance and has two refuges. One in stone housing 40 persons and one wooden slightly warmer. Don't forget your sleeping bag or hammocks. Cost is $3 per person. When we went up the mountain in July 2003 the Refugio was deserted because of the eruptions and danger. Altough there were rumours the refuge has been destroyed by the eruptions it's still standing and OK. About the volcanic activity (in Spanish):clickhere Or at Volcanodiscovery Hi there, perhaps you may consider adding this page to those others in the Cordillera Oriental range in addition to just the "Ecuador's Big 10" list? Cheers, Baarb. To bad, but you're not allowed to climb Tungurahua at the moment (February 2010). In fact, it's illegal to do so. They've even put signs at the bottom of the volcano. In September 2016 Tungurahua area officially reopened, there are rangers up there at the entrance and you can climb - same regulations apply as for other Ecuadorian mountains >5k. Refuge is open but nobody is there, you have to bring your own food etc. but can sleep there (mattresses) and use the stove (gas) for USD 5,-- per person.	https://www.summitpost.org/tungurahua/152901
One of the most frequent questions that we hear from our customers is “How do I paint the needles on my gauges?” As a result, we have prepared to following tutorial to point you in the right direction. Setup: Start with a clean, well lit work area. We recommend placing down a sheet of cardboard or old newspaper in case of any accidents. Arrange your work area with all of your supplies within reach. Begin by shaking the paint until it is mixed. The paint has stainless steel balls in the container to help break up any solids that have formed together and mix the paint. Once the mixing balls are moving freely in the paint, in normally requires an additional minute or two to fully mix the paint. If you are shaking one of our paint vials, use caution and do not shake violently. The vials are glass and can easily break if shaken too hard. We recommend leaving the paint in card package (if possible) and shaking it by holding the sides of the card package. Do not use objects to stir the paint! Shaking the paint is the correct way to fully integrate the pigment solids properly in the paint. Using an object like a toothpick or other stirring tool, can contaminate the paint and will not effectively mix the pigment. Furthermore, wooden objects like toothpicks can absorb part of the liquid leaving more solids than liquid in your paint container. Once the paint is mixed, set it aside. Clean your brush in soap and water. Even if it is new, it can have chemicals or substances in the brush that can contaminate the paint and lead to poor results. Wash with warm soapy water and pat dry with a paper towel. Do not use Q-tips or other substitutes to apply the paint. A good quality brush is the only way to get good results. The better the brush quality, the better your results will be. Needle Preparation: Slide one of your index cards under the gauge needle. Then take your Q-tip and dip it into the denatured alcohol. Carefully clean the needle to remove any substances that may be on the surface. Avoid contact the gauge face! We find that the letters and characters screened on the face of the gauges can easily be damaged. Even a damp cloth can remove the screen printed characters from the face of your gauges. Repeat this cleaning process on each of the gauge needles that you intend to paint. Repeat the process 2-3 times. Allow the alcohol to fully evaporate before painting the needles. Next, apply a small amount of paint to one of the needles. If the paint does not stick, the needles are not properly prepped. Please note, the paint will normally appear nearly transparent for the first coat. that does not mean that it is not adhering properly. Immediately, clean the paint from the needle. If the paint is not adhering properly, this indicates that the needles are not clean enough. Repeat the steps in the previous paragraph until the needles are clean and able to hold paint. Some needles may have a glossy finish. In such cases, the needles may need to be lightly sanded with a very fine grit sandpaper (e.g. 1000+grit). The light sanding will produce a better surface for paint adhesion. After light sanding, treat again with alcohol to remove any dust or debris. Repeat these steps until needles hold paint. Painting: Now, we are ready to paint the needles. First, mix the paint again. It should mix easily this time. The paint has a high fluorescent solids content, so it is important to make sure that all of the solids have mixed back into the paint. The solids are the pigment that makes the paint glow, so it is important that they are fully mixed. Slide an index card under the needle that you wish to paint. This will catch any paint that would otherwise land on your gauge. Put a small amount of paint on the tip of the brush and apply the paint to the gauge needle. Start in the center of the needle and pull the brush in long strokes toward the tip of the needle. Apply a very thin coat of paint. The first coat will appear nearly transparent. It will most likely not look good after the first coat. Most needles take 3-5 thin coats. Allow at 3-5 minutes between coats for the paint to setup. If you are changing the color of the gauge needles, you will likely need more coats and more drying time between coats. The paint may seem to have a slight texture to it and/or you may be able to see some minor brush strokes. This will fade as the paint dries. If you find any small unmixed clumps of pigment while painting, be sure to carefully remove them from the needle using the brush. After applying the required number of coats of paint to your needles, allow them time to dry. We suggest leaving the index cards in place until the paint is dry. Clean Up: Clean your brush with warm water and soap and pat dry with a paper towel. Clean any paint from the rim of the container, wash the lid and reseal the paint. Be sure to store at room temperature and away from sunlight for maximum shelf life. Reassemble your gauge cluster and enjoy your newly restored gauge needles.	https://www.hipoparts.com/how-to-paint-gauge-needles/
Hey everyone, hope you are having an amazing day today. Today, I will show you a way to make a distinctive dish, hina matsuri sushi cake. It is one of my favorites food recipes. This time, I am going to make it a bit tasty. This is gonna smell and look delicious. Made using all natural ingredients, with some plum vinegar, nori seaweed, and seasoned sushi rice, these sushi rolls are a feast for the eyes. On this day, we set up hina dolls (hina-ningyo) since they are believed to protect girls from bad luck. To begin with this particular recipe, we must prepare a few components. You can have hina matsuri sushi cake using 17 ingredients and 5 steps. Here is how you cook it. The ingredients needed to make Hina Matsuri Sushi Cake- - Take 14 cm cake pan For the sushi rice- - Get 1 cut Grilled salmon - Prepare 2 eggs' worth Kinshi-tamago (thinly sliced omelet) - Prepare 1 Shirasu (boiled and dried baby sardines) - Take Note, The ● ingredients are for the dolls - Prepare 2 ●Quail egg - Take 2 sheets ●Seasoned nori - Make ready 1 slice ●Ham - Get 1 dash ●Cucumber - Get 1 dash of pink ●Kamaboko - Make ready 2 ●Toothpicks - Take 1 Thinly sliced salt-water boiled carrot (a small amount) - Prepare 1 slice Sliced cheese - Prepare 1 thinly sliced Cucumber, rubbed with salt - Prepare 3 slice Smoked salmon - Take 1 Salmon roe - Get 1 small amount Denbu (sweet pink-colored semi-dried fish flakes) Instructions to make Hina Matsuri Sushi Cake- - [Preparation] Lightly rub the cucumber with salt, then wash. Boil the quail eggs and carrot in salt water. Make sushi rice (I use rice + powdered sushi vinegar). Prepare the kinshi tamago (thinly shredded egg omelet). Grill the salmon and flake it. - Line the cake pan with plastic wrap, then layer the kinshi tamago, sushi rice, salmon and shirasu and then the sushi rice into the pan while lightly pressing down with each addition. If you line the pan with the plastic wrap, the sushi cake won't break easily, but if the shape starts to break apart, it's easier to adjust if you put plastic wrap on top. - Cut the ● ham in half. Then, as shown in the picture, lay on top of the ham, hanging a bit over the edge (to make the overlaid collar). Wrap the quail egg up. If you use a little bit of water on the nori, it will hold it together. Make slits in the ham and nori on the left and right sides so that the "kimono" can flare out as though the doll is sitting (bottom part of the picture). - Lay out the decorating ingredients on a plate. [Flower] Use a straw as a cutter. [Roses] Make by wrapping up smoked salmon. [Dolls] Use toothpicks to secure the fans and crowns to the quail eggs. (When attaching them to the cake, you can push the toothpicks deep inside) - Transfer the cake from Step 2 to a plate and remove the plastic wrap. Decorate with the ingredients from Step 4. Last, sprinkle the entire surface with the denbu to finish! There's no faces on the dolls, but if you like, it's fun to use nori to make the faces with your child! Your daily diet should have at least several of these super foods. First off, beans are excellent, and berries, particularly blueberries. Next, try adding several green foods such as broccoli, spinach, or green tea. Walnuts and whole cereals are a couple of other foods to include. In addition, you may wish to eat salmon, turkey, yogurt, soya, tomatoes, oranges, and pumpkins. Making these foods a usual part of your diet will eliminate your weight gain problems. Adopting a green living eating plan will give you exactly what you need to become healthy and fit. Your immune system will be much improved, and your body can be free of disease. You can anticipate a healthy future by modifying your food choices now. Sushi Cake for Hinamatsuri 寿司ケーキの作り方 (ひな祭りレシピ) - OCHIKERON - CREATE EAT HAPPY. On this day, we set up hina dolls (hina-ningyo) since they are believed to protect girls from bad luck. Hishi Mochi (diamond shaped rice cake) is usually presented with hina dolls. Made using all natural ingredients, with some plum vinegar, nori seaweed, and seasoned sushi rice, these sushi rolls are a feast for the eyes. Chirashi Sushi Cake & Temari Sushi This past Saturday was a special day for young girls in Japan. Selain disajikan untuk perayaan Hina Matsuri, hamaguri-zushi juga bisa dijadikan lauk untuk bento atau bekal makan siang. I wanted to make a very colorful cake-style chirashi sushi for the Hinamatsuri festivities. The most iconic image of this event is the displaying of beautiful hina. Hinamatsuri (雛祭り, Hina-matsuri), also called Doll's Day or Girls' Day, is a special day in Japan. There are no reviews for Matsuri Sushi, Japan yet. Be the first to write a review! On this day, families with girls wish their daughters a successful and happy life. Dolls are displayed in the house together with peach blossoms. The doll festival has its origin in a. Hina Matsuri is a celebration dedicated to girls, in which their parents pray for the good health and happiness of their daughters. This day corresponds to the time when peach blossoms start to bloom. That is why the celebration is also called Momo no Sekku (literally, the peach blossom festival). Hina Matsuri (Dolls Festival)- Read a feature article on the historical beginnings and traditions of Japan's Hina Matsuri or Dolls Festival. Every year, in mid-February, families with daughters all over Japan clear space in the home for a set of elaborately-made. So that’s going to wrap it up with this special food hina matsuri sushi cake recipe. Thanks so much for your time. I am confident you can make this at home. There is gonna be more interesting food at home recipes coming up. Don’t forget to save this page in your browser, and share it to your family, friends and colleague. Thank you for reading. Go on get cooking!	http://onbrokenwingsma.com/1100-easiest-way-to-make-favorite-hina-matsuri-sushi-cake/
The world’s largest butterfly survey has been launched by Chris Packham and wildlife charity Butterfly Conservation to help assess the health of our environment. The Big Butterfly Count attracted more than 113,000 people last year, and with many of us turning to nature during lockdown, 2020 is expected to be the biggest year yet. Taking part in the Big Butterfly Count is an easy way to give something positive back to nature. Those participating in the nationwide survey are asked to spend just 15 minutes in an outdoor space counting the butterflies they see during a three week period. The sunny spring weather has meant that almost all butterfly species have emerged early this summer, and as a result, Butterfly Conservation is expecting some interesting data this year. Chris Packham says: “While so many of us have had a bit more time to appreciate the nature on our doorsteps during the lockdown period, and learning about the natural world has been a mindful distraction from uncertainty, this is a real chance to do something positive and contribute to conserving nature. Butterflies and moths are key indicators of the health of our environment and anyone can help contribute to our understanding of these incredible creatures by taking part in the Big Butterfly Count. “The sightings you submit will be used to map and measure populations and the geographic spread of species across the UK. We’re asking everyone who has been given a helping hand from nature this year to return the favour.” The Big Butterfly Count runs from the 17th of July until the 9th of August and is open to all ages. You can submit your sightings online at www.bigbutterflycount.org or via the free Big Butterfly Count app.	https://www.discoveranimals.co.uk/news/join-big-butterfly-count-2020/
4.25 is a fraction. It is a fraction in decimal form rather than in the form of a ratio. However, that does not stop it being a fraction. In word form it is four and twenty five hundredths. Negative one with twenty-five zeros is negative ten septillion. five hours and twenty minute is not necessarily a fraction. It is, for example, 320 minutes, which is an integer. Seven and a quarterSeven point two fiveSeven and twenty-five hundredths Greater. 4/5 four and twenty-five hundredths = 425/100 -5.5 (negative five point five) Expressed as a proper fraction in its simplest form, (5/3)-3 = 27/125 or twenty-seven one hundred and twenty-fifths. Expressed as a decimal, this is equal to 0.216. 35% as a simplified fraction is seven twenty-fifths, or seven over twenty-five. Negative 3 and five sevenths expressed as an improper fraction is - 26/7. twenty five tenths = 25/10 = 5/2 (improper fraction) = 21/2 (mixed number) The answer is 25.5 (twenty five point five). That is "twenty nine point five, six, seven, five" 1/4 1/2 -5 - 25 = -30 Fourteen point two five. One four point two five Fourteen point twenty-five One four point twenty-five Fourteen and twenty-five hundredths. Fourteen and a quarter. two point twenty-five. 51 10/40 is greater.	https://math.answers.com/Q/What_is_negative_point_twenty_five_as_a_fraction
Expression of transforming growth factor beta(1), beta(3), and basic fibroblast growth factor in full-thickness skin wounds of equine limbs and thorax. To map the expression of transforming growth factor (TGF)-beta(1), TGF-beta(3), and basic fibroblast growth factor (bFGF) in full-thickness skin wounds of the horse. To determine whether their expression differs between limbs and thorax, to understand the pathogenesis of exuberant granulation tissue. Six wounds were created on one lateral metacarpal area and one midthoracic area of each horse. Sequential wound biopsies allowed comparison of the temporal expression of growth factors between limb and thoracic wounds. Four 2- to 4-year-old horses. Wounds were assessed grossly and histologically at 12 and 24 hours, and 2, 5, 10, and 14 days postoperatively. ELISAs were used to measure the growth factor concentrations of homogenates of wound biopsies taken at the same timepoints. TGF-beta(1) peaked at 24 hours in both locations and returned to baseline in thoracic wounds by 14 days but remained elevated in limb wounds for the duration of the study. Expression kinetics of TGF-beta(3) differed from those of TGF-beta(1). TGF-beta(3) concentrations gradually increased over time, showing a trend toward an earlier and higher peak in thoracic compared with limb wounds. bFGF expression kinetics resembled those of TGF-beta(1), but no statistically significant differences existed between limb and thoracic wounds. Growth factor expression is up-regulated during normal equine wound repair. TGF-beta(1) and TGF-beta(3) show a reciprocal temporal regulation. Statistically significant differences exist between limb and thoracic wounds with respect to TGF-beta(1) expression. The persistence of TGF-beta(1) expression in leg wounds may be related to the development of exuberant granulation tissue in this location, because TGF-beta(1) is profibrotic.
Note. It is to be observed that there is a decided difference in sound, except in the case of a, between the long and the short vowels. It is not merely a matter of quantity but also of quality. 6. In diphthongs (two-vowel sounds) both vowels are heard in a single syllable. Diphtongs Latin Examples ae as ai in aisle tae'-dae au as ou in out gau'-dĕt ei as ei in eight hei eu as ĕ'o͝o (a short e followed by a short u in one syllable) seu oe like oi in toil foe'-dŭs ui like o͝o'ĭ (a short u followed by a short i in one syllable. Cf. English we) cui, huic Note. Give all the vowels and diphthongs their proper sounds and do not slur over them in unaccented syllables, as is done in English. 7. Consonants are pronounced as in English, except that Consonants Latin Examples c is always like c in cat, never as in cent că'-dō, cĭ'-bŭs, cē'-nă g is always like g in get, never as in gem gĕ'-mō, gĭg'-nō i consonant is always like y in yes iăm, iŏ'-cŭs n before c, qu, or g is like ng in sing (compare the sound of n in anchor) ăn'-cŏ-ră (ang'-ko-ra) qu, gu, and sometimes su before a vowel have the sound of qw, gw, and sw. Here u has the value of consontant v and is not counted a vowel ĭn'-quĭt, quī, lĭn'-guă, săn'-guĭs, suā'dĕ-ō' s is like s in sea, never as in ease rǒ'-să, ĭs t is always like t in native, never as in nation ră'-tĭ-ō, nā'-tĭ-ō Consonants Latin Examples v is like w in wine never as in vine vī'-nǔm, vǐr x has the value of two consonants (cs or gs) and is like x in extract, not as in exact ěx'-trā, ěx-āc'-tǔs bs is like ps and bt like pt ǔrbs, ǒb-tǐ'-ně-ō ch, ph, and th are like c,p, t pǔl'-chěr, Phoe'-bē, thě-ā'-trǔm a. In combinations of consonants give each its distinct sound. Doubled consonants should be pronounced with a slight pause between the two sounds. Thus pronounce tt as in rat-trap, not as in rattle; pp as in hop-pole, not as in upper. Examples, mǐt'-tō, Ǎp'pǐ-ǔs, běl-lǔm. 11. The quantity of a vowel or a syllable is the time it takes to pronounce it. Correct pronunciation and accent depend upon the proper observance of quantity. 12. Quantity of Vowels. Vowels are either long ( ¯ ) or short ( ˘ ). In this book the long vowels are marked. Unmarked vowels are to be considered short. 1. A vowel is short before another vowel or h ; as pŏ-ē'-ta, tră'-hō. 2. A vowel is short before nt and nd, before final m or t, and, except in words of one syllable, before final l or r. Thus a'-mănt, a-măn'-dus, a-mā'-băm, a-mā'-băt, a'-ni-măl, a'-mŏr. 3. A vowel is long before nf, ns, nx, and nct. Thus īn'-fe-rō, re'-gēns, sān'-xī, sānc'-tus. 4. Diphthongs are always long, and are not marked. 13. Quantity of Syllables. Syllables are either long or short, and their quantity must be carefully distinguished from that of vowels. 1. A syllable is short, a. If it ends in a short vowel; as ă'-mō, pĭ'-grī. Note. In final syllables the short vowel may be followed by a final consonant. Thus the word mĕ-mŏ'-rĭ-ăm contains four short syllables. In the first three a short vowel ends the syllable, in the last the short vowel is followed by a final consonant. 2. A syllable is long, a. If it contains a long vowel or a diphthong, as cū'-rō, poe'-nae, aes-tā'-te. b. If it ends in a consonant which is followed by another consonant, as cor'-pus, mag'-nus. Note. The vowel in a long syllable may be either long or short, and should be pronounced accordingly. Thus in ter'-ra, in'-ter, the first syllable is long, but the vowel in each case is short and should be given the short sound. In words like saxum the first syllable is long because x has the value of two consonants (cs or gs). 3. In determining quantity h is not counted a consonant. Note. Give about twice as much time to the long syllables as to the short ones. It takes about as long to pronounce a short vowel plus a consonant as it does to pronounce a long vowel or a diphthong, and so these quantities are considered equally long. For example, it takes about as long to say cǔr'-rō as it does cū'-rō, and so each of these first syllables is long. Compare mǒr-lis and mō'-lis, a-mis'-sī and ā-mā'-sī. 14. Words of two syllables are accented on the first, as mēn'-sa, Cae'-sar. 15. Words of more than two syllables are accented on the penult if the penult is long. If the penult is short, accent the antepenult. Thus mo-nē'-mus, re'-gi-tur, a-gri'-co-la, a-man'-dus. Note. Observe that the position of the accent is determined by the length of the syllable and not by the length of the vowel in the syllable. (Cf.§ 13.2, Note.) 16. Certain little words called enclit'ics[5] which have no separate existence, are added to and pronounced with a preceding word. The most common are que, and; -ve, or; and -ne, the question sign. The syllable before an enclitic takes the accent, regardless of its quantity. Thus populus'que, dea'que, rēgna've, audit 'ne. 17. To read Latin well is not so difficult, if you begin right. Correct habits of reading should be formed now. Notice the quantities carefully, especially the quantity of the penult, to insure your getting the accent on the right syllable. (Cf. § 15.) Give every vowel its proper sound and every syllable its proper length. Then bear in mind that we should read Latin as we read English, in phrases rather than in separate words. Group together words that are closely connected in thoughts. No good reader halts at the end of each word. 18. Read the stanzas of the following poem by Longfellow, one at a time, first the English and then the Latin version. The syllables inclosed in parentheses are to be slurred or omitted to secure smoothness of meter. ↑N.B. The sounds of the letters are best learned by hearing them correctly pronounced. The matter in this section is, therefore, intended for reference rather than for assignment as a lesson. As a first step it is suggested that the teacher pronounce the examples in class, the pupils following.
Joints are cartilage surfaces that connect bones to each other. This cartilage allows our bones to glide smoothly against one another, allowing us painless movement. There are four joints in each finger, totaling 20 joints in each hand! Distal Interphalangeal Joint (DIP): The DIP joint is located at the tip of the finger, just before the finger nail starts. Arthritis can develop at this joint, and it is also commonly fractured. Proximal Interphalangeal Joint (PIP): The PIP joint is the joint just below the DIP joint. It is located below the top two bones of the finger and allows the finger to bend and extend. This joint can become stiff easily after injury. Metacarpophalangeal Joint (MCP): The MP joint is where the hand bone meets the finger bone, referred to as the “knuckle.” These joints are very important, allowing us to bend/flex and spread our fingers. Carpometacarpal Joint (CMC Joint): The CMC joint is located at the bottom of the hand bone. This joint varies in each finger. For example, in the index finger, it has little motion. In the small finger, it has a lot of motion. Injuries and problems with this joint are uncommon. The thumb joints are a little different than the other finger joints. To learn more about the thumb joints and more about the finger joints, visit our online Anatomy section.	http://blog.handcare.org/blog/2017/10/26/anatomy-101-finger-joints/
DESCRIPTION: Double fronted freehold property located to the North of Blackpool Centre close to the Promenade and just off Dickson Road. 4 self contained flats (3 x 1 bed & 1 x 2 bed) let at £100 per week each. The present owner takes single mature gentleman and has established long term tenants. Rental income £20,800 per annum. Central heating included in rents from landlord supply, electricity is coin meters. Individual Council Tax bills. LOCATION: On the Southerly frontage of Carshalton Road which is off Dickson Road to the rear of the Promenade Grand Hotel (ex Hilton). ACCOMMODATION Ground Floor: Flat 1: Lounge 3.7m x 4.3m (16sqm), kitchen with fitted units 3.2m x 3.9m (13sqm); shower and wc; utility room (landlords boiler); bedroom 2.2m x 2.8m; bedroom 4m x 4m with en suite shower &wc; Flat 2: Lounge 5.2m x 4m; bedroom 4m x 3.5m; shower and wc 1.7m x 3.1m; First Floor: Flat 3: Lounge 2.9m x 4.3; bedroom 3.2m x 4m; kitchen 2m x 2.7m; shower and wc; Flat 4: Lounge 2.9m x 4.3m; bedroom 4m x 3.9m with en suite shower and wc; kitchen 2m x 3.3.m; Exterior: Rear yard. RENT SCHEDULE: All 4 flats let at £100 per week, total £400 per week (£20,800 per annum). SERVICES: All mains services connected, L2 alarm, central heating (landlord pays bill), £1 coin meters for electric. Viewing STRICTLY BY PRIOR TELEPHONE APPOINTMENT THROUGH OUR OFFICE.	https://www.kaysestates.co.uk/properties/permanent-investment-flats/?pervious_url=https%3A%2F%2Fwww.kaysestates.co.uk%2F
On April 6, 2020, Tapplock, Inc., a Canadian maker of internet-connected smart locks, entered into a settlement with the Federal Trade Commission (“FTC”) to resolve allegations that the company deceived consumers by falsely claiming that it had implemented reasonable steps to secure user data and that its locks were “unbreakable.” The FTC alleged that these representations amounted to deceptive conduct under Section 5 of the FTC Act. In its press release accompanying the settlement, the FTC provided guidance for IoT companies regarding the design and implementation of privacy and security measures for “smart” devices, as discussed further below in this post. Among other allegations that the Company lacked reasonable security measures, the FTC’s complaint noted that the company had not: - adopted and implemented written data security standards, policies, procedures or practices; - identified reasonably foreseeable risks, such as through vulnerability or penetration testing; - implemented privacy and security guidance or training for employees responsible for software design, testing, and approval; or - sufficiently detected and prevented users from bypassing authentication procedures to gain access to Tapplock’s API information about user accounts. The 20-year settlement prohibits Tapplock from making misrepresentations about the security of its products or with respect to its handling of personal information. Moreover, under the settlement’s requirements, Tapplock must implement a comprehensive security program. Among other requirements, this security program must include: - documentation of the security program and evaluations to the board of directors or similar corporate governing body; - designation of a qualified employee to coordinate the security program; - implementation of safeguards to control risks involved with the internet-connected locks, which may include: employee training, technical measures, and access controls; and - biennial, independent privacy assessments. In its press release regarding the settlement, the FTC provides advice for IoT businesses regarding their privacy and security practices. Specifically, the FTC recommends that IoT companies: - implement “security by design” by implementing security measures in products at the outset of product design and development efforts and by conducting vulnerability and penetration testing before product release; - encourage a culture of security through written security standards and designating a senior official responsible for security; - design products with secure authentication in mind, which the FTC says “is a must in the Internet of Things” due to the potential for a vulnerable IoT device to allow broader access to the network to which it is connected; - take advantage of advice and guidance from security experts to properly implement security measures, such as encryption; and - protect interfaces between your product and other devices or services, which may prevent security weaknesses at the point where a service communicates with your device. The FTC’s authority extends to foreign defendants—such as the Canadian Tapplock—in certain circumstances. Section 5 of the FTC Act empowers the agency to enforce “unfair or deceptive acts or practices in or affecting commerce.” In 2006, Congress passed the Undertaking Spam, Spyware, and Fraud Enforcement with Enforcers beyond Borders Act of 2006 (“U.S. Safe Web Act of 2006”), which amends the FTC Act and clarifies that unfair or deceptive practices includes acts or practices “involving foreign commerce” that “(i) cause or are likely to cause reasonably foreseeable injury within the United States, or (ii) involve material conduct occurring in the United States.” The FTC’s guidance for IoT companies will be important for companies to consider as they design, manufacture, and provide IoT devices. Regular updates on developments related to IoT and cybersecurity can be found on Covington’s Internet of Things website.	https://www.insideprivacy.com/consumer-protection/iot-update-ftc-settles-with-smart-lock-manufacturer-and-provides-guidance-for-iot-companies/
Every year on September 30, we celebrate International Translation Day. Why on that date? Because it coincides with the Feast of St. Jerome, who was one of the first translators to translate the Bible into Latin. International Translation Day is about celebrating translators and the work they do. To mark the occasion, I’ve compiled 10 interesting facts on the fascinating world of translation! 1) The 5 most translated languages in the world are English, French, German, Russian and Italian. 2) The Bible, which can be read in nearly 650 languages, is thought to be the most translated publication (and at least one of its books has been translated into 3,225 languages). Next is the United Nations’ Universal Declaration of Human Rights, which is available in more than 500 languages. 3) Certain literary classics have also been translated into many languages, such as The Adventures of Pinocchio (available in 260 languages) and The Little Prince (available in 300 languages). Translated into 70 languages, Harry Potter still has a way to go! 4) About 330,000 people practise translation as a profession, and that doesn’t include those who do it informally. 5) According to a UNESCO database called “Index Translationum”, which lists all of the books translated in the world, the top 3 most translated authors are Agatha Christie, Jules Verne and William Shakespeare. 6) The English verb “translate” comes from “translatus,” a form of the Latin verb “transferre,” meaning “to bring over, carry over.” 7) The translation profession is more than 2,000 years old! That’s right: the Old Testament is thought to have been translated into Greek in the 3rd century BC, in which case it would be the oldest recorded translation. 8) Scientific knowledge has long been shared through translation. For example, Émilie de Breteuil, marquise du Châtelet, an 18th-century physicist, was the first person to translate Isaac Newton’s law of universal gravitation into French. 9) Translators are known for inventing alphabets, the precious tools that allow us to share our knowledge. Mesrop Machtots invented the Armenian and Albanian alphabets. Saint Cyril invented the Cyrillic alphabet, used today to write Russian, Bulgarian and Serbian, among many other languages. And a little closer to home, we have James Evans in Manitoba, who invented Cree syllabics, which are used to write the most widely spoken Indigenous language in Canada. 10) Translators play a key role in providing access to foreign literature. Portuguese writer José Saramago, recipient of the Nobel Prize in Literature, expressed this reality so well when he said, “Writers make national literature, while translators make universal literature.” What do you think of Saramago’s quote? Have you discovered authors you love through translated works? If so, share them with us in the comments below!	https://www.tdntranslation.com/resources/translation-blogs/10-interesting-facts-on-translation-and-translators.html
MÉMOIRE X.O. CALVADOS Unique union of the single and double distillation, Mémoire X.O. is made from calvados of at least 15 years. After being aged separately, the selected eaux-de-vie have acquired a common memory in oak barrels for long years. The cellar master has put his mastery in creating a delicate and distinguished composition with powerful notes of apple and candied fruits. Colour : Crimson amber. Nose : Ripe apples, vanilla, tobacco, leather, clove, black pepper. Palette : Candied orange, apricot, nutmeg, cigar, butterscotch. Finish : Elegant, woody, candied fruits and spices.	https://www.thelittlewhiskyshop.co.uk/product/pere-magloire-calvados-xo-memoire/
NimYAML Introduction NimYAML is a pure YAML implementation for Nim. It is able to read from and write to YAML character streams, and to serialize from and construct to native Nim types. It exclusively supports YAML 1.2. Source code can be found on GitHub. You can install it with Nimble: nimble install yaml Quickstart Dumping reference types and cyclic structures code.nim out.yaml import yaml . serialization , streams type Node = ref NodeObj NodeObj = object name : string left , right : Node var node1 , node2 , node3 : Node new ( node1 ) ; new ( node2 ) ; new ( node3 ) node1 . name = "Node 1" node2 . name = "Node 2" node3 . name = "Node 3" node1 . left = node2 node1 . right = node3 node2 . right = node3 node3 . left = node1 var s = newFileStream ( "out.yaml" , fmWrite ) dump ( node1 , s ) s . close ( ) %YAML 1.2 %TAG !n! tag:nimyaml.org,2016: --- &a name : Node 1 left : name : Node 2 left : ~ right : &b name : Node 3 left : a right : ~ right : b Loading reference types and cyclic structures code.nim in.yaml import yaml . serialization , streams type Node = ref NodeObj NodeObj = object name : string left , right : Node var node1 : Node var s = newFileStream ( "in.yaml" ) load ( s , node1 ) s . close ( ) %YAML 1.2 %TAG !n! tag:nimyaml.org,2016: --- &a name : Node 1 left : name : Node 2 left : ~ right : &b name : Node 3 left : a right : ~ right : b Defining a custom tag uri for a type code.nim out.yaml import yaml , streams type Mob = object level , experience : int32 drops : seq [ string ] setTagUri ( Mob , "!Mob" ) setTagUri ( seq [ string ] , "!Drops" ) var mob = Mob ( level : 42 , experience : 1800 , drops : @ [ "Sword of Mob Slaying" ] ) var s = newFileStream ( "out.yaml" , fmWrite ) dump ( mob , s , tagStyle = tsAll ) s . close ( ) %YAML 1.2 %TAG !n! tag:nimyaml.org,2016: --- level : 42 experience : 1800 drops : [ Sword of Mob Slaying ] Customizing output style code.nim out.yaml import yaml . serialization , yaml . presenter , streams type Person = object name : string age : int32 var personList = newSeq [ Person ] ( ) personList . add ( Person ( name : "Karl Koch" , age : 23 ) ) personList . add ( Person ( name : "Peter Pan" , age : 12 ) ) var s = newFileStream ( "out.yaml" , fmWrite ) dump ( personList , s , options = defineOptions ( style = psCanonical , indentationStep = 3 , newlines = nlLF , outputVersion = ov1_1 ) ) s . close ( ) %YAML 1.1 %TAG !n! tag:nimyaml.org,2016: --- [ { ? " name" : " Karl Koch" , ? " age" : " 23" } , { ? " name" : " Peter Pan" , ? " age" : " 12" } ] Loading Nim objects from JSON code.nim in.yaml import yaml . serialization , streams type Person = object name : string age : int32 var personList : seq [ Person ] var s = newFileStream ( "in.yaml" ) load ( s , personList ) s . close ( ) [ { " name" : " Karl Koch" , " age" : 23 } , { " name" : " Peter Pan" , " age" : 12 } ] Dumping Nim objects as YAML code.nim out.yaml import yaml . serialization , streams type Person = object name : string age : int32 var personList = newSeq [ Person ] ( ) personList . add ( Person ( name : "Karl Koch" , age : 23 ) ) personList . add ( Person ( name : "Peter Pan" , age : 12 ) ) var s = newFileStream ( "out.yaml" , fmWrite ) dump ( personList , s ) s . close ( ) %YAML 1.2 %TAG !n! tag:nimyaml.org,2016: --- - name : Karl Koch age : 23 - name : Peter Pan age : 12 Processing a Sequence of Heterogeneous Items … With variant objects code.nim in.yaml import yaml , streams type Person = object name : string ContainerKind = enum ckString , ckInt , ckBool , ckPerson , ckNone Container { . implicit . } = object case kind : ContainerKind of ckString : strVal : string of ckInt : intVal : int of ckBool : boolVal : bool of ckPerson : personVal : Person of ckNone : discard setTagUri ( Person , nimTag ( "demo:Person" ) ) var list : seq [ Container ] var s = newFileStream ( "in.yaml" ) load ( s , list ) s . close ( ) assert ( list [ 0 ] . kind == ckString ) assert ( list [ 0 ] . strVal == "this is a string" ) %YAML 1.2 %TAG !n! tag:nimyaml.org,2016: --- - this is a string - 42 - false - 23 - { name : Trillian } - … With the Sequential API code.nim in.yaml import yaml , streams type Person = object name : string setTagUri ( Person , nimTag ( "demo:Person" ) , yTagPerson ) var s = newFileStream ( "in.yaml" , fmRead ) context = newConstructionContext ( ) parser = newYamlParser ( serializationTagLibrary ) events = parser . parse ( s ) assert events . next ( ) . kind == yamlStartDoc assert events . next ( ) . kind == yamlStartSeq var nextEvent = events . peek ( ) while nextEvent . kind != yamlEndSeq : var curTag = nextEvent . tag ( ) if curTag == yTagQuestionMark : assert nextEvent . kind == yamlScalar case guessType ( nextEvent . scalarContent ) of yTypeInteger : curTag = yTagInteger of yTypeBoolTrue , yTypeBoolFalse : curTag = yTagBoolean of yTypeUnknown : curTag = yTagString else : assert false , "Type not supported!" elif curTag == yTagExclamationMark : curTag = yTagString case curTag of yTagString : var s : string events . constructChild ( context , s ) echo "got string: " , s of yTagInteger : var i : int32 events . constructChild ( context , i ) echo "got integer: " , i of yTagBoolean : var b : bool events . constructChild ( context , b ) echo "got boolean: " , b of yTagPerson : var p : Person events . constructChild ( context , p ) echo "got Person with name: " , p . name else : assert false , "unsupported tag: " & $ curTag nextEvent = events . peek ( ) assert events . next ( ) . kind == yamlEndSeq assert events . next ( ) . kind == yamlEndDoc assert events . finished ( ) s . close ( ) %YAML 1.2 %TAG !n! tag:nimyaml.org,2016: --- - this is a string - 42 - false - 23 - { name : Trillian } Loading Nim objects from YAML code.nim in.yaml import yaml . serialization , streams type Person = object name : string age : int32 var personList : seq [ Person ] var s = newFileStream ( "in.yaml" ) load ( s , personList ) s . close ( ) %YAML 1.2 --- - { name : Karl Koch , age : 23 } - { name : Peter Pan , age : 12 } Dumping Nim objects as JSON
Don’t forget to save us in your favourites and follow using the button to receive new post alerts. ‘Stitching Favourite Journeys’ Janet Browne In March, we were treated to a huge selection of Janet’s work, as she talked through not only her journey with textiles and embroidery but her many individual journeys she continues to capture and represent in style! Janet is local to Halifax and says she remembers seeing the textile dust along the streets and the smell of lanolin from the mills, so it was in her blood from the start. At school, she found a love of colour in her studies but was advised to reduce this a little and consider the benefits of neutrals and monotone in terms of design. Janet brought her map books and sketchbooks showing how this concentrated her thoughts such as looking through areas to the landscape beyond, and the development of symbols which she continues to apply to her maps today. Walks are broken down into sections, taking a snapshot every few hindered paces. These include quick sketches, symbols and words. Some pieces are pleated, representing the fact that you cannot see the whole landscape at any one point in a journey. When in town traffic, she plots stops in order to capture something of interest or significance. Some of the journeys became closer to home, and Janet’s garden has been a recurring subject for her map making. He shape of the maps show the shape of the journey, for example up and down a hill, or the wiggly shape the walk or ride took her. An earlier piece of work: And a more recent drawn design: Janet briefly described how she works from the reverse of a piece, adding small pieces of fabric as needed to stitch the tiny elements which make her map. Layers form a ‘sandwich’ of hand-dyed material, wadding and dressmaker’s tissue with her drawn design. Janet is currently taking tiny elements from larger maps and expanding these into new pieces, such as an allotment, a garden or creating the birds she has seen on a journey. Everything she includes has been seen by her. Here are one or two pieces created on Janet’s workshop. There are more on our Facebook page (link below).	https://halifaxembroiderersguild.com/2019/03/16/the-journey-begins/
GDP (or Gross Domestic Product) and GNP (Gross National Product) measure the size and strength of an economy but are calculated and used in different ways. \|Gross Domestic Product\|\|Gross National Product\| \|An estimated value of the total worth of a country’s production and services, within its boundary, by its nationals and foreigners, calculated over the course on one year.\|\|An estimated value of the total worth of production and services, by citizens of a country, on its land or on foreign land, calculated over the course on one year.\| \|GDP = consumption + investment + (government spending) + (exports − imports).\|\|GNP = GDP + NR (Net income inflow from assets abroad or Net Income Receipts) — NP (Net payment outflow to foreign assets).\| \|Business, Economic Forecasting.\|\|Business, Economic Forecasting.\| \|To see the strength of a country’s local economy.\|\|To see how the nationals of a country are doing economically.\| \|Total value of products & Services produced within the territorial boundary of a country.\|\|Total value of Goods and Services produced by all nationals of a country (whether within or outside the country).\| \|Qatar ($102,785)\|\|Luxembourg ($45,360).\| \|Malawi ($242).\|\|Mozambique ($80).\| \|USA ($17.42 Trillion in 2014).\|\|USA (~ $11.5 Trillion in 2005).\| GDP stands for Gross Domestic Product, the total worth estimated in currency values of a nation’s production in a given year, including service sector, research, and development. That translates to a sum of all industrial production, work, sales, business and service sector activity in the country. Usually this is calculated over a period of one year, but there may be analysis of short and long term trends to be used for economic forecast. Gross Domestic Product can also be calculated on a per capita (or per person) basis to give a relative example of the economic development of nations. GNP Definition GNP stands for Gross National Product. In general terms, GNP means the total of all business production and service sector industry in a country plus its gain on overseas investment. In some cases GNP will also be calculated by subtracting the capital gains of foreign nationals or companies earned domestically. Through GNP an accurate portrait of a nation’s yearly economy can be analyzed and studied for trends since GNP calculates the total income of all the nationals of a country. This gives a far more realistic picture than the income of foreign nationals in the country as it is more reliable and permanent in nature. Gross National Product can also be calculated on a per capita basis to demonstrate the consumer buying power of an individual from a particular country, and an estimate of average wealth, wages, and ownership distribution in a society. Here is a video of economist Phil Holden explaining the difference between GNP and GDP and talking about how they are measured and how accurate they are. Calculation GDP of a country is defined as the total market value of all final goods and services produced within a country in a given period of time (usually a calendar year). It is also considered the sum of value added at every stage of production (the intermediate stages) of all final goods and services produced within a country in a given period of time. The most common approach to measuring and understanding GDP is the expenditure method: GDP = consumption + investment + (government spending) + (exports – imports), or, GDP = C + I + G + (X-M) How GNP is calculated There are various ways of calculating GNP numbers. The expenditure approach determines aggregate demand, or Gross National Expenditure, by summing consumption, investment, government expenditure and net exports. The income approach and the closely related output approach sum wages, rents,interest, profits, non income charges, and net foreign factor income earned. The three methods yield the same result because total expenditures on goods and services (GNE) is equal to the value of goods and services produced (GNP) which is equal to the total income paid to the factors that produced the goods and services (GNI). Expenditure Approach to calculating GNP:GNP = GDP + NR (Net income from assets abroad (Net Income Receipts)). Applications of GDP and GNP numbers GDP and GNP figures are both calculated on a per capita basis to give a portrait of a country's economic development. GDP (or Gross Domestic Product) may be compared directly with GNP (or Gross National Product), to see the relationship between a country's export business and local economy. A region's GDP is one of the ways of measuring the size of its local economy whereas the GNP measures the overall economic strength of a country. These figures can also be used to analyze the distribution of wealth throughout a society, or the average purchasing power of an individual in the country etc. Increase in exports of a country will lead to increase in both GDP and GNP of the country. Correspondingly, increase in imports will decrease GDP and GNP. However, sometimes increase in exports might only lead to increase in GDP and not GNP. The exact relationship will depend on the nationality status of the company doing the export or import. E.g. if Microsoft Corporation has a 100% owned subsidiary in India, and that office exports US$2 Billion worth of services India, then US$2 Billion will be added to the GDP of India. However, it will not be added to the GNP figure since the export is done by a US company and not an Indian company. Criticism GDP is perhaps the most widely used metric to measure the health of economies. But some economists have argued that GDP is a flawed metric because it does not measure the economic well being of society. For example, it's possible that GDP is going up but median income going down and poverty rate increasing. GDP also does not measure environmental impact of growth, nor sustainability. Other important metrics include health of the population, infant mortality rates, and malnutrition rates, none of which are captured by GDP. Here's Nobel laureate Joseph Stiglitz offering a criticism of GDP. And at about the 4:45 mark, he talks about the difference between GDP and GNP: Stiglitz says that around 1990, GDP supplanted GNP as the primary measure of economic progress. He says that GNP measures the income of the people within the country whereas GDP measures economic activity in the country. If economic activity occurs in the country but the income from this activity accrues to foreigners, it will still be counted in GDP but not in GNP. He cites the example of privatized mining. Often the state gets a royalty of 1-2% but the income from privatized, foreign-owned mines accrues largely to shareholders. (Also see Stiglitz's article: GDP Fetishism). Social Progress Index The Social Progress Index was designed to measure non-economic indicators of well-being such as literacy rates, child mortality rates, shelter, access to water etc. The Economist plotted SPI data against per capita GDP to see which countries are «punching above their weight» in terms of social progress. SPI (Social Progress Index) vs. per capita GDP. Source: The Economist The chart reveals interesting insights about the effect or correlation of GDP on well-being in society. In general, the higher the per capita GDP, the higher the SPI. This is represented by the red line that plots the «average» curve. Countries above the red line are those where social progress indicators are better than per capita GDP would suggest. For example, Iran and Costa Rica have similar per capita GDP. However, Costa Rica performs significantly better than Iran on measures of social progress. Another example contrasts Brazil and UAE. Both are similar in their SPI scores even though UAE has a significantly higher GDP per person. Examples: U.S. and Ireland In 2010, U.S. GDP was $14.59 trillion. In the same year, the GNP was $14.64 trillion. The numbers for the U.S. are not very divergent because U.S. income receipts and payments are roughly in balance. On the other hand, Ireland GDP in 2010 was $211.39 billion and GNP $149.54 billion. References Share this comparison: If you read this far, you should follow us: «Gross Domestic Product (GDP) vs Gross National Product (GNP).» Diffen.com. Diffen LLC, n.d. Web. 23 Nov 2020. < > Источник: https://www.diffen.com/difference/GDP_vs_GNP Валовой внутренний продукт (ВВП) (Gross Domestic Product). Определение, значение термина — глоссарий Альпари Термин ВВП используется при проведении оценки экономического состояния рынка на мировом уровне или отдельно взятого государства. Появилось понятие в 1934 году, его предложил использовать американский экономист Саймон Смит. Значение ВВП отражается в национальной или иностранной валюте в зависимости от цели, для которой оно определяется. Как расшифровывается ВВП простыми словами Произошел термин от фразы «Валовой внутренний продукт» или сокращенно «ВВП» (от англ. Gross Domestic Product, GDP). Данный показатель отражает коммерческую стоимость всего объема изготовленных за год товаров и оказанных услуг. В расчет берутся все отрасли экономики, включая экспортные, связанные с потреблением и накоплением товарно-материальных ценностей. До начала применения этого показателя в 1930-х годах оценку экономического состояния страны никто не проводил. ВВП считается ключевым макроэкономическим показателем государства. Выделяют три разновидности ВВП: - Номинальный. Не учитывает наличие и уровень инфляции. - Реальный. Вычисляется с учетом инфляционных процессов. - По паритету покупательной способности на единицу населения. В последнем случае расчеты ведутся, исходя из валового продукта и населения страны. Но во всех вариантах исчисления учитываются товары / услуги без исключения: автомобили и бензин, путевки и конфеты, бытовая техника и стратегические запасы зерна. Показатель не зависит от того, применяются ли учитываемые товары в производстве, реализуются ли они конечному потребителю или экспортируются в другие страны. Номинальный ВВП Период, за который рассчитывается валовый внутренний продукт, равен календарному (бюджетному) году. Это происходит из-за особенностей государственного обеспечения регионов. В расчет берется, как затратная часть, так и налоговые / иные сборы с коммерческих предприятий, позволяющие пополнять бюджет. Производят вычисления по утвержденным методикам, что гарантирует точность, сравнимость результатов за несколько периодов. Номинальное значение ВВП вычисляется простым суммированием валового продукта. Номинальный ВВП рассчитывается и определяется, исходя из цен нынешнего года. При этом учитываются изменения стоимости каждого товара / услуги и реальное производство (фактически выпущенная продукция, выполненные работы). Номинальное значение получить проще, но оно не позволяет проводить сравнительный анализ между странами. Также затрудняется сравнение периодов, т. к. на реальное значение ВВП влияет инфляция, значительно изменяющаяся по значению от года к году. Реальный ВВП Более информативным оказывается значение реального ВВП. Именно такое значение дает точное определение периода экономики в государстве — наблюдается кризисный спад или сохраняется научно-технический рост. Реальный ВВП не зависит от курсовых колебаний национальной валюты в отношении доллара США, иных иностранных валют. Формула вычисления: Реальный ВВП = Номинальный ВВП / ОУЦ, где ОУЦ — общий уровень цен, рассчитываемый по средневзвешенной стоимости услуг и продукции на конкретный момент времени. Показатель позволяет учитывать имеющуюся инфляцию и сравнивать благосостояние населения страны в разные периоды одного года или одни и те же периоды нескольких лет. Вычисление показателя ВВП требует точных данных по валовому производству. Иногда формулу расчета реального ВВП переделывают — соотношение номинального ВВП к индексу цен. Последнее значение определяется делением цен текущего года на цены базисного года (умноженное на 100%). Вместо индекса цен можно брать показатель ИПЦ (индекс потребительских цен). Он определяется на базе стоимости той продукции, что включена в состав потребительской корзины. Ключевой особенностью значения реального ВВП является расчет на базе фактически произведенных товаров без учета оказанных услуг. Используются только стоимостные показания изготовителей, доступные после передачи данных в органы статистики. Любые показатели, используемые при вычислении ВВП, обычно доступны частным организациям и государственным органам власти. Как рассчитывается ВВП Размер ВВП РФ можно узнать на официальном сайте Росстата, где публикуются данные различных периодов. Применяются три основных способа расчетов, результаты которых в итоге оказываются одинаковыми. Расчетами занимаются экономисты органов статистики. Методики основаны на вычислениях: - По расходам. В расчет берутся потребительские, государственные, экспортные и инвестиционные виды расходов. Здесь в большей мере учитывается обеспечение бюджетной сферы и военных нужд государства, внедрение нововведений в производстве и методах подготовки персонала. - По доходам. По-другому способ называется распределительным. В итоговый показатель включены: национальный доход вместе с косвенными налогами, доход от экспорта и амортизационные отчисления. В дополнение к основной задаче показывает наличие дефицита или излишков обеспечения производственных предприятий. - По добавленной стоимости. По-другому производственный метод. Суммируется объем добавленной стоимости по всем существующим отраслям. На значение влияет как сам уровень наценки, так и затратная часть, т. е. повысить показатели можно лишь за счет уменьшения себестоимости производства. Способ вычисления выбирается, исходя из конкретных целей, имеющихся показателей по экономике. В зависимости от цели валовый внутренний продукт может рассчитываться в: - Национальной валюте (для России — это рубли). - Иностранной валюте по текущему биржевому курсу (в отношении рубля). - Долларах США. Последнее популярно при сравнении показателей нескольких государств. Осуществляется последовательный перевод национальной валюты каждой страны в одну, используемую повсеместно. В результате появляется возможность сравнительной оценки стран, составления общих рейтингов, их разделения на категории по состоянию экономики. В каких целях используется показатель ВВП Наиболее известным применением показателя валового внутреннего продукта является оценка экономического роста. ВВП может представляться как в абсолютном виде, так и в расчете на единицу населения государства. Во втором случае рост / падение уровня населения играет существенную роль наравне с производственными факторами. Благодаря оценке ВВП легко выявить рост или падение экономики. Валовый внутренний продукт позволяет оценивать следующие показатели: - Национальный доход государства. - Эффективность хозяйственной деятельности. - Степень активности субъектов экономики. - Направление развития экономики страны. Благодаря оценке абсолютного значения ВВП в определенные периоды легко понять динамику, качество изменений в экономике. Рост населения в государстве однозначно влияет на уровень жизни отрицательно, если валовый продукт остается на одном и том же значении. При тенденции снижения рождаемости / повышения смертности уровень жизни отдельно взятого гражданина возрастает. Значение показателя ВВП (абсолютное) напрямую влияет на ценность национальной валюты на мировом рынке, ее нахождение в корзине валют, используемой для вычисления различных финансовых индексов вроде DXY (Индекс доллара США). Высокое значение воспринимается как показатель богатого положения страны в целом. Внутри же уровень жизни может колебаться кардинально: обеспеченные люди становятся еще более богатыми, а бедные — получают очередное ухудшение в уровне своей жизни. Поэтому во время сравнения учитывают возможность анализа лишь макроэкономического состояния экономики. Какие показатели рассчитывают вместе с ВВП По одному лишь значению ВВП достаточно трудно давать объективную оценку экономической ситуации в стране, поэтому при подготовке аналитических и сравнительных отчетов берут во внимание еще несколько показателей. Объективной считается оценка, учитывающая несколько финансовых показателей. Наиболее полезны следующие величины: - Валовый национальный продукт (ВНП). Отражает стоимость всех ценных бумаг на рынке государства, эмитированных резидентами страны. - Чистый национальный продукт. Вычисляется путем вычитания из показателя ВНП затрат на амортизацию (субсидирование направлений производства с истощенным запасом ресурсов). - Фактический ВВП. Значение с учетом вероятной неполной занятости населения и отражения реализованных экономических показателей. - Потенциальный ВВП. Расчет ведется, исходя из полной занятости и возможностей производственных, иных мощностей. При изучении перечисленных показателей выявляется истинное благополучие населения, включая возможность разделения результатов по отдельным регионам. Когда речь идет о сравнении различных государств, такой способ менее подходит, т. к. каждый из исследуемых субъектов может обладать определенными особенностями, несочетающимися с другими показателями. Иногда показатели разделяют на интенсивные и экстенсивные. К первым относится рост динамики модернизации производства, совершенствование управленческой деятельности, повышение уровня сотрудников, распределение ресурсов между различными отраслями. Ко вторым обычно относят земельные, природные, трудовые ресурсы. Речь идет о привлечении дополнительного количества работников, повышении уровня технической оснащенности средствами автоматизации. Показатели ВВП России и Европы Сопоставление значений валового внутреннего продукта различных стран осуществляется международными организациями вроде ООН, МВФ. Сравнительная оценка охватывает большое количество государств различного уровня в развитии экономики, что позволяет составлять общий рейтинг их благосостояния по итогам вычислений. Показатели ВВП некоторых стран взаимосвязаны. Речь идет о наличии торговых отношений, предполагающих взаимный экспорт больших объемов продукции. Снижение его уровня с одной стороны уменьшает валовый продукт экспортера, с другой стороны — стимулирует собственное производство стран, куда перестали поставляться товары. Это относится и к сектору государственных закупок. Чем меньше власти стимулируют производство, тем больше теряется стабильность ВВП, т. к. показатель попадает под влияние коммерческих структур, иногда преследующих другие цели, несовпадающие с государственными. Повысить данный уровень можно ограничением импорта и принудительным переводом рынка на отечественные продукты. Источник: https://alpari.com/ru/beginner/glossary/gross-domestic-product/ Finance & Development Updated: February 24, 2020 Finance & Development Tim Callen When it is growing, especially if inflation is not a problem, workers and businesses are generally better off than when it is not Many professions commonly use abbreviations. To doctors, accountants, and baseball players, the letters MRI (magnetic resonance imaging), GAAP (generally accepted accounting principles), and ERA (earned run average), respectively, need no explanation. To someone unfamiliar with these fields, however, without an explanation these initialisms are a stumbling block to a better understanding of the subject at hand. Economics is no different. Economists use many abbreviations. One of the most common is GDP, which stands for gross domestic product. It is often cited in newspapers, on the television news, and in reports by governments, central banks, and the business community. It has become widely used as a reference point for the health of national and global economies. When GDP is growing, especially if inflation is not a problem, workers and businesses are generally better off than when it is not. Measuring GDP GDP measures the monetary value of final goods and services—that is, those that are bought by the final user—produced in a country in a given period of time (say a quarter or a year). It counts all of the output generated within the borders of a country. GDP is composed of goods and services produced for sale in the market and also includes some nonmarket production, such as defense or education services provided by the government. An alternative concept, gross national product, or GNP, counts all the output of the residents of a country. So if a German-owned company has a factory in the United States, the output of this factory would be included in U.S. GDP, but in German GNP. Not all productive activity is included in GDP. For example, unpaid work (such as that performed in the home or by volunteers) and black-market activities are not included because they are difficult to measure and value accurately. That means, for example, that a baker who produces a loaf of bread for a customer would contribute to GDP, but would not contribute to GDP if he baked the same loaf for his family (although the ingredients he purchased would be counted). Moreover, “gross” domestic product takes no account of the “wear and tear” on the machinery, buildings, and so on (the so-called capital stock) that are used in producing the output. If this depletion of the capital stock, called depreciation, is subtracted from GDP we get net domestic product. Theoretically, GDP can be viewed in three different ways: ● The production approach sums the “value-added” at each stage of production, where value-added is defined as total sales less the value of intermediate inputs into the production process. For example, flour would be an intermediate input and bread the final product; or an architect’s services would be an intermediate input and the building the final product. ● The expenditure approach adds up the value of purchases made by final users—for example, the consumption of food, televisions, and medical services by households; the investments in machinery by companies; and the purchases of goods and services by the government and foreigners. ● The income approach sums the incomes generated by production—for example, the compensation employees receive and the operating surplus of companies (roughly sales less costs). GDP in a country is usually calculated by the national statistical agency, which compiles the information from a large number of sources. In making the calculations, however, most countries follow established international standards. The international standard for measuring GDP is contained in the System of National Accounts, 1993, compiled by the International Monetary Fund, the European Commission, the Organization for Economic Cooperation and Development, the United Nations, and the World Bank. Real GDP One thing people want to know about an economy is whether its total output of goods and services is growing or shrinking. But because GDP is collected at current, or nominal, prices, one cannot compare two periods without making adjustments for inflation. To determine “real” GDP, its nominal value must be adjusted to take into account price changes to allow us to see whether the value of output has gone up because more is being produced or simply because prices have increased. A statistical tool called the price deflator is used to adjust GDP from nominal to constant prices. GDP is important because it gives information about the size of the economy and how an economy is performing. The growth rate of real GDP is often used as an indicator of the general health of the economy. In broad terms, an increase in real GDP is interpreted as a sign that the economy is doing well. When real GDP is growing strongly, employment is ly to be increasing as companies hire more workers for their factories and people have more money in their pockets. When GDP is shrinking, as it did in many countries during the recent global economic crisis, employment often declines. In some cases, GDP may be growing, but not fast enough to create a sufficient number of jobs for those seeking them. But real GDP growth does move in cycles over time. Economies are sometimes in periods of boom, and sometimes in periods of slow growth or even recession (with the latter often defined as two consecutive quarters during which output declines). In the United States, for example, there were six recessions of varying length and severity between 1950 and 2011. The National Bureau of Economic Research makes the call on the dates of U.S. business cycles. Comparing GDPs of two countries GDP is measured in the currency of the country in question. That requires adjustment when trying to compare the value of output in two countries using different currencies. The usual method is to convert the value of GDP of each country into U.S. dollars and then compare them. Conversion to dollars can be done either using market exchange rates—those that prevail in the foreign exchange market—or purchasing power parity (PPP) exchange rates. The PPP exchange rate is the rate at which the currency of one country would have to be converted into that of another to purchase the same amount of goods and services in each country. There is a large gap between market and PPP-based exchange rates in emerging market and developing countries. For most emerging market and developing countries, the ratio of the market and PPP U.S. dollar exchange rates is between 2 and 4. This is because nontraded goods and services tend to be cheaper in low-income than in high-income countries—for example, a haircut in New York is more expensive than in Bishkek—even when the cost of making tradable goods, such as machinery, across two countries is the same. For advanced economies, market and PPP exchange rates tend to be much closer. These differences mean that emerging market and developing countries have a higher estimated dollar GDP when the PPP exchange rate is used. The IMF publishes an array of GDP data on its website (www.imf.org). International institutions such as the IMF also calculate global and regional real GDP growth. These give an idea of how quickly or slowly the world economy or the economies in a particular region of the world are growing. The aggregates are constructed as weighted averages of the GDP in individual countries, with weights reflecting each country’s share of GDP in the group (with PPP exchange rates used to determine the appropriate weights). What GDP does not reveal It is also important to understand what GDP cannot tell us. GDP is not a measure of the overall standard of living or well-being of a country. Although changes in the output of goods and services per person (GDP per capita) are often used as a measure of whether the average citizen in a country is better or worse off, it does not capture things that may be deemed important to general well-being. So, for example, increased output may come at the cost of environmental damage or other external costs such as noise. Or it might involve the reduction of leisure time or the depletion of nonrenewable natural resources. The quality of life may also depend on the distribution of GDP among the residents of a country, not just the overall level. To try to account for such factors, the United Nations computes a Human Development Index, which ranks countries not only GDP per capita, but on other factors, such as life expectancy, literacy, and school enrollment. Other attempts have been made to account for some of the shortcomings of GDP, such as the Genuine Progress Indicator and the Gross National Happiness Index, but these too have their critics. Tim Callen is an Assistant Director in the IMF’s External Relations Department. Источник: https://www.imf.org/external/pubs/ft/fandd/basics/gdp.htm GDP Ranked by Country 2020 Gross Domestic Product (GDP) is the monetary market value of all final goods and services made within a country during a specific period. GDP helps to provide a snapshot of a country’s economy and can be calculated using expenditures, production, or incomes. World GDP The world GDP is the added total of the gross national income for every country in the world. Gross national income takes a country’s GDP, adds the value of income from imports, and subtracts the value of money from exports. The value of gross national income, GNI, differs from that of GDP because it reflects the impact of domestic and international trade. When the GNIs of every country in the world are added together, the value of imports and exports are in balance. The world economy consists of 193 economies, with the United States being the largest. As per World Bank estimates, the nominal world GDP in 2017 was $80,683.79 billion. In 2018, the nominal world GDP was $84,835.46 billion in 2018, and it’s projected to be $88,081.13 billion in 2019. In 2018, the growth rate for the world GDP was 3.6%. Nominal GDP vs. PPP GDP To compare GDPs around the world, currencies must be converted so that they’re consistent across all countries. There are two main systems of common currency conversion: nominal and PPP. These two approaches to GDP estimation have separate strengths and are generally used for different reasons. Nominal GDP is useful for large-scope GDP comparison, either for a country or region or on an international scale. The nominal GDP of an area is determined using up-to-date market prices and shifts according to inflation. By incorporating an area’s inflation rate in the GDP calculation, nominal GDP can indicate when prices rise in an economy. The rate of price increases in an economy is also factored into nominal GDP. The main downfall of nominal GDP is that it doesn’t account for the living standards in a country — it focuses only on economic growth and performance. Also, generally speaking, nominal GDP can differ significantly from year to year depending on variations in the exchange rate. PPP stands for purchasing power parity. PPP GDP is used to measure both the economic growth and living standards in a country, making it a useful tool in global comparisons. The PPP approach uses exchange rates to convert one country’s currency into the other. Then, using a consistent amount of money, the quantity of goods and services that may be purchased in the countries is compared. For example, PPP may compare the cost of a car in France to the cost of a car in Japan (after using the exchange rate to convert yen to Euros, or vice versa) to analyze the difference in GDP and cost of living between these nations. PPP GDP stays relatively stable from year to year and isn’t significantly impacted by shifts in the exchange rate. PPP GDP can be faulted for the fact that it doesn’t incorporate discrepancies in quality between goods and services in different countries. In general, it’s less exact than nominal GDP and often hinges on estimates rather than calculations. As such, the nominal GDP is typically used to measure and compare the size of national economies. Nominal GDP Rankings by Country What are the largest economies in the world? According to the International Monetary Fund, these are the highest ranking countries in the world in nominal GDP: - United States (GDP: 20.49 trillion) - China (GDP: 13.4 trillion) - Japan: (GDP: 4.97 trillion) - Germany: (GDP: 4.00 trillion) - United Kingdom: (GDP: 2.83 trillion) - France: (GDP: 2.78 trillion) - India: (GDP: 2.72 trillion) - Italy: (GDP: 2.07 trillion) - Brazil: (GDP: 1.87 trillion) - Canada: (GDP: 1.71 trillion) The Largest Economies in the World The three largest economies in the world as measured by nominal GDP are the United States, China, and Japan. Economic growth and prosperity are impacted by a wide array of factors, namely investment in workforce education, production output (as determined by investment in physical capital), natural resources, and entrepreneurship. The economies of the U. S., China, and Japan all have a unique combination of these factors that have led to economic growth over time, as outlined below. United States The United States has been the world’s largest economy since 1871. The nominal GDP for the United States is $21.44 trillion. The U.S. GDP (PPP) is also $21.44 trillion. Additionally, the United States is ranked second in the world for the approximate value of natural resources. In 2016, the U.S. had an estimated natural resource value of $45 trillion. Several factors contribute to the U.S.’s powerful economy. The U.S. is known globally for cultivating a society that supports and encourages entrepreneurship, which encourages innovation and, in turn, leads to economic growth. The growing population in the U.S. has helped diversify the workforce. The U.S. is also one of the leading manufacturing industries in the world, coming only second to China. The U.S. dollar is also the most widely used currency for global transactions. China As the second-largest economy in the world, China has seen an average growth rate of 9.52% between 1989 and 2019. China is the second-largest economy considering nominal GDP, at $14.14 trillion, and the largest using GDP (PPP), which is $27.31 trillion. China has approximately $23 trillion in natural resources, 90% of which are rare earth metals and coal. China’s economic reform program of 1978 was a large success and resulted in the rise in average economic growth from 6% to over 9%. The reform program emphasized the creation of private and rural businesses, easing the state regulations on prices, and investment in workforce education and industrial output. Another driving force behind the growth of China’s economy is worker efficiency. Japan Japan has the third-largest economy in the world with a GDP of $5.15 trillion. Japan’s GDP (PPP) is $5.75 trillion. Japan’s economy is market-driven so businesses, production, and prices shift according to consumer demand, not governmental action. While the 2008 financial crisis took a hit on the Japanese economy and has stunted its growth since then, it is expected that the 2020 Olympics will give it a boost. The Japanese economy’s strength comes from its electronic goods industry, which is the largest in the world, and its automobile industry, which is the third-largest in the world. Going forward, the Japanese economy faces some large challenges such as a declining population and an ever-increasing debt that, as of 2017, is 236% of its GDP. Germany The German economy is the fourth-largest in the world with a GDP of $4.0 trillion. Germany has a GDP (PPP) of $4.44 trillion and a per capita GDP of $46,560, the 18th –highest in the world. Germany’s highly developed social market economy is Europe’s largest and strongest economy and has one of the most skilled workforces. According to the International Monetary Fund, Germany accounted for 28% of the euro area economy. Germany’s major industries are car manufacture, machinery, household equipment, and chemicals. Because of its dependency on capital good exports, the economy had a significant setback post-2008 financial crisis. The German economy is currently in the middle of its fourth industrial revolution due to the Internet and the digital age. Industry 4. 0 is the term used for this transformation, which embraces solutions, processes, and technologies and describes the use of IT and a high degree of system networking in factories. India India’s economy is the fifth-largest in the world with a GDP of $2.94 trillion, overtaking the UK and France in 2019 to take the fifth spot. India’s GDP (PPP) is $10. 51 trillion, exceeding that of Japan and Germany. Due to India’s high population, India’s GDP per capita is $2,170 (for comparison, the U.S. is $62,794). India’s real GDP growth, however, is expected to weaken for the third straight year from 7.5% to 5%. India is developing into an open-market economy from its previous autarkic policies. India’s economic liberalization began in the early 1990s and included industrial deregulation, reduced control on foreign trade and investment, and privatization of state-owned enterprises. These measures have helped India accelerate economic growth. India’s service sector is the fast-growing sector in the world accounting for 60% of the economy and 28% of employment. Manufacturing and agriculture are two other significant sectors of the economy. United Kingdom The United Kingdom, which has a $2.83 trillion GDP, is the sixth-largest economy in the world. In terms of GDP purchasing power parity, the UK is in the ninth spot with a GDP (PPP) of The UK is ranked 23rd for GDP per capita which is $42,558. The UK is expected to fall to the seventh-largest economy by 2023 with a GDP of $3.27 trillion. In 2016, the UK was the tenth-largest exporter of goods in the world, exporting goods to 160 countries worldwide. In the 18th century, the United Kingdom was the first country to industrialize. The service sector dominates the UK economy, contributing about 80% of GDP, particularly the financial services industry. London is the second-largest financial center in the world. Manufacturing and agriculture are the second- and third-largest sectors in the United Kingdom. Britain’s aerospace industry is the second-largest in the world and its pharmaceutical industry is the tenth-largest. France France is the third-largest economy in Europe (behind Germany and the UK) and the seventh-largest economy in the world. France has a nominal GDP of $2.71 trillion. France’s GDP per capita is $42,877. 56, the 19th highest in the world, and GDP (PPP) is $2.96 trillion. According to World Bank, France has unfortunately experienced high unemployment rates in recent years: a 10% unemployment rate was recorded for 2014, 2015, and 2016 and it declined to 9.681% in 2017. France’s economy is a diversified free-market-oriented economy. The chemical industry is a key sector for France, as well as agriculture and tourism. France accounts for about one-third of all agricultural land in the European Union and is the sixth-largest agricultural producer and the second-largest agricultural exporter in the world, behind the United States. France is the most visited destination in the world. Additionally, France ranks 5th in the Fortune Global 500 behind the United States, China, Japan, and Germany with 28 of the 500 biggest companies. Italy With a nominal GDP of $1.99 trillion, Italy is the eighth-largest economy in the world. In terms of GDP (PPP) Italy’s economy is worth $2.40 trillion and its per capita GDP is $34,260.34. Italy’s economy is expected to expand to $2.26 trillion by 2023. Unfortunately, Italy is experiencing a relatively high unemployment rate of 9.7% and a debt at 132% of GDP. Fortunately, Italy’s exports are helping to recover the economy. Italy is the eighth-largest exporter in the world, conducting 59% of its trade with other European Union countries. Before World War II, Italy was primarily an agricultural economy and has now transformed into one of the world’s most advanced nations. Italy is the second-largest exporter in the European Union, behind Germany, and has a significant trade surplus from exporting machinery, vehicles, food, clothing, luxury goods, and more. Brazil Brazil has the ninth-largest economy in the world and the largest in Latin America with a nominal GDP of $1.85 trillion. Brazil is also the largest and most populous nation in Latin America. Brazil has the world’s 73rd highest per capita GDP of $8,967 and a GDP (PPP) of $2.40 trillion. The country has an estimated $21. 8 trillion in natural resources, which includes vast amounts of timber, uranium, gold, and iron. Brazil is a developing free-market economy. From 2000 to 2012, Brazil was one of the fastest-growing major economies in the world. Brazil, however, has one of the most unequal economies in the world. In 2017, the economic crisis, corruption, and lack of public policies increased the poverty rate and many became homeless. Six billionaires alone in Brazil are richer than more than 100 million of the poorest Brazilians. Canada Canada has the tenth-largest economy in the world with a nominal GDP of $1.73 trillion. Canada’s per capita GDP of $46,260.71 is ranked 20th globally while its GDP (PPP) of $1.84 trillion is ranked 17th globally. Canada’s GDP is expected to rise to $2.13 trillion by 2023. Canada has the fourth-highest estimated value of natural resources of $33.2 trillion. Canada is considered an energy superpower due to its abundant natural resources such as petroleum and natural gas. According to the Corruption Perceptions Index, Canada is one of the least corrupt countries in the world and one of the world’s top ten trading countries. Canada ranks above the United States on the Index of Economic Freedom and experiences a relatively low level of income disparity. IMF data from the April 2018 IMF World Economic Outlook database. UN data from the July 2018 World Development Indicators. GDP is in trillions of US dollars. Источник:	https://investingd.ru/gdp-gross-domestic-product.html
Designed to meet two sets of challenges, transitional and street skating, the 16-foot kidney-shaped skate bowl creates a challenge for the advanced transitional skater. The bowl transistions from a 7ft. verticle to an 11 ft. deep well. The plaza contains four staircases with hand rails, five ramps and 100- feet of ledges. Funding for the 15,000 square foot skate park was in part due to a grant obtained from the State as part of the Local Parks and Recreation Fund (LPRF). The City was awarded a $100,000 matching grant in July 2005. Skate Park Designer SITE Design Group, Inc. Skate Park Contractor x Skate Park Size 15,000 Sq. Ft.	https://www.sitedesigngroup.com/tennessee-franklin-jim-warren
Beautiful 243 acre Ranch in Southwest Colorado. Less than 30 minutes to Historic Durango Colorado. This Ranch has a one mile border with the San Juan National Forest and boasts a manicured blend of Ponderosa Pine forest and pristine meadows. The acreage has a total of 6 ponds and 2 of the larger ponds have a variety of fish. The largest pond has a island perfect for relaxing, paddle boarding, swimming, or picnics. There are ample water rights plus mineral rights that will transfer with the sale. Two excellent water wells are permitted and in place. Electricity is into the major section of the property along with phone and internet service. Abundant wildlife will be enjoyed on a regular basis including elk, deer, turkey and waterfowl. Great private hunting property. There is a old post and log barn and a newer 10 stall horse barn. There is also an older mobile home with a septic system that will transfer as well but is not considered in the value. The property was surveyed in 2012 and the boundary is well marked with orange t-posts and signage every 100 or less. There is ATV access to nearly the entire 3 mile length of border along with multiple trails and roads built throughout the interior of this property. The main mile driveway access from the highway is a deeded easement and the infrastructure of roads and trails within the property give access to a large selection of great building sites. \|MLS#\|\|762247\| \|Style\|\|Farm or Ranch\| \|Sub-Type\|\|Resale\| \|Status\|\|Active\| \|Year\|\|1973\| \|Days On Site\|\|24\| \|Bed(s)\|\|2\| \|Bath(s)\|\|1 Total\| \|SQ Feet\|\|1,000\| \|Car Port\|\|Yes\| \|Story(s)\|\|1 Story\| \|Acreage\|\|243.00\| \|Water Front\|\|Yes\| \|Property View\|\|Yes\| \|Taxes\|\|420\| \|Available Amenities\|\|Electricity,Phone,Cable TV,Propane-Tank Owned\| Area Information \|State\|\|CO\| \|County\|\|La Plata\| \|City\|\|Bayfield\| \|Zip Code\|\|81122\| \|Area\|\|Bayfield Rural\| \|Middle School\|\|Bayfield 6-8\| \|High School\|\|Bayfield 9-12\| \|Listing VS Median (in 81122)\| Price: $2,860,000 low \| high Med: $269,950 100% of Average SqFt: 1,000 low \| high Med: 720 78% of Average $ / SqFt: $2,860 low \| high Med: $94 100% of Average On Site: 24 low \| high Med:	https://www.realestatedurango.com/for-sale/49243-e-hwy-160-bayfield-co-81122/119-169635
Bid on a percentage of public performance royalties (both writer's share and publisher's share) generated from a collection of songs by Sheldon Reynolds. The writer's share of royalties is distributed by SESAC and the publisher's share is distributed by Reyshel Music. $15,800 Auction Closed Historic royalty income is no indication of future royalty income. Future royalty income is dependent upon future sales and licensing revenue generated by the sound recordings or compositions associated with this listing.	https://auctions.royaltyexchange.com/auctions/sheldon-reynolds/?state=closed
--- abstract: 'We associate [convex regions]{} in $\r^n$ to $\m$-primary graded sequences of subspaces, in particular $\m$-primary graded sequences of ideals, in a large class of local algebras (including analytically irreducible local domains). These [convex regions]{} encode information about Samuel multiplicities. This is in the spirit of the theory of Gröbner bases and Newton polyhedra on one hand, and the theory of Newton-Okounkov bodies for linear systems on the other hand. We use this to give a new proof, as well as a generalization of a Brunn-Minkowski inequality for multiplicities due to Teissier and Rees-Sharp.' address: - 'Department of Mathematics, School of Arts and Sciences, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, U.S.A.' - 'Department of Mathematics, University of Toronto, Toronto, Canada; Moscow Independent University; Institute for Systems Analysis, Russian Academy of Sciences' author: - Kiumars Kaveh - 'A. G. Khovanskii' title: Convex bodies and multiplicities of ideals --- [^1] [^2] Introduction {#introduction .unnumbered} ============ The purpose of this note is to employ, in the local case, techniques from the theory of semigroups of integral points and Newton-Okounkov bodies (for the global case) and to obtain new results as well as new proofs of some previously known results about multiplicities of ideals in local rings. Let $R = \mathcal{O}_{X, p}$ be the local ring of a point $p$ on an $n$-dimensional irreducible algebraic variety $X$ over an algebraically closed field $\k$. Let $\m$ denote the maximal ideal of $R$ and let $\a$ be an $\m$-primary ideal, i.e. $\a$ is an ideal containing a power of the maximal ideal $\m$. Geometrically speaking, $\a$ is $\m$-primary if its zero set (around $p$) is the single point $p$ itself. Let $f_1, \ldots, f_n$ be $n$ generic elements in $\a$. The [multiplicity]{} $e(\a)$ of the ideal $\a$ is the intersection multiplicity, at the origin, of the hypersurfaces $H_i = \{ x \mid f(x) = 0\}$, $i=1, \ldots, n$ (it can be shown that this number is independent of the choice of the $f_i$). According to Hilbert-Samuel’s theorem, the multiplicity $e(\a)$ is equal to: $$n!~\lim_{k \to \infty} \frac{\dim_\k(R/\a^k)}{k^n}.$$ (This result is analogous to Hilbert’s theorem on the Hilbert function and degree of a projective variety.) More generally, let $R$ be an $n$-dimensional Noetherian local domain over $\k$ (where $\k$ is isomorphic to the residue field $R/\m$ and $\m$ is the maximal ideal). Let $\a$ be an $\m$-primary ideal of $R$. Since $\a$ contains a power of the maximal ideal $\m$, $R/\a$ is finite dimensional regarded as a vector space over $\k$. The [Hilbert-Samuel function]{} of the $\m$-primary ideal $\a$ is defined by: $$H_\a(k) = \dim_\k(R/\a^k).$$ For large values of $k$, $H_\a(k)$ coincides with a polynomial of degree $n$ called the [Hilbert-Samuel polynomial]{} of $\a$. The [Samuel multiplicity]{}, $e(\a)$ of $\a$ is defined to be the leading coefficient of $H_\a(k)$ multiplied by $n!$. It is well-known that the Samuel multiplicity satisfies a Brunn-Minkowski inequality [@Teissier1; @RS]. That is, for any two $\m$-primary ideals $\a$, $\b \in R$ we have: $$\label{equ-Brunn-Mink-intro} e(\a)^{1/n} + e(\b)^{1/n} \leq e(\a\b)^{1/n}.$$ More generally we define multiplicity for [$\m$-primary graded sequences of subspaces]{}. That is, a sequence $\a_1, \a_2, \ldots$ of $\k$-subspaces in $R$ such that for all $k, m$ we have $\a_k \a_m \subset \a_{k+m}$, and [ $\a_1$ contains a power of the maximal ideal $\m$]{} (Definition \[def-graded-seq-subspace\]). We recall that if $\a, \b$ are two $\k$-subspaces of $R$, $\a\b$ denotes the $\k$-span of all the $xy$ where $x \in \a$ and $y \in \b$. In particular, a graded sequence $\a_\bullet$ where each $\a_k$ is an $\m$-primary ideal, is an $\m$-primary graded sequence of subspaces. We call such $\a_\bullet$ an [$\m$-primary graded sequence of ideals]{}. For an $\m$-primary graded sequence of subspaces we define multiplicity $e(\a_\bullet)$ to be: $$\label{equ-multi-graded-seq-ideals-intro} e(\a_\bullet) = \limsup_{k} \frac{\dim_\k(R/\a_k)}{k^n}.$$ (It is not a priori clear that the limit exists.) We will use convex geometric arguments to prove the existence of the limit in and a generalization of to $\m$-primary graded sequences of subspaces, for a large class of local domains $R$. [Let us briefly discuss the convex geometry part of the story]{}. Let $\C$ be a closed strongly convex cone with apex at the origin (i.e. $\C$ is a convex cone and does not contain any line). We call a closed convex set $\Gamma \subset \C$, a [$\C$-convex region]{} if for any $x \in \Gamma$ we have $x + \C \subset \Gamma$. We say that $\Gamma$ is [cobounded]{} if $\C \setminus \Gamma$ is bounded. It is easy to verify that the set of cobounded $\C$-convex regions is closed under addition (Minkowski sum of convex sets) and multiplication with a positive real number. For a cobounded $\C$-convex region $\Gamma$ we call the volume of the bounded region $\C \setminus \Gamma$ the [covolume of $\Gamma$]{} and denote it by $\covol(\Gamma)$. Also we refer to $\C \setminus \Gamma$ as a [$\C$-coconvex body]{}. (Instead of working with convex regions one can alternatively work with coconvex bodies.) In [@Askold-Vladlen; @Askold-Vladlen2], similar to convex bodies and their volumes (and mixed volumes), the authors develop a theory of convex regions and their covolumes (and mixed covolumes). Moreover they prove an analogue of the Alexandrov-Fenchel inequality for mixed covolumes (see Theorem \[thm-alexandrov-fenchel-covolume\]). The usual Alexandrov-Fenchel inequality is an important inequality about mixed volumes of convex bodies in $\r^n$ and generalizes the classical isoperimetric inequality and the Brunn-Minkowski inequality. In a similar way, the result in [@Askold-Vladlen] implies a Brunn-Minkowski inequality for covolumes, that is, for any two cobounded $\C$-convex regions $\Gamma_1$, $\Gamma_2$ where $\C$ is an $n$-dimensional cone, we have: $$\label{equ-intro-BM-inequ-covol} \covol(\Gamma_1)^{1/n} + \covol(\Gamma_2)^{1/n} \geq \covol(\Gamma_1 + \Gamma_2)^{1/n}.$$ We associate convex regions to $\m$-primary graded sequences of subspaces (in particular $\m$-primary ideals) and use the inequality to prove the Brunn-Minkowski inequality for multiplicities. To associate a convex region to a graded sequence of subspaces we need a valuation on the ring $R$. We will assume that there is a valuation $v$ on $R$ with values in $\z^n$ (with respect to a total order on $\z^n$ respecting addition) such that the residue field of $v$ is $\k$, and moreover the following conditions (i)-(ii) hold [^3]. We call such $v$ a [good valuation]{} on $R$ (Definition \[def-good-valuation\]): \(i) Let $\S = v(R \setminus \{0\}) \cup \{0\}$ be the value semigroup of $(R, v)$. Let $\C = C(\S)$ be the closure of the convex hull of $\S$. It is a closed convex cone with apex at the origin. We assume that $\C$ is a strongly convex cone. Let $\ell: \r^n \to \r$ be a linear function. For any $a \in \r$ let $\ell_{\geq a}$ denote the half-space $\{ x \mid \ell(x) \geq a\}$. Since the cone $\C = C(\S)$ associated to the semigroup $\S$ is assumed to be strongly convex we can find a linear function $\ell$ such that the cone $\C$ lies in $\ell_{\geq 0}$ and it intersects the hyperplane $\ell^{-1}(0)$ only at the origin. \(ii) We assume there exists $r_0 > 0$ and a linear function $\ell$ as above such that for any $f \in R$, if $\ell(v(f)) \geq kr_0$ for some $k>0$ then $f \in \m^k$. Let $\M_k = v(\m^k \setminus \{0\})$ denote the image of $\m^k$ under the valuation $v$. The condition (ii) in particular implies that for any $k>0$ we have $\M_k \cap \ell_{\geq kr_0} = \S \cap \ell_{\geq kr_0}$. As an example, let $R=\k[x_1, \ldots, x_n]_{(0)}$ be the algebra of polynomials localized at the maximal ideal $(x_1, \ldots, x_n)$. Then the map $v$ which associates to a polynomial its lowest exponent (with respect to some term order) defines a [good valuation]{} on $R$ and the value semigroup $\S$ coincides with the semigroup $\z_{\geq 0}^n$, that is, the semigroup of all the integral points in the positive orthant $\C = \r_{\geq 0}^n$. In the same fashion any regular local ring has a good valuation, as well as the local ring of a toroidal singularity (Example \[ex-good-val-toric-local-ring\] and Theorem \[th-good-val-reg-local-ring\]). More generally, in Section \[sec-valuation-ideal\] we see that an analytically irreducible local domain $R$ has a good valuation (Theorem \[th-good-val-S/R\] and Theorem \[th-good-val-analytically-irr-sing\]; see also [@Cutkosky1 Theorem 4.2 and Lemma 4.3]). A local ring $R$ is said to be analytically irreducible if its completion is an integral domain. Regular local rings and local rings of toroidal singularities are analytically irreducible. (We should point out that in the first version of the paper we had addressed only the case where $R$ is a regular local ring or the local ring of a toroidal singularity.) [Given a good $\z^n$-valued valuation $v$ on the domain $R$, we associate the (strongly) convex cone $\C = \C(R) \subset \r^n$ to the domain $R$ which is the closure of convex hull of the value semigroup $\S$. Then to each $\m$-primary graded sequence of subspaces $\a_\bullet$ in $R$ we associate a convex region $\Gamma(\a_\bullet) \subset \C$, such that the set $\C \setminus \Gamma(\a_\bullet)$ is bounded (Definition \[def-Gamma-I\]).]{} The main result of the manuscript (Theorem \[th-multi-ideal-covol\]) is that the limit in (\[equ-multi-graded-seq-ideals-intro\]) exists and: $$\label{equ-intro-main} e(\a_\bullet) = n!~\covol(\Gamma(\a_\bullet)).$$ [The inequality and the Brunn-Minkowski inequality for covolumes (see or Corollary \[cor-Brunn-Mink-covol\]) are the main ingredients in proving a generalization of the inequality to $\m$-primary graded sequences of subspaces (Corollary \[cor-Brunn-Mink-multi\]).]{} [ We would like to point out that the construction of $\Gamma(\a_\bullet)$ is an analogue of the construction of the Newton-Okounkov body of a linear system on an algebraic variety (see [@Okounkov-log-concave], [@Okounkov-Brunn-Mink], [@KKh-Annals], [@LM]). In fact, the approach and results in the present paper are analogous to the approach and results in [@KKh-Annals] regarding the asymptotic behavior of Hilbert functions of a general class of graded algebras. In the present manuscript we also deal with certain graded algebras (i.e. $\m$-primary graded sequences of subspaces) but instead of dimension of graded pieces we are interested in the codimension (i.e. dimension of $R / \a_k$), that is why in our main theorem (Theorem \[th-multi-ideal-covol\]) the covolume of a convex region appears as opposed to the volume of a convex body ([@KKh-Annals Theorem 2.31]). Also our Theorem \[th-multi-ideal-covol\] generalizes [@KKh-Annals Corollary 3.2] which gives a formula for the degree of a projective variety $X$ in terms of the volume of its corresponding Newton-Okounkov body, because the Hilbert function of a projective variety $X$ can be regarded as the Hilbert-Samuel function of the affine cone over $X$ at the origin.]{} On the other hand, the construction of $\Gamma(\a)$ generalizes the notion of the Newton diagram of a power series (see [@Kushnirenko] and [@AVG Section 12.7]). To a monomial ideal in a polynomial ring (or a power series ring), i.e. an ideal generated by monomials, one can associate its (unbounded) [Newton polyhedron]{}. It is the convex hull of the exponents of the monomials appearing in the ideal. The [Newton diagram]{} of a monomial ideal is the union of the bounded faces of the Newton polyhedron. One can see that for a monomial ideal $\a$, the convex region $\Gamma(\a)$ coincides with its Newton polyhedron (Theorem \[th-covol-monomial\]). The main theorem in this manuscript (Theorem \[th-multi-ideal-covol\]) for the case of monomial ideals recovers the local case of the well-known theorem of Bernstein-Kushnirenko, about computing the multiplicity at the origin of a system $f_1(x) = \cdots = f_n(x) = 0$ where the $f_i$ are generic functions from $\m$-primary monomial ideals (see Section \[sec-monomial-ideal\] and [@AVG Section 12.7]). [ Another immediate corollary of is the following: let $\a$ be an $\m$-primary ideal in $R = \k[x_1, \ldots, x_n]_{(0)}$. Fix a term order on $\z^n$ and for each $k > 0$ let $\In(\a^k)$ denote the initial ideal of the ideal $\a^k$ (generated by the lowest terms of elements of $\a^k$). Then the sequence of numbers $$\frac{e(\In(\a^k))}{k^n}$$ is decreasing and converges to $e(\a)$ as $k \to \infty$ (Corollary \[cor-Lech\]). ]{} The Brunn-Minkowski inequality proved in this paper [ is closely related to]{} the more general Alexandrov-Fenchel inequality for mixed multiplicities. Take $\m$-primary ideals $\a_1, \ldots, \a_n$ in a local ring $R = \mathcal{O}_{X, p}$ of a point $p$ on an $n$-dimensional algebraic variety $X$. The [mixed multiplicity]{} $e(\a_1, \ldots, \a_n)$ is equal to the intersection multiplicity, at the origin, of the hypersurfaces $H_i = \{ x \mid f_i(x) = 0\}$, $i = 1, \ldots, n$, where each $f_i$ is a generic function from $\a_i$. Alternatively one can define the mixed multiplicity as the polarization of the Hilbert-Samuel multiplicity $e(\a)$, i.e. it is the unique function $e(\a_1, \ldots, \a_n)$ which is invariant under permuting the arguments, is multi-additive with respect to product of ideals, and for any $\m$-primary ideal $\a$ the mixed multiplicity $e(\a, \ldots, \a)$ coincides with $e(\a)$. In fact, in the above the $\a_i$ need not be ideals and it suffices for them to be $\m$-primary subspaces. The Alexandrov-Fenchel inequality is the following inequality among the mixed multiplicities of the $\a_i$: $$\label{equ-Alex-Fenchel-mixed-multi} e(\a_1, \a_1, \a_3, \ldots, \a_n) e(\a_2, \a_2, \a_3, \ldots, \a_n) \geq e(\a_1, \a_2, \a_3, \ldots, \a_n)^2$$ [When $n = \dim R = 2$ it is easy to see that the Brunn-Minkowski inequality and the Alexandrov-Fenchl inequality are equivalent. By a reduction of dimension theorem for mixed multiplicities one can get a proof of the Alexandrov-Fenchel inequality from the Brunn-Minkowski inequality for $\dim(R)=2$. The Brunn-Minkowski inequality was originally proved in [@Teissier1; @RS].]{} In a recent paper [@KKh-mixed-multi] we give a simple proof of the Alexandrov-Fenchel inequality for mixed multiplicities of ideals using arguments similar to but different from those of this paper. This then implies an Alexandrov-Fenchel inequality for covolumes of convex regions (in a similar way that in [@KKh-Annals] and in [@Askold-BZ] the authors obtain an alternative proof of the usual Alexandrov-Fenchel inequality for volumes of convex bodies from similar inequalities for intersection numbers of divisors on algebraic varieties). We would like to point out that the Alexandrov-Fenchel inequality in [@Askold-Vladlen] for covolumes of coconvex bodies is related to an analogue of this inequality for convex bodies in higher dimensional hyperbolic space (or higher dimensional Minkowski space-time). [ From this point of view, the Alexandrov-Fenchel inequality has been proved for certain coconvex bodies in [@Fillastre].]{} After the first version of this note was completed we learned about the recent papers [@Cutkosky1; @Cutkosky2] and [@Fulger] which establish the existence of limit (\[equ-multi-graded-seq-ideals-intro\]) in more general settings. We would also like to mention the paper of Teissier [@Teissier2] which discusses Newton polyhedron of a power series, and notes the relationship/analogy between notions from local commutative algebra and convex geometry. [ Also we were notified that, for ideals in a polynomial ring, ideas similar to construction of $\Gamma(I_\bullet)$ appears in [@Mustata] were the highest term of polynomials is used instead of a valuation. Moreover, in [@Mustata Corollary 1.9] the Brunn-Minkowski-inequality for multiplicities of graded sequences of $\m$-primary ideals is proved for regular local rings using Teissier’s Brunn-Minkowski .]{} [Finally as the final version of this manuscript was being prepared for publication, the preprint of D. Cutksoky [@Cutkosky3] appeared in arXiv.org in which the author uses similar methods to prove Brunn-Minkowski inequality for graded sequences of $\m$-primary ideals in local domains.]{} And few words about the organization of the paper: Section \[sec-mixed-vol\] recalls basic background material about volumes/mixed volumes of convex bodies. Section \[sec-convex-diag\] is about convex regions and their covolumes/mixed covolumes, which we can think of as a local version of the theory of mixed volumes of convex bodies. In Sections \[sec-semigp-int\] and \[sec-semigp-ideal\] we associate a convex region to primary sequences of subsets in a semigroup and prove the main combinatorial result required later (Definitions \[def-Gamma-I-semigroup\] and Theorem \[th-vol-Gamma-semigroup\]). [ In Section \[sec-multi-ideal\] we recall some basic definitions and facts from commutative algebra about multiplicities of $\m$-primary ideals (and subspaces) in local rings. The next section (Section \[sec-monomial-ideal\]) discusses the case of monomial ideals and the Bernstein-Kushnirenko theorem. Finally in Section \[sec-valuation-ideal\], using a valuation on the ring $R$, we associate a convex region $\Gamma(\a_\bullet)$ to an $\m$-primary graded sequence of subspaces $\a_\bullet$ and prove the main results of this note (Theorem \[th-multi-ideal-covol\] and Corollary \[cor-Brunn-Mink-multi\]).]{}\ [Acknowledgement.]{} The first author would like to thank Dale Cutkosky, Vladlen Timorin and Javid Validashti for helpful discussions. We are also thankful to Bernard Teissier, Dale Cutkosky, Francois Fillastre and Mircea Mustaţă for informing us about their interesting papers [@Teissier2], [@Cutkosky1; @Cutkosky2], [@Fillastre] and [@Mustata]. Mixed volume of convex bodies {#sec-mixed-vol} ============================= The collection of all convex bodies in $\r^n$ is a cone, that is, we can add convex bodies and multiply a convex body with a positive number. For two convex bodies $\Delta_1, \Delta_2 \subset \r^n$, their (Minkowski) sum $\Delta_1 + \Delta_2$ is $\{ x + y \mid x \in \Delta_1, y \in \Delta_2\}$. Let $\vol$ denote the $n$-dimensional volume in $\r^n$ with respect to the standard Euclidean metric. The function $\vol$ is a homogeneous polynomial of degree $n$ on the cone of convex bodies, i.e. its restriction to each finite dimensional section of the cone is a homogeneous polynomial of degree $n$. In other words, for any collection of convex bodies $\Delta_1, \ldots, \Delta_r$, the function: $$P_{\Delta_1, \ldots, \Delta_r}(\lambda_1, \ldots, \lambda_r) = \vol(\lambda_1\Delta_1 + \cdots + \lambda_r\Delta_r),$$ is a homogeneous polynomial of degree $n$ in $\lambda_1, \ldots, \lambda_r$. By definition the [mixed volume]{} $V(\Delta_1,\dots,\Delta_n)$ of an $n$-tuple $(\Delta_1,\dots,\Delta_n)$ of convex bodies is the coefficient of the monomial $\lambda_1 \cdots \lambda_n$ in the polynomial $P_{\Delta_1, \ldots, \Delta_n}(\lambda_1, \ldots, \lambda_n)$ divided by $n!$. This definition implies that mixed volume is the [polarization]{} of the volume polynomial, that is, it is the unique function on the $n$-tuples of convex bodies satisfying the following: - (Symmetry) $V$ is symmetric with respect to permuting the bodies $\Delta_1, \ldots, \Delta_n$. - (Multi-linearity) It is [linear]{} in each argument with respect to the Minkowski sum. [The linearity]{} in first argument means that for convex bodies $\Delta_1'$, $\Delta_1'', \Delta_2,\dots,\Delta_n$, and real numbers $\lambda', \lambda'' \geq 0$ we have: $$V(\lambda'\Delta_1'+\lambda''\Delta_1'', \Delta_2, \dots, \Delta_n)=\lambda'V(\Delta_1', \Delta_2, \dots, \Delta_n) + \lambda''V(\Delta_1'', \Delta_2, \dots, \Delta_n).$$ - (Relation with volume) On the diagonal it coincides with the volume, i.e. if $\Delta_1 =\cdots=\Delta_n=\Delta$, then $V(\Delta_1,\ldots, \Delta_n)=\vol(\Delta)$. The following inequality attributed to Alexandrov and Fenchel is important and very useful in convex geometry (see [@BZ]): \[thm-alexandrov-fenchel\] Let $\Delta_1, \ldots, \Delta_n$ be convex bodies in $\r^n$. Then $$V(\Delta_1, \Delta_1, \Delta_3, \ldots, \Delta_n) V(\Delta_2, \Delta_2, \Delta_3, \ldots, \Delta_n) \leq V^2(\Delta_1, \Delta_2, \ldots, \Delta_n).$$ In dimension $2$, this inequality is elementary. We call it the [generalized isoperimetric inequality]{}, because when $\Delta_2$ is the unit ball it coincides with the classical isoperimetric inequality. The celebrated [Brunn-Minkowski inequality]{} concerns volume of convex bodies in $\r^n$. [It is an easy corollary of the Alexandrov-Fenchel inequality. (For $n=2$ it is equivalent to the Alexandrov-Fenchel inequality.)]{} \[th-Brunn-Mink\] Let $\Delta_1$, $\Delta_2$ be convex bodies in $\r^n$. Then $$\vol^{1/n}(\Delta_1) + \vol^{1/n}(\Delta_2)\leq \vol^{1/n}(\Delta_1+\Delta_2).$$ Mixed covolume of convex regions {#sec-convex-diag} ================================ Let $\C$ be a strongly convex closed $n$-dimensional cone in $\r^n$ with apex at the origin. (A convex cone is strongly convex if it does not contain any lines through the origin.) We are interested in closed convex subsets of $\C$ which have bounded complement. \[def-convex-region\] We call a closed convex subset $\Gamma \subset \C$ a [$\C$-convex region]{} (or simply a convex region when the cone $\C$ is understood from the context) if for any $x \in \Gamma$ and $y \in \C$ we have $x + y \in \Gamma$. Moreover, we say that a convex region $\Gamma$ is [cobounded]{} (alternatively we say $\Gamma$ is [inscribed]{} in $\C$) if the complement $\C \setminus \Gamma$ is bounded. In this case the volume of $\C \setminus \Gamma$ is finite which we call the [covolume of $\Gamma$]{} and denote it by $\covol(\Gamma)$. [One also refers to $\C \setminus \Gamma$ as a [$\C$-coconvex body]{}.]{} The collection of $\C$-convex regions (respectively cobounded regions) is closed under the Minkowski sum and multiplication by positive scalars. Similar to the volume of convex bodies, one proves that the covolume of convex regions is a homogeneous polynomial [@Askold-Vladlen]. More precisely: \[th-covol-polynomial\] Let $\Gamma_1, \ldots, \Gamma_r$ be cobounded $\C$-convex regions in the cone $\C$. Then the function $$P_{\Gamma_1, \ldots, \Gamma_r}(\lambda_1, \ldots, \lambda_r) = \covol(\lambda_1\Gamma_1 + \cdots + \lambda_r\Gamma_r),$$ is a homogeneous polynomial of degree $n$ in the $\lambda_i$. As in the case of convex bodies, one uses the above theorem to define mixed covolume of cobounded regions. By definition the [mixed covolume]{} $CV(\Gamma_1, \ldots, \Gamma_n)$ of an $n$-tuple $(\Gamma_1,\dots,\Gamma_n)$ of cobounded convex regions is the coefficient of the monomial $\lambda_1 \cdots \lambda_n$ in the polynomial $P_{\Gamma_1,\ldots, \Gamma_n}(\lambda_1, \ldots, \lambda_n)$ divided by $n!$. That is, mixed covolume is the unique function on the $n$-tuples of cobounded regions satisfying the following: - (Symmetry) $CV$ is symmetric with respect to permuting the regions $\Gamma_1, \ldots, \Gamma_n$. - (Multi-linearity) It is linear in each argument with respect to the Minkowski sum. - (Relation with covolume) For any cobounded region $\Gamma \subset \C$: $$CV(\Gamma, \ldots, \Gamma)=\covol(\Gamma).$$ The mixed covolume satisfies an Alexandrov-Fenchel inequality [@Askold-Vladlen]. Note that the inequality is reversed compared to the Alexandrov-Fenchel for mixed volumes of convex bodies. \[thm-alexandrov-fenchel-covolume\] Let $\Gamma_1, \ldots, \Gamma_n$ be cobounded $\C$-convex regions. Then: $$CV(\Gamma_1, \Gamma_1, \Gamma_3, \ldots, \Gamma_n) CV(\Gamma_2, \Gamma_2, \Gamma_3, \ldots, \Gamma_n) \geq CV^2(\Gamma_1, \Gamma_2, \Gamma_3, \ldots, \Gamma_n).$$ The (reversed) Alexandrov-Fenchel inequality implies a (reversed) Brunn-Minkowski inequality. [ (For $n=2$ it is equivalent to the Alexandrov-Fenchel inequality.)]{} \[cor-Brunn-Mink-covol\] Let $\Gamma_1$, $\Gamma_2$ be cobounded $\C$-convex regions. Then: $$\covol^{1/n}(\Gamma_1) + \covol^{1/n}(\Gamma_2) \geq \covol^{1/n}(\Gamma_1 + \Gamma_2).$$ Semigroups of integral points {#sec-semigp-int} ============================= In this section we recall some general facts from [@KKh-Annals] about the asymptotic behavior of semigroups of integral points. Let $S \subset \z^n \times \z_{\geq 0}$ be an additive semigroup. Let $\pi: \r^n \times \r \to \r$ denote the projection onto the second factor, and let $S_k = S \cap \pi^{-1}(k)$ be the set of points in $S$ at level $k$. For simplicity, assume $S_1 \neq \emptyset$ and that $S$ generates the whole lattice $\z^{n+1}$. Define the function $H_S$ by: $$H_S(k) = \#S_k.$$ We call $H_S$ the [Hilbert function of the semigroup $S$]{}. We wish to describe the asymptotic behavior of $H_S$ as $k \to \infty$. Let $C(S)$ be the closure of the convex hull of $S \cup \{0\}$, that is, the smallest closed convex cone (with apex at the origin) containing $S$. We call the projection of the convex set $C(S) \cap \pi^{-1}(1)$ to $\r^n$ (under the projection onto the first factor $(x, 1) \mapsto x$), the [Newton-Okounkov convex set of the semigroup $S$]{} and denote it by $\Delta(S)$. In other words, $$\Delta(S) = \overline{\conv(\bigcup_{k>0} \{x/k \mid (x, k) \in S_k\})}.$$ [ (In fact, one can show that taking convex hull in the above definition is not necessary.)]{} If $C(S) \cap \pi^{-1}(0) = \{0\}$ then $\Delta(S)$ is compact and hence a convex body. The Newton-Okounkov convex set $\Delta(S)$ is responsible for the asymptotic behavior of the Hilbert function of $S$ (see [@KKh-Annals Corollary 1.16]): \[th-semigp-NO\] The limit $$\lim_{k \to \infty} \frac{H_S(k)}{k^n},$$ exists and is equal to $\vol(\Delta(S))$. Primary sequences in a semigroup and convex regions {#sec-semigp-ideal} =================================================== In this section we discuss the notion of a primary graded sequence of subsets in a semigroup and describe its asymptotic behavior using Theorem \[th-semigp-NO\]. In section \[sec-valuation-ideal\] we will employ this to describe the asymptotic behavior of the Hilbert-Samuel function of a graded sequence of $\m$-primary ideals in a local domain. Let $\S \subset \z^n$ be an additive semigroup containing the origin. Without loss of generality we assume that $\S$ generates the whole $\z^n$. Let as above $\C = C(\S)$ denote the cone of $\S$ i.e. the closure of convex hull of $\S = \S \cup \{0\}$. We also assume that $\C$ is a strongly convex cone, i.e. it does not contain any lines through the origin. For two subsets $\I, \J \subset \S$, the sum $\I + \J$ is the set $\{x+y \mid x \in \I,~ y \in \J\}$. For any integer $k > 0$, by the product $k\I$ we mean $\I + \cdots + \I$ ($k$ times). \[def-graded-seq-semigp-ideal\] A [graded sequence of subsets]{} in $\S$ is a sequence $\I_\bullet = (\I_1, \I_2, \ldots)$ of subsets such that for any $k, m > 0$ we have $\I_k + \I_m \subset \I_{k+m}$. \[ex-powers-of-semigp-ideal\] Let $\I \subset \S$. Then the sequence $\I_\bullet$ defined by $\I_k = k \I$ is clearly a graded sequence of subsets. Let $\I'_\bullet$, $\I''_\bullet$ be graded sequences of subsets. Then the sequence $\I_\bullet = \I'_\bullet + \I''_\bullet$ defined by $$\I_k = \I'_k + \I''_k,$$ is also a graded sequence of subsets which we call the [sum of sequences $\I'_\bullet$ and $\I''_\bullet$]{}. Let $\ell: \r^n \to \r$ be a linear function. For any $a \in \r$ let $\ell_{\geq a}$ (respectively $\ell_{> a}$) denote the half-space $\{ x \mid \ell(x) \geq a\}$ (respectively $\{ x \mid \ell(x) > a\}$), and similarly for $\ell_{\leq a}$ and $\ell_{< a}$. By assumption the cone $\C = C(S)$ associated to the semigroup $\S$ is strongly convex. Thus we can find a linear function $\ell$ such that the cone $\C = C(\S)$ lies in $\ell_{\geq 0}$ and it intersects the hyperplane $\ell^{-1}(0)$ only at the origin. Let us fix such a linear function $\ell$. We will be interested in graded sequences of subsets $\I_\bullet$ satisfying the following condition: \[def-primary-sequence\] [We say that a graded sequence of subsets $\I_\bullet$ is [primary]{} if there exists an integer $t_0 > 0$ such that for any integer $k >0$ we have: ]{} $$\label{equ-primary-semigroup-ideal} \I_k \cap \ell_{\geq kt_0} = \S \cap \ell_{\geq kt_0}.$$ One verifies that if $\ell'$ is another linear function such that $\C$ lies in $\ell'_{\geq 0}$ and it intersects the hyperplane $\ell'^{-1}(0)$ only at the origin then it automatically satisfies with perhaps a different constant $t'_0 > 0$. Hence the condition of being a primary graded sequence does not depend on the linear function $\ell$. Nevertheless when we refer to a primary graded sequence $\I_\bullet$, choices of a linear function $\ell$ and an integer $t_0 > 0$ are implied. \[prop-I\_k-cofinite\] Let $\I_\bullet$ be a primary graded sequence. Then for all $k > 0$, the set $\S \setminus \I_k$ is finite. Since $\C$ intersects $\ell^{-1}(0)$ only at the origin it follows that for any $k > 0$ the set $\C \cap \ell_{< kt_0}$ is bounded which implies that $\S \cap \ell_{< kt_0}$ is finite. But by , $\S \setminus \I_k \subset \S \cap \ell_{<kt_0}$ and hence is finite. \[def-Hilbert-Samuel-semigroup\] Let $\I_\bullet$ be a primary graded sequence. Define the function $H_{\I_\bullet}$ by: $$H_{\I_\bullet}(k) = \#(\S \setminus \I_k).$$ (Note that by Proposition \[prop-I\_k-cofinite\] this number is finite for all $k>0$.) We call it the [Hilbert-Samuel function of $\I_\bullet$]{}. To a primary graded sequence of subsets $\I_\bullet$ we can associate a $\C$-convex region $\Gamma(\I_\bullet)$ (see Definition \[def-convex-region\]). This convex set encodes information about the asymptotic behavior of the Hilbert-Samuel function of $\I_\bullet$. \[def-Gamma-I-semigroup\] Let $\I_\bullet$ be a primary graded sequence of subsets. Define the convex set $\Gamma(\I_\bullet)$ by $$\Gamma(\I_\bullet) = \overline{\conv(\bigcup_{k>0} \{x/k \mid x \in \I_k\})}.$$ It is an unbounded convex set in $\C$. [ (As in Section \[sec-semigp-int\], one can show that taking convex hull in the above definition is not necessary.)]{} Let $\I_\bullet$ be a primary graded sequence. Then $\Gamma = \Gamma(\I_\bullet)$ is a $\C$-convex region in the cone $\C$, i.e. for any $x \in \Gamma$, $x+\C \subset \Gamma$. Moreover, the region $\Gamma$ is cobounded i.e. $\C \setminus \Gamma$ is bounded. [ Let $\ell$ and $t_0$ be as in Definition \[def-primary-sequence\]. From the definitions it follows that the region $\Gamma$ contains $\C \cap \ell_{\geq t_0}$. Thus $(\C \setminus \Gamma) \subset (\C \cap \ell_{< t_0})$ and hence is bounded. Next let $x \in \Gamma$. Since $x+\C \subset \C$, $\Gamma$ contains the set $(x+\C) \cap \ell_{\geq t_0}$. But the convex hull of $x$ and $(x+\C) \cap \ell_{\geq t_0}$ is $x+\C$. Thus $x+\C \subset \Gamma$ because $\Gamma$ is convex.]{} The following is an important example of a primary graded sequence in a semigroup $\S$. \[prop-polyhedral-primary-semigroup-ideal\] Let $\C$ be an $n$-dimensional strongly convex rational polyhedral cone in $\r^n$ and let $\S = \C \cap \z^n$. Also let $\I \subset \S$ be a subset such that $\S \setminus \I$ is finite. Then the sequence $\I_\bullet$ defined by $\I_k := k * \I$ is a primary graded sequence and $\Gamma(\I_\bullet) = \conv(\I)$. [Since $\S \setminus \I$ is finite there exists $t_1>0$ such that $\S \cap \ell_{\geq t_1} \subset \I$. Put $\M_1 = \S \cap \ell_{\geq t_1}$. Because $\C$ is a rational polyhedral cone, $\M_1$ is a finitely generated semigroup. Let $v_1, \ldots, v_s$ be semigroup generators for $\M_1$. Let $t_0 > 0$ be bigger than all the $\ell(v_i)$. For $k>0$ take $x \in \S \cap \ell_{\geq kt_0} \subset \M_1$. Then $x = \sum_{i=1}^s c_i v_i$ for $c_i \in \z_{\geq 0}$. Thus $kt_0 \leq \ell(x) = \sum_i c_i \ell(v_i) \leq (\sum_i c_i)t_0$. This implies that $k \leq \sum_i c_i$ and hence $(\sum_i c_i) \M_1 \subset k * \M_1$. It follows that $x \in k\M_1$. That is, $\S \cap \ell_{\geq kt_0} = (k \M_1) \cap \ell_{\geq kt_0} \subset (k * \I) \cap \ell_{\geq kt_0}$ and hence $\S \cap \ell_{\geq kt_0} = (k\I) \cap \ell_{\geq kt_0}$ as required. The assertion $\Gamma(\I_\bullet) = \conv(\I_\bullet)$ follows from the observation that $\conv(k \I) = k ~ \conv(\I)$.]{} The following is our main result about the asymptotic behavior of a primary graded sequence. \[th-vol-Gamma-semigroup\] Let $\I_\bullet$ be a primary graded sequence. Then $$\lim_{k \to \infty} \frac{H_{\I_\bullet}(k)}{k^n}$$ exists and is equal to $\covol(\Gamma(\I_\bullet))$. Let $t_0 > 0$ be as in Definition \[def-primary-sequence\]. Then for all $k>0$ we have $\S \cap \ell_{\geq kt_0} = \I_k \cap \ell_{\geq kt_0}.$ [ Moreover take $t_0$ to be large enough so that the finite set $\I_1 \cap \ell_{< kt_0}$ generates the lattice $\z^n$ (this is possible because $\S$ and hence $\I_1$ generate $\z^n$).]{} Consider $$\tilde{S} = \{(x, k) \mid x \in \I_k \cap \ell_{< kt_0} \}.$$ $$\tilde{T} = \{(x, k) \mid x \in \S \cap \ell_{< kt_0} \}.$$ $\tilde{S}$ and $\tilde{T}$ are semigroups in $\z^n \times \z_{\geq 0}$ and we have $\tilde{S} \subset \tilde{T}$. From the definition it follows that both of the groups generated by $\tilde{S}$ and $\tilde{T}$ are $\z^{n+1}$. Also the Newton-Okounkov bodies of $\tilde{S}$ and $\tilde{T}$ are: $$\Delta(\tilde{S}) = \Gamma(\I_\bullet) \cap \Delta(t_0),$$ $$\Delta(\tilde{T}) = \Delta(t_0),$$ where $\Delta(t_0) = \C \cap \ell_{\leq t_0}$. Since $\S \cap \ell_{\geq kt_0} = \I_k \cap \ell_{\geq kt_0}$, we have: $$\S \setminus \I_k = \tilde{T}_k \setminus \tilde{S}_k,$$ Here as usual $\tilde{S}_k = \{(x, k) \mid (x, k) \in \tilde{S}\}$ (respectively $\tilde{T}_k$) denotes the set of points in $\tilde{S}$ (respectively $\tilde{T}$) at level $k$. Hence $$H_{\I_\bullet}(k) = \#\tilde{T}_k - \#\tilde{S}_k.$$ By Theorem \[th-semigp-NO\] we have: $$\lim_{k \to \infty} \frac{\#\tilde{S}_k}{k^n} = \vol(\Delta(\tilde{S})),$$ $$\lim_{k \to \infty} \frac{\#\tilde{T}_k}{k^n} = \vol(\Delta(t_0)).$$ Thus $$\lim_{k \to \infty} \frac{\#(\S \setminus \I_k)}{k^n} = \vol(\Delta(t_0)) - \vol(\Delta(\tilde{S})).$$ On the other hand, we have: $$\Delta(t_0) \setminus \Delta(\tilde{S}) = \C \setminus \Gamma(\I_\bullet),$$ and hence $\vol(\Delta(t_0)) - \vol(\Delta(\tilde{S})) = \covol(\Gamma(\I_\bullet)).$ This finishes the proof. \[def-multiplicity-semigp-ideal\] For a primary graded sequence $\I_\bullet$ we denote $\lim_{k \to \infty} H_{\I_\bullet}(k) / k^n$ by $e(\I_\bullet)$. Motivated by commutative algebra, we call it the [multiplicity of $\I_\bullet$]{}. The following additivity property is straightforward from definition. \[prop-additivity-semigroup-ideal\] Let $\I_\bullet'$, $\I_\bullet''$ be primary graded sequences. We have: $$\Gamma(\I_\bullet') + \Gamma(\I_\bullet'') = \Gamma(\I_\bullet' + \I_\bullet'').$$ [ For any $k>0$ we have $\conv(\I'_k) + \conv(\I''_k) = \conv(\I'_k + \I''_k)$. From this the required equality $\Gamma(\I_\bullet') + \Gamma(\I_\bullet'') = \Gamma(\I_\bullet' + \I_\bullet'')$ follows.]{} Let $\I_{1, \bullet}, \ldots, \I_{n, \bullet}$ be $n$ primary graded sequences. Define the function $P_{\I_{1, \bullet}, \ldots, \I_{n, \bullet}}: \n^n \to \n$ by: $$P_{\I_{1, \bullet}, \ldots, \I_{n, \bullet}}(k_1, \ldots, k_n) = e(k_1\I_{1, \bullet} + \cdots + k_n\I_{n, \bullet}).$$ \[th-mixed-multi-poly-semigp\] The function $P_{\I_{1, \bullet}, \ldots, \I_{n, \bullet}}$ is a homogeneous polynomial of degree $n$ in $k_1, \ldots, k_n$. Follows immediately from Proposition \[prop-additivity-semigroup-ideal\], Theorem \[th-vol-Gamma-semigroup\] and Theorem \[th-covol-polynomial\]. \[def-mixed-multi-semigroup\] [Let $\I_{1, \bullet}, \ldots, \I_{n, \bullet}$ be primary graded sequences. Define the [mixed multiplicity]{} $e(\I_{1, \bullet}, \ldots, \I_{n, \bullet})$ to be the coefficient of $k_1 \cdots k_n$ in the polynomial $P_{\I_{1, \bullet}, \ldots, \I_{n, \bullet}}$ divided by $n!$.]{} From Theorem \[th-vol-Gamma-semigroup\] and Proposition \[prop-additivity-semigroup-ideal\] we have the following corollary: $$e(\I_{1, \bullet}, \ldots, \I_{n, \bullet}) = n!~CV(\Gamma(\I_{1, \bullet}), \ldots, \Gamma(\I_{n, \bullet})),$$ where as before $CV$ denotes the mixed covolume of cobounded regions. From Theorem \[thm-alexandrov-fenchel-covolume\] and Corollary \[cor-Brunn-Mink-covol\] we then obtain: \[cor-AF-semigroup\] For primary graded sequences $\I_{1, \bullet}, \ldots, \I_{n, \bullet}$ in the semigroup $\S$ we have: $$e(\I_{1, \bullet}, \I_{1, \bullet}, \I_{3, \bullet}, \ldots, \I_{n, \bullet}) e(\I_{2, \bullet}, \I_{2, \bullet}, \I_{3, \bullet}, \ldots, \I_{n, \bullet}) \geq e(\I_{1, \bullet}, \I_{2, \bullet}, \I_{3, \bullet}, \ldots, \I_{n, \bullet})^2.$$ [ Let $\I_\bullet$, $\J_\bullet$ be primary graded sequences in the semigroup $\S$. We have: $$e^{1/n}(\I_\bullet) + e^{1/n}(\J_\bullet) \geq e^{1/n}(\I_\bullet + \J_\bullet).$$]{} Multiplicities of ideals and subspaces in local rings {#sec-multi-ideal} ===================================================== Let $R$ be a Noetherian local domain of Krull dimension $n$ over a field $\k$, and with maximal ideal $\m$. We also assume that the residue field $R/\m$ is $\k$. \[ex-local-ring-of-subvariety\] Let $X$ be an irreducible variety of dimension $n$ over $\k$, and let $p$ be a point in $X$. Then the local ring $R = \mathcal{O}_{X, p}$ (consisting of rational functions on $X$ which are regular in a neighborhood of $p$) is a Noetherian local domain of Krull dimension $n$ over $\k$. The ideal $\m$ consists of functions which vanish at $p$. If $\a, \b \subset R$ are two $\k$-subspaces then by $\a\b$ we denote the $\k$-span of all the $xy$ where $x \in \a$ and $y \in \b$. Note that if $\a, \b$ are ideals in $R$ then $\a\b$ coincides with the product of $\a$ and $\b$ as ideals. \[def-graded-seq-subspace\] - A $\k$-subspace $\a$ in $R$ is called [$\m$-primary]{} if it contains a power of the maximal ideal $\m$. - A [graded sequence of subspaces]{} is a sequence $\a_\bullet = (\a_1, \a_2, \ldots)$ of $\k$-subspaces in $R$ such that for all $k, m>0$ we have $\a_k \a_m \subset \a_{k+m}$. We call $\a_\bullet$, $\m$-primary if moreover $\a_1$ is $\m$-primary. It then follows that every $\a_k$ is $\m$-primary and hence $\dim_{\k}(R/\a_k)$ is finite. (If each $\a_k$ is an $\m$-primary ideal in $R$ we call $\a_\bullet$ an [$\m$-primary graded sequence of ideals]{}.) When $\k$ is algebraically closed, an ideal $\a$ in $R = \mathcal{O}_{X, p}$ is $\m$-primary if the subvariety it defines around $p$ coincides with the single point $p$ itself. \[ex-powers-of-semigp-ideal\] Let $\a$ be an $\m$-primary subspace. Then the sequence $\a_\bullet$ defined by $\a_k = \a^k$ is an $\m$-primary graded sequence of subspaces. Let $\a_\bullet$, $\b_\bullet$ be $\m$-primary graded sequences of subspaces. Then the sequence $\mathfrak{c}_\bullet = \a_\bullet \b_\bullet$ defined by $$\mathfrak{c}_k = \a_k \b_k,$$ is also an $\m$-primary graded sequence of subspaces which we call the [product of $\a_\bullet$ and $\b_\bullet$]{}. \[def-Hilbert-Samuel-semigroup\] Let $\a_\bullet$ be an $\m$-primary graded sequence of subspaces. Define the function $H_{\a_\bullet}$ by: $$H_{\a_\bullet}(k) = \dim_\k(R / \a_k).$$ We call it the [Hilbert-Samuel function of $\a_\bullet$]{}. The [Hilbert-Samuel function]{} $H_\a(k)$ of an $\m$-primary subspace $\a$ is the Hilbert-Samuel function of the sequence $\a_\bullet = (\a, \a^2, \ldots)$. That is, $H_\a(k) = \dim_\k(R / \a^k)$. \[rem-Hilbert-Samuel-poly\] For an $\m$-primary ideal $\a$ it is well-known that, for sufficiently large values of $k$, the Hilbert-Samuel function $H_\a$ coincides with a polynomial of degree $n$ called the [Hilbert-Samuel polynomial of $\a$]{} ([@SZ]). \[def-multiplicity\] Let $\a_\bullet$ be an $\m$-primary graded sequence of subspaces. We define the [multiplicity $e(\a_\bullet)$]{} to be: $$e(\a_\bullet) = n!~\limsup_{k} \frac{H_{\a_\bullet}(k)}{k^n}.$$ (It is not a priori clear that the limit exists.) The multiplicity $e(\a)$ of an $\m$-primary ideal $\a$ is the multiplicity of its associated sequence $(\a, \a^2, \ldots)$. That is: $$e(\a) = n!~\lim_{k \to \infty} \frac{H_{\a}(k)}{k^n}.$$ [ (Note that by Remark \[rem-Hilbert-Samuel-poly\] the limit exists in this case.)]{} The notion of multiplicity comes from the following basic example: \[ex-meaning-multiplicity\] Let $\a$ be an $\m$-primary subspace in the local ring $R = \mathcal{O}_{X, p}$ of a [ point]{} $p$ in an irreducible variety $X$ over an algebraically closed filed $\k$. Let $f_1, \ldots, f_n$ be generic elements in $\a$. Then the multiplicity $e(\a)$ is equal to the intersection multiplicity at $p$ of the hypersurfaces $H_i = \{ x \mid f_i(x) = 0\}$, $i=1, \ldots, n$. In Section \[sec-valuation-ideal\] we use the material in Section \[sec-semigp-ideal\] to give a formula for $e(\a_\bullet)$ in terms of covolume of a convex region. One can also define the notion of mixed multiplicity for $\m$-primary ideals as the polarization of the Hilbert-Samuel multiplicity $e(\a)$, i.e. it is the unique function $e(\a_1, \ldots, \a_n)$ which is invariant under permuting the arguments, is multi-additive with respect to product, and for any $\m$-primary ideal $\a$ the mixed multiplicity $e(\a, \ldots, \a)$ coincides with $e(\a)$. In fact one can show that in the above definition of mixed multiplicity the $\a_i$ need not be ideals and it suffices for them to be $\m$-primary subspaces. Similar to multiplicity we have the following geometric meaning for the notion of mixed multiplicity when $R = \mathcal{O}_{X, p}$ is the local ring of a point $p$ on an $n$-dimensional algebraic variety $X$. Take $\m$-primary subspaces $\a_1, \ldots, \a_n$ in $R$. The mixed multiplicity $e(\a_1, \ldots, \a_n)$ is equal to the intersection multiplicity, at the origin, of the hypersurfaces $H_i = \{ x \mid f_i(x) = 0\}$, $i = 1, \ldots, n$, where each $f_i$ is a generic function from the $\a_i$. Case of monomial ideals and Newton polyhedra {#sec-monomial-ideal} ============================================ In this section we discuss the case of monomial ideals. It is related to the classical notion of Newton polyhedron of a power series in $n$ variables. We will see that our Theorem \[th-vol-Gamma-semigroup\] in this case immediately recovers (and generalizes) the local version of celebrated theorem of Bernstein-Kushnirenko ([@Kushnirenko] and [@AVG Section 12.7]). Let $R$ be the local ring of an affine toric variety at its torus fixed point. The algebra $R$ can be realized as follows: Let $\C \subset \r^n$ be an $n$-dimensional strongly convex rational polyhedral cone with apex at the origin, that is, $\C$ is an $n$-dimensional convex cone generated by a finite number of rational vectors and it does not contain any lines through the origin. Consider the semigroup algebra over $\k$ of the semigroup of integral points $\S = \C \cap \z^n$. In other words, consider the algebra of Laurent polynomials consisting of all the $f$ of the form $f = \sum_{\alpha \in \C \cap \z^n} c_\alpha x^\alpha$, where we have used the shorthand notation $x = (x_1, \ldots, x_n)$, $\alpha = (a_1, \ldots, a_n)$ and $x^\alpha = x_1^{a_1} \cdots x_n^{a_n}$. Let $R$ be the localization of this Laurent polynomial algebra at the maximal ideal $\m$ generated by non-constant monomials. (Similarly instead of $R$ we can take its completion at the maximal ideal $\m$ which is an algebra of power series .) \[def-Newton-polyhedron\] Let $\a$ be an $\m$-primary monomial ideal in $R$, that is, an $\m$-primary ideal generated by monomials. To $\a$ we can associate a subset $\I(\a) \subset \C \cap \z^n$ by $$\I(\a) = \{\alpha \mid x^\alpha \in \a\}.$$ The convex hull $\Gamma(\a)$ of $\I(\a)$ is usually called the [Newton polyhedron]{} of the monomial ideal $\a$. It is a convex unbounded polyhedron in $\C$, moreover it is a $\C$-convex region. The [Newton diagram of $\a$]{} is the union of bounded faces of its Newton polyhedron. \[rem-semigp-ideal-monomial-ideal\] It is easy to see that if $\a$ is an ideal in $R$ then $\I = \I(\a)$ is a semigroup ideal in $\S = \C \cap \z^n$, that is, if $x \in \I$ and $y \in \S$ then $x+y \in \I$. Let $\a$ be an $\m$-primary monomial ideal. Then for any $k>0$ we have $\I(\a^k) = k * \I(\a)$. It follows from Proposition \[prop-polyhedral-primary-semigroup-ideal\] that $\I_\bullet$ defined by $\I_k = k * \I(\a)$ is a primary graded sequence in $\S = \C \cap \z^n$ and the convex region $\Gamma(\I_\bullet)$ associated to $\I_\bullet$ coincides with the Newton polyhedron $\Gamma(\a) = \conv(\I(\a))$ defined above. More generally, let $\a_\bullet$ be an $\m$-primary graded sequence of monomial ideals in $R$. Associate a graded sequence $\I_\bullet = \I_\bullet(\a_\bullet)$ in $\S$ to $\a_\bullet$ by: $$\I_k = \I(\a_k) = \{ \alpha \mid x^\alpha \in \a_k\}.$$ \[prop-convex-region-monomial-ideal\] Let $\a_\bullet$ be an $\m$-primary graded sequence of monomial ideals. Then the graded sequence $\I_\bullet = \I(\a_\bullet)$ is a primary graded sequence in the sense of Definition \[def-primary-sequence\]. As usual let $\ell$ be a linear function such that $\C$ lies in $\ell_{\geq 0}$ and intersects $\ell^{-1}(0)$ only at the origin. Let $\M = \S \setminus \{0\} = \I(\m)$. Then for all $k>0$, $k\M = \I(\m^k)$. As in the proof of Proposition \[prop-polyhedral-primary-semigroup-ideal\] we can find $t'_0 >0$ such that for all $k>0$, $\S \cap \ell_{\geq kt'_0} = (k\M) \cap \ell_{\geq kt'_0}$. Since $\a_\bullet$ is $\m$-primary there exists $m>0$ such that $\m^m \subset \a_1$ and hence for all $k>0$ we have $\m^{km} \subset \a_{k}$. Thus $\S \cap \ell_{\geq kmt'_0} = (km\M) \cap \ell_{\geq kmt'_0} \subset \I(\a_k) \cap \ell_{\geq kmt'_0}$, which implies that $\S \cap \ell_{\geq kmt'_0} = \I(\a_k) \cap \ell_{\geq kmt'_0}$. That is, $\I_\bullet$ is a primary graded sequence in $\S$ with $t_0 = mt'_0$. Let $\Gamma(\a_\bullet)$ denote the convex region associated to the primary graded sequence $\I_\bullet = \I(\a_\bullet)$ (Definition \[def-Gamma-I-semigroup\]). We make the following important observation that $\a_\bullet \mapsto \Gamma(\a_\bullet)$ is additive with respect to the product of graded sequences of monomial ideals. \[prop-Newton-poly-additive\] Let $\a_\bullet, \b_\bullet$ be $\m$-primary graded sequences of monomial ideals in $R$. Then $\I(\a_\bullet \b_\bullet) = \I(\a_\bullet) + \I(\b_\bullet)$. It follows from Proposition \[prop-additivity-semigroup-ideal\] that: $$\Gamma(\a_\bullet \b_\bullet) = \Gamma(\a_\bullet) + \Gamma(\b_\bullet).$$ From Proposition \[prop-convex-region-monomial-ideal\], Proposition \[prop-Newton-poly-additive\] and Theorem \[th-vol-Gamma-semigroup\] we readily obtain the following. \[th-covol-monomial\] Let $\a_\bullet$ be an $\m$-primary graded sequence of monomial ideals in $R$. Then: $$e(\a_\bullet) = n!~ \covol(\Gamma(\a_\bullet)).$$ In particular, if $\a$ is an $\m$-primary monomial ideal then: $$e(\a) = n!~ \covol(\Gamma(\a)).$$ Here $\Gamma(\a)$ is the Newton polyhedron of $\a$ i.e. the convex hull of $\I(\a)$. \[th-mixed-multi-mixed-covol-monomial\] Let $\a_{1}, \ldots, \a_{n}$ be $\m$-primary monomial ideals in $R$. Then the mixed multiplicity $e(\a_{1}, \ldots, \a_{n})$ is given by: $$e(\a_{1}, \ldots, \a_{n}) = n!~CV(\Gamma(\a_{1}), \ldots, \Gamma(\a_{n})),$$ where as before $CV$ denotes the mixed covolume of cobounded convex regions. \[rem-mixed-multi-graded-seq-monomial\] Using Theorem \[th-mixed-multi-poly-semigp\] one can define the mixed multiplicity of $\m$-primary graded sequences of monomial ideals. Then Theorem \[th-mixed-multi-mixed-covol-monomial\] can immediately be extended to mixed multiplicities of $\m$-primary graded sequences of monomial ideals. One knows that the mixed multiplicity of an $n$-tuple $(\a_1, \ldots, \a_n)$ of $\m$-primary subspaces in $R$ gives the intersection multiplicity, at the origin, of hypersurfacs $H_i = \{ x \mid f_i(x) = 0\}$, $i=1, \ldots, n$, where each $f_i$ is a generic element from the subspace $\a_i$. Theorem \[th-mixed-multi-mixed-covol-monomial\] then gives the following corollary. \[cor-local-BK-toric-sing\] Let $\a_1, \ldots, \a_n$ be $\m$-primary monomial ideals in $R$. Consider a system of equations $f_1(x) = \cdots = f_n(x) = 0$ where each $f_i$ is a generic element from $\a_i$. Then the intersection multiplicity at the origin of this system is equal to $n!~\covol(\Gamma(\a_1), \ldots, \Gamma(\a_n)).$ - When $R$ is the algebra of polynomials $\k[x_1, \ldots, x_n]_{(0)}$ localized at the origin (or the algebra of power series localized at the origin), i.e. the case corresponding to the local ring of a smooth affine toric variety, Corollary \[cor-local-BK-toric-sing\] is the local version of the classical Bernstein-Kushnirenko theorem ([@AVG Section 12.7]). - As opposed to the proof above, the original proof of the Kushnirenko theorem is quite involved. - Corollary \[cor-local-BK-toric-sing\] has been known to the second author since the early 90’s (cf. [@Askold-finite-sums]), although as far as the authors know it has not been published. Main results {#sec-valuation-ideal} ============ Let $R$ be a domain over a field $\k$. Equip the group $\z^n$ with a total order respecting addition. \[def-valuation\] A [valuation]{} $v: R \setminus\{0\} \to \z^n$ is a function satisfying: 1. For all $0 \neq f, g \in R$, $v(fg) = v(f) + v(g)$. 2. For all $0 \neq f,g \in R$ with $f+g \neq 0$ we have $v(f+g) \geq \min(v(f), v(g))$. (One then shows that when $v(f) \neq v(g)$, $v(f+g) = \min(v(f), v(g))$.) 3. For all $0 \neq \lambda \in \k$, $v(\lambda) = 0$. We say that $v$ has [one-dimensional leaves]{} if whenever $v(f) = v(g)$, there is $0 \neq \lambda \in \k$ with $v(g + \lambda f) > v(g)$. [ From definition $\S = v(R \setminus \{0\}) \cup \{0\}$ is an additive subsemigroup of $\z^n$ which we call the [value semigroup of $(R, v)$]{}. Any valuation on $R$ extends to the field of fractions $K$ of $R$ by defining $v(f/g) = v(f) - v(g)$. The set $R_v = \{0 \neq f \in K \mid v(f) \geq 0\} \cup \{0\}$ is a subring of $K$ called the [valuation ring of $v$]{}. Also $\m_v = \{0 \neq f \in K \mid v(f) > 0\} \cup \{0\}$ is a maximal ideal in $R_v$. The field $R_v / \m_v$ is called the [residue field of $v$]{}. One can see that $v$ has one-dimensional leaves if and only if the residue field of $v$ is $\k$.]{} \[def-semigp-ideal-of-ideal\] For a subspace $\a$ in $R$ define $\I = \I(\a) \subset \S$ by: $$\I = \{ v(f) \mid f \in \a \setminus \{0\} \}.$$ Similarly, for a graded sequence of subspaces $\a_\bullet$ in $R$, define $\I_\bullet = \I(\a_\bullet)$ by: $$\I_k = \I(\a_k) = \{ v(f) \mid f \in \a_k \setminus \{0\} \}.$$ For the rest of the paper we assume that $R$ is a Notherian local domain of dimension $n$ such that $R$ is an algebra over a field $\k$ isomorphic to the residue field $R/\m$, where $\m$ is the maximal ideal of $R$. Moreover, we assume that $R$ has a [good]{} valuation in the following sense: \[def-good-valuation\] We say that a $\z^n$-valued valuation $v$ on $R$ with one-dimensional leaves is [good]{} if the following hold: [(i)]{} The cone $C(\S)$ associated to the value semigroup $\S = v(R \setminus \{0\}) \cup \{0\}$ is a strongly convex cone (recall that $C(\S)$ is the closure of convex hull of $\S$). It implies that there is a linear function $\ell: \r^n \to \r$ such that $C(\S)$ lies in $\ell_{\geq 0}$ and intersects $\ell^{-1}(0)$ only at the origin. [(ii)]{} We assume there exists $r_0 > 0$ and a linear function $\ell$ as above such that for any $f \in R$ if $\ell(v(f)) \geq kr_0$ for some $k>0$ then $f \in \m^k$. The condition (ii) in particular implies that for any $k>0$ we have: $$\I(\m^k) \cap \ell_{\geq kr_0} = \S \cap \ell_{\geq kr_0}.$$ In other words, the sequence $\M_\bullet$ given by $\M_k = \I(\m^k)$ is a primary graded sequence in the value semigroup $\S$. The following is a generalization of Proposition \[prop-convex-region-monomial-ideal\]. \[prop-good-val-primary-ideal\] Let $v$ be a good valuation on $R$. Let $\a_\bullet$ be an $\m$-primary graded sequence of subspaces in $R$. Then the associated graded sequence $\I_\bullet = \I(\a_\bullet)$ is a primary graded sequence in the value semigroup $\S$ in the sense of Definition \[def-primary-sequence\]. Let $r_0>0$ be as in Definition \[def-good-valuation\] and let $m>0$ be such that $\m^m \subset \a_1$. Then for $k>0$ we have $\m^{km} \subset \a_{k}$. From Definition \[def-good-valuation\] for all $k>0$, $\S \cap \ell_{\geq kmr_0} = \I(\m^{km}) \cap \ell_{\geq kmr_0} \subset \I(\a_k) \cap \ell_{\geq kmr_0}$, which implies that $\S \cap \ell_{\geq kmr_0} = \I(\a_k) \cap \ell_{\geq kmr_0}$. That is, $\I_\bullet$ is a primary graded sequence with $t_0 = mr_0$. \[ex-good-val-toric-local-ring\] As in Section \[sec-monomial-ideal\] let $R$ be the local ring of an affine toric variety at its torus fixed point: Take $\C \subset \r^n$ to be an $n$-dimensional strongly convex rational polyhedral cone with apex at the origin. Consider the algebra of Laurent polynomials consisting of all the $f$ of the form $f = \sum_{\alpha \in \C \cap \z^n} c_\alpha x^\alpha$. Then $R$ is the localization of this algebra at the maximal ideal generated by non-constant monomials. Take a total order on $\z^n$ which respects addition and such that the semigroup $\S = \C \cap \z^n$ is well-ordered. We also require that if $\ell(\alpha) > \ell(\beta)$ then $\alpha > \beta$, for any $\alpha, \beta \in \z^n$. Such a total order can be constructed as follows: pick linear functions $\ell_2, \ldots, \ell_n$ on $\r^n$ such that $\ell, \ell_2, \ldots, \ell_n$ are linearly independent, and for each $i$ the cone $\C$ lies in $(\ell_{i})_{\geq 0}$. Given $\alpha, \beta \in \z^n$ define $\alpha > \beta$ if $\ell(\alpha) > \ell(\beta)$, or $\ell(\alpha) = \ell(\beta)$ and $\ell_2(\alpha) > \ell_2(\beta)$, and so on. Now one defines a (lowest term) valuation $v$ on the algebra $R$ with values in $\S = \C \cap \z^n$ as follows: For $f = \sum_{\alpha \in \S} c_\alpha x^\alpha$ put: $$v(f) = \min\{ \alpha \mid c_\alpha \neq 0\}.$$ Clearly $v$ extends to the field of fractions of Laurent polynomials and in particular to $R$ ($v$ can be defined similarly for Laurent series as well). It is easy to see that $v$ is a valuation with one-dimensional leaves on $R$. Let us show that it is moreover a good valuation. Take $0 \neq f \in R$. Without loss of generality we can take $f$ to be a Laurent series $f = \sum_{\alpha \in \S} c_\alpha x^\alpha$. Applying Proposition \[prop-polyhedral-primary-semigroup-ideal\] to the sequence $\I_\bullet$, where $\I_k = \{ \alpha \mid x^\alpha \in \m^k\}$, we know that there exists $r_0>0$ with the following property: if for some $\alpha \in \S$ we have $\ell(\alpha) \geq kr_0$ then $x^\alpha \in \m^k$. On the other hand, if $\alpha \leq \beta$ then $\ell(\alpha) \leq \ell(\beta)$. Thus $\ell(\beta) \geq kr_0$ and hence $x^\beta \in \m^k$. It follows that if $\ell(v(f)) \geq kr_0$ then all the nonzero monomials in $f$ lie in $\m^k$ and hence $f \in \m^k$. This proves the claim that $v$ is a good valuation on $R$. The arguments in Example \[ex-good-val-toric-local-ring\] in particular show the following: \[th-good-val-reg-local-ring\] If $R$ is a regular local ring then $R$ has a good valuation. More generally, one has: \[th-good-val-S/R\] Suppose $R$ is an analytically irreducible local domain (i.e. the completion of $R$ has no zero divisors). Moreover, suppose that there exists a regular local ring $S$ containing $R$ such that $S$ is [ essentially of finite type]{} over $R$, $R$ and $S$ have the same quotient field and the residue field map $R/\m_R \to S/\m_S$ is an isomorphism. Then $R$ has a good valuation. By Theorem \[th-good-val-reg-local-ring\], $S$ has a good valuation. By the linear Zariski subspace theorem in [@Hubl Theorem 1] or [@Cutkosky1 Lemma 4.3] $v_{\|R}$ is a good valuation for $R$ too. Using Theorem \[th-good-val-S/R\] and as in [@Cutkosky1 Theorem 5.2] we have the following: \[th-good-val-analytically-irr-sing\] Let $R$ be an analytically irreducible local domain over $\k$. Then $R$ has a good valuation. Let $\I = \I(\a)$ be the subset of integral points associated to an $\m$-primary subspace $\a$ in $R$. Then we have: $$\dim_\k(R / \a) = \#( \S \setminus \I).$$ Take $m>0$ with $\m^m \subset \a$ and let $r_0$ and $\ell$ be as in Definition \[def-good-valuation\]. If $\ell(v(f)) > mr_0$ then $f \in \m^{m} \subset \a$. Thus the set of valuations of elements in $R \setminus \a$ is bounded. In particular $\S \setminus \I$ is finite. Let $B$ be a finite subset of $R$ such that all the $v(b)$, $b \in B$, are distinct and $\{v(b) \mid b \in B\}$ coincides with $\S \setminus \I$. Among the set of elements in $R$ that are not in the span of $\a$ and $B$ take $f$ with maximum $v(f)$. If $v(f) = v(b)$ for some $b \in B$, then we can subtract a multiple of $b$ from $f$ getting an element $g$ with $v(g) > v(f)$ which contradicts the choice of $f$. Similarly $v(f)$ can not lie in $\I$ otherwise we can subtract an element of $\a$ from $f$ to arrive at a similar contradiction. This shows that the set of images of elements of $B$ in $R/\a$ is a $\k$-vector space basis for $R/\a$ which proves the proposition. Let $\a_\bullet$ be an $\m$-primary graded sequence of subspaces in $R$ and put $\I_\bullet = \I(\a_\bullet)$. We then have: $$e(\a_\bullet) = e(\I_\bullet).$$ \[def-Gamma-I\] To the sequence of subspaces $\a_\bullet$ we associate a $\C$-convex region $\Gamma(\a_\bullet)$, which is the convex region $\Gamma(\I_\bullet)$ associated to the primary sequence $\I_\bullet = \I(\a_\bullet)$. The convex region $\Gamma(\a_\bullet)$ depends on the choice of the valuation $v$. By definition the convex region $\Gamma(\a)$ associated to an $\m$-primary subspace $\a$ is the convex region associated to the sequence of subspaces $(\a, \a^2, \a^3, \ldots)$. \[th-multi-ideal-covol\] Let $\a_\bullet$ be an $\m$-primary graded sequence of subspaces in $R$. Then: $$e(\a_\bullet) = \lim_{k \to \infty} \frac{H_{\a_\bullet}(k)}{k^n} = n!~ \covol(\Gamma(\a_\bullet)).$$ In particular, if $\a$ is an $\m$-primary ideal we have $e(\a) = n!~\covol(\Gamma(\a))$. The following superadditivity follows from Proposition \[prop-additivity-semigroup-ideal\]. Note that $\I(\a_\bullet) + \I(\b_\bullet) \subset \I(\a_\bullet \b_\bullet)$ (cf. Proposition \[prop-Newton-poly-additive\]). \[prop-superadd-ideal\] Let $\a_\bullet$, $\b_\bullet$ be two $\m$-primary graded sequences of subspaces in $R$. We have: $$\Gamma(\a_\bullet) + \Gamma(\b_\bullet) \subset \Gamma(\a_\bullet \b_\bullet).$$ From Theorem \[th-multi-ideal-covol\], Proposition \[prop-superadd-ideal\] and Corollary \[cor-Brunn-Mink-covol\] we readily obtain: (Brunn-Minkowski for multiplicities) \[cor-Brunn-Mink-multi\] Let $\a_\bullet$, $\b_\bullet$ be two $\m$-primary graded sequences of subspaces in $R$. Then: $$\label{equ-Brunn-Mink-graded-seq-ideals} e(\a_\bullet)^{1/n} + e(\b_\bullet)^{1/n} \geq e(\a_\bullet \b_\bullet)^{1/n}.$$ \[rem-BM-non-local-analytic-irreducible\] By Theorem \[th-good-val-analytically-irr-sing\] and Corollary \[cor-Brunn-Mink-multi\] we obtain the Brunn-Minkowski inequality for an analytically irreducible local domain $R$. But in fact the assumption that $R$ is analytically irreducible is not necessary: Suppose $R$ is not necessarily analytically irreducible. First by a reduction theorem the statement can be reduced to $\dim R = n = 2$. In dimension $2$, the inequality implies that the mixed multiplicity of ideals $e(\cdot, \cdot)$, regarded as a bilinear form on the (multiplicative) semigroup of $\m$-primary graded sequences of ideals, is negative semidefinite restricted to each local analytic irreducible component. But the sum of negative semidefinite forms is again negative semidefinite which implies that Corollary \[cor-Brunn-Mink-multi\] should hold for $R$ itself. As another corollary of Theorem \[th-multi-ideal-covol\] we can immediately obtain inequalities between the multiplicity of an $\m$-primary ideal, multiplicity of its associated initial ideal and its length. Let $R$ be a regular local ring of dimension $n$ with a good valuation (as in Example \[ex-good-val-toric-local-ring\] and Theorem \[th-good-val-reg-local-ring\]). \[cor-Lech\] Let $\a$ be an $\m$-primary ideal in $R$ and let $\In(\a)$ denote the initial ideal of $\a$, that is, the monomial ideal in the polynomial algebra localized at the origin $\k[x_1, \ldots, x_n]_{(0)}$ corresponding to the semigroup ideal $\I(\a)$. We have: $$e(\a) \leq e(\In(\a)) \leq n!~ \dim_\k(R / \a).$$ More generally, if $\In(\a^k)$ denote the monomial ideal in $\k[x_1, \ldots, x_n]_{(0)}$ corresponding to the semigroup ideal $\I(\a^k)$ then the sequence of numbers $$\frac{e(\In(\a^k))}{k^n}$$ is decreasing and converges to $e(\a)$ as $k \to \infty$. From definition one shows that $\Gamma(\In(\a))$ is the convex hull of $\I(\a)$ (see Theorem \[th-covol-monomial\]). It easily follows that $$\I(\a) \subset \Gamma(\In(\a)) \subset \Gamma(\a).$$ We now notice that $\dim_\k(R / \a)$ is the number of integral points in $\S \setminus \I(\a)$ that in turn is bigger than or equal to the volume of $\R \setminus \Gamma(\In(\a))$ and hence the volume of $\R \setminus \Gamma(\a)$. More generally, from the definition of $\Gamma(\a)$ we have an increasing sequence of convex regions: $$\Gamma(\In(\a)) \subset (1/2)\Gamma(\In(\a^2)) \subset \cdots \subset \Gamma(\a) = \overline{\bigcup_{k=1}^\infty (1/k)\Gamma(\In(\a^k))}.$$ Now from Theorem \[th-multi-ideal-covol\] we have $e(\a) = n!~\covol(\Gamma(\a))$ and for each $k$, $e(\In(\a^k)) = n!~\covol(\Gamma(\In(\a^k)))$. This finishes the proof. The inequality $e(\a) \leq n!~\dim_\k(R / \a)$ is a special case of an inequality of Lech [@Lech Theorem 3]. See also Lemma 1.3 in [@FEM]. [99]{} Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N. [Singularities of differentiable maps]{}. Modern Birkhäuser Classics. Birkhäuser/Springer, New York, 2012. Burago, Yu. D.; Zalgaller, V. A. [Geometric inequalities]{}. Translated from the Russian by A. B. Sosinskiĭ. Grundlehren der Mathematischen Wissenschaften, 285. Springer Series in Soviet Mathematics (1988). Cutkosky, S. D. [Multiplicities Associated to Graded Families of Ideals]{}. arXiv:1206.4077. To appear in Journal of Algebra and Number Theory. Cutkosky, S. D. [Multiplicities of graded families of linear series and ideals]{}. arXiv:1301.5613. Cutkosky, S. D. [Asymptotic multiplicities]{}. arXiv:1311.1432. de Fernex, T.; Ein, L.; Mustaţă, M. [Multiplicities and log canonical threshold]{}. J. Algebraic Geom. 13 (2004), no. 3, 603Ð615. Fillastre, F.; [Fuchsian convex bodies: basics of BrunnÐMinkowski theory]{}. Geom. Funct. Anal. 23 (2013), no. 1, 295–333. Fulger, M. [Local volumes on normal algebraic varieties]{}. arXiv:1105.2981. Hübl, R. [Completions of local morphisms and valuations]{}. Math. Z. 236 (2001), no. 1, 201–214. Kaveh, K.; Khovanskii, A. G. [Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory]{}. Annals of Mathematics, 176 (2012), 1–54. Kaveh, K.; Khovanskii, A. G. [On mixed multiplicities of ideals]{}. arXiv:1310.7979. Khovanskii, A. G. [Algebra and mixed volumes]{}. Appendix 3 in: Burago, Yu. D.; Zalgaller, V. A. [Geometric inequalities]{}. Translated from the Russian by A. B. Sosinskii. Grundlehren der Mathematischen Wissenschaften, 285. Springer Series in Soviet Mathematics (1988). Khovanskii, A. G. [Newton polyhedron, Hilbert polynomial and sums of finite sets]{}. (Russian) Funktsional. Anal. i Prilozhen. 26 (1992), no. 4, 57–63, 96; translation in Funct. Anal. Appl. 26 (1992), no. 4, 276–281. Khovanskii, A. G.; Timorin, V. [Alexandrov-Fenchel inequality for coconvex bodies]{}. arXiv:1305.4484. Khovanskii, A. G.; Timorin, V. [On the theory of coconvex bodies]{}. arXiv:1308.1781. Kushnirenko, A. G. [Polyedres de Newton et nombres de Milnor]{}. (French) Invent. Math. 32 (1976), no. 1, 1–31. Lazarsfeld, R.; Mustaţă, M. [Convex bodies associated to linear series]{}. Ann. Sci. Ec. Norm. Super. (4) 42 (2009), no. 5, 783Ð835. Lech, C. [Inequalities related to certain couples of local rings]{}. Acta Math. 112 (1964), 69–89. Mustaţă, M. [On multiplicities of graded sequences of ideals]{}. J. of Algebra 256 (2002), 229–249. Okounkov, A. [Brunn-Minkowski inequality for multiplicities]{}. Invent. Math. 125 (1996), no. 3, 405–411. Okounkov, A. [Why would multiplicities be log-concave?]{} The orbit method in geometry and physics (Marseille, 2000), 329–347, Progr. Math., 213, Birkhäuser Boston, Boston, MA, 2003. Rees, D.; Sharp, R. Y. [On a theorem of B. Teissier on multiplicities of ideals in local rings]{}. J. London Math. Soc. (2) 18 (1978), no. 3, 449Ð463. Samuel, P.; Zariski, O. [Commutative algebra]{}. Vol. II. Reprint of the 1960 edition. Graduate Texts in Mathematics, Vol. 29. Teissier, B. [Sur une inegalite pour les multiplicites]{}, (Appendix to a paper by D. Eisenbud and H. Levine). Ann. Math. 106 (1977), 38Ð44. Teissier, B. [Jacobian Newton Polyhedra and equisingularity]{}. arXiv:1203.5595. [^1]: The first author is partially supported by a Simons Foundation Collaboration Grants for Mathematicians (Grant ID: 210099) and a National Science Foundation (Grant ID: 1200581). [^2]: The second author is partially supported by the Canadian Grant N 156833-12. [^3]: In [@KKh-Annals] a valuation $v$ with values in $\z^n$ and residue field $\k$ is called a [valuation with one-dimensional leaves (see Definition \[def-valuation\]).*]{}
BACKGROUND OF THE INVENTION The invention relates to an attenuation arrangement comprising a step attenuator having an input, through which first and second voltage terminals are arranged in cascade with a voltage divider having an output, said attenuation arrangement also comprising a control circuit having a control input for controlling the step attenuator and the controllable voltage divider, the step attenuator having, connected between the input and a reference terminal, a series arrangement of attenuator elements which are connectable in parallel with the two voltage terminals. Such an attenuation arrangement is known from Dutch patent application No. 300875, which has been laid open to public inspection. The step attenuator of the known attenuation arrangement distributes an input voltage applied between the input and the reference terminal over the attenuation elements in a plurality of voltage increments. By connecting one of these attenuation elements in parallel with the two voltage terminals, the voltage increment present across this attenuation element is applied to the controllable voltage divider via the voltage terminals. Herein a voltage value located within the range of the voltage increment is connected to the output of the controllable voltage divider. An increase in the output voltage over the range of the voltage increments of two adjacent attenuation elements is realized in the known attenuation arrangement as follows. After the highest position of the controllable voltage divider has been attained, wherein the highest voltage value of the voltage increment applied between the two voltage terminals is connected to the output, the two ends of the relevant attenuation element are disconnected from the two voltage terminals under the control of the control circuit and the two corresponding ends of the next attenuation element are connected to the two voltage terminals. The controllable voltage divider is now adjusted from the last-mentioned highest position to the lowest position, wherein the lowest voltage value of the voltage increment between the two voltage terminals is connected to the output. The output voltage can now increase further over the range of the higher voltage increment. A decrease of the output voltage over the range of two adjacent attenuation elements is realized by reversing the above method. The switching actions which must be performed to switch from one voltage increment to the other produce voltage peaks which may become apparent in an unacceptable manner. When this attenuation arrangement is used as, for example, the volume control in wireless sets, these voltage peaks become audible as plopping sounds SUMMARY OF THE INVENTION It is an object of the invention to provide an attenuation arrangement wherein a variation in the output voltage over two consecutive voltage increments is realized with a minimum number of switching actions, switching peaks being prevented from occurring. According to the invention an attenuation arrangement of the type mentioned in the opening paragraph is therefore characterized in that when switching the two voltage terminals from a first to an adjacent second attenuation element under the control of the control circuit, the connection of the common junction between the two attenuation elements and one of the two voltage terminals is maintained and the connection from the other voltage terminal to the end of the first attenuation element located opposite this junction is switched over to the end of the second attenuation element located opposite this junction. When using this measure according to the invention, the switching actions are limited to breaking the connection of the end of the first attenuation element located opposite said junction and one of the two voltage terminals and connecting the end of the second attenuation element, also located opposite this junction, to the last- mentioned voltage terminal. In response thereto the polarity of the voltage between the two voltage terminals changes, resulting, the position of the controllable voltage divider not being changed, in that the lowest voltage value of the voltage increment connected between the voltage terminals or viceversa is now connected to the output instead of the highest voltage value. Consequently, a switch-over in the controllable voltage divider can now be omitted. A further advantage of the maintained connection of the common junction between the two attenuation elements and one of the two voltage terminals during switching is that the voltage at this voltage terminal, this voltage being connected to an output terminal as the output voltage via the voltage divider, is kept at a fixed value, thus preventing switching peaks. A preferred embodiment of an attenuation arrangement according to the invention is characterized in that the controllable voltage divider comprises a series arrangement of attenuation elements connected between the two voltage terminals, each end of the attenuation elements being individually connectable to an output of the attenuation arrangement under the control of the control circuit. When this measure is used the controllable voltage divider has a discrete implementation and the control thereof can be realised by means of a simple control circuit. A still further preferred embodiment of such an attenuation arrangement is characterized in that the control input of the control circuit comprises first and second control terminals coupled to a pulse generator, these elements being connected in a switchable manner to an adding device for adding the pulses applied to the first control terminal and for subtracting the pulses applied to the second control terminal, the adding device being connected to a switching control device for converting the output signal of the adding device into a switching control signal for the step attenuator and the controllable voltage divider, a pulse train applied to the adding device via one of the two control terminals producing an increase and a pulse train applied to the adding device via the other control terminal producing a decrease of the output voltage of the attenuation arrangement. When using this measure the whole attenuation arrangement is integratable. Another preferred embodiment of an attenuation arrangement according to the invention is characterized in that the attenuation elements of the step attenuator divides a voltage applied between the input and the reference terminal into partial voltages which have a mutual logarithmic relationship. When using this measure a logarithmic voltage variation takes place which is desired, inter alia for controlling the volume of audio signals. DESCRIPTION OF THE DRAWINGS The invention will now be further explained by way of non- limitative example with reference to the Figures shown in the drawings. Herein: FIG. 1 shows a circuit diagram of the step attenuator and the controllable voltage divider of the attenuation arrangement according to the invention; FIG. 1a shows a logarithmic step attenuator; FIG. 1b shows a construction of an attenuation arrangement wherein a plurality of step attenuators and a controllable voltage divider are arranged in cascade; and FIG. 2 shows a practical embodiment of an integratable attenuation arrangement according to the invention. DESCRIPTION OF THE PREFERRED EMBODIMENT FIG. 1 shows a step attenuator 1 which is arranged in cascade with a controllable voltage divider 2 via first and second voltage terminals 5 and 6. The step attenuator 1 comprises an input 3 and a reference terminal 4 which is coupled to ground, a voltage source 8 being connected therebetween and comprises a series arrangement of resistors 11 to 15 inclusive, operating as attenuation elements and connected between the input 3 and the reference terminal 4. The resistor 11 is connectable between the voltage terminals 5 and 6 by means of switches 21 and 22, the resistor 12 by means of switches 23 and 22, the resistor 13 by means of switches 23 and 24, the resistor 14 by means of the switches 25 and 24 and the resistor 15 by means of the switches 25 and 26. The controllable voltage divider 2 comprises a series arrangement, connected between the voltage sources 5 and 6, of resistors 31 to 36, inclusive, which operate as attenuation elements. The ends of these resistors 31 to 36, inclusive, are individually connected in a switchable manner to an output 7 of the controllable voltage divider 2 via a parallel arrangement of switches 41 to 47, inclusive respectively. The output voltage is measured at the output 7 with respect to an output reference terminal 9 connected to the reference terminal 4. At a voltage V of the voltage source 8 and equal resistors 11 to 15, inclusive, the voltage terminal 6 is connected to ground and the voltage terminal 5 to a voltage of 1/5 V by closing the switches 25 and 26. The output voltage at the output 7 can now be brought to a voltage value in the range from 0 to 1/5 V inclusive, by closing one of the switches 41 to 47, inclusive. When resistors 31 to 36, inclusive, are equal, the output voltage can be brought to 1/30 V, for example by closing switch 46. By opening switch 46 and closing switch 47 the output voltage is brought to 2/30 V. In this manner the output voltage can increase in steps of 1/30 V to the maximum value (1/5 V) of the voltage increment between the voltage terminals 5 and 6. The switch 41 is then closed and the switches 42 to 47, inclusive, are opened. A further increase in the output voltage is possible by transferring the voltage increment of the next resistor 14 to the voltage terminals 5 and 6. According to the invention this is realised by opening the switch 26 and closing the switch 24, the switch 25 remaining in the closed condition. As a result thereof a voltage of 1/5 V remains connected to the voltage terminal 5 whereas a voltage of 2/5 V is now applied to the voltage terminal 6. The controllable voltage divider 2 which connected the highest voltage value (1/5 V) of the preceding voltage increment to the output 7, the switch 41 being in the closed condition, now passes, the switch 41 remaining in the closed condition, the lowest voltage value (1/5 V) of the new voltage increment (1/5 V-2/5 V). The polarity of the voltage between the voltage terminals 5 and 6 is now opposite to the polarity of the preceding voltage increment. The output voltage can now further increase in increments of 1/30 V until 2/5 V by sequentially switching the switches 42 to 47, inclusive. A further increase in the output voltage after 2/5 V has been reached on closing of the switch 47 is rendered possible by transferring the voltage increment of the next resistor 13 to the voltage terminals 5 and 6. According to the invention this is realised by opening the switch 25 and closing the switch 23. The polarity of the voltage between the voltage terminals 5 and 6 changes: a voltage of 3/5 V is connected to the voltage terminal 5 and a voltage of 2/5 V is connected to the voltage terminal 6. The output voltage can now further increase over this voltage increment from 2/5 V to 3/5 V in steps of 1/30 V by sequentially switching the switches 46 to 41, inclusive. In the above-described manner it is possible to have the output voltage increase in steps of 1/30 V over the following voltage increments (3/5 V- 4/5 V) and (4/5 V-5/5 V). The output voltage is decreased by performing the switching actions in the inverse sequence. The controllable voltage divider 2 which subdivides the voltage applied to the voltage terminals 5 and 6 into discrete voltage levels, may now be replaced by an analog voltage divider, for example a potentiometer or a circuit as described in the Netherlands patent application OA No. 1,802,973. Such a construction, not shown, has the advantage that, with the above-described switching mode in the step attenuator no switching actions occur in the controllable voltage divider so that switching peaks are completely avoided. FIG. 1a shows a step attenuator 1' having a logarithmic voltage division function. The elements corresponding to the elements of the step attenuator shown in the preceding figure have been given the same reference numerals. The step attenuator 1' comprises resistors 27, 28, 29 and 30, one end of which is connected to the reference terminal 4, the other end being connected to the common junctions of the resistors 11 and 12, 12 and 13, 13 and 14 and 14 and 15, respectively. When the resistors 15 and 27 to 30, inclusive, have a value of 2 R and the resistors 11 to 14, inclusive, a value of R, the total value of resistors 15 and 30 becomes R, which also applies to the total value of the resistors 14, 15, 29 and 30; 13, 14, 15, 28, 29 and 30; 12, 13, 14, 15, 27, 28, 29 and 30. If the input 30 is brought to a voltage value V and the reference voltage 4 is connected to ground a voltage of 1/2 V is present at the common junction of the resistors 11, 12, and 27; a voltage 1/4 V at the common junction of the resistors 12, 13 and 28; a voltage 1/8 V at the common junction of the resistors 13, 14 and 29; a voltage 1/16 V at the common junction of the resistors 14, 15 and 30. In the said sequence the voltage at the junctions decreases by a factor of 1/2. Such a voltage distribution is advantageous for, for example, volume controls in audio amplifiers. FIG. 1b shows a cascade arrangement of step attenuators 1a to 1k, inclusive, and the controllable voltage divider 2, the step attenuators 1a to 1k being identical to the step attenuator 1. The voltage source 8 is connected between input 3k and the reference terminal 4k of the step attenuator. The reference terminal 4k is connected to ground and is coupled to the output reference terminal 9. Assuming that in an internal switching configuration as shown in FIG. 1, there are N.sub.o attenuation elements in the controllable voltage divider 2, and N.sub.1 to N.sub.k, inclusive, attenuation elements in the step attenuators 1a to 1k, inclusive, respectively, N. sub.O to N.sub.k, inclusive, being equal to or greater than 3, it is possible to realise with such a cascade arrangement ##EQU1## different voltage levels by means of ##EQU2## switches. A maximum number of different voltage levels is achieved with a minimum number of switches when N.sub.j =N.sub.j+1 =2 (j=o . . . -k-1). The number of different voltage levels is 1+2.sup.k, the number of switches 2k+1. FIG. 2 shows an integratable attenuation arrangement according to the invention wherein the terminals 3 to 7 correspond to terminals of FIG. 1, having the same reference numerals. The step attenuator 1 and the controllable voltage divider 2 have the same function as those in FIG. 1. The control of the step attenuator 1 and the controllable voltage divider 2 is realised by means of a control circuit comprising a change-over switch 86, a clock pulse generator 83, which is coupled to an adding device 90 via a blocking circuit A, a cycle indicator 92 coupled between an output of the adding device 90 and an input of the blocking circuit A and a switching control device 91, which is also coupled to the output of the adding device 90 and is also connected to the step attenuator 1 and the controllable voltage divider 2 for applying switching control signals thereto. The above-mentioned circuits and devices are implemented by means of integrated circuits full details of which are included in the Philips Data Handbook "Semiconductors and Integrated Circuits", part 6, May 1976. The connecting terminals of these integrated circuits are denoted in the present Figure by means of a letter and an index. This index denotes the number of the terminal of the relevant connecting terminal as mentioned in said Philips Data Handbook. The voltage supply terminals as well as the connecting terminals which are not relevant to the said control have not been shown. The blocking circuit A comprises an integrated circuit of the type HEF 4012 P, which includes two NANDgates A' and A" having input terminals A. sub.2 to A.sub.4, inclusive, and A.sub.9 to A.sub.11, inclusive, respectively. The input terminals A.sub.2 and A.sub.10 are connected to ground via resistors 85 and 84, respectively, and are interchangeably connected to a positive supply voltage via the change- over switch 86; in a position I of the change-over switch 86 the input terminal A.sub.10 is connected to the supply voltage, in a position II the input terminal A. sub.2 is connected to the supply voltage. The input terminals A.sub.4 and A.sub.9 function as the first and second control terminals of the control circuit and are coupled to an output of the clock pulse generator 83, which produces positive pulses with a frequency of approximately 10 Hz. The input terminals A.sub.3 and A.sub. 11 are connected to an output of the cycle indicator 92. The adding device 90 comprises two integrated circuits B and C of the type HEF 40193 P having input terminals B.sub.4, B.sub.5 and C.sub. 4, C. sub. 5 and output terminals B.sub.3, B.sub.2, B.sub.6, B.sub.7 and C. sub. 3, C.sub. 2, C.sub.6, respectively. One end of the integrated circuit B is coupled to the output terminals A.sub.1 and A.sub.13 of the NAND- gates A' and A" via the input terminals B.sub.5 and B.sub.4, the other end to the input terminals C.sub.5 and C.sub.4 of the integrated circuit C via borrow bit output terminals B.sub.12 and B.sub.13. The sum of the pulses applied to the input terminal B.sub.5 is presented in the form of a binary-decimal number to the output terminals B.sub.3, B. sub.2, B.sub.6, B.sub.7, C.sub.3, C.sub.2 and C.sub.6, the bit significance increasing in this sequence from 2° to 2.sup.6. A high terminal voltage, to be denoted 1-voltage hereinafter, corresponds to the binary value 1 and a low terminal voltage, to be denoted 0-voltage hereinafter, corresponds to the binary value 0. Pulses applied to the input terminal B.sub.4 reduce the binary-decimal number at the last- mentioned output terminals. The cycle indicator 92 comprises an integrated NAND-gate D of the type HEF 4068 P, having input terminals D.sub.2 to D.sub.5, inclusive and D. sub.9 to D.sub.11, inclusive and an output terminal D.sub.13, as well as an integrated NOR-gate E of the type HEF 4078 P, having input terminals E. sub.2 to E. sub.5, inclusive and E.sub.9 to E.sub.11, inclusive and an output terminal E.sub. 13. The output terminal D.sub.13 is coupled to the input terminal A.sub.3 of the NAND-gate A', the output terminal E.sub.13 to the input terminal A.sub.11 of the NAND-gate A" via an inverter 81. The input terminal D.sub.5 of the NAND-gate D is coupled to the output terminal B.sub.7 of the adding device 90 via an inverter 80, the other input terminals D.sub.2 to D.sub.4, inclusive and D.sub.9 to D. sub.11, inclusive are coupled to the output terminals B.sub. 3, B. sub.2, B.sub.6, C.sub.3, C.sub.2 and C.sub.6, respectively. The input terminals E.sub.2 to E.sub.5, inclusive and E.sub.9 to E.sub.11, inclusive, of the NOR-gate E are also coupled to the output terminals B. sub.3, B.sub.2, B. sub.6, B. sub.7, C.sub.3, C.sub.2, C.sub.6, respectively, of the counting device 90. For a numerical value of the binary-decimal number at the output terminals B.sub.3, B.sub.2, B.sub.6, B.sub.7, C.sub.3, C.sub.2 and C.sub. 6 of the adding device 90 in the range from 1 to 118, inclusive, (that is to say from 1000000 to 0110111), the 1-voltage and the 0-voltage are connected to the output terminals D.sub.13 and E.sub.13, respectively, of the cycle indicator 92. The 0-voltage at the output terminal E.sub.13 is converted into the 1-voltage by the inverter 81, this voltage being applied to the input terminal A.sub. 11 of the NAND-gate A". In position II of the change-over switch 86 an 1-voltage is also applied to the input terminal A.sub.2 of this NAND-gate A'. A 0- voltage is then present at the input terminal A.sub.10 of the NAND-gate A", so that this gate is kept in the closed condition. The NAND-gate A' is conductive for the positive clock pulses of the clock pulse generator 83, which are applied to the input terminal A.sub.4. These clock pulses appear at the output terminal A.sub.1, are applied to the input terminal B.sub.5 of the adding device 90 and added to the binary-decimal number at the output terminals thereof. When this binary- decimal number reaches the value 119 (that is to say 1110111), then the 0-voltage appears at the output terminal D.sub.13, so that the NAND-gate A' is blocked. The transfer of further clock pulses from the clock pulse generator 83 to the adding device 90 is then blocked. If, thereafter, the change-over switch 86 is switched to position I, then a 0-voltage is applied to the input terminal A.sub.2 of the NAND- gate A' in response to which this gate is blocked, also for other values of the said binary-decimal number. A 1-voltage is applied to the input terminal A.sub. 10 of the NAND-gate A" via the change-over switch 86. A 1- voltage is applied to the input terminal A.sub.11 via the inverter 81. The NAND-gate A" is then conductive for the clock pulses of the clock pulse generator 83 and it passes these clock pulses to the input terminal B.sub.4 of the adding device 90 via the output terminal A. sub. 13. The clock pulses reduce the numerical value of the binary- decimal number at the output terminals B.sub.3, B.sub.2, B. sub.6, B.sub.7, C.sub. 3, C.sub. 2 and C.sub.6. When the value 0 is reached (that is to say 0000000) then there appears at the output terminal E.sub.13 the 1- voltage which is converted into the 0-voltage in the inverter 81. This 0- voltage blocks the NAND-gate A", in response to which the transfer of further clock pulses to the adding device 90 is blocked. Consequently the counting cycle comprises 120 different numerical values from 0000000 to 1110111, respectively, the direction of the counting cycle being determined by the position of change-over switch 86. The switching control device 91 comprises an integrated buffer circuit G of the type HEF 40097 P and an integrated inverting buffer circuit F of the type HEF 40098 P for the generation of one switching control signal for the controllable voltage divider 2, as well as an integrated adding circuit H of the type HEF 4008 P for the generation of a switching control signal for the step attenuator 1. The controllable voltage divider 2 comprises an integrated multiplex circuit I of the type HEF 4051 P and the step attenuator 1 comprises two multiplex circuits K and L, also of the type HEF 4051 P. The multiplex circuit I has input terminals I.sub.9 to I.sub.11, inclusive and output terminals I.sub.1 to I.sub.5, inclusive and I.sub. 12 to I.sub. 15, inclusive, the output terminal I.sub.3 being internally through-connected to one of the said further output terminals I.sub.1, I. sub.2, I.sub.4, I. sub.5 and I.sub.12 to I.sub.15, inclusive, in dependence on the numerical value of the binary signal at the input terminals I.sub.11, I.sub.10 and I.sub.9 (from 000 to 111). The multiplex circuit K, having input terminals K.sub.9 to K. sub. 11, inclusive, and output terminals K.sub.1 to K.sub.5, inclusive and K. sub. 12 to K.sub.15, inclusive, and also the multiplex circuit L, having input terminals L.sub.9 to L.sub.11, inclusive and output terminals L. sub.1 to L.sub.5, inclusive and L.sub.12 to L.sub.15, inclusive, operate in a similar manner. Resistors 50 to 56, inclusive, are connected between the output terminals I.sub.4 and I.sub.2, I.sub.2 and I.sub.5, I.sub.5 and I.sub.1, I.sub.1 and I.sub.12, I.sub.12 and I.sub.15, I.sub.15 and I.sub.14, I. sub.14 and I. sub.13, respectively. The output terminal I.sub.3 is connected to an output 7 of the attenuation arrangement. In the integrated multiplex circuit I the same switching functions have been realised as those obtained by means of the switches 41 to 47, inclusive, of the controllable voltage divider shown in FIG. 1. The resistors 60 to 74, inclusive, are connected between the output terminals K.sub.13 and L.sub.13, L.sub.13 and K.sub.14, K.sub.14 and L. sub.14, L.sub.14 and K.sub.15, K.sub.15 and L.sub.15, L.sub.15 and K.sub. 12, K. sub.12 and L.sub.12, L.sub.12 and K.sub.1, K.sub.1 and L.sub. 1, L. sub.1 and K.sub.5, K.sub.5 and L.sub.5, L.sub.5 and K.sub.2, K.sub. 2 and L.sub.2, L.sub.2 and K.sub.4, K.sub.4 and L.sub.4, respectively. The output terminal K.sub.3 is coupled to the output terminal I.sub.13 via the voltage terminal 5 and the output terminal L.sub.3 to the output terminal I.sub.4 via the voltage terminal 6. The output terminal K.sub. 13 is connected to a positive voltage V to be distributed, via the input 3 of the attenuation arrangement, and the output terminal L.sub.4 is connected to ground via the reference terminal 4. The same switching functions are realised in the integrated multiplex circuit K as those obtained by means of the switches 21, 23 and 25 of the step attenuator 1 of FIG. 1, the same switching functions being realised in the integrated multiplex circuit L as those obtained by means of the switches 22, 24 and 26 of this step attenuator. The output terminals B.sub.3, B.sub.2, B.sub.6 and B.sub.7 of the adding device 90 are connected to input terminals G.sub.2, G.sub.4, G. sub.6 and G.sub.1, respectively, of buffer circuit G and to input terminals F.sub.2, F.sub.4, F.sub.6 and F.sub.1, respectively, of the inverting buffer circuit F, an inverter 82 being included between the output terminal B.sub.7 and the input terminal F.sub.1. Output terminals G.sub.3, G.sub.5 and G.sub.7 of the buffer circuit G and output terminal F.sub.2, F.sub.5 and F.sub.7 of the inverting buffer circuit F are coupled to the input terminals I.sub.11, I.sub.10 and I.sub.9, respectively, of the multiplex circuit I. The buffer circuit G transfers the voltage applied to the input terminals G.sub.2, G.sub.4 and G.sub.6 to the output terminals G.sub.3, G. sub.5 and G.sub.7, respectively, when an 0-voltage is present at the input terminal G.sub.1. When this 0-voltage is changed into a 1-voltage, the voltage transfer from the input to the output terminals is blocked. The inverting buffer circuit F operates in the same manner but transfers the voltage at the input terminals F.sub.2, F.sub.4 and F.sub.6 in the inverted version to the output terminals F.sub.3, F.sub.5 and F.sub.7, respectively. The output terminals B.sub.7, C.sub.3, C.sub.2 and C.sub.6 of the adding device 90 are coupled to input terminals H.sub.7, H.sub.5, H. sub. 3 and H.sub.1, respectively, of the adding circuit H. The output terminals C. sub.3, C.sub.2 and C.sub.6 are also coupled to the input terminals K.sub.9, K. sub.10 and K.sub.11, respectively, of the multiplex circuit K. Output terminals H.sub.11, H.sub.12 and H.sub.13 of the adding circuit H are connected to input terminals L.sub.11, L.sub.10 and L.sub.9, respectively, of the multiplex circuit L. By connecting the input terminal H.sub.6 to the 1-voltage and the input terminals H.sub.4, H.sub. 2 and H.sub.15 to a 0-voltage (ground), a binary signal 1000 is applied to further input terminals H.sub.6, H.sub.4, H.sub.2 and H.sub. 15 of the adding circuit H. The adding circuit H produces at the output terminals H. sub.11, H.sub.12 and H.sub.13 the most significant bits of the sum of the binary-decimal number present at the input terminals H. sub.7, H.sub. 5, H. sub.3, H.sub.1 and the (constant) binary-decimal number 1000, applied to the input terminals H.sub.6, H.sub.4, H.sub.2 and H.sub.15. The binary-decimal number at the input terminals I.sub.11, I. sub. 10 and I.sub.9 of the multiplex circuit I alternately cycles through the number cycle from 0 to 7, inclusive (000 to 111, inclusive) and from 7 to 0, inclusive, (111 to 000, inclusive), at a continuous increase or decrease of the binary number at the output terminals B.sub.3, B.sub.2, B. sub.6 and B.sub.7. In this manner one of the output terminals I.sub.4, I. sub.2, I.sub.5, I.sub.1, I.sub.12, I.sub.15, I.sub.14, I.sub.13 is successively connected to the output terminal I.sub.3, in this sequence or in the inversed sequence. This results in a similar switching mode as the switching mode of the controllable voltage divider 2 shown in FIG. 1. After each cycle of the numbers 0 to 7, inclusive, the step attenuator 1 must be switched-over, namely such that a switching action is alternately performed in the multiplex circuit K and in the multiplex circuit L. Such a switching mode is realised by using the binary-decimal number at the output terminals B.sub.7, C.sub.3, C.sub.2 and C.sub.6 as the switching control signal for the step attenuator 1. When this binary- decimal number changes from odd to even, the connection from the output terminal K.sub.3 to one of the output terminals K.sub.4, K.sub.2, K.sub. 5, K.sub.1, K.sub.12 and K.sub.15 is switched to a next output terminal. The binary-decimal number at the three most significant input terminals H. sub.5, H.sub.3 and H.sub.1 then determines which one of the said output terminals is connected to the output terminal K.sub.3. When the said binary-decimal number changes from even to odd, then the connection from the output terminal L.sub.3 to one of the output terminals L.sub.1, L.sub. 2, L.sub.4, L.sub.5, L.sub.12 to L.sub.15, inclusive, is switched to a next output terminal. Here the three most significant bits of the sum of the binary-decimal number at the input terminals H.sub.7, H.sub.5, H.sub. 3, H.sub.1 and the binary-decimal number 1000 at the input terminals H. sub.6, H.sub.4, H.sub.2 and H.sub.5 determine which one of the said output terminals L.sub.1, L.sub.2, L.sub.4, L.sub.5, L.sub.12 to L.sub.15, inclusive, is connected to the output terminal L.sub.3. The switching mode thus obtained corresponds to the switching mode performed in the step attenuator shown in FIG. 1. In position I of the change-over switch 86 of voltage at the output 7 decreases with respect to the reference terminal 4 from a value between 0 and +V to 0 Volt in not more than 105 voltage increments. In position II of the change-over switch 86, the voltage at the output increases with respect to the reference terminal 4 from a value between 0 and +V to V volt, of course also in not more than 105 voltage increments. By means of a different mutual connection of the integrated circuits it is alternatively possible to have the output voltage at the output 7 increase in position I of the change-over switch 86 and to have it decrease in position II. Should an analog voltage increase or decrease be preferred, then the discrete controllable voltage divider 2 must be replaced by an analog controllable voltage divider, not shown, as described in, for example, Netherlands patent application OA No. 7,802,973. The multiplex circuits F and G must then be followed by a digital-to-analog converter to obtain an analog control signal for such an analog controllable voltage divider.

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